Nonlinear8708/... · 2008. 9. 25. · nonlinear optics. Therefore, whenev er p ossible, the notations and con v en tions used in this thesis are in tended to follo w the standard

VKThe Nonlinear Optics

of

Magneto-Optic Media

Fredrik Jonsson

s1

s2

s3; w3

w1

w2

The Royal Institute of Technology

Department of Physics

Stockholm 2000

The Nonlinear Optics

of

Magneto-Optic Media

Fredrik Jonsson

VDoctoral Thesis

The Royal Institute of Technology

Department of Physics -Optics

Stockholm 2000

Akademisk avhandling som med tillst�and av Kungliga Tekniska H�ogskolan framl�agges tilloffentlig granskning f�or avl�aggande av teknologie doktorsexamen fredagen den 26 maj2000 kl 10.00 i Sal D3, Lindstedtsv�agen 5, Kungl Tekniska H�ogskolan, Stockholm.

ISBN 91-7170-575-9TRITA-FYS 2208ISSN 0280-316X

Copyright c Fredrik Jonsson 2000Typeset in 9/12/16 pt Computer Modern using TEX and METAPOST

Printed by KTH H�ogskoletryckeriet, Stockholm 2000

Fredrik Jonsson, The Nonlinear Optics of Magneto-Optic Media

Department of Physics -Optics, The Royal Institute of Technology,SE-100 44, Stockholm, Sweden

Abstract

Magneto-optics is the discipline in physics concerned with the interaction of optical waveswith matter under in uence of magnetic �elds. The introductory part of the thesis devel-

ops a consistent treatise of the fundamentals of magneto-optical light-matter interactions.The original research of this thesis presents theoretical studies for two groups of nonlin-ear optical phenomena, within the coexistence of linear and nonlinear magneto-optical

interactions.The �rst group of studies in nonlinear optics focus on the third and fourth order

optical and magneto-optical interactions governing optical Kerr-e�ect and photoinduced

Faraday rotation. A Fabry-P�erot interferometer �lled with a magneto-optic dielectric,possessing optical Kerr-e�ect, is studied in the Faraday con�guration under the action of astatic magnetic �eld, in whose presence linear and photoinduced Faraday rotation appear.

It is shown that by exploiting the nonreciprocity of the magneto-optical interactions ofthe medium, the cavity shows a multistable behaviour at constant input light intensity,opening for a polarization state controlled switching device, being tunable by means of

the static magnetic �eld.Furthermore, unidirectional wave propagation in nonlinear magneto-optic media being

anisotropic in the nonlinear regime is studied. The evolution of the polarization state of

the beam is presented in terms of closed trajectories on the Poincar�e sphere, in a redu-ced Stokes vector description, and solutions for the corresponding eigenpolarizations arepresented.

The second group of studies in nonlinear optics focus on the second and third or-der optical and magneto-optical interactions governing the process of parametric genera-tion. Optical parametric interactions of unidirectionally propagating �elds are studied in

magneto-optic media, in the Faraday con�guration. In the analysis, �rst and second orderoptical interactions and second and third order magneto-optical interactions are taken intoaccount in the constitutive relations and subsequent wave propagation. It is shown that

phase-matching can be obtained for circularly polarized �elds by exploiting the arti�ciallyinduced gyrotropy of the medium.

The theory of a magneto-optical parametric oscillator is presented, in a large-signalregime, for a singly resonant con�guration. The pump threshold intensity and intracavity

signal-to-pump ratios for circularly polarized waves are presented as functions of phase-mismatch, and it is shown that tunable phase-matching can be obtained by means ofvarying the externally applied static magnetic �eld.

The studies presented in the thesis open for applications within the �elds of opticalswitching and logics, and provide novel schemes for phase-matching in optical parametricampli�cation and oscillation.

Keywords: Magneto-optical devices, optical bistability, optical parametric generation

ISBN 91-7170-575-9 � TRITA-FYS 2208 � ISSN 0280-316X

iii

iv

Preface

While nonlinear optical phenomena in a vast majority are studied within the frameworkof the electric-dipole approximation, that is to say, under the approximation that only the

electric dipole moments of the molecules participate in the interaction with light, one canin many aspects gain from an extension to magnetic dipolar interactions.

The electric dipole approximation is a perfectly satisfactory approximation for most

practical cases, since interactions such as magnetic dipolar or electric quadrupolar inter-actions almost always are very weak in comparison. However, the electric dipole approx-imation fails when it comes to describing the in uence of a magnetic �eld on the optical

properties of matter, and even though the magnetic dipolar e�ects usually are small, com-pared to the electric dipolar ones, many applications can be found whenever control ofthe phase, polarization state, or propagation of light is a key issue.

It is the aim with this thesis to explain some of the concepts of the magneto-opticale�ects and the underlying basics, and in particular to show how the magnetic dipolar

interactions can be applied in fusion with nonlinear optics, to provide new componentsfor optical switching and logics, optical parametric ampli�cation, and optical parametricoscillation, all of them featuring the possibility of being tunable by means of an externally

applied static magnetic �eld.

In a time when a majority of the research within electromagnetism in some way or an-other relate to applications within the �eld of information technology, the studies presented

in this thesis are no exception to the rule. Of the �ve publications that are reproduced atthe end of the thesis, each of them presents theory for components which control { or arecontrolled by { the polarization state of light, with potential applications within optical

switching for telecommunication.

Outline of the thesis

While most applications of magneto-optics are developed for light-matter interaction in a

linear optical regime, in which case the induced electric polarization density of the mediumis linearly related to the electric �eld of the light, the studies presented in this thesisare rather concerned with the applications of magneto-optical phenomena in nonlinear

optics. Therefore, whenever possible, the notations and conventions used in this thesis areintended to follow the standard ones as used in nonlinear optics, in the spirit of Butcherand Cotter [1]. In order to avoid confusion in the notation of higher-order susceptibilities

involving magneto-optical interactions, a style resembling that of Kielich and Zawodny [2]has been adopted. However, for the sake of clarity and self-consistency, the notations andsymbols used in this thesis are listed in Appendix A.

v

vi Preface

The thesis is split into two parts; the �rst one serves as an introduction to the funda-

mentals of magneto-optical light-matter interaction, providing the link to elementa foundin textbooks on classical mechanics and quantum mechanics.

The second part consists of �ve publications covering the theoretical modelling of

nonlinear optical phenomena such as optical bistability, optical parametric generationand ampli�cation, and optical parametric oscillation, in magneto-optic media.

The structure of the �rst part of the thesis is as follows:

� In Chapter 1, a brief introduction to magneto-optics is presented, as well as a shorthistorical review covering di�erent areas of magneto-optics.

� In Chapter 2, an all-classical analogy to the one-electron oscillator is introduced,phenomenologically explaining the basic concepts of linear magneto-optics, such asthe Faraday e�ect and Cotton-Mouton e�ect, from the Lorentz forces acting on the

system when set into motion by a time-harmonic electromagnetic wave.

� In Chapter 3, magneto-optical light-matter interactions are discussed in parallel to

the all-optical interactions. Starting from a perturbation analysis of the densityoperator of an ensemble of molecules in the presence of magnetic �elds, the originsof linear and nonlinear magneto-optical e�ects are presented in terms of the sus-

ceptibility formalism of nonlinear optics. Furthermore, in this chapter the spatialtransformation properties and symmetry considerations of magneto-optics are intro-duced, and similarities and di�erences to the classical susceptibilities of all-optical

nonlinear optics are discussed.

� In Chapter 4, the constitutive relations are inserted into Maxwell's equations, and the

characteristics of nonlinear wave propagation in magneto-optic media are presented.The representation of the polarization state of light in terms of Stokes parametersis reviewed, and the concept of the Poincar�e sphere is introduced.

� Finally, in Chapter 5, the original research work is discussed.

In Appendix C, algorithms for analysis of symmetry properties of arbitrary-rank tensors

are enclosed. In the writing of this thesis, I for a while considered whether to present thesealgorithms in pseudocode rather than in the language-speci�c form of MapleV1 code ashere used. However, it is my belief that the listed MapleV routines serve their purpose

just as well as any pseudocode, and they may easily be ported to other programminglanguages, either for symbolic or strictly numerical manipulations. An advantage withnon-pseudocode is also that anyone interested in trying out the routines may just feed them

into the computer for immediate use; examples of usage are presented in Section 3.3.4.

1Maple and MapleV are registered trademarks of Waterloo Maple Software.

Preface vii

Acknowledgements

The studies of nonlinear optical phenomena in magneto-optic media, presented in this

thesis, were performed during 1996{2000 at the Royal Institute of Technology, Departmentof Physics -Optics, Stockholm. In the progress of the present work, a lot of people havein some way or another encouraged and supported me. However, there are some of them

that I especially would like to mention:First of all { Prof. Christos Flytzanis, Ecole Polytechnique, France, for all inspiring

discussions and for being my mentor during these years. Without his continuous support,

guidance, and never ending enthusiasm for Physics, the work presented in this thesis wouldhave evolved at a considerably slower pace.

I thank Prof. Klaus Biedermann for providing me the hospitality of the Department ofPhysics -Optics, KTH, and for his encouragement and advice in writing this thesis, and

I would also like to express my gratitude to the G�oran Gustafsson foundation for payingmy salary.

I am indebted to Dr. G�oran Manneberg for all inspiring discussions I have had with

him, within linear as well as nonlinear optics. He should also be acknowledged for beingthe best physics teacher I ever had. I also want to express my gratitude to Dr. PeterUnsbo, for taking his time reading my manuscripts produced during these years, greatly

improving their quality.My special thanks to Prof. Nail Akhmediev, at the Optical Sciences Centre, Australian

National University, Canberra, and to Prof. Dietmar Fr�ohlich, at the University of Dort-

mund, Germany, for providing me insights in Lagrangian mechanics applied to nonlinearoptics. Prof. Akhmediev is, by the way, the one who in the �rst place introduced me tothe Stokes parameter space in nonlinear optics.

Dr. G�oran Lindblad, Department of Theoretical Physics, KTH, is acknowledged forproviding useful comments on the text in Chapter 3, regarding techniques for reductionof tensor elements.

Finally, all my love to �Asa, for always �nding me the light again, whenever it was faraway.

Stockholm, April 2000

Fredrik Jonsson

viii

List of publications

The thesis consists of an introductory treatise on the fundamentals of nonlinear magneto-optics, and of the original research work presented in the following publications:

I. Fredrik Jonsson and Christos Flytzanis, \Polarization state controlled multistability

of a nonlinear magneto-optic cavity", Phys. Rev. Lett. 82, 1426 (1999).

II. Fredrik Jonsson, \Polarization state evolution and eigenpolarizations of nonlinearmagneto-optics", submitted to J. Opt. Soc.Am.B.

III. Fredrik Jonsson and Christos Flytzanis, \Optical parametric generation and phase-matching in magneto-optic media", Opt. Lett. 24, 1514 (1999).

IV. Fredrik Jonsson and Christos Flytzanis, \Polarization state dependence of opticalparametric processes in arti�cially gyrotropic media", to appear in J.Opt. A: Pure

Appl. Opt. 2, No. 4 (2000)

V. Fredrik Jonsson and Christos Flytzanis, \Magneto-optical parametric oscillation",submitted to Opt. Lett.

Additional publications in the �eld of nonlinear magneto-optics, not being part of this

thesis:

I. M. Haddad, F. Jonsson, R. Frey, and C. Flytzanis. \Nonlinear Optical Gyrotropy",

accepted for publication in Nonlinear Optics.

II. M. Haddad, P. Leishing, F. Jonsson, J. Cibert, R. Frey, and C. Flytzanis. \Ro-

tation Faraday photo-induite dans les puits quantiques des semi-conducteurs semi-magn�etiques", Ann. Phys. Fr. 23, 189 (1998).

III. M. Haddad, F. Jonsson, R. Frey and C. Flytzanis, \Nonlinear Optical Gyrotropy",QELS'99 (Quantum Electronics and Laser Science Conference) Baltimore, USA (23-28 May 1999).

IV. M. Haddad, F. Jonsson, R. Frey and C. Flytzanis, \Nonlinear optical gyrotropy andnonreciprocity" NOMA (4th Mediterranean Workshop on Novel Optical Materials)Certraro, Italy (4-10 June 1999).

V. F. Jonsson, M. Haddad, R. Frey, and C. Flytzanis, \Nonlinear Magneto-opticalCavities", International Workshop on Nonlinear Magneto-Optics, Cardi�, UK (24-26 June, 1999).

ix

x List of publications

VI. F. Jonsson and C. Flytzanis, \Phase matching for optical parametric processes in

arti�cially gyrotropic media", Workshop on Applications of Nonlinear Optical Phe-

nomena and Related Industrial Perspectives (2nd Annual Meeting of the COST Ac-tion P2) Amal�, Italy (6-9 October 1999).

VII. F. Jonsson, R. Frey, and C. Flytzanis, \Polarization state sensitive nonlinear ef-fects in optical active and chiral molecular systems", ICONO'5, 5:th International

Conference on Organic Nonlinear Optics, Davos, Switzerland (12-16 March 2000).

VIII. F. Jonsson and C. Flytzanis, \Nonlinear magneto-optics applied to optical bistability

and parametric ampli�cation", to be presented at Northern Optics 2000 and EOSAM

2000 (Annual Meeting of the European Optical Society), Uppsala, Sweden (6-8 June2000).

IX. F. Jonsson and C. Flytzanis, \Optical parametric oscillation in magneto-optic me-dia", submitted to PELS-2000, Polarisation E�ects in Lasers, Spectroscopy and

Optoelectronics, Interdisciplinary International Conference, Southampton, UK (6-9September, 2000), satellite conference of CLEO/IQEC Europe 2000.

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

I Magneto-optics 1

1 Introduction 3

1.1 Basics of magneto-optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Brief history of magneto-optics . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Linear optical regime . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Nonlinear optical regime . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 The MiniDisc technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Classical mechanics of magneto-optics 7

2.1 Magneto-optics in the one-electron oscillator model . . . . . . . . . . . . . 7

2.1.1 The all-classical spring model . . . . . . . . . . . . . . . . . . . . . 8

2.1.2 Motion in a time-harmonic electromagnetic �eld . . . . . . . . . . 9

2.2 The polarization density of the medium . . . . . . . . . . . . . . . . . . . 11

2.2.1 The refractive index of the medium . . . . . . . . . . . . . . . . . . 12

2.2.2 The Faraday e�ect . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.3 The Cotton-Mouton e�ect . . . . . . . . . . . . . . . . . . . . . . . 13

3 Nonlinear magneto-optics 15

3.1 Magneto-optical light-matter interaction . . . . . . . . . . . . . . . . . . . 15

3.1.1 General outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.2 Time-evolution of the density operator in magneto-optics . . . . . 17

3.2 The susceptibility tensors of magneto-optics . . . . . . . . . . . . . . . . . 22

3.2.1 First order interaction . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.2 Second order interaction . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.3 Third order interaction . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.4 Fourth order interaction . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.5 Higher order interactions . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Symmetries of magneto-optical susceptibilities . . . . . . . . . . . . . . . . 29

3.3.1 Spatial transformation properties . . . . . . . . . . . . . . . . . . . 293.3.2 Susceptibilities in rotated reference frames . . . . . . . . . . . . . . 31

3.3.3 Neumann's principle . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.4 Isotropic media revisited . . . . . . . . . . . . . . . . . . . . . . . . 34

xi

xii Contents

3.3.5 Anisotropic media . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Wave propagation in magneto-optic media 39

4.1 On the separation of electric and magnetic polarization . . . . . . . . . . 394.2 The nonlinear wave equation in magneto-optic media . . . . . . . . . . . . 41

4.2.1 Quasimonochromatic light . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.2 Monochromatic light . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2.3 Monochromatic light in isotropic magneto-optic media . . . . . . . 46

4.3 Representation of the electromagnetic �eld . . . . . . . . . . . . . . . . . . 47

4.3.1 Circularly polarized basis vectors . . . . . . . . . . . . . . . . . . . 474.3.2 Stokes parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 Discussion of original research work 53

5.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.5 Paper V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

A List of symbols 59

B Tables of magneto-optical susceptibilities 63

C Routines for analysis of arbitrary rank tensors in rotated frames 73

Bibliography 79

II Papers I{V 83

Part I

Magneto-optics

1

2

Chapter 1

Introduction

\Qu. 26. Have not the Rays of Light several sides, endued with several

original Properties?"

{ ISAAC NEWTON, Opticks (1730)

1.1 Basics of magneto-optics

Magneto-optics is the discipline in physics concerned with the interaction of optical waveswith matter under in uence of magnetic �elds. Being an old science, with roots in the 19:thcentury, magneto-optics has been extensively studied within the linear optical regime, in

which commersially available devices such as optical isolators, magneto-optical switches,and magneto-optical information storage devices are examples of applications of today.Nevertheless, in the nonlinear optical regime it still is an open �eld of research, with many

applications still to be explored.

Within the �eld of magneto-optics, two optical phenomena are of general importance:the Faraday e�ect, which denotes the rotation of the polarization ellipse of the light as is

propagates collinearly with an externally applied static magnetic �eld, and the Cotton-Mouton e�ect, in which the static magnetic �eld is applied orthogonally to the directionof propagation of the light, causing a linear birefringence of the medium.

The Faraday e�ect can be seen as a lifting of the degeneracy of left and right circularlypolarized components of the light, traceable to the Zeeman splitting of magnetic sublevelsof molecules in a static magnetic �eld, causing the orthogonally circularly polarized com-

ponents to propagate with di�erent velocities in the medium. This di�erence in velocityof propagation causes the polarization ellipse of the light to rotate as the light propagates,and the e�ect is linear in the static magnetic �eld. In the nonlinear optical regime the

term photoinduced Faraday rotation refers to the intensity dependent modi�cation of thelinear Faraday rotation.

1.2 Brief history of magneto-optics

1.2.1 Linear optical regime

In 1845, Michael Faraday [3, 4] made the discovery that lead glass, when exposed to astatic magnetic �eld in the direction of propagation of linearly polarized light, causes the

plane of polarization of the light to rotate around the axis of propagation. With this event,the �eld of magneto-optics was initiated, and some years later, in 1888, Kerr discoveredthat when plane-polarized light is re ected at normal incidence from a polished pole of

3

4 Chapter 1. Introduction

an electromagnet, it becomes elliptically polarized with its major axis of the polarization

ellipse rotated with respect to the axis of polarization of the incident light. In 1907, Cottonand Mouton [5, 6] found that liquids in a static magnetic �eld, applied orthogonally tothe direction of propagation of the light, become birefringent for linearly polarized light.

Prokhorov, Smolenski�� and Ageev [7] have reviewed the theory and applications ofthin-�lm magneto-optics. Akhmediev [8] investigated spatial dispersion in magneto-optic

media. Barron [9, 10] and Rinard and Calvert [11] have discussed the space inversion andtime reversal (reciprocity) symmetries of the linear Faraday e�ect, and the impact of theFaraday e�ect on multiple scattering of light by spherical particles has been studied byMartinez and Maynard [12].

Only recently, Lee et al. [13] presented theory for optical pulse transmission of a linearFabry-P�erot cavity in the presence of a static magnetic �eld, including the e�ects of

Faraday rotation and the resulting di�erential cavity detuning between left and rightcircularly polarized light.

An excellent review of the chronological development of magneto-optics, including asection on the progress of nonlinear magneto-optical phenomena as well, is given by Za-wodny [14].

1.2.2 Nonlinear optical regime

Tran and Patel [15] have investigated free-carrier magneto-optical e�ects in far-infrareddi�erence-frequency generation in semiconductors. Rabin and Bey [16, 17] investigated the

theory of second harmonic generation with a linearly polarized pump wave in gyrotropicmedia.

Kielich et al. [2, 18, 19] have extensively studied the theory of nonlinear Faradayrotation and its potential applications. Malinowski et al. [20] have investigated reciprocityconditions for nonlocal optical interactions. Chen has investigated magneto-optical e�ects

in Caesium, in the Faraday [21, 22] as well as the Voigt [23] con�guration (See Figs. 2.2and 2.3). Nelson and Chen [24] have developed a Lagrangian theory of magnetic dielectrics.The magneto-electric e�ect [25] has been studied by Hornreich [26].

Akhmanov et al. [27] investigated photoinduced optical activity in isotropic nonlin-ear spatially dispersive media, and provided a phenomenological interpretation of thephotoinduced Faraday rotation in terms of an analogy with photoinduced optical activ-

ity. Nersinyan, Sarkisyan and Tabiryan [28, 29] have investigated the nonlinear e�ectsin magneto-optics, such as photoinduced Faraday rotation and parametric interactions.Boardman and Xie [30] have developed Lagrangian theory for the propagation of spatial

solitons in optical Kerr-media including linear magneto-optical interaction. Karpman hasstudied solitons in media with linear magneto-optical interaction [31]. Wabnitz et al. [32]have investigated the possibility of obtaining soliton mode locking in isotropic media by

using the photoinduced Faraday rotation, and Unsbo and Flytzanis [33] have investigateddegenerate four-wave mixing in nonlinear gyrotropic media, considering photoinducedFaraday rotation as well as optical activity.

The Faraday rotation of semimagnetic semiconductors has been extensively studied,in a linear as well as nonlinear optical regime, by Buss et al. [34, 35, 36, 37], Hugonnard-

Bruy�ere et al. [38, 39, 40], Pankoke et al. [41], Leisching et al. [42], and by Haddad [43, 44].Reviews of studies of photoinduced Faraday rotation in semimagnetic semiconductors aregiven by Buss et al. [36, 45], and by Tziligakis et al. [46].

1.3. The MiniDisc technology 5

Westin et al. [47] have investigated polarization ip- op operation of nonlinear gy-

rotropic resonators. Zheludev [48] has provided a thorough review of polarization multi-stable phenomena in nonlinear optics, containing some points related to gyrotropic media.

Pershan et al. [49] provided theory for the inverse Faraday e�ect, and optically induced

static magnetization in the optically resonant regime has been analysed by Wozniak, bothin terms of the inverse Faraday e�ect [50] and inverse magnetochiral birefringence [51].

1.3 The MiniDisc technology

Being a commersially available device using magneto-optic technology, the MiniDisc1

recorder is an example of current applications of magneto-optics. In the high density

magneto-optic disc drive systems of Sony, such as the technology used for the MiniDisc,the magneto-optic recording phase is performed by shining a focused beam from a DC-powered semiconductor laser onto the medium while modulating an externally applied

magnetic �eld over a region being enclosed by the light spot.Whenever the light beam heats the medium above the Curie temperature [52], any

spontaneous magnetization of the medium vanishes, and by modulating the direction of

the externally applied magnetic �eld, information is written to the disc. As the mediumis cooled below the Curie temperature, the orientations of the magnetic dipole momentsof the medium are �xed, and the disc becomes insensitive for magnetic �elds, at least on

a moderate scale of �eld intensities.Information retrieval is achieved by again focusing a linearly polarized beam of light

from a DC-powered semiconductor laser, being much weaker in intensity than the one

used for heating the medium, onto the disc. Due to magneto-optical Kerr-e�ect, beinga magneto-optical analogy to the DC Kerr-e�ect in optics, the magnetic state of mattercauses the re ected beam to be slightly elliptically polarized, with ellipticity and ori-

entation of the polarization state determined by the orientation of the magnetic dipolemoments in the magnetic layer of the disc.

The magneto-optic recording techniques of today use optics only by means of heating

the medium, while modulation of the magnetic �eld is achieved by alternating the currentthrough a coil. It should be emphasized that while the limit of information storage densityin the CompactDisc technology is determined by the di�raction limited spot size both in

the recording and information retrieval process, the storage density of the MiniDisc is inthe writing process not limited by the size of the di�raction-limited beam of light, butrather by the extent of the region to which the magnetic �eld is applied.

1http://www.minidisc.com/

6

Chapter 2

Classical mechanics of magneto-optics

\7498. Today worked with lines of magnetic force, passing them across

di�erent bodies (transparent in di�erent directions) and at the same time

passing a polarized ray of light through them and afterwards examining the

ray by a Nichol's Eyepiece or other means."

{MICHAEL FARADAY, Faraday's diary (13 Sept. 1845)

The fundaments of linear magneto-optics, such as Faraday rotation and the

Cotton-Mouton e�ect, are illustrated with an all-classical model of a linear elec-

tric dipole oscillator, exposed to a time-harmonic electromagnetic �eld in the

presence of a static magnetic �eld. The resulting electric polarization density

of an ensemble of oscillators is presented, and the concept of magneto-optical

susceptibilities are introduced.

2.1 Magneto-optics in the one-electron oscillator model

Among the simplest models of interaction between light and matter is the all-classical

one-electron oscillator, consisting of a negatively charged particle (electron) with massme,mutually interacting with a positively charged particle (proton) with mass mp, throughattractive Coulomb forces. Exposed to a static magnetic �eld and a rapidly oscillating

optical �eld, this simple model provides a qualitative picture of the underlying mechanismsof arti�cially induced gyrotropy in the linear optical regime.

In the one-electron oscillator model, several levels of approximations may be appliedto the problem, with increasing algebraic complexity. At the �rst level of approximation,the proton is assumed to be �xed in space, with the electron free to oscillate around the

proton. Quite generally, at least within the scope of linear optics, the restoring springforce which con�nes the electron can be assumed to be linear with the displacementdistance of the electron from the central position. Providing the very basic models of

the concept of refractive index and optical dispersion, this model has been applied bynumerous authors, such as Akhmanov [53], Feynman [54], and Born and Wolf [55]. Whena static magnetic �eld is applied to the system, the Lorentz forces acting on the electron

will lift the degeneracy of circular electron orbits around the direction of the �eld, andwhen solving the resulting equation of motion for the electron, the all-classical modelof Faraday rotation appears [56]. In this model, one equation of motion, for the time

evolution of the position of the electron, is to be solved.

