Dissertations in Forestry and Natural Sciences · covers a new intrinsic aspect of wave–particle duality of the photon, having no correspondence within the scalar quantum treatment

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PUBLICATIONS OF THE UNIVERSITY OF EASTERN FINLAND

Dissertations in Forestry and Natural Sciences

ISBN 978-952-61-2357-8ISSN 1798-5668


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252

ANDREAS NORRMAN

ELECTROMAGNETIC COHERENCE OF OPTICAL SURFACE AND QUANTUM LIGHT FIELDS


This thesis considers surface-plasmon

polaritons (SPPs), partially coherent optical surface fields, and complementarity in

vector-light photon interference. Novel SPPs, including a long-range higher-order metal-

slab mode, are predicted. Generation, partial polarization, and electromagnetic coherence

of polychromatic SPPs and evanescent light fields are also examined. Polarization

modulation of vectorial quantum light is explored to uncover a new intrinsic aspect of

photon wave–particle duality.

ANDREAS NORRMAN

ANDREAS NORRMAN

Electromagnetic Coherenceof Optical Surface andQuantum Light Fields

Publications of the University of Eastern FinlandDissertations in Forestry and Natural Sciences

No 252

Academic DissertationTo be presented by permission of the Faculty of Science and Forestry for publicexamination in the Auditorium AG100 in Agora Building at the University of

Eastern Finland, Joensuu, on December 16, 2016,at 12 o’clock noon.

Institute of Photonics

Grano Oy

Jyväskylä, 2016

Editors: Prof. Pertti Pasanen, Prof. Jukka Tuomela,

Prof. Matti Vornanen, Prof. Pekka Toivanen

Distribution:

University of Eastern Finland Library / Sales of publications

[email protected]

http://www.uef.fi/kirjasto

ISBN: 978-952-61-2357-8 (printed)

ISSNL: 1798-5668

ISSN: 1798-5668

ISBN: 978-952-61-2358-5 (pdf)

ISSNL: 1798-5668

ISSN: 1798-5676

Author’s address: University of Eastern FinlandInstitute of PhotonicsP. O. Box 11180101 JOENSUUFINLANDemail: [email protected]

Supervisors: Professor Ari T. Friberg, Ph.D., D.Sc. (Tech)University of Eastern FinlandInstitute of PhotonicsP. O. Box 11180101 JOENSUUFINLANDemail: [email protected]

Associate Professor Tero Setala, D.Sc. (Tech)University of Eastern FinlandInstitute of PhotonicsP. O. Box 11180101 JOENSUUFINLANDemail: [email protected]

Reviewers: Professor Taco D. Visser, Ph.D.Vrije Universiteit AmsterdamDepartment of Physics and AstronomyDe Boelelaan 10811081 HV AMSTERDAMTHE NETHERLANDSemail: [email protected]

Professor Goery Genty, D.Sc. (Tech)Tampere University of TechnologyDepartment of PhysicsP. O. Box 69233101 TAMPEREFINLANDemail: [email protected]

Opponent: Professor P. Scott Carney, Ph.D.University of Illinois at Urbana–ChampaignDepartment of Electrical and Computer Engineering405 North Mathews AvenueURBANA, IL 61801USAemail: [email protected]

Grano Oy

Jyväskylä, 2016

Editors: Prof. Pertti Pasanen, Prof. Jukka Tuomela,

Prof. Matti Vornanen, Prof. Pekka Toivanen

Distribution:

University of Eastern Finland Library / Sales of publications

[email protected]

http://www.uef.fi/kirjasto

ISBN: 978-952-61-2357-8 (printed)

ISSNL: 1798-5668

ISSN: 1798-5668

ISBN: 978-952-61-2358-5 (pdf)

ISSNL: 1798-5668

ISSN: 1798-5676

Author’s address: University of Eastern FinlandInstitute of PhotonicsP. O. Box 11180101 JOENSUUFINLANDemail: [email protected]

Supervisors: Professor Ari T. Friberg, Ph.D., D.Sc. (Tech)University of Eastern FinlandInstitute of PhotonicsP. O. Box 11180101 JOENSUUFINLANDemail: [email protected]

Associate Professor Tero Setala, D.Sc. (Tech)University of Eastern FinlandInstitute of PhotonicsP. O. Box 11180101 JOENSUUFINLANDemail: [email protected]

Reviewers: Professor Taco D. Visser, Ph.D.Vrije Universiteit AmsterdamDepartment of Physics and AstronomyDe Boelelaan 10811081 HV AMSTERDAMTHE NETHERLANDSemail: [email protected]

Professor Goery Genty, D.Sc. (Tech)Tampere University of TechnologyDepartment of PhysicsP. O. Box 69233101 TAMPEREFINLANDemail: [email protected]

Opponent: Professor P. Scott Carney, Ph.D.University of Illinois at Urbana–ChampaignDepartment of Electrical and Computer Engineering405 North Mathews AvenueURBANA, IL 61801USAemail: [email protected]

ABSTRACT

This thesis encompasses fundamental theoretical research on plas-monics, electromagnetic coherence, and quantum complementarity.Three main topics are covered: surface-plasmon polaritons (SPPs),partial coherence of optical surface fields, and complementarity invectorial photon interference.

Existence of new SPP modes at planar single-interface and thin-film geometries are predicted. It is further shown, for the first time,how a higher-order metal-slab mode, commonly not regarded use-ful, may turn into a strongly confined long-range surface mode inmany situations where the fundamental long-range SPP does notexist and the single-interface SPP propagation is negligible.

Partially coherent SPP fields are studied and a scheme to tailorthe vectorial coherence of polychromatic SPPs in the Kretschmannsetup by controlling the correlations of the source light is advanced.Generation and coherence of purely evanescent light fields are alsoexamined. It is demonstrated that, for such fields, the coherencelength in air can be notably shorter than the free-space wavelength.The analysis also reveals the possibility to excite an evanescent fieldthat shares the polarization properties of blackbody radiation, yetwith tunable coherence qualities.

Polarization modulation in double-pinhole photon interferenceis explored to derive two general complementarity relations amongdistinguishability and visibility associated with genuine vector-lightquantum fields of arbitrary state. The established framework un-covers a new intrinsic aspect of wave–particle duality of the photon,having no correspondence within the scalar quantum treatment.

Universal Decimal Classification: 53.01, 537.8, 535-6Keywords: theoretical physics; nanophotonics; plasmonics; optical surfacefields; surface-plasmon polaritons; light coherence; light polarization; pho-ton interference; quantum complementarity; wave–particle dualityAsiasanat: teoreettinen fysiikka; nanofotoniikka; plasmoniikka; optiset pin-takentat; koherenssi; polarisaatio; kvanttivalo; aalto-hiukkasdualismi

Preface

The research summarized in this dissertation started nearly fiveyears ago, at the end of 2011, when I began my doctoral studieson electromagnetic nanophotonics in the Department of AppliedPhysics at Aalto University, Espoo, Finland. However, as a result ofvarious factors, our theory group relocated in the beginning of 2013to the Institute of Photonics at the University of Eastern Finland,Joensuu, Finland, which turned this project into quite a colorfulroller-coaster journey. In particular, due to personal reasons, I havehad to carry out my research mainly from my home apartment inHelsinki on my own. The 450 km distance to my supervisors, not tomention the birth of my daughter seven months after the move, hasundeniably caused some real challenges and moments of despairfrom time to time. Even so, now in hindsight, I believe that thesecircumstances have also taught me to do and to take responsibilityof individual research in a positive manner. Of course, this wouldnot be the case without the constant encouragement and trust of mysupervisors, or the support of other colleagues, friends, and familymembers. I would therefore now take the opportunity to thank allthese wonderful people.

First, I would like to express my sincerest gratitude to Prof. AriT. Friberg, my supervisor, mentor, and friend, for his invaluableguidance and persistent support during all these years. His excep-tional enthusiasm and integrity towards scientific research, his con-stant and untiring devotion to research projects, as well as his abil-ity to challenge and to educate students’ thinking are simply un-equalled. Our in-depth conversations, covering science, art, sport,politics, religion, and life in general, have been very intriguing; ournumerous tough matches on the squash court have been extremelyenjoyable; our uncountable dusk-till-dawn adventures, naturallyaccompanied by a certain refreshing yellowish beverage, have beenabsolutely memorable. I am also indebted to my other supervisor,

ABSTRACT

This thesis encompasses fundamental theoretical research on plas-monics, electromagnetic coherence, and quantum complementarity.Three main topics are covered: surface-plasmon polaritons (SPPs),partial coherence of optical surface fields, and complementarity invectorial photon interference.

Existence of new SPP modes at planar single-interface and thin-film geometries are predicted. It is further shown, for the first time,how a higher-order metal-slab mode, commonly not regarded use-ful, may turn into a strongly confined long-range surface mode inmany situations where the fundamental long-range SPP does notexist and the single-interface SPP propagation is negligible.

Partially coherent SPP fields are studied and a scheme to tailorthe vectorial coherence of polychromatic SPPs in the Kretschmannsetup by controlling the correlations of the source light is advanced.Generation and coherence of purely evanescent light fields are alsoexamined. It is demonstrated that, for such fields, the coherencelength in air can be notably shorter than the free-space wavelength.The analysis also reveals the possibility to excite an evanescent fieldthat shares the polarization properties of blackbody radiation, yetwith tunable coherence qualities.

Polarization modulation in double-pinhole photon interferenceis explored to derive two general complementarity relations amongdistinguishability and visibility associated with genuine vector-lightquantum fields of arbitrary state. The established framework un-covers a new intrinsic aspect of wave–particle duality of the photon,having no correspondence within the scalar quantum treatment.

Universal Decimal Classification: 53.01, 537.8, 535-6Keywords: theoretical physics; nanophotonics; plasmonics; optical surfacefields; surface-plasmon polaritons; light coherence; light polarization; pho-ton interference; quantum complementarity; wave–particle dualityAsiasanat: teoreettinen fysiikka; nanofotoniikka; plasmoniikka; optiset pin-takentat; koherenssi; polarisaatio; kvanttivalo; aalto-hiukkasdualismi

Preface

The research summarized in this dissertation started nearly fiveyears ago, at the end of 2011, when I began my doctoral studieson electromagnetic nanophotonics in the Department of AppliedPhysics at Aalto University, Espoo, Finland. However, as a result ofvarious factors, our theory group relocated in the beginning of 2013to the Institute of Photonics at the University of Eastern Finland,Joensuu, Finland, which turned this project into quite a colorfulroller-coaster journey. In particular, due to personal reasons, I havehad to carry out my research mainly from my home apartment inHelsinki on my own. The 450 km distance to my supervisors, not tomention the birth of my daughter seven months after the move, hasundeniably caused some real challenges and moments of despairfrom time to time. Even so, now in hindsight, I believe that thesecircumstances have also taught me to do and to take responsibilityof individual research in a positive manner. Of course, this wouldnot be the case without the constant encouragement and trust of mysupervisors, or the support of other colleagues, friends, and familymembers. I would therefore now take the opportunity to thank allthese wonderful people.

First, I would like to express my sincerest gratitude to Prof. AriT. Friberg, my supervisor, mentor, and friend, for his invaluableguidance and persistent support during all these years. His excep-tional enthusiasm and integrity towards scientific research, his con-stant and untiring devotion to research projects, as well as his abil-ity to challenge and to educate students’ thinking are simply un-equalled. Our in-depth conversations, covering science, art, sport,politics, religion, and life in general, have been very intriguing; ournumerous tough matches on the squash court have been extremelyenjoyable; our uncountable dusk-till-dawn adventures, naturallyaccompanied by a certain refreshing yellowish beverage, have beenabsolutely memorable. I am also indebted to my other supervisor,

Prof. Tero Setala, for his valuable instructions, ingenious wits, andactive involvement, not only at the office level, but also throughextracurricular late-night activities. Overall, I want to thank bothsupervisors for contributing to a stimulating and open academicand social environment. It has been an honor and privilege to workwith them all these years.

I also want to extend my gratitude to my other coauthors, Prof.Sergey A. Ponomarenko and Dr. Kasimir Blomstedt, for provid-ing broad expertise, important insights, and essential contributionsduring the course of this thesis. The perceptive comments and sug-gestions by the reviewers, Prof. Taco D. Visser and Prof. GoeryGenty, are highly appreciated. In addition, I am grateful to Prof. P.Scott Carney for accepting to be my opponent on short notice.

During my time at Aalto University and the University of East-ern Finland I have been fortunate to make acquaintance with othertalented individuals as well. Special thanks go to my former fel-low colleagues, Dr. Timo Hakkarainen, Dr. Timo Voipio, and M.Sc.Henri Kellock, with whom I have had the pleasure to share manyfruitful discussions and occasions over the years, both inside andoutside the university. My present workmates, Dr. Lasse-PetteriLeppanen and M.Sc. Matias Koivurova, have also had a positiveinfluence on my well-being. For efficiency at the bureaucratic level,Dr. Noora Heikkila, Ms. Hannele Karppinen, Ms. Katri Mustonen,and Ms. Marita Ratilainen deserve my sincerest gratitude.

Personal grants from the Jenny and Antti Wihuri Foundation,the Emil Aaltonen Foundation, and the Finnish Foundation forTechnology Promotion are gratefully acknowledged. Furthermore,I thank the Academy of Finland for funding my research throughvarious projects.

I am also indebted to my inspiring soulmates, Alexander, Erik,Hannes, Janek, Jukka-Pekka, Jussi, Markku, Martin, Nikolas, Ras-mus, Sampo, Simo, and Veli-Matti, of course not to disregard theirbetter halves, who have strongly enriched my life and provided mewith (sometimes much needed) nonscientific counterbalance, eachin their own unique way.

Last, but not least, I owe my heartfelt gratefulness to my belovedparents for their endless empathy, kindness, and presence through-out my entire life. Above all, my deepest thankfulness and love goto my best friend and life companion, my wonderful wife Jenni, aswell as to my most precious jewel, my sweet daughter Lily. With-out their unwavering patience and support at moments of darknessand desperation this thesis would not exist. The least I can do is todedicate this dissertation to them.

For my family.

Helsinki, November 24, 2016 Andreas Norrman

Prof. Tero Setala, for his valuable instructions, ingenious wits, andactive involvement, not only at the office level, but also throughextracurricular late-night activities. Overall, I want to thank bothsupervisors for contributing to a stimulating and open academicand social environment. It has been an honor and privilege to workwith them all these years.

I also want to extend my gratitude to my other coauthors, Prof.Sergey A. Ponomarenko and Dr. Kasimir Blomstedt, for provid-ing broad expertise, important insights, and essential contributionsduring the course of this thesis. The perceptive comments and sug-gestions by the reviewers, Prof. Taco D. Visser and Prof. GoeryGenty, are highly appreciated. In addition, I am grateful to Prof. P.Scott Carney for accepting to be my opponent on short notice.

During my time at Aalto University and the University of East-ern Finland I have been fortunate to make acquaintance with othertalented individuals as well. Special thanks go to my former fel-low colleagues, Dr. Timo Hakkarainen, Dr. Timo Voipio, and M.Sc.Henri Kellock, with whom I have had the pleasure to share manyfruitful discussions and occasions over the years, both inside andoutside the university. My present workmates, Dr. Lasse-PetteriLeppanen and M.Sc. Matias Koivurova, have also had a positiveinfluence on my well-being. For efficiency at the bureaucratic level,Dr. Noora Heikkila, Ms. Hannele Karppinen, Ms. Katri Mustonen,and Ms. Marita Ratilainen deserve my sincerest gratitude.

Personal grants from the Jenny and Antti Wihuri Foundation,the Emil Aaltonen Foundation, and the Finnish Foundation forTechnology Promotion are gratefully acknowledged. Furthermore,I thank the Academy of Finland for funding my research throughvarious projects.

I am also indebted to my inspiring soulmates, Alexander, Erik,Hannes, Janek, Jukka-Pekka, Jussi, Markku, Martin, Nikolas, Ras-mus, Sampo, Simo, and Veli-Matti, of course not to disregard theirbetter halves, who have strongly enriched my life and provided mewith (sometimes much needed) nonscientific counterbalance, eachin their own unique way.

Last, but not least, I owe my heartfelt gratefulness to my belovedparents for their endless empathy, kindness, and presence through-out my entire life. Above all, my deepest thankfulness and love goto my best friend and life companion, my wonderful wife Jenni, aswell as to my most precious jewel, my sweet daughter Lily. With-out their unwavering patience and support at moments of darknessand desperation this thesis would not exist. The least I can do is todedicate this dissertation to them.

For my family.

Helsinki, November 24, 2016 Andreas Norrman

LIST OF PUBLICATIONS

This thesis consists of an overview of the author’s work in the fieldof electromagnetic nanophotonics and the following selection of theauthor’s publications:

I A. Norrman, T. Setala, and A. T. Friberg, “Exact surface-plas-mon polariton solutions at a lossy interface,” Opt. Lett. 38,1119–1121 (2013).

II A. Norrman, T. Setala, and A. T. Friberg, “Surface-plasmonpolariton solutions at a lossy slab in a symmetric surround-ing,” Opt. Express 22, 4628–4648 (2014).

III A. Norrman, T. Setala, and A. T. Friberg, “Long-range higher-order surface-plasmon polaritons,” Phys. Rev. A 90, 053849(2014).

IV A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Partiallycoherent surface plasmon polaritons,” submitted (2016).

V A. Norrman, T. Setala, and A. T. Friberg, “Partial coherenceand polarization of a two-mode surface-plasmon polaritonfield at a metallic nanoslab,” Opt. Express 23, 20696–20714(2015).

VI A. Norrman, T. Setala, and A. T. Friberg, “Partial spatial co-herence and partial polarization in random evanescent fieldson lossless interfaces,” J. Opt. Soc. Am. A 28, 391–400 (2011).

VII A. Norrman, T. Setala, and A. T. Friberg, “Generation andelectromagnetic coherence of unpolarized three-componentlight fields,” Opt. Lett. 40, 5216–5219 (2015).

VIII A. Norrman, K. Blomstedt, T. Setala, and A. T. Friberg, “Com-plementarity and polarization modulation in photon interfer-ence,” submitted (2016).

Throughout the overview, these publications will be referred to byRoman numerals.

The results reported in Publications I–VI and VIII have been pre-sented in the following international conferences:

1. 5th EOS Topical Meeting on Advanced Imaging Techniques(Engelberg, Switzerland, 2010).

2. 9th Euro-American Workshop on Information Optics (Helsin-ki, Finland, 2010).

3. Nordic Physics Days 2011 (Helsinki, Finland, 2011).

4. 22nd General Congress of the International Commission forOptics (Puebla, Mexico, 2011).

5. 18th Microoptics Conference (Tokyo, Japan, 2013).

6. 1st Joensuu Conference on Coherence and Random Polariza-tion: Electromagnetic Optics with Random Light (Joensuu,Finland, 2014).

7. Northern Optics & Photonics 2015 (Lappeenranta, Finland,2015).

8. Frontiers in Optics / Laser Science 2016 (Rochester, NY, USA,2016).

9. OSA Incubator on Emerging Connections: Quantum and Clas-sical Optics (Washington, DC, USA, 2016).

LIST OF PUBLICATIONS

This thesis consists of an overview of the author’s work in the fieldof electromagnetic nanophotonics and the following selection of theauthor’s publications:

I A. Norrman, T. Setala, and A. T. Friberg, “Exact surface-plas-mon polariton solutions at a lossy interface,” Opt. Lett. 38,1119–1121 (2013).

II A. Norrman, T. Setala, and A. T. Friberg, “Surface-plasmonpolariton solutions at a lossy slab in a symmetric surround-ing,” Opt. Express 22, 4628–4648 (2014).

III A. Norrman, T. Setala, and A. T. Friberg, “Long-range higher-order surface-plasmon polaritons,” Phys. Rev. A 90, 053849(2014).

IV A. Norrman, S. A. Ponomarenko, and A. T. Friberg, “Partiallycoherent surface plasmon polaritons,” submitted (2016).

V A. Norrman, T. Setala, and A. T. Friberg, “Partial coherenceand polarization of a two-mode surface-plasmon polaritonfield at a metallic nanoslab,” Opt. Express 23, 20696–20714(2015).

VI A. Norrman, T. Setala, and A. T. Friberg, “Partial spatial co-herence and partial polarization in random evanescent fieldson lossless interfaces,” J. Opt. Soc. Am. A 28, 391–400 (2011).

VII A. Norrman, T. Setala, and A. T. Friberg, “Generation andelectromagnetic coherence of unpolarized three-componentlight fields,” Opt. Lett. 40, 5216–5219 (2015).

VIII A. Norrman, K. Blomstedt, T. Setala, and A. T. Friberg, “Com-plementarity and polarization modulation in photon interfer-ence,” submitted (2016).

Throughout the overview, these publications will be referred to byRoman numerals.

The results reported in Publications I–VI and VIII have been pre-sented in the following international conferences:

1. 5th EOS Topical Meeting on Advanced Imaging Techniques(Engelberg, Switzerland, 2010).

2. 9th Euro-American Workshop on Information Optics (Helsin-ki, Finland, 2010).

3. Nordic Physics Days 2011 (Helsinki, Finland, 2011).

4. 22nd General Congress of the International Commission forOptics (Puebla, Mexico, 2011).

5. 18th Microoptics Conference (Tokyo, Japan, 2013).

6. 1st Joensuu Conference on Coherence and Random Polariza-tion: Electromagnetic Optics with Random Light (Joensuu,Finland, 2014).

7. Northern Optics & Photonics 2015 (Lappeenranta, Finland,2015).

8. Frontiers in Optics / Laser Science 2016 (Rochester, NY, USA,2016).

9. OSA Incubator on Emerging Connections: Quantum and Clas-sical Optics (Washington, DC, USA, 2016).

AUTHOR’S CONTRIBUTION

The author has had a central role in all aspects of the research workreported in this thesis. The author carried out the majority of themathematical derivations and the numerical calculations in all thepublications. The research subjects arose from discussions with thecoauthors, with the author contributing significantly to the origi-nal ideas and to the interpretation of the results. In particular, thenotions of the long-range higher-order surface-plasmon polaritonsand the complementarity relations for quantum vector light camefrom the author. The author wrote the first drafts of all publications,which subsequently were finalized together with the coauthors.

Contents

1 INTRODUCTION 11.1 Plasmonics, electromagnetic coherence, and quantum

complementarity . . . . . . . . . . . . . . . . . . . . . 11.2 Scope of the thesis . . . . . . . . . . . . . . . . . . . . 6

2 ELECTROMAGNETIC SURFACE WAVES 92.1 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . 92.2 Field characterization . . . . . . . . . . . . . . . . . . . 10

2.2.1 Representation . . . . . . . . . . . . . . . . . . 112.2.2 Propagation . . . . . . . . . . . . . . . . . . . . 122.2.3 Energy flow . . . . . . . . . . . . . . . . . . . . 14

2.3 Existence of electromagnetic surface modes . . . . . . 152.4 Fresnel coefficients and existence conditions . . . . . 17

3 SURFACE-PLASMON POLARITONS 213.1 Single-interface modes . . . . . . . . . . . . . . . . . . 21

3.1.1 Approximate solutions vs. exact solutions . . 223.1.2 Mode types SPP I and SPP II . . . . . . . . . . 233.1.3 Flow of energy . . . . . . . . . . . . . . . . . . 24

3.2 Metal-slab modes . . . . . . . . . . . . . . . . . . . . . 263.2.1 Mode class M1 . . . . . . . . . . . . . . . . . . 283.2.2 Mode class M2 . . . . . . . . . . . . . . . . . . 293.2.3 Mode class M3 . . . . . . . . . . . . . . . . . . 313.2.4 Forward- and backward-propagating modes . 37

3.3 Long-range modes . . . . . . . . . . . . . . . . . . . . 383.3.1 Mode interchanges . . . . . . . . . . . . . . . . 393.3.2 Long-range higher-order modes . . . . . . . . 41

4 PARTIALLY COHERENTSURFACE-PLASMON POLARITONS 454.1 Single-interface geometry . . . . . . . . . . . . . . . . 45

AUTHOR’S CONTRIBUTION

The author has had a central role in all aspects of the research workreported in this thesis. The author carried out the majority of themathematical derivations and the numerical calculations in all thepublications. The research subjects arose from discussions with thecoauthors, with the author contributing significantly to the origi-nal ideas and to the interpretation of the results. In particular, thenotions of the long-range higher-order surface-plasmon polaritonsand the complementarity relations for quantum vector light camefrom the author. The author wrote the first drafts of all publications,which subsequently were finalized together with the coauthors.

Contents

1 INTRODUCTION 11.1 Plasmonics, electromagnetic coherence, and quantum

complementarity . . . . . . . . . . . . . . . . . . . . . 11.2 Scope of the thesis . . . . . . . . . . . . . . . . . . . . 6

2 ELECTROMAGNETIC SURFACE WAVES 92.1 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . 92.2 Field characterization . . . . . . . . . . . . . . . . . . . 10

2.2.1 Representation . . . . . . . . . . . . . . . . . . 112.2.2 Propagation . . . . . . . . . . . . . . . . . . . . 122.2.3 Energy flow . . . . . . . . . . . . . . . . . . . . 14

2.3 Existence of electromagnetic surface modes . . . . . . 152.4 Fresnel coefficients and existence conditions . . . . . 17

3 SURFACE-PLASMON POLARITONS 213.1 Single-interface modes . . . . . . . . . . . . . . . . . . 21

3.1.1 Approximate solutions vs. exact solutions . . 223.1.2 Mode types SPP I and SPP II . . . . . . . . . . 233.1.3 Flow of energy . . . . . . . . . . . . . . . . . . 24

3.2 Metal-slab modes . . . . . . . . . . . . . . . . . . . . . 263.2.1 Mode class M1 . . . . . . . . . . . . . . . . . . 283.2.2 Mode class M2 . . . . . . . . . . . . . . . . . . 293.2.3 Mode class M3 . . . . . . . . . . . . . . . . . . 313.2.4 Forward- and backward-propagating modes . 37

3.3 Long-range modes . . . . . . . . . . . . . . . . . . . . 383.3.1 Mode interchanges . . . . . . . . . . . . . . . . 393.3.2 Long-range higher-order modes . . . . . . . . 41

4 PARTIALLY COHERENTSURFACE-PLASMON POLARITONS 454.1 Single-interface geometry . . . . . . . . . . . . . . . . 45

4.1.1 Polychromatic surface-plasmon polaritons . . 464.1.2 Narrowband and broadband fields . . . . . . 484.1.3 Plasmon coherence engineering . . . . . . . . 48

4.2 Metal-slab geometry . . . . . . . . . . . . . . . . . . . 504.2.1 Degree of coherence . . . . . . . . . . . . . . . 524.2.2 Local and global coherence length . . . . . . . 544.2.3 Degree of polarization . . . . . . . . . . . . . . 55

5 ELECTROMAGNETIC COHERENCE OFEVANESCENT LIGHT FIELDS 575.1 Evanescent wave in total internal reflection . . . . . . 575.2 Random evanescent fields . . . . . . . . . . . . . . . . 59

5.2.1 Subwavelength coherence lengths . . . . . . . 615.2.2 Genuine 3D-polarized states . . . . . . . . . . 62

5.3 3D-unpolarized evanescent fields . . . . . . . . . . . . 625.3.1 Generation . . . . . . . . . . . . . . . . . . . . . 635.3.2 Degree of coherence . . . . . . . . . . . . . . . 64

6 COMPLEMENTARITY IN PHOTON INTERFERENCE 676.1 Coherence of vectorial quantum light . . . . . . . . . 686.2 Photon interference law . . . . . . . . . . . . . . . . . 706.3 Visibility and distinguishability . . . . . . . . . . . . . 726.4 Weak and strong complementarity . . . . . . . . . . . 746.5 Wave–particle duality of the photon . . . . . . . . . . 75

7 CONCLUSIONS 797.1 Summary of main results . . . . . . . . . . . . . . . . 797.2 Future prospects . . . . . . . . . . . . . . . . . . . . . 83

A CLASSICAL THEORY OFELECTROMAGNETIC COHERENCE 85A.1 Nonstationary fields . . . . . . . . . . . . . . . . . . . 86A.2 Stationary fields . . . . . . . . . . . . . . . . . . . . . . 87A.3 Degree of polarization . . . . . . . . . . . . . . . . . . 89

REFERENCES 93

1 Introduction

The rapid progress in nano-optics and photonics during the last twodecades has opened a whole new realm of possibilities for interdis-ciplinary science and technology [1–4]. Groundbreaking discoveriesin such engrossing areas as superlens imaging [5,6], transformationoptics [7–9], optical cloaking [10–12], and ghost imaging [13–15],entail extraordinary physical phenomena with fascinating potentialapplications for technoscientific research and engineering. At thesame time, the advent of near-field optics [16, 17], strongly domi-nated by the manipulation and utilization of evanescent waves atsubwavelength dimensions [18], has allowed to surpass the tradi-tional diffraction limit and plays a vital role in the design and man-ufacture of various optoelectronic nanocomponents. In particular,tailoring novel composite materials with unforeseen precision en-ables fabrication of hitherto unrealizable structures to exploit light–matter interactions of diverse nature at microscopic and nanoscopicscales [19–23].

1.1 PLASMONICS, ELECTROMAGNETIC COHERENCE,AND QUANTUM COMPLEMENTARITY

Modern plasmonics [24–27], constituting an exceptionally rich areaof nanophotonics [28–30], offers methods for subwavelength lightcontrol over dimensions as small as a few nanometers [31–33], withnumerous applications in biomedical and chemical sensing [34–36],detectors and emitters [37,38], nanophotonic devices [39–41], lasers[42–44], complex materials [45–47], nonlinear and quantum inter-actions [48, 49], as well as light shaping and holography [50–53].Optical coherence theory, in turn, forms an important discipline inall areas of electromagnetic light physics, by providing the basicmeans to understand spectral distribution, propagation, interfer-ence, interactions, polarization, and other fundamental characteris-

Dissertations in Forestry and Natural Sciences No 252 1

4.1.1 Polychromatic surface-plasmon polaritons . . 464.1.2 Narrowband and broadband fields . . . . . . 484.1.3 Plasmon coherence engineering . . . . . . . . 48

4.2 Metal-slab geometry . . . . . . . . . . . . . . . . . . . 504.2.1 Degree of coherence . . . . . . . . . . . . . . . 524.2.2 Local and global coherence length . . . . . . . 544.2.3 Degree of polarization . . . . . . . . . . . . . . 55

5 ELECTROMAGNETIC COHERENCE OFEVANESCENT LIGHT FIELDS 575.1 Evanescent wave in total internal reflection . . . . . . 575.2 Random evanescent fields . . . . . . . . . . . . . . . . 59

5.2.1 Subwavelength coherence lengths . . . . . . . 615.2.2 Genuine 3D-polarized states . . . . . . . . . . 62

5.3 3D-unpolarized evanescent fields . . . . . . . . . . . . 625.3.1 Generation . . . . . . . . . . . . . . . . . . . . . 635.3.2 Degree of coherence . . . . . . . . . . . . . . . 64

6 COMPLEMENTARITY IN PHOTON INTERFERENCE 676.1 Coherence of vectorial quantum light . . . . . . . . . 686.2 Photon interference law . . . . . . . . . . . . . . . . . 706.3 Visibility and distinguishability . . . . . . . . . . . . . 726.4 Weak and strong complementarity . . . . . . . . . . . 746.5 Wave–particle duality of the photon . . . . . . . . . . 75

7 CONCLUSIONS 797.1 Summary of main results . . . . . . . . . . . . . . . . 797.2 Future prospects . . . . . . . . . . . . . . . . . . . . . 83

A CLASSICAL THEORY OFELECTROMAGNETIC COHERENCE 85A.1 Nonstationary fields . . . . . . . . . . . . . . . . . . . 86A.2 Stationary fields . . . . . . . . . . . . . . . . . . . . . . 87A.3 Degree of polarization . . . . . . . . . . . . . . . . . . 89

REFERENCES 93

1 Introduction

The rapid progress in nano-optics and photonics during the last twodecades has opened a whole new realm of possibilities for interdis-ciplinary science and technology [1–4]. Groundbreaking discoveriesin such engrossing areas as superlens imaging [5,6], transformationoptics [7–9], optical cloaking [10–12], and ghost imaging [13–15],entail extraordinary physical phenomena with fascinating potentialapplications for technoscientific research and engineering. At thesame time, the advent of near-field optics [16, 17], strongly domi-nated by the manipulation and utilization of evanescent waves atsubwavelength dimensions [18], has allowed to surpass the tradi-tional diffraction limit and plays a vital role in the design and man-ufacture of various optoelectronic nanocomponents. In particular,tailoring novel composite materials with unforeseen precision en-ables fabrication of hitherto unrealizable structures to exploit light–matter interactions of diverse nature at microscopic and nanoscopicscales [19–23].

1.1 PLASMONICS, ELECTROMAGNETIC COHERENCE,AND QUANTUM COMPLEMENTARITY

Modern plasmonics [24–27], constituting an exceptionally rich areaof nanophotonics [28–30], offers methods for subwavelength lightcontrol over dimensions as small as a few nanometers [31–33], withnumerous applications in biomedical and chemical sensing [34–36],detectors and emitters [37,38], nanophotonic devices [39–41], lasers[42–44], complex materials [45–47], nonlinear and quantum inter-actions [48, 49], as well as light shaping and holography [50–53].Optical coherence theory, in turn, forms an important discipline inall areas of electromagnetic light physics, by providing the basicmeans to understand spectral distribution, propagation, interfer-ence, interactions, polarization, and other fundamental characteris-


Andreas Norrman: Electromagnetic Coherence of Optical Surface andQuantum Light Fields

tics of both classical and quantum wave fields [54–59].

The success of plasmonics in optical physics is mainly due tothe celebrated surface-plasmon polariton (SPP), a hybridized excita-tion between light and collective charge-density oscillations thattypically appear at metal–dielectric interfaces. The seemingly ear-liest recorded observation associated with SPPs, although not rec-ognized at that time, is the spectral anomaly discovered by Woodin 1902 when studying light diffraction from a metal grating [60].Despite major efforts and contributions [61–63], Wood’s anomaly,as it is now called, remained unexplained for nearly 40 years, untilFano in 1941 gave it a proper description in terms of ‘superficialwaves’ [64]. Later during the 1940s, experiments performed withfast electrons impinging on thin metal films revealed unexpectedpeaks in the measured energy-loss spectrum [65–67]. In the early1950s, Pines and Bohm, attributed for coining the phrase ‘plasmon’,provided an explanation for these observations by realizing that thelong-range Coulomb interaction between valence electrons in met-als could result in longitudinal collective plasma oscillations [68,69].Interest towards plasma excitations started to grow in the late 1950s.Especially, the pioneering theoretical investigations made by Ritchiein 1957 led to the prediction of a self-sustained surface-collective ex-citation [70], whose existence was experimentally verified two yearslater by Powell and Swan [71, 72], and subsequently by Stern andFerrell [73], who were the first to describe the new excitation as a‘surface plasmon’.

The earliest publication of prism-coupled excitation of an SPPby means of optical evanescent waves would appear to be that ofTurbadar from 1959 [74], only a year after Hopfield had inventedthe term ‘polariton’ [75]. Unfortunately, Turbadar did not explic-itly state that he had actually excited an SPP, whereupon the creditfor prism-coupled SPP generation has been given to Otto [76] andto Kretschmann and Raether [77] for their experiments performedindependently in 1968. Also the important theoretical contribu-tions of Kliewer and Fuchs [78] and of Economou [79] from the late1960s deserve mentioning. In the following two decades SPPs were

2 Dissertations in Forestry and Natural Sciences No 252

Introduction

extensively studied by several groups, leading to the discovery ofthe long-range SPP [80, 81] and eventually culminating in a semi-nal text by Raether in 1988 [82], which still remains an importantlandmark in the field. Over the next ten years the research evolvedsomewhat slowly, until an extraordinary strong light transmissionthrough subwavelength hole arrays due to plasmon excitations wasobserved by Ebbesen and co-workers in 1998 [83], which sparked anexplosion of interest and spawned the field of modern plasmonicsas we recognize it today.

To date, plasmonics has chiefly dealt with spatially and spec-trally totally coherent (and polarized) SPPs. Surface plasmons arenonetheless known to greatly alter the spectrum, polarization, andspatial coherence of optical near fields. For instance, some timeago it was demonstrated that the correlation length in a fluctuatingthermal near field can extend over several tens of wavelengths ifsurface plasmons are present [84]. Likewise, a thermal broadbandnear field may become essentially quasimonochromatic [85], andhighly polarized [86], when surface plasmons are involved. Bodiesin thermal equilibrium may emit radiation in the form of spatiallycoherent beam lobes of directionally dependent spectra if gratingsare fabricated on their surface [87]. Surface plasmons also play akey role in modifying the coherence properties of fields transmit-ted through periodic hole arrays, gratings, and slits in thin metalfilms [88–90]. These findings illustrate that, depending on the pre-vailing circumstances, the presence of surface plasmons may havea significant effect on the coherence and polarization characteris-tics of electromagnetic near fields. However, a randomly fluctuat-ing near field, with or without the influence of surface plasmons,is generally a three-component, non-beamlike field, necessitating ap-propriate theoretical methods for its statistical analysis.

The rigorous, classical theory of optical coherence in the space–time domain was established during the 1950s, primarily throughthe efforts of Wolf, when the polarization matrix as well as the elec-tromagnetic coherence matrices were introduced, together with theequations that govern their behavior [91–95]. Two decades later,



tics of both classical and quantum wave fields [54–59].

The success of plasmonics in optical physics is mainly due tothe celebrated surface-plasmon polariton (SPP), a hybridized excita-tion between light and collective charge-density oscillations thattypically appear at metal–dielectric interfaces. The seemingly ear-liest recorded observation associated with SPPs, although not rec-ognized at that time, is the spectral anomaly discovered by Woodin 1902 when studying light diffraction from a metal grating [60].Despite major efforts and contributions [61–63], Wood’s anomaly,as it is now called, remained unexplained for nearly 40 years, untilFano in 1941 gave it a proper description in terms of ‘superficialwaves’ [64]. Later during the 1940s, experiments performed withfast electrons impinging on thin metal films revealed unexpectedpeaks in the measured energy-loss spectrum [65–67]. In the early1950s, Pines and Bohm, attributed for coining the phrase ‘plasmon’,provided an explanation for these observations by realizing that thelong-range Coulomb interaction between valence electrons in met-als could result in longitudinal collective plasma oscillations [68,69].Interest towards plasma excitations started to grow in the late 1950s.Especially, the pioneering theoretical investigations made by Ritchiein 1957 led to the prediction of a self-sustained surface-collective ex-citation [70], whose existence was experimentally verified two yearslater by Powell and Swan [71, 72], and subsequently by Stern andFerrell [73], who were the first to describe the new excitation as a‘surface plasmon’.

The earliest publication of prism-coupled excitation of an SPPby means of optical evanescent waves would appear to be that ofTurbadar from 1959 [74], only a year after Hopfield had inventedthe term ‘polariton’ [75]. Unfortunately, Turbadar did not explic-itly state that he had actually excited an SPP, whereupon the creditfor prism-coupled SPP generation has been given to Otto [76] andto Kretschmann and Raether [77] for their experiments performedindependently in 1968. Also the important theoretical contribu-tions of Kliewer and Fuchs [78] and of Economou [79] from the late1960s deserve mentioning. In the following two decades SPPs were


Introduction

extensively studied by several groups, leading to the discovery ofthe long-range SPP [80, 81] and eventually culminating in a semi-nal text by Raether in 1988 [82], which still remains an importantlandmark in the field. Over the next ten years the research evolvedsomewhat slowly, until an extraordinary strong light transmissionthrough subwavelength hole arrays due to plasmon excitations wasobserved by Ebbesen and co-workers in 1998 [83], which sparked anexplosion of interest and spawned the field of modern plasmonicsas we recognize it today.

To date, plasmonics has chiefly dealt with spatially and spec-trally totally coherent (and polarized) SPPs. Surface plasmons arenonetheless known to greatly alter the spectrum, polarization, andspatial coherence of optical near fields. For instance, some timeago it was demonstrated that the correlation length in a fluctuatingthermal near field can extend over several tens of wavelengths ifsurface plasmons are present [84]. Likewise, a thermal broadbandnear field may become essentially quasimonochromatic [85], andhighly polarized [86], when surface plasmons are involved. Bodiesin thermal equilibrium may emit radiation in the form of spatiallycoherent beam lobes of directionally dependent spectra if gratingsare fabricated on their surface [87]. Surface plasmons also play akey role in modifying the coherence properties of fields transmit-ted through periodic hole arrays, gratings, and slits in thin metalfilms [88–90]. These findings illustrate that, depending on the pre-vailing circumstances, the presence of surface plasmons may havea significant effect on the coherence and polarization characteris-tics of electromagnetic near fields. However, a randomly fluctuat-ing near field, with or without the influence of surface plasmons,is generally a three-component, non-beamlike field, necessitating ap-propriate theoretical methods for its statistical analysis.

The rigorous, classical theory of optical coherence in the space–time domain was established during the 1950s, primarily throughthe efforts of Wolf, when the polarization matrix as well as the elec-tromagnetic coherence matrices were introduced, together with theequations that govern their behavior [91–95]. Two decades later,



the foundations of classical optical coherence theory in the space–frequency domain were put forward by Mandel and Wolf [96, 97].An alternative representation, first for scalar light [98–100] and onlyrelatively recently for vectorial light [101,102], was established. Thefrequency-domain formulation may in some sense be viewed as amore fundamental theory, as it yields insight into the inner (spec-tral) structure of the coherence of light. The spectral coherencetheory is also advantageous when light–matter interactions are in-vestigated. Whereas these treatments concern stationary light fields,i.e., light fields for which the character of the random fluctuationsdoes not change with time, the advent of pulsed lasers, supercon-tinuum light, and ultrafast detectors have in recent years awakeninterest, and highlighted the need, to study optical coherence andpolarization of nonstationary light [103–108].

The fundamental measures to characterize any partially coher-ent and partially polarized light field are the degree of coherence andthe degree of polarization. Conventionally, these two quantities havebeen restricted, respectively, to scalar light and to two-component(2D) fields [54]. However, recent advances in nano-optics [2] andhigh-numerical-aperture imaging systems [109–111] have accentu-ated the importance to extend these concepts for arbitrary three-component (3D) light fields. Partial coherence of vectorial light maybe assessed by the electromagnetic degree of coherence [112–114],which describes the strength of correlations that exist between allthe orthogonal components of the electric field at a pair of points.In addition, it is invariant under unitary transformations, a naturalrequirement [115], and most importantly, amounts to equivalencebetween complete coherence and factorization of the coherence ma-trix (see below). We emphasize, though, that other coherence mea-sures have been suggested [116–123], with different physical moti-vations, implications, and mathematical properties [124]. The 3Ddegree of polarization, in turn, can be constructed via an expansionof the polarization matrix in terms of the Gell-Mann matrices andthe generalized Stokes parameters [125], in analogy with the Paulispin matrices and the usual Stokes parameters for the customary


Introduction

2D degree of polarization [54]. In this representation, the degreeof polarization can be interpreted as the square root of the averageof the normalized correlations squared among the three orthogonalelectric-field components in a reference frame where the diagonalelements of the polarization matrix are equal. However, also othermeans to address partial polarization of 3D-light fields have beenproposed [126–132], and compared [133, 134].

