## Abstract

Photonic band structures of surface modes on structured metal surfaces with periodic holes in them are obtained using the full-vectorial finite-difference time-domain method. Surface modes do exist for almost all investigated lattice types with any hole size, shape and depth. The results for a square lattice of wax-filled box holes in brass are also in very good agreement with the experimental results. We also show that, for structured surfaces with holes of finite depth, the holes might act as cavities. Thus there exist propagating coupled cavity modes with low group velocity, confined at the surface and decaying exponentially into the dielectric above.

©2005 Optical Society of America

## 1. Introduction

A surface plasmon (SP) is a collective oscillation of the electrons at the interface between a metal and a dielectric. Surface plasmons give rise to surface-plasmon-waves (SPWs), which are propagating electromagnetic waves bound at the metal-dielectric interface. Nevertheless, an interface between a perfect electric conductor (PEC) and a dielectric usually does not support surface waves since electromagnetic fields can not penetrate into the perfect conductor. Even so, it has been shown that both one-dimensional (1D) [1, 2, 3, 4] and two-dimensional(2D)[5, 6] structured PEC surfaces (i.e., corrugated PEC plane) can support surface waves, as the holes allow the electromagnetic fields to penetrate into the structure. Furthermore, it was shown [5] that these structured surfaces have many properties in common with the electron plasma in a metal surface, in particular, the dispersion relation of the surface waves has a plasmonic response (i.e., the structure can be optically described as an effective medium with a dielectric function of plasmon form). Such so-called designer-surface-plasmons in structured brass surfaces have been experimentally verified [7] in the microwave regime, in which metals can approximately treated as perfect conductors. These surface waves, together with conventional (planar-surface) surface waves at the metal-dielectric interface, are actually responsible for some extraordinary metal optics, e.g., high transmission through subwavelength holes in metallic films [8, 9].

The dispersion relation of surface modes in structured PEC surfaces was presented in Refs. [5, 6 ] with the assumption that the feature size of the surface structure is much smaller than the light wavelength. However, a rigorous, full-vectorial theoretical study for the dispersion relation of surface waves in 2D structured PEC surfaces has not been performed yet. In the present paper, we illustrate the dispersion relations (i.e., photonic band structures) for such surface modes using the three-dimensional (3D) finite-difference time-domain (FDTD) method [10], which in principle provides a rigorous full-vectorial solutions to Maxwell’s equations. Different lattice types of periodic holes on surfaces are investigated. The influence of hole depth is also considered. Our results add evidence to surface waves existing on almost all structured PEC surfaces.

## 2. Results and discussion

First consider a PEC surface with a square array of holes, which have a square cross section *a*×*a*, and a distance *d* (the lattice constant), as shown in Fig. 1(a). The depth of the holes in the conductor is *h*. Only one single unit cell in the *x*-*y* plane is used to compute the photonic band structures of surface modes. Bloch boundary conditions are applied at the *x* and *y* boundaries of the unit cell [11, 12]. The height (in the *z*-direction) of the computational domain is chosen to be large enough (10*d* is used in most of our computations below) to ensure the existence of the surface modes. Perfectly matched layers [13] are used in the *z*-direction in order to absorb radiating modes. 20 grid points for each lattice constant *d* are used in FDTD computations.

The calculated photonic band structures for surface modes of such an air-hole structured PEC surface with *a* = 0.875*d* and *h* = 1.0*d* are shown as blue curves in Fig. 2(a). The shaded gray region indicates all possible modes in the air background. The lower boundary of this region is the air light line. Only those mode below the light line are truly surface modes, since they cannot couple with modes in the bulk background (i.e., they are confined at the surface and decay exponentially into air) [14]. On the other hand, the holes in the PEC act as waveguides and hence there is cutoff frequency below which no propagating modes are allowed. The cutoff frequency for holes of square cross section *a*×*a* is given by

where *c* is the speed of light in vacuum and *ε _{h}* and

*μ*are the relative permittivity and permeability of the material filling the holes. This cut-off frequency is equivalent to the effective “surface plasma” frequency

_{h}*ω*of the structured surface [5]. For the present case, the normalized cut-off frequency is (

_{pl}*ω*/2

_{c}d*πc*)=0.571, shown in Fig. 2(a) as the red dashed line. The line divides the mode region below the light line into two regions: (1) region with modes truly confined at the surface and decaying exponentially both into air and holes; (2) region with modes decaying exponentially into air but coupled to hole waveguide modes. For shallow holes with

*h*= 1 .0

*d*, one can see from Fig. 2(a) that only true surface modes exists. Figure 3 shows the field cross sections of the surface mode at the

*X*point. It clearly demonstrates the mode is confined in the

*z*-direction (decaying exponentially both into air and holes) meanwhile propagation in the

*x*-

*y*plane (along the

*x*-direction for the present case).

