spoof surface plasmon revision - KAIST€¦ · C. Kittel, Introduction to Solid State Physics, 7th ed. (John Wiley, 1996 ... photonic Fermi surfaces by plasmon tomography,” Appl

Propagation of spoof surface plasmon onmetallic square lattice: bending and

splitting of self-collimated beams

Kap-Joong Kim,1 Jae-Eun Kim,1 Hae Yong Park,1 Yong-Hee Lee,1,4

Seong-Han Kim,2 Sun-Goo Lee,2 and Chul-Sik Kee2,3,∗

1Department of Physics, KAIST, Daejon 305-701, South Korea2Ultra-Intense Laser Laboratory, APRI, GIST, Gwangju 500-712, South Korea

3Center for Subwavelegth Optics, Seoul 151-747, South [email protected]∗[email protected]

Abstract: The propagation characteristics of spoof surface plasmonmodes are studied in both real and reciprocal spaces. From the metallicsquare lattice, we obtain constant frequency contours by directly measuringelectric fields in the microwave frequency regime. The anisotropy of themeasured constant frequency contour supports the presence of the negativerefraction and the self-collimation which are confirmed from measuredelectric fields. Additionally, we demonstrate the spoof surface plasmonbeam splitter in which the splitting ratio of the self-collimated beam iscontrolled by varying the height of rods.

© 2014 Optical Society of America

OCIS codes: (240.6680) Surface plasmons; (240.6690) Surface waves; (250.5403) Plasmonics;(160.3918) Metamaterials.

References and links1. H. Raether,Surface Plasmons (Springer, 1988).2. J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,”

Science305, 847–848 (2004).3. A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science

308, 670–672 (2005).4. H. J. Rance, I. R. Hooper, A. P. Hibbins, and J. R. Sambles, “Structurally dictated anisotropic designer surface

plasmons,” Appl. Phys. Lett.99, 181107 (2011).5. S. J. Berry, T. Campbell, A. P. Hibbins, and J. R. Sambles, “Surface wave resonances supported on a square array

of square metallic pillars,” Appl. Phys. Lett.100, 101107 (2012).6. C. R. Williams, S. R. Andrews, S. A. Maier, A. I. Fernandez-Dominguez, L. Martin Moreno, and F. J. Garcia-

Vidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Pho-tonics2, 175–179 (2008).

7. J. T. Shen, P. B. Catrysse, and S. H. Fan, “Mechanism for designing metallic metamaterials with a high index ofrefraction,” Phys. Rev. Lett.94, 197401 (2005).

8. J. Shin, J.-T. Shen, P. B. Catrysse, and S. Fan, “Cut-through metal slit array as an anisotropic metamaterial film,”IEEE J. Sel. Top. Quantum Electron.12, 1116–1121 (2006).

9. C. Kittel, Introduction to Solid State Physics, 7th ed. (John Wiley, 1996).10. G. R. Fowles,Introduction to Modern Optics (Dover, 1989).11. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating

phenomena in photonic crystals,” Appl. Phys. Lett.74, 1212–1214 (1999).12. P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacic, G. S. Petrich, J. D. Joannopoulos, L. A.

Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensionalphotonic crystal,” Nat. Mater.5, 93–96 (2006).

#201357 - $15.00 USD Received 18 Nov 2013; revised 6 Feb 2014; accepted 7 Feb 2014; published 13 Feb 2014(C) 2014 OSA 24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.004050 | OPTICS EXPRESS 4050

13. B. Stein, E. Devaux, C. Genet, and T. W. Ebbesen, “Self-collimation of surface plasmon beams,” Opt. Lett.37,1916–1918 (2010).

14. S.-H. Kim, T.-T. Kim, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Experimental demonstration of self-collimation of spoof surface plasmons,” Phys. Rev. B83, 165109 (2011).