Moving on to the next level of approximation, the bound proton-electron pair may

be considered as constituting a two-body central force problem of classical mechanics, inwhich one may assume a �xed center of mass of the system, around which the proton aswell as the electron are free to oscillate. In this level of approximation, by introducing the

7

8 Chapter 2. Classical mechanics of magneto-optics

me; �e

mp; +e

x

yz

re

rp

Figure 2.1. Schematic picture of the classical one-electron oscillator system,consisting of a bound electron-proton pair under in uence of a central force�eld modelled by a mechanical spring force with spring constant ke. As thesystem is set into motion by an externally applied time-harmonic electro-magnetic �eld in the presence of a static magnetic �eld, the Lorentz forcesacting on the system will govern magneto-optical e�ects such as the Faradayrotation and Cotton-Mouton e�ect.

concept of reduced mass for the two moving particles, the equations of motion for the twoparticles can be reduced to one equation of motion, for the evolution of the electric dipolemoment of the system.

The third level of approximation which may be identi�ed is when the center of massis allowed to oscillate as well, in which case an equation of motion for the center of massappears in addition to the one for the evolution of the electric dipole moment. Throughout

the following discussion, this model will be the one used, and similarities and di�erencesto the results as would have been obtained in the other models will be pointed out alongthe way.

In each of the models, nonlinearities of the restoring central force �eld may be intro-duced as to include nonlinear interactions; such models have been used in the �rst level ofapproximation by Alexiewicz and Kasprowicz-Kielich [57] in the discussion of dispersion

and absorption in nonlinear molecular polarizabilities.

It should be emphasized that the spring model, as now will be introduced, gives anidentical form of the set of nonzero elements of the susceptibility tensors, as compared

with those obtained in Chapter 3, where a quantummechanical analysis of magneto-opticsis performed in a more strict manner; with this in mind, the spring model serves well asa qualitative description of the e�ects found in linear magneto-optics.

2.1.1 The all-classical spring model

Throughout this analysis, the wavelength of the electromagnetic �eld will be assumed tobe su�ciently large in order to neglect any spatial variations of the �elds over the spatial

2.1. Magneto-optics in the one-electron oscillator model 9

extent of the oscillator system. In this model, the central force �eld is modelled by a

mechanical spring force with spring constant ke, as shown schematically in Fig. 2.1, andthe all-classical Newton's equations of motion for the electron and nucleus are

me�re = F(L)e + F

(C)e + F

(D)e ; (2.1a)

mp�rp = F(L)p + F

(C)p + F

(D)p ; (2.1b)

where the Lorentz forces acting on the particles are given by

F(L)e = �e[E(r; t) + _re �B(r; t)];

F(L)p = +e[E(r; t) + _rp �B(r; t)];

constituting the interaction between the electromagnetic �eld and the particles. Using thelinear mechanical spring model, the attractive central force �eld is modelled as

F(C)e = �F(C)

p = �ke(re � rp):

Since no linear direction of oscillation has preference, this central force �eld also expresses

the intrinsic isotropy of the system. In this classical oscillator model, the damping forcesare modelled as

F(D)e = �me�e _re; F

(D)p = �mp�p _rp;

where �e and �p are heuristically introduced damping constants of the system, conformingto an extension of the damping of the linear oscillator model in [53] to three dimensions.

By introducing the center of mass R = (mere +mprp)=(me + mp) and the reduced

mass [58] mr = memp=(me +mp) of the system, the equation of motion for the electricdipole moment � = �e(re � rp) is obtained from (2.1) as

��+ �� _�+ (ke=mr)�+(e=mr)(1�me=mp)

(1 +me=mp)_��B(r; t)

= (e2=mr)[E(r; t) + _R�B(r; t)] + e(�e � �p) _R;

(2.2)

with �� = (mp�e+me�p)=(me+mp) being the e�ective damping constant of the oscillatingdipole, and where the motion of the center of mass is described by

�R+ �R _R = [ _��B(r; t) + (mr=e)(�e � �p) _�]=(me +mp); (2.3)

where �R = (me�e+mp�p)=(me +mp) describes the e�ective damping for motion of thecentre of mass. The Eqs. (2.2) and (2.3) are valid for any temporally varying electromag-netic �elds, and they do so far not rely on any assumptions of the relative masses of the

charged particles.

2.1.2 Motion in a time-harmonic electromagnetic �eld

The electromagnetic �eld will now be assumed to consist of a time-harmonic part related tothe light, and an additional term governing the static magnetic �eld; in complex notation

the �elds are taken as E(r; t) = Re[E! exp (�i!t)], B(r; t) = B0 +Re[B! exp (�i!t)]. Byassuming jB!j � jB0j, i. e. neglecting the weak magneto-optical self-coupling to the light,the equation of motion (2.3) for the center of mass is easily integrated, and as the solution


for R(t) is inserted into Eq. (2.2), the equation for steady-state harmonic motion of the

electric dipole moment � = Re[�!exp (�i!t)] becomes

f0(!)�! � if1(!)�! �B0 � f2(!)(�! �B0)B0 = (e2=mr)E!; (2.4)

where the functions f0(!), f1(!), and f2(!) are de�ned as

f0(!) = (2 � !2 � i!��)

�1 +

mr

(me +mp)

(4!2L � (�e � �p)2)

(2 � !2 � i!��)

!

(! + i�R)

�; (2.5a)

f1(!) = (e=mr)!

�(1�me=mp)

(1 +me=mp)� i

2mr

(me +mp)

(�e � �p)

(! + i�R)

�; (2.5b)

f2(!) = (e=mr)2 mr

(me +mp)

!

(! + i�R); (2.5c)

with the resonance frequency of the system determined by 2 = ke=mr, and where theLarmor frequency [59] !L = ejB0j=(2mr) was introduced. The resulting equation of mo-

tion (2.4) constitutes a linear algebraic system for the complex components of the electricdipole moment, which can be solved for in a straightforward manner.

The last term of the left-hand side of Eq. (2.4) is a direct consequence from the inclusionof the motion of the center of mass of the system. For most practical purposes, theapproximation me � mp may now be applied to the problem, in which case the bracketed

expressions in the right-hand sides of Eqs. (2.5) are approximated by unity. However, thisapproximation only serves as to reduce the size of the expressions, and is not essential forsolving the equation of motion, Eq. (2.4), which is valid for arbitrary masses me and mp.

Whether or not the approximation is applied, Eq. (2.4) is straightforward to solve for theelectric dipole moment �

!, to yield the solution

�i

! = (ee)

ijEj

! + (eem)

ijkEj

!Bk

0 + (eemm)

ijklEj

!Bk

0Bl

0; (2.6)

where summation over repeated indices i; j; k; l = x; y; z is implied, and where the nonzero

elements of the tensors (ee), (eem), and (eemm) are expressed in terms of the func-

tions (2.5) as

(ee)xx =

(ee)yy =

(ee)zz

=e2

mr

f20 (!)

[f20 (!)� f21 (!)jB0j2][f0(!)� f2(!)jB0j2] ; (2.7a)

(eem)xyz =

(eem)zxy =

(eem)yzx = � (eem)zyx = � (eem)xzy = � (eem)yxz

= ie2

mr

f0(!)f1(!)

[f20 (!)� f21 (!)jB0j2][f0(!)� f2(!)jB0j2] ; (2.7b)

(eemm)xxxx =

(eemm)yyyy =

(eemm)zzzz =

(eemm)xxyy +

(eemm)xyxy +

(eemm)xyyx ; (2.7c)

2.2. The polarization density of the medium 11

(eemm)xxyy =

(eemm)yyxx =

(eemm)yyzz =

(eemm)zzyy =

(eemm)zzxx =

(eemm)xxzz

= � e2

mr

f0(!)f2(!)

[f20 (!)� f21 (!)jB0j2][f0(!)� f2(!)jB0j2] ; (2.7d)

(eemm)xyxy =

(eemm)yxyx =

(eemm)zxzx =

(eemm)xzxz =

(eemm)yzyz =

(eemm)zyzy

= (eemm)xyyx =

(eemm)yxxy =

(eemm)zxxz =

(eemm)xzzx =

(eemm)yzzy =

(eemm)zyyz

=1

2

e2

mr

f0(!)f2(!)� f21 (!)

[f20 (!)� f21 (!)jB0j2][f0(!)� f2(!)jB0j2] : (2.7e)

The tensors (2.7) correspond to the linear electric dipole micropolarizabilities [1] of an

isotropic medium, here they are extended to include magneto-optical interactions of sec-ond and third orders; in Chapter 3 similar forms are derived using quantum-mechanicalperturbation analysis and applying symmetry considerations for isotropic media. The form

of the solution for the electric dipole moment, as presented in Eq. (2.6), illustrates a con-ceptual problem of magneto-optics; within the terminology of optics the solution clearlyshows a linear optical response, that is to say, the electric dipole moment is linear in the

electric �eld strength. However, the contributions of the �rst, second, and third terms areof zeroth, �rst, and second order in the static magnetic �eld, and hence the electric dipolemoment is nonlinear in the static magnetic �eld. In order to avoid any potential confusion

in this respect, the terminology of \nonlinear magneto-optics" is throughout this thesisdevoted exclusively to phenomena being nonlinear in the electric �eld strength, but ofarbitrary nonzero order in the magnetic �eld.

Using the symmetry properties of the nonzero tensor elements in Eqs. (2.7), an alter-

native form of expressing the solution (2.6) for the electric dipole moment is

�!=

(ee)xx E! +

(eem)xyz E! �B0 +

(eemm)xxyy (B0 �B0)E!

+ ( (eemm)xyxy + (eemm)xyyx )(E! �B0)B0:

(2.8)

In the right-hand side of Eq. (2.8), the third term derives directly from the motion of the

center of mass of the oscillator, while the other terms are found within the approximationof a center of mass being in rest as well.

2.2 The polarization density of the medium

For an ensemble of N identical and mutually noninteracting electric dipole oscillators

per unit volume, the electric polarization density is from Eq. (2.8) given as P(r; t) =N Re[�

!exp (�i!t)]. Equivalently, this polarization density can be expressed as P(r; t) =

Re[P! exp (�i!t)], with the components of the temporal envelope P! given by

Pi

! = "0[�(ee)

ijEj

! + �(eem)

ijkEj

!Bk

0 + �(eemm)

ijklEj

!Bk

0Bl

0]; (2.9)

where the standard notation for the spring oscillator equivalents to the optical and magneto-optical susceptibilities was employed; in terms of the micropolarizabilities, these tensors


are given as1

�(ee)

ij= (N="0)

(ee)

ij; �

(eem)

ijk= (N="0)

(eem)

ijk; �

(eemm)

ijkl= (N="0)

(eemm)

ijkl: (2.10)

Whenever the angular frequency ! of the electric �eld is moved far away from the resonancefrequency of the system, the tensor elements of (ee) and (eemm) are all real-valued,while the elements of (eem) are entirely imaginary. In this nonresonant regime, the electric

polarization density is expressed in terms of real �eld quantities as

P(r; t) = "0

��1E(r; t) + �2

@E(r; t)

@t�B0

+ �03(B0 �B0)

@2E(r; t)

@t2+ �3

�B0 � @

2E(r; t)

@t2

�B0

�;

(2.11)

where

�1 =N

"0

e2

mr

(2 � !2)

[(2 � !2)2 � 4!2L!2]; (2.12a)

�2 = �N"0

e3

m2r

1

[(2 � !2)2 � 4!2L!2]; (2.12b)

�03 =

N

"0

e4

m3r

(mr=mp)

[(2 � !2)2 � 4!2L!2]!2

; (2.12c)

�3 =N

"0

e4

m3r

[1� (mr=mp)(2 � !

2)]

[(2 � !2)2 � 4!2L!2](2 � !2)!2

: (2.12d)

The obtained form (2.11) exactly corresponds to the spatially local general expression2

for the linear polarization density incorporating up to quadratic powers of the magnetic�eld, as obtained from symmetry arguments of the transformation properties of electricand magnetic �elds.

2.2.1 The refractive index of the medium

In the right-hand side of Eq. (2.11), the �rst and third terms determine the induced electricdipole moment in the direction of the electric �eld; the former as the all-electric alignmentof the dipole moment, and the latter as its magnetic �eld-induced modi�cation. In this

mechanical model of matter, these mechanisms both contribute to the non-birefringentpart of the refractive index of the isotropic medium.

2.2.2 The Faraday e�ect

The second term of the right-hand side of Eq. (2.11), being linear in the electric as well

as the magnetic �eld, introduces a birefringence for circularly polarized light, causing thepolarization ellipse of the propagating light to gyrate around the direction of the staticmagnetic �eld [25]. This arti�cially induced gyrotropy of the medium is classically denoted

as Faraday rotation. The common setup for exploiting this e�ect is shown in Fig. 2.2,which throughout the thesis is denoted as the Faraday con�guration.

1C. f. Sec. 3.3 and Appendix C, Table C.1, for tabulated nonzero elements of linear magneto-opticalsusceptibility tensors of isotropic media.

2E. g. Jackson [60], Eq. (6.149).

2.2. The polarization density of the medium 13

B0E(r; t)

z

Figure 2.2. Schematic picture of a setup in the Faraday con�guration, withthe static magnetic �eld B0 applied in parallel to the direction of propagationof the electromagnetic �eld.

B0

E(r; t)

z

Figure 2.3. Schematic picture of a setup in the Voigt con�guration, with thestatic magnetic �eld B0 applied orthogonally to the direction of propagationof the electromagnetic �eld.

For a wave propagating in the positive z-direction of a medium described by the re-fractive index n and the magnetic �eld-induced gyration constant , as discussed in more

detail in Chapter 4, the refractive indices experienced by the circularly polarized com-ponents of the light will typically be n+ = n + =(2n) for left circular polarization, andn� = n� =(2n) for right circular polarization.

2.2.3 The Cotton-Mouton e�ect

Finally, the fourth term of the right-hand side of Eq. (2.11), being quadratic in the static

magnetic �eld, introduces a preference for the polarization density of the medium in thedirection of the static magnetic �eld. If the static magnetic �eld is applied orthogonally tothe direction of propagation of the light, as in the Voigt con�guration shown in Fig. 2.3,

only the component of the electric �eld being parallel to the magnetic �eld will throughthis term contribute to electric polarization of the medium; hence a birefringence forlinearly polarized light arises.

The name of this setup derives from the fact that, even though Cotton and Mouton [5, 6]made their discovery of magnetic �eld-induced birefringence for linearly polarized light in

1907 for liquids, Voigt [61] used this con�guration already in 1902 in the discovery ofbirefringence for linearly polarized light in vapours subject to a static magnetic �eld.Since the Voigt e�ect is small [56] compared to the one found by Cotton and Mouton, the


latter one is the most commonly found in the literature; nevertheless the con�guration,

for historical reasons, is named after Voigt.

Chapter 3

Nonlinear magneto-optics

\2199. The Law, therefore, by which an electric current acts on a ray

of light is easily expressed. When an electric current passes round a ray

of polarized light in a plane perpendicular to the ray, it causes the ray to

revolve on its axis, as long as it is under the in uence of the current, in the

same direction as that in which the current is passing."

{MICHAEL FARADAY, Phil. Mag. 28 (1846)

The constitutive relations of light-matter interaction are here developed from a

quantum mechanical description in terms of a perturbation analysis of the den-

sity operator for a system of identical and mutually noninteracting molecules.

In terms of various-order contributions to the perturbation series of the den-

sity operator of the system, the magneto-optical susceptibility tensors are in-

troduced, and their spatial symmetry properties are discussed in terms of the

transformation properties of polar and axial tensors, under proper and im-

proper rotations of the coordinate system of the medium. As an illustration of

the symmetry properties, the elements of the second and third order magneto-

optical susceptibility tensors are for isotropic media reduced to their minimum

set of nonzero and independent elements.

3.1 Magneto-optical light-matter interaction

In nonlinear optics, one central item of interest is the calculation of the electric polarizationdensity of the medium, induced by externally applied electromagnetic �elds. In this section

a perturbation analysis of the density operator of an ensemble of molecules, exposed tooptical waves under in uence of magnetic �elds is performed.

3.1.1 General outline

Without entering any assumptions regarding the particular form of interaction betweenthe electromagnetic �eld and matter, the general technique of obtaining the solution forthe density operator of a quantum-mechanical system, by means of perturbation analysis,

will now be outlined. We consider a general Hamiltonian of the form

H = H0 + HI(t) + HR; (3.1)

in which H0 is the Hamiltonian describing the unperturbed system at thermal equilibrium,

HI(t) is the interaction Hamiltonian, describing the perturbation of the system causedby light{matter interaction, and HR is an operator which phenomenologically describesthe relaxation processes, bringing the ensemble back into thermal equilibrium whenever

15

16 Chapter 3. Nonlinear magneto-optics

external forces are absent [1, 62]. The evolution in time of the density operator � of the

system is governed by the general equation of motion

i~d�

dt= H�� H = [H; �]; (3.2)

where the conventional bracket notation for the commutator of two operators was in-troduced. The equation of motion can now be solved by employing the technique of

perturbation analysis. In the following analysis the relaxation processes will be assumedto follow a phenomenological model, in which the density operator relaxes toward thermalequilibrium as

[HR; �] = i~�(�(0) � �);

where �(0) = � exp(�H0=kBT ) is the quasi-static density operator of the system in thermalequilibrium, with � being a normalization constant ensuring the probabilities to sum upto unity, Tr [�(0)] = 1. The matrix elements of the operator � are constants considered as

being independent of the interaction Hamiltonian HI(t).

The density operator is now expressed as a power series (perturbation series) in the

interaction Hamiltonian,

� = �(0) + �

(1)(t) + �(2)(t) + : : :+ �

(n)(t) + : : : ;

where each of the terms �(k)(t) contains the interaction Hamiltonian to the power of k,

i. e. �(1)(t) is linear in HI(t), �(2)(t) is quadratic in HI(t), etc. By using the perturbation

series for the density operator, the equation of motion (3.2) separates into the system

i~d�

(0)

dt= [H0; �

(0)];

i~d�

(1)(t)

dt= [H0; �

(1)(t)] + [HI(t); �(0)]� i~��(1)(t);

i~d�

(2)(t)

dt= [H0; �

(2)(t)] + [HI(t); �(1)(t)]� i~��(2)(t);

...

i~d�

(n)(t)

dt= [H0; �

(n)(t)] + [HI(t); �(n�1)(t)]� i~��(n)(t);

which under the boundary conditions �(n)(�1) = 0, n = 1; 2; : : : , provides a set of suc-cessive solutions for �(k)(t), k = 1; 2; : : : . By de�ning the unperturbed time-development

operator U0(t) = exp(�iH0t=~) and introducing the abbreviated notation

H0I(t) = U0(�t)HI(t)U0(t)

3.1. Magneto-optical light-matter interaction 17

for the interaction Hamiltonian in the interaction picture, one obtains the equation of

motion of the perturbation terms of the density operator in the interaction picture as

i~d

dt[U0(�t)�(1)(t)U0(t)] = [H 0

I(t); U0(�t)�(0)U0(t)]� i~U0(�t)��(1)(t)U0(t);

i~d

dt[U0(�t)�(2)(t)U0(t)] = [H 0

I(t); U0(�t)�(1)(t)U0(t)]� i~U0(�t)��(2)(t)U0(t);...

i~d

dt[U0(�t)�(n)(t)U0(t)] = [H 0

I(t); U0(�t)�(n�1)(t)U0(t)]� i~U0(�t)��(n)(t)U0(t):

The solutions for the matrix elements �(n)

ab(t) = haj�(n)(t)jbi of the n:th order perturbation

term of the density operator are then obtained recursively from the lower-order terms as

�(n)

ab(t) =

1

i~exp[�i(ab � i�ab)t]

tZ�1

[H 0I(�); U0(��)�(n�1)(� )U0(�)]ab exp (�ab� ) d�;

(3.3)

which successively can be solved for n = 1; 2; : : : , and where the transition frequencyab = (Ea � Eb )=~ was de�ned, where Ea and Eb are the energies of the system at states

jai and jbi, respectively.

3.1.2 Time-evolution of the density operator in magneto-optics

For simplicity, the electric quadrupolar interactions will from now on be neglected in thesubsequent analysis. Considering a system ofM identical, independent and mutually non-interacting molecules within the volume V , with �� and m� being the components of the

quantum-mechanical electric and magnetic dipole operators of the individual molecules,respectively, the molecular interaction Hamiltonian HI(t) is in the Schr�odinger picturegiven by [63]

HI(t) = ��E�(r; t)� m�B�(r; t);

and the n:th order molecular density operator hence, from Eq. (3.3), obeys the equation

�(n)

ab(t) =

1

i~exp (�iabt� �abt)

tZ�1

[H0I(� ); U0(��)�(n�1)(� )U0(� )]ab exp (�ab�) d�:

(3.4)


By expressing the involved electric and magnetic �elds in terms of their complex temporal

Fourier transforms1 E(r; !) and B(r; !), using

E(r; t) =

1Z�1

E(r; !) exp (�i!t) d!; B(r; t) =

1Z�1

B(r; !) exp (�i!t) d!; (3.5)

and by expanding the commutator in Eq. (3.4), some algebraic rearrangement gives therecursive relation

�(n)

ab(t) = � 1

i~

1Z�1

exp [�i(ab � i�ab)t] (3.6)

�Xc

n tZ�1

exp [i(ab � !n � i�ab)� ]�(n�1)

cb(�) d� [��acE�(r; !n) +m

�

acB�(r; !n)]

�tZ

�1

exp [i(ab � !n � i�ab)� ]�(n�1)ac (�) d� [��

cbE�(r; !n) +m

�

cbB�(r; !n)]

od!n:

The matrix elements of the n:th order density operator of the individual molecules cannow be obtained from Eq. (3.6) for n = 1; 2; : : : ; n by expanding the commutator and

performing a straightforward integration in a recursive manner. The matrix elements ofthe �rst order perturbation are given in an explicit form as

�(1)

ab(t) =

1

~

1Z�1

(�(0)

bb� �

(0)aa )

(ab � ! � i�ab)[��

abE�(r; !) +m

�

abB�(r; !)] exp (�i!t) d!; (3.7a)

where summation over repeated indices � is implied. For future purposes, it turns out to

be convenient to rewrite this in the somewhat simpli�ed form

�(1)

ab(t) =

1Z�1

[S(e)�

ab(!)E�(r; !) + S

(m)�

ab(!)B�(r; !)] exp (�i!t) d!; (3.8)

where

S(e)�

ab(!) =

1

~

(�(0)

bb� �

(0)aa )

(ab � ! � i�ab)��

ab; (3.9a)

S(m)�

ab(!) =

1

~

(�(0)

bb� �

(0)aa )

(ab � ! � i�ab)m

�

ab; (3.9b)

are the �rst-order response functions (rank-one tensors), determining the strengths withwhich the electric and magnetic �elds a�ect the time-evolution of the density operator as

1The reality condition for E(r; t) and B(r; t) requires E(r; !) = E�(r;�!) and B(r; !) = B�(r;�!);the real �elds are related to these quantities through the inverse transform

E(r; !) =1

2�

1Z

�1

E(r; t) exp (i!t) dt; B(r; !) =1

2�

1Z

�1

B(r; t) exp (i!t) dt:


functions of the frequency distributions of the electric and magnetic �elds. Proceeding in

a similar manner, the second-order perturbation of the ab:th matrix element of the densityoperator yields

�(2)

ab(t) =

1Z�1

1Z�1

�S(ee)�

ab(!1; !2)E�(r; !1)E (r; !2)

+ S(em)�

ab(!1; !2)E�(r; !1)B (r; !2)

+ S(me)�

ab(!1; !2)B�(r; !1)E (r; !2) (3.10)

+ S(mm)�

ab(!1; !2)B�(r; !1)B (r; !2)

exp [�i(!1 + !2)t] d!1 d!2;

where the second-order response functions (rank-two tensors) are

S(ee)�

ab(!1; !2) =

1

~

Xc

[S(e)�

cb(!1)�

ac � S(e)�ac (!1)�

cb]

(ab � !1 � !2 � i�ab); (3.11a)

S(em)�

ab(!1; !2) =

1

~

Xc

[S(e)�

cb(!1)m

ac � S(e)�ac (!1)m

cb]

(ab � !1 � !2 � i�ab); (3.11b)

S(me)�

ab(!1; !2) =

1

~

Xc

[S(m)�

cb(!1)�

ac � S(m)�ac (!1)�

cb]

(ab � !1 � !2 � i�ab); (3.11c)

S(mm)�

ab(!1; !2) =

1

~

Xc

[S(m)�

cb(!1)m

ac � S(m)�ac (!1)m

cb]

(ab � !1 � !2 � i�ab): (3.11d)

In Eq. (3.10) the degeneracy of terms involving mixed electric and magnetic �eld compo-nents may be eliminated by simply interchanging dummy indices � and variables ofintegration !1 !2 in the third term. This implies that, when forming the sum in the

expression for the density operator elements, only the three distinguishable combinationsof electric and magnetic �elds are explicitly encountered.

