In this work, we investigate the propagation of designer surface plasmons in planar perfect electric conductor structures that are subject to a parabolic graded-index distribution. A three-dimensional, fully vectorial finite-difference time-domain method was used to engineer a structure with a parabolic effective group index by modulating the dielectric constant of the structure’s square holes. Using this structure in our simulations, the lateral confinement of propagating designer surface plasmons is demonstrated. Focusing, collimation and waveguiding of designer plasmons in the lateral direction is realized by changing the width of the source beam. Our findings contribute to applications of designer surface plasmons that require energy concentration, diffusion, guiding, and beam aperture modification within planar perfect electric conductors.
©2009 Optical Society of America
Extraordinary optical transmission (EOT) is the enhanced transmission of incident electromagnetic radiation through sub-wavelength-scale holes in metallic films. [1, 2] EOT in real metals is based on two types of surface waves: 1) conventional surface plasmons (SPs) that are excited at a metal-dielectric interface, and 2) surface-bound states that exist on structured perfect electric conductor (PEC) surfaces. Although a PEC generally does not support any surface-bound states since an electromagnetic field cannot penetrate the surface, highly localized surface-bound states appear when the PEC is periodically modulated with arrays of sub-wavelength square or circular holes. Both theoretical [3, 4, 5, 6] and experimental [7, 8, 9, 10] studies suggest that surface-bound states and SPs exhibit similar dispersion relationships. Due to the similarity such surface-bound states are referred to as ‘spoof’ or ‘designer’ surface plasmons (DSPs).
Structured PEC surfaces and the excited DSPs have recently garnered interest within the photonics community, as a new platform to engineer surface-bound states of a wide frequency range. [11, 12, 13, 14] An important example is the guiding of terahertz-range radiation [15, 16] in the form of DSPs. This enables the application of terahertz plasmonics  to near-field imaging, sensing, and spectroscopy. A prime advantage of DSP is that, unlike conventional SP, the propagation of these waves can be controlled by engineering the material-independent, perceived group index.  PEC structures have been engineered to guide DSPs of specific terahertz-range frequencies. For example, periodically corrugated metal wires have been developed for guiding and focusing terahertz-frequency pulses [18, 19, 20]. By placing two optimized metallic grating structures on opposite sides of a narrow slit, Gan et al. were able to selectively guide terahertz-range waves along the two desired directions.  Similarly, metallic gratings have been found to significantly decrease the group velocity near the cutoff frequency.  Gan et al.  extended this notion by using graded metallic gratings to slow the propagation of wide-bandwidth, terahertz-range DSPs.
Successful applications of the propagation of DSPs on planar structures include waveguiding, imaging and sensing. Such applications require both vertical and lateral confinement of DSPs. To realize the latter, Oh et al.  exploited the anisotropic variation of the effective group indices of DSPs with frequency to achieve self-collimation . Ruan et al.  achieved focusing/imaging by negative refraction. These methods work only at wavelengths comparable to the periodic spacing of the apertures where the isofrequencies have zero or negative curvature. Another approach to realize lateral confinement was proposed by Maier et al., in which a defect mode was realized by gradually increasing the hole size, leading to evanescent decay along the lateral direction. 
In this study, we demonstrate the lateral confinement of DSPs by engineering planar PEC structures as parabolic graded media. A lens with a parabolic gradient in its refractive index, N, transverse to the propagation of incident light, will cause the beam to propagate in a periodic fashion, as illustrated in Fig. 1(a). [28, 29] The parabolic gradient forces the beam to propagate periodically and can be explained by Snell’s law. Electromagnetic energy emanating from a line source is focused in the first quarter pitch and later expanded/collimated in the second quarter pitch of the lens (the opposite is true for a point source). A similar effect has been observed for bulk waves in graded metamaterials of negative refractive indices  as well as for photonic crystals [31, 32, 33]. However, to-date no one has reported the propagation of surface waves like DSPs for perfect electric conductors using parabolic gradient index media. Such engineered structures would enable lateral confinement of DSPs at frequencies in the isotropic dispersion region, and would function as on-chip optical devices after the DSPs were excited and being guided.
To establish such a parabolic gradient in the effective group index along the direction transverse to the propagation of DSPs, we modulated either the size of the holes a or the dielectric constant εh of holes in the PEC structure, as shown in Fig. 1(b)–(c). Using a three-dimensional (3D) finite-difference time-domain (FDTD) method, we confirmed the effect of the parabolic gradient on the propagation of DSPs by achieving focusing, collimation, and wave-guiding (depending on the width of the light source). Our FDTD simulations agreed with analytic models from standard Gaussian optics.
