Propagation of spoof surface plasmon on metallic square lattice : bending and splitting of self-collimated beams

The propagation characteristics of spoof surface plasmon modes are studied in both real and reciprocal spaces. From the metallic square lattice, we obtain constant frequency contours by directly measuring electric fields in the microwave frequency regime. The anisotropy of the measured constant frequency contour supports the presence of the negative refraction and the self-collimation which are confirmed from measured electric fields. Additionally, we demonstrate the spoof surface plasmon beam splitter in which the splitting ratio of the self-collimated beam is controlled 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 links 1. 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. GarciaVidal, “Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces,” Nat. Photonics2, 175–179 (2008). 7. J. T. Shen, P. B. Catrysse, and S. H. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” 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. Soljačić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic 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 selfcollimation of spoof surface plasmons,” Phys. Rev. B 83, 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 metamaterials,” Phys. Rev. A77, 053821 (2008). 17. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096–R10099 (1998). 18. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B65, 201104 (2002). 19. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Negative refraction by photonic crystals,” Nature423, 604–605 (2003). 20. H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96, 073907 (2006). 21. E. Verhagen, R. de Waele, L. Kuipers, and A. Polman, “Three-dimensional negative index of refraction at optical frequencies 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 negative refraction,” Phys. Rev. Lett. 105, 266804 (2010). 23. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Subwavelength resolution in a twodimensional 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. E 70, 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 selfcollimated 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. Express 16, 4270–4277 (2008).


Introduction
In the visible wavelength regime, electromagnetic (EM) surface waves can be formed at the interface between the dielectric and the metal (i.e., surface plasmons [1]).At terahertz and microwave frequencies, there are no surface waves which are tightly bound at the metal-dielectric interface because metals can often be treated as perfect electric conductors.However, recent studies have shown that surface waves on the structured metal with arrays of sub-wavelength holes or rods are strongly confined at these frequencies.The surface wave is commonly called as a spoof or designer surface plasmon [2][3][4][5].In the terahertz regime, versatile approaches can be employed to control spoof surface plasmon modes since the dispersion relation can be designed by engineering the structure.Recently, guiding devices based on the spoof surface plasmon 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 films with periodic cut-through slits at low frequencies [7,8].Hence, unlike the hole array, by varying geometrical parameters, the rod array can be regarded as a material with high effective refractive index.Such capability can be advantageous for the miniaturization of photonic devices.Moreover, anisotropic dispersion characteristics can lead to unusual propagation on the rod array.Certain characteristics of the EM wave propagation can be described and understood more clearly from its constant frequency surfaces in the reciprocal space [9].The propagation direction of an EM wave is identical to that of the group velocity, v g = ∇ k ω(k), which means that the group velocity is normal to the constant frequency surfaces [10].Specially, in artificial periodic structures, the constant frequency contour (CFC) has been employed in understanding unusual light propagation phenomena, i.e., self-collimation [11][12][13][14], directive emission [15,16], superprism [17], negative refraction [18][19][20][21][22], and sub-wavelength imaging [23][24][25].
In plasmonic crystals, the leakage radiation setup [26][27][28][29] or an imaging scatterometer [30] were commonly employed to map CFCs.Recently in the microwave regime, phase resolved measurement techniques were employed to obtain CFCs and dispersion relations along only limited directions [4,5].However, for insightful understanding of the surface wave propagation on structured metals, the dispersion relations along all symmetric directions are necessary.In this paper, we experimentally obtain CFCs of two-dimensional (2D) metallic square lattices (MSLs) by scanning over the MSL.We demonstrate the negative refraction and the self-collimated propagation of spoof surface plasmon.We further demonstrate the bending and splitting of self-collimated beams by varying the height of rods.The physical system investigated here consists of a square lattice of copper rods in air, having a width w = 6 mm, height h = 10 mm, and period a = 10 mm. Figure 1  lation [31], the MSL is designed to produce a flat region along the frequency contour and to obtain the negative refraction around 6 GHz.The band structure shows a surface wave band gap ranging from 6.35 to 10.88 GHz as shown in Fig. 1(b).Since, in the microwave regime, metals act as perfect electric conductors (PECs), the copper is modeled as a PEC in the simulation.