Around the time of the invention of the laser, influenced by therevolutionary experiment performed in 1956 by Hanbury Brownand Twiss [135], Glauber formulated the quantum theory of opticalcoherence in terms of nth-order correlation functions [136]. Higher-order correlations have gained major importance in quantum op-tics [137], since they can supply information about the nonclassi-cal nature of light. Furthermore, such correlations have a crucialposition in the characterization of quantum polarization [138], es-pecially in addressing the degree(s) of polarization for quantizedlight, a subject which is in constant development [139]. At the heartof Glauber’s seminal work is the equivalence between complete co-herence and the factorization of the coherence matrix into a productof two vectors. This foundation provides a more general defini-tion for complete coherence than the original definition from 1938of Zernike for classical scalar light [140], which connects completecoherence with full intensity visibility in Young’s interference ex-periment [141]. An essential requirement for a completely coherentfield, according to the definition of Glauber, is that it must be totallypolarized. Overall, as with the part played by Wolf (and Mandel)in the context of classical optics [142], the importance of Glauber’s(and Mandel’s) contributions for quantum optics is hard to overes-timate [143].

The principle of complementarity [144], stating that quantum ob-jects share mutually exclusive characteristics, has had a major sig-nificance for the foundations of physics and a profound impact onthe interpretation of the fundamental nature of reality [145, 146].The arguable most recognized manifestation of complementarity inphysics is the wave–particle duality, which places a trade-off for the



the foundations of classical optical coherence theory in the space–frequency domain were put forward by Mandel and Wolf [96, 97].An alternative representation, first for scalar light [98–100] and onlyrelatively recently for vectorial light [101,102], was established. Thefrequency-domain formulation may in some sense be viewed as amore fundamental theory, as it yields insight into the inner (spec-tral) structure of the coherence of light. The spectral coherencetheory is also advantageous when light–matter interactions are in-vestigated. Whereas these treatments concern stationary light fields,i.e., light fields for which the character of the random fluctuationsdoes not change with time, the advent of pulsed lasers, supercon-tinuum light, and ultrafast detectors have in recent years awakeninterest, and highlighted the need, to study optical coherence andpolarization of nonstationary light [103–108].

The fundamental measures to characterize any partially coher-ent and partially polarized light field are the degree of coherence andthe degree of polarization. Conventionally, these two quantities havebeen restricted, respectively, to scalar light and to two-component(2D) fields [54]. However, recent advances in nano-optics [2] andhigh-numerical-aperture imaging systems [109–111] have accentu-ated the importance to extend these concepts for arbitrary three-component (3D) light fields. Partial coherence of vectorial light maybe assessed by the electromagnetic degree of coherence [112–114],which describes the strength of correlations that exist between allthe orthogonal components of the electric field at a pair of points.In addition, it is invariant under unitary transformations, a naturalrequirement [115], and most importantly, amounts to equivalencebetween complete coherence and factorization of the coherence ma-trix (see below). We emphasize, though, that other coherence mea-sures have been suggested [116–123], with different physical moti-vations, implications, and mathematical properties [124]. The 3Ddegree of polarization, in turn, can be constructed via an expansionof the polarization matrix in terms of the Gell-Mann matrices andthe generalized Stokes parameters [125], in analogy with the Paulispin matrices and the usual Stokes parameters for the customary


Introduction

2D degree of polarization [54]. In this representation, the degreeof polarization can be interpreted as the square root of the averageof the normalized correlations squared among the three orthogonalelectric-field components in a reference frame where the diagonalelements of the polarization matrix are equal. However, also othermeans to address partial polarization of 3D-light fields have beenproposed [126–132], and compared [133, 134].

Around the time of the invention of the laser, influenced by therevolutionary experiment performed in 1956 by Hanbury Brownand Twiss [135], Glauber formulated the quantum theory of opticalcoherence in terms of nth-order correlation functions [136]. Higher-order correlations have gained major importance in quantum op-tics [137], since they can supply information about the nonclassi-cal nature of light. Furthermore, such correlations have a crucialposition in the characterization of quantum polarization [138], es-pecially in addressing the degree(s) of polarization for quantizedlight, a subject which is in constant development [139]. At the heartof Glauber’s seminal work is the equivalence between complete co-herence and the factorization of the coherence matrix into a productof two vectors. This foundation provides a more general defini-tion for complete coherence than the original definition from 1938of Zernike for classical scalar light [140], which connects completecoherence with full intensity visibility in Young’s interference ex-periment [141]. An essential requirement for a completely coherentfield, according to the definition of Glauber, is that it must be totallypolarized. Overall, as with the part played by Wolf (and Mandel)in the context of classical optics [142], the importance of Glauber’s(and Mandel’s) contributions for quantum optics is hard to overes-timate [143].

The principle of complementarity [144], stating that quantum ob-jects share mutually exclusive characteristics, has had a major sig-nificance for the foundations of physics and a profound impact onthe interpretation of the fundamental nature of reality [145, 146].The arguable most recognized manifestation of complementarity inphysics is the wave–particle duality, which places a trade-off for the



wave and particle qualities of quantum systems [147–150]. In thethird volume of his famous lecture series [151], Feynman declaresthat this dual wave–particle behavior “... has in it the heart of quan-tum mechanics. In reality, it contains the only mystery.” The dualityhas been given quantitative expressions in two-way interferometry[152–154], such as the celebrated double-slit (or double-pinhole) ex-periment [155,156], according to which path predictability and pathdistinguishability, representing different kinds of ’which-path infor-mation’, are complementary with the visibility of intensity fringes.In the case of photons, however, interference does not necessarilyappear only as intensity fringes, but also, or exclusively, as polariza-tion contrasts [157–159]. How complementarity is manifested, andquantified, under such polarization modulation, has not been consid-ered before.

1.2 SCOPE OF THE THESIS

This thesis encompasses fundamental theoretical research on sev-eral topics in electromagnetic nanophotonics. The subjects coveredcan be classified into three main categories: rigorous modal studiesof SPPs (Publications I–III), partially coherent optical surface fields(Publications IV–VII), and complementarity in vector-light photoninterference (Publication VIII). Below we give a brief overview ofthe central aspects concerning these themes; in the chapters thatfollow we address the motivations, results, and implications of ourresearch in more detail.

Publications I–III deal with novel SPP modes at planar single-interface and metal-slab geometries. In Publication I, by utilizing arigorous electromagnetic treatment, we demonstrate that the stan-dard approximate approach that is frequently used to characterizeSPPs at a single boundary can lead to false predictions even in situa-tions where it is supposed to be valid. As a main result, we predict anew type of backward-propagating SPP mode that does not followfrom the approximate analysis. Publication II establishes a unifiedframework and classification for all possible mode solutions, in-


Introduction

cluding sets of entirely new ones, at a lossy metal slab in a symmet-ric and lossless surrounding. While previous works have focusedmerely on the region outside the slab, we investigate the propaga-tion and energy-flow features of the modes also within the film. Itis found that the various modes may appear either as forward- orbackward-propagating waves inside the slab. In Publication III, weshow how a higher-order metal-slab mode, commonly presumedto have little practical importance, can turn into a strongly confinedlong-range surface mode in circumstances where the fundamentallong-range SPP does not exist and the propagation length of thesingle-interface SPP is minuscule. This discovery, which is encoun-tered for a broad range of materials and bandwidths, but whichseems not to have been observed or even suggested before, formsthe culmination of our modal studies.

Publications IV and V consider partially coherent SPP fields ata planar surface and on a metal slab. In publication IV, we formu-late a theory for partially coherent polychromatic SPPs excited ata metal–air interface in the Kretschmann configuration. The for-malism covers stationary as well as nonstationary SPP fields of anyspectra. The key result is establishing a generic scheme to tailor theelectromagnetic coherence of such polychromatic SPPs by control-ling the coherence state of the light source. The main objective ofPublication V is to examine the fundamental ranges that the spec-tral degrees of coherence and polarization of a stationary two-modefield composed of the long-range and short-range SPPs at a metallicnanoslab can attain, regardless of the excitation method. We alsoexplore how the degrees are influenced when the media, frequency,and film thickness are changed.

Publications VI and VII concern the generation, partial polar-ization, and spatial coherence of stationary, purely evanescent lightfields at a lossless dielectric surface. The analysis in Publication VIshows that, for such fields, the coherence length in air can attainvalues notably shorter than the free-space wavelength, in contrastto the common view that blackbody radiation exhibits the shortestcoherence length (about half a light’s wavelength). Publication VI



wave and particle qualities of quantum systems [147–150]. In thethird volume of his famous lecture series [151], Feynman declaresthat this dual wave–particle behavior “... has in it the heart of quan-tum mechanics. In reality, it contains the only mystery.” The dualityhas been given quantitative expressions in two-way interferometry[152–154], such as the celebrated double-slit (or double-pinhole) ex-periment [155,156], according to which path predictability and pathdistinguishability, representing different kinds of ’which-path infor-mation’, are complementary with the visibility of intensity fringes.In the case of photons, however, interference does not necessarilyappear only as intensity fringes, but also, or exclusively, as polariza-tion contrasts [157–159]. How complementarity is manifested, andquantified, under such polarization modulation, has not been consid-ered before.

1.2 SCOPE OF THE THESIS

This thesis encompasses fundamental theoretical research on sev-eral topics in electromagnetic nanophotonics. The subjects coveredcan be classified into three main categories: rigorous modal studiesof SPPs (Publications I–III), partially coherent optical surface fields(Publications IV–VII), and complementarity in vector-light photoninterference (Publication VIII). Below we give a brief overview ofthe central aspects concerning these themes; in the chapters thatfollow we address the motivations, results, and implications of ourresearch in more detail.

Publications I–III deal with novel SPP modes at planar single-interface and metal-slab geometries. In Publication I, by utilizing arigorous electromagnetic treatment, we demonstrate that the stan-dard approximate approach that is frequently used to characterizeSPPs at a single boundary can lead to false predictions even in situa-tions where it is supposed to be valid. As a main result, we predict anew type of backward-propagating SPP mode that does not followfrom the approximate analysis. Publication II establishes a unifiedframework and classification for all possible mode solutions, in-


Introduction

cluding sets of entirely new ones, at a lossy metal slab in a symmet-ric and lossless surrounding. While previous works have focusedmerely on the region outside the slab, we investigate the propaga-tion and energy-flow features of the modes also within the film. Itis found that the various modes may appear either as forward- orbackward-propagating waves inside the slab. In Publication III, weshow how a higher-order metal-slab mode, commonly presumedto have little practical importance, can turn into a strongly confinedlong-range surface mode in circumstances where the fundamentallong-range SPP does not exist and the propagation length of thesingle-interface SPP is minuscule. This discovery, which is encoun-tered for a broad range of materials and bandwidths, but whichseems not to have been observed or even suggested before, formsthe culmination of our modal studies.

Publications IV and V consider partially coherent SPP fields ata planar surface and on a metal slab. In publication IV, we formu-late a theory for partially coherent polychromatic SPPs excited ata metal–air interface in the Kretschmann configuration. The for-malism covers stationary as well as nonstationary SPP fields of anyspectra. The key result is establishing a generic scheme to tailor theelectromagnetic coherence of such polychromatic SPPs by control-ling the coherence state of the light source. The main objective ofPublication V is to examine the fundamental ranges that the spec-tral degrees of coherence and polarization of a stationary two-modefield composed of the long-range and short-range SPPs at a metallicnanoslab can attain, regardless of the excitation method. We alsoexplore how the degrees are influenced when the media, frequency,and film thickness are changed.

Publications VI and VII concern the generation, partial polar-ization, and spatial coherence of stationary, purely evanescent lightfields at a lossless dielectric surface. The analysis in Publication VIshows that, for such fields, the coherence length in air can attainvalues notably shorter than the free-space wavelength, in contrastto the common view that blackbody radiation exhibits the shortestcoherence length (about half a light’s wavelength). Publication VI



also demonstrates that evanescent fields are generally genuine 3D-polarized fields, highlighting the need for full 3D polarization treat-ment. In Publication VII, motivated by the results of PublicationVI, we explore conditions for controlled excitation of completely3D-unpolarized evanescent light fields in specific multibeam illu-mination setups and investigate their electromagnetic coherence.The results reveal the possibility to excite an evanescent field thatshares the polarization properties of blackbody radiation, yet withtunable coherence characteristics.

Publication VIII, dealing with quantum coherence and formingan essential part of this thesis, concerns complementarity and po-larization modulation in double-pinhole photon interference. Asa main contribution, we derive two general complementarity rela-tions for genuine vector-light quantum fields of arbitrary state. Thecomplementarity relations are shown to reflect two distinct, funda-mental aspects of wave–particle duality of the photon, having nocorrespondence in scalar quantum theory. In particular, we demon-strate that, contrary to scalar light, for pure single-photon vectorlight the a priori which-path information does not attach to the in-tensity visibility, but to a generalized visibility, which takes into ac-count the polarization-state modulation as well. It is also elucidatedthat in the general case such complementarity is a manifestation ofcomplete coherence, not of quantum-state purity.

Since many topics of this thesis involve electromagnetic surfacewaves at planar interfaces, we begin by introducing the basic for-malism for characterizing such fields in Chap. 2. The novel SPPmodes are presented in Chap. 3 (Publications I–III), whereas inChap. 4 we address partially coherent SPP fields (Publications IVand V). Chapter 5 concerns electromagnetic coherence and polar-ization of purely evanescent light fields (Publications VI and VII).Complementarity and polarization modulation in vectorial dual-pinhole photon interference are discussed in Chap. 6 (PublicationVIII). Finally, Chap. 7 summarizes the main conclusions and futureprospects of this work. A brief account of classical electromagneticcoherence theory is provided in Appendix A.


2 Electromagnetic surfacewaves

Despite its broad versatility, the SPP is just a certain species in therich family of surface polaritons [160], manifested also via phonons,excitons, and magnons, among others. Surface polaritons, in turn,constitute a part of the more general class of electromagnetic surfacewaves (ESWs) [161], whose attractiveness comes from their intrinsicunique capability of strong confinement and long-range guidanceof electromagnetic energy along the supporting interface. In thischapter, we introduce the basic formalism to characterize ESWs inhomogeneous and isotropic media. The formalism is extensivelyemployed in Publications I–VII.

2.1 NOMENCLATURE

In its simplest form, an ESW is a field that propagates along a pla-nar interface with an exponentially decaying amplitude away fromthe surface. Long before Fano introduced his superficial waves forthe optics community, ESWs had been studied by a completely dif-ferent physics community. In 1907, while analyzing radio-wavepropagation parallel to Earth’s surface, Zenneck found solutionsof Maxwell’s equations representing ESWs at a flat interface sep-arating two homogeneous media of different permittivities [162].Although Zenneck showed that Maxwell’s equations allow the exis-tence of such ESWs, he did not examine the excitation process of thefields. Two years later Sommerfeld published an influential paper,in which he investigated rigorously the field generated by a verticaldipole near a conductive plane [163]. He divided the dipole fieldinto two parts, a ‘space wave’ and a ‘surface wave’, and concludedthat the latter dominates near the boundary and goes over into thatpredicted by Zenneck as the distance is increased. However, the



also demonstrates that evanescent fields are generally genuine 3D-polarized fields, highlighting the need for full 3D polarization treat-ment. In Publication VII, motivated by the results of PublicationVI, we explore conditions for controlled excitation of completely3D-unpolarized evanescent light fields in specific multibeam illu-mination setups and investigate their electromagnetic coherence.The results reveal the possibility to excite an evanescent field thatshares the polarization properties of blackbody radiation, yet withtunable coherence characteristics.

Publication VIII, dealing with quantum coherence and formingan essential part of this thesis, concerns complementarity and po-larization modulation in double-pinhole photon interference. Asa main contribution, we derive two general complementarity rela-tions for genuine vector-light quantum fields of arbitrary state. Thecomplementarity relations are shown to reflect two distinct, funda-mental aspects of wave–particle duality of the photon, having nocorrespondence in scalar quantum theory. In particular, we demon-strate that, contrary to scalar light, for pure single-photon vectorlight the a priori which-path information does not attach to the in-tensity visibility, but to a generalized visibility, which takes into ac-count the polarization-state modulation as well. It is also elucidatedthat in the general case such complementarity is a manifestation ofcomplete coherence, not of quantum-state purity.

Since many topics of this thesis involve electromagnetic surfacewaves at planar interfaces, we begin by introducing the basic for-malism for characterizing such fields in Chap. 2. The novel SPPmodes are presented in Chap. 3 (Publications I–III), whereas inChap. 4 we address partially coherent SPP fields (Publications IVand V). Chapter 5 concerns electromagnetic coherence and polar-ization of purely evanescent light fields (Publications VI and VII).Complementarity and polarization modulation in vectorial dual-pinhole photon interference are discussed in Chap. 6 (PublicationVIII). Finally, Chap. 7 summarizes the main conclusions and futureprospects of this work. A brief account of classical electromagneticcoherence theory is provided in Appendix A.


2 Electromagnetic surfacewaves

Despite its broad versatility, the SPP is just a certain species in therich family of surface polaritons [160], manifested also via phonons,excitons, and magnons, among others. Surface polaritons, in turn,constitute a part of the more general class of electromagnetic surfacewaves (ESWs) [161], whose attractiveness comes from their intrinsicunique capability of strong confinement and long-range guidanceof electromagnetic energy along the supporting interface. In thischapter, we introduce the basic formalism to characterize ESWs inhomogeneous and isotropic media. The formalism is extensivelyemployed in Publications I–VII.

2.1 NOMENCLATURE

In its simplest form, an ESW is a field that propagates along a pla-nar interface with an exponentially decaying amplitude away fromthe surface. Long before Fano introduced his superficial waves forthe optics community, ESWs had been studied by a completely dif-ferent physics community. In 1907, while analyzing radio-wavepropagation parallel to Earth’s surface, Zenneck found solutionsof Maxwell’s equations representing ESWs at a flat interface sep-arating two homogeneous media of different permittivities [162].Although Zenneck showed that Maxwell’s equations allow the exis-tence of such ESWs, he did not examine the excitation process of thefields. Two years later Sommerfeld published an influential paper,in which he investigated rigorously the field generated by a verticaldipole near a conductive plane [163]. He divided the dipole fieldinto two parts, a ‘space wave’ and a ‘surface wave’, and concludedthat the latter dominates near the boundary and goes over into thatpredicted by Zenneck as the distance is increased. However, the



original paper contained an erroneous calculation which was latercorrected and extended by himself [164] and others [165–169].

Sommerfeld’s original paper left its mark on the radio-engineer-ing community, where the Zenneck wave and the Sommerfeld wavehave often been used as synonyms. Even more terms have been es-tablished over the years, which has caused misunderstandings andconfusion from time to time [170,171]. The extensive nomenclaturefor various ESWs serves as a good example: Zenneck wave, Som-merfeld wave, Norton surface wave, ground wave, improper mode,and lateral wave [171]. In addition, contradictory results concern-ing Zenneck waves have been published and no consensus on theirexistence seems yet to exist [172]. The problematic terminology isnot restricted to radio-wave physics only, but also concerns opticalplasmonics, where surface plasmons, surface-plasmon polaritons,Fano waves, and even Zenneck waves are often mixed and used todescribe same or different phenomena. With the incarnation of suchexotic ESWs as Tamm waves, Dyakonov waves, and Dyakonov–Tammwaves [161], not to disregard spoof modes [173–175], the taxonomyhas become even more involved. Nevertheless, it is important tounderstand that, regardless of the complex nomenclature and thedifferent physical origins of these various ESWs, their essential elec-tromagnetic character is the same.

2.2 FIELD CHARACTERIZATION

From a fundamental point of view, the complete description of anelectromagnetic field requires that both the electric field and themagnetic field are taken into account [176]. Nonetheless, since theinteraction between light and matter takes predominantly place viathe electric field (principally through the dipole moment) [177], wemainly focus on the electric part. As another starting point wetake all the involved media to be linear, homogeneous, isotropic,stationary in time, spatially nondispersive, free of sources, and pas-sive. The electromagnetic response of the medium then generallydepends on the (angular) frequency ω through a complex-valued


Electromagnetic surface waves

(relative) electric permittivity ϵr(ω) and a complex-valued (relative)magnetic permeability µr(ω), accounting to both temporal disper-sion as well as absorption. Alternatively, as is common in opticalphysics [177], the materials can be characterized by the refractiveindex n(ω) =

√ϵr(ω)µr(ω). Both conventions are alternately em-

ployed throughout this thesis.

2.2.1 Representation

Let us consider a monochromatic electromagnetic plane wave inconnection of a planar interface. The wave vector is taken to beof the form k(ω) = k∥(ω)e∥ + k⊥(ω)e⊥, where k∥(ω) and k⊥(ω)

are complex numbers, whereas e∥ and e⊥ are real-valued unit vec-tors lying parallel and perpendicularly to the surface, respectively.In this case k(ω) and e⊥ span a plane analogous to the ‘plane ofincidence’ encountered in the context of light propagation acrossa boundary [177]; yet, since we are dealing with ESWs, we preferthe more neutral term ‘propagation plane’. The electric field, at aspace–time point (r, t), can thus be decomposed into an s-polarizedpart (which is perpendicular to the propagation plane) and a p-polarized part (which is parallel to it) as

E(r, t) = [Es(ω)s(ω) + Ep(ω)p(ω)]ei[k(ω)·r−ωt], (2.1)

where Es(ω) and Ep(ω) are the (complex-valued) amplitudes of thes-polarized and the p-polarized field components, respectively. Thecorresponding unit polarization vectors, s(ω) and p(ω), obeying

k(ω) · s(ω) = 0, k(ω) · p(ω) = 0, (2.2)

as required by Maxwell’s equations [176, 177], are constructed as

s(ω) = e⊥ × e∥, p(ω) = k(ω)× s(ω); k(ω) =k(ω)

|k(ω)| . (2.3)

In this way k(ω), s(ω), p(ω) constitutes a right-handed and unit-normalized vector triad.



original paper contained an erroneous calculation which was latercorrected and extended by himself [164] and others [165–169].

Sommerfeld’s original paper left its mark on the radio-engineer-ing community, where the Zenneck wave and the Sommerfeld wavehave often been used as synonyms. Even more terms have been es-tablished over the years, which has caused misunderstandings andconfusion from time to time [170,171]. The extensive nomenclaturefor various ESWs serves as a good example: Zenneck wave, Som-merfeld wave, Norton surface wave, ground wave, improper mode,and lateral wave [171]. In addition, contradictory results concern-ing Zenneck waves have been published and no consensus on theirexistence seems yet to exist [172]. The problematic terminology isnot restricted to radio-wave physics only, but also concerns opticalplasmonics, where surface plasmons, surface-plasmon polaritons,Fano waves, and even Zenneck waves are often mixed and used todescribe same or different phenomena. With the incarnation of suchexotic ESWs as Tamm waves, Dyakonov waves, and Dyakonov–Tammwaves [161], not to disregard spoof modes [173–175], the taxonomyhas become even more involved. Nevertheless, it is important tounderstand that, regardless of the complex nomenclature and thedifferent physical origins of these various ESWs, their essential elec-tromagnetic character is the same.

2.2 FIELD CHARACTERIZATION

From a fundamental point of view, the complete description of anelectromagnetic field requires that both the electric field and themagnetic field are taken into account [176]. Nonetheless, since theinteraction between light and matter takes predominantly place viathe electric field (principally through the dipole moment) [177], wemainly focus on the electric part. As another starting point wetake all the involved media to be linear, homogeneous, isotropic,stationary in time, spatially nondispersive, free of sources, and pas-sive. The electromagnetic response of the medium then generallydepends on the (angular) frequency ω through a complex-valued



(relative) electric permittivity ϵr(ω) and a complex-valued (relative)magnetic permeability µr(ω), accounting to both temporal disper-sion as well as absorption. Alternatively, as is common in opticalphysics [177], the materials can be characterized by the refractiveindex n(ω) =

√ϵr(ω)µr(ω). Both conventions are alternately em-

ployed throughout this thesis.

2.2.1 Representation

Let us consider a monochromatic electromagnetic plane wave inconnection of a planar interface. The wave vector is taken to beof the form k(ω) = k∥(ω)e∥ + k⊥(ω)e⊥, where k∥(ω) and k⊥(ω)

are complex numbers, whereas e∥ and e⊥ are real-valued unit vec-tors lying parallel and perpendicularly to the surface, respectively.In this case k(ω) and e⊥ span a plane analogous to the ‘plane ofincidence’ encountered in the context of light propagation acrossa boundary [177]; yet, since we are dealing with ESWs, we preferthe more neutral term ‘propagation plane’. The electric field, at aspace–time point (r, t), can thus be decomposed into an s-polarizedpart (which is perpendicular to the propagation plane) and a p-polarized part (which is parallel to it) as

E(r, t) = [Es(ω)s(ω) + Ep(ω)p(ω)]ei[k(ω)·r−ωt], (2.1)

where Es(ω) and Ep(ω) are the (complex-valued) amplitudes of thes-polarized and the p-polarized field components, respectively. Thecorresponding unit polarization vectors, s(ω) and p(ω), obeying

k(ω) · s(ω) = 0, k(ω) · p(ω) = 0, (2.2)

as required by Maxwell’s equations [176, 177], are constructed as

s(ω) = e⊥ × e∥, p(ω) = k(ω)× s(ω); k(ω) =k(ω)

|k(ω)| . (2.3)

In this way k(ω), s(ω), p(ω) constitutes a right-handed and unit-normalized vector triad.



We stress that, instead of the length |k(ω)| in Eq. (2.3), the wavevector is customarily normalized with respect to the wave numberk(ω) = k0(ω)n(ω) [2, 176–179], where k0(ω) is the free-space wavenumber. Such a choice, however, does not yield a k(ω) of unitlength in the case of a true complex-valued wave vector, for which|k(ω)| = k(ω). The virtue of using |k(ω)| instead of k(ω) in thenormalization is actually not merely mathematical, but rather phys-ical, as is explained in Sec. 2.4. Another point that must be empha-sized is that, regardless of the normalization, and even if s(ω) is (bydefinition) perpendicular to both k(ω) as well as p(ω), the vectorsk(ω) and p(ω) are principally not mutually orthogonal when thewave vector is complex [180], unlike sometimes stated [178]. Hencethe triad k(ω), s(ω), p(ω) is generally semi-orthogonal.

2.2.2 Propagation

To elucidate the propagation characteristics of an ESW, we adoptthe notation k(ω) = k′(ω) + ik′′(ω) for the wave vector, where thereal part k′(ω) = k′∥(ω)e∥ + k′⊥(ω)e⊥ describes the phase move-ment, while the imaginary part k′′(ω) = k′′∥(ω)e∥ + k′′⊥(ω)e⊥ ac-counts to the confinement of the field. The existence of an ESW re-quires, by definition, that at least k′′⊥(ω) = 0 and k′∥(ω) = 0. In thespecial case that k′⊥(ω) = k′′∥(ω) = 0 the ESW is pure, i.e., the fieldis purely evanescent perpendicularly to the boundary and strictlypropagating along it without attenuation. In general, though, bothk′⊥(ω) and k′′∥(ω) are nonzero, thereby resulting in quasibound orpseudo ESWs [181], which are neither purely evanescent away fromthe surface, nor solely propagating along it.

The condition k(ω) · k(ω) = k20(ω)n2(ω) enables to determine

the magnitudes and the relative directions of k′(ω) and k′′(ω). Inthe presence of absorption, we find that

|k′(ω)| > |k′′(ω)|, if |n′(ω)| > n′′(ω), (2.4)

|k′(ω)| = |k′′(ω)|, if |n′(ω)| = n′′(ω), (2.5)

|k′(ω)| < |k′′(ω)|, if |n′(ω)| < n′′(ω), (2.6)



where n′(ω) and n′′(ω) are, respectively, the real and imaginaryparts of the refractive index, with n′′(ω) > 0 [180], and

α < π/2, if n′(ω) > 0, (2.7)

α = π/2, if n′(ω) = 0, (2.8)

α > π/2, if n′(ω) < 0, (2.9)

in which α ∈ [0, π] is the angle between k′(ω) and k′′(ω). Further,

k′⊥(ω)k′′⊥(ω) < 0 ⇔ k′∥(ω)k′′∥(ω) > k20(ω)n′(ω)n′′(ω), (2.10)

k′⊥(ω) = 0 ⇔ k′∥(ω)k′′∥(ω) = k20(ω)n′(ω)n′′(ω), (2.11)

k′⊥(ω)k′′⊥(ω) > 0 ⇔ k′∥(ω)k′′∥(ω) < k20(ω)n′(ω)n′′(ω), (2.12)

where Eq. (2.10) corresponds to the situation in which the directionsof phase movement and amplitude attenuation are antiparallel per-pendicularly to the boundary, while in the case of Eq. (2.12) theyare collinear. At the transition point, Eq. (2.11), the ESW is purelyevanescent transversally to the surface.

In a lossless medium, one instead always has |k′(ω)| > |k′′(ω)|and α = π/2, with the latter signifying orthogonality between thewavefront advancement and field attenuation. Moreover,

k′⊥(ω)k′′⊥(ω) = −k′∥(ω)k′′∥(ω), if n′′(ω) = 0, (2.13)

stating that if the directions of phase propagation and amplitudedecay are the same parallel to the surface, then they will be oppositeperpendicularly to it (and vice versa).

Eventually, the basic physical quantities characterizing the fieldpropagation of an ESW are [161]

Propagation length : l∥(ω) = 1/|k′′∥(ω)|, (2.14)

Penetration depth : l⊥(ω) = 1/|k′′⊥(ω)|, (2.15)

Surface wavelength : Λ(ω) = 2π/|k′∥(ω)|, (2.16)

Surface phase velocity : vp(ω) = ω/k′∥(ω), (2.17)

Surface group velocity : vg(ω) = ∂ω/∂k′∥(ω), (2.18)

which are frequently encountered later on.



We stress that, instead of the length |k(ω)| in Eq. (2.3), the wavevector is customarily normalized with respect to the wave numberk(ω) = k0(ω)n(ω) [2, 176–179], where k0(ω) is the free-space wavenumber. Such a choice, however, does not yield a k(ω) of unitlength in the case of a true complex-valued wave vector, for which|k(ω)| = k(ω). The virtue of using |k(ω)| instead of k(ω) in thenormalization is actually not merely mathematical, but rather phys-ical, as is explained in Sec. 2.4. Another point that must be empha-sized is that, regardless of the normalization, and even if s(ω) is (bydefinition) perpendicular to both k(ω) as well as p(ω), the vectorsk(ω) and p(ω) are principally not mutually orthogonal when thewave vector is complex [180], unlike sometimes stated [178]. Hencethe triad k(ω), s(ω), p(ω) is generally semi-orthogonal.

2.2.2 Propagation

To elucidate the propagation characteristics of an ESW, we adoptthe notation k(ω) = k′(ω) + ik′′(ω) for the wave vector, where thereal part k′(ω) = k′∥(ω)e∥ + k′⊥(ω)e⊥ describes the phase move-ment, while the imaginary part k′′(ω) = k′′∥(ω)e∥ + k′′⊥(ω)e⊥ ac-counts to the confinement of the field. The existence of an ESW re-quires, by definition, that at least k′′⊥(ω) = 0 and k′∥(ω) = 0. In thespecial case that k′⊥(ω) = k′′∥(ω) = 0 the ESW is pure, i.e., the fieldis purely evanescent perpendicularly to the boundary and strictlypropagating along it without attenuation. In general, though, bothk′⊥(ω) and k′′∥(ω) are nonzero, thereby resulting in quasibound orpseudo ESWs [181], which are neither purely evanescent away fromthe surface, nor solely propagating along it.

The condition k(ω) · k(ω) = k20(ω)n2(ω) enables to determine

the magnitudes and the relative directions of k′(ω) and k′′(ω). Inthe presence of absorption, we find that

|k′(ω)| > |k′′(ω)|, if |n′(ω)| > n′′(ω), (2.4)

|k′(ω)| = |k′′(ω)|, if |n′(ω)| = n′′(ω), (2.5)

|k′(ω)| < |k′′(ω)|, if |n′(ω)| < n′′(ω), (2.6)



where n′(ω) and n′′(ω) are, respectively, the real and imaginaryparts of the refractive index, with n′′(ω) > 0 [180], and

α < π/2, if n′(ω) > 0, (2.7)

α = π/2, if n′(ω) = 0, (2.8)

α > π/2, if n′(ω) < 0, (2.9)

in which α ∈ [0, π] is the angle between k′(ω) and k′′(ω). Further,

k′⊥(ω)k′′⊥(ω) < 0 ⇔ k′∥(ω)k′′∥(ω) > k20(ω)n′(ω)n′′(ω), (2.10)

k′⊥(ω) = 0 ⇔ k′∥(ω)k′′∥(ω) = k20(ω)n′(ω)n′′(ω), (2.11)

k′⊥(ω)k′′⊥(ω) > 0 ⇔ k′∥(ω)k′′∥(ω) < k20(ω)n′(ω)n′′(ω), (2.12)

where Eq. (2.10) corresponds to the situation in which the directionsof phase movement and amplitude attenuation are antiparallel per-pendicularly to the boundary, while in the case of Eq. (2.12) theyare collinear. At the transition point, Eq. (2.11), the ESW is purelyevanescent transversally to the surface.

In a lossless medium, one instead always has |k′(ω)| > |k′′(ω)|and α = π/2, with the latter signifying orthogonality between thewavefront advancement and field attenuation. Moreover,

k′⊥(ω)k′′⊥(ω) = −k′∥(ω)k′′∥(ω), if n′′(ω) = 0, (2.13)

stating that if the directions of phase propagation and amplitudedecay are the same parallel to the surface, then they will be oppositeperpendicularly to it (and vice versa).

Eventually, the basic physical quantities characterizing the fieldpropagation of an ESW are [161]

Propagation length : l∥(ω) = 1/|k′′∥(ω)|, (2.14)

Penetration depth : l⊥(ω) = 1/|k′′⊥(ω)|, (2.15)

Surface wavelength : Λ(ω) = 2π/|k′∥(ω)|, (2.16)

Surface phase velocity : vp(ω) = ω/k′∥(ω), (2.17)

Surface group velocity : vg(ω) = ∂ω/∂k′∥(ω), (2.18)

which are frequently encountered later on.



2.2.3 Energy flow

To assess the (net) flux of electromagnetic energy, we make use ofthe time-averaged Poynting vector for harmonic fields [2, 176],

S(r, ω) =12[E(r, t)× H∗(r, t)]′, (2.19)

where H(r, t) is the magnetic field, the asterix denotes complex con-jugation, and the prime stands for the real part. The Poynting vec-tor itself may, with certain caution, be interpreted as describing thedirection of the energy flux, whereas the integral of S(r, ω) takenover a closed surface amounts to the total energy flowing throughthe boundary of the considered volume. For an ESW, according toEqs. (2.1)–(2.3) and (2.19),

S(r, ω) =ϵ0c

2k0(ω)

[Ss(ω) + Sp(ω) + 2iSsp(ω)

µ∗r (ω)

]′e−2k′′(ω)·r, (2.20)

with ϵ0 and c being the vacuum permittivity and the speed of light,respectively, and

Ss(ω) = |Es(ω)|2k∗(ω), (2.21)

Sp(ω) = |Ep(ω)|2 k2∗(ω)

|k(ω)|2 k(ω), (2.22)

Ssp(ω) = E∗s (ω)Ep(ω)

[k∗∥(ω)k⊥(ω)]′′

|k(ω)| s(ω), (2.23)

where the double prime in Eq. (2.23) stands for the imaginary part.From Eqs. (2.20)–(2.23) we make three main observations.

Firstly, regardless of the polarization, the Poynting vector decaysexponentially at twice the rate of the field in the direction deter-mined by k′′(ω). Secondly, in a nonabsorptive medium, when theESW is fully s polarized or completely p polarized, the energy-flowdirection is either parallel or antiparallel to that of phase propaga-tion, specified by k′(ω). In our studies, fields whose wavefrontsmove parallel to S(r, ω) are regarded as forward-propagating waves,while those fields for which k′(ω) is opposite to the energy trans-fer are referred to as backward-propagating waves. Thirdly, in the case



that the ESW contains both s- and p-polarized contributions, thePoynting vector may have a component in a direction orthogonal tok′(ω). A nonzero energy flux perpendicularly to the propagationplane is not necessarily an artifact, but has been viewed as a man-ifestation of the inertial effect of the photon spin [17, 18, 182], alsoknown as the spin-Hall effect of light [183,184]. This prominent fea-ture is responsible for the Imbert–Fedorov effect [185,186], in whicha small transverse phase shift with respect to the propagation planeis detected when a light beam is totally internally reflected at an in-terface [187]. The phenomenon is akin to the Goos–Hanchen effectinvolving a longitudinal phase shift [188].

2.3 EXISTENCE OF ELECTROMAGNETIC SURFACE MODES

Up to this point, we have been dealing with a rather arbitrary andindependent ESW. According to Maxwell’s equations, however, theESW is unavoidably coupled to another field on the other side ofthe interface via the electromagnetic boundary conditions. This factplaces constraints on the existence of the ESW.

In this section, we consider surface-bound solutions for a con-figuration in which there is one wave, as given by Eq. (2.1), on eachside of the interface. Such delicate, self-sustained ESW solutionscan be regarded as representing electromagnetic surface modes (ESMs)of the system. Without loss of generality, we take the boundary to

)ω(r2µ,)ω(r2ǫ

)ω(r1µ,)ω(r1ǫ

z

x

)t,r(2E

)t,r(1E

Figure 2.1: Illustration of the geometry and notation for two electric plane waves at a flatinterface (z=0) between two media of relative electric permittivities ϵr1(ω) and ϵr2(ω) aswell as relative magnetic permeabilities µr1(ω) and µr2(ω).



2.2.3 Energy flow

To assess the (net) flux of electromagnetic energy, we make use ofthe time-averaged Poynting vector for harmonic fields [2, 176],

S(r, ω) =12[E(r, t)× H∗(r, t)]′, (2.19)

where H(r, t) is the magnetic field, the asterix denotes complex con-jugation, and the prime stands for the real part. The Poynting vec-tor itself may, with certain caution, be interpreted as describing thedirection of the energy flux, whereas the integral of S(r, ω) takenover a closed surface amounts to the total energy flowing throughthe boundary of the considered volume. For an ESW, according toEqs. (2.1)–(2.3) and (2.19),

S(r, ω) =ϵ0c

2k0(ω)

[Ss(ω) + Sp(ω) + 2iSsp(ω)

µ∗r (ω)

]′e−2k′′(ω)·r, (2.20)

with ϵ0 and c being the vacuum permittivity and the speed of light,respectively, and

Ss(ω) = |Es(ω)|2k∗(ω), (2.21)

Sp(ω) = |Ep(ω)|2 k2∗(ω)

|k(ω)|2 k(ω), (2.22)

Ssp(ω) = E∗s (ω)Ep(ω)

[k∗∥(ω)k⊥(ω)]′′

|k(ω)| s(ω), (2.23)

where the double prime in Eq. (2.23) stands for the imaginary part.From Eqs. (2.20)–(2.23) we make three main observations.

Firstly, regardless of the polarization, the Poynting vector decaysexponentially at twice the rate of the field in the direction deter-mined by k′′(ω). Secondly, in a nonabsorptive medium, when theESW is fully s polarized or completely p polarized, the energy-flowdirection is either parallel or antiparallel to that of phase propaga-tion, specified by k′(ω). In our studies, fields whose wavefrontsmove parallel to S(r, ω) are regarded as forward-propagating waves,while those fields for which k′(ω) is opposite to the energy trans-fer are referred to as backward-propagating waves. Thirdly, in the case



that the ESW contains both s- and p-polarized contributions, thePoynting vector may have a component in a direction orthogonal tok′(ω). A nonzero energy flux perpendicularly to the propagationplane is not necessarily an artifact, but has been viewed as a man-ifestation of the inertial effect of the photon spin [17, 18, 182], alsoknown as the spin-Hall effect of light [183,184]. This prominent fea-ture is responsible for the Imbert–Fedorov effect [185,186], in whicha small transverse phase shift with respect to the propagation planeis detected when a light beam is totally internally reflected at an in-terface [187]. The phenomenon is akin to the Goos–Hanchen effectinvolving a longitudinal phase shift [188].

2.3 EXISTENCE OF ELECTROMAGNETIC SURFACE MODES

Up to this point, we have been dealing with a rather arbitrary andindependent ESW. According to Maxwell’s equations, however, theESW is unavoidably coupled to another field on the other side ofthe interface via the electromagnetic boundary conditions. This factplaces constraints on the existence of the ESW.

In this section, we consider surface-bound solutions for a con-figuration in which there is one wave, as given by Eq. (2.1), on eachside of the interface. Such delicate, self-sustained ESW solutionscan be regarded as representing electromagnetic surface modes (ESMs)of the system. Without loss of generality, we take the boundary to

)ω(r2µ,)ω(r2ǫ

)ω(r1µ,)ω(r1ǫ

z

x

)t,r(2E

)t,r(1E

Figure 2.1: Illustration of the geometry and notation for two electric plane waves at a flatinterface (z=0) between two media of relative electric permittivities ϵr1(ω) and ϵr2(ω) aswell as relative magnetic permeabilities µr1(ω) and µr2(ω).



lie in the Cartesian xy plane (z = 0) and the wave propagation to bealong the x axis, as illustrated in Fig. 2.1, whereby k∥(ω) → kx(ω)

and k⊥(ω) → kz(ω). Under these conditions, the s-polarized andp-polarized field components have to satisfy

s polarization : µr2(ω)kz1(ω) = µr1(ω)kz2(ω), (2.24)

p polarization : ϵr2(ω)kz1(ω) = ϵr1(ω)kz2(ω), (2.25)

where the subscripts 1 and 2 have been introduced to distinguishthe half-space regions z < 0 (medium 1) and z > 0 (medium 2), re-spectively. Equations (2.24) and (2.25), which are found to be sym-metric with respect to the interchange of ϵr(ω) and µr(ω), spec-ify the exact conditions that the two fields at the planar interfacemust fulfill in order to exist. Therefore, we refer to Eqs. (2.24) and(2.25) as the existence conditions for the two-wave system depicted inFig. 2.1. When Eqs. (2.24) and (2.25) are further accompanied by thesurface-bound requirements k′′z1(ω) < 0 and k′′z2(ω) > 0, we end upwith the existence conditions for ESMs.