2. Design of the PEC structure
For a PEC structure with arrays of square holes having a lattice constant d, hole depth h, hole size a, and a dielectric constant εh, one can approximate the isotropic dispersion relation where λ >> d>a, as :
where kx is the propagation constant, ko = ω/c, S = 2a√2/πd, and . Using this equation one can solve for ω and differentiate with respect to kx to obtain an expression for the effective group index:
where ωpl is the cutoff frequency such that . A is given by
Figure 2 shows the dispersion relations for various εh, and a calculated per Eq. 1. The frequency and the wavevectors are presented in normalized units of and , respectively. An increase in a or εh, will lead to a decrease in the cutoff frequency, ωpl, thus decreasing the range of propagating modes. In order to realize periodic focusing of DSPs, the effective group index encountered by the DSPs should be parabolically modulated in the transverse direction. In principle, a change in either a, h or εh, can be used to engineer the parabolic variation of Ng. For the application of this concept in the terahertz range, where a is in the range of 100’s of μm, structured PEC material can be achieved by patterning a polymer using standard lithography and then electroplating the PEC material. Changes in a and h within PEC structures can be achieved by controlling the polymer pattern and time of electroplating, respectively. To control the dielectric constants, various refractive index oils can be filled into different holes using a precision fluid dispensing system coupled to a micro-positioner. However, resolving the small changes in a or h in a three-dimensional FDTD simulations is computationally challenging; we therefore focused our investigation on changing εh. Because both types of PEC structures can be fabricated and have similar behavior (as confirmed from the dispersion curves shown in Fig. 2), we believe that knowledge obtained from one structure can be applied to the other. Although one could calculate the dispersion relationships for various εh, using Eq. 1, a mismatch at wavelengths comparable to the feature size is expected since the expression was derived for feature sizes much smaller than the wavelength of light.  Therefore, to obtain a more-accurate dispersion relation and modulation of εh, with parabolic variation in Ng, we used a 3D FDTD method  to solve Maxwell’s equations for homogeneous structures of different εh.
The dispersion relations were obtained by considering a single unit cell, as shown in the inset of Fig. 3(a), with Bloch periodic boundary conditions in the X and Y directions and perfectly matching layers (PMLs) along the Z direction. A grid size of δx=δy=δz=d/15 was used in the calculations. Excitation was achieved with a wide-band Gaussian source arbitrarily placed within the unit cell. Harmonic inversion of time signals  was used to extract the resonance modes for different εh with fixed d, a = 0.85d, and h = 1.2d. The FDTD calculations were performed using the open software package MEEP, wherein sub-pixel smoothing was used for increased accuracy. 
Figure 3(a) shows the dispersion relations calculated for different εh, ranging from 1.25 to 3 in increments of 0.25. Keeping in view the practical importance of the proposed work in THz plasmonics research, we used an operating frequency of 1.122 THz, which for a hole depth of d = 100μm yields an operating frequency of 0.3742 normalized units. For this operating frequency, the variation of Ng with respect to different εh was obtained by numerical differentiation and is shown in Fig. 3(b). The variation of Ng can be approximated with an exponential increase, as shown by the best fit (solid line) to the data points obtained by FDTD (dotted line).
Having established the relationship between Ng and εh at a certain operating frequency, we proceeded to design the graded index media and simulate via FDTD the propagation of designer plasmons. The 3D model used for FDTD consisted of a rectangular box of size 83d×23d×7d which included a PML thickness of 1d in each dimension, as shown in Fig. 4(a)–(c). To achieve 2D focusing and collimation of surface waves in the X-Y plane, we chose an effective group index that changed parabolically along the transverse direction of the propagation. The effective group index was highest at the center (Y=0) and varied along the positive and negative Y direction (-10d to 10d) as:
where No is the group index at Y = 0 and α is the gradient coefficient. By choosing No=1.1077 and Ng=1.014 at Y = ±10d, we obtained a gradient with α=0.04031. The variation of Ng with these parameters is shown in Fig. 4(d). To realize this group index change, the dielectric constant of the holes in the PEC-structured surface was altered by changing εh (based on a nearly exponential relationship between Ng and εh). The group index change is shown in Fig. 3(b). The exact distribution of εh along the transverse direction is shown in Fig. 4(d).
3. Results and discussion
The propagation of Gaussian beams with different beam widths in free-space parabolic graded lens was thoroughly studied by Gomez-Reino et al.  It has been shown that Gaussian beams with smaller beam widths expand and collimate in the first quarter pitch of the lens. Similarly, a Gaussian beam with a larger beam width gets focused and the beam width is decreased. At a certain beam width, when collimation and focusing effects cancel each other out, light will propagate at the same beam width through the structure. Our investigations focused on extending this physical mechanism from free-space optics to surface plasmons.