15. S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev.Lett. 89, 213902 (2002).

16. Y. Yuan, L. Shen, L. Ran, T. Jiang, J. Huangfu, and J. A. Kong, “Directive emission based on anisotropic meta-materials,” Phys. Rev. A77, 053821 (2008).

17. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phe-nomena in photonic crystals,” Phys. Rev. B58, R10096–R10099 (1998).

18. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negativeeffective index,” Phys. Rev. B65, 201104 (2002).

19. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Negative refraction by photoniccrystals,” Nature423, 604–605 (2003).

20. H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metalstructure,” Phys. Rev. Lett.96, 073907 (2006).

21. E. Verhagen, R. de Waele, L. Kuipers, and A. Polman, “Three-dimensional negative index of refraction at opticalfrequencies by coupling plasmonic waveguides,” Phys. Rev. Lett.105, 223901 (2010).

22. B. Stein, J. Y. Laluet, E. Devaux, C. Genet, and T. W. Ebbesen, “Surface plasmon mode steering and negativerefraction,” Phys. Rev. Lett.105, 266804 (2010).

23. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Subwavelength resolution in a two-dimensional photonic-crystal-based superlens,” Phys. Rev. Lett.91, 207401 (2003).

24. P. A. Belov, C. R. Simovski, and P. Ikonen, “Canalization of subwavelength images by electromagnetic crystals,”Phys. Rev. B71, 193105 (2005).

25. X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater.7, 435–441 (2008).26. B. Stein, J. Y. Laluet, E. Devaux, C. Genet, and T. W. Ebbesen, “Surface plasmon mode steering and negative

refraction,” Phys. Rev. Lett.105, 266804 (2010).27. B. Stein, E. Devaux, C. Genet, and T. W. Ebbesen, “Self-collimation of surface plasmon beams,” Opt. Lett.37,

1916–1918 (2012).28. C. J. Regan, A. Krishnan, R. Lopez-Boada, L. Grave de Peralta, and A. A. Bernussi, “Direct observation of

photonic Fermi surfaces by plasmon tomography,” Appl. Phys. Lett.98, 151113 (2011).29. C. J. Regan, L. Grave de Peralta, and A. A. Bernussi, “Equifrequency curve dispersion in dielectric-loaded

plasmonic crystals,” J. Appl. Phys.111, 073105 (2012).30. T. J. Constant, A. P. Hibbins, A. J. Lethbridge, J. R. Sambles, E. K. Stone, and P. Vukusic, “Direct mapping of

surface plasmon dispersion using imaging scatterometry,” Appl. Phys. Lett.102, 251107 (2013).31. A. Taflove and S. C. Hagness,Computational Electrodynamics: The Finite-Difference Time-Domain Method,

(Artech House, 2000).32. J. D. Jackson,Classical Electrodynamics (John Wiley, 1999).33. X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystal,” Appl. Phys. Lett.83, 3251

(2003).34. X. Yu and S. Fan, “Anomalous reflections at photonic crystal surfaces,” Phys. Rev. E70, 055601 (2004).35. S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Line-defect-induced bending and splitting of selfcol-

limated beams in two-dimensional photonic crystals,” Appl. Phys. Lett.87, 181106 (2005).36. S.-G. Lee, J.-S. Choi, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Reflection minimization at two-dimensional photonic

crystal interfaces,” Opt. Express16, 4270–4277 (2008).

1. Introduction

In the visible wavelength regime, electromagnetic (EM) surface waves can be formed at theinterface between the dielectric and the metal (i.e., surface plasmons [1]). At terahertz and mi-crowave frequencies, there are no surface waves which are tightly bound at the metal-dielectricinterface because metals can often be treated as perfect electric conductors. However, recentstudies have shown that surface waves on the structured metal with arrays of sub-wavelengthholes or rods are strongly confined at these frequencies. The surface wave is commonly calledas a spoof or designer surface plasmon [2–5]. In the terahertz regime, versatile approachescan be employed to control spoof surface plasmon modes since the dispersion relation can bedesigned by engineering the structure. Recently, guiding devices based on the spoof surfaceplasmon have been demonstrated [6].