For most natural materials (dielectrics, metals, semiconductors)at optical frequencies, µr(ω) assumes up to a very good accuracythe value of unity [180]. In this case Eq. (2.24) states that kz1(ω) =

kz2(ω), implying that the surface-bound requirements k′′z1(ω) < 0and k′′z2(ω) > 0 cannot both be satisfied, thereby excluding the ex-istence of s-polarized ESMs (it should be mentioned, though, thatmetamaterials as well as nonhomogeneous and anisotropic mediapermit s-polarized ESMs within the optical domain [161, 189]). Forp polarization, on the other hand, there is no such proscription, andafter setting µr1(ω) = µr2(ω) = 1 as well as using the fact that thetangential wave-vector component is continuous across the bound-ary, viz., kx1(ω) = kx2(ω) = kx(ω), Eq. (2.25) yields

kx(ω) = k0(ω)

√ϵr1(ω)ϵr2(ω)

ϵr1(ω) + ϵr2(ω), (2.26)

kzα(ω) = k0(ω)ϵrα(ω)√

ϵr1(ω) + ϵr2(ω), α ∈ 1, 2. (2.27)



The relations above are well-established and specify the basic phys-ical properties [Eqs. (2.14)–(2.18)] of p-polarized ESMs [161].

In some studies, the basic starting point is to neglect absorp-tion, i.e., ϵ′′r1(ω) = ϵ′′r2(ω) = 0, whereupon Eqs. (2.26) and (2.27)lead to surface-bound solutions only when ϵ′r1(ω)ϵ′r2(ω) < 0 andϵ′r1(ω) + ϵ′r2(ω) < 0. Under these conditions the ESMs are pure andsometimes called Fano waves [161,172,190]. However, a more realis-tic scenario involves losses, at least in one of the media, renderingthe situation quite different. In particular, the inclusion of absorp-tion results in quasibound ESMs, since the wave-vector componentsgenerally become to include both real and imaginary parts. Yet, al-beit the pureness is lost as a result of the losses, what is perhapsless known is that the strict requirements on ϵr1(ω) and ϵr2(ω) alsovanish. In fact, when absorption is present, the interface supports(quasibound) ESMs for almost any value of ϵr1(ω) and ϵr2(ω) [191].

2.4 FRESNEL COEFFICIENTS ANDEXISTENCE CONDITIONS

Let us next consider three monochromatic plane waves, constructedaccording to Eqs. (2.1)–(2.3), at a planar boundary as illustrated inFig. 2.2. The setup is otherwise the same as in Fig. 2.1, but now withtwo fields in medium 1 and one field in medium 2. To distinguishthe waves in medium 1, we introduce the superscripts (1) and (2)

)ω(r2µ,)ω(r2ǫ

)ω(r1µ,)ω(r1ǫ

z

x

)t,r(2E

)t,r(1

)1(E )t,r(

1

)2(E

Figure 2.2: Illustration of the geometry and notation for three electric plane waves at a flatinterface (z=0) between two media of relative electric permittivities ϵr1(ω) and ϵr2(ω) aswell as relative magnetic permeabilities µr1(ω) and µr2(ω) (cf. Fig. 2.1).



lie in the Cartesian xy plane (z = 0) and the wave propagation to bealong the x axis, as illustrated in Fig. 2.1, whereby k∥(ω) → kx(ω)

and k⊥(ω) → kz(ω). Under these conditions, the s-polarized andp-polarized field components have to satisfy

s polarization : µr2(ω)kz1(ω) = µr1(ω)kz2(ω), (2.24)

p polarization : ϵr2(ω)kz1(ω) = ϵr1(ω)kz2(ω), (2.25)

where the subscripts 1 and 2 have been introduced to distinguishthe half-space regions z < 0 (medium 1) and z > 0 (medium 2), re-spectively. Equations (2.24) and (2.25), which are found to be sym-metric with respect to the interchange of ϵr(ω) and µr(ω), spec-ify the exact conditions that the two fields at the planar interfacemust fulfill in order to exist. Therefore, we refer to Eqs. (2.24) and(2.25) as the existence conditions for the two-wave system depicted inFig. 2.1. When Eqs. (2.24) and (2.25) are further accompanied by thesurface-bound requirements k′′z1(ω) < 0 and k′′z2(ω) > 0, we end upwith the existence conditions for ESMs.

For most natural materials (dielectrics, metals, semiconductors)at optical frequencies, µr(ω) assumes up to a very good accuracythe value of unity [180]. In this case Eq. (2.24) states that kz1(ω) =

kz2(ω), implying that the surface-bound requirements k′′z1(ω) < 0and k′′z2(ω) > 0 cannot both be satisfied, thereby excluding the ex-istence of s-polarized ESMs (it should be mentioned, though, thatmetamaterials as well as nonhomogeneous and anisotropic mediapermit s-polarized ESMs within the optical domain [161, 189]). Forp polarization, on the other hand, there is no such proscription, andafter setting µr1(ω) = µr2(ω) = 1 as well as using the fact that thetangential wave-vector component is continuous across the bound-ary, viz., kx1(ω) = kx2(ω) = kx(ω), Eq. (2.25) yields

kx(ω) = k0(ω)

√ϵr1(ω)ϵr2(ω)

ϵr1(ω) + ϵr2(ω), (2.26)


ϵr1(ω) + ϵr2(ω), α ∈ 1, 2. (2.27)



The relations above are well-established and specify the basic phys-ical properties [Eqs. (2.14)–(2.18)] of p-polarized ESMs [161].

In some studies, the basic starting point is to neglect absorp-tion, i.e., ϵ′′r1(ω) = ϵ′′r2(ω) = 0, whereupon Eqs. (2.26) and (2.27)lead to surface-bound solutions only when ϵ′r1(ω)ϵ′r2(ω) < 0 andϵ′r1(ω) + ϵ′r2(ω) < 0. Under these conditions the ESMs are pure andsometimes called Fano waves [161,172,190]. However, a more realis-tic scenario involves losses, at least in one of the media, renderingthe situation quite different. In particular, the inclusion of absorp-tion results in quasibound ESMs, since the wave-vector componentsgenerally become to include both real and imaginary parts. Yet, al-beit the pureness is lost as a result of the losses, what is perhapsless known is that the strict requirements on ϵr1(ω) and ϵr2(ω) alsovanish. In fact, when absorption is present, the interface supports(quasibound) ESMs for almost any value of ϵr1(ω) and ϵr2(ω) [191].

2.4 FRESNEL COEFFICIENTS ANDEXISTENCE CONDITIONS

Let us next consider three monochromatic plane waves, constructedaccording to Eqs. (2.1)–(2.3), at a planar boundary as illustrated inFig. 2.2. The setup is otherwise the same as in Fig. 2.1, but now withtwo fields in medium 1 and one field in medium 2. To distinguishthe waves in medium 1, we introduce the superscripts (1) and (2)

)ω(r2µ,)ω(r2ǫ

)ω(r1µ,)ω(r1ǫ

z

x

)t,r(2E

)t,r(1

)1(E )t,r(

1

)2(E

Figure 2.2: Illustration of the geometry and notation for three electric plane waves at a flatinterface (z=0) between two media of relative electric permittivities ϵr1(ω) and ϵr2(ω) aswell as relative magnetic permeabilities µr1(ω) and µr2(ω) (cf. Fig. 2.1).



to these fields. From the electromagnetic boundary conditions wethen find that the Fresnel reflection coefficients, with respect to thes- and p-polarized amplitudes of the field E(1)

1 (r, t), become

rs(ω) =µr2(ω)k(1)z1 (ω)− µr1(ω)kz2(ω)

µr2(ω)k(1)z1 (ω) + µr1(ω)kz2(ω), (2.28)

rp(ω) =ϵr2(ω)k(1)z1 (ω)− ϵr1(ω)kz2(ω)

ϵr2(ω)k(1)z1 (ω) + ϵr1(ω)kz2(ω), (2.29)

while the transmission coefficients read as

ts(ω) =2µr2(ω)k(1)z1 (ω)

µr2(ω)k(1)z1 (ω) + µr1(ω)kz2(ω), (2.30)

tp(ω) =2ϵr1(ω)k(1)z1 (ω)

ϵr2(ω)k(1)z1 (ω) + ϵr1(ω)kz2(ω)

|k2(ω)||k1(ω)| , (2.31)

where |k1(ω)| = |k(1)1 (ω)| = |k(2)

1 (ω)| in Eq. (2.31). We emphasizethat the derivation of Eqs. (2.28)–(2.31) assumes nothing more thank(2)z1 (ω) = −k(1)z1 (ω) in medium 1.

As reported in Publication VI, the Fresnel transmission coeffi-cient tp(ω) in Eq. (2.31) differs from that given in most of the lit-erature [1, 2, 17, 18, 25, 176, 177] if the wave vectors k1(ω) and/ork2(ω) contain imaginary parts. The difference stems from the defi-nition of the basis vector k(ω) given by Eq. (2.3); instead of the unitvector k(ω) = k(ω)/|k(ω)|, many works use k(ω) = k(ω)/k(ω)

which is not normalized to unity for a complex-valued k(ω), asdiscussed in Sec. 2.2.1. The advantage of k(ω) = k(ω)/|k(ω)| isthat it always preserves the physical meaning of the transmissioncoefficient as being the ratio between the complex field amplitudeson the two opposite sides of the interface. For k(ω) = k(ω)/k(ω),on the other hand, this is only true for purely propagating waves inlossless media having real-valued wave vectors.

The Fresnel coefficients, Eqs. (2.28) and (2.30), and correspond-



ingly Eqs. (2.29) and (2.31), have singularities, respectively, at

−µr2(ω)k(1)z1 (ω) = µr1(ω)kz2(ω), (2.32)

−ϵr2(ω)k(1)z1 (ω) = ϵr1(ω)kz2(ω), (2.33)

which we refer to as pole conditions. Under these circumstances thefield E(1)

1 (r, t) becomes vanishingly small compared to E(2)1 (r, t) and

E2(r, t). The zeros of the reflection coefficients in Eqs. (2.28) and(2.29), on the other hand, are found at

µr2(ω)k(1)z1 (ω) = µr1(ω)kz2(ω), (2.34)

ϵr2(ω)k(1)z1 (ω) = ϵr1(ω)kz2(ω), (2.35)

respectively, corresponding to situations for which the field E(2)1 (r, t)

vanishes. We may hence interpret Eqs. (2.34) and (2.35) as general-izations of the standard Brewster angle [177], and consequently callthem Brewster conditions.

Let us consider the p-polarized case. Physically the pole andBrewster conditions represent a similar phenomenon in the sensethat if Eq. (2.33) or Eq. (2.35) is satisfied, the three-wave configura-tion in Fig. 2.2 reduces to the two-wave system in Fig. 2.1. Further-more, we observe that Eqs. (2.25), (2.33), and (2.35) are mathemati-cally of similar form [analogous connection is found for Eqs. (2.24),(2.32), and (2.34) representing s polarization]. Thus the existencecondition, the pole condition, and the Brewster condition describethe same physical two-field situation, but from slightly differentpoints of view.

SPPs have frequently been interpreted to correspond to the polecondition [2,25,192,193], while Zenneck waves have been associatedwith the Brewster condition [170, 194]. However, both wave typescan in fact be regarded as representing either the pole condition orthe Brewster condition. Note that we could equally well have de-fined the reflection and transmission coefficients with respect to thefield E(2)

1 (r, t). The expressions in Eqs. (2.28)–(2.35) would remainthe same, except for the change of the superscript (1) to (2). Since



to these fields. From the electromagnetic boundary conditions wethen find that the Fresnel reflection coefficients, with respect to thes- and p-polarized amplitudes of the field E(1)

1 (r, t), become

rs(ω) =µr2(ω)k(1)z1 (ω)− µr1(ω)kz2(ω)

µr2(ω)k(1)z1 (ω) + µr1(ω)kz2(ω), (2.28)

rp(ω) =ϵr2(ω)k(1)z1 (ω)− ϵr1(ω)kz2(ω)

ϵr2(ω)k(1)z1 (ω) + ϵr1(ω)kz2(ω), (2.29)

while the transmission coefficients read as

ts(ω) =2µr2(ω)k(1)z1 (ω)

µr2(ω)k(1)z1 (ω) + µr1(ω)kz2(ω), (2.30)

tp(ω) =2ϵr1(ω)k(1)z1 (ω)

ϵr2(ω)k(1)z1 (ω) + ϵr1(ω)kz2(ω)

|k2(ω)||k1(ω)| , (2.31)

where |k1(ω)| = |k(1)1 (ω)| = |k(2)

1 (ω)| in Eq. (2.31). We emphasizethat the derivation of Eqs. (2.28)–(2.31) assumes nothing more thank(2)z1 (ω) = −k(1)z1 (ω) in medium 1.

As reported in Publication VI, the Fresnel transmission coeffi-cient tp(ω) in Eq. (2.31) differs from that given in most of the lit-erature [1, 2, 17, 18, 25, 176, 177] if the wave vectors k1(ω) and/ork2(ω) contain imaginary parts. The difference stems from the defi-nition of the basis vector k(ω) given by Eq. (2.3); instead of the unitvector k(ω) = k(ω)/|k(ω)|, many works use k(ω) = k(ω)/k(ω)

which is not normalized to unity for a complex-valued k(ω), asdiscussed in Sec. 2.2.1. The advantage of k(ω) = k(ω)/|k(ω)| isthat it always preserves the physical meaning of the transmissioncoefficient as being the ratio between the complex field amplitudeson the two opposite sides of the interface. For k(ω) = k(ω)/k(ω),on the other hand, this is only true for purely propagating waves inlossless media having real-valued wave vectors.

The Fresnel coefficients, Eqs. (2.28) and (2.30), and correspond-



ingly Eqs. (2.29) and (2.31), have singularities, respectively, at

−µr2(ω)k(1)z1 (ω) = µr1(ω)kz2(ω), (2.32)

−ϵr2(ω)k(1)z1 (ω) = ϵr1(ω)kz2(ω), (2.33)

which we refer to as pole conditions. Under these circumstances thefield E(1)

1 (r, t) becomes vanishingly small compared to E(2)1 (r, t) and

E2(r, t). The zeros of the reflection coefficients in Eqs. (2.28) and(2.29), on the other hand, are found at

µr2(ω)k(1)z1 (ω) = µr1(ω)kz2(ω), (2.34)

ϵr2(ω)k(1)z1 (ω) = ϵr1(ω)kz2(ω), (2.35)

respectively, corresponding to situations for which the field E(2)1 (r, t)

vanishes. We may hence interpret Eqs. (2.34) and (2.35) as general-izations of the standard Brewster angle [177], and consequently callthem Brewster conditions.

Let us consider the p-polarized case. Physically the pole andBrewster conditions represent a similar phenomenon in the sensethat if Eq. (2.33) or Eq. (2.35) is satisfied, the three-wave configura-tion in Fig. 2.2 reduces to the two-wave system in Fig. 2.1. Further-more, we observe that Eqs. (2.25), (2.33), and (2.35) are mathemati-cally of similar form [analogous connection is found for Eqs. (2.24),(2.32), and (2.34) representing s polarization]. Thus the existencecondition, the pole condition, and the Brewster condition describethe same physical two-field situation, but from slightly differentpoints of view.

SPPs have frequently been interpreted to correspond to the polecondition [2,25,192,193], while Zenneck waves have been associatedwith the Brewster condition [170, 194]. However, both wave typescan in fact be regarded as representing either the pole condition orthe Brewster condition. Note that we could equally well have de-fined the reflection and transmission coefficients with respect to thefield E(2)

1 (r, t). The expressions in Eqs. (2.28)–(2.35) would remainthe same, except for the change of the superscript (1) to (2). Since



k(2)z1 (ω) = −k(1)z1 (ω), the pole and Brewster conditions in Eqs. (2.32)–(2.35) would then simply interchange their roles, i.e., the pole con-dition of the field E(1)

1 (r, t) is the same as the Brewster conditionof the wave E(2)

1 (r, t), and vice versa. Hence the pole and Brewsterconditions depend on the interpretation which of the two fields inmedium 1 is considered as the ‘incident’ wave and which as the ‘re-flected’ wave (a division that is not necessarily unambiguous whendealing with complex-valued wave vectors). Figure 2.2 might sug-gest that the field E(1)

1 (r, t) represents an ‘incident’ wave, whereasthe field E(2)

1 (r, t) stands for a ‘reflected’ wave. This is, however,not the case, but the purpose of Fig. 2.2 is merely to provide visualsupport for our discussion. In reality, we have not at any point spec-ified the absolute directions of either k(1)z1 (ω) or k(2)z1 (ω); only theirrelative directions have been determined via k(2)z1 (ω) = −k(1)z1 (ω).


3 Surface-plasmon polaritons

Ever since the foundations, many theoretical studies on SPPs haveconcerned idealized, lossless metals, usually by considering theconduction electrons as an undamped free-electron gas. Absorp-tion is, however, an integral part of most real metals in the opti-cal domain [195], mainly due to scattering processes and interbandtransitions. This is important to take into account if one aims fora deeper understanding as well as to improve the agreement be-tween theory and experiments. Particular attention should also bepaid when introducing any approximations into the SPP-field anal-ysis, since this can lead to solutions which are no longer admissi-ble [181]. At the same time, field solutions arising from a rigoroustreatment may not appear, or even have any correspondence, withina simplificative framework [191, 196–198].

Although the fabric for the fundamental properties of SPPs atplanar (and rough) interfaces was established nearly 30 years ago[82], recent studies have indicated that there still remain issues thatcould benefit from further critical analysis [181,196–199]. This chap-ter is an account of the novel single-interface SPP, metal-slab, andlong-range modes that are predicted in Publications I–III, in whichrigorous electromagnetic theory and empirical data for the materialparameters are employed.

3.1 SINGLE-INTERFACE MODES

The conceptually most fundamental SPP modes are those supportedby a single planar interface (see Fig. 2.1), whose propagation charac-teristics are completely specified by Eqs. (2.26) and (2.27). Whereasfor many dielectrics (and some semiconductors) ϵr(ω) can be takenas real-valued and positive, many metals, especially noble metals,possess losses and a negative real part of the relative permittivitywithin the visible spectrum [195]. Accordingly, we choose medium



k(2)z1 (ω) = −k(1)z1 (ω), the pole and Brewster conditions in Eqs. (2.32)–(2.35) would then simply interchange their roles, i.e., the pole con-dition of the field E(1)

1 (r, t) is the same as the Brewster conditionof the wave E(2)

1 (r, t), and vice versa. Hence the pole and Brewsterconditions depend on the interpretation which of the two fields inmedium 1 is considered as the ‘incident’ wave and which as the ‘re-flected’ wave (a division that is not necessarily unambiguous whendealing with complex-valued wave vectors). Figure 2.2 might sug-gest that the field E(1)

1 (r, t) represents an ‘incident’ wave, whereasthe field E(2)

1 (r, t) stands for a ‘reflected’ wave. This is, however,not the case, but the purpose of Fig. 2.2 is merely to provide visualsupport for our discussion. In reality, we have not at any point spec-ified the absolute directions of either k(1)z1 (ω) or k(2)z1 (ω); only theirrelative directions have been determined via k(2)z1 (ω) = −k(1)z1 (ω).


3 Surface-plasmon polaritons

Ever since the foundations, many theoretical studies on SPPs haveconcerned idealized, lossless metals, usually by considering theconduction electrons as an undamped free-electron gas. Absorp-tion is, however, an integral part of most real metals in the opti-cal domain [195], mainly due to scattering processes and interbandtransitions. This is important to take into account if one aims fora deeper understanding as well as to improve the agreement be-tween theory and experiments. Particular attention should also bepaid when introducing any approximations into the SPP-field anal-ysis, since this can lead to solutions which are no longer admissi-ble [181]. At the same time, field solutions arising from a rigoroustreatment may not appear, or even have any correspondence, withina simplificative framework [191, 196–198].

Although the fabric for the fundamental properties of SPPs atplanar (and rough) interfaces was established nearly 30 years ago[82], recent studies have indicated that there still remain issues thatcould benefit from further critical analysis [181,196–199]. This chap-ter is an account of the novel single-interface SPP, metal-slab, andlong-range modes that are predicted in Publications I–III, in whichrigorous electromagnetic theory and empirical data for the materialparameters are employed.

3.1 SINGLE-INTERFACE MODES

The conceptually most fundamental SPP modes are those supportedby a single planar interface (see Fig. 2.1), whose propagation charac-teristics are completely specified by Eqs. (2.26) and (2.27). Whereasfor many dielectrics (and some semiconductors) ϵr(ω) can be takenas real-valued and positive, many metals, especially noble metals,possess losses and a negative real part of the relative permittivitywithin the visible spectrum [195]. Accordingly, we choose medium



1 to represent the absorptive metal with ϵ′r1(ω) < 0 and ϵ′′r1(ω) > 0,while ϵr2(ω) of the lossless medium 2 is a positive real number.Then again, both media are taken to be nonmagnetic, as is applica-ble to most natural materials in the optical regime.

3.1.1 Approximate solutions vs. exact solutions

Discussions on SPP propagation at a planar interface under theinfluence of metal absorption can be found in standard textbooks[2, 82]. Such treatments, however, employ approximate expressionsfor the wave-vector components [Eqs. (A7)–(A9) in Publication I]which rely on the assumption that ϵ′′r1(ω) ≪ |ϵ′r1(ω)|. Another con-straint embedded in those approximations is that |ϵ′r1(ω)| > ϵr2(ω),leading to a cut-off frequency similar to that occurring in the loss-less free-electron model. At the cut-off frequency, where |ϵ′r1(ω)| =ϵr2(ω), both the real and imaginary parts of the wave vector di-verge, so that the mode becomes sort of a ‘frozen’ and infinitelystrongly localized surface ‘spot’. In contrast, such singularities arenever present in the exact expressions [Eqs. (3)–(5) in Publication I],neither do they prohibit mode solutions in the ‘forbidden’ regionwhere |ϵ′r1(ω)| < ϵr2(ω).

The rigorous expressions can, in fact, entail rather peculiar phys-ical features in regimes beyond the scope of the approximate frame-work. For instance, when |ϵ′r1(ω)| = ϵr2(ω) and ϵ′′r1(ω) ≫ |ϵ′r1(ω)|,the exact formulas yield a mode solution for which

kx(ω) ≈ k0(ω)√

ϵr2(ω)[1 + i

ϵr2(ω)

2ϵ′′r1(ω)

]. (3.1)

Surprisingly, the real part of kx(ω) in Eq. (3.1), being equal to thewave number in the region z > 0, suggests that the mode couldbe excited directly by light incident from medium 2. The imaginarypart, in turn, indicates that the damping of the mode is inverselyproportional to the losses, i.e., the larger the absorption, the longerthe propagation. Other curiosities emerging in the rigorous frame-work, which at first sight may appear puzzling, are the seeminglyinfinite and negative group velocities due to backbent dispersion


Surface-plasmon polaritons

curves [196]. However, superluminal (including infinite) as wellas negative group velocities are encountered in situations involvingstrong absorption [200], and in such cases the group velocity shouldnot be associated with the signal velocity [176].

3.1.2 Mode types SPP I and SPP II

In Publication I, based on an exact treatment, we demonstrate thatthe approximate approach discussed above, frequently used to as-sess SPP propagation [29, 30, 201], may lead to inaccurate conclu-sions even in situations where it is supposed to hold. Yet, the mainresult of Publication I is the prediction of a new type of backward-propagating SPP mode that does not follow from the customaryapproximate analysis.

To address the subject, in the following we constrain ourselvesto fields which attenuate in the positive x direction, i.e., k′′x(ω) > 0,but stress that the main conclusions are also valid for k′′x(ω) < 0.As shown rigorously in Publication I, under these conditions

k′x(ω) > 0, k′z2(ω) < 0, (3.2)

indicating that along the x axis the phase movement is in the samepositive direction as the amplitude attenuation, while along the zaxis in the lossless medium 2 the wavefronts propagate towards thesurface. The situation is more involved for k′z1(ω) within the metal,for which the exact treatment reveals a material dependency:

k′z1(ω) < 0, if ϵr2(ω) < ϵ′′r1(ω), (3.3)

k′z1(ω) < 0, if [ϵ′r1(ω) + ϵr2(ω)]2 > ϵ2r2(ω)− ϵ′′2r1 (ω), (3.4)

k′z1(ω) = 0, if [ϵ′r1(ω) + ϵr2(ω)]2 = ϵ2r2(ω)− ϵ′′2r1 (ω), (3.5)

k′z1(ω) > 0, if [ϵ′r1(ω) + ϵr2(ω)]2 < ϵ2r2(ω)− ϵ′′2r1 (ω). (3.6)

We may thereby define two different types of SPPs in medium 1 as

SPP I : k′z1(ω) < 0, SPP II : k′z1(ω) > 0, (3.7)

which are illustrated in Fig. 3.1. The situation of k′z1(ω) = 0, wherethe field in the metal is purely evanescent in the z direction and



1 to represent the absorptive metal with ϵ′r1(ω) < 0 and ϵ′′r1(ω) > 0,while ϵr2(ω) of the lossless medium 2 is a positive real number.Then again, both media are taken to be nonmagnetic, as is applica-ble to most natural materials in the optical regime.

3.1.1 Approximate solutions vs. exact solutions

Discussions on SPP propagation at a planar interface under theinfluence of metal absorption can be found in standard textbooks[2, 82]. Such treatments, however, employ approximate expressionsfor the wave-vector components [Eqs. (A7)–(A9) in Publication I]which rely on the assumption that ϵ′′r1(ω) ≪ |ϵ′r1(ω)|. Another con-straint embedded in those approximations is that |ϵ′r1(ω)| > ϵr2(ω),leading to a cut-off frequency similar to that occurring in the loss-less free-electron model. At the cut-off frequency, where |ϵ′r1(ω)| =ϵr2(ω), both the real and imaginary parts of the wave vector di-verge, so that the mode becomes sort of a ‘frozen’ and infinitelystrongly localized surface ‘spot’. In contrast, such singularities arenever present in the exact expressions [Eqs. (3)–(5) in Publication I],neither do they prohibit mode solutions in the ‘forbidden’ regionwhere |ϵ′r1(ω)| < ϵr2(ω).

The rigorous expressions can, in fact, entail rather peculiar phys-ical features in regimes beyond the scope of the approximate frame-work. For instance, when |ϵ′r1(ω)| = ϵr2(ω) and ϵ′′r1(ω) ≫ |ϵ′r1(ω)|,the exact formulas yield a mode solution for which

kx(ω) ≈ k0(ω)√

ϵr2(ω)[1 + i

ϵr2(ω)

2ϵ′′r1(ω)

]. (3.1)

Surprisingly, the real part of kx(ω) in Eq. (3.1), being equal to thewave number in the region z > 0, suggests that the mode couldbe excited directly by light incident from medium 2. The imaginarypart, in turn, indicates that the damping of the mode is inverselyproportional to the losses, i.e., the larger the absorption, the longerthe propagation. Other curiosities emerging in the rigorous frame-work, which at first sight may appear puzzling, are the seeminglyinfinite and negative group velocities due to backbent dispersion



curves [196]. However, superluminal (including infinite) as wellas negative group velocities are encountered in situations involvingstrong absorption [200], and in such cases the group velocity shouldnot be associated with the signal velocity [176].

3.1.2 Mode types SPP I and SPP II

In Publication I, based on an exact treatment, we demonstrate thatthe approximate approach discussed above, frequently used to as-sess SPP propagation [29, 30, 201], may lead to inaccurate conclu-sions even in situations where it is supposed to hold. Yet, the mainresult of Publication I is the prediction of a new type of backward-propagating SPP mode that does not follow from the customaryapproximate analysis.

To address the subject, in the following we constrain ourselvesto fields which attenuate in the positive x direction, i.e., k′′x(ω) > 0,but stress that the main conclusions are also valid for k′′x(ω) < 0.As shown rigorously in Publication I, under these conditions

k′x(ω) > 0, k′z2(ω) < 0, (3.2)

indicating that along the x axis the phase movement is in the samepositive direction as the amplitude attenuation, while along the zaxis in the lossless medium 2 the wavefronts propagate towards thesurface. The situation is more involved for k′z1(ω) within the metal,for which the exact treatment reveals a material dependency:

k′z1(ω) < 0, if ϵr2(ω) < ϵ′′r1(ω), (3.3)

k′z1(ω) < 0, if [ϵ′r1(ω) + ϵr2(ω)]2 > ϵ2r2(ω)− ϵ′′2r1 (ω), (3.4)

k′z1(ω) = 0, if [ϵ′r1(ω) + ϵr2(ω)]2 = ϵ2r2(ω)− ϵ′′2r1 (ω), (3.5)

k′z1(ω) > 0, if [ϵ′r1(ω) + ϵr2(ω)]2 < ϵ2r2(ω)− ϵ′′2r1 (ω). (3.6)

We may thereby define two different types of SPPs in medium 1 as

SPP I : k′z1(ω) < 0, SPP II : k′z1(ω) > 0, (3.7)

which are illustrated in Fig. 3.1. The situation of k′z1(ω) = 0, wherethe field in the metal is purely evanescent in the z direction and



SPP I SPP II

Figure 3.1: Illustration of the directions of phase movement (black arrows) and field atten-uation (solid-red curves) for SPP I (left) and SPP II (right) decaying to the right.

the wavefronts advance only along the surface, is a transition pointbetween SPP I and SPP II.

According to the approximate analysis, on the other hand, lean-ing on the assumptions ϵ′′r1(ω) ≪ |ϵ′r1(ω)| and |ϵ′r1(ω)| > ϵr2(ω),one always has k′z1(ω) < 0, corresponding to SPP I. The existence ofSPP II, a prediction of the rigorous analysis, is thereby completelyexcluded within the approximate framework. There is another se-vere issue with the approximate approach. To see this, we considera Ag–GaP interface at the free-space wavelength λ0(ω) = 632.8 nmfor which ϵr1(ω) = −15.85 + i1.08 [195] and ϵr2(ω) = 11.01 [202].Now ϵ′′r1(ω) ≪ |ϵ′r1(ω)| and |ϵ′r1(ω)| > ϵr2(ω), as required in theapproximate treatment, whereupon we should obtain k′z1(ω) < 0.However, by looking at Eqs. (3.3)–(3.6), derived by rigorous means,we find that these material parameter values fall under Eq. (3.6),expressing that k′z1(ω) > 0. Thus the approximate treatment doesnot only exclude SPP II, but it may also lead to false physical pre-dictions in situations where it is supposed to be valid.

3.1.3 Flow of energy

To gain further insight into SPP propagation based on an exacttreatment, we consider the energy flow of the (p-polarized) SPPsin terms of the Poynting vector given by Eq. (2.20), which yields

S(r, ω) ∝ [ϵ′r(ω)k′(ω) + ϵ′′r (ω)k′′(ω)]e−2k′′ ·r. (3.8)

For ϵ′r(ω) > 0 and ϵ′′r (ω) = 0, the relation above states that S(r, ω)

is parallel to k′(ω) and thus, according to Eq. (3.2), the energy



transfer in the region z > 0 is tilted toward the surface in the samedirection as the wavefronts propagate. Consequently, only forward-propagating SPPs are encountered in medium 2, as is characteristicof electromagnetic plane waves in dielectrics. Within the metal thesituation is more involved, due to absorption, whereupon it is il-lustrative to consider the energy flow separately along the z and xaxes.

As concluded in Publication I, the z component of the Poynt-ing vector is always negative within the metal, i.e., Sz1(r, ω) < 0,indicating that energy is transported away from the interface (andabsorbed) in medium 1. Interestingly, this is true even in the case ofEq. (3.5) with k′1z(ω) = 0, for which the field is purely evanescentperpendicularly to the surface. In view of Eq. (3.7) we then find thatSPP I is a forward-propagating wave, whereas SPP II is a backward-propagating wave with respect to the z axis. As pointed out in theprevious section, the backward-propagating SPP II is not met in thestandard approximate framework, but is exclusively a consequenceof the rigorous treatment. Regarding the x component, the exactanalysis reveals the possibility of energy transfer in both directions:

Sx1(r, ω) > 0, if |ϵ′r1(ω)| < ϵr2(ω)/2, (3.9)

Sx1(r, ω) < 0, if |ϵ′r1(ω)| > ϵr2(ω)/2, (3.10)

which is completely independent of ϵ′′r1(ω). Hence, since k′x(ω) > 0according to Eq. (3.2), a similar forward–backward behavior occursalso along the surface. At the transition point |ϵ′r1(ω)| = ϵr2(ω)/2,where Sx1(r, ω) = 0, no energy is transferred along the boundary.The approximate treatment, on the other hand, requiring |ϵ′r1(ω)| >ϵr2(ω), allows energy flow in only one direction.

It is important to understand that the physical mechanisms forthe appearance of forward–backward propagation along the z andx axes are very different: in the former the behavior arises from thechange of direction in the phase movement, while in the latter thefeature is caused by the change in the energy-flow direction. Aswe will discuss in the next chapter, also metal-slab modes possesssimilar field-propagation properties.



SPP I SPP II

Figure 3.1: Illustration of the directions of phase movement (black arrows) and field atten-uation (solid-red curves) for SPP I (left) and SPP II (right) decaying to the right.

the wavefronts advance only along the surface, is a transition pointbetween SPP I and SPP II.

According to the approximate analysis, on the other hand, lean-ing on the assumptions ϵ′′r1(ω) ≪ |ϵ′r1(ω)| and |ϵ′r1(ω)| > ϵr2(ω),one always has k′z1(ω) < 0, corresponding to SPP I. The existence ofSPP II, a prediction of the rigorous analysis, is thereby completelyexcluded within the approximate framework. There is another se-vere issue with the approximate approach. To see this, we considera Ag–GaP interface at the free-space wavelength λ0(ω) = 632.8 nmfor which ϵr1(ω) = −15.85 + i1.08 [195] and ϵr2(ω) = 11.01 [202].Now ϵ′′r1(ω) ≪ |ϵ′r1(ω)| and |ϵ′r1(ω)| > ϵr2(ω), as required in theapproximate treatment, whereupon we should obtain k′z1(ω) < 0.However, by looking at Eqs. (3.3)–(3.6), derived by rigorous means,we find that these material parameter values fall under Eq. (3.6),expressing that k′z1(ω) > 0. Thus the approximate treatment doesnot only exclude SPP II, but it may also lead to false physical pre-dictions in situations where it is supposed to be valid.

3.1.3 Flow of energy

To gain further insight into SPP propagation based on an exacttreatment, we consider the energy flow of the (p-polarized) SPPsin terms of the Poynting vector given by Eq. (2.20), which yields

S(r, ω) ∝ [ϵ′r(ω)k′(ω) + ϵ′′r (ω)k′′(ω)]e−2k′′ ·r. (3.8)

For ϵ′r(ω) > 0 and ϵ′′r (ω) = 0, the relation above states that S(r, ω)

is parallel to k′(ω) and thus, according to Eq. (3.2), the energy



transfer in the region z > 0 is tilted toward the surface in the samedirection as the wavefronts propagate. Consequently, only forward-propagating SPPs are encountered in medium 2, as is characteristicof electromagnetic plane waves in dielectrics. Within the metal thesituation is more involved, due to absorption, whereupon it is il-lustrative to consider the energy flow separately along the z and xaxes.

As concluded in Publication I, the z component of the Poynt-ing vector is always negative within the metal, i.e., Sz1(r, ω) < 0,indicating that energy is transported away from the interface (andabsorbed) in medium 1. Interestingly, this is true even in the case ofEq. (3.5) with k′1z(ω) = 0, for which the field is purely evanescentperpendicularly to the surface. In view of Eq. (3.7) we then find thatSPP I is a forward-propagating wave, whereas SPP II is a backward-propagating wave with respect to the z axis. As pointed out in theprevious section, the backward-propagating SPP II is not met in thestandard approximate framework, but is exclusively a consequenceof the rigorous treatment. Regarding the x component, the exactanalysis reveals the possibility of energy transfer in both directions:

Sx1(r, ω) > 0, if |ϵ′r1(ω)| < ϵr2(ω)/2, (3.9)

Sx1(r, ω) < 0, if |ϵ′r1(ω)| > ϵr2(ω)/2, (3.10)

which is completely independent of ϵ′′r1(ω). Hence, since k′x(ω) > 0according to Eq. (3.2), a similar forward–backward behavior occursalso along the surface. At the transition point |ϵ′r1(ω)| = ϵr2(ω)/2,where Sx1(r, ω) = 0, no energy is transferred along the boundary.The approximate treatment, on the other hand, requiring |ϵ′r1(ω)| >ϵr2(ω), allows energy flow in only one direction.

It is important to understand that the physical mechanisms forthe appearance of forward–backward propagation along the z andx axes are very different: in the former the behavior arises from thechange of direction in the phase movement, while in the latter thefeature is caused by the change in the energy-flow direction. Aswe will discuss in the next chapter, also metal-slab modes possesssimilar field-propagation properties.



3.2 METAL-SLAB MODES

What makes a thin metal slab with two boundaries fundamentallyvery different from a single metal surface is that, much like a dielec-tric waveguide, it can support an unlimited number of modes [198].Moreover, not only from a fundamental, but also from an applica-tion point of view, the thickness of the slab provides an importantdegree of freedom, not possessed by the single-interface geometry,which can be utilized to modify various physical properties of themodes. How the slab thickness (among other parameters) affectsthe coherence and polarization of an SPP field consisting of the twowell-known symmetric and antisymmetric modes [203], the subjectof Publication V, is discussed in the next chapter.

This section summarizes the results of Publication II, in whicha rigorous theoretical formulation based on electromagnetic planewaves is utilized to construct a unified framework and identifica-tion for all possible mode solutions at an absorptive (nonmagnetic)metal slab in a symmetric and lossless surrounding. The mode so-lutions are categorized into three main classes and divided furtherinto subspecies depending on their field profile and propagationcharacteristics. It turns out that the various modes appear not onlyas bound waves, but also as fields with an exponentially growingamplitude away from the supporting surface. We refer to thesetransversally growing fields as leaky waves, following the nomen-clature used by Burke, Stegeman, and Tamir [203]. Although anexponentially growing field amplitude is somewhat dubious, leakywaves can be practically meaningful in a transient sense over lim-ited regions of space [203]. Furthermore, whereas many works havefocused only on the region outside the film, we investigate the fieldcharacteristics also inside the slab. Energy-flow considerations re-veal that the various modes manifest themselves both as forward-and backward-propagating waves within the film.

To be specific, the geometry that we consider (see Fig. 3.2) en-compasses an absorptive metal film of thickness d, characterized bya complex-valued relative permittivity ϵr1(ω) = ϵ′r1(ω) + iϵ′′r1(ω) in



)ω(r2ǫ

)ω(r2ǫ

)ω(r1ǫ

z

)t,r(1

)1(E

)t,r(2

)1(E

)t,r(2

)2(E

)t,r(1

)2(E

x

d

Figure 3.2: Illustration of the geometry and notation for electric plane waves at a lossymetal film (|z| < d/2) of thickness d surrounded by a nonabsorptive medium (|z| ≥ d/2),possessing relative permittivities ϵr1(ω) (complex) and ϵr2(ω) (real), respectively.

the region |z| < d/2, with ϵ′r1(ω) < 0 and ϵ′′r1(ω) > 0, which is sur-rounded on both sides by a lossless medium possessing a real andpositive permittivity ϵr2(ω). The spatial part of the (p-polarized)electric field outside the slab reads as

E2(r, ω) =

E(1)2 (ω)p(1)

2 (ω)eik(1)2 (ω)·r, z ≥ d/2,

E(2)2 (ω)p(2)

2 (ω)eik(2)2 (ω)·r, z ≤ −d/2,

(3.11)

while the (p-polarized) field inside the film is expressed as

E1(r, ω) = E(1)1 (ω)p(1)

1 (ω)eik(1)1 (ω)·r + E(2)

1 (ω)p(2)1 (ω)eik(2)

1 (ω)·r. (3.12)

The superscripts (1) and (2) have been introduced to separate thewaves in each medium and the unit polarization vectors are con-structed according to Eq. (2.3).

As is customary, we thus examine a system in which there aretwo plane waves within and one plane wave on each side of the slab.However, unlike in previous studies, we place no restrictions on thewave-vector directions inside or outside the slab. More precisely,whereas the tangential components of the wave vectors are continu-ous across the boundaries, i.e., k(β)

xα (ω) = kx(ω) for all α, β ∈ 1, 2,we make no assumptions on the transverse components, for whichone has two options in each medium, namely k(2)zα (ω) = ±k(1)zα (ω).



3.2 METAL-SLAB MODES

What makes a thin metal slab with two boundaries fundamentallyvery different from a single metal surface is that, much like a dielec-tric waveguide, it can support an unlimited number of modes [198].Moreover, not only from a fundamental, but also from an applica-tion point of view, the thickness of the slab provides an importantdegree of freedom, not possessed by the single-interface geometry,which can be utilized to modify various physical properties of themodes. How the slab thickness (among other parameters) affectsthe coherence and polarization of an SPP field consisting of the twowell-known symmetric and antisymmetric modes [203], the subjectof Publication V, is discussed in the next chapter.

This section summarizes the results of Publication II, in whicha rigorous theoretical formulation based on electromagnetic planewaves is utilized to construct a unified framework and identifica-tion for all possible mode solutions at an absorptive (nonmagnetic)metal slab in a symmetric and lossless surrounding. The mode so-lutions are categorized into three main classes and divided furtherinto subspecies depending on their field profile and propagationcharacteristics. It turns out that the various modes appear not onlyas bound waves, but also as fields with an exponentially growingamplitude away from the supporting surface. We refer to thesetransversally growing fields as leaky waves, following the nomen-clature used by Burke, Stegeman, and Tamir [203]. Although anexponentially growing field amplitude is somewhat dubious, leakywaves can be practically meaningful in a transient sense over lim-ited regions of space [203]. Furthermore, whereas many works havefocused only on the region outside the film, we investigate the fieldcharacteristics also inside the slab. Energy-flow considerations re-veal that the various modes manifest themselves both as forward-and backward-propagating waves within the film.