Proceeding in a similar manner, the matrix elements of the third order perturbation

term of the molecular density operator become

�(3)

ab(t) =

1Z�1

1Z�1

1Z�1

�S(eee)� �

ab(!1; !2; !3)E�(r; !1)E (r; !2)E�(r; !3)

+ S(eem)� �

ab(!1; !2; !3)E�(r; !1)E (r; !2)B�(r; !3)

+ S(eme)� �

ab(!1; !2; !3)E�(r; !1)B (r; !2)E�(r; !3)

+ S(emm)� �

ab(!1; !2; !3)E�(r; !1)B (r; !2)B�(r; !3)

+ S(mee)� �

ab(!1; !2; !3)B�(r; !1)E (r; !2)E�(r; !3)

+ S(mem)� �

ab(!1; !2; !3)B�(r; !1)E (r; !2)B�(r; !3)

+ S(mme)� �

ab(!1; !2; !3)B�(r; !1)B (r; !2)E�(r; !3)

+ S(mmm)� �

ab(!1; !2; !3)B�(r; !1)B (r; !2)B�(r; !3)

�� exp [�i(!1 + !2 + !3)t] d!1 d!2 d!3;

(3.12)


where the third-order response functions (rank-three tensors) are de�ned as

S(eee)� �

ab(!1; !2; !3) =

1

~

Xd

[S(ee)�

db(!1; !2)�

�

ad � S(ee)�

ad(!1; !2)�

�

db]

(ab � !1 � !2 � !3 � i�ab); (3.13a)

S(eem)� �

ab(!1; !2; !3) =

1

~

Xd

[S(ee)�

db(!1; !2)m

�

ad � S(ee)�

ad(!1; !2)m

�

db]

(ab � !1 � !2 � !3 � i�ab); (3.13b)

S(eme)� �

ab(!1; !2; !3) =

1

~

Xd

[S(em)�

db(!1; !2)�

�

ad � S(em)�

ad(!1; !2)�

�

db]

(ab � !1 � !2 � !3 � i�ab); (3.13c)

S(emm)� �

ab(!1; !2; !3) =

1

~

Xd

[S(em)�

db(!1; !2)m

�

ad � S(em)�

ad(!1; !2)m

�

db]

(ab � !1 � !2 � !3 � i�ab); (3.13d)

S(mee)� �

ab(!1; !2; !3) =

1

~

Xd

[S(me)�

db(!1; !2)�

�

ad � S(me)�

ad(!1; !2)�

�

db]

(ab � !1 � !2 � !3 � i�ab); (3.13e)

S(mem)� �

ab(!1; !2; !3) =

1

~

Xd

[S(me)�

db(!1; !2)m

�

ad � S(me)�

ad(!1; !2)m

�

db]

(ab � !1 � !2 � !3 � i�ab); (3.13f)

S(mme)� �

ab(!1; !2; !3) =

1

~

Xd

[S(mm)�

db(!1; !2)�

�

ad � S(mm)�

ad(!1; !2)�

�

db]

(ab � !1 � !2 � !3 � i�ab); (3.13g)

S(mmm)� �

ab(!1; !2; !3) =

1

~

Xd

[S(mm)�

db(!1; !2)m

�

ad � S(mm)�

ad(!1; !2)m

�

db]

(ab � !1 � !2 � !3 � i�ab): (3.13h)

Removal of mixed terms in Eq. (3.12) being degenerate in the powers of electric and

magnetic �elds is performed in similar to the second-order perturbation, by permutatingdummy indices and variables of integration in degenerate terms.

Finally, the matrix elements of the fourth order perturbation term of the molecular

density operator become

�(4)

ab(t) =

1Z�1

1Z�1

1Z�1

1Z�1

�S(eeee)� ��

ab(!1; !2; !3; !4)E�(r; !1)E (r; !2)E�(r; !3)E�(r; !4)

+ S(eeem)� ��

ab(!1; !2; !3; !4)E�(r; !1)E (r; !2)E�(r; !3)B�(r; !4)

+ S(eeme)� ��

ab(!1; !2; !3; !4)E�(r; !1)E (r; !2)B�(r; !3)E�(r; !4)

+ S(eemm)� ��

ab(!1; !2; !3; !4)E�(r; !1)E (r; !2)B�(r; !3)B�(r; !4)

+ S(emee)� ��

ab(!1; !2; !3; !4)E�(r; !1)B (r; !2)E�(r; !3)E�(r; !4)

+ S(emem)� ��

ab(!1; !2; !3; !4)E�(r; !1)B (r; !2)E�(r; !3)B�(r; !4)

+ S(emme)� ��

ab(!1; !2; !3; !4)E�(r; !1)B (r; !2)B�(r; !3)E�(r; !4)

+ S(emmm)� ��

ab(!1; !2; !3; !4)E�(r; !1)B (r; !2)B�(r; !3)B�(r; !4) (3.14)

+ S(meee)� ��

ab(!1; !2; !3; !4)B�(r; !1)E (r; !2)E�(r; !3)E�(r; !4)

+ S(meem)� ��

ab(!1; !2; !3; !4)B�(r; !1)E (r; !2)E�(r; !3)B�(r; !4)

+ S(meme)� ��

ab(!1; !2; !3; !4)B�(r; !1)E (r; !2)B�(r; !3)E�(r; !4)


+ S(memm)� ��

ab(!1; !2; !3; !4)B�(r; !1)E (r; !2)B�(r; !3)B�(r; !4)

+ S(mmee)� ��

ab(!1; !2; !3; !4)B�(r; !1)B (r; !2)E�(r; !3)E�(r; !4)

+ S(mmem)� ��

ab(!1; !2; !3; !4)B�(r; !1)B (r; !2)E�(r; !3)B�(r; !4)

+ S(mmme)� ��

ab(!1; !2; !3; !4)B�(r; !1)B (r; !2)B�(r; !3)E�(r; !4)

+ S(mmmm)� ��

ab(!1; !2; !3; !4)B�(r; !1)B (r; !2)B�(r; !3)B�(r; !4)

�� exp [�i(!1 + !2 + !3 + !4)t] d!1 d!2 d!3 d!4;

where the fourth-order response functions (rank-four tensors) are de�ned as

S(eeee)� ��

ab(!1; !2; !3; !4)

=1

~

Xf

[S(eee)� �

fb(!1; !2; !3)�

�

af � S(eee)� �

af(!1; !2; !3)�

�

fb]

(ab � !1 � !2 � !3 � !4 � i�ab); (3.15a)

S(eeem)� ��

ab(!1; !2; !3; !4)

=1

~

Xf

[S(eee)� �

fb(!1; !2; !3)m

�

af � S(eee)� �

af(!1; !2; !3)m

�

fb]

(ab � !1 � !2 � !3 � !4 � i�ab); (3.15b)

S(eeme)� ��

ab(!1; !2; !3; !4)

=1

~

Xf

[S(eem)� �

fb(!1; !2; !3)�

�

af � S(eem)� �

af(!1; !2; !3)�

�

fb]

(ab � !1 � !2 � !3 � !4 � i�ab); (3.15c)

S(eemm)� ��

ab(!1; !2; !3; !4)

=1

~

Xf

[S(eem)� �

fb(!1; !2; !3)m

�

af � S(eem)� �

af(!1; !2; !3)m

�

fb]

(ab � !1 � !2 � !3 � !4 � i�ab); (3.15d)

S(emee)� ��

ab(!1; !2; !3; !4)

=1

~

Xf

[S(eme)� �

fb(!1; !2; !3)�

�

af � S(eme)� �

af(!1; !2; !3)�

�

fb]

(ab � !1 � !2 � !3 � !4 � i�ab); (3.15e)

S(emem)� ��

ab(!1; !2; !3; !4)

=1

~

Xf

[S(eme)� �

fb(!1; !2; !3)m

�

af � S(eme)� �

af(!1; !2; !3)m

�

fb]

(ab � !1 � !2 � !3 � !4 � i�ab); (3.15f)

S(emme)� ��

ab(!1; !2; !3; !4)

=1

~

Xf

[S(emm)� �

fb(!1; !2; !3)�

�

af � S(emm)� �

af(!1; !2; !3)�

�

fb]

(ab � !1 � !2 � !3 � !4 � i�ab); (3.15g)

S(emmm)� ��

ab(!1; !2; !3; !4)

=1

~

Xf

[S(emm)� �

fb(!1; !2; !3)m

�

af � S(emm)� �

af(!1; !2; !3)m

�

fb]

(ab � !1 � !2 � !3 � !4 � i�ab); (3.15h)


S(meee)� ��

ab(!1; !2; !3; !4)

=1

~

Xf

[S(mee)� �

fb(!1; !2; !3)�

�

af � S(mee)� �

af(!1; !2; !3)�

�

fb]

(ab � !1 � !2 � !3 � !4 � i�ab); (3.15i)

S(meem)� ��

ab(!1; !2; !3; !4)

=1

~

Xf

[S(mee)� �

fb(!1; !2; !3)m

�

af � S(mee)� �

af(!1; !2; !3)m

�

fb]

(ab � !1 � !2 � !3 � !4 � i�ab); (3.15j)

S(meme)� ��

ab(!1; !2; !3; !4)

=1

~

Xf

[S(mem)� �

fb(!1; !2; !3)�

�

af � S(mem)� �

af(!1; !2; !3)�

�

fb]

(ab � !1 � !2 � !3 � !4 � i�ab); (3.15k)

S(memm)� ��

ab(!1; !2; !3; !4)

=1

~

Xf

[S(mem)� �

fb(!1; !2; !3)m

�

af � S(mem)� �

af(!1; !2; !3)m

�

fb]

(ab � !1 � !2 � !3 � !4 � i�ab); (3.15l)

S(mmee)� ��

ab(!1; !2; !3; !4)

=1

~

Xf

[S(mme)� �

fb(!1; !2; !3)�

�

af � S(mme)� �

af(!1; !2; !3)�

�

fb]

(ab � !1 � !2 � !3 � !4 � i�ab); (3.15m)

S(mmem)� ��

ab(!1; !2; !3; !4)

=1

~

Xf

[S(mme)� �

fb(!1; !2; !3)m

�

af � S(mme)� �

af(!1; !2; !3)m

�

fb]

(ab � !1 � !2 � !3 � !4 � i�ab); (3.15n)

S(mmme)� ��

ab(!1; !2; !3; !4)

=1

~

Xf

[S(mmm)� �

fb(!1; !2; !3)�

�

af � S(mmm)� �

af(!1; !2; !3)�

�

fb]

(ab � !1 � !2 � !3 � !4 � i�ab); (3.15o)

S(mmmm)� ��

ab(!1; !2; !3; !4)

=1

~

Xf

[S(mmm)� �

fb(!1; !2; !3)m

�

af � S(mmm)� �

af(!1; !2; !3)m

�

fb]

(ab � !1 � !2 � !3 � !4 � i�ab): (3.15p)

To recapitulate, we have now obtained an approximative solution for the time-evolution

of the matrix elements of the density operator, given in terms of the perturbation series

�ab(t) = �(0)

ab+ �

(1)

ab(t) + �

(2)

ab(t) + �

(3)

ab(t) + �

(4)

ab(t);

where the �rst-, second-, third-, and fourth-order terms are given by Eqs. (3.8), (3.10),(3.12), and (3.14), respectively.

3.2 The susceptibility tensors of magneto-optics

The electric polarization density of the medium is now calculated from the mean valueof the electric dipole operator of the ensemble of identical and mutually noninteracting

3.2. The susceptibility tensors of magneto-optics 23

molecules. For such a system, the electric polarization density (electric dipole moment

per unit volume)

P(r; t) =

1Xn=0

e�P(n)� (r; t);

and magnetization (magnetic dipole moment per unit volume)

M(r; t) =

1Xn=0

e�M(n)� (r; t);

are obtained from the various-order contributions of the perturbation series of the density

operator as

P(n)� (r; t) = N Tr [�(n)��] = N

Xa

Xb

��

ba�(n)

ab(t); (3.16a)

M(n)� (r; t) = N Tr [�(n)m�] = N

Xa

Xb

m�

ba�(n)

ab(t); (3.16b)

where N = M=V denotes the number density of molecules, and where �(n)

ab(t), as previ-

ously, is the n:th order term of the molecular density operator.

3.2.1 First order interaction

Using Eqs. (3.8) and (3.16a), the �rst order term of the electric polarization density isobtained as

P(1)� (r; t) = "0

1Z�1

[�(ee)

��(�!;!)E�(r; !) + �

(em)

��(�!;!)B�(r; !)] exp (�i!t) d!; (3.17)

where the introduced elements of the �rst order susceptibilities (rank-two tensors) areexpressed in terms of the response functions of Eqs. (3.9) as

�(ee)

��(�!;!) = N

"0

Xa

Xb

��

baS(e)�

ab(!); (3.18a)

�(em)

��(�!;!) = N

"0

Xa

Xb

��

baS(m)�

ab(!); (3.18b)

with the frequency arguments of the susceptibilities written as conforming to the stan-

dard notation in nonlinear optics, e. g. that of Butcher and Cotter [1]. The susceptibilitygiven by Eq. (3.18a) gives the part of the electric polarization density being linear inthe electric �eld, determining the �eld-independent refractive index and linear absorption

due to electric dipolar interactions, while the susceptibility given by Eq. (3.18b) gives themagnetic-�eld induced electric polarization density of the medium, the so-called magneto-electric e�ect [25].


Similar to the electric polarization density, the magnetization of the medium can be

written as

M(1)� (r; t) =

1

�0

1Z�1

[�(me)

��(�!;!)E�(r; !) + �

(mm)

��(�!;!)B�(r; !)] exp (�i!t) d!;

(3.19)

where the �rst-order magnetic susceptibilities are de�ned as

�(me)

��(�!;!) = N�0

Xa

Xb

m�

baS(e)�

ab(!); (3.20a)

�(mm)

��(�!;!) = N�0

Xa

Xb

m�

baS(m)�

ab(!): (3.20b)

The susceptibility given in Eq. (3.20b) is the linear magnetic susceptibility [52], while thesusceptibility given in Eq. (3.20b) gives the electric-�eld induced magnetization of the

medium, also being classi�ed as a magneto-electric e�ect [25].

3.2.2 Second order interaction

Since the second-order light-matter interaction, as generally described by the elementsof the second-order term (3.10) of the density operator, contains mixed terms which, as

previously mentioned in Sec. 3.1.2, are degenerate, it is in order to form a consequentconvention for the inclusion of magneto-optical e�ects in the susceptibility formalism de-sirable to symmetrize the terms in the density matrix, in such a way that only distinct

combinations of electric and magnetic �eld components appear. This can be done in astraightforward manner by interchanging dummy indices � and variables of integra-tion !1 !2 in the third term of Eq. (3.10).

Similarly, in order to symmetrize the �rst and fourth terms of Eq. (3.10), an interchangeof dummy indices and variables of integration gives that the response functions can bereplaced by half the sum of the response functions obtained by performing the 2! = 2

pairwise permutations of (�; !1) and ( ; !2).

Taking these symmetrization operations into account, the second order contribution tothe electric polarization density is obtained as

P(2)� (r; t) = "0

1Z�1

1Z�1

��(eee)

�� (�!�;!1; !2)E�(r; !1)E (r; !2)

+ �(eem)

�� (�!�;!1; !2)E�(r; !1)B (r; !2)

+ �(emm)

�� (�!�;!1; !2)B�(r; !1)B (r; !2)

�exp [�i(!1 + !2)t] d!1 d!2;

(3.21)


where the second-order susceptibility tensors are expressed in terms of the response func-

tions of Eqs. (3.11) as

�(eee)

�� (�!�;!1; !2) = N

"0

1

2

Xa

Xb

��

ba[S(ee)�

ab(!1; !2) + S

(ee) �

ab(!2; !1)]; (3.22a)

�(eem)

�� (�!�;!1; !2) = N

"0

Xa

Xb

��

ba[S(em)�

ab(!1; !2) + S

(me) �

ab(!2; !1)]; (3.22b)

�(emm)

�� (�!�;!1; !2) = N

"0

1

2

Xa

Xb

��

ba[S(mm)�

ab(!1; !2) + S

(mm) �

ab(!2; !1)]: (3.22c)

Similar symmetrization schemes will be applied to higher-order susceptibilities; in orderto reduce the increasing algebraic complexity of the resulting expressions, a simpli�ednotation will be used from now on, as

�(eee)

�� (�!�;!1; !2) = N

"0

1

2S2

Xa

Xb

��

baS(ee)�

ab(!1; !2); (3.23a)

�(eem)

�� (�!�;!1; !2) = N

"0S2

Xa

Xb

��

baS(em)�

ab(!1; !2); (3.23b)

�(emm)

�� (�!�;!1; !2) = N

"0

1

2S2

Xa

Xb

��

baS(mm)�

ab(!1; !2): (3.23c)

In Eqs. (3.23) the symmetrization operator S2 indicates that the expression following itto the right is to be summed over the 2! = 2 pairwise permutations of (�; !1) and ( ; !2),

under simultaneous permutation of the respective superscripts in the response functions.Using the symmetrized form, the susceptibility tensors possess the intrinsic permutationsymmetries [1]

�(eee)

�� (�!�;!1; !2) = �

(eee)

� �(�!�;!2; !1);

�(emm)

�� (�!�;!1; !2) = �

(emm)

� �(�!�;!2; !1);

which are direct consequences of the symmetrization operations as previously performed.

The second-order all-optical susceptibility given by Eq. (3.23a) describes for nonzero

frequecy argements !1 and !2 sum- and di�erence-frequency generation, or, in the degen-erate case !1 = !2, the process of second-harmonic generation. In the case of a staticmagnetic �eld, i. e. !2 = 0, the susceptibility �(eem)(�!;!; 0) given by Eq. (3.23b) de-

scribes the linear part of the Faraday rotation.

From Eqs. (3.10) and (3.16b), the second order magnetization of the medium is similarlyobtained in a symmetrized form as

M(2)� (r; t) =

1

�0

1Z�1

1Z�1

��(mee)

�� (�!�;!1; !2)E�(r; !1)E (r; !2)

+ �(mem)

�� (�!�;!1; !2)E�(r; !1)B (r; !2)

+ �(mmm)

�� (�!�;!1; !2)B�(r; !1)B (r; !2)

�exp [�i(!1 + !2)t] d!1 d!2;


with the second order magnetic susceptibilities de�ned as

�(mee)

�� (�!�;!1; !2) = N�0

1

2S2

Xa

Xb

m�

baS(ee)�

ab(!1; !2); (3.24a)

�(mem)

�� (�!�;!1; !2) = N�0S2

Xa

Xb

m�

baS(em)�

ab(!1; !2); (3.24b)

�(mmm)

�� (�!�;!1; !2) = N�0

1

2S2

Xa

Xb

m�

baS(mm)�

ab(!1; !2): (3.24c)

For the special case of !2 = �!1, the magnetic susceptibility in Eq. (3.24a) for isotropicmedia describes the static magnetization of the medium in the presence of a circularlypolarized electric �eld, commonly denoted as the inverse Faraday e�ect [25, 49].

3.2.3 Third order interaction

Taking the symmetrization operations into account, the third order electric polarization

density is obtained as

P(3)� (r; t) = "0

1Z�1

1Z�1

1Z�1

��(eeee)

�� (�!�;!1; !2; !3)E�(r; !1)E (r; !2)E�(r; !3)

+ �(eeem)

�� (�!�;!1; !2; !3)E�(r; !1)E (r; !2)B�(r; !3)

+ �(eemm)

�� (�!�;!1; !2; !3)E�(r; !1)B (r; !2)B�(r; !3)

+ �(emmm)

�� (�!�;!1; !2; !3)B�(r; !1)B (r; !2)B�(r; !3)

�� exp [�i(!1 + !2 + !3)t] d!1 d!2 d!3;

(3.25)

where the third order susceptibility tensors are expressed in terms of the response functionsof Eqs. (3.13), in the symmetrized form

�(eeee)

�� (�!�;!1; !2; !3) = N

"0

1

3!S3

Xa

Xb

��

baS(eee)� �

ab(!1; !2; !3); (3.26a)

�(eeem)

�� (�!�;!1; !2; !3) = N

"0

1

2!S3

Xa

Xb

��

baS(eem)� �

ab(!1; !2; !3); (3.26b)

�(eemm)

�� (�!�;!1; !2; !3) = N

"0

1

2!S3

Xa

Xb

��

baS(emm)� �

ab(!1; !2; !3); (3.26c)

�(emmm)

�� (�!�;!1; !2; !3) = N

"0

1

3!S3

Xa

Xb

��

baS(mmm)� �

ab(!1; !2; !3): (3.26d)

In analogy to the symmetrization operator for the second order interaction, the sym-

metrization operator S3 indicates that the expression following it to the right is to besummed over the 3! = 6 permutations of the pairs (�; !1), ( ; !2), and (�; !3), undersimultaneous permutation of the superscripts in the respective response functions.

For the susceptibility tensors described by the elements in Eqs. (3.26a) and (3.26d), theintrinsic permutation symmetries ensure them to be invariant under any of the 3! permu-tations of the pairs (�; !1), ( ; !2), and (�; !3), while the magneto-optical susceptibilities


described by the elements in Eqs. (3.26b) and (3.26c) possess the intrinsic permutation

symmetries

�(eeem)

�� (�!�;!1; !2; !3) = �

(eeem)

�� (�!�;!2; !1; !3);

�(eemm)

�� (�!�;!1; !2; !3) = �

(eemm)

�� (�!�;!1; !3; !2):

For a static magnetic �eld, i. e. !3 = 0, the susceptibility �(eeem)(�!�;!1; !2; 0) givenby Eq. (3.26b) describes parametric interactions, such as sum- and di�erence-frequencygeneration, induced by a static magnetic �eld, as analysed in Papers III{V. For the degen-

erate case !1 = !2 = !, !3 = 0, the susceptibility �(eeem)(�!�;!;!; 0) describes second-harmonic generation induced by a static magnetic �eld. For a static magnetic �eld, with!1 = !, !2 = !3 = 0, the magneto-optical susceptibility tensor �(eemm)(�!;!; 0; 0), givenby Eq. (3.26c), describes the Cotton-Mouton e�ect.

3.2.4 Fourth order interaction

Taking the symmetrization operations into account, the fourth order contribution to theelectric polarization density is obtained from Eqs. (3.14) and (3.16a) as

P(4)� (r; t) = "0

1Z�1

1Z�1

1Z�1

1Z�1

��(eeeee)

�� (�!�;!1; !2; !3; !4)E�(r; !1)E (r; !2)E�(r; !3)E�(r; !4)

�(eeeem)

�� (�!�;!1; !2; !3; !4)E�(r; !1)E (r; !2)E�(r; !3)B�(r; !4)

�(eeemm)

�� (�!�;!1; !2; !3; !4)E�(r; !1)E (r; !2)B�(r; !3)B�(r; !4)

�(eemmm)

�� (�!�;!1; !2; !3; !4)E�(r; !1)B (r; !2)B�(r; !3)B�(r; !4)

�(emmmm)

�� (�!�;!1; !2; !3; !4)B�(r; !1)B (r; !2)B�(r; !3)B�(r; !4)

�� exp [�i(!1 + !2 + !3 + !4)t] d!1 d!2 d!3 d!4;

(3.27)

where the fourth order susceptibility tensors are expressed in terms of the response func-tions of Eqs. (3.15), in the symmetrized form

�(eeeee)

�� (�!�;!1; !2; !3; !4) = N

"0

1

4!S4

Xa

Xb

��

baS(eeee)� ��

ab(!1; !2; !3; !4); (3.28a)

�(eeeem)

�� (�!�;!1; !2; !3; !4) = N

"0

1

3!S4

Xa

Xb

��

baS(eeem)� ��

ab(!1; !2; !3; !4); (3.28b)

�(eeemm)

�� (�!�;!1; !2; !3; !4) = N

"0

1

2!2!S4

Xa

Xb

��

baS(eemm)� ��

ab(!1; !2; !3; !4);

(3.28c)

�(eemmm)

�� (�!�;!1; !2; !3; !4) = N

"0

1

3!S4

Xa

Xb

��

baS(emmm)� ��

ab(!1; !2; !3; !4):

(3.28d)


�(emmmm)

�� (�!�;!1; !2; !3; !4) = N

"0

1

4!S4

Xa

Xb

��

baS(mmmm)� ��

ab(!1; !2; !3; !4):

(3.28e)

In analogy to the symmetrization operator for lower order interactions, the symmetrizationoperator S4 indicates that the expression following it to the right is to be summed over the4! = 24 permutations of the pairs (�; !1), ( ; !2), (�; !3), and (�; !4), under simultaneouspermutation of the superscripts in the respective response functions. As a consequence

of the symmetrization, the all-optical susceptibility tensor in Eq. (3.28a) is left invariantunder any of the 4! pairwise permutations of (�; !1), ( ; !2), (�; !3), and (�;!4), whilethe magneto-optical susceptibility tensor in Eq. (3.28b) is left invariant under any of

the 3! permutations of (�; !1), ( ; !2), (�; !3). The magneto-optical susceptibility tensor(3.28c) is left invariant under any of the 2! permutations of (�; !1), ( ; !2), and the 2!permutations of (�; !3), (�; !4). The intrinsic permutation symmetry of the magneto-

optical susceptibility tensor (3.28e) is identical to the one all-optical one (3.28a).

For a static magnetic �eld, i. e. !4 = 0, and frequency arguments !1 = !2 = !3 = !

the susceptibility �(eeeem)(�!�;!; !; !; 0) given by Eq. (3.28b) describes third-harmonicgeneration induced by a static magnetic �eld, while for frequency arguments !1 = !2 =

�!3 = ! the susceptibility �(eeeem)(�!�;!; !;�!; 0) describes photoinduced Faraday

rotation, as analysed in Papers I{II. For the case !1 = !2 6= !3, !4 = 0, the susceptibility�(eeeem)(�!�;!1; !1; !3; 0) describes sum- and di�erence-frequency mixing induced by a

static magnetic �eld.

3.2.5 Higher order interactions

Following the schemes as outlined for interactions up to fourth order, a generalization ofthe algorithm for �nding the form of the n:th order magneto-optical susceptibility for the

description of interaction of m electric �eld components, with frequencies !1; : : : ; !m, and(n�m) magnetic �elds components, with frequencies !m+1; : : : ; !n, yields

�(ee��em��m)

��1��m�m+1��n(�!�;!1; : : : ; !m; !m+1; : : : ; !n) =

N

"0

1

m!(n�m)!

�Sn

Xa

Xb

��

baS(e��em��m)�1��m�m+1��n

ab(!1; : : : ; !m; !m+1; : : : ; !n);

(3.29)

where the operator Sn indicates that the expression following it to the right is to be

summed over all n! permutations of the n triplets f(e; �1; !1), (e; �2; !2), : : : , (e; �m; !m),(m; �m+1; !m+1), : : : , (m; �n; !n)g. The susceptibility tensor described by the elementsabove possesses the intrinsic permutation symmetry that it is left invariant under any of

the m! possible permutations of (�j ; !j) and (�k; !k) for 1 � fj; kg � m, or the (n�m)!possible permutations for m+ 1 � fj; kg � n.