Sample and experiment setup
To measure the electric field distribution, we employ a near-field scanning system in microwave frequency regime.The experimental setup consists of a vector network analyzer (HP 8720C) and a pair of two monopole antennas (stripped coaxial antennas) arranged in the zdirection 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 in Fig. 2(b).The source antenna is designed to generate both far-and near-fields, where the source antenna acts as a point source produced by an oscillating electric dipole in the z-direction [32].Therefore, the transverse-magnetically polarized radiation emitted from the monopole antenna can couple with the surface wave.The z-component of the electric fields of the surface wave can be coupled back into the detector monopole antenna.The detector antenna is mounted on an xy-motorized stage to measure the S-parameters at any given point.

Theory
To obtain a CFC for a 2D periodic system, the electric field distribution with all supported modes in the system at a frequency of ω has to be collected.The electric field E(r)e iωt at a position vector r and with a time dependence e iωt can be obtained from the S-parameter as E(r)e iωt = S 21 (r)E src (r 0 )e iωt , where E src (r 0 ) is the electric field of a source at a position vector r 0 .When the system size is sufficiently larger than wavelength corresponding to ω, this system can be closely approximated to an infinite lattice.Thus, the electric field can be defined at a single wavevector k as E k,G (r) = c k,G e i(G+k)•r , where G is a lattice vector of the periodic system and c k,G are the Fourier coefficients.In the reciprocal space, the electric field E S (G + k ′ − k) can be obtained by a 2D Fourier transform, S E k ′ ,G (r)e −ik•r d 2 r, where S 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 lim S→inf E S (G + k ′ − k) ∝ δ (G + k ′ − k).This means that the transformed field is zero everywhere except at k = G + k ′ .Thus, CFCs in the periodic system 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 an area of the electric field distribution is finite.However, when the size of the system is much larger than operating wavelengths, resonance peaks in finite and infinite cases become almost identical.Therefore, the peaks of the Fourier-transformed fields can be justifiably used for determination of the experimental CFC.The MSL for the measurement is composed of 20 √ 2 a × 20 √ 2 a as shown in Fig. 2(a).To minimize the reflection at the MSL/air boundaries, the sample is surrounded by the microwave absorber (Fig. 2(c)) with -20 dB reflectance at normal incidence.The source antenna is placed at z = 1 mm above the MSL and at 1.5 √ 2 a from an absorber boundary to excite the all surface waves as shown in Fig. 1(b).The phase and amplitude of the collected signal expressed by S-parameter S 21 are recorded using a vector network analyzer connected to two monopole antennas.We collect each S-parameter of 11,449 points in the step of 2 mm along the xy-direction on the plane z = 2 mm above the MSL.The frequency is varied from 3 to 6.5 GHz with an interval of 0.005 GHz.
From the S-parameters, the measured electric field distribution at 6.13 GHz is shown in Fig. 3(a), where the scanning area is 212 mm × 212 mm as shown in Fig. 2(a).From this field distribution with zero padding, typical Fourier-transformed electric fields are obtained.Considering the symmetry of the square lattice, we select peaks in the transformed fields within a part of the first BZ.The measured CFCs at selected frequencies (Fig. 3(c)) and the measured dispersion of the first band at all frequencies (Fig. 3(d)) show excellent agreement with the predictions from the 3D FDTD simulation.
To observe the refraction of incident waves from air into the MSL, a point source is placed at 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 the z = 2 mm plane above the rods array.The scanning area is 298 mm × 360 mm.In this structure, the curvature of the CFCs changes from concave to convex through the flat region as shown in Fig. 3(c).Thus, we classify three types of wave propagation, which are experimentally shown in Figs.4(a), 4(b), and 4(c).The first type is the positive refraction (Fig. 4(a)) showing the divergent propagation with upward wavefronts at 5.9 GHz.The second is the self-collimation (Fig. 4(b)) showing the collimated propagation without diffraction in the direction normal to flat CFCs at 6.06 GHz.It is important to note that although the wave from the point source diverges, the collimated beam measured along xy-plane has an approximately unchanging width.The last type is the negative refraction showing convergent propagation with the downward wavefronts at 6.2 GHz.This gives rise to the negative refraction as shown in Fig. 4(c).