To be specific, the geometry that we consider (see Fig. 3.2) en-compasses an absorptive metal film of thickness d, characterized bya complex-valued relative permittivity ϵr1(ω) = ϵ′r1(ω) + iϵ′′r1(ω) in



)ω(r2ǫ

)ω(r2ǫ

)ω(r1ǫ

z

)t,r(1

)1(E

)t,r(2

)1(E

)t,r(2

)2(E

)t,r(1

)2(E

x

d

Figure 3.2: Illustration of the geometry and notation for electric plane waves at a lossymetal film (|z| < d/2) of thickness d surrounded by a nonabsorptive medium (|z| ≥ d/2),possessing relative permittivities ϵr1(ω) (complex) and ϵr2(ω) (real), respectively.

the region |z| < d/2, with ϵ′r1(ω) < 0 and ϵ′′r1(ω) > 0, which is sur-rounded on both sides by a lossless medium possessing a real andpositive permittivity ϵr2(ω). The spatial part of the (p-polarized)electric field outside the slab reads as

E2(r, ω) =

E(1)2 (ω)p(1)

2 (ω)eik(1)2 (ω)·r, z ≥ d/2,

E(2)2 (ω)p(2)

2 (ω)eik(2)2 (ω)·r, z ≤ −d/2,

(3.11)

while the (p-polarized) field inside the film is expressed as

E1(r, ω) = E(1)1 (ω)p(1)

1 (ω)eik(1)1 (ω)·r + E(2)

1 (ω)p(2)1 (ω)eik(2)

1 (ω)·r. (3.12)

The superscripts (1) and (2) have been introduced to separate thewaves in each medium and the unit polarization vectors are con-structed according to Eq. (2.3).

As is customary, we thus examine a system in which there aretwo plane waves within and one plane wave on each side of the slab.However, unlike in previous studies, we place no restrictions on thewave-vector directions inside or outside the slab. More precisely,whereas the tangential components of the wave vectors are continu-ous across the boundaries, i.e., k(β)

xα (ω) = kx(ω) for all α, β ∈ 1, 2,we make no assumptions on the transverse components, for whichone has two options in each medium, namely k(2)zα (ω) = ±k(1)zα (ω).



As shown in Publication II, under these circumstances Maxwell’sequations permit three different mode-solution classes, which are

M1 : k(1)z1 (ω) = k(2)z1 (ω), k(1)z2 (ω) = k(2)z2 (ω), (3.13)

M2 : k(1)z1 (ω) = −k(2)z1 (ω), k(1)z2 (ω) = k(2)z2 (ω), (3.14)

M3 : k(1)z1 (ω) = −k(2)z1 (ω), k(1)z2 (ω) = −k(2)z2 (ω). (3.15)

Note that the cases k(1)z1 (ω) = k(2)z1 (ω) and k(1)z2 (ω) = −k(2)z2 (ω) arenot allowed. We next review the field characteristics of each typeseparately, to which end the notation k(1)zα (ω) = kzα(ω) is adopted.

3.2.1 Mode class M1

The first class, M1, standing for the case where the two waves insidethe slab coincide, are characterized by exactly the same wave-vectorcomponents as those of SPPs at a single interface,

kx(ω) = k0(ω)

√ϵr1(ω)ϵr2(ω)

ϵr1(ω) + ϵr2(ω), (3.16)


ϵr1(ω) + ϵr2(ω), α ∈ 1, 2. (3.17)

Yet, since in the present two-boundary situation k(1)z2 (ω) = k(2)z2 (ω),the class M1 always contains one bound wave and one leaky waveoutside the film. That the wave-vector components of the M1 modesare the same as in the single-interface case, and thereby completelyindependent of the slab thickness, might at first glance be some-what puzzling. After a second thought, however, this is quite intu-itive, because as only one plane wave exists on each side of the twointerfaces, the same single-interface existence condition [Eq. (2.25)]can be met at both boundaries. The M1 modes may thus be inter-preted as corresponding to conditions for which the Fresnel reflec-tion coefficient goes to zero outside as well as inside the film (seeSec. 2.4).

Regarding field propagation, as we show in Publication II,

k′x(ω)k′′x(ω) > 0, k′z2(ω)k′′z2(ω) < 0, (3.18)



indicating that the directions of phase propagation and amplitudeattenuation are the same along the x axis, but along the z axis theyare opposite in the region |z| ≥ d/2, respectively. For k1z(ω) in theregion |z| < d/2 we instead obtain three possibilities, which leadus to classify three different types of M1 modes as

M1I : k′z1(ω)k′′z1(ω) > 0, (3.19)

M1II : k′z1(ω) = 0, (3.20)

M1III : k′z1(ω)k′′z1(ω) < 0. (3.21)

M1I represents modes whose phases move in the same direction asthe fields decay, while for M1III the phase advancement is oppo-site to that of attenuation (along the z axis). M1II stands for thecase where the field is purely evanescent in the z direction and thewavefronts propagate only along the x axis.

In addition, it turns out that the imaginary part of the normalwave-vector components satisfy

k′′z1(ω)k′′z2(ω) < 0, (3.22)

i.e., they have opposite signs. Equations (3.18)–(3.22) then allow al-together twelve different combinations of field propagation for theM1 modes, of which those six corresponding to fields attenuatingin the positive x direction are illustrated in Fig. 3.3. We observe inparticular that all scenarios involve bound waves at one interface,but leaky waves at the other.

3.2.2 Mode class M2

The M2 class, defined via Eq. (3.14), represents the general case forwhich the wave vectors in the regions on the two sides of the slabare identical and there are two transversally counter-propagatingwaves within the slab (for M1 only one wave exists inside the slab).The M2 mode solutions are thereby a generalization of the propa-gating plane waves that on interference at a dielectric plane-parallelplate produce no reflected field [177].



As shown in Publication II, under these circumstances Maxwell’sequations permit three different mode-solution classes, which are

M1 : k(1)z1 (ω) = k(2)z1 (ω), k(1)z2 (ω) = k(2)z2 (ω), (3.13)

M2 : k(1)z1 (ω) = −k(2)z1 (ω), k(1)z2 (ω) = k(2)z2 (ω), (3.14)

M3 : k(1)z1 (ω) = −k(2)z1 (ω), k(1)z2 (ω) = −k(2)z2 (ω). (3.15)

Note that the cases k(1)z1 (ω) = k(2)z1 (ω) and k(1)z2 (ω) = −k(2)z2 (ω) arenot allowed. We next review the field characteristics of each typeseparately, to which end the notation k(1)zα (ω) = kzα(ω) is adopted.

3.2.1 Mode class M1

The first class, M1, standing for the case where the two waves insidethe slab coincide, are characterized by exactly the same wave-vectorcomponents as those of SPPs at a single interface,

kx(ω) = k0(ω)

√ϵr1(ω)ϵr2(ω)

ϵr1(ω) + ϵr2(ω), (3.16)


ϵr1(ω) + ϵr2(ω), α ∈ 1, 2. (3.17)

Yet, since in the present two-boundary situation k(1)z2 (ω) = k(2)z2 (ω),the class M1 always contains one bound wave and one leaky waveoutside the film. That the wave-vector components of the M1 modesare the same as in the single-interface case, and thereby completelyindependent of the slab thickness, might at first glance be some-what puzzling. After a second thought, however, this is quite intu-itive, because as only one plane wave exists on each side of the twointerfaces, the same single-interface existence condition [Eq. (2.25)]can be met at both boundaries. The M1 modes may thus be inter-preted as corresponding to conditions for which the Fresnel reflec-tion coefficient goes to zero outside as well as inside the film (seeSec. 2.4).

Regarding field propagation, as we show in Publication II,

k′x(ω)k′′x(ω) > 0, k′z2(ω)k′′z2(ω) < 0, (3.18)



indicating that the directions of phase propagation and amplitudeattenuation are the same along the x axis, but along the z axis theyare opposite in the region |z| ≥ d/2, respectively. For k1z(ω) in theregion |z| < d/2 we instead obtain three possibilities, which leadus to classify three different types of M1 modes as

M1I : k′z1(ω)k′′z1(ω) > 0, (3.19)

M1II : k′z1(ω) = 0, (3.20)

M1III : k′z1(ω)k′′z1(ω) < 0. (3.21)

M1I represents modes whose phases move in the same direction asthe fields decay, while for M1III the phase advancement is oppo-site to that of attenuation (along the z axis). M1II stands for thecase where the field is purely evanescent in the z direction and thewavefronts propagate only along the x axis.

In addition, it turns out that the imaginary part of the normalwave-vector components satisfy

k′′z1(ω)k′′z2(ω) < 0, (3.22)

i.e., they have opposite signs. Equations (3.18)–(3.22) then allow al-together twelve different combinations of field propagation for theM1 modes, of which those six corresponding to fields attenuatingin the positive x direction are illustrated in Fig. 3.3. We observe inparticular that all scenarios involve bound waves at one interface,but leaky waves at the other.

3.2.2 Mode class M2

The M2 class, defined via Eq. (3.14), represents the general case forwhich the wave vectors in the regions on the two sides of the slabare identical and there are two transversally counter-propagatingwaves within the slab (for M1 only one wave exists inside the slab).The M2 mode solutions are thereby a generalization of the propa-gating plane waves that on interference at a dielectric plane-parallelplate produce no reflected field [177].



IM1

IM1

IIM1

IIM1

IIIM1

IIIM1

Figure 3.3: Illustration of the possible directions of phase movement (black arrows) andfield attenuation (solid-red curves) for M1 modes decaying to the right. The left, middle,and right columns represent M1I [Eq. (3.19)], M1II [Eq. (3.20)], and M1III [Eq. (3.21)],respectively. The graphs in the bottom row are mirror images of those in the top row.

The electromagnetic boundary conditions dictate that the M2modes come in two species with different field profiles, denoted bythe subscripts + and −, whose wave-vector components read as

kx(ω) = k0(ω)

√ϵr1(ω)−

[ m±π

k0(ω)d]2, (3.23)

kz1(ω) = m±(π

d

), (3.24)

kz2(ω) = k0(ω)

√ϵr2(ω)− ϵr1(ω) +

[ m±π

k0(ω)d]2, (3.25)

where m+ (m−) is an even (odd) integer. Qualities that make the M2modes very different from those of the M1 class are that Eqs. (3.23)–(3.25) stand for an infinite number of modes and depend on theslab thickness, which Eqs. (3.16) and (3.17) instead do not. Otherprominent properties of the M2 modes are that kz1(ω) is purelyreal [even though ϵr1(ω) is complex] and fully independent of thematerial parameters, while kx(ω) does not depend on ϵr2(ω), also



Figure 3.4: Illustration of the possible directions of phase movement (black arrows) andfield attenuation (solid-red curves) for M2 modes decaying to the right. The graph to theright is a mirror image of that to the left.

at variance with the M1 solutions.There are, however, similarities between the two classes. One is

that also the M2 class contains a bound wave and a leaky wave inthe surrounding, since the same complex-valued kz2(ω) is encoun-tered in both regions outside the slab. In addition,

k′x(ω)k′′x(ω) > 0, k′z2(ω)k′′z2(ω) < 0, (3.26)

whereby the propagation of the M2 modes along the x axis andfor |z| ≥ d/2 is similar to that of the M1 modes [Eq. (3.18)]. Yet,the situation is different for |z| < d/2 because, as Eq. (3.24) shows,kz1(ω) is purely real and therefore the fields are neither decayingnor growing in the z direction. Moreover, for the M2 class thereare two transversally counter-propagating waves within the slaband, consequently, we cannot identify the direction of the total-fieldphase movement along the z axis inside the slab.

Eventually, we end up with four different field-propagation pos-sibilities for the M2 modes; those two representing waves decayingin the positive x direction are illustrated in Fig. 3.4.

3.2.3 Mode class M3

The last class, M3, as defined through Eq. (3.15), is the convention-ally studied scenario, involving electric fields whose normal com-ponents (or magnetic fields) are either symmetric or antisymmetric



IM1

IM1

IIM1

IIM1

IIIM1

IIIM1

Figure 3.3: Illustration of the possible directions of phase movement (black arrows) andfield attenuation (solid-red curves) for M1 modes decaying to the right. The left, middle,and right columns represent M1I [Eq. (3.19)], M1II [Eq. (3.20)], and M1III [Eq. (3.21)],respectively. The graphs in the bottom row are mirror images of those in the top row.

The electromagnetic boundary conditions dictate that the M2modes come in two species with different field profiles, denoted bythe subscripts + and −, whose wave-vector components read as

kx(ω) = k0(ω)

√ϵr1(ω)−

[ m±π

k0(ω)d]2, (3.23)

kz1(ω) = m±(π

d

), (3.24)

kz2(ω) = k0(ω)

√ϵr2(ω)− ϵr1(ω) +

[ m±π

k0(ω)d]2, (3.25)

where m+ (m−) is an even (odd) integer. Qualities that make the M2modes very different from those of the M1 class are that Eqs. (3.23)–(3.25) stand for an infinite number of modes and depend on theslab thickness, which Eqs. (3.16) and (3.17) instead do not. Otherprominent properties of the M2 modes are that kz1(ω) is purelyreal [even though ϵr1(ω) is complex] and fully independent of thematerial parameters, while kx(ω) does not depend on ϵr2(ω), also



Figure 3.4: Illustration of the possible directions of phase movement (black arrows) andfield attenuation (solid-red curves) for M2 modes decaying to the right. The graph to theright is a mirror image of that to the left.

at variance with the M1 solutions.There are, however, similarities between the two classes. One is

that also the M2 class contains a bound wave and a leaky wave inthe surrounding, since the same complex-valued kz2(ω) is encoun-tered in both regions outside the slab. In addition,

k′x(ω)k′′x(ω) > 0, k′z2(ω)k′′z2(ω) < 0, (3.26)

whereby the propagation of the M2 modes along the x axis andfor |z| ≥ d/2 is similar to that of the M1 modes [Eq. (3.18)]. Yet,the situation is different for |z| < d/2 because, as Eq. (3.24) shows,kz1(ω) is purely real and therefore the fields are neither decayingnor growing in the z direction. Moreover, for the M2 class thereare two transversally counter-propagating waves within the slaband, consequently, we cannot identify the direction of the total-fieldphase movement along the z axis inside the slab.

Eventually, we end up with four different field-propagation pos-sibilities for the M2 modes; those two representing waves decayingin the positive x direction are illustrated in Fig. 3.4.

3.2.3 Mode class M3

The last class, M3, as defined through Eq. (3.15), is the convention-ally studied scenario, involving electric fields whose normal com-ponents (or magnetic fields) are either symmetric or antisymmetric



with respect to z = 0, and which obey

Symmetric :ϵr1(ω)

ϵr2(ω)

kz2(ω)

kz1(ω)= tanh

[12

ikz1(ω)d], (3.27)

Antisymmetric :ϵr2(ω)

ϵr1(ω)

kz1(ω)

kz2(ω)= coth

[12

ikz1(ω)d]. (3.28)

Equations (3.27) and (3.28), both standing for an infinite number ofmodes for any chosen media, frequency, and film thickness [198],are transcendental equations for kx(ω) and generally require nu-merical methods to solve. Yet, as outlined in Publication II, all ofthese mode solutions can be divided (by analytical means) into twosets depending on their behavior in the limit d → ∞.

One of the sets corresponds to the wave-vector components

kx(ω) → k0(ω)

√ϵr1(ω)ϵr2(ω)

ϵr1(ω) + ϵr2(ω), (3.29)

kzα(ω) → k0(ω)ϵrα(ω)√

ϵr1(ω) + ϵr2(ω), α ∈ 1, 2, (3.30)

which are exactly those obtained for SPPs at a single interface. Themodes associated with the solutions of Eqs. (3.27) and (3.28) thatapproach the limits of Eqs. (3.29) and (3.30) as d → ∞ are there-fore regarded (at any d) as fundamental modes (FMs) [198]. To put itthe other way around, the FMs are associated with those fields thatarise from the coupling between the SPPs supported by the indi-vidual interfaces of the slab. As concluded in Publication II, thereare only two FMs, one symmetric and one antisymmetric, for anychosen media, frequency, and slab thickness.

The other set represents modes for which in the limit d → ∞

kx(ω) → k0(ω)√

ϵr1(ω), (3.31)

kz1(ω) → 0, (3.32)

kz2(ω) → k0(ω)√

ϵr2(ω)− ϵr1(ω), (3.33)

having no correspondence at a single boundary. We thereby refer tothese fields (at any d) as higher-order modes (HOMs). Unlike with the



FMs, there are an infinite number of symmetric and antisymmetricHOMs for any chosen media, frequency, and slab thickness [198].Yet, since the value kz1(ω) = 0 represents the scenario for whichthe electromagnetic field vanishes both inside and outside the film[198], all the HOMs disappear when d → ∞.

Below we examine the general field-propagation characteristicsof the FMs and HOMs separately, but before that we want to dis-cuss an important aspect concerning all the M3 modes. First of all,the situation outside the slab is now different from that of M1 andM2, because for the class M3 the waves on each side are both ei-ther bound or leaky [owing to k(1)z2 (ω) = −k(2)z2 (ω)]. However, thereis another fundamental difference between the classes. Unlike fre-quently asserted [203], we emphasize that for the M3 modes bothsigns in kz2(ω) = ±[k2

2(ω) − k2x(ω)]1/2, where k2(ω) is the wave

number in the surrounding, are not allowed for a fixed kx(ω), sinceEqs. (3.27) and (3.28) are not invariant with respect to the changeof sign of kz2(ω). This implies that for a given bound (leaky) so-lution Eqs. (3.27) and (3.28) do not admit the corresponding leaky(bound) mode. In other words, for any particular tangential wave-vector component kx(ω), Maxwell’s equations permit for the classM3 only a bound or a leaky mode, but not both. This property isat variance with the M1 and M2 classes, for which both signs aresimultaneously allowed for the transverse wave-vector componentoutside the film.

Fundamental modes

The numerical analysis of Publication II suggests that only boundFMs with k′′z2(ω) > 0 are allowed by Maxwell’s equations, whereasleaky FMs possessing k′′z2(ω) < 0 do not exist. Furthermore, ourstudies indicate that k′x(ω) and k′′x(ω) always have the same sign.Consequently, according to Eq. (2.13), we then have

k′x(ω)k′′x(ω) > 0, k′z2(ω) < 0, (3.34)

implying that the directions of wavefront propagation and ampli-tude attenuation of FMs are the same along the x axis (similarly to



with respect to z = 0, and which obey

Symmetric :ϵr1(ω)

ϵr2(ω)

kz2(ω)

kz1(ω)= tanh

[12

ikz1(ω)d], (3.27)

Antisymmetric :ϵr2(ω)

ϵr1(ω)

kz1(ω)

kz2(ω)= coth

[12

ikz1(ω)d]. (3.28)

Equations (3.27) and (3.28), both standing for an infinite number ofmodes for any chosen media, frequency, and film thickness [198],are transcendental equations for kx(ω) and generally require nu-merical methods to solve. Yet, as outlined in Publication II, all ofthese mode solutions can be divided (by analytical means) into twosets depending on their behavior in the limit d → ∞.

One of the sets corresponds to the wave-vector components

kx(ω) → k0(ω)

√ϵr1(ω)ϵr2(ω)

ϵr1(ω) + ϵr2(ω), (3.29)

kzα(ω) → k0(ω)ϵrα(ω)√

ϵr1(ω) + ϵr2(ω), α ∈ 1, 2, (3.30)

which are exactly those obtained for SPPs at a single interface. Themodes associated with the solutions of Eqs. (3.27) and (3.28) thatapproach the limits of Eqs. (3.29) and (3.30) as d → ∞ are there-fore regarded (at any d) as fundamental modes (FMs) [198]. To put itthe other way around, the FMs are associated with those fields thatarise from the coupling between the SPPs supported by the indi-vidual interfaces of the slab. As concluded in Publication II, thereare only two FMs, one symmetric and one antisymmetric, for anychosen media, frequency, and slab thickness.

The other set represents modes for which in the limit d → ∞

kx(ω) → k0(ω)√

ϵr1(ω), (3.31)

kz1(ω) → 0, (3.32)

kz2(ω) → k0(ω)√

ϵr2(ω)− ϵr1(ω), (3.33)

having no correspondence at a single boundary. We thereby refer tothese fields (at any d) as higher-order modes (HOMs). Unlike with the



FMs, there are an infinite number of symmetric and antisymmetricHOMs for any chosen media, frequency, and slab thickness [198].Yet, since the value kz1(ω) = 0 represents the scenario for whichthe electromagnetic field vanishes both inside and outside the film[198], all the HOMs disappear when d → ∞.

Below we examine the general field-propagation characteristicsof the FMs and HOMs separately, but before that we want to dis-cuss an important aspect concerning all the M3 modes. First of all,the situation outside the slab is now different from that of M1 andM2, because for the class M3 the waves on each side are both ei-ther bound or leaky [owing to k(1)z2 (ω) = −k(2)z2 (ω)]. However, thereis another fundamental difference between the classes. Unlike fre-quently asserted [203], we emphasize that for the M3 modes bothsigns in kz2(ω) = ±[k2

2(ω) − k2x(ω)]1/2, where k2(ω) is the wave

number in the surrounding, are not allowed for a fixed kx(ω), sinceEqs. (3.27) and (3.28) are not invariant with respect to the changeof sign of kz2(ω). This implies that for a given bound (leaky) so-lution Eqs. (3.27) and (3.28) do not admit the corresponding leaky(bound) mode. In other words, for any particular tangential wave-vector component kx(ω), Maxwell’s equations permit for the classM3 only a bound or a leaky mode, but not both. This property isat variance with the M1 and M2 classes, for which both signs aresimultaneously allowed for the transverse wave-vector componentoutside the film.

Fundamental modes

The numerical analysis of Publication II suggests that only boundFMs with k′′z2(ω) > 0 are allowed by Maxwell’s equations, whereasleaky FMs possessing k′′z2(ω) < 0 do not exist. Furthermore, ourstudies indicate that k′x(ω) and k′′x(ω) always have the same sign.Consequently, according to Eq. (2.13), we then have

k′x(ω)k′′x(ω) > 0, k′z2(ω) < 0, (3.34)

implying that the directions of wavefront propagation and ampli-tude attenuation of FMs are the same along the x axis (similarly to



Figure 3.5: Illustration of the possible directions of phase movement (black arrows) andfield attenuation (solid-red curves) for FMs decaying to the right. The left, middle, andright graphs correspond to the first, second, and third condition of Eq. (3.35), respectively.

the M1 and M2 modes) and the phases move towards the surfacesoutside the slab. Regarding kz1(ω), it turns out that all of the cases

k′z1(ω)k′′z1(ω) > 0, k′z1(ω) = 0, k′z1(ω)k′′z1(ω) < 0, (3.35)

are possible [the situation with k′′z1(ω) = 0 does not occur], where-upon the behavior of kz1(ω) for |z| < d/2 of the FMs to some extentresembles that of the M1 modes [Eqs. (3.19)–(3.21)]. Nevertheless,the M3 class encompasses two transversally counter-propagatingwaves within the slab, making the phase motion along the z axisambiguous (cf. M2 class).

The results above establish six separate combinations of fieldpropagation for the FMs, of which those three representing wavesdecaying in the positive x direction are illustrated in Fig. 3.5 (notethe absence of leaky FMs).

Higher-order modes

As demonstrated in Publication II, the HOMs occur in two distinctspecies. One of these, which we call HOMIs, are fields that manifestthemselves exclusively as bound waves outside the slab, regardlessof the film thickness. In other words,

HOMI : k′′z2(ω) > 0, ∀d. (3.36)



Figure 3.6: Illustration of the directions of phase movement (black arrows) and field atten-uation (solid-red curves) for HOMIs decaying to the right.

The wave propagation of the HOMIs along the x axis and for |z| ≥d/2 is similar to that of the FMs [Eq. (3.34)], i.e.,

k′x(ω)k′′x(ω) > 0, k′z2(ω) < 0, (3.37)

whereas for kz1(ω) inside the film one only has [cf. Eq. (3.35)]

k′z1(ω)k′′z1(ω) < 0. (3.38)

Figure 3.6 illustrates the total wave-propagation behavior of HOMIsattenuating in the positive x direction.

The other species, HOMII, is defined via

HOMII :

k′′z2(ω) > 0, if d < dc,

k′′z2(ω) = 0, if d = dc,

k′′z2(ω) < 0, if d > dc,

(3.39)

where dc is a critical thickness that depends on the particular mode,media, as well as frequency. Thus the HOMIIs can be either bound(d < dc), or leaky (d > dc), or strictly propagating in opposite direc-tions along the z axis outside the slab (d = dc); the only mode typein the M3 class with this kind of property.

The analysis in Publication II shows that the HOMIIs obey

k′x(ω)k′′x(ω) < 0, k′z2(ω) > 0, if d < dc, (3.40)

k′′x(ω) = 0, k′z2(ω) > 0, if d = dc, (3.41)

k′x(ω)k′′x(ω) > 0, k′z2(ω) > 0, if d > dc. (3.42)



Figure 3.5: Illustration of the possible directions of phase movement (black arrows) andfield attenuation (solid-red curves) for FMs decaying to the right. The left, middle, andright graphs correspond to the first, second, and third condition of Eq. (3.35), respectively.

the M1 and M2 modes) and the phases move towards the surfacesoutside the slab. Regarding kz1(ω), it turns out that all of the cases

k′z1(ω)k′′z1(ω) > 0, k′z1(ω) = 0, k′z1(ω)k′′z1(ω) < 0, (3.35)

are possible [the situation with k′′z1(ω) = 0 does not occur], where-upon the behavior of kz1(ω) for |z| < d/2 of the FMs to some extentresembles that of the M1 modes [Eqs. (3.19)–(3.21)]. Nevertheless,the M3 class encompasses two transversally counter-propagatingwaves within the slab, making the phase motion along the z axisambiguous (cf. M2 class).

The results above establish six separate combinations of fieldpropagation for the FMs, of which those three representing wavesdecaying in the positive x direction are illustrated in Fig. 3.5 (notethe absence of leaky FMs).

Higher-order modes

As demonstrated in Publication II, the HOMs occur in two distinctspecies. One of these, which we call HOMIs, are fields that manifestthemselves exclusively as bound waves outside the slab, regardlessof the film thickness. In other words,

HOMI : k′′z2(ω) > 0, ∀d. (3.36)



Figure 3.6: Illustration of the directions of phase movement (black arrows) and field atten-uation (solid-red curves) for HOMIs decaying to the right.

The wave propagation of the HOMIs along the x axis and for |z| ≥d/2 is similar to that of the FMs [Eq. (3.34)], i.e.,

k′x(ω)k′′x(ω) > 0, k′z2(ω) < 0, (3.37)

whereas for kz1(ω) inside the film one only has [cf. Eq. (3.35)]

k′z1(ω)k′′z1(ω) < 0. (3.38)

Figure 3.6 illustrates the total wave-propagation behavior of HOMIsattenuating in the positive x direction.

The other species, HOMII, is defined via

HOMII :

k′′z2(ω) > 0, if d < dc,

k′′z2(ω) = 0, if d = dc,

k′′z2(ω) < 0, if d > dc,

(3.39)

where dc is a critical thickness that depends on the particular mode,media, as well as frequency. Thus the HOMIIs can be either bound(d < dc), or leaky (d > dc), or strictly propagating in opposite direc-tions along the z axis outside the slab (d = dc); the only mode typein the M3 class with this kind of property.

The analysis in Publication II shows that the HOMIIs obey

k′x(ω)k′′x(ω) < 0, k′z2(ω) > 0, if d < dc, (3.40)

k′′x(ω) = 0, k′z2(ω) > 0, if d = dc, (3.41)

k′x(ω)k′′x(ω) > 0, k′z2(ω) > 0, if d > dc. (3.42)



cd>dcd=dd<d c

Figure 3.7: Illustration of the possible directions of phase motion (black arrows) and fieldattenuation (solid-red curves) for HOMIIs decaying to the right, when the slab thicknessd is smaller than (left), equal to (middle), and larger than (right) the critical thickness dc.

Equations (3.40)–(3.42) especially indicate that the HOMIIs possessk′z2(ω) > 0, stating that the wavefronts move away from the slab for|z| ≥ d/2, in contrast to the FMs [Eq. (3.34)] and HOMIs [Eq. (3.37)].Regarding kx(ω), the first case, Eq. (3.40), representing (bound)waves for which the phase motion and amplitude attenuation areopposite along the surfaces, and the second one, Eq. (3.41), stand-ing for fields that are purely evanescent in the x direction (in bothmedia), are situations not met earlier. The last scenario, Eq. (3.42),corresponds to (leaky) waves for which the behavior of kx(ω) isanalogous to that of the FMs [Eq. (3.34)] and HOMIs [Eq. (3.37)].When it comes to kz1(ω) of the HOMIIs, our investigations in Pub-lication II indicate that

k′z1(ω)k′′z1(ω) > 0, ∀d, (3.43)

contrary to the situation with the HOMIs [Eq. (3.38)].In Fig. 3.7 we have summarized the three possible cases of field

propagation for HOMIIs attenuating along the positive x axis. Fig-ure 3.7 illustrates three particular qualities that make the HOMIIsvery different from the other metal-slab modes. Firstly, as d < dc

(left graph), the phases advance in the negative x direction eventhough the waves attenuate in the positive direction. Secondly,when d = dc (middle graph), the fields become purely evanescentwith no phase movement at all along the x axis (in both media) andstrictly propagating along the z axis in the surrounding. Thirdly,



for d > dc (right graph), the HOMIIs outside the slab turn leaky,with the wavefronts tilted away from the interfaces.

3.2.4 Forward- and backward-propagating modes

As a final point in Publication II, we investigate the flow of energyof the various metal-slab modes. According to Eq. (2.20), regardlessof the mode type, the Poynting vector in the lossless surroundingis always parallel to the real part of the wave vector and decays attwice the rate of the field in the direction specified by the imag-inary part. Hence only forward propagation (FP) occurs for theM1–M3 modes outside the slab and their energy-flow behavior (inthat region) is illustrated by the black arrows and solid-red curvesin Figs. 3.3–3.7. The situation is significantly more involved insidethe absorptive film, where also backward propagation (BP) is pos-sible and where the energy-flow behavior depends strongly on theparticular mode type.

In Table 3.1 we have summarized the results on FP and BP forthe M1–M3 modes within the slab. It is seen that along the x axisboth FP and BP are possible for all modes species in the M1 andM3 classes, while for those in M2 exclusively FP occurs. Further weobserve that M1I and M1III are, respectively, the only mode typesfor which FP and BP are found (and defined) in the z direction. The

No

M1 M2 M3

I II IIISymmetric Antisymmetric

FM FMIHOM IIHOM IIHOMIHOM

Yes

Yes

Yes

Yes

Yes

Yes Yes Yes Yes Yes

Yes No No Yes Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

No Yes

FP

BP

axisx

axisx

axisz

axisz

Table 3.1: Summary of the possibility of FP and BP for the M1–M3 modes inside theslab. The yellow boxes correspond to situations where the FP–BP behavior arises from thechange of direction in the energy flow, while the blue boxes represent cases in which thebehavior is caused by the change in the phase direction. The grey areas stand for scenariosfor which FP or BP is not defined.



cd>dcd=dd<d c

Figure 3.7: Illustration of the possible directions of phase motion (black arrows) and fieldattenuation (solid-red curves) for HOMIIs decaying to the right, when the slab thicknessd is smaller than (left), equal to (middle), and larger than (right) the critical thickness dc.

Equations (3.40)–(3.42) especially indicate that the HOMIIs possessk′z2(ω) > 0, stating that the wavefronts move away from the slab for|z| ≥ d/2, in contrast to the FMs [Eq. (3.34)] and HOMIs [Eq. (3.37)].Regarding kx(ω), the first case, Eq. (3.40), representing (bound)waves for which the phase motion and amplitude attenuation areopposite along the surfaces, and the second one, Eq. (3.41), stand-ing for fields that are purely evanescent in the x direction (in bothmedia), are situations not met earlier. The last scenario, Eq. (3.42),corresponds to (leaky) waves for which the behavior of kx(ω) isanalogous to that of the FMs [Eq. (3.34)] and HOMIs [Eq. (3.37)].When it comes to kz1(ω) of the HOMIIs, our investigations in Pub-lication II indicate that

k′z1(ω)k′′z1(ω) > 0, ∀d, (3.43)

contrary to the situation with the HOMIs [Eq. (3.38)].In Fig. 3.7 we have summarized the three possible cases of field

propagation for HOMIIs attenuating along the positive x axis. Fig-ure 3.7 illustrates three particular qualities that make the HOMIIsvery different from the other metal-slab modes. Firstly, as d < dc

(left graph), the phases advance in the negative x direction eventhough the waves attenuate in the positive direction. Secondly,when d = dc (middle graph), the fields become purely evanescentwith no phase movement at all along the x axis (in both media) andstrictly propagating along the z axis in the surrounding. Thirdly,



for d > dc (right graph), the HOMIIs outside the slab turn leaky,with the wavefronts tilted away from the interfaces.

3.2.4 Forward- and backward-propagating modes

As a final point in Publication II, we investigate the flow of energyof the various metal-slab modes. According to Eq. (2.20), regardlessof the mode type, the Poynting vector in the lossless surroundingis always parallel to the real part of the wave vector and decays attwice the rate of the field in the direction specified by the imag-inary part. Hence only forward propagation (FP) occurs for theM1–M3 modes outside the slab and their energy-flow behavior (inthat region) is illustrated by the black arrows and solid-red curvesin Figs. 3.3–3.7. The situation is significantly more involved insidethe absorptive film, where also backward propagation (BP) is pos-sible and where the energy-flow behavior depends strongly on theparticular mode type.

In Table 3.1 we have summarized the results on FP and BP forthe M1–M3 modes within the slab. It is seen that along the x axisboth FP and BP are possible for all modes species in the M1 andM3 classes, while for those in M2 exclusively FP occurs. Further weobserve that M1I and M1III are, respectively, the only mode typesfor which FP and BP are found (and defined) in the z direction. The

No

M1 M2 M3

I II IIISymmetric Antisymmetric

FM FMIHOM IIHOM IIHOMIHOM

Yes

Yes

Yes

Yes

Yes

Yes Yes Yes Yes Yes

Yes No No Yes Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

No Yes

FP

BP

axisx

axisx

axisz

axisz

Table 3.1: Summary of the possibility of FP and BP for the M1–M3 modes inside theslab. The yellow boxes correspond to situations where the FP–BP behavior arises from thechange of direction in the energy flow, while the blue boxes represent cases in which thebehavior is caused by the change in the phase direction. The grey areas stand for scenariosfor which FP or BP is not defined.



HOMIIs, on the other hand, are the only mode species for which theFP–BP behavior is caused by the change of direction in the phasemovement; for all other cases the feature stems from the change inthe energy-flow direction.

3.3 LONG-RANGE MODES

Although the single-interface SPP exhibits many useful properties,it is also characterized by a relatively high propagation loss, whichlimits its feasibility especially for waveguide purposes. The metal-slab geometry, on the other hand, encompasses the salient qualitythat it can support a surface mode having a much longer propaga-tion range [191, 196, 198, 203]. This particular long-range mode, alsoknown as the long-range surface-plasmon polariton (LRSPP) [24,25,30,32, 80, 81, 161], is a family member of the symmetric M3 modes. Itcorresponds to the solution of Eq. (3.27) for which

kx(ω) → k0(ω)√

ϵr2(ω), (3.44)

kz1(ω) → k0(ω)√

ϵr1(ω)− ϵr2(ω), (3.45)

kz2(ω) → 0, (3.46)

in the limit d → ∞, representing a field that is vertically polar-ized outside the slab and of infinite extent along the x axis. If dis small enough, the LRSPP may attain a propagation length evenseveral hundreds or thousands of times greater than that of the re-spective single-interface SPP, which makes it highly attractive forSPP-based waveguide applications and integrated optical compo-nents [81, 204].

Unfortunately, albeit the propagation range extension of the LR-SPP mitigates the limitation of the single-interface SPP, usually thiscomes at the expense of reduced field confinement as the field getsforced out of the metal and spreads progressively into the sur-rounding with decreasing film thickness [81]. The extended prop-agation length may, on the other hand, outweigh the reduced sur-face confinement. Depending on the situation, e.g., the operating



frequency, some materials are more advantageous than others foroptimizing the trade-off between long propagation distance andstrong surface localization, but no single material seems to offersuperior performance for all applications [205]. The LRSPP is alsovery sensitive to surface properties, and when d → 0, effects suchas surface roughness and grain size in the film start playing in-creased roles [82]. In addition, as the thickness decreases below acertain threshold value, the slab becomes islandized due to the for-mation of voids, and the material parameters start to differ fromtheir bulk values [81]. Recently, however, methods which allow thefabrication of uniform, highly smooth (surface roughness around0.2 nm), ultra-thin (d ≈ 5 nm), low-loss films have been demon-strated [206, 207].

3.3.1 Mode interchanges

To date, the LRSPP has exclusively been associated with the (sym-metric) FM, defined by Eqs. (3.29) and (3.30), which originate fromthe coupling between the SPPs supported by the individual bound-aries of the slab [24, 25, 30, 80–82, 161, 191, 196, 198, 203, 204]. Thismight be the reason why traditionally only the FM of the M3 classhas acquired attention and physical importance, whereas the HOMsspecified via Eqs. (3.31)–(3.33), having no analogue in the single-interface geometry and typically possessing propagation distancesof only a few nanometers, are rarely encountered in the literature[198]. Indeed, owing to their extremely short propagation lengths,the HOMs do not appear (at first glance) to have any other prac-tical significance than in matching the boundary conditions uponSPP end launching [198].

These widely held viewpoints are challenged in Publication III,in which we demonstrate the transformation of a HOM, normallynot regarded to be useful, into a strongly confined long-range modein circumstances where the fundamental LRSPP does not exist andthe propagation length of the single-interface SPP is negligible. Inthis process, a peculiar mode transition takes place between a HOM



HOMIIs, on the other hand, are the only mode species for which theFP–BP behavior is caused by the change of direction in the phasemovement; for all other cases the feature stems from the change inthe energy-flow direction.

3.3 LONG-RANGE MODES

Although the single-interface SPP exhibits many useful properties,it is also characterized by a relatively high propagation loss, whichlimits its feasibility especially for waveguide purposes. The metal-slab geometry, on the other hand, encompasses the salient qualitythat it can support a surface mode having a much longer propaga-tion range [191, 196, 198, 203]. This particular long-range mode, alsoknown as the long-range surface-plasmon polariton (LRSPP) [24,25,30,32, 80, 81, 161], is a family member of the symmetric M3 modes. Itcorresponds to the solution of Eq. (3.27) for which

kx(ω) → k0(ω)√

ϵr2(ω), (3.44)

kz1(ω) → k0(ω)√

ϵr1(ω)− ϵr2(ω), (3.45)

kz2(ω) → 0, (3.46)

in the limit d → ∞, representing a field that is vertically polar-ized outside the slab and of infinite extent along the x axis. If dis small enough, the LRSPP may attain a propagation length evenseveral hundreds or thousands of times greater than that of the re-spective single-interface SPP, which makes it highly attractive forSPP-based waveguide applications and integrated optical compo-nents [81, 204].

Unfortunately, albeit the propagation range extension of the LR-SPP mitigates the limitation of the single-interface SPP, usually thiscomes at the expense of reduced field confinement as the field getsforced out of the metal and spreads progressively into the sur-rounding with decreasing film thickness [81]. The extended prop-agation length may, on the other hand, outweigh the reduced sur-face confinement. Depending on the situation, e.g., the operating



frequency, some materials are more advantageous than others foroptimizing the trade-off between long propagation distance andstrong surface localization, but no single material seems to offersuperior performance for all applications [205]. The LRSPP is alsovery sensitive to surface properties, and when d → 0, effects suchas surface roughness and grain size in the film start playing in-creased roles [82]. In addition, as the thickness decreases below acertain threshold value, the slab becomes islandized due to the for-mation of voids, and the material parameters start to differ fromtheir bulk values [81]. Recently, however, methods which allow thefabrication of uniform, highly smooth (surface roughness around0.2 nm), ultra-thin (d ≈ 5 nm), low-loss films have been demon-strated [206, 207].

3.3.1 Mode interchanges

To date, the LRSPP has exclusively been associated with the (sym-metric) FM, defined by Eqs. (3.29) and (3.30), which originate fromthe coupling between the SPPs supported by the individual bound-aries of the slab [24, 25, 30, 80–82, 161, 191, 196, 198, 203, 204]. Thismight be the reason why traditionally only the FM of the M3 classhas acquired attention and physical importance, whereas the HOMsspecified via Eqs. (3.31)–(3.33), having no analogue in the single-interface geometry and typically possessing propagation distancesof only a few nanometers, are rarely encountered in the literature[198]. Indeed, owing to their extremely short propagation lengths,the HOMs do not appear (at first glance) to have any other prac-tical significance than in matching the boundary conditions uponSPP end launching [198].

These widely held viewpoints are challenged in Publication III,in which we demonstrate the transformation of a HOM, normallynot regarded to be useful, into a strongly confined long-range modein circumstances where the fundamental LRSPP does not exist andthe propagation length of the single-interface SPP is negligible. Inthis process, a peculiar mode transition takes place between a HOM



0 8040 120 160

12080 160120800.4

0.2

0.6

0.8

0.07

0.08

0.09

1.0

0.0200

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0 8040 120 160

1600.4

0.2

0.6

0.8

0.07

0.08

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0.0200

d [nm]

0 8040 120 160

12080 1600.4

0.2

0.6

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0 8040 120 160

12080 1600.4

0.2

0.6

0.8

0.07

0.08

0.09

1.0

0.0200

d [nm]

)ω(

0λ

/)ω(

xl

)ω(

0λ

/)ω(

xl

)ω(

0λ

/)ω(

xl

)ω(

0λ

/)ω(

xl

Figure 3.8: Propagation length lx(ω) of the FM (solid-blue curves) and the lowest-orderHOM (dotted-red curves) at a Ag slab surrounded by SiO2 as a function of the filmthickness d for selected free-space wavelengths λ0(ω): 355 nm (top left), 353 nm (topright), 352 nm (bottom left), and 350 nm (bottom right). The insets give a close-up view ofthe FM–HOM interchange. The horizontal dashed-black lines corresponding to the single-interface SPP are included for reference. The relative permittivities for Ag and SiO2 areobtained from the empirical data of [195].

(to be more specific, a bound and nonradiative HOMI) and the FM,whereby the HOM assumes the capability of long-range guidancewhile the fundamental LRSPP vanishes.

As an example how the FM–HOM interchange is manifested inthe propagation length lx(ω) = 1/k′′x(ω) of the fields, we consider acase involving a Ag slab surrounded by SiO2. Figure 3.8 illustratesthe behavior of lx(ω) for the FM and the lowest-order HOM as afunction of d in the near-ultraviolet regime. The figure shows that,whereas for the free-space wavelength λ0(ω) = 355 nm (top left)we still have the customary scenario in which the FM (solid-bluecurve) evolves into the LRSPP as d gets small, for λ0(ω) = 350 nm(bottom right) the HOM (dotted-red curve) has become to stand



for the LRSPP. This extraordinary transition happens around d ≈110 nm between λ0(ω) = 353 nm (top right) and λ0(ω) = 352 nm(bottom left), which is demonstrated in detail by the insets. Moreprecisely, the propagation length for λ0(ω) = 353 nm below d ≈110 nm corresponding to the FM gets interchanged with that of theHOM when λ0(ω) = 352 nm, and vice versa.