The resulting n:th order electric polarization density is then given in a form containingthe sum of all distinct combinations of various-order powers of electric and magnetic �elds,

3.3. Symmetries of magneto-optical susceptibilities 29

as

P(n)� (r; t) = "0

1Z�1

1Z�1

: : :

1Z�1

1Z�1

[�(eee��ee)

��1�2��n(�!�;!1; !2; : : : ; !n)E�1

(r; !1) : : : E�n�1(r; !n�1)E�n

(r; !n)

+ �(eee��em)

��1�2��n(�!�;!1; !2; : : : ; !n)E�1

(r; !1) : : : E�n�1(r; !n�1)B�n

(r; !n)

+ : : :

+ �(emm��mm)

��1�2��n(�!�;!1; !2; : : : ; !n)B�1 (r; !1) : : : B�n�1

(r; !n�1)B�n(r; !n)]

� exp [�i(!1 + !2 + : : : + !n�1 + !n)t] d!1 d!2 : : : d!n�1 d!n:

(3.30)

This expression is valid for any spectral distribution of the electric and magnetic �elds,within the validity of the perturbation analysis of the density operator. Later on, however,the form (3.30) will be simpli�ed by just considering magneto-optical contributions from

static magnetic �elds.

3.3 Symmetries of magneto-optical susceptibilities

So far the developed description of magneto-optical light-matter interaction is expressedin a coordinate system shared between the medium and the laboratory frame. In stan-

dard texts on nonlinear optics, the spatial symmetries of the optical susceptibilities aretreated in terms of regular coordinate transformations such as rotations and inversionsof spatial coordinates, usually within the framework of light-matter interactions exclu-

sively described by electric dipolar interactions. Since the electric �eld and polarizationdensity both are polar vectors [60, 64] that transform in the same way as regular spatialcoordinates under arbitrary rotation or inversion of the coordinate system, the analysis of

spatial symmeries of the medium in the electric dipole approximation is performed in astraightforward manner [1].

However, since magneto-optics involves electric- as well as magnetic-dipolar interac-tions, which behave in di�erent ways under inversion operations, some extra care must be

taken in the analysis of spatial tranformation properties of the electromagnetic quantitiesin magneto-optic media.

3.3.1 Spatial transformation properties

Basically, a rotation in three dimensions is a linear transformation of the spatial coordi-nates of a point in such a way that the sum of the squares of the coordinates remains

invariant. In the subsequent analysis, a vector taken in the \original" coordinate systemwill be denoted A = A�e�, while the same vector taken in the rotated system yieldsA = A

0ae

0a, where e� and e0a are the sets of unit vectors in the directions of the coordinate

axes of respective coordinate system. Typically, these coordinate systems will be the in-trinsic one of the medium, e. g. the crystal reference frame, and any conveniently chosenlaboratory system, within which setups for experiments or applications are designed.


x

y

z

7�!

(a)

x0

y0

z0

x

y

z

7�!

(b)

x0

y0

z0

Figure 3.1. The coordinate transformations (a) x = (x; y; z) 7! x0 =(�x;�y; z), constituting a proper rotation around the z-axis, and (b) thespace inversion x 7! x0 = �x, an improper rotation corresponding to, for ex-ample, a rotation around the z-axis followed by an inversion in the xy-plane.

In terms of the original set of coordinates x�, the transformed set of coordinates x0aare given from the linear transformation

x0a = Ra�x� (3.31)

where the real elements Ra� = e0a � e� are the direction cosines of the angles between thecoordinate axes of the primed and unprimed coordinate systems, and where summationover repeated indices is implied. The requirement of the transformation (3.31) to leave

lengths invariant, i. e. x0ax0a = x�x�, puts the restriction

Ra�Ra� = �� (3.32)

on the coe�cients of the transformation, where �� is the Kronecker delta function.The requirement (3.32) implies the square of the determinant of the transformation

matrix to equal to unity; rotations for which det (R) = 1 are denoted as being properrotations, while those for which det (R) = �1 are denoted as improper rotations, i. e. aproper rotation in combination with a coordinate inversion. It should be emphasized

that proper rotations leave right-handed coordinate systems right-handed, while improperrotations turn right-handed coordinate systems into left-handed ones and vice versa, asillustrated in Fig. 3.1.

Under subject of such a generalized rotation, the so-called polar vector quantities, suchas the electric �eld E(r; t) or electric polarization density P(r; t), transform according to

A0a = Ra�A�; (3.33)

while the corresponding transformation law for the axial or pseudovectors, such as the

magnetic �eld B(r; t) or magnetization M(r; t), yields

A0a = det (R)Ra�A�: (3.34)


X

Y

Z

x

y

z

Figure 3.2. Illustration of proper rotation of the laboratory reference frame(x; y; z) relative to the crystal frame (X;Y;Z), by means of xa = Ra�X�

with det (R) = 1.

The inverse relations of the transformation laws (3.33) and (3.34) yield, for polar vectorquantities,

A� = A0aRa�; (3.35)

and for axial vector quantities

A� = det (R)A0aRa�; (3.36)

respectively. In a fashion similar to the vector quantities, higher-rank tensors are classi-�ed [64] as being polar or, equivalently, true tensors if they transform according to

A0a1a2��an

= Ra1�1Ra2�2 � � �Ran�nA�1�2��n ; (3.37)

while tensors transforming according to

A0a1a2��an

= det (R)Ra1�1Ra2�2 � � �Ran�nA�1�2��n ; (3.38)

are classi�ed as being axial or, equivalently, pseudotensors.

3.3.2 Susceptibilities in rotated reference frames

Having stated the transformation properties of polar and axial vectors, the propertiesof the electric polarization density involving magneto-optical interactions can now beanalysed by insertion of the electric and magnetic �elds, expressed in a coordinate system

rotated by means of the generalized rotations described by Eqs. (3.33) and (3.34), intothe constitutive relations for the various-order contributions to the polarization densityas derived in Sec. 3.2.


For the �rst order interactions, the electric polarization density from Eq. (3.17) becomes

P(1)a

0(r; t) = "0

1Z�1

[�(ee)

ab

0(�!;!)E0b(r; !) + �(em)

ab

0(�!;!)B0b(r; !)] exp (�i!t) d!;

where the �rst-order susceptibilities in the rotated reference frame are given in terms ofthe susceptibilities (3.19) in the frame of the medium as

�(ee)

ab

0(�!;!) = Ra�Rb��(ee)

��(�!;!); (3.39a)

�(em)

ab

0(�!;!) = det (R)Ra�Rb��(em)

��(�!;!): (3.39b)

By de�nition, the �rst order susceptibility in Eq. (3.39a) is hence a polar tensor, whilethe one in Eq. (3.39b) is an axial tensor, both of them being of rank two.

Proceeding in a similar manner, the second-order electric polarization density in therotated laboratory frame is obtained from Eq. (3.21) as

P(2)a

0(r; t) = "0

1Z�1

1Z�1

��(eee)

abc

0(�!�;!1; !2)E0b(r; !1)E0c(r; !2)

+ �(eem)

abc

0(�!�;!1; !2)E0b(r; !1)B0c(r; !2)

+ �(emm)

abc

0(�!�;!1; !2)B0b(r; !1)B

0c(r; !2)

�exp [�i(!1 + !2)t] d!1 d!2;

where the second-order susceptibilities in the rotated reference frame are given in termsof the susceptibilities (3.22) in the frame of the medium as

�(eee)

abc

0(�!�;!1; !2) = Ra�Rb�Rc �(eee)

�� (�!�;!1; !2); (3.40a)

�(eem)

abc

0(�!�;!1; !2) = det (R)Ra�Rb�Rc �(eem)

�� (�!�;!1; !2); (3.40b)

�(emm)

abc

0(�!�;!1; !2) = Ra�Rb�Rc �(emm)

�� (�!�;!1; !2); (3.40c)

where we in Eq. (3.40c) used [det (R)]2 = 1 for proper as well as improper rotations.From these results one concludes that the second-order susceptibilities �(eee) and �(emm)

are rank-three polar tensors, while the second-order magneto-optical susceptibility �(eem)

is a rank-three axial tensor. The di�erence in tranformation characteristics of these tensorsis the very reason for the tensor components of the second order optical and magneto-

optical susceptbilities, which both are rank-three tensors, not generally sharing the sameset of nonzero tensor elements for a given point-symmetry group of the medium.

For higher order interactions, the transformation of the n:th order magneto-opticalsusceptibility governing the interaction ofm electric �eld components and (n�m) magnetic

�eld components yield

�(eee��em��m)

ab1b2��bn

0 = [det (R)](n�m)Ra�Rb1�1Rb2�2 � � �Rbm�m

� � �Rbn�n�(eee��em��m)

��1�2��n; (3.41)

from which it is found that any magneto-optical susceptibility tensor governing the interac-tion of an even number of magnetic �eld components with an arbitrary number of electric�eld components will be a polar tensor, while those governing interactions involving an


odd number of magnetic �eld components will be axial. Hence the third-order optical and

magneto-optical susceptibility tensors �(eeee) and �(eemm) both are polar rank-four ten-sors, sharing the same set of nonzero tensor elements, while �(eeem) is an axial rank-fourtensor. Regarding fourth-order interactions, the optical and magneto-optical susceptibility

tensors �(eeeee), �(eeemm), and �(emmmm) are all polar rank-�ve tensors, sharing the sameset of nonzero tensor elements, while �(eeeem) and �(eemmm) are axial rank-�ve tensors.

3.3.3 Neumann's principle

The reduction of the elements of magneto-optical susceptibility tensors of arbitrary or-der can be performed by applying Neumann's principle [64] and the method of directinspection. Neumann's principle states that any type of symmetry which is exhibited by

the point symmetry group of the medium is possessed by every physical property of themedium.

For polar tensors, the requirement is that if any proper or improper rotation, described

by the matrix R, of the coordinate system is a symmetry operation of the medium,then the elements of the polar tensor must be left invariant under the correspondingtransformation (3.37), i. e. A0a1a2��an = Aa1a2��an . Using the transformation (3.37) this

requirement is explicitly stated by the system of equations

Aa1a2��an = Ra1�1Ra2�2 � � �Ran�nA�1�2��n ; (3.42)

for a1; a2; : : : ; an taking all possible values.

Similarly, for axial tensors the requirement is that if any proper or improper rota-tion, described by the matrix R, of the coordinate system is a symmetry operation of themedium, then the elements of the axial tensor must be left invariant under the correspond-

ing transformation (3.38), i. e. A0a1a2��an = Aa1a2��an . Using the transformation (3.38)this requirement is explicitly stated by the system of equations

Aa1a2��an = det (R)Ra1�1Ra2�2 � � �Ran�nA�1�2��n ; (3.43)

for a1; a2; : : : ; an taking all possible values. As an illustration of the impact of Neumann'sprinciple on the optical and magneto-optical susceptibilities, the item of inversion sym-metry of a medium will now be considered.

In nonlinear optics, a well known fact [1, 65] is that any all-optical susceptibility tensordescribing an interaction of even order vanishes in media possessing inversion symmetry.This is a direct consequence of Neumann's principle, and prohibits processes such as sec-

ond harmonic generation, optical parametric ampli�cation, and all higher even-orderedharmonic generations in media possessing a center of inversion, in the electric dipole ap-proximation. Recapitulating that these optical processes are described by susceptibilities

being polar tensors of odd rank, this is easily seen from Eq. (3.42) which, if the spatialinversion operation x0a = �xa (i. e. Ra� = ��a�) is a symmetry operation of the medium,takes the form

�(eee��e)

ab1b2��bn= (�1)(n+1)�(eee��e)

ab1b2��bn;

and hence all tensor elements of any all-optical susceptibility tensor of even order n, i. e. ofodd rank (n+ 1), must vanish.


Noteworthy is that similar arguments apply to axial tensors, such as some of the

magneto-optical susceptibilities, as well. Considering, for example, the n:th order magneto-optical susceptibility tensor of Eq. (3.29) governing a magneto-optical interaction involvingm electric �eld components and (n �m) magnetic �eld components, its tensor elements

are under any proper or improper rotation described by the transformation (3.41). Undersubject of a symmetry operation of the medium, constituted by any proper or improperrotation, the invariance of the tensor elements of this magneto-optical susceptibility re-

quires


ab1b2��bn= [det (R)](n�m)

Ra�Rb1�1Rb2�2

� � �Rbm�m� � �Rbn�n


��1�2��n;

that is to say if the inversion operation, described by Rab = ��ab, is a symmetry operationof the medium, then the tensor elements of the susceptibility must obey


ab1b2��bn= (�1)(n�m)(�1)(n+1)�(eee��em��m)

ab1b2��bn

= (�1)(m+1)�(eee��em��m)

ab1b2��bn:

This implies that the elements of any magneto-optical susceptibility tensor governing aninteraction involving an even number of electric �elds must vanish identically in a medium

possessing inversion symmetry, irregardless of the number of magnetic �elds involved inthe process. Thus, just as in the all-optical case, any magnetic �eld-induced contributionsto processes such as second harmonic generation, parametric ampli�cation, and highereven-ordered harmonic generations are prohibited in media with a center of inversion.

Is should be emphasized that even though the second order all-optical susceptibilitytensor �(eee) = �

(eee)(�!�;!1; !2) vanishes in media possessing inversion symmetry, aspreviously shown, the elements of the second order magneto-optical susceptibility tensor

�(eem) = �

(eem)(�!�;!1; !2) are nonzero in the same medium.

3.3.4 Isotropic media revisited

The spring model analysed in Chapter 2 clearly constitutes a model for magneto-opticalinteractions in an isotropic geometry, in which any proper rotation of the laboratory

frame leaves the obtained properties such as electric dipole moment and a polarizationdensity of an ensemble of spring oscillators invariant. In particular, the equivalents tothe micropolarizabilities in the spring model, given by Eqs. (2.7), were found to possess

a minimum set of nonzero tensor elements. This set of nonzero tensor elements is alsoshared by the susceptibilities (2.10) which would describe the polarization density of anensemble of identical spring oscillators.

Applying Neumann's principle, the symmetry properties of the quantum-mechanically

derived optical and magneto-optical susceptibility tensors will now be addressed for iso-tropic media. For isotropic media there are, per de�nition, an in�nite number of symmetryoperations. This implies that one, at least in principle, may consider the system of equa-

tions for the tensor elements that would arise from an arbitrary rotation, symbolicallydescribed by for example the Euler angles (�; �; ), and solve the system for nontrivialsolutions under the requirement that the solutions should be independent of (�; �; ).

However elegant this approach may seem, for practical reasons it is far more cumber-some, in an algebraic sense, than just considering a few properly chosen rotations. As willbe shown below, the elements of the second order magneto-optical susceptibility tensor


�(eem) are reduced to their minimum set of six nonzero elements, of which only one is in-

dependent, by considering invariance under just four di�erent rotations of the coordinatesystem.

In isotropic media, the �rst order all-optical rank-two susceptibility tensor is diagonal

with all elements equal; this is shown in numerous standard texts [55] in optics and willnot be a main issue in this discussion.

For simplicity, the discussion will start with analysis of the symmetries of the elements

of the (axial) second order magneto-optical susceptibility tensor �(eem) in isotropic media,as being crucial for the Faraday rotation. Reduction schemes for susceptibilities in non-linear optics within the electric dipole approximation are given in more detail by Butcherand Cotter [1], and by Popov et al. [66].

In the subsequent steps of reduction of the general form of the tensor, the analysiswould in MapleV start by loading the routines listed in Appendix C and initiating ageneral form of the 33 = 27 elements of �(eem), with an independent constant assigned to

each of the respective elements of the tensor, by executing

read `tensrot.m`;

rank:=3:

chi:=[seq(a[k],k=1..(3^rank))]:

disptens(chi,rank);

The �rst step in the reduction of �(eem) is to consider invariance of its tensor elementsunder rotation of the coordinate system by an angle2 � = � around the z-axis; this

constitutes a proper rotation described by the matrix

R =

0@�1 0 0

0 �1 0

0 0 1

1A : (3.44)

By either performing the calculations by hand or using the enclosed MapleV routines,direct inspection of the overdetermined system of equations obtained from Eq. (3.43),stating invariance of the tensor elements under this particular rotation, requires the fol-

lowing tensor elements to be zero:

�(eem)xxx = �

(eem)yyy = �

(eem)xyy = �

(eem)yxy = �

(eem)yyx = �

(eem)xzz = �

(eem)zxz = �

(eem)zzx

= �(eem)yxx = �

(eem)xyx = �

(eem)xxy = �

(eem)yzz = �

(eem)zyz = �

(eem)zzy = 0;

and hence the number of nonzero elements is reduced to (27 � 14) = 13 elements. Using

the MapleV routines enclosed enclosed in Appendix C, this partial reduction of tensorelements is readily veri�ed by executing the MapleV code sequences

R:=rotmatrix(Pi,0,0);

newchi:=tensrot(chi,R,rank):

sys:={seq((chi[k]=newchi[k]),k=1..(3^rank))}:

vars:={seq(a[k],k=1..(3^rank))}:

solve(sys,vars);

2In this section all angles of rotation fully conform to Goldstein's [58] convention of Euler angles asused in classical mechanics.


Next, the invariance of the elements of the susceptibility tensor under rotation of the

coordinate system by an angle � = � around the x-axis is considered; this constitutes aproper rotation described by the matrix

R =

0@1 0 00 �1 0

0 0 �1

1A ; (3.45)

from which the following elements are required to vanish, in addition to the previouslyeliminated ones:

�(eem)zzz = �

(eem)zxx = �

(eem)xzx = �

(eem)xxz = �

(eem)zyy = �

(eem)yzy = �

(eem)yyz = 0;

hence leaving (13� 7) = 6 nonzero elements of �(eem), as can be veri�ed by executing thesequences

R:=rotmatrix(0,Pi,0);

newchi:=tensrot(chi,R,rank):

sys:={seq((chi[k]=newchi[k]),k=1..(3^rank))}:

vars:={seq(a[k],k=1..(3^rank))}:

solve(sys,vars);

The invariance of the elements of the susceptibility tensor under rotation of the coordinate

system by an angle � = �=2 around the z-axis, a proper rotation described by the matrix

R =

0@ 0 1 0�1 0 0

0 0 1

1A ; (3.46)

imposes the three restrictions

�(eem)xyz = ��(eem)yxz ; �

(eem)zxy = ��(eem)zyx ; �

(eem)yzx = ��(eem)xzy ;

on the remaining elements, hence reducing the number of independent elements to three.Finally, invariance of the elements of �(eem) under rotation of the coordinate system byan angle � = �=2 around the x-axis, a proper rotation being described by the matrix

R =

0@1 0 00 0 10 �1 0

1A ; (3.47)

imposes the restrictions

�(eem)yzx = ��(eem)zyx ; �

(eem)zxy = ��(eem)yxz ;

hence leaving only one independent element.

To conclude, there are six remaining nonzero elements of �(eem), of which only one isindependent:

�(eem)xyz = �

(eem)yzx = �

(eem)zxy = ��(eem)xzy = ��(eem)yxz = ��(eem)zyx : (3.48)

In order to generally verify that no further reductions of this tensor are possible, the

following MapleV code sequences symbolically simplify the tensor that results from anarbitrary proper rotation, decribed by arbitrary Euler angles (�; �; ), of the tensor givenby Eq. (3.48),


chi:=linalg[vector](3^rank):

for k from 1 to 3^rank do chi[k]:=0; od:

chi[indx([1,2,3])]:=c:

chi[indx([2,3,1])]:=c:

chi[indx([3,1,2])]:=c:

chi[indx([1,3,2])]:=-c:

chi[indx([2,1,3])]:=-c:

chi[indx([3,2,1])]:=-c:

disptens(chi,rank);

R:=rotmatrix('phi','theta','psi');

newchi := tensrot(chi,R,rank):

disptens(newchi,rank);

and as can be readily veri�ed, the obtained form (3.48) of the elements of the second-ordermagneto-optic susceptibility tensor are left invariant under arbitrary proper rotations,

stating the isotropy of the medium. The argument of non-reducibility of �(eem) is thenclosed by observing that the inversion operation Rab = ��ab when inserted into Eq. (3.43)

just leaves the identity �(eem)

abc= �

(eem)

abcfor any values of a; b; c.

Notice that the obtained set of nonzero elements of �(eem) is valid for any frequencyarguments, hence the involved magnetic �eld not necessarily has to be a static one; the

susceptibilities might just as well be applied to analysis of the self-interaction of the electric�eld of the optical wave with its own, rapidly oscillating magnetic �eld, however weak.However, for such an interaction, the terminology of linear optics is no longer appropriate,

since the magnetic �eld of the light is directly related to the electric �eld by Maxwell'sequations, hence this interaction should rather be considered as belonging to nonlinearoptics.

The reduction of tensor elements of the next higher order magneto-optical susceptibility

tensor �(eemm) can be analysed in a similar manner by direct inspection. However, usingthat �(eemm) is a polar rank-four tensor, the reduction scheme follows identically to thethird-order all-optical susceptibility tensor �(eeee); for isotropic media its 21 nonzero tensor

elements, of which only three elements are independent, are listed by Butcher and Cotterin Appendix 3 of Ref. [1]. The tensors �(eeee) and �

(eemm) will hence generally sharethe same set of nonzero elements, for arbitrary frequencies of the involved electric and

magnetic �eld, within the validity of the susceptibility formalism.In the case of an interaction where a static magnetic �eld is considered as being the

dominant contributor to magneto-optical e�ects, the frequency arguments of �(eemm) are

taken as �(eemm)(�!;!; 0; 0), and the tensor hence describes a linear optical interaction.Observing that the elements of this tensor, by using intrinsic permutation symmetry inthe last two indices, must satisfy

�(eemm)

�� (�!;!; 0; 0) = �

(eemm)

�� (�!;!; 0; 0); (3.49)

an additional restriction is put on its set of nonzero elements, resulting in 21 nonzeroelements of which only two are independent.

To summarize, the set of nonzero tensor elements of the optical and magneto-opticalsusceptibilities, describing interactions which are linear in the electric �eld, are given as

�(ee)xx = �

(ee)yy = �

(ee)zz (3.50a)

�(eem)xyz = �

(eem)zxy = �

(eem)yzx = ��(eem)zyx = ��(eem)xzy = ��(eem)yxz (3.50b)


�(eemm)xxxx = �

(eemm)yyyy = �

(eemm)zzzz = �

(eemm)xxyy + �

(eemm)xyxy + �

(eemm)xyyx (3.50c)

�(eemm)xxyy = �

(eemm)yyxx = �

(eemm)yyzz = �

(eemm)zzyy = �

(eemm)zzxx = �

(eemm)xxzz (3.50d)

�(eemm)xyxy = �

(eemm)yxyx = �

(eemm)zxzx = �

(eemm)xzxz = �

(eemm)yzyz = �

(eemm)zyzy

= �(eemm)xyyx = �

(eemm)yxxy = �

(eemm)zxxz = �

(eemm)xzzx = �

(eemm)yzzy = �

(eemm)zyyz : (3.50e)

As can be seem by comparing these sets of nonzero tensor components with the onesobtained for the micropolarizabilities in the all-classical oscillator model in Chapter 2,Eqs. (2.7) and (2.10), the sets are identical.

For isotropic media, the method of direct inspection may be somewhat more cumber-some than other techniques such as, for example, the technique of isotropic averaging, asdescribed by Wagni�ere [67, 68]. However, in non-isotropic media the method of direct

inspection is often straightforward to apply whenever just a few sets of crystallographicpoint symmetry classes are to be analysed.

3.3.5 Anisotropic media

Returning to the general forms of the susceptibilities for media belonging to arbitrarypoint symmetry group, the issue of magneto-optical interactions in anisotropic media willnow be addressed.

Considering analysis of wave propagation in an arbitrary direction of the crystal frame,the transformation connecting the laboratory and crystal coordinate systems is alwaysdescribed by a proper rotation for which det (R) = 1, as illustrated in Fig. 3.2. The

transformation law for any proper rotation of the n:th order magneto-optical susceptibilitytensor as described by Eq. (3.30) hence yields


ab1b2��bn

0 = Ra�Rb1�1Rb2�2 � � �Rbm�m� � �Rbn�n


��1�2��n: (3.51)

For any given set of nonzero elements of the susceptibility tensor in a reference frame(eX ; eY ; eZ) of the medium, the corresponding susceptibility elements in the rotated lab-

oratory frame (ex; ey; ez) are then easily calculated by means of a straightforward matrix-tensor multiplication. The transformation law (3.51) is identical in form to the one fortransformation of the all-optical n:th order susceptibility in the electric-dipole approxi-

mation [1], and can conveniently be used for analysis of wave propagation in anisotropicmagneto-optic media with arbitrary direction of propagation relative the coordinate axesof the reference frame of the medium.

Kielich and Zawodny [2] have applied group-theoretical methods for the calculationsof nonzero and independent elements of the magneto-optical susceptibility tensors for allcrystallographic point symmetry classes, up to and including fourth order magneto-optical

interactions being linear in the magnetic �eld. Their results are tabulated in Appendix Bof this thesis.

The routines for calculation of arbitrary-rank tensor components in rotated coordinate

frames, enclosed in Appendix C, have been used in the calculations of optical and magneto-optical susceptibilities for nonlinear wave propagation in the (111)-direction of a crystalof point symmetry �43m, as used in Papers III{V of this thesis, and for veri�cation of

the results by Kielich and Zawodny [2] for crystallographic point-symmetry groups beingrelevant to Papers I{V.

Chapter 4

Wave propagation in magneto-optic media

\When a ray of circularly-polarized light falls on a medium under the action

of magnetic force, its propagation within the medium is a�ected by the

relation of the direction of rotation of the light to the direction of the

magnetic force."