In particular, the rod array on conducting ground plane was shown to support propagating


transverse electromagnetic modes [5]. These modes are similar to that supported in metal filmswith periodic cut-through slits at low frequencies [7, 8]. Hence, unlike the hole array, by vary-ing geometrical parameters, the rod array can be regarded as a material with high effectiverefractive index. Such capability can be advantageous for the miniaturization of photonic de-vices. Moreover, anisotropic dispersion characteristics can lead to unusual propagation on therod array.

Certain characteristics of the EM wave propagation can be described and understood moreclearly from its constant frequency surfaces in the reciprocal space [9]. The propagation di-rection of an EM wave is identical to that of the group velocity,vg = ∇kω(k), which meansthat the group velocity is normal to the constant frequency surfaces [10]. Specially, in artificialperiodic structures, the constant frequency contour (CFC) has been employed in understandingunusual light propagation phenomena, i.e., self-collimation [11–14], directive emission [15,16],superprism [17], negative refraction [18–22], and sub-wavelength imaging [23–25].

In plasmonic crystals, the leakage radiation setup [26–29] or an imaging scatterometer [30]were commonly employed to map CFCs. Recently in the microwave regime, phase resolvedmeasurement techniques were employed to obtain CFCs and dispersion relations along onlylimited directions [4, 5]. However, for insightful understanding of the surface wave propaga-tion on structured metals, the dispersion relations along all symmetric directions are neces-sary. In this paper, we experimentally obtain CFCs of two-dimensional (2D) metallic squarelattices (MSLs) by scanning over the MSL. We demonstrate the negative refraction and theself-collimated propagation of spoof surface plasmon. We further demonstrate the bending andsplitting of self-collimated beams by varying the height of rods.

2. Sample and experiment setup

(a)

(c)

0

5

10

15

20

(b)

Γ X ΓM

Fre

qu

ency

(G

Hz)

Fig. 1. (a) A sample of the square copper rod (w = 6 mm, h = 10 mm, anda = 10 mm)on the copper plate. (b) A band structure for the MSL, where a red line, gray lines, and ablack line correspond to the first band, higher bands, and the light line respectively. Thegray region indicated a surface wave band gap of the MSL. The region above the light lineindicates the radiation modes in air. (c) Schematic of the MSL.

The physical system investigated here consists of a square lattice of copper rods in air, havinga widthw = 6 mm, heighth = 10 mm, and perioda = 10 mm. Figure 1(a) shows the schematicdiagram of a unit cell of the MSL. The MSL is placed on the flat cooper plate as shown inFig. 1(c). By performing three-dimensional (3D) finite-difference time-domain (FDTD) simu-


(a) (b)

(c)

Fig. 2. (a) Experimental configuration in which two monopole antennas (light blue) used asa source and a detector, respectively. The sample is surrounded by the microwave absorber(blue). The dashed-line box corresponds to the scanning area. Schematic of (b) monopoleantenna and (c) microwave absorber.

lation [31], the MSL is designed to produce a flat region along the frequency contour and toobtain the negative refraction around 6 GHz. The band structure shows a surface wave band gapranging from 6.35 to 10.88 GHz as shown in Fig. 1(b). Since, in the microwave regime, metalsact as perfect electric conductors (PECs), the copper is modeled as a PEC in the simulation.