Similar flips are also possible between the FM and the second-order, third-order, etc., HOM, implying that several HOMs mayhave a larger propagation length than that of the FM. The switchesaffect not only the propagation length, but also other physical prop-erties of the modes, such as the surface wavelength, dispersion, thepenetration depths into the slab and the surrounding, and the po-larization state, as demonstrated in Publication III.

3.3.2 Long-range higher-order modes

The appearance of such a FM–HOM crossover in which the lowest-order HOM evolves into a long-range mode is the main result ofPublication III. At the transition frequency, the propagation lengthof the HOM may experience even a remarkable thousandfold en-largement. Phenomena of this type, where the HOMs, contrary tocommon belief, acquire long-range wave guidance and thus practi-cal significance, seem not to have been observed or even suggestedbefore. It is important to understand that, unlike with the funda-mental LRSPP, the long-range HOM does not emerge from the cou-pling between the SPPs supported by the individual boundaries ofthe slab. The origin of the long-range HOM is completely differ-ent: for a thick film there are no HOMs (they vanish as the slabthickness gets large and hence have no single-boundary correspon-dence), but when the thickness is reduced the HOMs start to showup and, as the thickness is small enough, one of the HOMs turnsinto a long-range mode. Thus the long-range HOM does not fol-low from the coupling of the two single-interface SPPs on each sideof the film (their propagation lengths are generally negligible inthose situations where the long-range HOM exists), but is exclu-



0 8040 120 160

12080 160120800.4

0.2

0.6

0.8

0.07

0.08

0.09

1.0

0.0200

d [nm]

0 8040 120 160

1600.4

0.2

0.6

0.8

0.07

0.08

0.09

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0.0200

d [nm]

0 8040 120 160

12080 1600.4

0.2

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0.08

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0 8040 120 160

12080 1600.4

0.2

0.6

0.8

0.07

0.08

0.09

1.0

0.0200

d [nm]

)ω(

0λ

/)ω(

xl

)ω(

0λ

/)ω(

xl

)ω(

0λ

/)ω(

xl

)ω(

0λ

/)ω(

xl

Figure 3.8: Propagation length lx(ω) of the FM (solid-blue curves) and the lowest-orderHOM (dotted-red curves) at a Ag slab surrounded by SiO2 as a function of the filmthickness d for selected free-space wavelengths λ0(ω): 355 nm (top left), 353 nm (topright), 352 nm (bottom left), and 350 nm (bottom right). The insets give a close-up view ofthe FM–HOM interchange. The horizontal dashed-black lines corresponding to the single-interface SPP are included for reference. The relative permittivities for Ag and SiO2 areobtained from the empirical data of [195].

(to be more specific, a bound and nonradiative HOMI) and the FM,whereby the HOM assumes the capability of long-range guidancewhile the fundamental LRSPP vanishes.

As an example how the FM–HOM interchange is manifested inthe propagation length lx(ω) = 1/k′′x(ω) of the fields, we consider acase involving a Ag slab surrounded by SiO2. Figure 3.8 illustratesthe behavior of lx(ω) for the FM and the lowest-order HOM as afunction of d in the near-ultraviolet regime. The figure shows that,whereas for the free-space wavelength λ0(ω) = 355 nm (top left)we still have the customary scenario in which the FM (solid-bluecurve) evolves into the LRSPP as d gets small, for λ0(ω) = 350 nm(bottom right) the HOM (dotted-red curve) has become to stand



for the LRSPP. This extraordinary transition happens around d ≈110 nm between λ0(ω) = 353 nm (top right) and λ0(ω) = 352 nm(bottom left), which is demonstrated in detail by the insets. Moreprecisely, the propagation length for λ0(ω) = 353 nm below d ≈110 nm corresponding to the FM gets interchanged with that of theHOM when λ0(ω) = 352 nm, and vice versa.

Similar flips are also possible between the FM and the second-order, third-order, etc., HOM, implying that several HOMs mayhave a larger propagation length than that of the FM. The switchesaffect not only the propagation length, but also other physical prop-erties of the modes, such as the surface wavelength, dispersion, thepenetration depths into the slab and the surrounding, and the po-larization state, as demonstrated in Publication III.

3.3.2 Long-range higher-order modes

The appearance of such a FM–HOM crossover in which the lowest-order HOM evolves into a long-range mode is the main result ofPublication III. At the transition frequency, the propagation lengthof the HOM may experience even a remarkable thousandfold en-largement. Phenomena of this type, where the HOMs, contrary tocommon belief, acquire long-range wave guidance and thus practi-cal significance, seem not to have been observed or even suggestedbefore. It is important to understand that, unlike with the funda-mental LRSPP, the long-range HOM does not emerge from the cou-pling between the SPPs supported by the individual boundaries ofthe slab. The origin of the long-range HOM is completely differ-ent: for a thick film there are no HOMs (they vanish as the slabthickness gets large and hence have no single-boundary correspon-dence), but when the thickness is reduced the HOMs start to showup and, as the thickness is small enough, one of the HOMs turnsinto a long-range mode. Thus the long-range HOM does not fol-low from the coupling of the two single-interface SPPs on each sideof the film (their propagation lengths are generally negligible inthose situations where the long-range HOM exists), but is exclu-



sively a consequence of the slab geometry and seems to come outfrom ‘nothing’.

Furthermore, while the extended propagation range of the FMis generally associated with the loss of field confinement [81], forthe HOM the situation can be different. As shown in PublicationIII, the long-range HOM may have a stronger surface confinementthan the respective single-interface SPP even when the propagationlength of the former is over 100 times larger than that of the latter.In fact, the analysis in Publication III reveals that the propagationrange extension can, remarkably, even enhance the field localizationin some cases, thus suggesting that simultaneous optimization ofstrong field confinement and long-range propagation is possible forthe HOMs.

The long-range HOMs may occur for several different materialsand frequency ranges. Examples of some of these situations arepresented in Fig. 3.9 for free-space wavelengths extending from ex-treme ultraviolet to near infrared. Although it is not entirely clearwhy these FM–HOM interchanges occur, or under what circum-stances the HOM adopts long-range behavior, we have found thata long-range HOM is supported when

2ϵ′′r1(ω) |ϵ′r1(ω)| ϵr2(ω). (3.47)

In particular, the condition above requires that |ϵ′r1(ω)| ϵr2(ω); aregime which is commonly regarded as ‘forbidden’ in plasmonics.

100 200 300 400 500 600 700 800

]nm) [ω(0λ

2Ag/SiO

Ag/GaP

Cu/GaP

Au/GaPAg/ZnONa/vacuumAl/vacuum

2Al/SiO 2Na/SiO Na/ZnO Na/GaPNa/GaP

Figure 3.9: Examples of some materials and bandwidths for which the lowest-order HOMamounts to the LRSPP. The relative permittivities for SiO2, Ag, Au, Cu, and Na arefrom [195], and those for GaP, ZnO, and Al are from [202], [208], and [209], respectively.



Further one observes from Eq. (3.47) that a surrounding possessinga relatively small ϵr2(ω) demands low values for both ϵ′r1(ω) andϵ′′r1(ω), while a larger ϵr2(ω) allows more variation for the metal.The ongoing and open-ended search for better and alternative plas-monic materials [210–212] provides an excellent opportunity to ex-tend the borders for these highly localized long-range HOMs.



sively a consequence of the slab geometry and seems to come outfrom ‘nothing’.

Furthermore, while the extended propagation range of the FMis generally associated with the loss of field confinement [81], forthe HOM the situation can be different. As shown in PublicationIII, the long-range HOM may have a stronger surface confinementthan the respective single-interface SPP even when the propagationlength of the former is over 100 times larger than that of the latter.In fact, the analysis in Publication III reveals that the propagationrange extension can, remarkably, even enhance the field localizationin some cases, thus suggesting that simultaneous optimization ofstrong field confinement and long-range propagation is possible forthe HOMs.

The long-range HOMs may occur for several different materialsand frequency ranges. Examples of some of these situations arepresented in Fig. 3.9 for free-space wavelengths extending from ex-treme ultraviolet to near infrared. Although it is not entirely clearwhy these FM–HOM interchanges occur, or under what circum-stances the HOM adopts long-range behavior, we have found thata long-range HOM is supported when

2ϵ′′r1(ω) |ϵ′r1(ω)| ϵr2(ω). (3.47)

In particular, the condition above requires that |ϵ′r1(ω)| ϵr2(ω); aregime which is commonly regarded as ‘forbidden’ in plasmonics.

100 200 300 400 500 600 700 800

]nm) [ω(0λ

2Ag/SiO

Ag/GaP

Cu/GaP

Au/GaPAg/ZnONa/vacuumAl/vacuum

2Al/SiO 2Na/SiO Na/ZnO Na/GaPNa/GaP

Figure 3.9: Examples of some materials and bandwidths for which the lowest-order HOMamounts to the LRSPP. The relative permittivities for SiO2, Ag, Au, Cu, and Na arefrom [195], and those for GaP, ZnO, and Al are from [202], [208], and [209], respectively.



Further one observes from Eq. (3.47) that a surrounding possessinga relatively small ϵr2(ω) demands low values for both ϵ′r1(ω) andϵ′′r1(ω), while a larger ϵr2(ω) allows more variation for the metal.The ongoing and open-ended search for better and alternative plas-monic materials [210–212] provides an excellent opportunity to ex-tend the borders for these highly localized long-range HOMs.




4 Partially coherent surface-plasmon polaritons

So far, we have dealt with fully monochromatic and therefore alsocompletely deterministic ESWs. In reality, however, every electro-magnetic field found in nature exhibits at least some degree of ran-dom fluctuations, caused by scattering processes within the involvedmedium or, ultimately, by the indeterministic quantum-physicalatomic transitions in the field sources. These intrinsic, random fluc-tuations are manifested as partial coherence and partial polarization ofthe electromagnetic field.

In this chapter, we examine partially coherent and partially po-larized SPPs. Physically, the coherence and polarization states in-fluence the interaction of SPPs with the surrounding and collectionsof nanoparticles located in close proximity of the interface. For ex-ample, SPPs propagating on the surfaces of a metallic nanolayer canform a highly sensitive interferometric biosensor [213]. Likewise, anSPP field with varying spatial coherence may excite a random setof molecules to radiate coherently [214]. Nanoparticle scattering isknown to depend on the polarization properties of the field, to theextent that the field’s polarization state can be fully deduced frommeasurements of the scattered radiation [215]. Studying the coher-ence and polarization characteristics of SPPs is hence important,not only from a foundational, but also from an application point ofview, as coherence and polarization offer indispensable degrees offreedom for manipulating diverse light–matter interactions.

4.1 SINGLE-INTERFACE GEOMETRY

To date, albeit SPPs have attracted a significant amount of interestin fundamental and applied sciences, most plasmonics research has




4 Partially coherent surface-plasmon polaritons

So far, we have dealt with fully monochromatic and therefore alsocompletely deterministic ESWs. In reality, however, every electro-magnetic field found in nature exhibits at least some degree of ran-dom fluctuations, caused by scattering processes within the involvedmedium or, ultimately, by the indeterministic quantum-physicalatomic transitions in the field sources. These intrinsic, random fluc-tuations are manifested as partial coherence and partial polarization ofthe electromagnetic field.

In this chapter, we examine partially coherent and partially po-larized SPPs. Physically, the coherence and polarization states in-fluence the interaction of SPPs with the surrounding and collectionsof nanoparticles located in close proximity of the interface. For ex-ample, SPPs propagating on the surfaces of a metallic nanolayer canform a highly sensitive interferometric biosensor [213]. Likewise, anSPP field with varying spatial coherence may excite a random setof molecules to radiate coherently [214]. Nanoparticle scattering isknown to depend on the polarization properties of the field, to theextent that the field’s polarization state can be fully deduced frommeasurements of the scattered radiation [215]. Studying the coher-ence and polarization characteristics of SPPs is hence important,not only from a foundational, but also from an application point ofview, as coherence and polarization offer indispensable degrees offreedom for manipulating diverse light–matter interactions.

4.1 SINGLE-INTERFACE GEOMETRY

To date, albeit SPPs have attracted a significant amount of interestin fundamental and applied sciences, most plasmonics research has



involved monochromatic and, consequently, fully coherent SPPs.Coherence-tailored polychromatic SPPs could nonetheless serve asversatile tools, e.g., for ultrashort optical pulse manipulation innanostructured optoelectronic circuits [31], controlled coupling oflight-emitting elements [214,216], plasmon continuum spectroscopy[217], and subwavelength white-light imaging [218]. At the sametime, exploring the statistical features and excitation mechanismsof polychromatic SPP fields presents a fundamental interest.

In publication IV, we develop a theory for partially coherentpolychromatic SPPs at a metal–air interface. The formalism coversstationary as well as nonstationary SPP fields of arbitrary spectra.As a main result, we formulate a framework to tailor the electro-magnetic coherence of such polychromatic SPPs in the Kretschmannsetup [24, 25] by controlling the correlations of the excitation light.The connection between the coherence state of the light source andthe ensuing SPP field establishes a novel paradigm in statisticalplasmonics, which we refer to as plasmon coherence engineering.

4.1.1 Polychromatic surface-plasmon polaritons

Let us consider polychromatic SPPs generated in the Kretschmannconfiguration, sketched in Fig. 4.1, with an absorptive and nonmag-netic metal film deposited on a glass prism. The film is taken to bethick enough so that any coupling between the metal-slab modes

Air SPP

Metal

Glass

Figure 4.1: Polychromatic SPP excitation in the Kretschmann coupling modality.


Partially coherentsurface-plasmon polaritons

can be neglected, whereupon the region near the metal–air surfacecan be treated as the semi-infinite half-space geometry in Fig. 2.1.The (p-polarized) polychromatic electric field in air then reads as

E(r, t) =∫ ω+

ω−E(ω)p(ω)ei[k(ω)·r−ωt]dω, (4.1)

where ω−,+ specify the frequency range, E(ω) is the spectral com-plex field amplitude of a monochromatic SPP at the origin (r = 0),and p(ω) is the respective polarization vector given by Eq. (2.3).Note that we have dropped the subscript 2 (referring to medium 2)in order to keep the notation simpler.

Now, let Eq. (4.1) represent a realization of the random electricfield. All coherence information of the polychormatic SPP field isthen encoded in the electric coherence matrix [Eq. (A.2)]

Γ(r1, t1; r2, t2) =∫ ω+

ω−

∫ ω+

ω−W(r1, ω1; r2, ω2)e−i(ω2t2−ω1t1)dω1dω2, (4.2)

in which we have the spectral electric coherence matrix [Eq. (A.4)]

W(r1, ω1; r2, ω2) = W(ω1, ω2)K(ω1, ω2)ei[k(ω2)·r2−k∗(ω1)·r1], (4.3)

including the spectral electric correlation function

W(ω1, ω2) = ⟨E∗(ω1)E(ω2)⟩, (4.4)

and the matrix

K(ω1, ω2) = [|k(ω1)||k(ω2)|]−1

×[

k∗z(ω1)kz(ω2) −k∗z(ω1)kx(ω2)

−k∗x(ω1)kz(ω2) k∗x(ω1)kx(ω2)

]. (4.5)

Equation (4.2) is general as it sets no restrictions on metal disper-sion, the spectrum of the field, or the spectral correlations; it coversany partially coherent polychromatic SPP field. The spectral cor-relation function W(ω1, ω2) of Eq. (4.4) is in this context essential,since it determines fully the space–frequency (and hence also thespace–time) coherence characteristic of the SPPs.



involved monochromatic and, consequently, fully coherent SPPs.Coherence-tailored polychromatic SPPs could nonetheless serve asversatile tools, e.g., for ultrashort optical pulse manipulation innanostructured optoelectronic circuits [31], controlled coupling oflight-emitting elements [214,216], plasmon continuum spectroscopy[217], and subwavelength white-light imaging [218]. At the sametime, exploring the statistical features and excitation mechanismsof polychromatic SPP fields presents a fundamental interest.

In publication IV, we develop a theory for partially coherentpolychromatic SPPs at a metal–air interface. The formalism coversstationary as well as nonstationary SPP fields of arbitrary spectra.As a main result, we formulate a framework to tailor the electro-magnetic coherence of such polychromatic SPPs in the Kretschmannsetup [24, 25] by controlling the correlations of the excitation light.The connection between the coherence state of the light source andthe ensuing SPP field establishes a novel paradigm in statisticalplasmonics, which we refer to as plasmon coherence engineering.

4.1.1 Polychromatic surface-plasmon polaritons

Let us consider polychromatic SPPs generated in the Kretschmannconfiguration, sketched in Fig. 4.1, with an absorptive and nonmag-netic metal film deposited on a glass prism. The film is taken to bethick enough so that any coupling between the metal-slab modes

Air SPP

Metal

Glass

Figure 4.1: Polychromatic SPP excitation in the Kretschmann coupling modality.



can be neglected, whereupon the region near the metal–air surfacecan be treated as the semi-infinite half-space geometry in Fig. 2.1.The (p-polarized) polychromatic electric field in air then reads as

E(r, t) =∫ ω+

ω−E(ω)p(ω)ei[k(ω)·r−ωt]dω, (4.1)

where ω−,+ specify the frequency range, E(ω) is the spectral com-plex field amplitude of a monochromatic SPP at the origin (r = 0),and p(ω) is the respective polarization vector given by Eq. (2.3).Note that we have dropped the subscript 2 (referring to medium 2)in order to keep the notation simpler.

Now, let Eq. (4.1) represent a realization of the random electricfield. All coherence information of the polychormatic SPP field isthen encoded in the electric coherence matrix [Eq. (A.2)]

Γ(r1, t1; r2, t2) =∫ ω+

ω−

∫ ω+

ω−W(r1, ω1; r2, ω2)e−i(ω2t2−ω1t1)dω1dω2, (4.2)

in which we have the spectral electric coherence matrix [Eq. (A.4)]

W(r1, ω1; r2, ω2) = W(ω1, ω2)K(ω1, ω2)ei[k(ω2)·r2−k∗(ω1)·r1], (4.3)

including the spectral electric correlation function

W(ω1, ω2) = ⟨E∗(ω1)E(ω2)⟩, (4.4)

and the matrix

K(ω1, ω2) = [|k(ω1)||k(ω2)|]−1

×[

k∗z(ω1)kz(ω2) −k∗z(ω1)kx(ω2)

−k∗x(ω1)kz(ω2) k∗x(ω1)kx(ω2)

]. (4.5)

Equation (4.2) is general as it sets no restrictions on metal disper-sion, the spectrum of the field, or the spectral correlations; it coversany partially coherent polychromatic SPP field. The spectral cor-relation function W(ω1, ω2) of Eq. (4.4) is in this context essential,since it determines fully the space–frequency (and hence also thespace–time) coherence characteristic of the SPPs.



4.1.2 Narrowband and broadband fields

As shown in Publication IV, in the case of a narrowband SPP fieldfor which metal dispersion can be neglected, all elements of theelectric coherence matrix in Eq. (4.2) have identical space–time de-pendence, i.e, the correlations among the field components prop-agate and attenuate in exactly the same way. Such polychromaticnarrowband SPP fields are virtually propagation invariant and alsostrictly polarized, which could facilitate nearly distortion-free infor-mation transfer in plasmonic networks.

For broadband spectra, then again, dispersion in the metal canno longer be ignored, and thus each element of Γ(r1, t1; r2, t2) hasto be treated separately (and numerically), whereupon the correla-tions between the electric-field components will in general have dif-ferent space–time evolutions. Yet, also the broadband SPP fields arehighly polarized, at least for metals and optical frequency rangesfor which SPP propagation is appreciable (such as Ag and Au in themid and lower frequency domains of the visible spectrum), sincethe polarization vectors of the SPP constituents are rather similar.For a stationary field having an ultra-wide spectrum, e.g., thermalradiation, the coherence length scale is only on the order of themean wavelength, but allowing nonzero correlations in W(ω1, ω2)

renders the polychromatic SPP field nonstationary and more coher-ent. At high levels of spectral correlations the SPPs become pulses.

4.1.3 Plasmon coherence engineering

The idea of plasmon coherence engineering is to prudently tailorW(ω1, ω2) of Eq. (4.4) into the wanted form by controlling the co-herence state of the light source. In Publication IV, as a main contri-bution, we establish exactly how the spatio–spectral statistical prop-erties of the stationary or pulsed excitation beam are to be tuned tocreate a polychromatic SPP field with the desired coherence char-acteristics.

To this end, we take the illumination light incident on the prismin the geometry of Fig. 4.1 to be a p-polarized, partially coherent,



Z

X

z

x

ω

0ω

θ∆

0θ

Figure 4.2: Plasmon coherence engineering with polychromatic beam illumination. Theangle θ0 between the XZ and xz frames corresponds to perfect phase matching among thecentral angular spectrum mode of frequency ω0 and the respective SPP. At every frequencywithin the excitation bandwidth, the angular spectrum wave of frequency ω = ω0 incidentat an angle ∆θ with respect to the Z axis generates the corresponding SPP.

polychromatic beam expressed by the angular spectrum representa-tion [2]. The electric-field amplitude of the angular spectrum modeof frequency ω is denoted by E(kX, ω), with kX being the tangen-tial wave-vector component in a coordinate frame XZ, where the Zaxis makes an angle θ0 with respect to the z axis of the xz frame(see Fig. 4.2). The second-order statistical properties of the incidentfield are then specified by the spectral electric correlation function

W(kX1, ω1; kX2, ω2) = ⟨E∗(kX1, ω1)E(kX2, ω2)⟩. (4.6)

We further choose θ0 such that in the xz frame the tangential wave-vector component of the beam mode of central frequency ω0 andkX = 0 within the angular spectrum exactly corresponds to k′x(ω0)

of the SPP obtained from Eq. (2.26), viz.,

n(ω0)ω0

csin θ0 = k′x(ω0). (4.7)

This condition, where n(ω0) is the refractive index of the prism,represents precise phase matching between the central illuminatingplane wave and the central SPP mode at the metal–air surface.

To ensure that an SPP mode is generated at every ω within thespectral excitation bandwidth, one must impose a similar matching



4.1.2 Narrowband and broadband fields

As shown in Publication IV, in the case of a narrowband SPP fieldfor which metal dispersion can be neglected, all elements of theelectric coherence matrix in Eq. (4.2) have identical space–time de-pendence, i.e, the correlations among the field components prop-agate and attenuate in exactly the same way. Such polychromaticnarrowband SPP fields are virtually propagation invariant and alsostrictly polarized, which could facilitate nearly distortion-free infor-mation transfer in plasmonic networks.

For broadband spectra, then again, dispersion in the metal canno longer be ignored, and thus each element of Γ(r1, t1; r2, t2) hasto be treated separately (and numerically), whereupon the correla-tions between the electric-field components will in general have dif-ferent space–time evolutions. Yet, also the broadband SPP fields arehighly polarized, at least for metals and optical frequency rangesfor which SPP propagation is appreciable (such as Ag and Au in themid and lower frequency domains of the visible spectrum), sincethe polarization vectors of the SPP constituents are rather similar.For a stationary field having an ultra-wide spectrum, e.g., thermalradiation, the coherence length scale is only on the order of themean wavelength, but allowing nonzero correlations in W(ω1, ω2)

renders the polychromatic SPP field nonstationary and more coher-ent. At high levels of spectral correlations the SPPs become pulses.

4.1.3 Plasmon coherence engineering

The idea of plasmon coherence engineering is to prudently tailorW(ω1, ω2) of Eq. (4.4) into the wanted form by controlling the co-herence state of the light source. In Publication IV, as a main contri-bution, we establish exactly how the spatio–spectral statistical prop-erties of the stationary or pulsed excitation beam are to be tuned tocreate a polychromatic SPP field with the desired coherence char-acteristics.

To this end, we take the illumination light incident on the prismin the geometry of Fig. 4.1 to be a p-polarized, partially coherent,



Z

X

z

x

ω

0ω

θ∆

0θ

Figure 4.2: Plasmon coherence engineering with polychromatic beam illumination. Theangle θ0 between the XZ and xz frames corresponds to perfect phase matching among thecentral angular spectrum mode of frequency ω0 and the respective SPP. At every frequencywithin the excitation bandwidth, the angular spectrum wave of frequency ω = ω0 incidentat an angle ∆θ with respect to the Z axis generates the corresponding SPP.

polychromatic beam expressed by the angular spectrum representa-tion [2]. The electric-field amplitude of the angular spectrum modeof frequency ω is denoted by E(kX, ω), with kX being the tangen-tial wave-vector component in a coordinate frame XZ, where the Zaxis makes an angle θ0 with respect to the z axis of the xz frame(see Fig. 4.2). The second-order statistical properties of the incidentfield are then specified by the spectral electric correlation function

W(kX1, ω1; kX2, ω2) = ⟨E∗(kX1, ω1)E(kX2, ω2)⟩. (4.6)

We further choose θ0 such that in the xz frame the tangential wave-vector component of the beam mode of central frequency ω0 andkX = 0 within the angular spectrum exactly corresponds to k′x(ω0)

of the SPP obtained from Eq. (2.26), viz.,

n(ω0)ω0

csin θ0 = k′x(ω0). (4.7)

This condition, where n(ω0) is the refractive index of the prism,represents precise phase matching between the central illuminatingplane wave and the central SPP mode at the metal–air surface.

To ensure that an SPP mode is generated at every ω within thespectral excitation bandwidth, one must impose a similar matching



condition for the other illumination plane waves as well. Supposethat for an arbitrary frequency ω = ω0 the angular spectrum modewith kX = n(ω)(ω/c) sin ∆θ, where ∆θ is the angle among the wavevector and the Z axis, couples to the respective SPP (see Fig. 4.2).As shown in Publication IV, for beamlike illumination this implies

kX =k′x(ω)− k′x(ω0)

cos θ0. (4.8)

At each frequency ω within the bandwidth the angular spectrumwave satisfying Eq. (4.8) thus excites the corresponding monochro-matic SPP mode. Now, since E(ω) ∝ E(kX, ω), with the exact cou-pling efficiency specified by the transmission coefficient of the slab,we get between the SPP correlation function [Eq. (4.4)] and the cor-relation function of the incident light [Eq. (4.6)] the relation

W(ω1, ω2) ∝ W[ k′x(ω1)− k′x(ω0)

cos θ0, ω1;

k′x(ω2)− k′x(ω0)

cos θ0, ω2

]. (4.9)

Equation (4.9) should be identified as an explicit (inverse) relation,which enables one to determine exactly (e.g., numerically by itera-tion when the metal dispersion is known) how the spectral correla-tion function of the illumination source has to be tuned in order toachieve any desired coherence properties for the ensuing polychro-matic SPP field. This result is the main contribution of PublicationIV and the crux of plasmon coherence engineering.

4.2 METAL-SLAB GEOMETRY

Contrary to the single-interface geometry, the metal-slab configura-tion illustrated in Fig. 3.2 is capable to sustain a multitude of modesat a given frequency. As the FMs of the M3 class are not only the mostwell-known, but also the most prominent modes in a broad rangeof applications, henceforth we focus on these two modes. Underusual conditions, the symmetric FM evolves into the LRSPP whenthe slab thickness decreases, while the antisymmetric FM acquiresa much smaller propagation length than that of the LRSPP and the



respective single-interface SPP. Therefore, the antisymmetric FM iscommonly termed the short-range surface-plasmon polariton (SRSPP).Owing to its unique capability of long-range guidance, the LRSPPhas over the years received considerably more attention and prac-tical significance than the SRSPP. Nevertheless, compared to theLRSPP, the SRSPP offers advantages as regards strong field con-finement [32], and several phenomena have been reported in whichthe SRSPP plays a crucial role, including plasmonic focusing [219],stimulated-amplification supported SPP propagation [220], extraor-dinarily low transmission through nanopatterned films [221], andplasmon-waveguide sensing [222]. Methods which allow efficientexcitation of the SRSPP, either simultaneously with [223] or with-out [224] the LRSPP, have also been presented.

Individually, the LRSPP and SRSPP are completely coherent andpolarized, but their superposition allows for the possibility of partialcoherence and partial polarization. To gain insight into the spatialcoherence and polarization properties of such a LRSPP–SRSPP su-perposition, it is natural to employ the degrees of coherence andpolarization, since these are the basic measures to characterize anypartially coherent and partially polarized light field. It is of partic-ular theoretical interest to investigate how much the two degreesmay vary, as the upper and lower ranges specify to which extentthe coherence and polarization of the two-mode field can be modi-fied (and utilized) in practice. The limits are primarily determinedby the mutual correlation between the modes, which in a practicalarrangement depends on the excitation process.

The main objective of Publication V is to examine the fundamen-tal ranges that the spectral degrees of coherence and polarization ofsuch a stationary LRSPP–SRSPP field above the metallic nanoslab inFig. 3.2 can attain, regardless of the excitation method. In addition,we explore how the two degrees vary within their extremal valueswhen the media, frequency, and film thickness are changed. Again,as we are considering the electric field only outside the film, to keepthe notation simpler, henceforward the subscript 2 in the field am-plitudes, wave vectors, etc., referring to medium 2 (surrounding) is



condition for the other illumination plane waves as well. Supposethat for an arbitrary frequency ω = ω0 the angular spectrum modewith kX = n(ω)(ω/c) sin ∆θ, where ∆θ is the angle among the wavevector and the Z axis, couples to the respective SPP (see Fig. 4.2).As shown in Publication IV, for beamlike illumination this implies

kX =k′x(ω)− k′x(ω0)

cos θ0. (4.8)

At each frequency ω within the bandwidth the angular spectrumwave satisfying Eq. (4.8) thus excites the corresponding monochro-matic SPP mode. Now, since E(ω) ∝ E(kX, ω), with the exact cou-pling efficiency specified by the transmission coefficient of the slab,we get between the SPP correlation function [Eq. (4.4)] and the cor-relation function of the incident light [Eq. (4.6)] the relation

W(ω1, ω2) ∝ W[ k′x(ω1)− k′x(ω0)

cos θ0, ω1;

k′x(ω2)− k′x(ω0)

cos θ0, ω2

]. (4.9)

Equation (4.9) should be identified as an explicit (inverse) relation,which enables one to determine exactly (e.g., numerically by itera-tion when the metal dispersion is known) how the spectral correla-tion function of the illumination source has to be tuned in order toachieve any desired coherence properties for the ensuing polychro-matic SPP field. This result is the main contribution of PublicationIV and the crux of plasmon coherence engineering.

4.2 METAL-SLAB GEOMETRY

Contrary to the single-interface geometry, the metal-slab configura-tion illustrated in Fig. 3.2 is capable to sustain a multitude of modesat a given frequency. As the FMs of the M3 class are not only the mostwell-known, but also the most prominent modes in a broad rangeof applications, henceforth we focus on these two modes. Underusual conditions, the symmetric FM evolves into the LRSPP whenthe slab thickness decreases, while the antisymmetric FM acquiresa much smaller propagation length than that of the LRSPP and the



respective single-interface SPP. Therefore, the antisymmetric FM iscommonly termed the short-range surface-plasmon polariton (SRSPP).Owing to its unique capability of long-range guidance, the LRSPPhas over the years received considerably more attention and prac-tical significance than the SRSPP. Nevertheless, compared to theLRSPP, the SRSPP offers advantages as regards strong field con-finement [32], and several phenomena have been reported in whichthe SRSPP plays a crucial role, including plasmonic focusing [219],stimulated-amplification supported SPP propagation [220], extraor-dinarily low transmission through nanopatterned films [221], andplasmon-waveguide sensing [222]. Methods which allow efficientexcitation of the SRSPP, either simultaneously with [223] or with-out [224] the LRSPP, have also been presented.

Individually, the LRSPP and SRSPP are completely coherent andpolarized, but their superposition allows for the possibility of partialcoherence and partial polarization. To gain insight into the spatialcoherence and polarization properties of such a LRSPP–SRSPP su-perposition, it is natural to employ the degrees of coherence andpolarization, since these are the basic measures to characterize anypartially coherent and partially polarized light field. It is of partic-ular theoretical interest to investigate how much the two degreesmay vary, as the upper and lower ranges specify to which extentthe coherence and polarization of the two-mode field can be modi-fied (and utilized) in practice. The limits are primarily determinedby the mutual correlation between the modes, which in a practicalarrangement depends on the excitation process.

The main objective of Publication V is to examine the fundamen-tal ranges that the spectral degrees of coherence and polarization ofsuch a stationary LRSPP–SRSPP field above the metallic nanoslab inFig. 3.2 can attain, regardless of the excitation method. In addition,we explore how the two degrees vary within their extremal valueswhen the media, frequency, and film thickness are changed. Again,as we are considering the electric field only outside the film, to keepthe notation simpler, henceforward the subscript 2 in the field am-plitudes, wave vectors, etc., referring to medium 2 (surrounding) is



suppressed. When the constituents are mutually fully correlated,the total field is completely coherent and polarized, thus establish-ing directly the upper limits for the degrees. To assess the lowerranges, we take the modes to be mutually uncorrelated, since anycorrelation among them is expected to yield a more coherent andpolarized field. For the same reason, the modes are considered tohave equal intensities at r1 = x1ex + (d/2)ez, which we define asthe excitation point.

4.2.1 Degree of coherence

Due to the vectorial nature of SPPs, for which the traditional scalardegree of coherence is inadequate, we employ the generalized vec-tor degree of coherence in Eq. (A.14) to investigate the spatial co-herence of the LRSPP–SRSPP field. Under the above assumptions,

µ(∆r, ω) =1√2

√1 + κ(ω)

cos [∆k′(ω) · ∆r]cosh [∆k′′(ω) · ∆r]

, (4.10)

in which ∆r = r2 − r1 is the separation between the observation andexcitation points, and

κ(ω) =p(+)∗(ω) · p(−)(ω)

2, (4.11)

∆k′(ω) = k(+)′(ω)− k(−)′(ω), (4.12)

∆k′′(ω) = k(+)′′(ω)− k(−)′′(ω), (4.13)

where the superscripts (+) and (−) refer to the LRSPP and SRSPP,respectively. The polarization term κ(ω) in Eq. (4.11) is bounded as

1/2 ≤ κ(ω) ≤ 1, (4.14)

with the lower (upper) limit corresponding to d → 0 (d → ∞). Wefurther emphasize that generally ∆k′(ω) · ∆k′′(ω) = 0.

The cosine in Eq. (4.10) indicates that, as a rule, the degree of co-herence oscillates, whereupon the LRSPP-SRSPP field may show ahigh (or a low) degree of coherence at certain locations even though



the two modes are mutually uncorrelated and hence do not inter-fere. The oscillation of µ(∆r, ω) originates from the fact that at spe-cific periodic distances the superposition is electromagnetically sim-ilar [225] to the total field at the excitation point (we neglect thedecay of the modes for the moment). This effect is akin to the cus-tomary oscillatory behavior of the (scalar) degree of coherence fortwo uncorrelated modes in a gas laser [226].

In addition to the oscillation, the hyperbolic cosine in Eq. (4.10)implies that µ(∆r, ω) generally also decays, which arises from theattenuation of the two modes owing to metal absorption. As longas ∆r is not perpendicular to ∆k′′(ω), we find from Eq. (4.10) thatµ(∆r, ω) → 1/

√2 when |∆r| → ∞. This value, representing partial

coherence and being independent of the material parameters, thefrequency of the field, or the thickness of the slab, is a consequenceof the different decay rates of the two modes: for a sufficientlylarge |∆r|, the mode with the lower decay rate (LRSPP) dominatesthe superposition and the mode with a higher decay rate (SRSPP)can be neglected. Hence, far away from the excitation region, thefield can practically be considered as a single LRSPP which attainsan essentially constant degree of coherence.

Concerning the (global) maximum and minimum of the degreeof coherence, we get from Eq. (4.10) that

µmax(ω) =√[1 + κ(ω)]/2, µmin(ω) =

√[1 − κ(ω)]/2, (4.15)

which set the fundamental limits for the domain in which µ(∆r, ω)

of the LRSPP–SRSPP field is restricted. Equation (4.15) shows thatµ2

max(ω) + µ2min(ω) = 1, indicating that an increase of one is accom-

panied with a decrease of the other. From Eqs. (4.11) and (4.15) onefurther finds that the extrema are bounded as

√3/2 ≤ µmax(ω) ≤ 1, 0 ≤ µmin(ω) ≤ 1/2, (4.16)

where the lower and upper limits of µmax(ω) [µmin(ω)] correspondto d → 0 (d → ∞) and d → ∞ (d → 0), respectively. Equation (4.16)especially demonstrates that, irrespective of the media, frequency,



suppressed. When the constituents are mutually fully correlated,the total field is completely coherent and polarized, thus establish-ing directly the upper limits for the degrees. To assess the lowerranges, we take the modes to be mutually uncorrelated, since anycorrelation among them is expected to yield a more coherent andpolarized field. For the same reason, the modes are considered tohave equal intensities at r1 = x1ex + (d/2)ez, which we define asthe excitation point.


Due to the vectorial nature of SPPs, for which the traditional scalardegree of coherence is inadequate, we employ the generalized vec-tor degree of coherence in Eq. (A.14) to investigate the spatial co-herence of the LRSPP–SRSPP field. Under the above assumptions,

µ(∆r, ω) =1√2

√1 + κ(ω)

cos [∆k′(ω) · ∆r]cosh [∆k′′(ω) · ∆r]

, (4.10)

in which ∆r = r2 − r1 is the separation between the observation andexcitation points, and

κ(ω) =p(+)∗(ω) · p(−)(ω)

2, (4.11)

∆k′(ω) = k(+)′(ω)− k(−)′(ω), (4.12)

∆k′′(ω) = k(+)′′(ω)− k(−)′′(ω), (4.13)

where the superscripts (+) and (−) refer to the LRSPP and SRSPP,respectively. The polarization term κ(ω) in Eq. (4.11) is bounded as

1/2 ≤ κ(ω) ≤ 1, (4.14)

with the lower (upper) limit corresponding to d → 0 (d → ∞). Wefurther emphasize that generally ∆k′(ω) · ∆k′′(ω) = 0.

The cosine in Eq. (4.10) indicates that, as a rule, the degree of co-herence oscillates, whereupon the LRSPP-SRSPP field may show ahigh (or a low) degree of coherence at certain locations even though



the two modes are mutually uncorrelated and hence do not inter-fere. The oscillation of µ(∆r, ω) originates from the fact that at spe-cific periodic distances the superposition is electromagnetically sim-ilar [225] to the total field at the excitation point (we neglect thedecay of the modes for the moment). This effect is akin to the cus-tomary oscillatory behavior of the (scalar) degree of coherence fortwo uncorrelated modes in a gas laser [226].

In addition to the oscillation, the hyperbolic cosine in Eq. (4.10)implies that µ(∆r, ω) generally also decays, which arises from theattenuation of the two modes owing to metal absorption. As longas ∆r is not perpendicular to ∆k′′(ω), we find from Eq. (4.10) thatµ(∆r, ω) → 1/

√2 when |∆r| → ∞. This value, representing partial

coherence and being independent of the material parameters, thefrequency of the field, or the thickness of the slab, is a consequenceof the different decay rates of the two modes: for a sufficientlylarge |∆r|, the mode with the lower decay rate (LRSPP) dominatesthe superposition and the mode with a higher decay rate (SRSPP)can be neglected. Hence, far away from the excitation region, thefield can practically be considered as a single LRSPP which attainsan essentially constant degree of coherence.

Concerning the (global) maximum and minimum of the degreeof coherence, we get from Eq. (4.10) that

µmax(ω) =√[1 + κ(ω)]/2, µmin(ω) =

√[1 − κ(ω)]/2, (4.15)

which set the fundamental limits for the domain in which µ(∆r, ω)

of the LRSPP–SRSPP field is restricted. Equation (4.15) shows thatµ2

max(ω) + µ2min(ω) = 1, indicating that an increase of one is accom-

panied with a decrease of the other. From Eqs. (4.11) and (4.15) onefurther finds that the extrema are bounded as

√3/2 ≤ µmax(ω) ≤ 1, 0 ≤ µmin(ω) ≤ 1/2, (4.16)

where the lower and upper limits of µmax(ω) [µmin(ω)] correspondto d → 0 (d → ∞) and d → ∞ (d → 0), respectively. Equation (4.16)especially demonstrates that, irrespective of the media, frequency,



and slab thickness, there are always regions for which the field dis-plays a quite high or a rather low degree of coherence.

4.2.2 Local and global coherence length

We notice from Eqs. (4.10) and (4.15) that the maximum of µ(∆r, ω)

is found at the point r1 = r2 where the two modes are generated.Furthermore, the minimum in Eq. (4.15) is not just the global, butalso the nearest minimum with respect to the excitation point [thereexists a sequence of local minima (and maxima) due to the oscilla-tory behavior of µ(∆r, ω)]. Therefore, besides the actual values, itis natural to investigate the distance between µmax(ω) and µmin(ω)

to get a rough estimation for the domain at the excitation regionin which the field is highly electromagnetically coherent (meaningthat all field components are strongly correlated). We refer to thisparticular distance as the local coherence length. As demonstrated inPublication V, depending on the materials and frequency, the localcoherence length can be of subwavelength order for ultra-thin films,while for larger slab thicknesses it can extend over several tens ofwavelengths.

Because of the property µ(∆r, ω) → 1/√

2 when |∆r| → ∞, itis reasonable to introduce an additional, global coherence length, asa distance between r1 and r2 over which µ(∆r, ω) drops from itsmaximum value at r1 = r2 to a particular number close to 1/

√2.

The ‘particular number’ is not unambiguous, but is chosen appro-priately for each situation. The global coherence length is thereby arough measure for the range beyond which the degree of coherencedoes not essentially change anymore, i.e., it marks the distance upto which µ(∆r, ω) oscillates. Physically, within the global coherencelength there are regions in which the SPP superposition at the twopoints may be highly correlated and regions where it may be ratheruncorrelated. At these locations, the SPP field would interact withnanoparticles in the vicinity of the surface in a coherent, or incoher-ent, manner. The investigation in Publication V reveals that, muchlike the local coherence length, the global coherence length can ex-



tend from subwavelength scales (thin films) to even thousands ofwavelengths (thick films).