Bending and splitting of self-collimated beams
If the interface-parallel component of the incident wavevector in the incident medium is larger than that of refracted waves in air, then the incident waves are totally reflected.In this case, the 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 the self-collimated propagating along ΓM direction is numerically found −0.06 k x 0.06 as shown in Fig. 5(a), where k x lies outside the CFC for air (Fig. 5(a)).Thus, the MSL/air interface (Fig. 5(b)) along ΓX direction behaves like a total internal reflector and can be used to bend the self-collimated beams as illustrated in Fig. 5(c) [33,34].Upon the total internal reflection, the electric fields in air decays exponentially.Therefore, the self-collimated beam can be reflected by removing rods and split into two self-collimated beams by varying degree of the tunneling of the electric fields [35].To investigate the possibility of beam splitting, we design a line defect structure which consists of smaller rods in a row along ΓX direction as shown in Fig. 6(a).A source antenna is used to generate self-collimated beams propagating along ΓM direction and placed at a distance of 5 √ 2 mm from the MSL.The time averaged power flow across the beam cross section is numerically obtained without back reflection from MSL boundaries at the far end.Figure 6(b) shows the reflected and transmitted powers normalized by the incident power at 6.05 GHz.The result shows that, by varying h d from 0 mm to 10 mm, the beam splitting controlled from 0.06 to 1.The sum of the transmitted and reflected power is between 0.97 and 0.98.Moreover, the beam splitter behaves as a 50:50 splitter when h d is 0.93a.
For experimental verification, we employ 20 √ 2 a × 20 √ 2 a with the line defect composed of 15 rods along ΓX direction.To observe the reflection at the MSL/air interface, the the scanning area is 298 mm × 360 mm as illustrated in Fig. 7(a).In the step of 2 mm on the z = 2 mm plane above the rods array, we measure S-parameters of 27,150 points.A point source is placed at a distance of 5 √ 2 mm from the MSL/air interface (Fig. 7(a)).Several different values of h d = 0, 5, 7, 8, 9, and 9.5 mm are tested for comparison purpose at 6.06 GHz.effects can be minimized [36].

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

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

Fig. 2 .
Fig. 2. (a) Experimental configuration in which two monopole antennas (light blue) used as a 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) monopole antenna and (c) microwave absorber.

Fig. 3 .
Fig. 3. (a) Measured distribution of E z at 6.13 GHz.(b) The magnitude distribution of the Fourier-transformed fields in the second BZ obtained from (a).The dashed-line box indicates the first BZ and the solid-line box is the area selected by the symmetry.White curves 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.
Figure 3(b) shows the magnitude distribution of the Fourier-transformed fields selected in the second 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 with k y > 0 are strong.

Fig. 5 .Fig. 6 .
Fig. 5. (a) A magnified view of calculated CFC in which the gray area indicates wavevector region for the self-collimated beam.The dashed-line indicates maximum magnitude of the MSL/air interface parallel component of wavevectors in air.(b) Calculated CFCs for air and the MSL at 6.05 GHz.Here arrows correspond to v g of the input beam and the reflected beam.A black line corresponds to the MSL/air interface.(c) Schematic of the bending structure where the input beam undergoes total internal reflection at the MSL/air interface.

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

Figure 7 (
b) shows total internal reflection of the self-collimated beam, where h d = 0 mm.Even with the line defect of only one period (1a) along ΓX direction, the electric field amplitude is most rapidly decayed into air gap.In Figs.7(c) and 7(d), the field distributions (h d = 5 and 9.5 mm) show splitting of self-collimated beams.The weak beams in the +x side are associated with the back reflection at the MSL/air boundaries.With proper anti-reflection layers, these unwanted back reflection #201357 -$15.00USD 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 4057