To measure the electric field distribution, we employ a near-field scanning system in mi-crowave frequency regime. The experimental setup consists of a vector network analyzer (HP8720C) and a pair of two monopole antennas (stripped coaxial antennas) arranged in thez-direction which excites and collects the surface waves, respectively, as illustrated in Fig. 2(a).The diameter of the monopole antenna is 1.6 mm and the length is 3 mm as illustrated inFig. 2(b). The source antenna is designed to generate both far- and near-fields, where the sourceantenna acts as a point source produced by an oscillating electric dipole in thez-direction [32].Therefore, the transverse-magnetically polarized radiation emitted from the monopole antennacan couple with the surface wave. Thez-component of the electric fields of the surface wavecan be coupled back into the detector monopole antenna. The detector antenna is mounted onanxy-motorized stage to measure the S-parameters at any given point.

3. Theory

To obtain a CFC for a 2D periodic system, the electric field distribution with all supportedmodes in the system at a frequency ofω has to be collected. The electric fieldE(r)eiωt ata position vectorr and with a time dependenceeiωt can be obtained from the S-parameterasE(r)eiωt = S21(r)Esrc(r0)eiωt , whereEsrc(r0) is the electric field of a source at a positionvector r0. When the system size is sufficiently larger than wavelength corresponding toω ,this system can be closely approximated to an infinite lattice. Thus, the electric field can bedefined at a single wavevectork asEk,G(r) = ck,Gei(G+k)·r, whereG is a lattice vector of theperiodic system andck,G are the Fourier coefficients. In the reciprocal space, the electric fieldES(G+k′−k) can be obtained by a 2D Fourier transform,

∫S Ek′,G(r)e−ik·rd2r, whereS is an

area of the electric field distribution. The electric field in the reciprocal space with infinity area


is described by a delta function written as limS→inf ES(G+k′−k) ∝ δ (G+k′−k). This meansthat the transformed field is zero everywhere except atk = G+k′. Thus, CFCs in the periodicsystem can be expressed by Fourier-transformed fields at each frequency.

In general, the electric field in reciprocal space is no longer the delta function because anarea of the electric field distribution is finite. However, when the size of the system is muchlarger than operating wavelengths, resonance peaks in finite and infinite cases become almostidentical. Therefore, the peaks of the Fourier-transformed fields can be justifiably used fordetermination of the experimental CFC.

4. Propagation of spoof surface plasmons

0.25

0.5

0

π2

2kya

/

x (cm)−10 0 100

y (c

m)

10

20

−1

0

1

(a)

(c)

0

1

−0.5π2 2k

xa/

0 0.5

(b)

1st BZ

3

5

7

Fre

qu

ency

(G

Hz)

Γ X ΓM

(d)

0

0.5

−0.5

π2 2kxa/

−0.5 0 0.5

π2

2kya

/

4.3 GHz

5.4 GHz

5.9 GHz

6.06 GHz

6.2 GHz

Γ

X

M

Fig. 3. (a) Measured distribution ofEz at 6.13 GHz. (b) The magnitude distribution ofthe Fourier-transformed fields in the second BZ obtained from (a). The dashed-line boxindicates the first BZ and the solid-line box is the area selected by the symmetry. Whitecurves indicate calculated CFCs at 6.13 GHz. The direction in (a) corresponds to that in (b)due to the symmetry of a square lattice. (c) Measured CFCs (red lines) and the calculation(black lines) at a few selected frequencies. The gray box corresponds to the solid-line box in(b). (d) The measured dispersion of the first band (red line) and the calculation superposed(black line). The gray area indicates the band gap.

The MSL for the measurement is composed of 20√

2 a×20√

2 a as shown in Fig. 2(a). Tominimize the reflection at the MSL/air boundaries, the sample is surrounded by the microwaveabsorber (Fig. 2(c)) with -20 dB reflectance at normal incidence. The source antenna is placedat z = 1 mm above the MSL and at 1.5

√2 a from an absorber boundary to excite the all surface


0

y (c

m)

10

20

x (cm)−10 0 10

(a)

x (cm)−10 0 10

(b)

x (cm)−10 0 10

(c)

0

−1

1

Fig. 4. Distributions ofEz of (a) the positive refraction (5.9 GHz), (b) self-collimation (6.06GHz), and (c) the negative refraction (6.2 GHz).

waves as shown in Fig. 1(b). The phase and amplitude of the collected signal expressed byS-parameterS21 are recorded using a vector network analyzer connected to two monopole an-tennas. We collect each S-parameter of 11,449 points in the step of 2 mm along thexy-directionon the planez = 2 mm above the MSL. The frequency is varied from 3 to 6.5 GHz with aninterval of 0.005 GHz.