4.2.3 Degree of polarization

Regarding the polarization degree of the LRSPP–SRSPP field, sincethe constituents have only two electric-field components, the con-ventional (2D) treatment is formally sufficient for analyzing the po-larization characteristics of the superposition. Yet, as in this case thedegree of polarization is defined in the plane parallel to the wavevectors (xz plane, where the modes are typically elliptically polar-ized), the situation differs substantially from that of beamlike wavefields for which the electric field is orthogonal to the wave vector.As shown in Publication V, when the two modes are uncorrelatedand have equal intensities at the excitation point, the 2D degree ofpolarization [Eq. (A.15)] for the SPP field above the slab becomes

P2D(r, ω) =

√1 − 1 − κ(ω)

cosh2[∆k′′(ω) · r], (4.17)

where r is the position vector measured from the excitation point.Unlike the degree of coherence in Eq. (4.10), the degree of po-

larization in Eq. (4.17) does not display an oscillatory term, but ischaracterized only by a hyperbolic cosine. Excluding the particu-lar direction along which P2D(r, ω) is constant, i.e., the one that isperpendicularly to ∆k′′(ω), we find from Eq. (4.17) that

P2D(r, ω) →

1, |r| → ∞,√κ(ω), |r| → 0,

(4.18)

which are verified to be, respectively, the maximum and minimumof P2D(r, ω). Consequently, since 1/2 ≤ κ(ω) ≤ 1 [Eq. (4.14)], weconclude that the degree of polarization for the LRSPP–SRSPP su-perposition is always bounded as

1/√

2 ≤ P2D(r, ω) ≤ 1. (4.19)



and slab thickness, there are always regions for which the field dis-plays a quite high or a rather low degree of coherence.

4.2.2 Local and global coherence length

We notice from Eqs. (4.10) and (4.15) that the maximum of µ(∆r, ω)

is found at the point r1 = r2 where the two modes are generated.Furthermore, the minimum in Eq. (4.15) is not just the global, butalso the nearest minimum with respect to the excitation point [thereexists a sequence of local minima (and maxima) due to the oscilla-tory behavior of µ(∆r, ω)]. Therefore, besides the actual values, itis natural to investigate the distance between µmax(ω) and µmin(ω)

to get a rough estimation for the domain at the excitation regionin which the field is highly electromagnetically coherent (meaningthat all field components are strongly correlated). We refer to thisparticular distance as the local coherence length. As demonstrated inPublication V, depending on the materials and frequency, the localcoherence length can be of subwavelength order for ultra-thin films,while for larger slab thicknesses it can extend over several tens ofwavelengths.

Because of the property µ(∆r, ω) → 1/√

2 when |∆r| → ∞, itis reasonable to introduce an additional, global coherence length, asa distance between r1 and r2 over which µ(∆r, ω) drops from itsmaximum value at r1 = r2 to a particular number close to 1/

√2.

The ‘particular number’ is not unambiguous, but is chosen appro-priately for each situation. The global coherence length is thereby arough measure for the range beyond which the degree of coherencedoes not essentially change anymore, i.e., it marks the distance upto which µ(∆r, ω) oscillates. Physically, within the global coherencelength there are regions in which the SPP superposition at the twopoints may be highly correlated and regions where it may be ratheruncorrelated. At these locations, the SPP field would interact withnanoparticles in the vicinity of the surface in a coherent, or incoher-ent, manner. The investigation in Publication V reveals that, muchlike the local coherence length, the global coherence length can ex-



tend from subwavelength scales (thin films) to even thousands ofwavelengths (thick films).

4.2.3 Degree of polarization

Regarding the polarization degree of the LRSPP–SRSPP field, sincethe constituents have only two electric-field components, the con-ventional (2D) treatment is formally sufficient for analyzing the po-larization characteristics of the superposition. Yet, as in this case thedegree of polarization is defined in the plane parallel to the wavevectors (xz plane, where the modes are typically elliptically polar-ized), the situation differs substantially from that of beamlike wavefields for which the electric field is orthogonal to the wave vector.As shown in Publication V, when the two modes are uncorrelatedand have equal intensities at the excitation point, the 2D degree ofpolarization [Eq. (A.15)] for the SPP field above the slab becomes

P2D(r, ω) =

√1 − 1 − κ(ω)

cosh2[∆k′′(ω) · r], (4.17)

where r is the position vector measured from the excitation point.Unlike the degree of coherence in Eq. (4.10), the degree of po-

larization in Eq. (4.17) does not display an oscillatory term, but ischaracterized only by a hyperbolic cosine. Excluding the particu-lar direction along which P2D(r, ω) is constant, i.e., the one that isperpendicularly to ∆k′′(ω), we find from Eq. (4.17) that

P2D(r, ω) →

1, |r| → ∞,√κ(ω), |r| → 0,

(4.18)

which are verified to be, respectively, the maximum and minimumof P2D(r, ω). Consequently, since 1/2 ≤ κ(ω) ≤ 1 [Eq. (4.14)], weconclude that the degree of polarization for the LRSPP–SRSPP su-perposition is always bounded as

1/√

2 ≤ P2D(r, ω) ≤ 1. (4.19)



The physical origin behind the maximum, representing a fully po-larized field, is quite apparent: when |r| gets large enough, the SR-SPP vanishes and the field can practically be considered as a singleLRSPP for which P2D(r, ω) = 1 regardless of the media, frequency,film thickness, and position. By the same token, as the contributionof the SRSPP to the degree of polarization cannot be ignored when|r| → 0, the minimum, standing for partial polarization, dependson the material parameters, frequency, as well as the slab thickness,according to Eqs. (3.27), (3.28), and (4.11).

Finally, our studies in Publication V indicate that generally thedegree of polarization of the LRSPP–SRSPP field is close to unity.Nonetheless, the analysis also suggests that for ultra-thin films,within subwavelength distances from the excitation point, the SPPsuperposition can be partially polarized and P2D(r, ω) may fluctu-ate. Increasing the operating frequency reduces the polarizationdegree, whereas varying the permittivity of the surrounding has anegligible effect on it.


5 Electromagnetic coherenceof evanescent light fields

Optical evanescent waves are a special type of (pure) ESWs, formedwhen a light field undergoes total internal reflection at a dielectricboundary [18]. When interacting with matter, evanescent waves en-able a phenomenon analogous to quantum mechanical tunnelingthrough a potential barrier [2, 18]. They also allow to study biolog-ical samples with a resolution well beyond the classical diffractionlimit [227–229] and play an important role in SPP excitation [24,25].Evanescent waves have therefore a pivotal position in nanophoton-ics and for the understanding of several optical phenomena that areconfined to subwavelength dimensions.

This chapter concerns the generation, partial polarization, andspatial coherence of statistically stationary, purely evanescent lightfields at a planar interface between two dielectric media. Contraryto SPPs, evanescent waves generally carry three orthogonal electric-field components, whereat a rigorous 3D treatment is required tofully describe their polarization state. Genuine 3D-polarized evanes-cent fields with tailored coherence properties could be utilized innear-field probing, single-molecule detection, particle trapping, andother polarization-sensitive surface-photonic applications.

5.1 EVANESCENT WAVE IN TOTAL INTERNAL REFLECTION

Let us consider a beam, represented as a monochromatic electro-magnetic plane wave, incident onto a planar interface (z = 0) be-tween two dielectric media (see Fig. 5.1). Both medium 1 (z < 0)and medium 2 (z > 0), having refractive indices n1(ω) and n2(ω),respectively, are taken lossless. The incoming wave, generally car-rying both an s-polarized and a p-polarized part, hits the boundary



The physical origin behind the maximum, representing a fully po-larized field, is quite apparent: when |r| gets large enough, the SR-SPP vanishes and the field can practically be considered as a singleLRSPP for which P2D(r, ω) = 1 regardless of the media, frequency,film thickness, and position. By the same token, as the contributionof the SRSPP to the degree of polarization cannot be ignored when|r| → 0, the minimum, standing for partial polarization, dependson the material parameters, frequency, as well as the slab thickness,according to Eqs. (3.27), (3.28), and (4.11).

Finally, our studies in Publication V indicate that generally thedegree of polarization of the LRSPP–SRSPP field is close to unity.Nonetheless, the analysis also suggests that for ultra-thin films,within subwavelength distances from the excitation point, the SPPsuperposition can be partially polarized and P2D(r, ω) may fluctu-ate. Increasing the operating frequency reduces the polarizationdegree, whereas varying the permittivity of the surrounding has anegligible effect on it.


5 Electromagnetic coherenceof evanescent light fields

Optical evanescent waves are a special type of (pure) ESWs, formedwhen a light field undergoes total internal reflection at a dielectricboundary [18]. When interacting with matter, evanescent waves en-able a phenomenon analogous to quantum mechanical tunnelingthrough a potential barrier [2, 18]. They also allow to study biolog-ical samples with a resolution well beyond the classical diffractionlimit [227–229] and play an important role in SPP excitation [24,25].Evanescent waves have therefore a pivotal position in nanophoton-ics and for the understanding of several optical phenomena that areconfined to subwavelength dimensions.

This chapter concerns the generation, partial polarization, andspatial coherence of statistically stationary, purely evanescent lightfields at a planar interface between two dielectric media. Contraryto SPPs, evanescent waves generally carry three orthogonal electric-field components, whereat a rigorous 3D treatment is required tofully describe their polarization state. Genuine 3D-polarized evanes-cent fields with tailored coherence properties could be utilized innear-field probing, single-molecule detection, particle trapping, andother polarization-sensitive surface-photonic applications.

5.1 EVANESCENT WAVE IN TOTAL INTERNAL REFLECTION

Let us consider a beam, represented as a monochromatic electro-magnetic plane wave, incident onto a planar interface (z = 0) be-tween two dielectric media (see Fig. 5.1). Both medium 1 (z < 0)and medium 2 (z > 0), having refractive indices n1(ω) and n2(ω),respectively, are taken lossless. The incoming wave, generally car-rying both an s-polarized and a p-polarized part, hits the boundary



)ω(ϕ

)ω(θ

)ω(2n

)ω(1n

z

y

x

Figure 5.1: Total internal reflection at a planar interface (z = 0) between two losslessdielectric media having refractive indices n1(ω) (z < 0) and n2(ω) (z > 0). The incidentbeam impinges the surface with an azimuthal angle φ(ω) at the angle of incidence θ(ω).

with an azimuthal angle 0 ≤ φ(ω) < 2π at the angle of incidence0 < θ(ω) < π/2. When θ(ω) > θc(ω), with θc(ω) = arcsin n−1(ω)

being the critical angle and n(ω) = n1(ω)/n2(ω) > 1, total inter-nal reflection takes place and the transmitted field in medium 2becomes an evanescent wave [18].

According to Eqs. (2.1)–(2.3), Maxwell’s equations, the bound-ary conditions, as well as Eqs. (2.30 and (2.31), the spatial part ofthe electric field for the evanescent wave takes on the form

E(r, ω) = [ts(ω)Es(ω)s(ω) + tp(ω)Ep(ω)p(ω)]eik(ω)·r, (5.1)

where Es(ω) and Ep(ω) are, respectively, the complex field ampli-tudes of the s- and p-polarized components of the incident light. InCartesian coordinates, the wave and polarization vectors read as

k(ω) = k1(ω)sin θ(ω)[cos φ(ω)ex + sin φ(ω)ey] + iγ(ω)ez, (5.2)

s(ω) = − sin φ(ω)ex + cos φ(ω)ey, (5.3)

p(ω) =−iγ(ω)[cos φ(ω)ex + sin φ(ω)ey] + sin θ(ω)ez√

sin2 θ(ω) + γ2(ω), (5.4)

in which k1(ω) is the wave number in medium 1 and

γ(ω) = n−1(ω)√[n(ω) sin θ(ω)]2 − 1. (5.5)


Electromagnetic coherence ofevanescent light fields

Eventually, the Fresnel transmission coefficients ts(ω) and tp(ω) forthe two polarizations are given by

ts(ω) =2 cos θ(ω)

cos θ(ω) + iγ(ω), (5.6)

tp(ω) =2n(ω) cos θ(ω)

√2n2(ω)γ2(ω) + 1

cos θ(ω) + in2(ω)γ(ω). (5.7)

We emphasize that Eqs. (5.1)–(5.7) are expressed solely in terms ofthe refractive indices and the parameters associated with the in-coming light. We note also that tp(ω) in Eq. (5.7) differs from theconventional expression [2,18] owing to our different normalizationof the wave vector (see Sec. 2.4).

The quantity γ(ω) defined in Eq. (5.5) can be interpreted as thedecay constant of the evanescent wave. We see that the larger theangle of incidence, the faster the wave decays with increasing dis-tance away from the surface [Eq. (2.15)]. Another essential quantitythat characterizes the evanescent wave is its wavelength [Eq. (2.16)],

Λ(ω) =λ0(ω)

n1(ω) sin θ(ω), (5.8)

which is readily verified to be bounded as

λ1(ω) < Λ(ω) < λ2(ω), (5.9)

where the lower and upper limits, λ1(ω) and λ2(ω), correspondingto θ(ω) = π/2 and θ(ω) = θc(ω), are the wavelengths in medium1 and 2, respectively. Equation (5.9) especially indicates that Λ(ω)

is always shorter than the wavelength of a propagating plane waveabove the surface, anticipating that, in random evanescent fields,also the coherence length can be shorter than λ2(ω).

5.2 RANDOM EVANESCENT FIELDS

Let us now consider the case in which several, stationary incidentbeams undergo total internal reflection in the modality of Fig. 5.1.



)ω(ϕ

)ω(θ

)ω(2n

)ω(1n

z

y

x

Figure 5.1: Total internal reflection at a planar interface (z = 0) between two losslessdielectric media having refractive indices n1(ω) (z < 0) and n2(ω) (z > 0). The incidentbeam impinges the surface with an azimuthal angle φ(ω) at the angle of incidence θ(ω).

with an azimuthal angle 0 ≤ φ(ω) < 2π at the angle of incidence0 < θ(ω) < π/2. When θ(ω) > θc(ω), with θc(ω) = arcsin n−1(ω)

being the critical angle and n(ω) = n1(ω)/n2(ω) > 1, total inter-nal reflection takes place and the transmitted field in medium 2becomes an evanescent wave [18].

According to Eqs. (2.1)–(2.3), Maxwell’s equations, the bound-ary conditions, as well as Eqs. (2.30 and (2.31), the spatial part ofthe electric field for the evanescent wave takes on the form

E(r, ω) = [ts(ω)Es(ω)s(ω) + tp(ω)Ep(ω)p(ω)]eik(ω)·r, (5.1)

where Es(ω) and Ep(ω) are, respectively, the complex field ampli-tudes of the s- and p-polarized components of the incident light. InCartesian coordinates, the wave and polarization vectors read as

k(ω) = k1(ω)sin θ(ω)[cos φ(ω)ex + sin φ(ω)ey] + iγ(ω)ez, (5.2)

s(ω) = − sin φ(ω)ex + cos φ(ω)ey, (5.3)

p(ω) =−iγ(ω)[cos φ(ω)ex + sin φ(ω)ey] + sin θ(ω)ez√

sin2 θ(ω) + γ2(ω), (5.4)

in which k1(ω) is the wave number in medium 1 and

γ(ω) = n−1(ω)√[n(ω) sin θ(ω)]2 − 1. (5.5)



Eventually, the Fresnel transmission coefficients ts(ω) and tp(ω) forthe two polarizations are given by

ts(ω) =2 cos θ(ω)

cos θ(ω) + iγ(ω), (5.6)

tp(ω) =2n(ω) cos θ(ω)

√2n2(ω)γ2(ω) + 1

cos θ(ω) + in2(ω)γ(ω). (5.7)

We emphasize that Eqs. (5.1)–(5.7) are expressed solely in terms ofthe refractive indices and the parameters associated with the in-coming light. We note also that tp(ω) in Eq. (5.7) differs from theconventional expression [2,18] owing to our different normalizationof the wave vector (see Sec. 2.4).

The quantity γ(ω) defined in Eq. (5.5) can be interpreted as thedecay constant of the evanescent wave. We see that the larger theangle of incidence, the faster the wave decays with increasing dis-tance away from the surface [Eq. (2.15)]. Another essential quantitythat characterizes the evanescent wave is its wavelength [Eq. (2.16)],

Λ(ω) =λ0(ω)

n1(ω) sin θ(ω), (5.8)

which is readily verified to be bounded as

λ1(ω) < Λ(ω) < λ2(ω), (5.9)

where the lower and upper limits, λ1(ω) and λ2(ω), correspondingto θ(ω) = π/2 and θ(ω) = θc(ω), are the wavelengths in medium1 and 2, respectively. Equation (5.9) especially indicates that Λ(ω)

is always shorter than the wavelength of a propagating plane waveabove the surface, anticipating that, in random evanescent fields,also the coherence length can be shorter than λ2(ω).

5.2 RANDOM EVANESCENT FIELDS

Let us now consider the case in which several, stationary incidentbeams undergo total internal reflection in the modality of Fig. 5.1.



The beams are allowed to have different polarization states (and de-grees), angles of incidence, and azimuthal angles. We let E(n)(r, ω),with n ∈ 1, 2, . . . , N, be the spatial part of a monochromatic real-ization of the evanescent wave generated by the nth incoming beam.The total evanescent field is then expressed by the sum

Etot(r, ω) =N

∑n=1

E(n)(r, ω), (5.10)

in which each constituent is constructed as in Eqs. (5.1)–(5.7).On taking an ensemble average over the realizations, the cross-

spectral density matrix [Eq. (A.11)] for the evanescent field becomes

W(r1, r2, ω) =N

∑m,n=1

W(mn)(ω)ei[k(n)(ω)·r2−k(m)∗(ω)·r1]. (5.11)

Here we have defined the matrix

Wmn(ω) = t(m)∗s (ω)t(n)s (ω)ϕ

(mn)ss (ω)s(m)∗(ω)s(n)T(ω)

+ t(m)∗p (ω)t(n)p (ω)ϕ

(mn)pp (ω)p(m)∗(ω)p(n)T(ω)

+ t(m)∗s (ω)t(n)p (ω)ϕ

(mn)sp (ω)s(m)∗(ω)p(n)T(ω)

+ t(m)∗p (ω)t(n)s (ω)ϕ

(mn)ps (ω)p(m)∗(ω)s(n)T(ω), (5.12)

where the superscript T denotes matrix transpose and the factors

ϕ(mn)µν (ω) = ⟨E(m)∗

µ (ω)E(n)ν (ω)⟩, (5.13)

with µ, ν ∈ s, p and m, n ∈ 1, 2, . . . , N, are related to the corre-lations among the incident light beams. For m = n, the quantitiesϕ(mn)µν (ω) are the elements of the 2 × 2 polarization matrix associ-

ated with the nth incident wave, while the cases m = n describe themutual correlations between different waves.

The corresponding spectral polarization matrix [Eq. (A.12)] isobtained by setting r1 = r2 = r in Eq. (5.11), viz.,

Φ(r, ω) =N

∑m,n=1

W(mn)(ω)ei[k(n)(ω)−k(m)∗(ω)]·r, (5.14)

which fully describes the polarization state of the evanescent field.



5.2.1 Subwavelength coherence lengths

The conventional wisdom in optics says that δ-correlated sourcesor blackbody radiators generate the spatially most incoherent wavefields, for which the coherence length, in a lossless medium, isroughly half a wavelength of the light [230, 231]. Recently, these ar-guments have been reassessed and it was shown that, in principle,a finite-sized source can produce a field whose coherence lengthwithin the source may be arbitrarily short, even in the absence ofabsorption [232]. In addition, it has been demonstrated that a ther-mal half-space source can generate an electromagnetic near fieldwhose longitudinal correlation length, due to absorption, may bemuch shorter than the light’s wavelength [84]. In Publication VI,we demonstrate that such subwavelength coherence lengths are alsoencountered in purely evanescent fields at lossless interfaces. As arule, the shortest coherence lengths are observed very close to thesurface for high refractive-index contrasts.

As an example, we consider the superposition of two s-polarizedevanescent waves in the immediate vicinity of the surface (z = 0),created by mutually uncorrelated beams sharing the same angle ofincidence θ(ω), but having opposite azimuthal angles φ(1)(ω) = 0and φ(2)(ω) = π. The amplitudes of the beams are adjusted so thatthe individual evanescent waves have equal intensities at z = 0. Itfollows from Eqs. (5.11) and (A.14) that for such a (fully polarized)evanescent field the degree of coherence becomes

µ(∆x, ω) =1√2

√1 + cos[2k1(ω) sin θ(ω)∆x], (5.15)

where ∆x = x2 − x1 is the distance between the observation pointsalong the x axis. Owing to statistical similarity, the degree of coher-ence in Eq. (5.15) oscillates sinusoidally between 1 and 0, whereatwe define the coherence length for the two-wave superposition asthe distance from a maximum to a nearby minimum, viz.,

lcoh(ω) =λ2(ω)

4n(ω) sin θ(ω). (5.16)



The beams are allowed to have different polarization states (and de-grees), angles of incidence, and azimuthal angles. We let E(n)(r, ω),with n ∈ 1, 2, . . . , N, be the spatial part of a monochromatic real-ization of the evanescent wave generated by the nth incoming beam.The total evanescent field is then expressed by the sum

Etot(r, ω) =N

∑n=1

E(n)(r, ω), (5.10)

in which each constituent is constructed as in Eqs. (5.1)–(5.7).On taking an ensemble average over the realizations, the cross-

spectral density matrix [Eq. (A.11)] for the evanescent field becomes

W(r1, r2, ω) =N

∑m,n=1

W(mn)(ω)ei[k(n)(ω)·r2−k(m)∗(ω)·r1]. (5.11)

Here we have defined the matrix

Wmn(ω) = t(m)∗s (ω)t(n)s (ω)ϕ

(mn)ss (ω)s(m)∗(ω)s(n)T(ω)

+ t(m)∗p (ω)t(n)p (ω)ϕ

(mn)pp (ω)p(m)∗(ω)p(n)T(ω)

+ t(m)∗s (ω)t(n)p (ω)ϕ

(mn)sp (ω)s(m)∗(ω)p(n)T(ω)

+ t(m)∗p (ω)t(n)s (ω)ϕ

(mn)ps (ω)p(m)∗(ω)s(n)T(ω), (5.12)

where the superscript T denotes matrix transpose and the factors

ϕ(mn)µν (ω) = ⟨E(m)∗

µ (ω)E(n)ν (ω)⟩, (5.13)

with µ, ν ∈ s, p and m, n ∈ 1, 2, . . . , N, are related to the corre-lations among the incident light beams. For m = n, the quantitiesϕ(mn)µν (ω) are the elements of the 2 × 2 polarization matrix associ-

ated with the nth incident wave, while the cases m = n describe themutual correlations between different waves.

The corresponding spectral polarization matrix [Eq. (A.12)] isobtained by setting r1 = r2 = r in Eq. (5.11), viz.,

Φ(r, ω) =N

∑m,n=1

W(mn)(ω)ei[k(n)(ω)−k(m)∗(ω)]·r, (5.14)

which fully describes the polarization state of the evanescent field.



5.2.1 Subwavelength coherence lengths

The conventional wisdom in optics says that δ-correlated sourcesor blackbody radiators generate the spatially most incoherent wavefields, for which the coherence length, in a lossless medium, isroughly half a wavelength of the light [230, 231]. Recently, these ar-guments have been reassessed and it was shown that, in principle,a finite-sized source can produce a field whose coherence lengthwithin the source may be arbitrarily short, even in the absence ofabsorption [232]. In addition, it has been demonstrated that a ther-mal half-space source can generate an electromagnetic near fieldwhose longitudinal correlation length, due to absorption, may bemuch shorter than the light’s wavelength [84]. In Publication VI,we demonstrate that such subwavelength coherence lengths are alsoencountered in purely evanescent fields at lossless interfaces. As arule, the shortest coherence lengths are observed very close to thesurface for high refractive-index contrasts.

As an example, we consider the superposition of two s-polarizedevanescent waves in the immediate vicinity of the surface (z = 0),created by mutually uncorrelated beams sharing the same angle ofincidence θ(ω), but having opposite azimuthal angles φ(1)(ω) = 0and φ(2)(ω) = π. The amplitudes of the beams are adjusted so thatthe individual evanescent waves have equal intensities at z = 0. Itfollows from Eqs. (5.11) and (A.14) that for such a (fully polarized)evanescent field the degree of coherence becomes

µ(∆x, ω) =1√2

√1 + cos[2k1(ω) sin θ(ω)∆x], (5.15)

where ∆x = x2 − x1 is the distance between the observation pointsalong the x axis. Owing to statistical similarity, the degree of coher-ence in Eq. (5.15) oscillates sinusoidally between 1 and 0, whereatwe define the coherence length for the two-wave superposition asthe distance from a maximum to a nearby minimum, viz.,

lcoh(ω) =λ2(ω)

4n(ω) sin θ(ω). (5.16)



We recognize at once that lcoh(ω) < λ2(ω)/4. For a high refractive-index-contrast surface, such as GaP and air with n(ω) ≈ 4 withinthe optical regime [195], the coherence length of the superpositionmay be as low as lcoh(ω) ≈ λ2(ω)/16.

5.2.2 Genuine 3D-polarized states

For 2D fields, such as beams, the 3D degree of polarization definedin Eq. (A.16) is invariably bounded as 1/2 ≤ P3D(r, ω) ≤ 1 [125].The lowest value, P3D(r, ω) = 1/2, is encountered for light that iscompletely unpolarized from the traditional 2D point of view. Val-ues within the range P3D(r, ω) < 1/2 are thereby clear signatures ofgenuine 3D fields which cannot be described with the conventionalformalism for the the degree of polarization [54].

It has been shown that light created by an optical system out ofa single, arbitrary polarized beam obeys P3D(r, ω) ≥ 1/2 [233]. Thisresult covers the situation of an evanescent wave being generatedby total internal reflection at a planar interface. Nevertheless, asdemonstrated in Publication VI, a superposition of beams may pro-duce an evanescent field with P3D(r, ω) < 1/2. In fact, for a typicalSiO2–air interface, already two partially polarized beams sharingthe same plane of incidence are sufficient for the excitation of a true3D evanescent field having P3D(r, ω) ≈ 1/4. This simple case high-lights that a rigorous and full 3D treatment is generally required todescribe the polarization state of evanescent fields.

5.3 3D-UNPOLARIZED EVANESCENT FIELDS

As the investigations in Publication VI indicate the prospect to gen-erate genuine 3D-polarized evanescent fields, one is tempted to ask:how to generate, if even possible, a fully 3D-unpolarized evanescentfield? While a completely 3D-unpolarized state is encountered forblackbody radiation [234, 235], situations in which totally unpolar-ized 3D evanescent fields can be generated by controlled meansand furnished with varying coherence properties have not previ-



ously been reported. In Publication VII, we investigate the gener-ation and electromagnetic coherence of unpolarized 3D evanescentlight fields in multibeam illumination at a planar dielectric inter-face. Our analysis reveals the feasibility to tailor evanescent fieldswith polarization qualities identical to those of universal blackbodyradiation, yet with tunable spatial coherence characteristics.

5.3.1 Generation

Let us first explore how such unconventional, fully unpolarized,genuine 3D evanescent fields could be generated above a dielec-tric surface under controllable circumstances. Because an unpolar-ized 3D light field is unambiguously represented by a polarizationmatrix that is proportional to the 3 × 3 identity matrix [125], wemust search for conditions that diagonalize Φ(r, ω) in Eq. (5.14).To this end, we employ a specific optical multibeam configurationin which the incoming waves are independent and have uncorre-lated s- and p-polarized components. Moreover, the incident beamsare taken to share the same angle of incidence θ(ω), with intensi-ties such that ⟨|E(n)

s (ω)|2⟩ = Is(ω) and ⟨|E(n)p (ω)|2⟩ = Ip(ω) for

all n ∈ 1, 2, . . . , N. In this case, the polarization matrix of theevanescent field depends only on the height above the surface, i.e.,Φ(r, ω) = Φ(z, ω), whereas the 3D degree of polarization is totallyposition independent, viz., P3D(r, ω) = P3D(ω).

To derive the conditions under which P3D(ω) = 0, we requirethe diagonal elements of Φ(z, ω) to be equal and the off-diagonalelements to be zero. Unfortunately, it turns out that P3D(ω) = 0cannot be achieved for the simplest case N = 2. Yet, as shown inPublication VII, if one utilizes two beams propagating in orthogonalazimuthal directions with Is(ω)/Ip(ω) = 2n2(ω), then P3D(ω) → 0when θ(ω) → θc(ω). So, with two beams, even though it is notpossible to generate a strictly 3D-unpolarized evanescent field, onecan create a nearly unpolarized 3D evanescent field by adjustingθ(ω) very close to the critical angle. This result has been confirmedto hold in the space–time domain too [236]. Things get different if



We recognize at once that lcoh(ω) < λ2(ω)/4. For a high refractive-index-contrast surface, such as GaP and air with n(ω) ≈ 4 withinthe optical regime [195], the coherence length of the superpositionmay be as low as lcoh(ω) ≈ λ2(ω)/16.

5.2.2 Genuine 3D-polarized states

For 2D fields, such as beams, the 3D degree of polarization definedin Eq. (A.16) is invariably bounded as 1/2 ≤ P3D(r, ω) ≤ 1 [125].The lowest value, P3D(r, ω) = 1/2, is encountered for light that iscompletely unpolarized from the traditional 2D point of view. Val-ues within the range P3D(r, ω) < 1/2 are thereby clear signatures ofgenuine 3D fields which cannot be described with the conventionalformalism for the the degree of polarization [54].

It has been shown that light created by an optical system out ofa single, arbitrary polarized beam obeys P3D(r, ω) ≥ 1/2 [233]. Thisresult covers the situation of an evanescent wave being generatedby total internal reflection at a planar interface. Nevertheless, asdemonstrated in Publication VI, a superposition of beams may pro-duce an evanescent field with P3D(r, ω) < 1/2. In fact, for a typicalSiO2–air interface, already two partially polarized beams sharingthe same plane of incidence are sufficient for the excitation of a true3D evanescent field having P3D(r, ω) ≈ 1/4. This simple case high-lights that a rigorous and full 3D treatment is generally required todescribe the polarization state of evanescent fields.

5.3 3D-UNPOLARIZED EVANESCENT FIELDS

As the investigations in Publication VI indicate the prospect to gen-erate genuine 3D-polarized evanescent fields, one is tempted to ask:how to generate, if even possible, a fully 3D-unpolarized evanescentfield? While a completely 3D-unpolarized state is encountered forblackbody radiation [234, 235], situations in which totally unpolar-ized 3D evanescent fields can be generated by controlled meansand furnished with varying coherence properties have not previ-



ously been reported. In Publication VII, we investigate the gener-ation and electromagnetic coherence of unpolarized 3D evanescentlight fields in multibeam illumination at a planar dielectric inter-face. Our analysis reveals the feasibility to tailor evanescent fieldswith polarization qualities identical to those of universal blackbodyradiation, yet with tunable spatial coherence characteristics.

5.3.1 Generation

Let us first explore how such unconventional, fully unpolarized,genuine 3D evanescent fields could be generated above a dielec-tric surface under controllable circumstances. Because an unpolar-ized 3D light field is unambiguously represented by a polarizationmatrix that is proportional to the 3 × 3 identity matrix [125], wemust search for conditions that diagonalize Φ(r, ω) in Eq. (5.14).To this end, we employ a specific optical multibeam configurationin which the incoming waves are independent and have uncorre-lated s- and p-polarized components. Moreover, the incident beamsare taken to share the same angle of incidence θ(ω), with intensi-ties such that ⟨|E(n)

s (ω)|2⟩ = Is(ω) and ⟨|E(n)p (ω)|2⟩ = Ip(ω) for

all n ∈ 1, 2, . . . , N. In this case, the polarization matrix of theevanescent field depends only on the height above the surface, i.e.,Φ(r, ω) = Φ(z, ω), whereas the 3D degree of polarization is totallyposition independent, viz., P3D(r, ω) = P3D(ω).

To derive the conditions under which P3D(ω) = 0, we requirethe diagonal elements of Φ(z, ω) to be equal and the off-diagonalelements to be zero. Unfortunately, it turns out that P3D(ω) = 0cannot be achieved for the simplest case N = 2. Yet, as shown inPublication VII, if one utilizes two beams propagating in orthogonalazimuthal directions with Is(ω)/Ip(ω) = 2n2(ω), then P3D(ω) → 0when θ(ω) → θc(ω). So, with two beams, even though it is notpossible to generate a strictly 3D-unpolarized evanescent field, onecan create a nearly unpolarized 3D evanescent field by adjustingθ(ω) very close to the critical angle. This result has been confirmedto hold in the space–time domain too [236]. Things get different if



we consider the scenario where N ≥ 3 and the incident beams areuniformly distributed, i.e., φ(n)(ω) = (n − 1)2π/N. For this case,as outlined in Publication VII, the off-diagonal elements vanish andΦ(z, ω) becomes a diagonal matrix when

Is(ω)

Ip(ω)=

[n2(ω)− 1][n2(ω) sin2 θ(ω) + 1]cos2 θ(ω) + n2(ω)[n2(ω) sin2 θ(ω)− 1]

. (5.17)

Equation (5.17), which is seen to be independent of N, is thus thecondition that results in P3D(ω) = 0. In other words, for any chosenθ(ω) > θc(ω) and n(ω) > 1, Eq. (5.17) determines precisely howthe intensities of the incident beams must be calibrated so that theevanescent field is strictly unpolarized in the 3D sense.


Besides conditions under which genuine 3D-unpolarized evanes-cent fields could be generated by manageable means, we also ana-lyze the spatial coherence of the fields in Publication VII. For the op-tical setup discussed above, the degree of coherence in Eq. (A.14) isindependent of z1 and z2 and depends only on ∆ρ = ∆xex + ∆yey,with ∆x = x2 − x1 and ∆y = y2 − y1, so that µ(r1, r2, ω) = µ(∆ρ, ω).In Fig. 5.2 we have plotted the spatial behavior of µ(∆ρ, ω) for a3D-unpolarized evanescent field above a SiO2–air interface excitedby different numbers of uniformly distributed incident beams withθ(ω) = π/3. The penetration depth of the field is roughly λ0(ω)/5.Figure 5.2 reveals peculiar subwavelength patterns in the degree ofcoherence, whose shapes depend on N. In particular, the structuresshow periodic rotational symmetries owing to the multibeam ex-citation setups, but not necessarily translational symmetries. Thisexample illustrates the possibility to excite an evanescent field thatshares the polarization properties of blackbody radiation, yet withradically different coherence characteristics, such as tunable sub-wavelength lattice-like structures in the degree of coherence.



0.28 0.43 0.58

0 1 2

0

1

2

0.20 0.39 0.58

0 1 2

0

1

2

0.20 0.39 0.58

0 1 2

0

1

2

0.18 0.38 0.58

0 1 2

0

1

2

2−

2−

1−

2−

1−

2−

1−

2−

1−

1−2− 1−

2− 1−2− 1−

)ω(

0λ

/y

∆

)ω(

0λ

/y

∆

)ω(

0λ

/y

∆

)ω(

0λ

/y

∆

)ω(0λ/x∆)ω(0λ/x∆

)ω(0λ/x∆)ω(0λ/x∆

)ω,ρ∆(µ)ω,ρ∆(µ


Figure 5.2: Spatial behavior of the degree of coherence µ(∆ρ, ω) for a 3D-unpolarizedevanescent field above a SiO2–air surface generated by N = 3 (top left), N = 4 (topright), N = 5 (bottom left), and N = 6 (bottom right) uniformly distributed and uncor-related incident beams with the angle of incidence θ(ω) = π/3. The refractive indices aren1(ω) = 1.5 and n2(ω) = 1, and λ0(ω) is the free-space wavelength.



we consider the scenario where N ≥ 3 and the incident beams areuniformly distributed, i.e., φ(n)(ω) = (n − 1)2π/N. For this case,as outlined in Publication VII, the off-diagonal elements vanish andΦ(z, ω) becomes a diagonal matrix when

Is(ω)

Ip(ω)=

[n2(ω)− 1][n2(ω) sin2 θ(ω) + 1]cos2 θ(ω) + n2(ω)[n2(ω) sin2 θ(ω)− 1]

. (5.17)

Equation (5.17), which is seen to be independent of N, is thus thecondition that results in P3D(ω) = 0. In other words, for any chosenθ(ω) > θc(ω) and n(ω) > 1, Eq. (5.17) determines precisely howthe intensities of the incident beams must be calibrated so that theevanescent field is strictly unpolarized in the 3D sense.


Besides conditions under which genuine 3D-unpolarized evanes-cent fields could be generated by manageable means, we also ana-lyze the spatial coherence of the fields in Publication VII. For the op-tical setup discussed above, the degree of coherence in Eq. (A.14) isindependent of z1 and z2 and depends only on ∆ρ = ∆xex + ∆yey,with ∆x = x2 − x1 and ∆y = y2 − y1, so that µ(r1, r2, ω) = µ(∆ρ, ω).In Fig. 5.2 we have plotted the spatial behavior of µ(∆ρ, ω) for a3D-unpolarized evanescent field above a SiO2–air interface excitedby different numbers of uniformly distributed incident beams withθ(ω) = π/3. The penetration depth of the field is roughly λ0(ω)/5.Figure 5.2 reveals peculiar subwavelength patterns in the degree ofcoherence, whose shapes depend on N. In particular, the structuresshow periodic rotational symmetries owing to the multibeam ex-citation setups, but not necessarily translational symmetries. Thisexample illustrates the possibility to excite an evanescent field thatshares the polarization properties of blackbody radiation, yet withradically different coherence characteristics, such as tunable sub-wavelength lattice-like structures in the degree of coherence.



0.28 0.43 0.58

0 1 2

0

1

2

0.20 0.39 0.58

0 1 2

0

1

2

0.20 0.39 0.58

0 1 2

0

1

2

0.18 0.38 0.58

0 1 2

0

1

2

2−

2−

1−

2−

1−

2−

1−

2−

1−

1−2− 1−

2− 1−2− 1−

)ω(

0λ

/y

∆

)ω(

0λ

/y

∆

)ω(

0λ

/y

∆

)ω(

0λ

/y

∆

)ω(0λ/x∆)ω(0λ/x∆

)ω(0λ/x∆)ω(0λ/x∆



Figure 5.2: Spatial behavior of the degree of coherence µ(∆ρ, ω) for a 3D-unpolarizedevanescent field above a SiO2–air surface generated by N = 3 (top left), N = 4 (topright), N = 5 (bottom left), and N = 6 (bottom right) uniformly distributed and uncor-related incident beams with the angle of incidence θ(ω) = π/3. The refractive indices aren1(ω) = 1.5 and n2(ω) = 1, and λ0(ω) is the free-space wavelength.




6 Complementarity in photoninterference

The principle of complementarity is a cornerstone in physics, declaringthat quantum systems inhold mutually exclusive properties [144].The arguable most prominent manifestation of complementarity isthe wave–particle duality, which restricts the coexistence of wave andparticle qualities of quantum objects [147–150]. In two-way inter-ferometry, such as the double-slit experiment or a Mach–Zehndersetup, the duality can be expressed through [152–154]

P2 + V2 ≤ 1, D2 + V2 ≤ 1, (6.1)

where P is the path predictability, quantifying the a priori ‘which-path information’ (WPI), D is the path distinguishability, represent-ing the available WPI stored in the system, and V is the intensityvisibility. For photons, however, interference does not necessarilyappear merely as intensity fringes, but also, or solely, as polarizationmodulation [157–159], a feature which the two relations in Eq. (6.1)do not account for. How complementarity is manifested under suchpolarization variation is therefore of fundamental interest.

In this chapter, by exploring polarization modulation in double-pinhole photon interference, we derive two general complementar-ity relations which cover genuine vectorial quantum-light fields ofarbitrary state. The complementarity relations are shown to reflecttwo different, intrinsic aspects of wave–particle duality of the pho-ton, having no correspondence in scalar quantum interferometry. Inparticular, it is demonstrated that, contrary to scalar light, for puresingle-photon vector light the a priori WPI does not couple to theintensity visibility, but to a generalized visibility, which also char-acterizes the variation of the polarization state. We also show thatfor general quantum light such complementarity is a manifestationof complete coherence, not of quantum-state purity.




6 Complementarity in photoninterference

The principle of complementarity is a cornerstone in physics, declaringthat quantum systems inhold mutually exclusive properties [144].The arguable most prominent manifestation of complementarity isthe wave–particle duality, which restricts the coexistence of wave andparticle qualities of quantum objects [147–150]. In two-way inter-ferometry, such as the double-slit experiment or a Mach–Zehndersetup, the duality can be expressed through [152–154]

P2 + V2 ≤ 1, D2 + V2 ≤ 1, (6.1)

where P is the path predictability, quantifying the a priori ‘which-path information’ (WPI), D is the path distinguishability, represent-ing the available WPI stored in the system, and V is the intensityvisibility. For photons, however, interference does not necessarilyappear merely as intensity fringes, but also, or solely, as polarizationmodulation [157–159], a feature which the two relations in Eq. (6.1)do not account for. How complementarity is manifested under suchpolarization variation is therefore of fundamental interest.

In this chapter, by exploring polarization modulation in double-pinhole photon interference, we derive two general complementar-ity relations which cover genuine vectorial quantum-light fields ofarbitrary state. The complementarity relations are shown to reflecttwo different, intrinsic aspects of wave–particle duality of the pho-ton, having no correspondence in scalar quantum interferometry. Inparticular, it is demonstrated that, contrary to scalar light, for puresingle-photon vector light the a priori WPI does not couple to theintensity visibility, but to a generalized visibility, which also char-acterizes the variation of the polarization state. We also show thatfor general quantum light such complementarity is a manifestationof complete coherence, not of quantum-state purity.



6.1 COHERENCE OF VECTORIAL QUANTUM LIGHT

In quantum theory of optical coherence [136], all information on thefirst-order statistical properties of a multicomponent and generallynonstationary quantized light field, at two space–time points x1 andx2, is encoded in the electric coherence matrix

G(x1, x2) = tr[ρE(−)(x1)E(+)(x2)]. (6.2)

Here E(+)(x) and E(−)(x) are the positive and negative frequencyparts of the total electric-field operator, ρ is the density operatorcharacterizing the quantum state, and tr denotes the trace. Mathe-matically, depending on whether the light is treated as a 2D or as a3D field, the electric coherence matrix in Eq. (6.2) is a 2 × 2 or 3 × 3matrix which satisfies the symmetry relation G†(x2, x1) = G(x1, x2)

[136], with the dagger representing conjugate transpose. Physically,the elements of G(x1, x2) describe the correlations between the or-thogonal field components at x1 and x2.

A complete specification of the coherence characteristics of lightrequires knowledge of all correlation orders [54]. Higher-order cor-relations are of major importance in quantum optics, as they canprovide information about the nonclassical properties of light [137].For instance, second-order correlation measurements are able todistinguish between light states with identical spectral distribu-tions but having different photon number distributions. The firstsuccessful demonstration of photon antibunching, offering a clearproof of the quantum nature of light, was made by Kimble, Dage-nais, and Mandel in 1977 by measuring such photon–photon corre-lations [237]. Yet, although higher-order correlations play a crucialrole in quantum optics, we focus exclusively on first-order correla-tions in this chapter, as they determine the intensity (and polariza-tion) variation in double-pinhole interference [54].