{ J. C. MAXWELL, Electricity and Magnetism (1891)

The constitutive relations as developed in Chapter 3 are inserted into Maxwell's

equations, and nonlinear wave propagation in magneto-optic media is dis-

cussed. The representation of the polarization state of light in terms of Stokes

parameters is reviewed, and the concept of the Poincar�e sphere is introduced.

4.1 On the separation of electric and magnetic polarization

In a vast majority of all studies in optics it is assumed that the medium is non-magnetized,that is to say it possesses zero magnetic dipole moment per unit volume. However, inmagneto-optics this assumption is often violated since the presence of a static magnetic

�eld eventually will cause the magnetic dipole moments of the molecules to line up alongthe �eld, hence magnetizing the medium. The source of the magnetization does notnecessarily have to be due to an externally applied magnetic �eld; it might just as well

originate from inverse Faraday e�ect whenever circularly polarized optical �elds are used.

The �rst aim in this chapter is therefore to formulate the wave-equation for electro-magnetic �elds in magnetized media, with starting point in Maxwell's equations and avery general form of the constitutive relations. Another item to clarify is the separation

of electric and magnetic polarization densities in magnetized media, as being a potentialsource of confusion1.

The starting point is the real and macroscopic form of Maxwell's equations [60], in SI-

units and in di�erential form, expressed solely in terms of the electromagnetic quantitiesE(r; t), B(r; t), D(r; t) and H(r; t):

r�E(r; t) = �@B(r; t)@t

; (Faraday's law) (4.1a)

r�H(r; t) = J(r; t) +@D(r; t)

@t; (Amp�ere's law) (4.1b)

r �B(r; t) = 0; (4.1c)

r �D(r; t) = �(r; t); (4.1d)

1To cite Boardman and Xie [30]: \The Russian literature makes it abundantly clear that material inwhich both polarization and magnetization are present presents a potential conceptual problem becausepolarization and magnetization are not uniquely separable."

39

40 Chapter 4. Wave propagation in magneto-optic media

where �(r; t) is the density of free charges, and J(r; t) the corresponding current density

of free charges. The constitutive relations are assumed to obey

D(r; t) = "0E(r; t) +P(r; t); (4.2a)

B(r; t) = �0[H(r; t) +M(r; t)]; (4.2b)

where P(r; t) = P[E(r; t);B(r; t)] is the macroscopic polarization density (electric dipolemoment per unit volume), and M(r; t) =M[E(r; t);B(r; t)] the magnetization (magneticdipole moment per unit volume) of the medium.

Here E(r; t) and B(r; t) are considered as the fundamental macroscopic electric andmagnetic �eld quantities; D(r; t) and H(r; t) are the corresponding derived �elds asso-

ciated with the state of matter, connected to E(r; t) and B(r; t) through the electricpolarization density P(r; t) and magnetization (magnetic polarization density) M(r; t)through the basic constitutive relations. In fact, Eqs. (4.2) form the very de�nitions [69]

of the polarization density and magnetization.

In Eqs. (4.1a) and (4.1b), the electric displacement D(r; t) and magnetic �eld intensityH(r; t) are now to be eliminated in favour of E(r; t), B(r; t), P(r; t) and M(r; t). Taking

the curl of Eqs. (4.1a) and (4.1b) after some straightforward rearrangement of the termsgives

r� [r�E(r; t)] +1

c2

@2E(r; t)

@t2= ��0 @

@t

�@P(r; t)

@t+r�M(r; t) + J(r; t)

�; (4.3a)

r� [r�B(r; t)] +1

c2

@2B(r; t)

@t2= �0r�

�@P(r; t)

@t+r�M(r; t) + J(r; t)

�: (4.3b)

Notice that in the steps leading to Eqs. (4.3), only Faraday's law (4.1a) and Amp�ere's

law (4.1b) were required, and hence the density of free charges �(r; t) never enters theequations explicitly. However, the density of free charges is always implicitly present interms of the free current density through the charge conservation law

r � J(r; t) + @�(r; t)

@t= 0;

which follows from Eqs. (4.1b) and (4.1d), using the vector identity r � (r� a) = 0.

Since the structure of interiors of the brackets in the right-hand sides of Eqs. (4.3a)

and (4.3b) are equal, the explicit appearance of, for example, the magnetization M(r,t)may be eliminated by incorporating the magnetization as an e�ective current density, as

Je�(r; t) = J(r; t) +r�M(r; t);

where the last term acts as a current density due to the magnetization, in addition tothe current density of free charges. In this case the constitutive relation for the magnetic

polarization density would be implicitly incorporated into the constitutive relations for thefree current density. The equivalence of a magnetized body and a distribution of currentdensity has been pointed out by Stratton [69].

Moreover, the bracketed source terms in the wave equations for the electric as well asmagnetic �elds have the same intrinsic form of \time derivative of electric polarization

4.2. The nonlinear wave equation in magneto-optic media 41

density plus rotation of magnetization", from which it follows that Eqs. (4.3) are left

invariant under the transformation

P(r; t)! P0(r; t) = P(r; t) +r� F(r; t);

M(r; t)!M0(r; t) =M(r; t)� @F(r; t)

@t;

(4.4)

where F(r; t) denotes any arbitrary di�erentiable vector function of time and space coordi-

nates. Thus, to a certain degree, an amount of arbitrariness is implied in the appearancesof the polarization density and magnetization as being source terms in the wave equationsgoverning the propagation of electric and magnetic �elds. This is not to be confused with

any ambiguity of the de�nitions of the electric and magnetic polarization densities; fromtheir de�nitions in Eqs. (4.2), they are perfectly �xed from a de�nitions point-of-view.

Clearly the transformation (4.4) leaves Eqs. (4.3) invariant, as well as the equations

they are derived from, i. e. Faraday's law (4.1a) and Amp�ere's law (4.1b). In order toverify that the full set of Maxwell's equations (4.1) also are left invariant under the trans-formation (4.4), notice that Eq. (4.1c) trivially is left invariant, and that Eq. (4.1d), by

using the vector identity r � [r � F(r; t)] = 0, is left invariant under the transformationas well.

Considering, for example, the case of linear optical activity in isotropic non-magnetized

media, for which the constitutive relations for the electric and magnetic polarization den-sities yield [25]

P(r; t) = "0

��1E(r; t) + �2r� @E(r; t)

@t

�; M(r; t) = 0;

the transformation (4.4) implies that the polarization density above might just as well be

replaced with the equivalent set of electric and magnetic polarization densities

P0(r; t) = "0�1E(r; t); M

0(r; t) = "0�2@2E(r; t)

@t2;

without altering any behaviour of the electric and magnetic �elds, being solutions of (4.3),since the wave equations they are described by clearly are left invariant.

4.2 The nonlinear wave equation in magneto-optic media

With starting point in the general description of wavepropagation, as stated in the previ-ous section, the mechanisms of dispersion will now be introduced, and linear and nonlinear

terms in the polarization density will be considered in somewhat more detail. For con-venience in the subsequent analysis, the electric polarization density is divided into twoparts,

P(r; t) = PL(r; t) +PNL(r; t); (4.5)

where P(L)(r; t) contains all terms being linear in the electric �eld, and P(NL)(r; t) allnonlinear source terms.

In most practical situations of magneto-optics, the static magnetic �eld can be consid-ered as much greater in magnitude than the weak, rapidly oscillating magnetic �eld of theoptical wave. Neglecting the magneto-optical self-coupling to the weak magnetic �eld of


the light wave, any occurence of B�(r; !) in the expressions for the polarization densities

will be replaced by its component at the center frequency at ! = 0, as

B�(r; !) = B�

0 �(!); (4.6)

where �(!) denotes the Dirac delta function, and where B�

0 = e� �B0 denote the Cartesiancomponents of the static magnetic �eld. Within this approximation, any susceptibilitycontaining two occurences of `e' in the superscript, in the notation as used in this thesis,

will describe a linear optical process. In addition, it will from now on be assumed thatthe source terms in the wave equations (4.3) due to magnetization of the medium arenegligible compared to the ones from the electric polarization density, and it will also be

assumed that the free charge and current densities are zero.By separating the electric polarization density into a linear and nonlinear part ac-

cording to Eq. (4.5), using the constitutive relations (3.17), (3.21), and (3.25), the wave-

equation (4.3a) for the electric �eld strength of the optical wave becomes

r� [r�E(r; t)] =1

c2e�

1Z�1

!2"��(!)E�(r; !) exp (�i!t) d! � �0

@2

@t2P(NL)(r; t); (4.7)

where the elements "��(!) of the e�ective dielectric tensor, including �rst order optical

and second and third order magneto-optical interactions, are de�ned as

"��(!) = �� + �(ee)

��(�!;!) + �

(eem)

�� (�!;!; 0)B

0

+ �(eemm)

�� (�!;!; 0; 0)B

0B�

0 :

(4.8)

Above the condition "��(�!) = "��(!) ensures the reality condition of the middle term

of Eq. (4.7) to hold.

4.2.1 Quasimonochromatic light

For a quasimonochromatic optical wave, the envelope of the wave is in the temporal

spectrum narrow around each of the carrier frequencies !k of the wave. In time domainthe electric �eld and polarization density are taken as

E(r; t) =Xk

Re [E!k(r; t) exp(�i!kt)]; (4.9a)

P(r; t) =Xk

Re [P!k(r; t) exp(�i!kt)]; (4.9b)

for !k � 0, with complex time-dependent envelopes E!k(r; t) and P!k

(r; t). In the fre-

quency domain, the electric �eld and polarization density described by Eqs. (4.9) are givenas

E(r; !) =1

2

Xk

[E!k(r; ! � !k) +E

�!k(r;�! � !k)]; (4.10a)

P(r; !) =1

2

Xk

[P!k(r; ! � !k) +P

�!k(r;�! � !k)]; (4.10b)

where E!k(r; !) and P!k

(r; !) denote the temporal Fourier transforms of the respectivecomplex time-dependent envelopes E!k

(r; t) and P!k(r; t), by means of Eq. (3.5). The


carrier frequencies !k of the optical wave, as appearing in Eqs. (4.9) and (4.10), are not

to be confused with the frequencies !k used as variables of integration in the previouslyderived expressions for the susceptibilities in Sec. 3.2. In order to reduce this potentialrisk of confusion, any variables of integration in the frequency domain will from now on

be used in a primed notation.

For quasimonochromatic light waves, the spectrum described by the temporal envelopes

E!k(r; !�!k) is considered as being narrow enough around the respective carrier frequen-

cies !k is order to justify Taylor expansions of the dielectric tensor in their vicinity. In-serting the quasimonochromatic �eld as given by Eqs. (4.9) into Eq. (4.7), and performing

a Taylor expansion of "�� after some straightforward algebra gives the wave equation

r� [r�E!k(r; t)] = e�

1Xm=0

im

m!�(m)

��(!k)

@m

@tm[e� �E!k

(r; t)]

� �0 exp (i!kt)@2

@t2

hP(NL)!k

(r; t) exp (�i!kt)i;

(4.11)

where the notation

�(m)

��(!k) =

1

c2

@m[!2"��(!)]

@!m

��!=!k

was employed. The series expansion of the permittivity in Eq. (4.9) corresponds to the

general expression for the dispersion given by Diels [70], here including the mechanismsfrom linear magneto-optic source terms in the electric polarization density as well.

4.2.2 Monochromatic light

For strictly monochromatic light, the envelopes of the wave are time-independent, at leastin the time scale of optical oscillations in the spectrum, and so the optical �eld in timedomain becomes

E(r; t) =Xk

Re [E!kexp(�i!kt)]; (4.12)

and, hence, the wave described by (4.10) in the frequency domain simply reduces to

E(r; !) =1

2

Xk

[E!k�(! � !k) +E

�!k�(�! � !k)]: (4.13)

In Eqs. (4.12) and (4.13) explicit notation of the spatial dependence of the temporal

envelopes E!k= E!k

(r) was omitted. From the electric �eld of the monochromatic opticalwave, the electric polarization density it induces can be calculated by direct insertion intoEqs. (3.21), (3.25), and (3.27), to yield, respectively, the second, third, and fourth order

electric polarization densities due to nonlinear optical and magneto-optical interactions. Itshould be emphasized that within the assumption of no magneto-optical interactions dueto self-coupling to the magnetic �eld of the optical wave, the terms containing the second

and third order magneto-optical susceptibilities �(eem)(�!;!; 0) and �(eemm)(�!;!; 0; 0)must be excluded from the nonlinear polarization density, since they are included in thee�ective permittivity "��(!).


Considering the general n:th order magneto-optical interaction involving m electric

�elds and (n�m) static magnetic �elds, insertion of Eq. (4.6) into the expression for then:th order electric polarization density, given by Eq. (3.30), gives

P(n)� (r; t) = "0B

�m+1

0 B�m+2

0 � � �B�n

0

1Z�1

1Z�1

� � �1Z

�1


��1�2��m�m+1��n(�!�;!01; !02; : : : ; !0m; 0; : : : ; 0)E�1

(r; !01)E�2(r; !02) � � �E�m

(r; !0m)

� exp [�i(!01 + !02 + : : :+ !

0m)t] d!

01 d!

02 : : : d!

0m:

(4.14)

Proceeding with monochromatic light, substitution of the monochromatic electric �eld

given by Eq. (4.13) into Eq. (4.14) gives

P(n)� (r; t) = "0B

�m+1

0 B�m+2

0 � � �B�n

0

Xk1

Xk2

� � �Xkm�1

Xkm

1Z�1

1Z�1

� � �1Z

�1


��1�2��m�m+1��n(�!�;!01; !02; : : : ; !0m; 0; : : : ; 0)

� �E�1!k1

�(!01 � !k1)E�2!k2

�(!02 � !k2) � � �E�m�1!km�1

�(!0m�1 � !km�1)E�m!km

�(!0m � !km)

+E�1!k1

�(!01 � !k1)E�2!k2

�(!02 � !k2) � � �E�m�1!km�1

�(!0m�1 � !km�1)E�m�!km

�(�!0m � !km)

+E�1!k1

�(!01 � !k1)E�2!k2

�(!02 � !k2 ) � � �E�m�1�

!km�1�(�!0m�1 � !km�1)E

�m!km

�(!0m � !km)

+E�1!k1

�(!01 � !k1)E�2!k2

�(!02 � !k2 ) � � �E�m�1�

!km�1�(�!0m�1 � !km�1)E

�m�!km

�(�!0m � !km)

+E�1�!k1

�(�!01 � !k1)E�2�!k2

�(�!02 � !k2) � � �E�m�1�

!km�1�(�!0m�1 � !km�1)E

�m�!km

�(�!0m � !km)

� exp [�i(!01 + !02 + : : :+ !

0m�1 + !

0m)t] d!

01 d!

02 : : : d!

0m�1 d!

0m;

(4.15)

where, for simplicity, the notation E�

!k= e� � E!k

was introduced. By performing thestraightforward integration in Eq. (4.15), the real and nonlinear electric polarization den-sity of the medium is obtained as

P(n)� (r; t) = "0

1

2mB�m+1

0 B�m+2

0 � � �B�n

0

Xk1

Xk2

� � �Xkm�1

Xkm


��1�2��m�m+1��n(�!�;!k1 ; !k2 ; : : : ; !km�1 ; !km ; 0; : : : ; 0)

�E�1!k1

E�2!k2

� � �E�m�1!km�1

E�m!km

exp [�i(!k1 + !k2 + : : : + !km�1 + !km)t]

+ �(eee��em��m)

��1�2��m�m+1��n(�!�;!k1 ; !k2 ; : : : ; !km�1 ;�!km ; 0; : : : ; 0)

�E�1!k1

E�2!k2

� � �E�m�1!km�1

E�m�!km

exp [�i(!k1 + !k2 + : : : + !km�1 � !km)t]


��1�2��m�m+1��n(�!�;!k1 ; !k2 ; : : : ;�!km�1 ; !km ; 0; : : : ; 0)

�E�1!k1

E�2!k2

� � �E�m�1�

!km�1E�m!km

exp [�i(!k1 + !k2 + : : : � !km�1 + !km)t]


��1�2��m�m+1��n(�!�;!k1 ; !k2 ; : : : ;�!km�1 ;�!km ; 0; : : : ; 0)

�E�1!k1

E�2!k2

� � �E�m�1�

!km�1E�m�!km

exp [�i(!k1 + !k2 + : : : � !km�1 � !km)t]


+ : : :


��1�2��m�m+1��n(�!�;�!k1 ;�!k2 ; : : : ;�!km�1 ;�!km ; 0; : : : ; 0)

�E�1�!k1

E�2�!k2

� � �E�m�1�

!km�1E�m�!km

exp [�i(�!k1 � !k2 � : : :� !km�1 � !km)t];

(4.16)

where the summation, as previously pointed out, is performed over all positive carrierfrequencies !k of the electric �eld, and where !� denotes the sum of all angular frequenciesin each of the particular optical or magneto-optical processes, as occuring in the list of

frequency arguments after each semicolon in the respective susceptibility. Paying attentionto any of the particular nonlinear optical processes in Eq. (4.16), each of them governingthe generation of optical waves at an angular frequency !�, the wave equation will now

be considered in more detail for monochromatic waves.As the electric �eld of the monochromatic optical wave, in time domain given by

Eq. (4.12), is inserted into the wave equation (4.11), the simpli�ed form

r� [r�E!� ] = (!�=c)2e�"��(!�)E

�

!�+ �0!

2�P

(NL)!�

; (4.17)

is obtained. In Eq. (4.17) the envelope of the nonlinear polarization density is readily

calculated from its real form, Eq. (4.16), as any set of nonlinear source terms giving riseto a particular frequency !�.

Focusing on a particular nonlinear interaction, say, the �rst term appearing in the

summation in Eq. (4.16) with all frequency arguments positive, the complex temporalenvelope of the nonlinear polarization density, related to the real and nonlinear electricpolarization density of the medium through

P(NL)(r; t) =

X�

Re [P(NL)!�

exp(�i!�t)]; (4.18)

is given as

P(NL)!�

=

1Xn=2

P(n)!�

(4.19)

where the various-order contributions to the electric polarization density are given by

1

2P(n)!�

= "01

2m

Xk1

Xk2

� � �Xkm�1

Xkm

e��(eee��em��m)

��1�2��m�m+1��n(�!�;!k1 ; !k2 ; : : : ; !km�1 ; !km ; 0; 0; : : : ; 0)

�E�1!k1

E�2!k2

� � �E�m�1!km�1

E�m!km

B�m+1

0 B�m+2

0 � � �B�n

0 ;

(4.20)

which for the special cases m = n describe all-optical interactions. Any of the other

interactions listed in Eq. (4.16) can be considered in a form similar to Eq. (4.20) simplyby replacing any of the positive angular frequencies !k with �!k, while simultaneouslyreplacing the corresponding electric �eld component E�

!kwith its complex conjugate E��

!k.

The factor of one half in the left-hand side of Eq. (4.20) only appears for a nonzero !�; for!� = 0, i. e. for any optically or magneto-optically induced DC contribution, this factor isreplaced by unity.


As the summation in Eq. (4.20) is performed, degenerate terms will appear multiple

times; in order to eliminate this behaviour, the polarization density for any combinationof the angular frequencies of the electric �eld can be taken as

P(n)!�

= "0K(!1; !2; : : : ; !m�1; !m)

e��(eee��em��m)

��1�2��m�m+1��n(�!�;!1; !2; : : : ; !m�1; !m; 0; 0; : : : ; 0)

�E�1!1E�2!2� � �E�m�1

!m�1E�m!mB�m+1

0 B�m+2

0 � � �B�n

0 ;

(4.21)

where, as previously, summation over repeated indices is implied, but where now onlysummation over distinct sets of frequencies, satisfying !� = !1+!2+ : : :+!m, is implied.In Eq. (4.21), the degeneration constant K = K(!1; !2; : : : ; !m�1; !m) is de�ned as K =

2l+s�mp, where p is the number of distinct permutations of the frequencies of the electric�eld components, m is the number of electric �eld components involved, s is the numberof the electric �eld components which appear at zero angular frequency, and l = 1 if

!� = 0, otherwise l = 0. The degeneration constant K is, due to the previous assumptionof the static magnetic �eld being dominant in any of the magneto-optical interactions,identical to the one in the electric dipole approximation, as conforming to the conventions

of Butcher and Cotter [1].

In fact, the assumption of the static magnetic �eld being dominant in the magneto-optical interactions is not essential for obtaining a form of the electric polarization densitysimilar to Eq. (4.21). Since the convention is to sum over all distinct magneto-optical

terms with all magnetic �eld components listed after the electric �eld components, inthe way the intrinsic permutation symmetries of the magneto-optical susceptibilities wereconstructed in Secs. 3.2, it can be seen that by allowing arbitrary frequency components

of a monochromatic magnetic �eld to enter the analysis leading to the expression (4.21),the impact would just be that the frequency arguments of the magnetic �elds would enterin Eq. (4.21), and that an additional degeneracy constant for the magnetic �elds would

appear as multiplied with K above, including the number of distinct permutations of thefrequencies of the magnetic �eld components, the number of magnetic �eld componentsinvolved, and the number of the magnetic �eld components appearing at zero angular

frequency; however excluding the term l which appears in K.

4.2.3 Monochromatic light in isotropic magneto-optic media

As an example of the wave equation (4.17) applied to a particular medium, the specialcase of an isotropic magneto-optic dielectric will now be considered. Taking onto accountthe speci�c form of the sets of nonzero elements of the magneto-optical susceptibilities, as

given by Eqs. (3.50) for isotropic media, the wave equation (4.17) for the electric �eld atthe center frequency ! becomes

r� [r�E!]� (n!=c)2E! = (!=c)2�(eem)xyz E! �B0 + (!=c)2�(eemm)xxyy (B0 �B0)E!

+ (!=c)2[�(eemm)xyxy + �(eemm)xyyx ](E! �B0)B0 + �0!

2P(NL)! ;

(4.22)

where n2 = 1 + �(ee)xx (�!;!) is the square of the linear refractive index resulting from

electric dipolar interactions. In the right-hand side of Eq. (4.22), the �rst, second, andthird term describe the Faraday e�ect, magnetic �eld induced contribution to the non-birefringent part of the refractive index, and the Cotton-Mouton e�ect, respectively.

4.3. Representation of the electromagnetic �eld 47

Depending on the geometry of the particular setup for wave propagation, usually only

one of the linear magneto-optic terms in the right-hand side of Eq. (4.22) will be takeninto account. For plane waves propagating in the geometry of the Faraday con�guration,as shown in Fig. 2.2, E! �B0 will be zero and the �rst term in the right-hand side will be

the dominant one.On the other hand, for plane waves propagating in the geometry of the Voigt con-

�guration, as shown in Fig. 2.3, E! � B0 will always be directed along the direction of

propagation, unless being zero as in the case when the static magnetic �eld is applied inparallel to the direction of polarization of a linearly polarized optical wave. The transverseoptical wave will, hence, in the Voigt con�guration be una�ected by the �rst term of the

right-hand side of Eq. (4.22), at least at a �rst level of approximation, and the third termwill instead dominate over the magneto-optical e�ects. In either case, the second termwill in principle always be present; however, this term is small compared to the others,

and can for most purposes be safely neglected in the analysis.

4.3 Representation of the electromagnetic �eld

Having stated the equations governing general nonlinear wave propagation in magneto-optic media, the representation of the polarization state of the light will now be reviewed.

While Cartesian basis vectors often are appropriately applied to problems in optics inwhich birefringence for linearly polarized light is present, magneto-optical phenomena inthe Faraday con�guration are often most conveniently described in a circularly polarized

basis. The reason is that in the presence of the arti�cially induced gyrotropy, a descrip-tion in a circularly polarized basis often enables separation of the di�erential equationsgoverning propagation of the orthogonal circularly polarized �eld-components, as being

eigenpolarizations [53].

4.3.1 Circularly polarized basis vectors

The polarization state of a plane monochromatic light wave, propagating along the z-axisof a Cartesian coordinate system, can be described by introducing the circularly polarizedbasis vectors, being de�ned as

e+ = (ex + iey)=p2; e� = (ex � iey)=

p2;

and from their de�nition, the circularly polarized basis vectors possess the properties

e�� e� = 1; e

�� e� = 0; e� � e� = �iez; e� � ez = �ie�:

In order to illustrate the application of this complex basis to optics, consider two plane

and monochromatic optical waves, counter-propagating along the z-axis of a Cartesiancoordinate system. The envelope of the electric �eld is separated into two terms, one forthe forward propagating wave and one for the backward propagating one, as

E! = Ef

! +Eb

!:

Each of the counterpropagating waves is then separated into left and right circularly

polarized components according to

Ef

! = (e+Ef

+ + e�Ef

�) exp (in!z=c);

Eb

! = (e�+Eb

+ + e��E

b

�) exp (�in!z=c);


and by insertion of the �eld envelope described above into the expression for the real,

physical electric �eld E(r; t) = Re [E! exp(�i!t)], one �nds that when looking into theoncoming wave propagating in the positive z-direction, the vector of the electric �eld ofthe wave describes for the term with magnitude jEf

+j counterclockwise motion, while it

for the term with magnitude jEf

�j describes clockwise motion. In the terminlogy of optics,

the component described by Ef

+ is denoted as being left circularly polarized, while the

component described by Ef

� is denoted as being right circularly polarized. This conventionconforms to the classical one as used in optics [60, 55].

Similarly, by looking into the oncoming wave propagating in the negative z-direction,the vector of the electric �eld of the wave describes for the term with magnitude jEb

+jcounterclockwise motion, while it for the term with magnitude jEb

�j describes clockwisemotion. Hence Eb

+ is the left circularly polarized component of the backward propagating

�eld, while Eb

� is the corresponding right circularly polarized component.