To investigate the efficiency of the proposed structure for lateral confinement of DSPs, we simulated the propagation of several DSPs that initiated from Gaussian beams of different beam widths relative to the lattice constant d. The use of Gaussian beams is appropriate in the context of THz plasmonics, since surface modes are excited on planar structures using Gaussian beams incident upon razor-blade-like structures. 
In simulation, the PEC was represented using a negative infinite dielectric constant and a grid size of δx=δy=δz =d/15 was used. The negative infinite dielectric constant creates the same effect as infinite conductivity by maintaining the electric field in the PEC structure at a constant value (zero in this case) Narrow-bandwidth TM-polarized (Ex, Hy, and Ez) Gaussian beams of different beam widths w at the operating frequency were launched in the X direction from X=-40d.
We first investigated the propagation of DSPs produced by a Gaussian beam of width 5d in a graded medium. As shown in Fig. 5(a)–(d), DSPs produced by such a wide source are focused as they reach X=0 (center of the structure) and then collimate after traveling 40d. This result was expected that per Gaussian optics, electromagnetic energy in a material of parabolic graded index is guided in an oscillatory fashion with a periodic focal point. The X-Z view shows the localization of the Ez field along the surface of DSPs and also shows that the intensity increases at the center of the structure (Y=0) due to the focusing of the DSPs. This increase is seen in the snapshots of Ez field in the Y-Z plane at X=-40d (Fig. 5(c)) and at X=0 (Fig. 5(d)). To qualitatively display the focusing effect, profiles of |Ez| along the transverse direction at X=-40d (dotted lines) and at X=0 (solid lines) are shown in Fig. 5(e). Focusing is evident from the increase in the intensity of Ez at Y=0 and from the lateral confinement. We also performed numerical simulations using similar source conditions and simulation geometries but with a uniform dielectric constant, and observed no focusing or collimation behavior (data not shown).
DSPs produced by Gaussian sources of smaller widths w = 2d undergo collimation instead of focusing in the first half of the structure, and then focus in the second half of the structure (Fig. 6). From Fig. 6(e), we see that the collimation of DSPs in the first half of the structure results in a spreading of the intensity along the Y direction. In the absence of the graded media, the DSPs propagate radially (data not shown here).
With both Gaussian sources beam widths (w = 5d or 2d), DSPs either get collimated (Fig. 5) or focused (Fig. 6) by traveling a distance approximately 40d. This distance matches closely with the quarter-pitch of graded lenses, , predicted by standard Gaussian optics. At intermediate widths along the incident Gaussian beam, collimation and focusing cancel out . DSPs are also guided along the length of the structure with no significant deviation in the beam width, as shown in Fig. 7(a)–(e). This behavior allows one to guide energy in the form of DSPs over long distances.
Since many THz applications require that waveguides can squeeze electromagnetic waves into sub-wavelength areas, it is important to find out how tight the confinement of guided mode can be achieved using graded media and what parameters determine the tightness. We know from standard Gaussian optics for free-space graded lens that the minimum width of the beam in the graded lens is given by the fundamental mode, . Based on the excellent match between the focusing lengths observed in our simulations and those observed in Gaussian optics, we infer that the confinement of spoof plasmons can be further improved by decreasing the wavelength or increasing the gradient or effective index at the center of the structure.
Graded media can also be used to control the propagation of surface plasmons at visible and IR domains using real metals. [38, 39, 40] Such an extension would require the consideration of absorption loss in real metals  due to dispersive nature  of real metals at visible frequencies. The effects of field penetration  in real metals should also be considered as it lowers the band positions observed in PEC structures.
Using FDTD simulations we have demonstrated that one can control the propagation of DSPs in structured PEC materials with arrays of square holes by exploiting graded index media. We found that graded index media can be realized by modulating either the size of the holes or the dielectric constant. Such a structure was engineered using FDTD simulations by modulating the dielectric constant of the structure’s square holes and shown to provide lateral confinement of propagating DSPs. Focusing, collimation, and waveguiding of designer plasmons in a lateral direction were realized by changing the width of the source beam. These results will promote high confinement waveguide and sensing research for THz designer surface plasmons.
This research was supported by the National Science Foundation (ECCS-0824183 and ECCS-0801922), the Air Force Office of Scientific Research (AFOSR), the Penn State Center for Nanoscale Science (MRSEC), start-up funds from the Pennsylvania State University (PSU), and a seed grant from PSU’s Research Computing and Cyberinfrastructure (a unit of Information Technology Services). The authors thank Jinjie Shi and Aitan Lawit for helpful discussions.
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