From the S-parameters, the measured electric field distribution at 6.13 GHz is shown inFig. 3(a), where the scanning area is 212 mm× 212 mm as shown in Fig. 2(a). From this fielddistribution with zero padding, typical Fourier-transformed electric fields are obtained. Fig-ure 3(b) shows the magnitude distribution of the Fourier-transformed fields selected in the sec-ond Brillouin zone (BZ), where the bright thick lines correspond to the guided surface waves.Because of the position of the point source, those guided surface waves withky > 0 are strong.Considering the symmetry of the square lattice, we select peaks in the transformed fields withina part of the first BZ. The measured CFCs at selected frequencies (Fig. 3(c)) and the measureddispersion of the first band at all frequencies (Fig. 3(d)) show excellent agreement with thepredictions from the 3D FDTD simulation.

To observe the refraction of incident waves from air into the MSL, a point source is placedat a distance of 5

√2 mm from the MSL as shown in Figs. 4(a), 4(b), and 4(c). S-parameters at

27,150 points are measured in the step of 2 mm on thez = 2 mm plane above the rods array.The scanning area is 298 mm× 360 mm. In this structure, the curvature of the CFCs changesfrom concave to convex through the flat region as shown in Fig. 3(c). Thus, we classify threetypes of wave propagation, which are experimentally shown in Figs. 4(a), 4(b), and 4(c). Thefirst type is the positive refraction (Fig. 4(a)) showing the divergent propagation with upwardwavefronts at 5.9 GHz. The second is the self-collimation (Fig. 4(b)) showing the collimatedpropagation without diffraction in the direction normal to flat CFCs at 6.06 GHz. It is importantto note that although the wave from the point source diverges, the collimated beam measuredalongxy-plane has an approximately unchanging width. The last type is the negative refractionshowing convergent propagation with the downward wavefronts at 6.2 GHz. This gives rise tothe negative refraction as shown in Fig. 4(c).

5. Bending and splitting of self-collimated beams

If the interface-parallel component of the incident wavevector in the incident medium is largerthan that of refracted waves in air, then the incident waves are totally reflected. In this case, the


π2 2kxa/

0

(c)

0.13−0.13

(a)

0.28

0.29π2

2kya/

0

0.5

−0.5

(b)

Air

Interface

MRA

π2 2kxa/

−0.5 0 0.5

π2

2kya/

Input

Reflection

Fig. 5. (a) A magnified view of calculated CFC in which the gray area indicates wavevectorregion for the self-collimated beam. The dashed-line indicates maximum magnitude of theMSL/air interface parallel component of wavevectors in air. (b) Calculated CFCs for air andthe MSL at 6.05 GHz. Here arrows correspond tovg of the input beam and the reflectedbeam. A black line corresponds to the MSL/air interface. (c) Schematic of the bendingstructure where the input beam undergoes total internal reflection at the MSL/air interface.

(a)

Total

Reflection

Transmission

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

No

rmal

ized

po

wer

(b)

hd (mm)

Fig. 6. (a) Schematic of an MSL beam splitter with the single line defect. (b) Total, re-flected, and transmitted powers are normalized by the input power as a function ofhd at6.05 GHz. Whenhd = 0.93a, a 50:50 splitter is obtained.

interface itself can be used as a perfect reflector.We find that the CFC shows a flat region in the vicinity of 6.05 GHz. And the self-collimation

phenomenon is observed experimentally as shown in Fig. 4(b). The wavevector region for theself-collimated propagating alongΓM direction is numerically found−0.06. kx . 0.06 asshown in Fig. 5(a), wherekx lies outside the CFC for air (Fig. 5(a)). Thus, the MSL/air interface(Fig. 5(b)) alongΓX direction behaves like a total internal reflector and can be used to bend theself-collimated beams as illustrated in Fig. 5(c) [33, 34]. Upon the total internal reflection, theelectric fields in air decays exponentially. Therefore, the self-collimated beam can be reflectedby removing rods and split into two self-collimated beams by varying degree of the tunnelingof the electric fields [35].