In general, the envelope components Ef

� and Eb

� will all be dependent of the prop-agation coordinate z, especially whenever absorption, optical birefringence, or nonlinear

interaction between the components comes into play. At any instant, the relative mag-nitude between the envelopes of the circularly polarized �eld components determines theellipticity of the polarization state of the beam; for Ef

+ 6= Ef

� = 0, the forward propagating

wave is completely left circularly polarized, while it for Ef

� 6= Ef

+ = 0 is completely right

circularly polarized. Whenever the magnitudes of the envelopes are equal, Ef

� = Ef

+ 6= 0,the wave is linearly polarized, with the direction of polarization determined by the relative

phase between the envelopes. Analogous arguments hold for the backward propagatingwave.

As a special case, consider the wave equation (4.22) for wave propagation in an isotropicmagneto-optic medium in the Faraday con�guration; in the in�nite plane-wave limit,

E! = E!(z), this equation yields

@2E!

@z2+ (n!=c)2E! + (!=c)2�(eem)xyz E! �B0 = ��0!2P(NL)

! ; (4.23)

where linear magneto-optic terms being quadratic in the static magnetic �eld have been

neglected, and where the static magnetic �eld is directed along the z-axis, B0 = B0ez. Fora forward propagating optical wave, the spatial dependence of the �eld envelope is takenaccording to the ansatz E! = [e+E

f

+(z) + e�Ef

�(z)] exp(in!z=c) which, when inserted

into Eq. (4.23) gives

@Ef

�

@z= �i !

2ncEf

� + i!

2"0nc(e�� P(NL)

! ) exp (�in!z=c); (4.24)

where the gyration constant = i�(eem)xyz B0 was de�ned, and where the slowly-varying

envelope approximation,

��@2E

f

�

@z2

��2n!c

@Ef

�

@z

��;was used. In order not to enter the abyss of phenomena potentially described by thenonlinear polarization density, this will be neglected for the sake of simplicity, and as a


result, Eq. (4.24) is immediately integrated to give the solution for the temporal envelope

of the forward propagating wave as

E! = e+Ef

+(0) exphi!

c

�n+

2n

�z

i+ e�E

f

�(0) exphi!

c

�n�

2n

�z

i: (4.25)

From Eq. (4.25) it follows that the circularly polarized components of the optical waveexperience di�erent e�ective refractive indices, with n+ = n + =(2n) for left circular

polarization, and n� = n � =(2n) for right circular polarization. Since this implies acontinuous change of the relative phase between the circularly polarized components, thepolarization ellipse of the light will rotate as the light propagates through the medium;

this illustrates the Faraday e�ect, as described in previous chapters.

Analysis of a backward propagating wave proceeds in a similar manner. Insertion ofthe ansatz E! = [e�+E

b

+(z) + e��Eb

�(z)] exp(�in!z=c) for a backward propagating �eldinto Eq. (4.23) gives

@Eb

�

@z= �i !

2ncEf

� � i!

2"0nc(e� �P(NL)

! ) exp (in!z=c); (4.26)

where the slowly-varying envelope approximation for the backward propagating �eld,

��@2Eb

�

@z2

��2n!c

@Eb

�

@z

��;

was used as well. When the nonlinear polarization density is neglected, Eq. (4.26) isintegrated to give the solution for the temporal envelope of the backward propagating

wave as

E! = e�+E

b

+(0) exph� i

!

c

�n�

2n

�z

i+ e

��E

b

�(0) exph� i

!

c

�n+

2n

�z

i: (4.27)

In Eq. (4.27) it is seen that the roles of the e�ective refractive indices have changed betweenthe right and left circularly polarized components for the backward propagating �eld, ascompared to the forward propagating one given by Eq. (4.25); now the left circularly

polarized component experiences n� as the refractive index, while the right circularlypolarized component experiences n+. This will cause the polarization ellipse to rotate inthe opposite sense with respect to the direction of propagation, compared to the rotation

of the polarization ellipse of the forward propagating wave. In contrary to the e�ect ofoptical activity, for which the polarization ellipse of a re ected ray of light will recover itsorientation on the way back, the magneto-optical e�ect will hence double the rotation of

the polarization ellipse of a re ected wave.

4.3.2 Stokes parameters

In order to describe the evolution of the intensity and polarization state of an optical

wave, it often turns out to be useful to apply a description in terms of the Stokes param-eters, which de�ne a transformation of the complex electric �eld into a three-dimensionalspace, in which the polarization state of the light is more easily interpreted. The Stokes


S1

S2

S3

x

y(LCP)

x

y(RCP)

x

y(lin: pol:)

x

y(lin: pol:)

x

y(ellip: pol:)

x

y(ellip: pol:)

Figure 4.1. The Poincar�e sphere, on which the Stokes vector S = (S1; S2; S3)provides a representation of the polarization state of the light. The auxiliary�gures show the paths traversed by the end point of the electric �eld vectoras the observer looks into the oncoming wave, propagating in the positive z-direction. At the upper pole of the sphere the light is left circularly polarized(LCP), while it at the lower pole is right circularly polarized (RCP), withthe latitude there in between determining the ellipticity of the polarizationstate. At the equator, linearly polarized states of the light are represented,and their orientations are determined by the meridian.

parameters [60] are for monochromatic light, propagating in the positive z-direction of aCartesian coordinate system, de�ned as

S0 = jex � E!j2 + jey �E!j2; S2 = 2Re[(ex � E!)�(ey � E!)];

S1 = jex � E!j2 � jey �E!j2; S3 = 2Im[(ex � E!)�(ey � E!)];

(4.28)

where S0 is a measure of the intensity of the wave, and S3 a measure of the ellipticity ofthe polarization state. The parameters S1 and S2 indicate the amount of power containedin the x- or y-directions and serve as to determine the orientation of the polarization

ellipse. An equivalent form of the Stokes parameters in a circularly polarized basis isreadily derived from Eqs. (4.28) as

S0 = je�+ �E!j2 + je�� E!j2; S1 = 2Re[(e�+ � E!)�(e�� E!)];

S3 = je�+ �E!j2 � je�� E!j2; S2 = 2Im[(e�+ � E!)�(e�� E!)]:

(4.29)

The four Stokes parameters are not all independent since they, for completely polarizedlight, obey the relation

S20 = S

21 + S

22 + S

23 ;

and, hence, for lossless wave propagation where S0 is constant, the evolution of the vectorformed by (S1; S2; S3) can be pictured as a trajectory in the Stokes parameter space,


on a sphere with radius S0. This sphere, as shown in Fig. 4.1, is classically denoted

as the Poincar�e sphere2, and it considerably simpli�es the interpretation of the Stokesparameters. For wave propagation in lossy media, one may instead consider the vectorformed by (S1=S0; S2=S0; S3=S0), which provides a similar interpretation of the evolution

of polarization state, on a sphere with unit radius, the so-called unitary Poincar�e sphere.As an example, consider the solution given by Eq. (4.25) for the forward propagating

wave in lossless linear magneto-optical media. For lossless media n and will be real, and

in terms of the Stokes parameters, the solution is expressed as

S0 = jEf

+(0)j2 + jEf

�(0)j2;S1 = jEf

+(0)Ef

�(0)j2 cos ['0 � ! z=(nc)];

S2 = jEf

+(0)Ef

�(0)j2 sin ['0 � ! z=(nc)];

S3 = jEf

+(0)j2 � jEf

�(0)j2:

In the Stokes parameter space, the reduced Stokes vector (S1; S2; S3) will hence traversecircular paths on the Poincar�e sphere as the light propagates. These circular paths are

parallel to the equatorial plane, and their latitude is determined by the value of theellipticity of the polarization state, S3=S0.

2After Henri Poincar�e, who introduced it in 1892 [71].

52

Chapter 5

Discussion of original research work

5.1 Paper I

Polarization state controlled multistability of a nonlinear magneto-optic cavity

Phys. Rev. Lett. 82, 1426 (1999)

A magneto-optic Fabry-P�erot interferometer, consisting of two plane-parallel partially

re ecting mirrors enclosing a nonlinear magneto-optic medium, is analysed in the Faradaycon�guration, with an externally static magnetic �eld applied collinearly with the directionof propagation of the light.

z

z = 0 z = L

EI�

EI+

ER�

ER+

ET�

ET+

H0

�(ee)�� , �

(eeee)�� ,

�(eem)�� , �

(eeeem)��

[isotropic]

M0 M1

Figure 5.1. The setup of the nonlinear magneto-optic Fabry-P�erot inter-ferometer in the Faraday con�guration, as studied in Paper I. The optical

susceptibility tensor �(eeee)(�!�;!; !;�!) describes the optical Kerr-e�ectgoverning all-optical ellipse rotation and the main contributions to intensity-dependent parts of the refractive index of the medium, while its magneto-

optical modi�cation �(eeeem)(�!�;!; !;�!; 0) describes phenomena such asthe photoinduced Faraday rotation.

The constitutive relations of the medium are taken as those for isotropic media pos-sessing optical Kerr-e�ect, and source terms in the polarization density governing linearas well as photoinduced Faraday rotation are taken into account in the analysis.

53

54 Chapter 5. Discussion of original research work

Within the slowly-varying envelope approximation and in�nite plane-wave limit, ana-

lytical solutions for arbitrary polarization state of the input light are presented in termsof Stokes parameters. In order to describe the state of tuning of the cavity, di�erentialdetuning angles are introduced, describing the relative intracavity roundtrip phaseshifts

between orthogonal circularly polarized components of the optical �eld.

For the input light being circularly polarized, exact solutions for the transmission ofthe cavity are obtained in terms of Jacobian elliptic functions, without need of applying

the slowly-varying envelope approximation.

One of the characteristics of magneto-optic media is the nonreciprocity of the mediumunder exposure of a static magnetic �eld; in this paper it is shown that the nonreciprocity

has the e�ective impact on the transmission of the cavity as a detuning being assym-metrical in the ellipticity of the polarization state of the input light beam. Neglectingthe photoinduced Faraday rotation of the polarization ellipse of the intracavity light, the

cavity for certain values of the di�erential detuning angle recovers the state where thetransmitted intensity becomes symmetric in the input ellipticity for a nonzero magnetic�eld, e�ectively cancelling the impacts imposed from the nonreciprocity of the medium.By tuning the cavity with the static magnetic �eld in such a way that the e�ects of

the Faraday rotation being linear in the electric �eld are eliminated, this con�gurationprovides a setup for study of the photoinduced Faraday rotation exclusively.

Applications of the assymmetrical impact of the nonreciprocity on intensity transmis-

sion characteristics can be identi�ed for multistable operation. Analysis of the transmittedintensity, as function of the ellipticity of the polarization state of the input light, revealsthat the cavity can be tuned in such a way that hysteresis loops appear at constant input

intensity, in which case the ellipticity of the polarization state of the input light may beused as the controlling parameter rather than the input intensity, as classically employedin multistable optical devices.

The theory presented in Paper I opens for applications of polarization controlled switch-ing and optical logics, with tuning of the components possible by means of varying theexternally applied static magnetic �eld.

5.2 Paper II

Polarization state evolution and eigenpolarizations of nonlinear magneto-optics

Submitted to J. Opt. Soc. Am.B

An investigation of unidirectional wavepropagation in non-isotropic and nonlinear magneto-

optic media is performed. The constitutive relations incorporate �rst and third orderoptical interactions, the latter governing the optical Kerr-e�ect, and second and fourthorder magneto-optical interactions, governing linear and photoinduced Faraday rotation.

The analysis is restricted to a set of cubic, hexagonal, and trigonal crystallographicpoint symmetry groups, for which the medium is isotropic in the linear optical regime(�(ee) and �(eem) isotropic) but anisotropic in the nonlinear source terms of the polariza-

tion density (�(eeee) and �(eeeem) anisotropic). The chosen set of point symmetry groupsencompass the structures of the commonly used semimagnetic semiconductors [72], mostof them being of zinc blende or wurtzite crystal structure, possessing some of the highest

rotatory abilities known to the present date [37, 44, 73].

The wave equation is cast into a vectorized nonlinear equation for the Stokes param-eters, in a form similar to Euler's equations of motion for the angular momentum of a

5.3. Paper III 55

z

z = 0 z = L

EI�

EI+

ET�

ET+

H0

�(ee)�� , �

(eeee)�� ,

�(eem)�� , �

(eeeem)��

[�43m, m3m, �6m2, : : :]

Figure 5.2. The setup as studied in Paper II, where unidirectional waveprop-agation is analysed in media belonging to cubic, hexagonal, and trigonal pointsymmetry groups, possessing optical Kerr-e�ect and linear and photoinducedFaraday rotation.

rotating rigid body in classical mechanics. In lossy media this form enables a straightfor-

ward interpretation of the changes of intensity and polarization state of the propagatingwave as corresponding to radial and tangential changes of the reduced Stokes vector.

In the transparent region of the anisotropic medium, implicit solutions for the en-velopes of the propagating optical �elds are obtained in terms of elliptic integrals, and

an interpretation of the solutions as describing the intersections of two surfaces in Stokesparameter space is presented. Describing the evolution of the polarization state of thelight, these intersections de�ne the closed trajectories on the Poincar�e sphere along which

the reduced Stokes vector moves as the light propagates.

As the externally applied static magnetic �eld is varied, these trajectories undergotopological changes, and three characteristic regions, with six, four, and two eigenpolar-izations, respectively, are identi�ed for an increasing magnitude of the externally applied

static magnetic �eld. Explicit formulae for the six possible eigenpolarizations of the opti-cal waves are presented, and changes of their properties under bifurcations are analysedby means of maps on the Poincar�e sphere.

The theory presented in Paper II opens for control of the stability of the polarizationstate of light by using an externally applied static magnetic �eld as controlling parameter.

5.3 Paper III

Optical parametric generation and phase-matching in magneto-optic media

Opt. Lett. 24, 1514 (1999)

In nonlinear optics, a large extent of the physics is connected to control of the phaseevolution of light, in parametric interactions being crucial for achieving phase-matching

of the interacting waves. Traditionally, phase-matching is for parametric optical processesachieved by means of using linearly polarized light and exploiting wave propagation inlinearly birefringent media.


z

z = 0 z = L

E�

!3(0), E�!2(0)

E+!3(0), E+

!2(0)

E�

!3(L), E�!2(L), E

�

!1(L)

E+!3(L), E+

!2(L), E+

!1(L)

H0

�(ee)�� , �

(eee)�� ,

�(eem)�� , �

(eeem)��

[�43m, 6mm]

Figure 5.3. The setup for unidirectional magneto-optical parametric ampli-�cation in the Faraday con�guration, as studied in Papers III and IV. The

second order optical susceptibility tensor �(eee)(�!1;2;!3;�!2;1) describesparametric ampli�cation, while its third order magneto-optical modi�cation

�(eeem)(�!1;2;!3;�!2;1; 0) describes the magnetic �eld-induced contribu-

tions to the parametric interaction.

In this paper, the possibility of obtaining phase-matching by exploiting arti�ciallyinduced circular birefringence is analysed. In order to simplify the algebraic analysisand allow for a straightforward interpretation of the selection rules for the parametric

process, wave propagation in the (111)-direction of a crystal of point-symmetry �43m inthe Faraday con�guration was chosen, as shown in Fig. 5.3. In this setup, no lineardirection of polarization has preference, and the constitutive relations are rotationally

symmetric around the direction of propagation of the light. It is shown that the thirdorder magneto-optical interactions appear as source terms being di�erentially distributedbetween sets of right and left circularly polarized waves in such a way that orthogonally

polarized sets experience the magneto-optically induced gain with equal magnitude butwith di�erent signs.

Within the nondepleted pump approximation, analytical solutions for the evolution ofintensity and polarization state of the idler wave are presented, and the signal-to-idlerconversion e�ciency as function of magnitude of the static magnetic �eld is analysed for

circularly polarized �elds. It is shown that phase-matching can be achieved by exploitingthe arti�cially induced circular birefringence, in a �xed collinear geometry where anywalk-o� e�ects are eliminated.

The theory presented in Paper III opens for devices for parametric ampli�cation with

tunable phase-matching obtained via an externally applied static magnetic �eld.

5.4 Paper IV

Polarization state dependence of optical parametric processes in arti�cially

gyrotropic media

To appear in J.Opt. A: Pure Appl.Opt. 2, No. 4 (2000)

Optical parametric ampli�cation in magneto-optic media is analysed in the same geometryas in Paper III. Solutions for the spatial evolution of the intensities and polarization states

5.5. Paper V 57

z

z = 0 z = L

EI�!3

EI+!3

ET�!3

, ET�!2

, ET�!1

ET+!3

, ET+!2

, ET+!1

H0

�(ee)�� , �

(eee)�� ,

�(eem)�� , �

(eeem)��

[�43m, 6mm]

M0 M1

Figure 5.4. The setup of the magneto-optic parametric oscillator as studiedin Paper V.

of the idler and signal are in the non-depleted pump approximation presented in a Stokesparameter description for arbitrary phase mismatch and dispersion. The in uences of the

magnitude of the static magnetic �eld on the polarization states of the signal and idler arediscussed; the latter being visualised as a parametric trajectory on the Poincar�e spherefor a varying magnetic �eld.

Selection rules for parametric interaction in magneto-optic media are presented, and it

is shown that in order for the parametric interaction to take place, a necessary conditionis that the pump and signal waves are orthogonally circularly polarized.

The Manley-Rowe relations are presented, taking magneto-optically induced contri-

butions to the parametric process into account, and a complete listing of the nonzeroelements of the second and third order optical and magneto-optical susceptibility tensors,taken in the laboratory reference frame, is presented.

5.5 Paper V

Magneto-Optical Parametric Oscillation

Submitted to Opt. Lett.

In this study, the constitutive relations as developed in Papers III and IV are applied

to the interior of a Fabry-P�erot interferometer, for which large-signal steady-state opticalparametric oscillation in the Faraday con�guration is analysed.

Allowing for a depleted pump, the nonlinear intracavity wave equation is within theslowly-varying envelope approximation solved analytically for the spatial distribution of

the idler, signal, and pump waves, in terms of Jacobian elliptic functions, and the keyperformances of the cavity, such as threshold pump power and intracavity signal-to-pumpratio, are analysed as functions of phase-mismatch.

As in the case of unidirectional parametric ampli�cation, the key issue with usingarti�cially induced gyrotropy for obtaining phase matching is that any walk-o� e�ects areeliminated, still with the possibility of a continuous tuning of the component.


While the generated idler in the case of parametric ampli�cation has its angular fre-

quency !1 �xed and equal for both circular polarization states, from the requirement!1 + !2 = !3, the magneto-optical parametric oscillator will typically generate two setsof frequencies, obeying the energy conservation requirement

!�

1 + !�

2 = !3;

where !3 is the angular frequency of the pump, !�1 the frequency of the left/right circularlypolarized component of the idler, and !�2 the frequency of the left/right circularly polarized

component of the signal. At perfect phase matching, the frequencies of the right circularlypolarized components of the idler and signal will generally di�er from the frequencies oftheir respective left circularly polarized components.

The theory presented in Paper V opens for a light source with frequency being tunableby an externally applied static magnetic �eld.

Appendix A

List of symbols

\Important references are given in boldface. Italicized numbers indicate

eeting references, whereas numbers in parentheses refer to mere implica-

tions or unwarranted extrapolations. Asterisks are used to identify particu-

larly distasteful passages."

{PETER SCHICKELE, The De�nitive Biography of P.D.Q. Bach (1976)

Conventions of notation

a Scalar quantity

a Vector or tensor quantitya� �-component of the vector aa�1�2:::�n �1�2 : : : �n:th element of the n:th-rank tensor a_a, �a First and second derivatives of a with respect to timea Quantum-mechanical operator, scalar quantitya Quantum-mechanical operator, vector quantity

a� �-component of operator aa�

mn mn:th matrix element of the operator a�; a�

mn = hmja�jni

List of symbols

B(r; t) Magnetic �eldB(r; !) Complex temporally Fourier-transformed magnetic �eldB0 Static magnetic �eld

B�

0 �-component of static magnetic �eld, B�

0 = e� �B0

c Speed of light in vacuum, �0"0c2 = 1

�(! � !k) The Dirac delta function

�� The Kronecker deltae The elementary chargee� Unit vector in x�-direction in Cartesian coordinate system

e+, e� Complex basis vectors for left and right circular polarization, respectivelyE(r; t) Electric �eld strengthE(r; !) Complex temporally Fourier-transformed electric �eld strength

E!(r; t) Complex time-dependent temporal envelope of quasi-monochromaticelectric �eld at angular frequency !; whenever the arguments are omit-ted, only spatial dependence is implied

E�

! �-component of temporal envelope of the electric �eld, E�

! = e� � E!

Ea Energy level of molecule in state jai"0 Electric permittivity constant of vacuum

�(r; t) Scalar potential of the electromagnetic �eldg Gyration vector

59

60 Appendix A. List of symbols

List of symbols (continued)

Magnetic �eld-induced gyration constant

(ee)

ijFirst-order electric dipole micropolarizability of one-electron oscillator

(eem)

ijkSecond-order electric dipole micropolarizability of one-electron oscillator

(eemm)

ijklThird-order electric dipole micropolarizability of one-electron oscillator

�e, �p Relaxation constants for motion of electron and nucleus, respectively�R, �� E�ective relaxation constants for motion of center of mass and electric

dipole moment of the one-electron oscillator, respectively

�, �ab Relaxation operator of a system, and its ab:th matrix element~ Planck's constant divided by 2�

H Hamiltonian operator of quantum-mechanical system

H0 Thermal-equilibrium Hamiltonian operator

HI(t) Interaction Hamiltonian

H0I(t) Interaction Hamiltonian in interaction picture

HI(t) Molecular interaction Hamiltonian

H0I(t) Molecular interaction Hamiltonian in interaction picture

J(r; t) Current density of free chargeske Spring constant of all-classical one-electron oscillatorkB Boltzmann's constantK Degeneration constant in nonlinear polarization density

me, mp Mass of electron and nucleus (proton), respectivelymr Reduced mass of electron and nucleus, mr = memp=(me +mp)M(r; t) Magnetization; magnetic dipole moment per unit volume

M(n)(r; t) n:th order term in perturbation analysis of magnetizationm�, m

�

ab �-component of magnetic dipole operator, and its ab:th matrix element

�0 Magnetic permeability constant of vacuum�, �

!Electric dipole moment of the classical one-electron oscillator

��, ��

ab �-component of electric dipole operator, and its ab:th matrix element

N Number density; number of molecules per unit volumeP(r; t) Electric polarization density; electric dipole moment per unit volumeP!(r; t) Temporal envelope of electric polarization density

P(n)(r; t) n:th order term in perturbation analysis of electric polarization densityre, rp Radius position vectors to electron and proton of one-electron oscillatorR Radius vector to center of mass of the one-electron oscillator [Ch. 2];

also matrix of rotation [Ch. 3]Ra� Elements of matrix governing the rotation of one coordinate system rel-

ative another

�(r; t) Density of free charges� Density operator of system

�(n)(t) n:th order perturbation term of the density operator

�(n)

ab(t) ab:th matrix element of n:th order perturbation of density operator

Sk Stokes parameters

S(e)�

abFirst-order electric dipole response function of the density operator

S(m)�

abFirst-order magnetic dipole response function of the density operator

61

List of symbols (continued)

S(ee)��

abSecond-order electric dipolar response functions of the density operator

S(em)��

ab

S(mm)��

abSecond-order magnetic dipolar response function of the density operator

S(eee)��

abThird-order electric dipolar response function of the density operator

S(eem)��

abThird-order response function governing coupling between two electricand one magnetic dipoles

S(emm)��

ab

S(mmm)��

abThird-order magnetic dipolar response function of the density operator

S(eeee)��

abFourth-order electric dipolar response functions of the density operator

......

S(mmmm)��

abFourth-order magnetic dipolar response functions of the density operator

Sn Symmetrization operator in n:th order susceptibilitiest Time variableT Absolute temperature

Tr(a) Trace of operator a; sum of all diagonal matrix elements of a

U0(t) Unperturbed time-development operator! Angular frequency!L The Larmor frequency; !L = ejB0j=(2mr)

!� Sum of all angular frequencies in a particular nonlinear optical processinvolving quasimonochromatic �elds, !� = !1 + !2 + : : :+ !n

Resonance frequency of the classical one-electron oscillator

ab Molecular transition frequency between states jai and jbi (with energylevels Ea and Eb , respectively)

�(ee)

��Elements of �rst-order optical susceptibility tensor

�(eee)

�� Elements of second-order optical susceptibility tensor

�(eeee)

�� Elements of third-order optical susceptibility tensor

�(em)

��Elements of �rst-order magneto-optical susceptibility tensor

�(eem)

�� Elements of second-order magneto-optical susceptibility tensor

�(eeem)

�� Elements of third-order magneto-optical susceptibility tensor

�(eeeem)

�� Elements of fourth-order magneto-optical susceptibility tensor

62

Appendix B

Tables of magneto-optical susceptibilities

In the tables N denotes the number of nonzero elements, andM the number of nonzero andindependent elements. The crystallographic point-symmetry classes are written according

to the international notation.

Table C.1. Nonzero elements of the second-order magneto-optical susceptibility tensor.Source: Kielich and Zawodny [2].