Motivated by the classical measure in Eq. (A.6), we may definethe degree of coherence for a general vector-light quantum field as

g(x1, x2) =∥G(x1, x2)∥F√

trG(x1, x1)trG(x2, x2), (6.3)


Complementarity in photon interference

where ∥ · ∥F is the Frobenius matrix norm. The quantity g(x1, x2) in-cludes all elements of G(x1, x2) and is a measure of the correlationsthat exist between all the orthogonal components of the quantizedelectric field at two space–time points. As its classical counterpart,also g(x1, x2) is real, invariant under unitary transformations, andbounded as 0 ≤ g(x1, x2) ≤ 1. Most importantly, following thesteps in [113,114], one can show that g(x1, x2) = 1 throughout somedomain Ω if, and only if, all the field components are fully corre-lated for all x1, x2 ∈ Ω. In this case G(x1, x2) factors in x1 and x2 andthe field is considered (first-order) fully coherent in Ω, consistentlywith Glauber’s definition of complete coherence [136]. Likewise, forg(x1, x2) = 0 no correlations occur between any of the components,and the photon field is incoherent. The range 0 < g(x1, x2) < 1corresponds to partial coherence.

For a 2D quantum-light field, say polarized in the xy plane, theelectric coherence matrix in Eq. (6.2) can be expressed as [158]

G(x1, x2) =12

3

∑j=0

Sj(x1, x2)σ j, (6.4)

where σ0 is the identity matrix, while σ1, σ2, and σ3 are the threePauli spin matrices [238]. The four complex-valued quantities

Sj(x1, x2) = tr[G(x1, x2)σ j], j ∈ 0, . . . , 3, (6.5)

offering an alternative, yet an equivalent way to represent the first-order coherence properties of the field, are quantum analogs of theclassical two-point Stokes parameters [239, 240]. For x1 = x2 = x,they become Sj(x) = tr[G(x, x)σ j], which give the expectation val-ues of the quantum Stokes operators [137], with the following inter-pretation: S0(x) is proportional to the (average) total photon num-ber, while S1(x), S2(x), and S3(x) give, respectively, the (average)differences between x- and y-polarized, +π/4- and −π/4-linearlypolarized, and right- and left-circularly polarized photons. Makinguse of Eqs. (6.3) and (6.4) one readily finds that

g2(x1, x2) =12

3

∑j=0

|sj(x1, x2)|2, (6.6)



6.1 COHERENCE OF VECTORIAL QUANTUM LIGHT

In quantum theory of optical coherence [136], all information on thefirst-order statistical properties of a multicomponent and generallynonstationary quantized light field, at two space–time points x1 andx2, is encoded in the electric coherence matrix

G(x1, x2) = tr[ρE(−)(x1)E(+)(x2)]. (6.2)

Here E(+)(x) and E(−)(x) are the positive and negative frequencyparts of the total electric-field operator, ρ is the density operatorcharacterizing the quantum state, and tr denotes the trace. Mathe-matically, depending on whether the light is treated as a 2D or as a3D field, the electric coherence matrix in Eq. (6.2) is a 2 × 2 or 3 × 3matrix which satisfies the symmetry relation G†(x2, x1) = G(x1, x2)

[136], with the dagger representing conjugate transpose. Physically,the elements of G(x1, x2) describe the correlations between the or-thogonal field components at x1 and x2.

A complete specification of the coherence characteristics of lightrequires knowledge of all correlation orders [54]. Higher-order cor-relations are of major importance in quantum optics, as they canprovide information about the nonclassical properties of light [137].For instance, second-order correlation measurements are able todistinguish between light states with identical spectral distribu-tions but having different photon number distributions. The firstsuccessful demonstration of photon antibunching, offering a clearproof of the quantum nature of light, was made by Kimble, Dage-nais, and Mandel in 1977 by measuring such photon–photon corre-lations [237]. Yet, although higher-order correlations play a crucialrole in quantum optics, we focus exclusively on first-order correla-tions in this chapter, as they determine the intensity (and polariza-tion) variation in double-pinhole interference [54].

Motivated by the classical measure in Eq. (A.6), we may definethe degree of coherence for a general vector-light quantum field as

g(x1, x2) =∥G(x1, x2)∥F√

trG(x1, x1)trG(x2, x2), (6.3)



where ∥ · ∥F is the Frobenius matrix norm. The quantity g(x1, x2) in-cludes all elements of G(x1, x2) and is a measure of the correlationsthat exist between all the orthogonal components of the quantizedelectric field at two space–time points. As its classical counterpart,also g(x1, x2) is real, invariant under unitary transformations, andbounded as 0 ≤ g(x1, x2) ≤ 1. Most importantly, following thesteps in [113,114], one can show that g(x1, x2) = 1 throughout somedomain Ω if, and only if, all the field components are fully corre-lated for all x1, x2 ∈ Ω. In this case G(x1, x2) factors in x1 and x2 andthe field is considered (first-order) fully coherent in Ω, consistentlywith Glauber’s definition of complete coherence [136]. Likewise, forg(x1, x2) = 0 no correlations occur between any of the components,and the photon field is incoherent. The range 0 < g(x1, x2) < 1corresponds to partial coherence.

For a 2D quantum-light field, say polarized in the xy plane, theelectric coherence matrix in Eq. (6.2) can be expressed as [158]

G(x1, x2) =12

3

∑j=0

Sj(x1, x2)σ j, (6.4)

where σ0 is the identity matrix, while σ1, σ2, and σ3 are the threePauli spin matrices [238]. The four complex-valued quantities

Sj(x1, x2) = tr[G(x1, x2)σ j], j ∈ 0, . . . , 3, (6.5)

offering an alternative, yet an equivalent way to represent the first-order coherence properties of the field, are quantum analogs of theclassical two-point Stokes parameters [239, 240]. For x1 = x2 = x,they become Sj(x) = tr[G(x, x)σ j], which give the expectation val-ues of the quantum Stokes operators [137], with the following inter-pretation: S0(x) is proportional to the (average) total photon num-ber, while S1(x), S2(x), and S3(x) give, respectively, the (average)differences between x- and y-polarized, +π/4- and −π/4-linearlypolarized, and right- and left-circularly polarized photons. Makinguse of Eqs. (6.3) and (6.4) one readily finds that

g2(x1, x2) =12

3

∑j=0

|sj(x1, x2)|2, (6.6)



involving the normalized two-point Stokes parameters

sj(x1, x2) =Sj(x1, x2)√

S0(x1)S0(x2), j ∈ 0, . . . , 3, (6.7)

which satisfy 0 ≤ |sj(x1, x2)| ≤ 1 for all j ∈ 0, . . . , 3.We emphasize that a representation similar to Eq. (6.4) may be

written for 3D quantum fields as well, with the expansion basisbeing the Gell-Mann matrices [238], as used to introduce the Stokesparameters for 3D classical fields [125].

6.2 PHOTON INTERFERENCE LAW

Let us now consider the double-pinhole interference experimentwith the vector nature of the photon field taken into account. Thetwo openings are located at r1 and r2 in an opaque screen A inthe xy plane and the emerging light (of angular frequency ω) isobserved on a screen B by a photodetector at position r and time t,as illustrated in Fig. 6.1. The openings are assumed to be so largethat boundary effects can be neglected, but so small that in eachthe field can be treated as uniform. Under these circumstances, by

2r

1r

r

B

A

Figure 6.1: Double-pinhole photon interference. Light impinges on a screen A with twoopenings located at r1 and r2. The emerging photons are investigated at a point r on theobservation screen B, where intensity fringes and polarization modulation may appear.



employing for both orthogonal field components the steps outlinedfor quantized scalar light in [241], the expression for the positivefrequency part of the electric-field operator at B can be written as

E(+)(r, t) = K(ω)e−iωt2

∑m=1

(amx ex + amyey)eik(ω)rm

rm, (6.8)

where K(ω) is a constant, ex and ey are the Cartesian unit vectors,k(ω) is the wave number, and rm is the distance from rm to r. More-over, the annihilation operators amµ, associated with the µ ∈ x, ypolarized radial modes emanating from pinhole m ∈ 1, 2, obeythe commutation relations

[amµ, anν] = [a†mµ, a†

nν] = 0, [amµ, a†nν] = δmnδµν, (6.9)

in which δ is the Kronecker delta.It is now straightforward to show that in the paraxial regime the

Stokes parameters in the observation plane B take on the forms

Sj(r) = S′j(r) + S′′

j (r) + 2[S′0(r)S

′′0 (r)]

1/2

× |sj(r1, r2)| cos [θj(r1, r2)− k(r1 − r2)], j ∈ 0, . . . , 3, (6.10)

where S′j(r) and S′′

j (r) are the Stokes parameters on B when, respec-tively, only the pinhole at r1 or r2 is open. Furthermore, sj(r1, r2)

is the normalized, equal-time, two-point Stokes parameter at thepinholes on A, given by Eq. (6.7), and θj(r1, r2) is its phase. Equa-tion (6.10), which is formally similar to the classical electromag-netic interference law [157–159], fully describes any effect that first-order coherence may introduce onto the intensity and the polariza-tion state of the photons in the double-pinhole configuration. Wethereby refer to Eq. (6.10) as the photon interference law. Governingany quantum light characterized by a density operator ρ, it espe-cially signifies that double-pinhole photon interference, resultingfrom spatial coherence at the two openings, does not necessarilymanifest itself solely as intensity fringes, but also, or exclusively, aspolarization-state variations on the observation screen.



involving the normalized two-point Stokes parameters

sj(x1, x2) =Sj(x1, x2)√

S0(x1)S0(x2), j ∈ 0, . . . , 3, (6.7)

which satisfy 0 ≤ |sj(x1, x2)| ≤ 1 for all j ∈ 0, . . . , 3.We emphasize that a representation similar to Eq. (6.4) may be

written for 3D quantum fields as well, with the expansion basisbeing the Gell-Mann matrices [238], as used to introduce the Stokesparameters for 3D classical fields [125].

6.2 PHOTON INTERFERENCE LAW

Let us now consider the double-pinhole interference experimentwith the vector nature of the photon field taken into account. Thetwo openings are located at r1 and r2 in an opaque screen A inthe xy plane and the emerging light (of angular frequency ω) isobserved on a screen B by a photodetector at position r and time t,as illustrated in Fig. 6.1. The openings are assumed to be so largethat boundary effects can be neglected, but so small that in eachthe field can be treated as uniform. Under these circumstances, by

2r

1r

r

B

A

Figure 6.1: Double-pinhole photon interference. Light impinges on a screen A with twoopenings located at r1 and r2. The emerging photons are investigated at a point r on theobservation screen B, where intensity fringes and polarization modulation may appear.



employing for both orthogonal field components the steps outlinedfor quantized scalar light in [241], the expression for the positivefrequency part of the electric-field operator at B can be written as

E(+)(r, t) = K(ω)e−iωt2

∑m=1

(amx ex + amyey)eik(ω)rm

rm, (6.8)

where K(ω) is a constant, ex and ey are the Cartesian unit vectors,k(ω) is the wave number, and rm is the distance from rm to r. More-over, the annihilation operators amµ, associated with the µ ∈ x, ypolarized radial modes emanating from pinhole m ∈ 1, 2, obeythe commutation relations

[amµ, anν] = [a†mµ, a†

nν] = 0, [amµ, a†nν] = δmnδµν, (6.9)

in which δ is the Kronecker delta.It is now straightforward to show that in the paraxial regime the

Stokes parameters in the observation plane B take on the forms

Sj(r) = S′j(r) + S′′

j (r) + 2[S′0(r)S

′′0 (r)]

1/2

× |sj(r1, r2)| cos [θj(r1, r2)− k(r1 − r2)], j ∈ 0, . . . , 3, (6.10)

where S′j(r) and S′′

j (r) are the Stokes parameters on B when, respec-tively, only the pinhole at r1 or r2 is open. Furthermore, sj(r1, r2)

is the normalized, equal-time, two-point Stokes parameter at thepinholes on A, given by Eq. (6.7), and θj(r1, r2) is its phase. Equa-tion (6.10), which is formally similar to the classical electromag-netic interference law [157–159], fully describes any effect that first-order coherence may introduce onto the intensity and the polariza-tion state of the photons in the double-pinhole configuration. Wethereby refer to Eq. (6.10) as the photon interference law. Governingany quantum light characterized by a density operator ρ, it espe-cially signifies that double-pinhole photon interference, resultingfrom spatial coherence at the two openings, does not necessarilymanifest itself solely as intensity fringes, but also, or exclusively, aspolarization-state variations on the observation screen.



6.3 VISIBILITY AND DISTINGUISHABILITY

As the photon interference law states that the Stokes parameters onscreen B are sinusoidally modulated by the correlations that prevailamong the electric-field components at the pinholes, we may definefour separate modulation contrasts (or visibilities) via

Vj(r) =max[Sj(r)]− min[Sj(r)]max[S0(r)] + min[S0(r)]

, j ∈0, . . . , 3, (6.11)

where max (min) stands for the maximum (minimum). The quan-tity V0(r) is the customary intensity visibility, whereas V1(r), V2(r),and V3(r) are polarization visibilities [157–159]. From Eq. (6.10) and(6.11) one then finds that

Vj(r) = C(r)|sj(r1, r2)|, j ∈ 0, . . . , 3, (6.12)

in which we have introduced the factor

C(r) =2[S′

0(r)S′′0 (r)]

1/2

S′0(r) + S′′

0 (r)≈ 2[S0(r1)S0(r2)]1/2

S0(r1) + S0(r2), (6.13)

with the approximation valid near the central axis where r1 ≈ r2.Since 0 ≤ C(r) ≤ 1, also 0 ≤ Vj(r) ≤ 1 for all j ∈ 0, . . . , 3.

In view of Eq. (6.12), it seems natural to try to find a single quan-tity that characterizes the intensity and the polarization visibilitiesof the photon field at the same time. Interestingly, on defining

V(r) =1√2

√V2

0 (r) + V21 (r) + V2

2 (r) + V23 (r), (6.14)

we observe from Eqs. (6.6), (6.12), and (6.14) that

V(r) = C(r)g(r1, r2). (6.15)

Equation (6.15) is recognized to be of the same form as the standardvisibility relation met in the scalar context [241], but now V(r) andg(r1, r2) replace the intensity visibility and the traditional degreeof coherence, respectively. It is also readily verified from Eq. (6.15)that 0 ≤ V(r) ≤ 1, with the lower limit taking place when C(r) = 0



or g(r1, r2) = 0, whereas the upper limit is reached if, and only if,C(r) = 1 and g(r1, r2) = 1. We are therefore tempted to interpretV(r) in Eq. (6.14), which includes both the intensity visibility andthe polarization-modulation contrasts, as the total visibility for thevectorial photon field. In particular, when the average number ofphotons passing through each pinhole is the same [S0(r1) = S0(r2)]the factor C(r) in Eq. (6.13) is unity, in which case Eq. (6.15) impliesthat V(r) = g(r1, r2), i.e., the total visibility on B is directly givenby the vectorial degree of coherence.

If g(r1, r2) = 0, viz., the photons are completely incoherent (nocorrelations exist between any of the components at the pinholes),then V(r) = 0, stating that neither intensity nor polarization modu-lations are observed on screen B. And vice versa, V(r) = 0 reflectsthe fact that g(r1, r2) = 0. Nonetheless, when the light exhibits par-tial coherence at the openings, in other words g(r1, r2) > 0, thenV(r) > 0 and at least one Sj(r) is modulated. And conversely, anyvariation in at least one of the Stokes parameters on B is a signatureof partially coherent photons at the pinholes. However, even in thecase of a fully coherent field, for which g(r1, r2) = 1, Eqs. (6.12) and(6.15) imply that one cannot have Vj(r) = 1 for all j ∈ 0, . . . , 3 atthe same time. Instead, at most two of the Stokes parameters mayexhibit maximum visibility simultaneously, in which case the othertwo are zero.

Next, to distinguish the light in the pinholes, we first introducethe intensity distinguishability through

D0(r1, r2) =|S0(r1)− S0(r2)|S0(r1) + S0(r2)

, (6.16)

which is bounded within the interval 0 ≤ D0(r1, r2) ≤ 1. The upperlimit, D0(r1, r2) = 1, corresponds to full intensity distinguishabilityand occurs if all photons pass through one pinhole only, in otherwords, if S0(r1) = 0 or S0(r2) = 0. The lower limit, D0(r1, r2) = 0,stands for complete intensity indistinguishability and takes placewhen there is (on average) an equal number of photons in the open-ings, viz., S0(r1) = S0(r2). The range 0 < D0(r1, r2) < 1 represents



6.3 VISIBILITY AND DISTINGUISHABILITY

As the photon interference law states that the Stokes parameters onscreen B are sinusoidally modulated by the correlations that prevailamong the electric-field components at the pinholes, we may definefour separate modulation contrasts (or visibilities) via

Vj(r) =max[Sj(r)]− min[Sj(r)]max[S0(r)] + min[S0(r)]

, j ∈0, . . . , 3, (6.11)

where max (min) stands for the maximum (minimum). The quan-tity V0(r) is the customary intensity visibility, whereas V1(r), V2(r),and V3(r) are polarization visibilities [157–159]. From Eq. (6.10) and(6.11) one then finds that

Vj(r) = C(r)|sj(r1, r2)|, j ∈ 0, . . . , 3, (6.12)

in which we have introduced the factor

C(r) =2[S′

0(r)S′′0 (r)]

1/2

S′0(r) + S′′

0 (r)≈ 2[S0(r1)S0(r2)]1/2

S0(r1) + S0(r2), (6.13)

with the approximation valid near the central axis where r1 ≈ r2.Since 0 ≤ C(r) ≤ 1, also 0 ≤ Vj(r) ≤ 1 for all j ∈ 0, . . . , 3.

In view of Eq. (6.12), it seems natural to try to find a single quan-tity that characterizes the intensity and the polarization visibilitiesof the photon field at the same time. Interestingly, on defining

V(r) =1√2

√V2

0 (r) + V21 (r) + V2

2 (r) + V23 (r), (6.14)

we observe from Eqs. (6.6), (6.12), and (6.14) that

V(r) = C(r)g(r1, r2). (6.15)

Equation (6.15) is recognized to be of the same form as the standardvisibility relation met in the scalar context [241], but now V(r) andg(r1, r2) replace the intensity visibility and the traditional degreeof coherence, respectively. It is also readily verified from Eq. (6.15)that 0 ≤ V(r) ≤ 1, with the lower limit taking place when C(r) = 0



or g(r1, r2) = 0, whereas the upper limit is reached if, and only if,C(r) = 1 and g(r1, r2) = 1. We are therefore tempted to interpretV(r) in Eq. (6.14), which includes both the intensity visibility andthe polarization-modulation contrasts, as the total visibility for thevectorial photon field. In particular, when the average number ofphotons passing through each pinhole is the same [S0(r1) = S0(r2)]the factor C(r) in Eq. (6.13) is unity, in which case Eq. (6.15) impliesthat V(r) = g(r1, r2), i.e., the total visibility on B is directly givenby the vectorial degree of coherence.

If g(r1, r2) = 0, viz., the photons are completely incoherent (nocorrelations exist between any of the components at the pinholes),then V(r) = 0, stating that neither intensity nor polarization modu-lations are observed on screen B. And vice versa, V(r) = 0 reflectsthe fact that g(r1, r2) = 0. Nonetheless, when the light exhibits par-tial coherence at the openings, in other words g(r1, r2) > 0, thenV(r) > 0 and at least one Sj(r) is modulated. And conversely, anyvariation in at least one of the Stokes parameters on B is a signatureof partially coherent photons at the pinholes. However, even in thecase of a fully coherent field, for which g(r1, r2) = 1, Eqs. (6.12) and(6.15) imply that one cannot have Vj(r) = 1 for all j ∈ 0, . . . , 3 atthe same time. Instead, at most two of the Stokes parameters mayexhibit maximum visibility simultaneously, in which case the othertwo are zero.

Next, to distinguish the light in the pinholes, we first introducethe intensity distinguishability through

D0(r1, r2) =|S0(r1)− S0(r2)|S0(r1) + S0(r2)

, (6.16)

which is bounded within the interval 0 ≤ D0(r1, r2) ≤ 1. The upperlimit, D0(r1, r2) = 1, corresponds to full intensity distinguishabilityand occurs if all photons pass through one pinhole only, in otherwords, if S0(r1) = 0 or S0(r2) = 0. The lower limit, D0(r1, r2) = 0,stands for complete intensity indistinguishability and takes placewhen there is (on average) an equal number of photons in the open-ings, viz., S0(r1) = S0(r2). The range 0 < D0(r1, r2) < 1 represents



partial intensity distinguishability. To further differentiate the lightin the pinholes, we define also the polarization distinguishability via

Dp(r1, r2) =|S(r1)− S(r2)|S0(r1) + S0(r2)

, (6.17)

where S(rm) = [S1(rm), S2(rm), S3(rm)] is the Poincare vector [55] inpinhole m ∈ 1, 2. Clearly 0 ≤ Dp(r1, r2) ≤ 1, with maximal polar-ization distinguishability, Dp(r1, r2) = 1, being possible only for or-thogonally fully polarized light, whereas total absence of polariza-tion distinguishability, Dp(r1, r2) = 0, occurs when S(r1) = S(r2).The intermediate values are signatures of partial polarization dis-tinguishability.

6.4 WEAK AND STRONG COMPLEMENTARITY

The stage is now set for quantifying complementarity for genuinevector-light quantum fields of arbitrary state in double-pinhole in-terference. As shown in Publication VIII,

D20(r1, r2) + V2(r) ≤ 1, (6.18)

D2p(r1, r2) + V2

0 (r) ≤ 1, (6.19)

establishing fundamental upper limits for D0(r1, r2) and V(r), aswell as for Dp(r1, r2) and V0(r), of a vectorial photon field. Note thatin the case of scalar light (corresponding to a uniformly polarizedlight field), for which V(r) → V0(r) and Dp(r1, r2) → D0(r1, r2),the two relations above merge into D2

0(r1, r2) + V20 (r) ≤ 1. Most

importantly, the analysis in Publication VIII reveals that

D20(r1, r2) + V2(r) = 1, if g(r1, r2) = 1, (6.20)

D2p(r1, r2) + V2

0 (r) = 1, if g(r1, r2) = 1, (6.21)

stating that when the light at the pinholes is completely coherent inthe full vector sense, intensity distinguishability and total visibility,as well as polarization distinguishability and intensity visibility, aremutually exclusive quantities.



Equations (6.18)–(6.19) and (6.20)–(6.21) constitute the main re-sults of Publication VIII. The former, making no restrictions on thequantum state involved, are associated with weak complementarity,because the parameters can in principle vary independently of eachother as long as the sums do not exceed unity. For the latter, on theother hand, a variation of one parameter always changes the otherso that the sums strictly equal unity, whereupon these relations areregarded as representing strong complementarity.

It is important to understand that strong complementarity is nota manifestation of quantum-state purity, but of complete coherence.To see this, we consider the pure four-photon state |ψ⟩ = |1, 1, 1, 1⟩,where the notation |n1x, n1y, n2x, n2y⟩ = |n1x⟩1x |n1y⟩1y |n2x⟩2x |n2y⟩2yhas been adopted, nmµ being the number of photons in the | · ⟩mµ

mode. In this case Eqs. (6.7) and (6.12) result in Vj(r) = 0 for allj ∈ 0, . . . , 3, whereupon neither intensity nor polarization modu-lations are observed. Equations (6.16) and (6.17), on the other hand,yield D0(r1, r2) = 0 and Dp(r1, r2) = 0, so there is no intensity orpolarization distinguishability either. Hence D2

0(r1, r2) + V2(r) = 0and D2

p(r1, r2) + V20 (r) = 0, contradicting strong complementarity.

A profound quality that concerns all the complementarity rela-tions (6.18)–(6.21) is that they govern, not only quantum light, butalso classical light, either scalar of vectorial in nature, since the def-initions for the visibilities and distinguishabilities do not care aboutwhether the electromagnetic field is quantized or classical.

6.5 WAVE–PARTICLE DUALITY OF THE PHOTON

Although the complementarity relations (6.18)–(6.21) cover light ofany state, it is of particular (and fundamental) interest to investi-gate how complementarity is manifested for a single photon in thepresence of polarization modulation. Let us therefore examine indetail the case of an arbitrary, pure single-photon state

|ψ⟩ = c1x |1, 0, 0, 0⟩+ c1y |0, 1, 0, 0⟩+ c2x |0, 0, 1, 0⟩+ c2y |0, 0, 0, 1⟩ , (6.22)



partial intensity distinguishability. To further differentiate the lightin the pinholes, we define also the polarization distinguishability via

Dp(r1, r2) =|S(r1)− S(r2)|S0(r1) + S0(r2)

, (6.17)

where S(rm) = [S1(rm), S2(rm), S3(rm)] is the Poincare vector [55] inpinhole m ∈ 1, 2. Clearly 0 ≤ Dp(r1, r2) ≤ 1, with maximal polar-ization distinguishability, Dp(r1, r2) = 1, being possible only for or-thogonally fully polarized light, whereas total absence of polariza-tion distinguishability, Dp(r1, r2) = 0, occurs when S(r1) = S(r2).The intermediate values are signatures of partial polarization dis-tinguishability.

6.4 WEAK AND STRONG COMPLEMENTARITY

The stage is now set for quantifying complementarity for genuinevector-light quantum fields of arbitrary state in double-pinhole in-terference. As shown in Publication VIII,

D20(r1, r2) + V2(r) ≤ 1, (6.18)

D2p(r1, r2) + V2

0 (r) ≤ 1, (6.19)

establishing fundamental upper limits for D0(r1, r2) and V(r), aswell as for Dp(r1, r2) and V0(r), of a vectorial photon field. Note thatin the case of scalar light (corresponding to a uniformly polarizedlight field), for which V(r) → V0(r) and Dp(r1, r2) → D0(r1, r2),the two relations above merge into D2

0(r1, r2) + V20 (r) ≤ 1. Most

importantly, the analysis in Publication VIII reveals that

D20(r1, r2) + V2(r) = 1, if g(r1, r2) = 1, (6.20)

D2p(r1, r2) + V2

0 (r) = 1, if g(r1, r2) = 1, (6.21)

stating that when the light at the pinholes is completely coherent inthe full vector sense, intensity distinguishability and total visibility,as well as polarization distinguishability and intensity visibility, aremutually exclusive quantities.



Equations (6.18)–(6.19) and (6.20)–(6.21) constitute the main re-sults of Publication VIII. The former, making no restrictions on thequantum state involved, are associated with weak complementarity,because the parameters can in principle vary independently of eachother as long as the sums do not exceed unity. For the latter, on theother hand, a variation of one parameter always changes the otherso that the sums strictly equal unity, whereupon these relations areregarded as representing strong complementarity.

It is important to understand that strong complementarity is nota manifestation of quantum-state purity, but of complete coherence.To see this, we consider the pure four-photon state |ψ⟩ = |1, 1, 1, 1⟩,where the notation |n1x, n1y, n2x, n2y⟩ = |n1x⟩1x |n1y⟩1y |n2x⟩2x |n2y⟩2yhas been adopted, nmµ being the number of photons in the | · ⟩mµ

mode. In this case Eqs. (6.7) and (6.12) result in Vj(r) = 0 for allj ∈ 0, . . . , 3, whereupon neither intensity nor polarization modu-lations are observed. Equations (6.16) and (6.17), on the other hand,yield D0(r1, r2) = 0 and Dp(r1, r2) = 0, so there is no intensity orpolarization distinguishability either. Hence D2

0(r1, r2) + V2(r) = 0and D2

p(r1, r2) + V20 (r) = 0, contradicting strong complementarity.

A profound quality that concerns all the complementarity rela-tions (6.18)–(6.21) is that they govern, not only quantum light, butalso classical light, either scalar of vectorial in nature, since the def-initions for the visibilities and distinguishabilities do not care aboutwhether the electromagnetic field is quantized or classical.

6.5 WAVE–PARTICLE DUALITY OF THE PHOTON

Although the complementarity relations (6.18)–(6.21) cover light ofany state, it is of particular (and fundamental) interest to investi-gate how complementarity is manifested for a single photon in thepresence of polarization modulation. Let us therefore examine indetail the case of an arbitrary, pure single-photon state

|ψ⟩ = c1x |1, 0, 0, 0⟩+ c1y |0, 1, 0, 0⟩+ c2x |0, 0, 1, 0⟩+ c2y |0, 0, 0, 1⟩ , (6.22)



in which the coefficients, with |cmµ|2 giving the probability to findthe photon µ ∈ x, y polarized in pinhole m ∈ 1, 2, are normal-ized such that |c1x|2 + |c1y|2 + |c2x|2 + |c2y|2 = 1.

From Eqs. (6.12) and (6.22) we first obtain (for clarity, we adoptlower-case symbols for this single-photon case)

v0(r) = 2|c∗1xc2x + c∗1yc2y|, (6.23)

v1(r) = 2|c∗1xc2x − c∗1yc2y|, (6.24)

v2(r) = 2|c∗1xc2y + c∗1yc2x|, (6.25)

v3(r) = 2|c∗1xc2y − c∗1yc2x|, (6.26)

revealing that for the (pure) single-photon state (6.22) all four vis-ibilities may be nonzero. Equations (6.6), (6.7), and (6.22) indicatefurther that g(r1, r2) = 1, signifying that not only in the scalar con-text [241], but also within the general vector framework the (pure)one-photon field is always completely coherent. Eventually, owingto full coherence, the strong complementarity relations (6.20) and(6.21) hold for any single-photon state possessing the form (6.22).Indeed, concerning total visibility and intensity distinguishability,from Eqs. (6.13)–(6.16) and (6.22) we find that

v(r) = 2√

p1 p2, d0(r1, r2) = |p1 − p2|, (6.27)

where pm = |cmx|2 + |cmy|2 is the probability for the photon to passthrough pinhole m ∈ 1, 2, and clearly

d20(r1, r2) + v2(r) = 1. (6.28)

For the polarization distinguishability, Eqs. (6.17) and (6.22) yield

dp(r1, r2) =√

1 − 4|c∗1xc2x + c∗1yc2y|2, (6.29)

implying together with Eq. (6.23) that

d2p(r1, r2) + v2

0(r) = 1, (6.30)

also in full agreement with Eq. (6.21).



The complementarity relations (6.28) and (6.30) may be regardedas reflecting two distinct aspects of wave–particle duality of the pho-ton. In particular, we observe that d0(r1, r2) in Eq. (6.27) coincideswith the path predictability P of Eq. (6.1). For scalar light, this a pri-ori WPI of any pure one-photon state satisfies, using our notation,d2

0(r1, r2) + v20(r) = 1 [152, 153]. Nevertheless, such a relationship is

principally no longer valid for vectorial light (see below), becauseEq. (6.1) does not take into account the polarization modulation,which Eq. (6.28) instead does. In other words, within the generalvector framework the initial WPI becomes coupled with total vis-ibility, not just intensity visibility, revealing a novel fundamentalaspect of photon wave–particle duality. This finding, which has nocorrespondence in scalar quantum interferometry, is another majorresult of Publication VIII.

As an example, let us consider the scenario with c1y = c2x = 0and c1x = c2y = 1/

√2, which means that

|ψ⟩ = (|1, 0, 0, 0⟩+ |0, 0, 0, 1⟩)/√

2. (6.31)

The photon is now in a superposition of being x polarized at pin-hole 1 with probability p1 = 1/2 and y polarized at pinhole 2 withprobability p2 = 1/2. Making use of Eqs. (6.23), (6.27), and (6.31)one readily verifies that v0(r) = 0 and d0(r1, r2) = 0, whereupond2

0(r1, r2) + v20(r) = 0 although the state (6.31) is pure, contradict-

ing the first relation of Eq. (6.1), where the equality sign holds forpure states [152, 153]. Yet, albeit intensity distinguishability in thissituation is zero, from Eqs. (6.27) and (6.31) we find that total vis-ibility is maximal, viz., v(r) = 1, and hence the complementarityrelation (6.28) holds. Moreover, Eqs. (6.29) and (6.31) yield maximalpolarization distinguishability, dp(r1, r2) = 1, whereat the comple-mentarity relation (6.30) is also satisfied.

The fact that v0(r) = 0 and v(r) = 1 signals that the state (6.31)exhibits exclusively polarization modulation, and Eqs. (6.24)–(6.26)reveal that v1(r) = 0, v2(r) = 1, and v3(r) = 1. As S′

j(r) and S′′j (r) in

the photon interference law (6.10) are spatially slowly varying func-tions near the central axis on B, the respective normalized Stokes



in which the coefficients, with |cmµ|2 giving the probability to findthe photon µ ∈ x, y polarized in pinhole m ∈ 1, 2, are normal-ized such that |c1x|2 + |c1y|2 + |c2x|2 + |c2y|2 = 1.

From Eqs. (6.12) and (6.22) we first obtain (for clarity, we adoptlower-case symbols for this single-photon case)

v0(r) = 2|c∗1xc2x + c∗1yc2y|, (6.23)

v1(r) = 2|c∗1xc2x − c∗1yc2y|, (6.24)

v2(r) = 2|c∗1xc2y + c∗1yc2x|, (6.25)

v3(r) = 2|c∗1xc2y − c∗1yc2x|, (6.26)

revealing that for the (pure) single-photon state (6.22) all four vis-ibilities may be nonzero. Equations (6.6), (6.7), and (6.22) indicatefurther that g(r1, r2) = 1, signifying that not only in the scalar con-text [241], but also within the general vector framework the (pure)one-photon field is always completely coherent. Eventually, owingto full coherence, the strong complementarity relations (6.20) and(6.21) hold for any single-photon state possessing the form (6.22).Indeed, concerning total visibility and intensity distinguishability,from Eqs. (6.13)–(6.16) and (6.22) we find that

v(r) = 2√

p1 p2, d0(r1, r2) = |p1 − p2|, (6.27)

where pm = |cmx|2 + |cmy|2 is the probability for the photon to passthrough pinhole m ∈ 1, 2, and clearly

d20(r1, r2) + v2(r) = 1. (6.28)

For the polarization distinguishability, Eqs. (6.17) and (6.22) yield

dp(r1, r2) =√

1 − 4|c∗1xc2x + c∗1yc2y|2, (6.29)

implying together with Eq. (6.23) that

d2p(r1, r2) + v2

0(r) = 1, (6.30)

also in full agreement with Eq. (6.21).



The complementarity relations (6.28) and (6.30) may be regardedas reflecting two distinct aspects of wave–particle duality of the pho-ton. In particular, we observe that d0(r1, r2) in Eq. (6.27) coincideswith the path predictability P of Eq. (6.1). For scalar light, this a pri-ori WPI of any pure one-photon state satisfies, using our notation,d2

0(r1, r2) + v20(r) = 1 [152, 153]. Nevertheless, such a relationship is

principally no longer valid for vectorial light (see below), becauseEq. (6.1) does not take into account the polarization modulation,which Eq. (6.28) instead does. In other words, within the generalvector framework the initial WPI becomes coupled with total vis-ibility, not just intensity visibility, revealing a novel fundamentalaspect of photon wave–particle duality. This finding, which has nocorrespondence in scalar quantum interferometry, is another majorresult of Publication VIII.

As an example, let us consider the scenario with c1y = c2x = 0and c1x = c2y = 1/

√2, which means that

|ψ⟩ = (|1, 0, 0, 0⟩+ |0, 0, 0, 1⟩)/√

2. (6.31)

The photon is now in a superposition of being x polarized at pin-hole 1 with probability p1 = 1/2 and y polarized at pinhole 2 withprobability p2 = 1/2. Making use of Eqs. (6.23), (6.27), and (6.31)one readily verifies that v0(r) = 0 and d0(r1, r2) = 0, whereupond2

0(r1, r2) + v20(r) = 0 although the state (6.31) is pure, contradict-

ing the first relation of Eq. (6.1), where the equality sign holds forpure states [152, 153]. Yet, albeit intensity distinguishability in thissituation is zero, from Eqs. (6.27) and (6.31) we find that total vis-ibility is maximal, viz., v(r) = 1, and hence the complementarityrelation (6.28) holds. Moreover, Eqs. (6.29) and (6.31) yield maximalpolarization distinguishability, dp(r1, r2) = 1, whereat the comple-mentarity relation (6.30) is also satisfied.

The fact that v0(r) = 0 and v(r) = 1 signals that the state (6.31)exhibits exclusively polarization modulation, and Eqs. (6.24)–(6.26)reveal that v1(r) = 0, v2(r) = 1, and v3(r) = 1. As S′

j(r) and S′′j (r) in

the photon interference law (6.10) are spatially slowly varying func-tions near the central axis on B, the respective normalized Stokes



parameters, sj(r) = Sj(r)/S0(r) with j ∈ 0, . . . , 3, are found to be

s1(r) = 0, s2(r) = cos k∆r, s3(r) = sin k∆r, (6.32)

where ∆r = r1 − r2 and Eq. (6.7) has been used. Recalling the phys-ical meanings of the Stokes parameters, here s1(r) = 0 indicatesthat at every point on B the photon is equally likely x polarized ory polarized. More precisely, in repeated single-photon experimentsone would get an equal distribution of x- and y-polarized light atthe observation screen. The two oscillatory terms, s2(r) and s3(r),are more intriguing. In particular, at k∆r = 2lπ with l ∈ Z we findthat s2(r) = 1, whereas for k∆r = (2l + 1)π one has s2(r) = −1 [inboth situations s3(r) = 0]. The former (latter) condition signifiesthat if a −π/4-linear (+π/4-linear) polarizer is placed in front ofthe detector, a count signal is never obtained from those locations.On the other hand, for k∆r = (2l + 1

2 )π and k∆r = (2l − 12 )π we see

that s3(r) = 1 and s3(r) = −1 [now s2(r) = 0], respectively, withthe former (latter) indicating that at those places a left-circularly(right-circularly) polarized photon is never detected.

Finally, because the state (6.31) is orthogonally polarized in theopenings, the photon is also completely which-path marked; for in-stance, if we were to measure y-polarized (x-polarized) light, thenwhenever a count signal is obtained from the detector we know thatthe photon has passed the second (first) pinhole. The same concernsall orthogonal polarization states of the photon, and in such situa-tions dp(r1, r2) = 1. By contrast, when the polarization is uniform inthe openings, so that dp(r1, r2) = 0, there is no chance to obtain anypath information of the photon from a polarization measurement.Following these reasonings, in scenarios where 0 < dp(r1, r2) < 1one could gain partial knowledge about the photon’s path. We maytherefore identify dp(r1, r2) in Eq. (6.29) as quantifying the a posteri-ori WPI of the photon, i.e., the available WPI that can be extractedretrodictively by polarization measurements. Hence, for the puresingle-photon state (6.22), the polarization distinguishability sharessimilarity with the path distinguishability D of Eq. (6.1) [154].


7 Conclusions

In this thesis we have presented results relating to theoretical re-search in three topics of electromagnetic nanophotonics: novel SPPmodes (Publications I–III), partial coherence of optical surface fields(Publications IV–VII), and complementarity in genuine vector-lightphoton interference (Publication VIII). Below we summarize themain conclusions of our work and discuss potential future research.

7.1 SUMMARY OF MAIN RESULTS

In Publication I, we investigated by rigorous means single-interfaceSPP propagation in the presence of metal absorption. Such SPPs areconceptually the most fundamental ones in plasmonics and, sinceabsorption is an inherent element of metals in the optical regime,incorporation of losses is necessary. It was shown that the conven-tional approximate analysis that is frequently utilized to estimateSPP propagation may yield inaccurate conclusions even in caseswhere it is presumed to hold. Most importantly, our exact approachpredicted the existence of a new type of backward-propagating SPPmode, sharing similarity with fields encountered in metamaterials,which is totally excluded within the approximate treatment. Theseresults do not only underline the essential differences between therigorous and approximate frameworks, but also convey an impor-tant message: just accounting for losses is not sufficient to get reli-able results, attention is also to be paid when introducing simplifi-cations into the SPP-field analysis.

Publication II dealt with mode solutions at an absorptive metalfilm in a symmetric and lossless surrounding. The specific aim wasto identify all possible plane-wave mode solutions in such a geom-etry that follow from Maxwell’s equations and to classify them ac-cording to their field-propagation characteristics. In addition to theones reported in literature, sets of entirely new mode types were



parameters, sj(r) = Sj(r)/S0(r) with j ∈ 0, . . . , 3, are found to be

s1(r) = 0, s2(r) = cos k∆r, s3(r) = sin k∆r, (6.32)

where ∆r = r1 − r2 and Eq. (6.7) has been used. Recalling the phys-ical meanings of the Stokes parameters, here s1(r) = 0 indicatesthat at every point on B the photon is equally likely x polarized ory polarized. More precisely, in repeated single-photon experimentsone would get an equal distribution of x- and y-polarized light atthe observation screen. The two oscillatory terms, s2(r) and s3(r),are more intriguing. In particular, at k∆r = 2lπ with l ∈ Z we findthat s2(r) = 1, whereas for k∆r = (2l + 1)π one has s2(r) = −1 [inboth situations s3(r) = 0]. The former (latter) condition signifiesthat if a −π/4-linear (+π/4-linear) polarizer is placed in front ofthe detector, a count signal is never obtained from those locations.On the other hand, for k∆r = (2l + 1

2 )π and k∆r = (2l − 12 )π we see

that s3(r) = 1 and s3(r) = −1 [now s2(r) = 0], respectively, withthe former (latter) indicating that at those places a left-circularly(right-circularly) polarized photon is never detected.

Finally, because the state (6.31) is orthogonally polarized in theopenings, the photon is also completely which-path marked; for in-stance, if we were to measure y-polarized (x-polarized) light, thenwhenever a count signal is obtained from the detector we know thatthe photon has passed the second (first) pinhole. The same concernsall orthogonal polarization states of the photon, and in such situa-tions dp(r1, r2) = 1. By contrast, when the polarization is uniform inthe openings, so that dp(r1, r2) = 0, there is no chance to obtain anypath information of the photon from a polarization measurement.Following these reasonings, in scenarios where 0 < dp(r1, r2) < 1one could gain partial knowledge about the photon’s path. We maytherefore identify dp(r1, r2) in Eq. (6.29) as quantifying the a posteri-ori WPI of the photon, i.e., the available WPI that can be extractedretrodictively by polarization measurements. Hence, for the puresingle-photon state (6.22), the polarization distinguishability sharessimilarity with the path distinguishability D of Eq. (6.1) [154].


7 Conclusions

In this thesis we have presented results relating to theoretical re-search in three topics of electromagnetic nanophotonics: novel SPPmodes (Publications I–III), partial coherence of optical surface fields(Publications IV–VII), and complementarity in genuine vector-lightphoton interference (Publication VIII). Below we summarize themain conclusions of our work and discuss potential future research.