If holes are much deeper into the conductor, modes coupled to waveguide modes might appear. As an example, the photonic band structures for the hole depth *h* = 4.0*d* are shown in Fig. 2(b). The true surface modes below the cut-off does not change significantly as the hole depth increase. This is not surprising since these modes are well confined at the surface. However, several modes appear in the magenta region. For the present example, six bands appear. These six bans can be grouped into three. Bands in each group are very close to each other in frequency, in particular, degenerated at the *M* point. Figure 4 shows the *E _{x}* field cross sections of the modes at the

*M*point. Only one of the degenerated modes is shown here. It clearly shows in Fig. 4(b)–(d) that, for modes above the cut-off frequency, wave can penetrate into the holes deeply. It also hints that each individual hole acts as a box cavity with one side open. Each cavity support discrete number of cavity modes. All these cavity modes coupled to each other, and therefore form the bands in the magenta region. Fig. 2(b) also shows that all these bands in the magenta region are quite flat in frequency, i.e., with a very low group velocity. It is very similar to coupled-resonator optical waveguides structures, e.g., demonstrated in photonic crystals [15]. These extra-slow modes in structured metal surface might have some potential applications in optical devices by manipulating light propagation. It is also worth mentioning that, since these bands are the results of the resonant behavior, they can also be observed above the light line (thought not plotted here as they are radiating modes).

Our results for surface mode dispersions are also compared with the approximated analytical results. The dispersion relation of the surface modes along the Γ*X* direction obtained in Ref. [5] has the following surface plasmon dispersion form,

where *k _{x}* is the wave vector along the Γ

*X*direction. The dispersion calculated by Eq. 2 is shown in Fig. 2(a) as solid dots. It is in quite good agreement with the FDTD results at low frequencies (long wavelengths), however, it deviates at higher frequencies. This is simply due to the failure of the assumption

*a*<

*d*≪

*λ*

_{0}used for Eq. 2. While the FDTD method used in the present paper is a full-vectorial first-principle method for the Maxwell equations, thus, it in principle works for all frequency ranges (as long as the mesh grid for field discretization is fine enough).

To verify our computations, we also compare our calculated results with the experimental results in [7]. In the experiments, the structured surface is a square lattice of square holes in brass, which can be considered as a PEC since experiments were performed in the microwave regime. The lattice constant of the structured surface is *d* = 9.525*mm*. The side length of the holes is *a* = 6.960*mm* ≈ 0.73*d*. A one-dimensional (1D) array of cylindrical brass rods (radius *r* = 1.0*mm* ≈ 0.10*d*) with pitch 2*d* is positioned on the surface of the array of tubes. The rods are along the *y*-direction. One may refer to Ref. [7] for other structure details. Here we consider only the case for holes filled with wax (*ε* = 2.3). The calculated dispersion of the surface modes are shown in Fig. 5 for light propagating along (a) the *x*-direction and (b) the *y*-direction. For light propagating along the *x*-direction (Γ*X*), due to the existence of the 1D array of cylindrical rods with period 2*d*, the Brillouin zone (also the photonic band) is folded at the A point (*k _{x}* =

*π*/2

*d*). Therefore, part of the surface modes are folded into the radiation region. This is very important since for the experimental setup in [7], only those lossy surface modes in the radiation region will be excited. While for light propagating along the

*y*-direction (Γ

*X*′), the photonic bands are not folded, but lossy surface modes still exist. One can see from Fig. 5 that our calculated results are in very good agreement with the experimental results. Though not shown in the present paper, the agreement is also very good for the air-filled structure.

It is worth pointing out that the second band in Fig. 5(b) should asymptotically approach the cut-off frequency as *k _{y}* increases. The difference between the calculated limit and the cut-off frequency is due to the finite depth of holes in our computations [7].

Finally, we consider the case of cylindrical holes. Figure 6 shows the photonic band structures for structured surfaces with circular air holes in (a) a square lattice and (b) a triangular lattice. The holes have a radius of *R* = 0.475*d*, and a depth of *h* = 1.0*d*. The surface modes indeed exist for both structures.

Our further calculations also show that surface modes exist for almost all these lattice types with any hole size, shape and depth. However, if holes are relatively small as compared to the lattice constant, the dispersion of the surface modes will be very close to the light line of the background.

## 3. Conclusions

In conclusion, we have studied the surface modes in structured metal surfaces with holes using the full-vectorial FDTD method. The full band structures of the surface modes with respect to propagating wave vectors are obtained. We show that the surface modes do exist for all cases: square lattice of box holes, square lattice of cylindrical holes, and triangular lattice of cylindrical holes. The modes are truly confined at the surface and decaying exponentially both into the background and holes. For the case of square lattice of wax-filled holes in brass, our computational results are also in very good agreement with the experimental results. We also show that, for structured surfaces with holes of finite depth, there exist modes penetrating into the holes but decaying exponentially into the dielectric above the surface. These modes, having extra-low group velocities, can be considered as coupled cavity modes, where the holes with finite depth act as cavities.

## Acknowledgments

The author thanks Dr. A. P. Hibbins for providing experimental data, and Prof. L. Thylén and Prof. E. Berglind for many helpful discussions. This work is supported by the Swedish Foundation for Strategic Research (SSF) on INGVAR program, the SSF Strategic Research Center in Photonics, and the Swedish Research Council (VR) under Project No. 2003-5501.

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