Class N M Nonzero elements of �(eem)

�� = �

(eem)

�� (�!�;!1; !2)

1 27 27 �(eem)zzz , �

(eem)zxx , �

(eem)xxz , �

(eem)xzx , �

(eem)zyy , �

(eem)yyz , �

(eem)yzy

�1 �(eem)xxx , �

(eem)xxy , �

(eem)yxx , �

(eem)xyx , �

(eem)xyy , �

(eem)yyx , �

(eem)yxy

�(eem)yyy , �

(eem)xzz , �

(eem)zzx , �

(eem)zxz , �

(eem)yzz , �

(eem)zzy , �

(eem)zyz

�(eem)xyz , �

(eem)xzy , �

(eem)yxz , �

(eem)yzx , �

(eem)zxy , �

(eem)zyx

m 13 13 �(eem)zzz , �

(eem)zxx , �

(eem)xxz , �

(eem)xzx , �

(eem)zyy , �

(eem)yyz , �

(eem)yzy

2 �(eem)xyz , �

(eem)xzy , �

(eem)yxz , �

(eem)yzx , �

(eem)zxy , �

(eem)zyx

2=m

222 6 6 �(eem)xyz , �

(eem)xzy , �

(eem)yxz , �

(eem)yzx , �

(eem)zxy , �

(eem)zyx

mm2

mmm

4 13 7 �(eem)zzz

�4 �(eem)zxx = �

(eem)zyy , �

(eem)xxz = �

(eem)yyz , �

(eem)xzx = �

(eem)yzy

4=m �(eem)xyz = �

(eem)yxz , �

(eem)yzx = ��(eem)xzy , �

(eem)zxy = ��(eem)zyx

6�66=m

422 6 3 �(eem)xyz = ��(eem)yxz , �



4mm�42m4=mmm

6226mm�6m26=mmm

63

64 Appendix B. Tables of magneto-optical susceptibilities

Table C.1 (continued)

Class N M Nonzero elements of �(eem)

�� = �

(eem)

�� (�!�;!1; !2)

3 21 9 �(eem)zzz

�3 �(eem)zxx = �

(eem)zyy , �

(eem)xxz = �

(eem)yyz , �

(eem)xzx = �

(eem)yzy

�(eem)xyz = ��(eem)yxz , �



�(eem)xxx = ��(eem)xyy = ��(eem)yxy = ��(eem)yyx

�(eem)yyy = ��(eem)yxx = ��(eem)xyx = ��(eem)xxy

32 10 4 �(eem)xxx = ��(eem)xyy = ��(eem)yxy = ��(eem)yyx

3m �(eem)xyz = ��(eem)yxz , �



�3m

23 6 2 �(eem)xyz = �

(eem)yzx = �

(eem)zxy , �

(eem)xzy = �

(eem)yxz = �

(eem)zyx

m3�43m 6 1 �

(eem)xyz = �

(eem)yzx = �

(eem)zxy = ��(eem)xzy = ��(eem)yxz = ��(eem)zyx

m3m

432Isotr.

Table C.2. Nonzero elements of the third-order magneto-optical susceptibility tensor1.Source: Kielich and Zawodny [2].

Class N M Nonzero elements of �(eeem)

�� = �

(eeem)

�� (�!�;!1; !2; !3)

1 81 81 A = f�(eeem)xxxx , �(eeem)yyyy , �

(eeem)zzzz ,

�(eeem)xxyy , �

(eeem)yyxx , �

(eeem)xyxy , �

(eeem)yxyx , �

(eeem)xyyx , �

(eeem)yxxy ,

�(eeem)xxzz , �

(eeem)zzxx , �

(eeem)xzxz , �

(eeem)zxzx , �

(eeem)xzzx , �

(eeem)zxxz ,

�(eeem)yyzz , �

(eeem)zzyy , �

(eeem)yzyz , �

(eeem)zyzy , �

(eeem)yzzy , �

(eeem)zyyz g

B = f�(eeem)xxxy , �(eeem)xxyx , �

(eeem)xyxx , �

(eeem)yxxx ,

�(eeem)yyyx , �

(eeem)yyxy , �

(eeem)yxyy , �

(eeem)xyyy ,

�(eeem)xzzy , �

(eeem)xzyz , �

(eeem)xyzz , �

(eeem)yzzx , �

(eeem)yzxz , �

(eeem)yxzz ,

�(eeem)zzxy , �

(eeem)zxzy , �

(eeem)zxyz , �

(eeem)zzyx , �

(eeem)zyzx , �

(eeem)zyxz g

D = f�(eeem)xxxz , �(eeem)xxzx , �

(eeem)xzxx , �

(eeem)zxxx ,

�(eeem)zzzx , �

(eeem)zzxz , �

(eeem)zxzz , �

(eeem)xzzz ,

�(eeem)xyyz , �

(eeem)xyzy , �

(eeem)xzyy , �

(eeem)zyyx , �

(eeem)zyxy , �

(eeem)zxyy ,

�(eeem)yyxz , �

(eeem)yxyz , �

(eeem)yxzy , �

(eeem)yyzx , �

(eeem)yzyx , �

(eeem)yzxy g

E = f�(eeem)yyyz , �(eeem)yyzy , �

(eeem)yzyy , �

(eeem)zyyy ,

�(eeem)zzzy , �

(eeem)zzyz , �

(eeem)zyzz , �

(eeem)yzzz ,

�(eeem)yxxz , �

(eeem)yxzx , �

(eeem)yzxx , �

(eeem)zxxy , �

(eeem)zxyx , �

(eeem)zyxx ,

�(eeem)xxyz , �

(eeem)xyxz , �

(eeem)xyzx , �

(eeem)xxzy , �

(eeem)xzxy , �

(eeem)xzyx g

�1 0 0

m 40 40 D and E

2 41 41 A and B

2=m 0 0

1The crystallographic classes having nonzero elements of �(eeem)

�� exactly correspond to the crystal-

lographic classes having nonzero elements of the all-optical second-order susceptibility �(eee)

�� .

65



�� = �

(eeem)

�� (�!�;!1; !2; !3)

222 21 21 A

mm2 20 20 B

mmm 0 0

4 41 21 F = f�(eeem)xxxx = �(eeem)yyyy , �

(eeem)zzzz ,

�(eeem)xxyy = �

(eeem)yyxx , �

(eeem)xyxy = �

(eeem)yxyx , �

(eeem)xyyx = �

(eeem)yxxy ,

�(eeem)xxzz = �

(eeem)yyzz , �

(eeem)xzxz = �

(eeem)yzyz , �

(eeem)xzzx = �

(eeem)yzzy ,

�(eeem)zzxx = �

(eeem)zzyy , �

(eeem)zxzx = �

(eeem)zyzy , �

(eeem)zxxz = �

(eeem)zyyz g

G = f�(eeem)xxxy = ��(eeem)yyyx , �(eeem)yxxx = ��(eeem)xyyy ,

�(eeem)xxyx = ��(eeem)yyxy , �

(eeem)xyxx = ��(eeem)yxyy ,

�(eeem)xyzz = ��(eeem)yxzz , �

(eeem)xzyz = ��(eeem)yzxz ,

�(eeem)xzzy = ��(eeem)yzzx , �

(eeem)zxyz = ��(eeem)zyxz ,

�(eeem)zxzy = ��(eeem)zyzx , �

(eeem)zzxy = ��(eeem)zzyx g

�4 40 20 H = f�(eeem)xxxx = ��(eeem)yyyy , �(eeem)xxyy = ��(eeem)yyxx ,

�(eeem)xyxy = ��(eeem)yxyx , �

(eeem)xyyx = ��(eeem)yxxy ,

�(eeem)xxzz = ��(eeem)yyzz , �

(eeem)xzxz = ��(eeem)yzyz ,

�(eeem)xzzx = ��(eeem)yzzy , �

(eeem)zzxx = ��(eeem)zzyy ,

�(eeem)zxzx = ��(eeem)zyzy , �

(eeem)zxxz = ��(eeem)zyyz g

J = f�(eeem)xxxy = �(eeem)yyyx , �

(eeem)yxxx = �

(eeem)xyyy ,

�(eeem)xxyx = �

(eeem)yyxy , �

(eeem)xyxx = �

(eeem)yxyy ,

�(eeem)xyzz = �

(eeem)yxzz , �

(eeem)xzyz = �

(eeem)yzxz ,

�(eeem)xzzy = �

(eeem)yzzx , �

(eeem)zxyz = �

(eeem)zyxz ,

�(eeem)zxzy = �

(eeem)zyzx , �

(eeem)zzxy = �

(eeem)zzyx g

4=m 0 0

422 21 11 F

4mm 20 10 G

�42m 20 10 H

4=mmm 0 0

3 73 27 L = f�(eeem)zzzz , �(eeem)xxxx = �

(eeem)yyyy = �

(eeem)xxyy + �

(eeem)xyxy + �

(eeem)xyyx ,

�(eeem)xxyy = �

(eeem)yyxx , �

(eeem)xyxy = �

(eeem)yxyx , �

(eeem)xyyx = �

(eeem)yxxy ,

�(eeem)xxzz = �

(eeem)yyzz , �

(eeem)xzxz = �

(eeem)yzyz , �

(eeem)xzzx = �

(eeem)yzzy ,

�(eeem)zzxx = �

(eeem)zzyy , �

(eeem)zxzx = �

(eeem)zyzy , �

(eeem)zxxz = �

(eeem)zyyz g

M = f�(eeem)xxxy = ��(eeem)yyyx = �(�(eeem)xxyx + �(eeem)xyxx + �

(eeem)yxxx ),

�(eeem)xxyx = ��(eeem)yyxy , �

(eeem)xyxx = ��(eeem)yxyy ,

�(eeem)yxxx = ��(eeem)xyyy , �

(eeem)zzxy = ��(eeem)zzyx ,

�(eeem)zxyz = ��(eeem)zyxz , �

(eeem)xzzy = ��(eeem)yzzx ,

�(eeem)xyzz = ��(eeem)yxzz , �

(eeem)zxzy = ��(eeem)zyzx ,

�(eeem)xzyz = ��(eeem)yzxz g




�� = �

(eeem)

�� (�!�;!1; !2; !3)

P = f�(eeem)xxxz = ��(eeem)xyyz = ��(eeem)yxyz = ��(eeem)yyxz ,

�(eeem)xxzx = ��(eeem)xyzy = ��(eeem)yxzy = ��(eeem)yyzx ,

�(eeem)xzxx = ��(eeem)xzyy = ��(eeem)yzxy = ��(eeem)yzyx ,

�(eeem)zxxx = ��(eeem)zxyy = ��(eeem)zyxy = ��(eeem)zyyx g

Q = f�(eeem)yyyz = ��(eeem)yxxz = ��(eeem)xyxz = ��(eeem)xxyz ,

�(eeem)yyzy = ��(eeem)yxzx = ��(eeem)xyzx = ��(eeem)xxzy ,

�(eeem)yzyy = ��(eeem)yzxx = ��(eeem)xzyx = ��(eeem)xzxy ,

�(eeem)zyyy = ��(eeem)zyxx = ��(eeem)zxyx = ��(eeem)zxxy g

�3 0 0

32 37 14 L and Q

3m 36 13 M and P�3m 0 0

6 41 19 L and M�6 32 8 P and Q

6=m 0 0

622 21 10 L

6mm 20 9 M

�6m2 16 4 Q

6=mmm 0 0

23 21 7 �(eeem)xxxx = �

(eeem)yyyy = �

(eeem)zzzz

�(eeem)xxyy = �

(eeem)yyzz = �

(eeem)zzxx , �

(eeem)yyxx = �

(eeem)xxzz = �

(eeem)zzyy

�(eeem)xyxy = �

(eeem)yzyz = �

(eeem)zxzx , �

(eeem)yxyx = �

(eeem)xzxz = �

(eeem)zyzy

�(eeem)xyyx = �

(eeem)yzzy = �

(eeem)zxxz , �

(eeem)yxxy = �

(eeem)xzzx = �

(eeem)zyyz

m3 0 0

432 21 4 �(eeem)xxxx = �

(eeem)yyyy = �

(eeem)zzzz

�(eeem)xxyy = �

(eeem)yyzz = �

(eeem)zzxx = �

(eeem)yyxx = �

(eeem)xxzz = �

(eeem)zzyy

�(eeem)xyxy = �

(eeem)yzyz = �

(eeem)zxzx = �

(eeem)yxyx = �

(eeem)xzxz = �

(eeem)zyzy

�(eeem)xyyx = �

(eeem)yzzy = �

(eeem)zxxz = �

(eeem)yxxy = �

(eeem)xzzx = �

(eeem)zyyz

m3m 0 0�43m 18 3 �

(eeem)xxyy = �

(eeem)yyzz = �

(eeem)zzxx = ��(eeem)yyxx = ��(eeem)xxzz = ��(eeem)zzyy

�(eeem)xyxy = �

(eeem)yzyz = �

(eeem)zxzx = ��(eeem)yxyx = ��(eeem)xzxz = ��(eeem)zyzy

�(eeem)xyyx = �

(eeem)yzzy = �

(eeem)zxxz = ��(eeem)yxxy = ��(eeem)xzzx = ��(eeem)zyyz

Isotr.a 21 3 �(eeem)xxxx = �

(eeem)yyyy = �

(eeem)zzzz = �

(eeem)xxyy + �

(eeem)xyxy + �

(eeem)xyyx

�(eeem)xxyy = �

(eeem)yyxx = �

(eeem)xxzz = �

(eeem)zzxx = �

(eeem)yyzz = �

(eeem)zzyy

�(eeem)xyxy = �

(eeem)yxyx = �

(eeem)xzxz = �

(eeem)zxzx = �

(eeem)yzyz = �

(eeem)zyzy

�(eeem)xyyx = �

(eeem)yxxy = �

(eeem)xzzx = �

(eeem)zxxz = �

(eeem)yzzy = �

(eeem)zyyz

aThe listed nonzero tensor elements apply to isotropic media with no center of inversion. For isotropicmedia possessing inversion symmetry all tensor components are zero. C. f. Sec. 3.3.3.

67

Table C.3. Nonzero elements of the fourth-order magneto-optical susceptibility tensor.

Source: Kielich and Zawodny [2].

Class N M Nonzero elements of �(eeeem)

�� = �

(eeeem)

�� (�!�;!1; !2; !3; !4)

1 243 243 A1 = f�(eeeem)zzzzz ,�1 �

(eeeem)xxxxz , �

(eeeem)xxxzx , �

(eeeem)xxzxx , �

(eeeem)xzxxx , �

(eeeem)zxxxx ,

�(eeeem)yyyyz , �

(eeeem)yyyzy , �

(eeeem)yyzyy , �

(eeeem)yzyyy , �

(eeeem)zyyyy ,

�(eeeem)zzzxx , �

(eeeem)zzxzx , �

(eeeem)zzxxz , �

(eeeem)zxzxz , �

(eeeem)zxzzx ,

�(eeeem)zxxzz , �

(eeeem)xzzzx , �

(eeeem)xzzxz , �

(eeeem)xzxzz , �

(eeeem)xxzzz ,

�(eeeem)zzzyy , �

(eeeem)zzyzy , �

(eeeem)zzyyz , �

(eeeem)zyzyz , �

(eeeem)zyzzy ,

�(eeeem)zyyzz , �

(eeeem)yzzzy , �

(eeeem)yzzyz , �

(eeeem)yzyzz , �

(eeeem)yyzzz g

B1 = f�(eeeem)xxxyz , �(eeeem)xxxzy , �

(eeeem)xxyxz , �

(eeeem)xxyzx , �

(eeeem)xxzxy ,

�(eeeem)xxzyx , �

(eeeem)xyxxz , �

(eeeem)xyxzx , �

(eeeem)xyzxx , �

(eeeem)xzxxy ,

�(eeeem)xzxyx , �

(eeeem)xzyxx , �

(eeeem)yxxxz , �

(eeeem)yxxzx , �

(eeeem)yxzxx ,

�(eeeem)yzxxx , �

(eeeem)zxxxy , �

(eeeem)zxxyx , �

(eeeem)zxyxx , �

(eeeem)zyxxx ,

�(eeeem)yyyxz , �

(eeeem)yyyzx , �

(eeeem)yyxyz , �

(eeeem)yyxzy , �

(eeeem)yyzyx ,

�(eeeem)yyzxy , �

(eeeem)yxyyz , �

(eeeem)yxyzy , �

(eeeem)yxzyy , �

(eeeem)yzyyx ,

�(eeeem)yzyxy , �

(eeeem)yzxyy , �

(eeeem)xyyyz , �

(eeeem)xyyzy , �

(eeeem)xyzyy ,

�(eeeem)xzyyy , �

(eeeem)zyyyx , �

(eeeem)zyyxy , �

(eeeem)zyxyy , �

(eeeem)zxyyy ,

�(eeeem)zzzxy , �

(eeeem)zzzyx , �

(eeeem)zzxzy , �

(eeeem)zzxyz , �

(eeeem)zzyzx ,

�(eeeem)zzyxz , �

(eeeem)zxzzy , �

(eeeem)zxzyz , �

(eeeem)zxyzz , �

(eeeem)zyzzx ,

�(eeeem)zyzxz , �

(eeeem)zyxzz , �

(eeeem)xzzzy , �

(eeeem)xzzyz , �

(eeeem)xzyzz ,

�(eeeem)xyzzz , �

(eeeem)yzzzx , �

(eeeem)yzzxz , �

(eeeem)yzxzz , �

(eeeem)yxzzz g

D1 = f�(eeeem)xxyyz , �(eeeem)xxyzy , �

(eeeem)xxzyy , �

(eeeem)xyxyz , �

(eeeem)xyxzy ,

�(eeeem)xyyzx , �

(eeeem)xyyxz , �

(eeeem)xyzxy , �

(eeeem)xyzyx , �

(eeeem)xzxyy ,

�(eeeem)xzyxy , �

(eeeem)xzyyx , �

(eeeem)yxxyz , �

(eeeem)yxxzy , �

(eeeem)yxyzx ,

�(eeeem)yxyxz , �

(eeeem)yxzyx , �

(eeeem)yxzxy , �

(eeeem)yyxxz , �

(eeeem)yyxzx ,

�(eeeem)yyzxx , �

(eeeem)yzxxy , �

(eeeem)yzxyx , �

(eeeem)yzyxx , �

(eeeem)zxxyy ,

�(eeeem)zxyxy , �

(eeeem)zxyyx , �

(eeeem)zyyxx , �

(eeeem)zyxyx , �

(eeeem)zyxxy g

E1 = f�(eeeem)xxxxx , �(eeeem)yyyyy ,

�(eeeem)xxxxy , �

(eeeem)xxxyx , �

(eeeem)xxyxx , �

(eeeem)xyxxx , �

(eeeem)yxxxx ,

�(eeeem)yyyyx , �

(eeeem)yyyxy , �

(eeeem)yyxyy , �

(eeeem)yxyyy , �

(eeeem)xyyyy ,

�(eeeem)zzzzx , �

(eeeem)zzzxz , �

(eeeem)zzxzz , �

(eeeem)zxzzz , �

(eeeem)xzzzz ,

�(eeeem)zzzzy , �

(eeeem)zzzyz , �

(eeeem)zzyzz , �

(eeeem)zyzzz , �

(eeeem)yzzzz ,

�(eeeem)xxxyy , �

(eeeem)xxyxy , �

(eeeem)xxyyx , �

(eeeem)xyxyx , �

(eeeem)xyxxy ,

�(eeeem)xyyxx , �

(eeeem)yxxxy , �

(eeeem)yxxyx , �

(eeeem)yxyxx , �

(eeeem)yyxxx ,

�(eeeem)yyyxx , �

(eeeem)yyxyx , �

(eeeem)yyxxy , �

(eeeem)yxyxy , �

(eeeem)yxyyx ,

�(eeeem)yxxyy , �

(eeeem)xyyyx , �

(eeeem)xyyxy , �

(eeeem)xyxyy , �

(eeeem)xxyyy ,

�(eeeem)xxxzz , �

(eeeem)xxzxz , �

(eeeem)xxzzx , �

(eeeem)xzxzx , �

(eeeem)xzxxz ,

�(eeeem)xzzxx , �

(eeeem)zxxxz , �

(eeeem)zxxzx , �

(eeeem)zxzxx , �

(eeeem)zzxxx ,

�(eeeem)yyyzz , �

(eeeem)yyzyz , �

(eeeem)yyzzy , �

(eeeem)yzyzy , �

(eeeem)yzyyz ,

�(eeeem)yzzyy , �

(eeeem)zyyyz , �

(eeeem)zyyzy , �

(eeeem)zyzyy , �

(eeeem)zzyyy ,




�� = �

(eeeem)

�� (�!�;!1; !2; !3; !4)

�(eeeem)xxzzy , �

(eeeem)xxzyz , �

(eeeem)xxyzz , �

(eeeem)xzxzy , �

(eeeem)xzxyz ,

�(eeeem)xzzyx , �

(eeeem)xzzxy , �

(eeeem)xzyxz , �

(eeeem)xzyzx , �

(eeeem)xyxzz ,

�(eeeem)xyzxz , �

(eeeem)xyzzx , �

(eeeem)zxxzy , �

(eeeem)zxxyz , �

(eeeem)zxzyx ,

�(eeeem)zxzxy , �

(eeeem)zxyzx , �

(eeeem)zxyxz , �

(eeeem)zzxxy , �

(eeeem)zzxyx ,

�(eeeem)zzyxx , �

(eeeem)zyxxz , �

(eeeem)zyxzx , �

(eeeem)zyzxx , �

(eeeem)yxxzz ,

�(eeeem)yxzxz , �

(eeeem)yxzzx , �

(eeeem)yzzxx , �

(eeeem)yzxzx , �

(eeeem)yzxxz ,

�(eeeem)yyzzx , �

(eeeem)yyzxz , �

(eeeem)yyxzz , �

(eeeem)yzyzx , �

(eeeem)yzyxz ,

�(eeeem)yzzxy , �

(eeeem)yzzyx , �

(eeeem)yzxyz , �

(eeeem)yzxzy , �

(eeeem)yxyzz ,

�(eeeem)yxzyz , �

(eeeem)yxzzy , �

(eeeem)zyyzx , �

(eeeem)zyyxz , �

(eeeem)zyzxy ,

�(eeeem)zyzyx , �

(eeeem)zyxzy , �

(eeeem)zyxyz , �

(eeeem)zzyyx , �

(eeeem)zzyxy ,

�(eeeem)zzxyy , �

(eeeem)zxyyz , �

(eeeem)zxyzy , �

(eeeem)zxzyy , �

(eeeem)xyyzz ,

�(eeeem)xyzyz , �

(eeeem)xyzzy , �

(eeeem)xzzyy , �

(eeeem)xzyyz , �

(eeeem)xzyzy g

m 121 121 A1, B1, and D1

2

2=m

222 60 60 B1

mm2

mmm

4 121 61 �(eeeem)zzzzz ,

�4 �(eeeem)xxxxz = �

(eeeem)yyyyz , �

(eeeem)xxxzx = �

(eeeem)yyyzy , �

(eeeem)xxzxx = �

(eeeem)yyzyy

4=m �(eeeem)xzxxx = �

(eeeem)yzyyy , �

(eeeem)zxxxx = �

(eeeem)zyyyy , �

(eeeem)zzzxx = �

(eeeem)zzzyy

�(eeeem)zzxxz = �

(eeeem)zzyyz , �

(eeeem)zzxzx = �

(eeeem)zzyzy , �

(eeeem)zxzzx = �

(eeeem)zyzzy

�(eeeem)zxxzz = �

(eeeem)zyyzz , �

(eeeem)zxzxz = �

(eeeem)zyzyz , �

(eeeem)xxzzz = �

(eeeem)yyzzz

�(eeeem)xzxzz = �

(eeeem)yzyzz , �

(eeeem)xzzxz = �

(eeeem)yzzyz , �

(eeeem)xzzzx = �

(eeeem)yzzzy

�(eeeem)xxyyz = �

(eeeem)yyxxz , �

(eeeem)xxyzy = �

(eeeem)yyxzx , �

(eeeem)xxzyy = �

(eeeem)yyzxx

�(eeeem)xyxyz = �

(eeeem)yxyxz , �

(eeeem)xyxzy = �

(eeeem)yxyzx , �

(eeeem)xyyxz = �

(eeeem)yxxyz

�(eeeem)xyyzx = �

(eeeem)yxxzy , �

(eeeem)xyzyx = �

(eeeem)yxzxy , �

(eeeem)xyzxy = �

(eeeem)yxzyx

�(eeeem)xzyyx = �

(eeeem)yzxxy , �

(eeeem)xzxyy = �

(eeeem)yzyxx , �

(eeeem)xzyxy = �

(eeeem)yzxyx

�(eeeem)zxxyy = �

(eeeem)zyyxx , �

(eeeem)zxyxy = �

(eeeem)zyxyx , �

(eeeem)zxyyx = �

(eeeem)zyxxy

F1 = f�(eeeem)xxxyz = ��(eeeem)yyyxz , �(eeeem)xxxzy = ��(eeeem)yyyzx ,

�(eeeem)xxzxy = ��(eeeem)yyzyx , �

(eeeem)xxyxz = ��(eeeem)yyxyz ,

�(eeeem)xxyzx = ��(eeeem)yyxzy , �

(eeeem)xxzyx = ��(eeeem)yyzxy ,

�(eeeem)xzxyx = ��(eeeem)yzyxy , �

(eeeem)xzyxx = ��(eeeem)yzxyy ,

�(eeeem)xzxxy = ��(eeeem)yzyyx , �

(eeeem)xyzxx = ��(eeeem)yxzyy ,

�(eeeem)xyxzx = ��(eeeem)yxyzy , �

(eeeem)xyxxz = ��(eeeem)yxyyz ,

�(eeeem)yxxxz = ��(eeeem)xyyyz , �

(eeeem)yxxzx = ��(eeeem)xyyzy ,

�(eeeem)yxzxx = ��(eeeem)xyzyy , �

(eeeem)yzxxx = ��(eeeem)xzyyy ,

�(eeeem)zyxxx = ��(eeeem)zxyyy , �

(eeeem)zxxyx = ��(eeeem)zyyxy ,

�(eeeem)zxyxx = ��(eeeem)zyxyy , �

(eeeem)zxxxy = ��(eeeem)zyyyx ,

69



�� = �

(eeeem)

�� (�!�;!1; !2; !3; !4)