0

10

20

y (c

m)

−10 0 10x (cm)

0

10

20

y (c

m)

−10 0 10x (cm)

0

10

20

y (c

m)

−10 0 10x (cm)

0

1

0

1

0

1(d)(c)

(b)(a)

Fig. 7. (a) Schematic of a beam splitter with the line defect which consists of 15 rods ina row with the heighthd , where the dashed-line box indicates the scanning area. Selectedelectric field intensities withhd of (b) 0 , (c) 5, and (d) 9.5 mm. at 6.06 GHz

To investigate the possibility of beam splitting, we design a line defect structure which con-sists of smaller rods in a row alongΓX direction as shown in Fig. 6(a). A source antenna is usedto generate self-collimated beams propagating alongΓM direction and placed at a distance of5√

2 mm from the MSL. The time averaged power flow across the beam cross section is numer-ically obtained without back reflection from MSL boundaries at the far end. Figure 6(b) showsthe reflected and transmitted powers normalized by the incident power at 6.05 GHz. The resultshows that, by varyinghd from 0 mm to 10 mm, the beam splitting ratio is controlled from 0.06to 1. The sum of the transmitted and reflected power is between 0.97 and 0.98. Moreover, thebeam splitter behaves as a 50:50 splitter whenhd is 0.93a.

For experimental verification, we employ 20√

2 a×20√

2 a with the line defect composed of15 rods alongΓX direction. To observe the reflection at the MSL/air interface, the the scanningarea is 298 mm× 360 mm as illustrated in Fig. 7(a). In the step of 2 mm on thez = 2 mm planeabove the rods array, we measure S-parameters of 27,150 points. A point source is placed at adistance of 5

√2 mm from the MSL/air interface (Fig. 7(a)). Several different values ofhd = 0,

5, 7, 8, 9, and 9.5 mm are tested for comparison purpose at 6.06 GHz. Figure 7(b) shows totalinternal reflection of the self-collimated beam, wherehd = 0 mm. Even with the line defect ofonly one period (1a) alongΓX direction, the electric field amplitude is most rapidly decayedinto air gap. In Figs. 7(c) and 7(d), the field distributions (hd = 5 and 9.5 mm) show splitting ofself-collimated beams. The weak beams in the+x side are associated with the back reflectionat the MSL/air boundaries. With proper anti-reflection layers, these unwanted back reflection


effects can be minimized [36].

6. Conclusion

The constant frequency contours of spoof surface plasmons are obtained by scanning elec-tric field distributions over a square array of metallic rods. Positive refraction, self-collimatedpropagation, and negative refraction of spoof surface plasmons are directly observed on thesurface of the metallic rod array. In addition, we demonstrate the bending and splitting of self-collimated beams, which could be useful for terahertz and microwave circuitries. This near-fieldscanning scheme could be employed to study surface waves in non-periodic structures.

Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) grant funded bythe Ministry of Education, Science and Technology (NRF-2007-0093863 NRF-2008-0062257,and NRF-2012-040573) and a research program (Applications of Ultrashort Quantum BeamFacility) through a Grant provided by the GIST in 2013.


Documents

spoof surface plasmon revision - KAIST€¦ · C. Kittel, Introduction to Solid State Physics, 7th ed. (John Wiley, 1996 ... photonic Fermi surfaces by plasmon tomography,” Appl