7.1 SUMMARY OF MAIN RESULTS

In Publication I, we investigated by rigorous means single-interfaceSPP propagation in the presence of metal absorption. Such SPPs areconceptually the most fundamental ones in plasmonics and, sinceabsorption is an inherent element of metals in the optical regime,incorporation of losses is necessary. It was shown that the conven-tional approximate analysis that is frequently utilized to estimateSPP propagation may yield inaccurate conclusions even in caseswhere it is presumed to hold. Most importantly, our exact approachpredicted the existence of a new type of backward-propagating SPPmode, sharing similarity with fields encountered in metamaterials,which is totally excluded within the approximate treatment. Theseresults do not only underline the essential differences between therigorous and approximate frameworks, but also convey an impor-tant message: just accounting for losses is not sufficient to get reli-able results, attention is also to be paid when introducing simplifi-cations into the SPP-field analysis.

Publication II dealt with mode solutions at an absorptive metalfilm in a symmetric and lossless surrounding. The specific aim wasto identify all possible plane-wave mode solutions in such a geom-etry that follow from Maxwell’s equations and to classify them ac-cording to their field-propagation characteristics. In addition to theones reported in literature, sets of entirely new mode types were



found, manifested both as bound waves and leaky waves. Like-wise, we showed that some commonly accepted mode solutions arenot actually admitted by Maxwell’s equations. Moreover, whereasprevious studies have concentrated only on the region outside thefilm, we analyzed the properties of the various modes also withinthe slab. It was found that in the film both forward- and backward-propagating waves may occur, depending on the mode type and thematerial parameters. The mode classes can be interpreted as rep-resenting resonance conditions generalized from ones encounteredin conventional optics.

In Publication III, we demonstrated for the first time the conver-sion of a higher-order mode (HOM) at a thin metal slab, normallynot regarded useful, into a long-range surface mode in situationswhere the long-range fundamental mode (FM) does not exist andthe propagation distance of the single-interface SPP is negligible.The finding of this novel electromagnetic near-field phenomenonconstitutes the pinnacle of our modal studies. In this process, anunexpected mode interchange occurs between a HOM and the FM,at which point the propagation length of the HOM may experienceeven a thousandfold enlargement. The discovery of such a modecrossover, which may take place for many different material com-binations and frequency bandwidths, is anything but trivial, sinceit cannot be predicted from the mode dispersion relations. In addi-tion, unlike with the FM, the HOM’s long-range behavior does notoriginate from the coupling of the SPPs on the two slab boundariesand, remarkably, in some cases it comes with even increased sur-face confinement. Our results, which follow from rigorous electro-magnetic theory, thus provide deeper insights into the foundationsof subwavelength thin-film plasmonics and may find use in futurenanophotonics and optoelectronics.

A framework to customize the vectorial coherence of polychro-matic SPPs in the Kretschmann setup by controlling the correlationsof the excitation light was advanced in Publication IV. To this end,the general space–time coherence matrix, valid for stationary andnonstationary SPP fields of arbitrary spectra and spectral correla-


Conclusions

tions, was analytically determined. As a key result, we derived therelation between the correlation functions of the light source andthe SPP field, enabling one to ascertain the illumination coherenceto achieve the desired coherence state for the polychromatic SPPfield. We also showed that narrowband SPPs are virtually propaga-tion invariant and fully polarized. Quite surprisingly, even broad-band SPPs of widely variable coherence were revealed to possessa high degree of polarization, at least for metals and optical fre-quencies for which SPP propagation is appreciable. Publication IVestablishes a novel paradigm in statistical plasmonics, referred toas plasmon coherence engineering, which could be instrumentalfor sensor applications, interferometry, spectroscopy, and controllednanoparticle excitation.

Publication V concerned the spatial coherence and polarizationof a stationary two-mode SPP field consisting of the long-range andshort-range SPPs at a metallic nanofilm. These two SPP modes areprominent in thin-film plasmonics, and in our study the short-rangeSPP played a pivotal role facilitating coherence and polarizationmodifications. The main objective was to examine the fundamen-tal limits that the spectral degrees of coherence and polarizationof such a two-mode field can assume, irrespective of the excitationprocess, and how the degrees vary within their extremal valueswhen the media, frequency, and film thickness are altered. As fullcorrelation naturally yields complete coherence and polarization,we took the modes uncorrelated to assess the lower ranges. It wasshown that, due to electromagnetic similarity, such an SPP fieldalways exhibits regions of quite a high (low) degree of coherence.At these locations, the two-mode field would interact with nearbynanoparticles in a coherent (incoherent) manner. We also demon-strated that the coherence lengths may extend from subwavelengthscales to tens or hundreds of wavelengths. Finally, Publication Vindicated that with ultra-thin films the generally highly polarizedSPP field can be partially polarized within subwavelength distancesfrom the excitation region.

In Publication VI, we analyzed the electromagnetic coherence



found, manifested both as bound waves and leaky waves. Like-wise, we showed that some commonly accepted mode solutions arenot actually admitted by Maxwell’s equations. Moreover, whereasprevious studies have concentrated only on the region outside thefilm, we analyzed the properties of the various modes also withinthe slab. It was found that in the film both forward- and backward-propagating waves may occur, depending on the mode type and thematerial parameters. The mode classes can be interpreted as rep-resenting resonance conditions generalized from ones encounteredin conventional optics.

In Publication III, we demonstrated for the first time the conver-sion of a higher-order mode (HOM) at a thin metal slab, normallynot regarded useful, into a long-range surface mode in situationswhere the long-range fundamental mode (FM) does not exist andthe propagation distance of the single-interface SPP is negligible.The finding of this novel electromagnetic near-field phenomenonconstitutes the pinnacle of our modal studies. In this process, anunexpected mode interchange occurs between a HOM and the FM,at which point the propagation length of the HOM may experienceeven a thousandfold enlargement. The discovery of such a modecrossover, which may take place for many different material com-binations and frequency bandwidths, is anything but trivial, sinceit cannot be predicted from the mode dispersion relations. In addi-tion, unlike with the FM, the HOM’s long-range behavior does notoriginate from the coupling of the SPPs on the two slab boundariesand, remarkably, in some cases it comes with even increased sur-face confinement. Our results, which follow from rigorous electro-magnetic theory, thus provide deeper insights into the foundationsof subwavelength thin-film plasmonics and may find use in futurenanophotonics and optoelectronics.

A framework to customize the vectorial coherence of polychro-matic SPPs in the Kretschmann setup by controlling the correlationsof the excitation light was advanced in Publication IV. To this end,the general space–time coherence matrix, valid for stationary andnonstationary SPP fields of arbitrary spectra and spectral correla-


Conclusions

tions, was analytically determined. As a key result, we derived therelation between the correlation functions of the light source andthe SPP field, enabling one to ascertain the illumination coherenceto achieve the desired coherence state for the polychromatic SPPfield. We also showed that narrowband SPPs are virtually propaga-tion invariant and fully polarized. Quite surprisingly, even broad-band SPPs of widely variable coherence were revealed to possessa high degree of polarization, at least for metals and optical fre-quencies for which SPP propagation is appreciable. Publication IVestablishes a novel paradigm in statistical plasmonics, referred toas plasmon coherence engineering, which could be instrumentalfor sensor applications, interferometry, spectroscopy, and controllednanoparticle excitation.

Publication V concerned the spatial coherence and polarizationof a stationary two-mode SPP field consisting of the long-range andshort-range SPPs at a metallic nanofilm. These two SPP modes areprominent in thin-film plasmonics, and in our study the short-rangeSPP played a pivotal role facilitating coherence and polarizationmodifications. The main objective was to examine the fundamen-tal limits that the spectral degrees of coherence and polarizationof such a two-mode field can assume, irrespective of the excitationprocess, and how the degrees vary within their extremal valueswhen the media, frequency, and film thickness are altered. As fullcorrelation naturally yields complete coherence and polarization,we took the modes uncorrelated to assess the lower ranges. It wasshown that, due to electromagnetic similarity, such an SPP fieldalways exhibits regions of quite a high (low) degree of coherence.At these locations, the two-mode field would interact with nearbynanoparticles in a coherent (incoherent) manner. We also demon-strated that the coherence lengths may extend from subwavelengthscales to tens or hundreds of wavelengths. Finally, Publication Vindicated that with ultra-thin films the generally highly polarizedSPP field can be partially polarized within subwavelength distancesfrom the excitation region.

In Publication VI, we analyzed the electromagnetic coherence



and partial polarization of stationary, purely evanescent light fieldsgenerated in total internal reflection at a lossless dielectric interface.Employing the spectral degree of coherence, we showed that forsuch fields the coherence length in air can be significantly shorterthan the free-space wavelength, especially for high refractive-indexcontrasts. The coherence length is typically smallest in the immedi-ate vicinity of the surface, but may get very large already within awavelength from it. We also adopted the 3D degree of polarizationto demonstrate that already two beams are sufficient to generatea true 3D evanescent field which cannot be described by the con-ventional polarization theory. Publication VI revealed that, in gen-eral, the coherence and polarization properties of electromagneticsurface fields at subwavelength scales may, even in the absence ofabsorption, differ notably from those a wavelength or more awayfrom the supporting interface.

Motivated by these results, in Publication VII we explored con-ditions under which stationary, fully 3D-unpolarized evanescentlight fields could be generated by manageable means at a losslessdielectric surface in a configuration involving multiple illuminationbeams. It was shown that by using two incident beams it is possibleto excite a nearly unpolarized 3D evanescent field, but to achievea strictly 3D-unpolarized state requires at least a three-beam setup.We also investigated the spectral electromagnetic coherence of suchfields and demonstrated that their degrees of coherence may varyconsiderably, exhibiting diverse subwavelength lattice-like patterns,depending on the applied modality. These findings suggest the fea-sibility to customize evanescent light fields with polarization char-acteristics identical to those of universal blackbody radiation, yetwith adjustable coherence properties.

Finally, in Publication VIII, by examining polarization modu-lation in double-pinhole photon interference, we formulated twogeneral complementarity relations for genuine vectorial quantum-light fields of arbitrary state. To this end, we derived the photoninterference law and a generalized visibility relation, which char-acterize any effect that spatial coherence at the openings may have


Conclusions

on the number and polarization variations of the photons in theobservation plane. In addition, we introduced an intensity distin-guishability and a polarization distinguishability to differentiate thelight in the pinholes. The complementarity relations, establishinglinks between these quantities, were identified to reflect two sepa-rate features of wave–particle duality of the photon. In particular,we demonstrated that the a priori which-path information of single-photon vector light does not couple to the intensity visibility, but tothe total visibility, which accounts also for polarization modulation.This discovery reveals an intrinsic aspect of photon wave–particleduality, not reported earlier to our knowledge. It was also shownthat for a general quantum-light field such complementarity is amanifestation of complete coherence, not of quantum-state purity.Photon-polarization modulation thereby entails novel, fundamentalphysical facets of quantum complementarity.

7.2 FUTURE PROSPECTS

Altogether, in this thesis, we have addressed only a limited numberof research topics within the realms of plasmonics, electromagneticcoherence, and quantum complementarity. There are naturally stilla rich diversity of physics to explore and important open questionsthat deserve further investigation in all of these areas.

The novel metal-slab mode solutions derived in this work con-cerned a symmetric environment. What about an asymmetric metal-slab or a multi-surface geometry? Can a rigorous analysis adaptedfor such configurations also reveal new mode solutions, or evenwhole mode-solution classes, with features similar as or drasticallydifferent from the ones met in the symmetric case? Are mode inter-changes, analogous to those discovered in Publication III, encoun-tered in the asymmetric geometry too? There are also unansweredissues related to the symmetric setup. For instance, the reason whyjust one of the modes, either the higher-order or the fundamentalmode, evolves into a long-range mode is still unclear. This naturallyraises the question: could two, or even more, long-range modes ex-



and partial polarization of stationary, purely evanescent light fieldsgenerated in total internal reflection at a lossless dielectric interface.Employing the spectral degree of coherence, we showed that forsuch fields the coherence length in air can be significantly shorterthan the free-space wavelength, especially for high refractive-indexcontrasts. The coherence length is typically smallest in the immedi-ate vicinity of the surface, but may get very large already within awavelength from it. We also adopted the 3D degree of polarizationto demonstrate that already two beams are sufficient to generatea true 3D evanescent field which cannot be described by the con-ventional polarization theory. Publication VI revealed that, in gen-eral, the coherence and polarization properties of electromagneticsurface fields at subwavelength scales may, even in the absence ofabsorption, differ notably from those a wavelength or more awayfrom the supporting interface.

Motivated by these results, in Publication VII we explored con-ditions under which stationary, fully 3D-unpolarized evanescentlight fields could be generated by manageable means at a losslessdielectric surface in a configuration involving multiple illuminationbeams. It was shown that by using two incident beams it is possibleto excite a nearly unpolarized 3D evanescent field, but to achievea strictly 3D-unpolarized state requires at least a three-beam setup.We also investigated the spectral electromagnetic coherence of suchfields and demonstrated that their degrees of coherence may varyconsiderably, exhibiting diverse subwavelength lattice-like patterns,depending on the applied modality. These findings suggest the fea-sibility to customize evanescent light fields with polarization char-acteristics identical to those of universal blackbody radiation, yetwith adjustable coherence properties.

Finally, in Publication VIII, by examining polarization modu-lation in double-pinhole photon interference, we formulated twogeneral complementarity relations for genuine vectorial quantum-light fields of arbitrary state. To this end, we derived the photoninterference law and a generalized visibility relation, which char-acterize any effect that spatial coherence at the openings may have


Conclusions

on the number and polarization variations of the photons in theobservation plane. In addition, we introduced an intensity distin-guishability and a polarization distinguishability to differentiate thelight in the pinholes. The complementarity relations, establishinglinks between these quantities, were identified to reflect two sepa-rate features of wave–particle duality of the photon. In particular,we demonstrated that the a priori which-path information of single-photon vector light does not couple to the intensity visibility, but tothe total visibility, which accounts also for polarization modulation.This discovery reveals an intrinsic aspect of photon wave–particleduality, not reported earlier to our knowledge. It was also shownthat for a general quantum-light field such complementarity is amanifestation of complete coherence, not of quantum-state purity.Photon-polarization modulation thereby entails novel, fundamentalphysical facets of quantum complementarity.

7.2 FUTURE PROSPECTS

Altogether, in this thesis, we have addressed only a limited numberof research topics within the realms of plasmonics, electromagneticcoherence, and quantum complementarity. There are naturally stilla rich diversity of physics to explore and important open questionsthat deserve further investigation in all of these areas.

The novel metal-slab mode solutions derived in this work con-cerned a symmetric environment. What about an asymmetric metal-slab or a multi-surface geometry? Can a rigorous analysis adaptedfor such configurations also reveal new mode solutions, or evenwhole mode-solution classes, with features similar as or drasticallydifferent from the ones met in the symmetric case? Are mode inter-changes, analogous to those discovered in Publication III, encoun-tered in the asymmetric geometry too? There are also unansweredissues related to the symmetric setup. For instance, the reason whyjust one of the modes, either the higher-order or the fundamentalmode, evolves into a long-range mode is still unclear. This naturallyraises the question: could two, or even more, long-range modes ex-



ist simultaneously for given media, frequency, and slab thickness?The feasibility of multiple long-range modes would undoubtedlybe of interest in thin-film plasmonics.

Exploring the coherence and polarization characteristics of gen-uine three-component SPP fields, and their classical, semi-classical,and quantum interactions with nanostructures, constitutes a broadresearch area of fundamental importance. In particular, extendingthe scheme of plasmon coherence engineering presented in Publi-cation IV from one-dimensional SPP propagation to planar dimen-sions would lead to new, exotic, and highly versatile SPP fields, ne-cessitating rigorous 3D electromagnetic coherence and polarizationtheory. Naturally, designing and constructing the required excita-tion light sources of customized spatio–spectral coherence presenta whole problem area on its own. Using nanoscatterers or scanningnear-field optical microscopy could be a promising step towardsprobing such novel electromagnetic surface fields.

Although quantum optics and quantum information have a longhistory, studies on the role of polarization in quantum coherenceand quantum interference phenomena appear ominously absent.Will the photon interference law established in Publication VIII pre-dict new discoveries? For instance, is there a quantum analog to thedegree of polarization being a measure of polarization modulationin beam self-interference, a consequence of the classical electromag-netic interference law? In multi-pinhole photon interference involv-ing polarization modulation entirely new complementary featureswould in all likelihood appear. What these entities are, how they arequantified via complementarity relations, and what novel physicsthey reveal, remain a challenge. Overall, quantum complementaritywith polarization, and the emerging connections between classicaland quantum light, constitute rich and promising research areas.


A Classical theory of electro-magnetic coherence

This Appendix provides a brief overview of the basic concepts tocharacterize second-order electromagnetic coherence in classical op-tical fields. The formalism, which is used in Chaps. 4 and 5 to as-sess electromagnetic coherence of optical surface fields, covers non-stationary and stationary light, both in the space–time and space–frequency domains. We note that the terminology in classical coher-ence theory differs by a factor of two from that in quantum theoryof optical coherence; the correlation order in classical theory fol-lows the power of field amplitude, while in the quantum context itfollows the power of intensity (photons).

The second-order statistical properties of a classical electromag-netic field are specified by the electric, magnetic, and two mixed-field coherence matrices [54], which constitute the foundation forthe modern treatment of partial coherence (and polarization). Al-though a full description of the electromagnetic coherence of lightrequires that all four matrices are taken into account, it is often suffi-cient to consider only electric-field correlations, as optical processesare primarily manifested via the electric field [177]. Consequently,henceforth we focus solely on the electric field, which we represent,at a space–time point (r, t), as a complex analytic signal [54],

E(r, t) =∫ ∞

0E(r, ω)e−iωtdω, (A.1)

where the column vector E(r, ω) is the frequency-domain Fouriertransform of the actual real-valued field. The real and imaginaryparts of the complex analytic signal representation form a Hilberttransform pair, which is equivalent to the fact that E(r, t) is an an-alytic (as the name implies) and regular function in the lower halfof the complex plane with respect to t [54]. Moreover, it is a natural



ist simultaneously for given media, frequency, and slab thickness?The feasibility of multiple long-range modes would undoubtedlybe of interest in thin-film plasmonics.

Exploring the coherence and polarization characteristics of gen-uine three-component SPP fields, and their classical, semi-classical,and quantum interactions with nanostructures, constitutes a broadresearch area of fundamental importance. In particular, extendingthe scheme of plasmon coherence engineering presented in Publi-cation IV from one-dimensional SPP propagation to planar dimen-sions would lead to new, exotic, and highly versatile SPP fields, ne-cessitating rigorous 3D electromagnetic coherence and polarizationtheory. Naturally, designing and constructing the required excita-tion light sources of customized spatio–spectral coherence presenta whole problem area on its own. Using nanoscatterers or scanningnear-field optical microscopy could be a promising step towardsprobing such novel electromagnetic surface fields.

Although quantum optics and quantum information have a longhistory, studies on the role of polarization in quantum coherenceand quantum interference phenomena appear ominously absent.Will the photon interference law established in Publication VIII pre-dict new discoveries? For instance, is there a quantum analog to thedegree of polarization being a measure of polarization modulationin beam self-interference, a consequence of the classical electromag-netic interference law? In multi-pinhole photon interference involv-ing polarization modulation entirely new complementary featureswould in all likelihood appear. What these entities are, how they arequantified via complementarity relations, and what novel physicsthey reveal, remain a challenge. Overall, quantum complementaritywith polarization, and the emerging connections between classicaland quantum light, constitute rich and promising research areas.


A Classical theory of electro-magnetic coherence

This Appendix provides a brief overview of the basic concepts tocharacterize second-order electromagnetic coherence in classical op-tical fields. The formalism, which is used in Chaps. 4 and 5 to as-sess electromagnetic coherence of optical surface fields, covers non-stationary and stationary light, both in the space–time and space–frequency domains. We note that the terminology in classical coher-ence theory differs by a factor of two from that in quantum theoryof optical coherence; the correlation order in classical theory fol-lows the power of field amplitude, while in the quantum context itfollows the power of intensity (photons).

The second-order statistical properties of a classical electromag-netic field are specified by the electric, magnetic, and two mixed-field coherence matrices [54], which constitute the foundation forthe modern treatment of partial coherence (and polarization). Al-though a full description of the electromagnetic coherence of lightrequires that all four matrices are taken into account, it is often suffi-cient to consider only electric-field correlations, as optical processesare primarily manifested via the electric field [177]. Consequently,henceforth we focus solely on the electric field, which we represent,at a space–time point (r, t), as a complex analytic signal [54],

E(r, t) =∫ ∞

0E(r, ω)e−iωtdω, (A.1)

where the column vector E(r, ω) is the frequency-domain Fouriertransform of the actual real-valued field. The real and imaginaryparts of the complex analytic signal representation form a Hilberttransform pair, which is equivalent to the fact that E(r, t) is an an-alytic (as the name implies) and regular function in the lower halfof the complex plane with respect to t [54]. Moreover, it is a natural



generalization of the complex representation that is frequently usedin the context of real monochromatic fields and plays an importantrole in quantum optics.

A.1 NONSTATIONARY FIELDS

All information about the second-order coherence properties of anonstationary classical light field is in the space–time domain in-cluded in the electric coherence matrix [107]

Γ(r1, t1; r2, t2) = ⟨E∗(r1, t1)ET(r2, t2)⟩, (A.2)

where the asterix and superscript T stand for complex conjugationand matrix transpose, respectively, and the angle brackets denoteensemble averaging. Physically, the elements of Γ(r1, t1; r2, t2) en-compass the correlations among the orthogonal field components attwo space–time points (r1, t1) and (r2, t2). The polarization featuresof the nonstationary light are encoded in the polarization matrix

J(r, t) = Γ(r, t; r, t), (A.3)

where the diagonal elements give the average intensity of each or-thogonal component, and where the off-diagonal elements charac-terize the correlations prevailing between the orthogonal field com-ponents at a single space–time point.

In the space–frequency domain, the spectral electric coherencematrix of a nonstationary field is expressed directly via the Fouriertransform E(r, ω) of E(r, t) as [107]

W(r1, ω1; r2, ω2) = ⟨E∗(r1, ω1)ET(r2, ω2)⟩, (A.4)

whose elements describe the spatial correlations of the componentsat two frequencies ω1 and ω2. The single-point, single-frequencycorrelations are characterized by the spectral polarization matrix

Φ(r, ω) = W(r, ω; r, ω), (A.5)


Classical theory ofelectromagnetic coherence

with the diagonal elements yielding the intensity of each orthogo-nal field component, and with the off-diagonal elements specifyingthe correlations among the orthogonal components at (r, ω).

The partial coherence of a vector-light field is conveniently quan-tified in the space–time and space–frequency domains by the elec-tromagnetic temporal [112] and spectral [113] degrees of coherence.The physical basis for these measures, which amount to the totalityof correlations existing between all the orthogonal field componentsfor a pair of points, are the intensity visibility and the polarizationmodulation in Young’s interference experiment [158]. For nonsta-tionary light, they are defined, respectively, through

γ(r1, t1; r2, t2) =∥Γ(r1, t1; r2, t2)∥F√trJ(r1, t1)trJ(r2, t2)

, (A.6)

µ(r1, ω1; r2, ω2) =∥W(r1, ω1; r2, ω2)∥F√trΦ(r1, ω1)trΦ(r2, ω2)

, (A.7)

where ∥ · ∥F is the Frobenius matrix norm [238] and tr stands for thetrace. Both quantities are bounded between zero and unity, with thelatter (former) limit representing complete coherence (incoherence)and taking place if, and only if, all orthogonal field components be-tween (r1, t1) and (r2, t2) in the space–time domain and (r1, ω1) and(r2, ω2) in the space–frequency domain are fully correlated (uncor-related). In particular, γ(r1, t1; r2, t2) = 1 and µ(r1, ω1; r2, ω2) = 1in a domain are equivalent with the factorization of Γ(r1, t1; r2, t2)

and W(r1, ω1; r2, ω2), respectively, which is considered to be a fun-damental property of a completely coherent field [136].

We stress that other measures for characterizing partial coher-ence in vectorial light fields have been put forward [118–123], withdifferent mathematical properties and physical implications [124].

A.2 STATIONARY FIELDS

The second-order statistical properties of a stationary vector-lightfield, i.e., a field for which the character of the random fluctuationsdoes not change with time, but depend only on the time separation



generalization of the complex representation that is frequently usedin the context of real monochromatic fields and plays an importantrole in quantum optics.

A.1 NONSTATIONARY FIELDS

All information about the second-order coherence properties of anonstationary classical light field is in the space–time domain in-cluded in the electric coherence matrix [107]

Γ(r1, t1; r2, t2) = ⟨E∗(r1, t1)ET(r2, t2)⟩, (A.2)

where the asterix and superscript T stand for complex conjugationand matrix transpose, respectively, and the angle brackets denoteensemble averaging. Physically, the elements of Γ(r1, t1; r2, t2) en-compass the correlations among the orthogonal field components attwo space–time points (r1, t1) and (r2, t2). The polarization featuresof the nonstationary light are encoded in the polarization matrix

J(r, t) = Γ(r, t; r, t), (A.3)

where the diagonal elements give the average intensity of each or-thogonal component, and where the off-diagonal elements charac-terize the correlations prevailing between the orthogonal field com-ponents at a single space–time point.

In the space–frequency domain, the spectral electric coherencematrix of a nonstationary field is expressed directly via the Fouriertransform E(r, ω) of E(r, t) as [107]

W(r1, ω1; r2, ω2) = ⟨E∗(r1, ω1)ET(r2, ω2)⟩, (A.4)

whose elements describe the spatial correlations of the componentsat two frequencies ω1 and ω2. The single-point, single-frequencycorrelations are characterized by the spectral polarization matrix

Φ(r, ω) = W(r, ω; r, ω), (A.5)



with the diagonal elements yielding the intensity of each orthogo-nal field component, and with the off-diagonal elements specifyingthe correlations among the orthogonal components at (r, ω).

The partial coherence of a vector-light field is conveniently quan-tified in the space–time and space–frequency domains by the elec-tromagnetic temporal [112] and spectral [113] degrees of coherence.The physical basis for these measures, which amount to the totalityof correlations existing between all the orthogonal field componentsfor a pair of points, are the intensity visibility and the polarizationmodulation in Young’s interference experiment [158]. For nonsta-tionary light, they are defined, respectively, through

γ(r1, t1; r2, t2) =∥Γ(r1, t1; r2, t2)∥F√trJ(r1, t1)trJ(r2, t2)

, (A.6)

µ(r1, ω1; r2, ω2) =∥W(r1, ω1; r2, ω2)∥F√trΦ(r1, ω1)trΦ(r2, ω2)

, (A.7)

where ∥ · ∥F is the Frobenius matrix norm [238] and tr stands for thetrace. Both quantities are bounded between zero and unity, with thelatter (former) limit representing complete coherence (incoherence)and taking place if, and only if, all orthogonal field components be-tween (r1, t1) and (r2, t2) in the space–time domain and (r1, ω1) and(r2, ω2) in the space–frequency domain are fully correlated (uncor-related). In particular, γ(r1, t1; r2, t2) = 1 and µ(r1, ω1; r2, ω2) = 1in a domain are equivalent with the factorization of Γ(r1, t1; r2, t2)

and W(r1, ω1; r2, ω2), respectively, which is considered to be a fun-damental property of a completely coherent field [136].

We stress that other measures for characterizing partial coher-ence in vectorial light fields have been put forward [118–123], withdifferent mathematical properties and physical implications [124].

A.2 STATIONARY FIELDS

The second-order statistical properties of a stationary vector-lightfield, i.e., a field for which the character of the random fluctuationsdoes not change with time, but depend only on the time separation



τ = t2 − t1, are in the space–time domain completely specified bythe electric coherence matrix [54]

Γ(r1, r2, τ) = ⟨E∗(r1, t)ET(r2, t + τ)⟩, (A.8)

with the angle brackets denoting ensemble or time averaging. Thespace–time polarization matrix of the stationary light is given by

J(r) = Γ(r, r, 0), (A.9)

which, contrary to the nonstationary case, is time independent.As a stationary field is not square integrable over time, whereby

it does not have a Fourier transform, the transition from space–timedomain to space–frequency domain is not as straightforward as fornonstationary light. Nevertheless, in most relevant physical situa-tions the elements of Γ(r1, r2, τ) can be taken as square integrable,whereupon the spectral coherence matrix is defined as the Fouriertransform of Γ(r1, r2, τ) [54], viz.,

W(r1, r2, ω) =1

2π

∫ ∞

−∞Γ(r1, r2, τ)eiωτdτ. (A.10)

A distinguishing feature of the space–frequency domain represen-tation of a stationary field is that different frequency componentsare fully uncorrelated; spectral correlations induce nonstationarylight. Equation (A.10) and the fact that the frequencies are uncor-related constitute the generalized Wiener–Khintchine theorem [54].Moreover, albeit W(r1, r2, ω) is not directly defined as a correlationmatrix of the two fields, it has an important and useful property,namely, it can be expressed as a correlation matrix over an ensem-ble E(r, ω)e−iωt of monochromatic field realizations [102], i.e.,

W(r1, r2, ω) = ⟨E∗(r1, ω)ET(r2, ω)⟩. (A.11)

We point out that here E(r, ω) is not the Fourier transform of E(r, t).The representation in Eq. (A.11) is a fundamental result that enablesthe analysis of stationary light frequency by frequency.



The spectral polarization properties of the stationary light fieldare encoded in the spectral polarization matrix

Φ(r, ω) = W(r, r, ω). (A.12)

While the space–time domain polarization matrix J(r) is indepen-dent on time, the space–frequency polarization matrix Φ(r, ω) de-pends on the frequency.

Similarly to nonstationary light, it is favorable to assess the par-tial coherence of a stationary vector-light field by employing the cor-responding temporal and spectral degrees of electromagnetic coher-ence. For stationary light, they are given, respectively, by [112, 113]

γ(r1, r2, τ) =∥Γ(r1, r2, τ)∥F√

trJ(r1)trJ(r2), (A.13)

µ(r1, r2, ω) =∥W(r1, r2, ω)∥F√

trΦ(r1, ω)trΦ(r2, ω). (A.14)

As for nonstationary light, the quantities γ(r1, r2, τ) and µ(r1, r2, ω)

are nonnegative, invariant under unitary transformations, and at-tain their maximum value (unity) when the fields at points r1 andr2, and at time separation τ or frequency ω, are completely coher-ent. If these conditions hold in a domain, the matrices Γ(r1, r2, τ)

and W(r1, r2, ω) factorize [59, 113, 114].

A.3 DEGREE OF POLARIZATION

The degree of polarization is a measure that reflects the amount ofcorrelations prevailing between the orthogonal electric-field compo-nents of the fluctuating light field at a single point. The traditionalformulation of this quantity has been restricted to two-component(2D) light fields, such as beams and far fields, whose electric-fieldvector fluctuates approximately in a plane transverse to the propa-gation direction [54, 55]. Nevertheless, as highlighted by novel ad-vancements in nano-optics [2] and high-numerical-aperture imag-ing systems [109–111], there are situations in which this approxima-tion is no longer valid. Therefore, the concept of the degree of polar-ization must be extended to also include genuine three-component



τ = t2 − t1, are in the space–time domain completely specified bythe electric coherence matrix [54]

Γ(r1, r2, τ) = ⟨E∗(r1, t)ET(r2, t + τ)⟩, (A.8)

with the angle brackets denoting ensemble or time averaging. Thespace–time polarization matrix of the stationary light is given by

J(r) = Γ(r, r, 0), (A.9)

which, contrary to the nonstationary case, is time independent.As a stationary field is not square integrable over time, whereby

it does not have a Fourier transform, the transition from space–timedomain to space–frequency domain is not as straightforward as fornonstationary light. Nevertheless, in most relevant physical situa-tions the elements of Γ(r1, r2, τ) can be taken as square integrable,whereupon the spectral coherence matrix is defined as the Fouriertransform of Γ(r1, r2, τ) [54], viz.,

W(r1, r2, ω) =1

2π

∫ ∞

−∞Γ(r1, r2, τ)eiωτdτ. (A.10)

A distinguishing feature of the space–frequency domain represen-tation of a stationary field is that different frequency componentsare fully uncorrelated; spectral correlations induce nonstationarylight. Equation (A.10) and the fact that the frequencies are uncor-related constitute the generalized Wiener–Khintchine theorem [54].Moreover, albeit W(r1, r2, ω) is not directly defined as a correlationmatrix of the two fields, it has an important and useful property,namely, it can be expressed as a correlation matrix over an ensem-ble E(r, ω)e−iωt of monochromatic field realizations [102], i.e.,

W(r1, r2, ω) = ⟨E∗(r1, ω)ET(r2, ω)⟩. (A.11)

We point out that here E(r, ω) is not the Fourier transform of E(r, t).The representation in Eq. (A.11) is a fundamental result that enablesthe analysis of stationary light frequency by frequency.



The spectral polarization properties of the stationary light fieldare encoded in the spectral polarization matrix

Φ(r, ω) = W(r, r, ω). (A.12)

While the space–time domain polarization matrix J(r) is indepen-dent on time, the space–frequency polarization matrix Φ(r, ω) de-pends on the frequency.

Similarly to nonstationary light, it is favorable to assess the par-tial coherence of a stationary vector-light field by employing the cor-responding temporal and spectral degrees of electromagnetic coher-ence. For stationary light, they are given, respectively, by [112, 113]

γ(r1, r2, τ) =∥Γ(r1, r2, τ)∥F√

trJ(r1)trJ(r2), (A.13)

µ(r1, r2, ω) =∥W(r1, r2, ω)∥F√

trΦ(r1, ω)trΦ(r2, ω). (A.14)

As for nonstationary light, the quantities γ(r1, r2, τ) and µ(r1, r2, ω)

are nonnegative, invariant under unitary transformations, and at-tain their maximum value (unity) when the fields at points r1 andr2, and at time separation τ or frequency ω, are completely coher-ent. If these conditions hold in a domain, the matrices Γ(r1, r2, τ)

and W(r1, r2, ω) factorize [59, 113, 114].

A.3 DEGREE OF POLARIZATION

The degree of polarization is a measure that reflects the amount ofcorrelations prevailing between the orthogonal electric-field compo-nents of the fluctuating light field at a single point. The traditionalformulation of this quantity has been restricted to two-component(2D) light fields, such as beams and far fields, whose electric-fieldvector fluctuates approximately in a plane transverse to the propa-gation direction [54, 55]. Nevertheless, as highlighted by novel ad-vancements in nano-optics [2] and high-numerical-aperture imag-ing systems [109–111], there are situations in which this approxima-tion is no longer valid. Therefore, the concept of the degree of polar-ization must be extended to also include genuine three-component



(3D) light fields with wave fronts of arbitrary form. In the followingwe limit our discussion to stationary light in the space–frequencydomain, but emphasize that analogous considerations apply in thespace–time domain and for nonstationary light.

For a 2D-light field, the spectral polarization matrix Φ(r, ω) inEq. (A.12) can be uniquely expressed as a sum of two matrices, oneof which corresponds to fully unpolarized light, being proportionalto the 2× 2 identity matrix, and another one which represents com-pletely polarized light. In this case, the 2D degree of polarization,P2D(r, ω), is defined as the ratio of the spectral density of the polar-ized part to that of the total field, which can be written as [54, 55]

P2D(r, ω) = 2

√trΦ2(r, ω)

tr2Φ(r, ω)− 1

2. (A.15)

The 2D degree of polarization satisfies 0 ≤ P2D(r, ω) ≤ 1, with thelower and upper limits corresponding to totally unpolarized andcompletely polarized light, respectively, while the intermediate val-ues stand for partial polarization. In a coordinate frame where theintensities of the two orthogonal components are equal P2D(r, ω) co-incides with the absolute value of the correlation coefficient amongthe components. Moreover, P2D(r, ω) can be given an interferomet-ric interpretation as an ability of a light beam to exhibit polarizationmodulation when it is allowed to interfere with itself [159].

Whereas for 2D light the polarization matrix can be expressedunambiguously as a sum of two matrices, one representing unpo-larized light and the other fully polarized light, for 3D-light fieldssuch a decomposition does generally not exist [55,57]. Accordingly,another approach must be taken in order to define the degree ofpolarization for genuine 3D-light fields. The generalization of thedegree of polarization for 3D light can be obtained by consider-ing the expansion of Φ(r, ω) in terms of 3 × 3 Gell-Mann matricesand generalized Stokes parameters [125], although other methodsfor assessing partial polarization of 3D-light fields have been pro-posed [126–134]. In this approach, the resulting expression for the



3D degree of polarization, P3D(r, ω), is

P3D(r, ω) =32

√trΦ2(r, ω)

tr2Φ(r, ω)− 1

3. (A.16)

As its 2D counterpart, the 3D degree of polarization is boundedbetween zero and unity. The lower limit P3D(r, ω) = 0 represents3D-unpolarized light, encountered when the spectral densities ofthe components are the same and no correlation exists between anyof them. The maximum value, P3D(r, ω) = 1, corresponds to fullypolarized light and takes place if, and only if, all the componentsare mutually fully correlated. Any other value of P3D(r, ω) standsfor a partially polarized light field.

The physical meaning of P3D(r, ω) becomes apparent by writingit in a coordinate frame oriented in such a way that the diagonalelements of Φ(r, ω) are equal (an orientation which can always befound) [125]. In such a coordinate system the 3D degree of polariza-tion turns into a direct measure for the average correlations betweenthe orthogonal electric-field components, analogously to P2D(r, ω).In addition, if the 3 × 3 polarization matrix can be represented asa sum of two matrices corresponding to a fully unpolarized and acompletely polarized part, then P3D(r, ω) is the ratio of the intensityof the polarized part to the total field intensity [215].

Finally, we note that the 2D and 3D degrees of polarization areconnected to the spectral degree of coherence in Eq. (A.14) via

µ(r, r, ω) =12

√1 + P2D(r, ω), (A.17)

µ(r, r, ω) =13

√1 + 2P3D(r, ω), (A.18)

both stating that, for vectorial light, the equal-point degree of coher-ence is unity if, and only if, the light field is completely polarized.



(3D) light fields with wave fronts of arbitrary form. In the followingwe limit our discussion to stationary light in the space–frequencydomain, but emphasize that analogous considerations apply in thespace–time domain and for nonstationary light.

For a 2D-light field, the spectral polarization matrix Φ(r, ω) inEq. (A.12) can be uniquely expressed as a sum of two matrices, oneof which corresponds to fully unpolarized light, being proportionalto the 2× 2 identity matrix, and another one which represents com-pletely polarized light. In this case, the 2D degree of polarization,P2D(r, ω), is defined as the ratio of the spectral density of the polar-ized part to that of the total field, which can be written as [54, 55]

P2D(r, ω) = 2

√trΦ2(r, ω)

tr2Φ(r, ω)− 1

2. (A.15)

The 2D degree of polarization satisfies 0 ≤ P2D(r, ω) ≤ 1, with thelower and upper limits corresponding to totally unpolarized andcompletely polarized light, respectively, while the intermediate val-ues stand for partial polarization. In a coordinate frame where theintensities of the two orthogonal components are equal P2D(r, ω) co-incides with the absolute value of the correlation coefficient amongthe components. Moreover, P2D(r, ω) can be given an interferomet-ric interpretation as an ability of a light beam to exhibit polarizationmodulation when it is allowed to interfere with itself [159].

Whereas for 2D light the polarization matrix can be expressedunambiguously as a sum of two matrices, one representing unpo-larized light and the other fully polarized light, for 3D-light fieldssuch a decomposition does generally not exist [55,57]. Accordingly,another approach must be taken in order to define the degree ofpolarization for genuine 3D-light fields. The generalization of thedegree of polarization for 3D light can be obtained by consider-ing the expansion of Φ(r, ω) in terms of 3 × 3 Gell-Mann matricesand generalized Stokes parameters [125], although other methodsfor assessing partial polarization of 3D-light fields have been pro-posed [126–134]. In this approach, the resulting expression for the



3D degree of polarization, P3D(r, ω), is

P3D(r, ω) =32

√trΦ2(r, ω)

tr2Φ(r, ω)− 1

3. (A.16)

As its 2D counterpart, the 3D degree of polarization is boundedbetween zero and unity. The lower limit P3D(r, ω) = 0 represents3D-unpolarized light, encountered when the spectral densities ofthe components are the same and no correlation exists between anyof them. The maximum value, P3D(r, ω) = 1, corresponds to fullypolarized light and takes place if, and only if, all the componentsare mutually fully correlated. Any other value of P3D(r, ω) standsfor a partially polarized light field.

The physical meaning of P3D(r, ω) becomes apparent by writingit in a coordinate frame oriented in such a way that the diagonalelements of Φ(r, ω) are equal (an orientation which can always befound) [125]. In such a coordinate system the 3D degree of polariza-tion turns into a direct measure for the average correlations betweenthe orthogonal electric-field components, analogously to P2D(r, ω).In addition, if the 3 × 3 polarization matrix can be represented asa sum of two matrices corresponding to a fully unpolarized and acompletely polarized part, then P3D(r, ω) is the ratio of the intensityof the polarized part to the total field intensity [215].

Finally, we note that the 2D and 3D degrees of polarization areconnected to the spectral degree of coherence in Eq. (A.14) via

µ(r, r, ω) =12

√1 + P2D(r, ω), (A.17)

µ(r, r, ω) =13

√1 + 2P3D(r, ω), (A.18)

both stating that, for vectorial light, the equal-point degree of coher-ence is unity if, and only if, the light field is completely polarized.




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Paper I

A. Norrman, T. Setala, and A. T. Friberg“Exact surface-plasmon polariton solutions

at a lossy interface”Optics Letters

38, 1119–1121 (2013).

c⃝ 2013 The Optical Society. Reprinted with permission.


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ISBN 978-952-61-2357-8ISSN 1798-5668


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ANDREAS NORRMAN

ELECTROMAGNETIC COHERENCE OF OPTICAL SURFACE AND QUANTUM LIGHT FIELDS


This thesis considers surface-plasmon

polaritons (SPPs), partially coherent optical surface fields, and complementarity in

vector-light photon interference. Novel SPPs, including a long-range higher-order metal-

slab mode, are predicted. Generation, partial polarization, and electromagnetic coherence

of polychromatic SPPs and evanescent light fields are also examined. Polarization

modulation of vectorial quantum light is explored to uncover a new intrinsic aspect of

photon wave–particle duality.

ANDREAS NORRMAN

Documents

Dissertations in Forestry and Natural Sciences · covers a new intrinsic aspect of wave–particle duality of the photon, having no correspondence within the scalar quantum treatment