�(eeeem)zzzxy = ��(eeeem)zzzyx , �

(eeeem)zzyzx = ��(eeeem)zzxzy ,

�(eeeem)zzxyz = ��(eeeem)zzyxz , �

(eeeem)zyzxz = ��(eeeem)zxzyz ,

�(eeeem)zyxzz = ��(eeeem)zxyzz , �

(eeeem)zyzzx = ��(eeeem)zxzzy ,

�(eeeem)xzzzy = ��(eeeem)yzzzx , �

(eeeem)xzzyz = ��(eeeem)yzzxz ,

�(eeeem)xzyzz = ��(eeeem)yzxzz , �

(eeeem)xyzzz = ��(eeeem)yxzzz g

422 60 30 F1

4mm�42m4=mmm

3 233 81 G1 = f��(eeeem)xxxyz = �(eeeem)yyyxz = �

(eeeem)xxyxz + �

(eeeem)xyxxz + �

(eeeem)yxxxz ,

�3 ��(eeeem)xxxzy = �(eeeem)yyyzx = �

(eeeem)xxyzx + �

(eeeem)xyxzx + �

(eeeem)yxxzx ,

��(eeeem)xxzxy = �(eeeem)yyzyx = �

(eeeem)xxzyx + �

(eeeem)xyzxx + �

(eeeem)yxzxx ,

��(eeeem)xzxxy = �(eeeem)yzyyx = �

(eeeem)xzxyx + �

(eeeem)xzyxx + �

(eeeem)yzxxx ,

��(eeeem)zxxxy = �(eeeem)zyyyx = �

(eeeem)zxxyx + �

(eeeem)zxyxx + �

(eeeem)zyxxx ,

�(eeeem)xxyxz = ��(eeeem)yyxyz , �

(eeeem)xxyzx = ��(eeeem)yyxzy ,

�(eeeem)xxzyx = ��(eeeem)yyzxy , �

(eeeem)xzxyx = ��(eeeem)yzyxy ,

�(eeeem)zxxyx = ��(eeeem)zyyxy , �

(eeeem)yxxxz = ��(eeeem)xyyyz ,

�(eeeem)yxxzx = ��(eeeem)xyyzy , �

(eeeem)yxzxx = ��(eeeem)xyzyy ,

�(eeeem)yzxxx = ��(eeeem)xzyyy , �

(eeeem)zyxxx = ��(eeeem)zxyyy ,

�(eeeem)xyxxz = ��(eeeem)yxyyz , �

(eeeem)xyxzx = ��(eeeem)yxyzy ,

�(eeeem)xyzxx = ��(eeeem)yxzyy , �

(eeeem)xzyxx = ��(eeeem)yzxyy ,

�(eeeem)zxyxx = ��(eeeem)zyxyy , �

(eeeem)zzzxy = ��(eeeem)zzzyx ,

�(eeeem)zzyzx = ��(eeeem)zzxzy , �

(eeeem)zzxyz = ��(eeeem)zzyxz ,

�(eeeem)zyzxz = ��(eeeem)zxzyz , �

(eeeem)zyxzz = ��(eeeem)zxyzz ,

�(eeeem)zyzzx = ��(eeeem)zxzzy , �

(eeeem)xzzzy = ��(eeeem)yzzzx ,

�(eeeem)xzzyz = ��(eeeem)yzzxz , �

(eeeem)xzyzz = ��(eeeem)yzxzz ,

�(eeeem)xyzzz = ��(eeeem)yxzzz g

H1 = f�(eeeem)xxxzz = ��(eeeem)xyyzz = ��(eeeem)yxyzz = ��(eeeem)yyxzz ,

�(eeeem)xzzxx = ��(eeeem)xzzyy = ��(eeeem)yzzxy = ��(eeeem)yzzyx ,

�(eeeem)xxzxz = ��(eeeem)xyzyz = ��(eeeem)yxzyz = ��(eeeem)yyzxz ,

�(eeeem)zxxzx = ��(eeeem)zxyzy = ��(eeeem)zyxzy = ��(eeeem)zyyzx ,

�(eeeem)xxzzx = ��(eeeem)xyzzy = ��(eeeem)yxzzy = ��(eeeem)yyzzx ,

�(eeeem)zxxxz = ��(eeeem)zxyyz = ��(eeeem)zyxyz = ��(eeeem)zyyxz ,

�(eeeem)xzxxz = ��(eeeem)xzyyz = ��(eeeem)yzxyz = ��(eeeem)yzyxz ,

�(eeeem)zxzxx = ��(eeeem)zxzyy = ��(eeeem)zyzxy = ��(eeeem)zyzyx ,

�(eeeem)xzxzx = ��(eeeem)xzyzy = ��(eeeem)yzxzy = ��(eeeem)yzyzx ,

�(eeeem)zzxxx = ��(eeeem)zzxyy = ��(eeeem)zzyxy = ��(eeeem)zzyyx ,

�(eeeem)xxxxx = � 1

3(�

(eeeem)yyyyx + �

(eeeem)yyyxy + �

(eeeem)yyxyy

+�(eeeem)yxyyy + �

(eeeem)xyyyy ),




�� = �

(eeeem)

�� (�!�;!1; !2; !3; !4)

�(eeeem)xxxyy = 1

3(2�

(eeeem)yyyyx + 2�

(eeeem)yyyxy � �

(eeeem)yyxyy

��(eeeem)yxyyy � �(eeeem)xyyyy ),

�(eeeem)xxyxy = 1

3(2�

(eeeem)yyyyx � �

(eeeem)yyyxy + 2�

(eeeem)yyxyy


�(eeeem)xxyyx = 1

3(��(eeeem)yyyyx + 2�

(eeeem)yyyxy + 2�

(eeeem)yyxyy


�(eeeem)xyxxy = 1

3(2�



(eeeem)yyxyy

+2�(eeeem)yxyyy � �

(eeeem)xyyyy ),

�(eeeem)xyyxx = 1

3(��(eeeem)yyyyx � �

(eeeem)yyyxy + 2�

(eeeem)yyxyy


(eeeem)xyyyy ),

�(eeeem)xyxyx = 1



(eeeem)yyxyy


(eeeem)xyyyy ),

�(eeeem)yxxxy = 1

3(2�



(eeeem)yyxyy

��(eeeem)yxyyy + 2�(eeeem)xyyyy ),

�(eeeem)yxxyx = 1



(eeeem)yyxyy


�(eeeem)yxyxx = 1


(eeeem)yyyxy + 2�

(eeeem)yyxyy


�(eeeem)yyxxx = 1



(eeeem)yyxyy

+2�(eeeem)yxyyy + 2�

(eeeem)xyyyy ),

�(eeeem)yyyyx , �

(eeeem)yyyxy , �

(eeeem)yyxyy , �

(eeeem)yxyyy , �

(eeeem)xyyyy g

J1 = f�(eeeem)zzzzz ,

�(eeeem)zzzxx = �

(eeeem)zzzyy , �

(eeeem)zzxxz = �

(eeeem)zzyyz ,

�(eeeem)zzxzx = �

(eeeem)zzyzy , �

(eeeem)zxzzx = �

(eeeem)zyzzy ,

�(eeeem)zxxzz = �

(eeeem)zyyzz , �

(eeeem)zxzxz = �

(eeeem)zyzyz ,

�(eeeem)xxzzz = �

(eeeem)yyzzz , �

(eeeem)xzxzz = �

(eeeem)yzyzz ,

�(eeeem)xzzxz = �

(eeeem)yzzyz , �

(eeeem)xzzzx = �

(eeeem)yzzzy ,

�(eeeem)xxxxz = �

(eeeem)yyyyz = �

(eeeem)xxyyz + �

(eeeem)xyxyz + �

(eeeem)xyyxz ,

�(eeeem)xxxzx = �

(eeeem)yyyzy = �

(eeeem)xxyzy + �

(eeeem)xyxzy + �

(eeeem)xyyzx ,

�(eeeem)xxzxx = �

(eeeem)yyzyy = �

(eeeem)xxzyy + �

(eeeem)xyzxy + �

(eeeem)xyzyx ,

�(eeeem)xzxxx = �

(eeeem)yzyyy = �

(eeeem)xzxyy + �

(eeeem)xzyxy + �

(eeeem)xzyyx ,

�(eeeem)zxxxx = �

(eeeem)zyyyy = �

(eeeem)zxxyy + �

(eeeem)zxyxy + �

(eeeem)zxyyx ,

�(eeeem)xxyyz = �

(eeeem)yyxxz , �

(eeeem)xxyzy = �

(eeeem)yyxzx ,

�(eeeem)xxzyy = �

(eeeem)yyzxx , �

(eeeem)xyxyz = �

(eeeem)yxyxz ,

�(eeeem)xyxzy = �

(eeeem)yxyzx , �

(eeeem)xyyxz = �

(eeeem)yxxyz ,

�(eeeem)xyyzx = �

(eeeem)yxxzy , �

(eeeem)xyzyx = �

(eeeem)yxzxy ,

�(eeeem)xyzxy = �

(eeeem)yxzyx , �

(eeeem)xzyyx = �

(eeeem)yzxxy ,

�(eeeem)xzxyy = �

(eeeem)yzyxx , �

(eeeem)xzyxy = �

(eeeem)yzxyx ,

71



�� = �

(eeeem)

�� (�!�;!1; !2; !3; !4)

�(eeeem)zxxyy = �

(eeeem)zyyxx , �

(eeeem)zxyxy = �

(eeeem)zyxyx ,

�(eeeem)zxyyx = �

(eeeem)zyxxy g

L1 = f�(eeeem)yyyzz = ��(eeeem)yxxzz = ��(eeeem)xyxzz = ��(eeeem)xxyzz ,

�(eeeem)yzzyy = ��(eeeem)yzzxx = ��(eeeem)xzzyx = ��(eeeem)xzzxy ,

�(eeeem)yyzyz = ��(eeeem)yxzxz = ��(eeeem)xyzxz = ��(eeeem)xxzyz ,

�(eeeem)zyyzy = ��(eeeem)zyxzx = ��(eeeem)zxyzx = ��(eeeem)zxxzy ,

�(eeeem)yyzzy = ��(eeeem)yxzzx = ��(eeeem)xyzzx = ��(eeeem)xxzzy ,

�(eeeem)zyyyz = ��(eeeem)zyxxz = ��(eeeem)zxyxz = ��(eeeem)zxxyz ,

�(eeeem)yzyyz = ��(eeeem)yzxxz = ��(eeeem)xzyxz = ��(eeeem)xzxyz ,

�(eeeem)zyzyy = ��(eeeem)zyzxx = ��(eeeem)zxzyx = ��(eeeem)zxzxy ,

�(eeeem)yzyzy = ��(eeeem)yzxzx = ��(eeeem)xzyzx = ��(eeeem)xzxzy ,

�(eeeem)zzyyy = ��(eeeem)zzyxx = ��(eeeem)zzxyx = ��(eeeem)zzxxy ,

�(eeeem)yyyyy = � 1

3(�

(eeeem)xxxxy + �

(eeeem)xxxyx + �

(eeeem)xxyxx

+�(eeeem)xyxxx + �

(eeeem)yxxxx ),

�(eeeem)yyyxx = 1

3(2�

(eeeem)xxxxy + 2�

(eeeem)xxxyx � �

(eeeem)xxyxx

��(eeeem)xyxxx � �(eeeem)yxxxx ),

�(eeeem)yyxyx = 1

3(2�

(eeeem)xxxxy � �

(eeeem)xxxyx + 2�

(eeeem)xxyxx


�(eeeem)yyxxy = 1

3(��(eeeem)xxxxy + 2�

(eeeem)xxxyx + 2�

(eeeem)xxyxx


�(eeeem)yxyyx = 1

3(2�



(eeeem)xxyxx

+2�(eeeem)xyxxx � �

(eeeem)yxxxx ),

�(eeeem)yxxyy = 1

3(��(eeeem)xxxxy � �

(eeeem)xxxyx + 2�

(eeeem)xxyxx


(eeeem)yxxxx ),

�(eeeem)yxyxy = 1



(eeeem)xxyxx


(eeeem)yxxxx ),

�(eeeem)xyyyx = 1

3(2�



(eeeem)xxyxx

��(eeeem)xyxxx + 2�(eeeem)yxxxx ),

�(eeeem)xyyxy = 1



(eeeem)xxyxx


�(eeeem)xyxyy = 1


(eeeem)xxxyx + 2�

(eeeem)xxyxx


�(eeeem)xxyyy = 1



(eeeem)xxyxx

+2�(eeeem)xyxxx + 2�

(eeeem)yxxxx ),

�(eeeem)xxxxy , �

(eeeem)xxxyx , �

(eeeem)xxyxx , �

(eeeem)xyxxx , �

(eeeem)yxxxx g

32 116 40 G1 and H1

3m�3m




�� = �

(eeeem)

�� (�!�;!1; !2; !3; !4)

6 121 51 G1 and J1�6

6=m

622 60 25 G1

6mm�6m26=mmm

23 60 20 �(eeeem)xxxyz = �

(eeeem)yyyzx = �

(eeeem)zzzxy , �

(eeeem)xxxzy = �

(eeeem)yyyxz = �

(eeeem)zzzyx

m3 �(eeeem)xxzxy = �

(eeeem)yyxyz = �

(eeeem)zzyzx , �

(eeeem)xxyxz = �

(eeeem)yyzyx = �

(eeeem)zzxzy

�(eeeem)xxyzx = �

(eeeem)yyzxy = �

(eeeem)zzxyz , �

(eeeem)xzxyx = �

(eeeem)yxyzy = �

(eeeem)zyzxz

�(eeeem)xzxxy = �

(eeeem)yxyyz = �

(eeeem)zyzzx , �

(eeeem)xzyxx = �

(eeeem)yxzyy = �

(eeeem)zyxzz

�(eeeem)xyzxx = �

(eeeem)yzxyy = �

(eeeem)zxyzz , �

(eeeem)xxzyx = �

(eeeem)yyxzy = �

(eeeem)zzyxz

�(eeeem)xyxzx = �

(eeeem)yzyxy = �

(eeeem)zxzyz , �

(eeeem)xyxxz = �

(eeeem)yzyyx = �

(eeeem)zxzzy

�(eeeem)yxxxz = �

(eeeem)zyyyx = �

(eeeem)xzzzy , �

(eeeem)yxxzx = �

(eeeem)zyyxy = �

(eeeem)xzzyz

�(eeeem)yxzxx = �

(eeeem)zyxyy = �

(eeeem)xzyzz , �

(eeeem)yzxxx = �

(eeeem)zxyyy = �

(eeeem)xyzzz

�(eeeem)zyxxx = �

(eeeem)xzyyy = �

(eeeem)yxzzz , �

(eeeem)zxyxx = �

(eeeem)xyzyy = �

(eeeem)yzxzz

�(eeeem)zxxyx = �

(eeeem)xyyzy = �

(eeeem)yzzxz , �

(eeeem)zxxxy = �

(eeeem)xyyyz = �

(eeeem)yzzzx

432 60 10 �(eeeem)xxxyz = �

(eeeem)yyyzx = �

(eeeem)zzzxy = ��(eeeem)xxxzy = ��(eeeem)yyyxz = ��(eeeem)zzzyx

�43m �(eeeem)xxyxz = �

(eeeem)yyzyx = �

(eeeem)zzxzy = ��(eeeem)xxzxy = ��(eeeem)yyxyz = ��(eeeem)zzyzx

m3m �(eeeem)xyxxz = �

(eeeem)yzyyx = �

(eeeem)zxzzy = ��(eeeem)xzxxy = ��(eeeem)yxyyz = ��(eeeem)zyzzx


(eeeem)zyyyx = �

(eeeem)xzzzy = ��(eeeem)zxxxy = ��(eeeem)xyyyz = ��(eeeem)yzzzx


(eeeem)yyzxy = �

(eeeem)zzxyz = ��(eeeem)xxzyx = ��(eeeem)yyxzy = ��(eeeem)zzyxz


(eeeem)yzyxy = �

(eeeem)zxzyz = ��(eeeem)xzxyx = ��(eeeem)yxyzy = ��(eeeem)zyzxz

�(eeeem)yxxzx = �

(eeeem)zyyxy = �

(eeeem)xzzyz = ��(eeeem)zxxyx = ��(eeeem)xyyzy = ��(eeeem)yzzxz


(eeeem)yzxyy = �

(eeeem)zxyzz = ��(eeeem)xzyxx = ��(eeeem)yxzyy = ��(eeeem)zyxzz


(eeeem)zyxyy = �

(eeeem)xzyzz = ��(eeeem)zxyxx = ��(eeeem)xyzyy = ��(eeeem)yzxzz

�(eeeem)yzxxx = �

(eeeem)zxyyy = �

(eeeem)xyzzz = ��(eeeem)zyxxx = ��(eeeem)xzyyy = ��(eeeem)yxzzz

Isotr. 60 6 �(eeeem)xxxyz = �

(eeeem)yyyzx = �

(eeeem)zzzxy = ��(eeeem)xxxzy = ��(eeeem)yyyxz = ��(eeeem)zzzyx

= �(eeeem)xxyzx + �

(eeeem)xyxzx + �

(eeeem)yxxzx

�(eeeem)xxyxz = �

(eeeem)yyzyx = �

(eeeem)zzxzy = ��(eeeem)xxzxy = ��(eeeem)yyxyz = ��(eeeem)zzyzx

= ��(eeeem)xxyzx + �(eeeem)xyzxx + �

(eeeem)yxzxx

�(eeeem)xyxxz = �

(eeeem)yzyyx = �

(eeeem)zxzzy = ��(eeeem)xzxxy = ��(eeeem)yxyyz = ��(eeeem)zyzzx

= ��(eeeem)xyxzx � �(eeeem)xyzxx + �

(eeeem)yzxxx


(eeeem)zyyyx = �

(eeeem)xzzzy = ��(eeeem)zxxxy = ��(eeeem)xyyyz = ��(eeeem)yzzzx

= ��(eeeem)yxxzx � �(eeeem)yxzxx � �

(eeeem)yzxxx


(eeeem)yyzxy = �

(eeeem)zzxyz = ��(eeeem)yyxzy = ��(eeeem)zzyxz = ��(eeeem)xxzyx


(eeeem)yzyxy = �

(eeeem)zxzyz = ��(eeeem)xzxyx = ��(eeeem)yxyzy = ��(eeeem)zyzxz

�(eeeem)yxxzx = �

(eeeem)zyyxy = �

(eeeem)xzzyz = ��(eeeem)zxxyx = ��(eeeem)xyyzy = ��(eeeem)yzzxz


(eeeem)yzxyy = �

(eeeem)zxyzz = ��(eeeem)xzyxx = ��(eeeem)yxzyy = ��(eeeem)zyxzz


(eeeem)zyxyy = �

(eeeem)xzyzz = ��(eeeem)zxyxx = ��(eeeem)xyzyy = ��(eeeem)yzxzz

�(eeeem)yzxxx = �

(eeeem)zxyyy = �

(eeeem)xyzzz = ��(eeeem)zyxxx = ��(eeeem)xzyyy = ��(eeeem)yxzzz

Appendix C

Routines for analysis of arbitrary rank tensorsin rotated frames

Here algorithms for analytical calculation of nonzero tensor components of arbitrary rank

tensors are listed. The algorithms are implemented as routines in MapleV code, andexamples of their usage can be found in Sec. 3.3.4.

# File: ~/TeX/sustable/tensrot.map [MapleV code]

# Created: 24/3-1998 <[email protected]>

# Last change: 22/5-1999 <[email protected]>

# Copyright (C) 1998-2000, Fredrik Jonsson <[email protected]>

# Non-commercial copying welcome

restart:

Given a susceptibility tensor of arbitrary rank, in this case describing the non-

linear optical polarization of a medium, expressed in the coordinate system

(x; y; z) of the crystal, this MapleV program calculates the corresponding sus-

ceptibility tensor expressed in the laboratory coordinate system (x0; y0; z0), ro-

tated relative to the previous one.

The vectorized storage of the tensor (with �a priori unknown rank) is done in

the following manner, here illustrated with an example for tensors of rank 4:

Tensor index Vector index

[1; 1; 1; 1] 1[1; 1; 1; 2] 2

[1; 1; 1; 3] 3[1; 1; 2; 1] 4[1; 1; 2; 2] 5

......

[3; 3; 3; 1] 79

[3; 3; 3; 2] 80[3; 3; 3; 3] 81

The indx() routine takes one vector j as argument, containing a tensor index

multiple. From this tensor index multiple, the routine calculates the vector

73

74 Appendix C. Routines for analysis of arbitrary rank tensors in rotated frames

index where to store the corresponding tensor element, as according to the

example above. On exit, the routine returns the vector index as a numerical

value, between 1 and 3dim(j).

indx:=proc()

local N,i,ind,jvec;

jvec := args[1]; # Index N:tiple

N := linalg[vectdim](jvec); # Rank of tensor

ind := 0;

for i from 1 to N-1 do

ind := (ind+jvec[i]-1)*3;

od;

ind := ind + jvec[N];

RETURN(ind);

end:

The tensrotcomp() routine calculates the components of the tensor in the

Cartesian basis (x0; y0; z0), given the tensor expressed in the coordinate system

(x; y; z) and the rotation matrix R by which transformation between the two

coordinate systems is performed.

The routine takes three arguments as input:

1:st arg. The 'old' tensor or, in this case, the susceptibility tensor

in the reference frame of the crystal. The tensor may be

of arbitrary rank, and must be stored in the 'vectorized'

form described above. The tensor elements may be in

either numerical or symbolic form.

2:nd arg. Indices in the new, Cartesian, basis. The indices are

supposed to be stored in a N-dimensional vector j, with

N being the rank of the tensor passed as the 1:st arg.

3:rd arg. The matrix R describing the rotation of the coordinate

system. If r = (x; y; z) denotes the coordinates in the

frame of the crystal, and r0 = (x0; y0; z0) the coordinatesin the laboratory frame, the [3 � 3] rotation matrix de-

�nes the transformation between the two coordinate sys-

tems by r0 = R � r.On exit, the calculated tensor component is returned in numerical or symbolical

form, depending on the input.

tensrotcomp:=proc()

local comp,prod,oldtensor,R,N,i,j,k,m;

oldtensor := args[1]; # Elements in old basis

j := args[2]; # Indices of tensor in new basis

R := args[3]; # The 3D rotation matrix

N := linalg[vectdim](j); # Rank of the tensor

i := linalg[vector](N);

comp := 0;

75

for k from 1 to N do

i[k] := 1;

od;

for m from 1 to 3^N do

prod := oldtensor[indx(i)];


prod := prod*R[j[k],i[k]];

od:

comp := comp + prod;

k := N;

while (i[k] >= 3) do

i[k] := 1;

if (k > 1) then

k := k - 1;

fi:

od;

i[k] := i[k] + 1;

od;

RETURN(eval(comp));

end:

The tensrot() routine calculates all tensor elements in the rotated Cartesian

basis, given the tensor in the old Cartesian basis and the rotation matrix R.

The calculations are performed by making repeated calls to tensrotcomp(),

calculating the tensor elements for all possible indices of the tensor elements

in the new coordinate system.

The routine takes three arguments as input:

1:st arg. The set of tensor elements in the 'old' reference frame.

The tensor may be of arbitrary rank, and must be stored

in the 'vectorized' form described above. The tensor may

be in either numerical or symbolic form.

2:nd arg. The matrix R describing the rotation of the coordinate

system.

3:rd arg. The rank N of the tensor passed as 1:st argument,

N = ln(dim(1:st arg.))= ln 3.

On exit, the routine returns the tensor elements in the rotated coordinate sys-

tem, stored in the vectorized form described above.

tensrot:=proc()

local oldtensor,newtensor,R,N,i,k,m;

oldtensor := args[1]; # Elements in old coordinate system

R := args[2]; # The 3D rotation matrix

N := args[3]; # Rank of the tensor


newtensor := linalg[vector](3^N);


76 Appendix C. Routines for analysis of arbitrary rank tensors in rotated frames

i[k] := 1;

od;


newtensor[indx(i)]:=simplify(eval(tensrotcomp(oldtensor,i,R)));

k := N;


i[k] := 1;

if (k > 1) then

k := k - 1;

fi:

od;

i[k] := i[k] + 1;

od;

RETURN(eval(newtensor));

end:

The routine disptens() simply displays the nonzero components of a given

tensor, stored in the vectorized form described above.

The routine takes two arguments as input:

1:st arg. The tensor to display nonzero elements of.

2:nd arg. The rank N of the tensor passed as 1:st argument.

On exit, an empty string is returned.

disptens:=proc()

local tensor,R,N,i,k,m;

tensor := args[1];

N := args[2];



i[k] := 1;

od;


if (tensor[indx(i)] <> 0) then

print(i,tensor[indx(i)]);

fi;

k := N;


i[k] := 1;

if (k > 1) then

k := k - 1;

fi:

od;

i[k] := i[k] + 1;

od;

RETURN(``);

end:

77

De�ne the general transformation matrix for proper rotations. This one is

calculated from the regular Euler angles, found in most textbooks on Classical

Mechanics, in this case obtained from Herbert Goldstein's "Classical Mechan-

ics", 2:nd ed., (Addison-Wesley, 1980), ch. 4.4, pp. 146-147.

rotmatrix:=proc()

local phi,theta,psi,A,B,C,R;

phi := args[1]; # Rotation angle around z-axis

theta := args[2]; # Rotation angle around new x-axis

psi := args[3]; # Rotation angle around new z-axis

A:=linalg[matrix]([[ cos(psi), sin(psi), 0 ],

[-sin(psi), cos(psi), 0 ],

[ 0 , 0 , 1 ]]);

B:=linalg[matrix]([[ 1 , 0 , 0 ],

[ 0 , cos(theta), sin(theta)],

[ 0 ,-sin(theta), cos(theta)]]);

C:=linalg[matrix]([[ cos(phi), sin(phi), 0 ],

[-sin(phi), cos(phi), 0 ],

[ 0 , 0 , 1 ]]);

R:=linalg[multiply](A,B,C);

RETURN(eval(R));

end:

Finally, the following sequence creates the �le tensrot.m, which can be loaded

by other MapleV programs,

save `tensrot.m`;

78

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Part II

Papers I{V

83

84

Documents

Nonlinear8708/... · 2008. 9. 25. · nonlinear optics. Therefore, whenev er p ossible, the notations and con v en tions used in this thesis are in tended to follo w the standard