## Abstract

We demonstrate that surface waves in structured perfect electric conductor surfaces can be self-collimated using the finite-difference time-domain method. The self-collimation frequency is obtained from the equi-frequency contours of a perfect electric conductor patterned with an array of square holes. The field patterns of the self-collimated surface wave, obtained using the periodic boundary conditions, show that the surface waves propagate with almost no spreading. We also show that self-collimation phenomena can be observed for the hybrid surface plasmon waves in structured metal surfaces using the finite-difference time-domain method with the Drude model. It is shown that for a structured silver surface the self-collimation can be achieved at the frequencies in the infrared region.

© 2007 Optical Society of America

## 1. Introduction

Since Ebbesen *et al*. reported the observation of extraordinary transmission of light through a subwavelength array of holes in a metal film, the interest in the interaction of light and metal films has increased remarkably [1]. It motivated a lot of research on surface plasmon polariton(SPP) in micron-size metal structures, for example, surface plasmonic bandgap, surface plasmon sensors and surface plasmon waveguides [2, 3, 4, 5]. It had been believed that the existence of surface waves in a metal-dielectric interface is a unique property of metals which have negative dielectric constants. However, it was shown theoretically that a perfect electric conductor (PEC) with a periodic array of holes on its surface can support a surface wave which has a plasmonic response [6, 7, 8]. This was experimentally demonstrated in the microwave region by measuring the band structure of the surface wave [9]. Qiu calculated the band structure for the surface wave using the finite-difference time-domain(FDTD) method [10], and compared the simulation results with the theoretical and experimental dispersion relations [11]. The surface wave mode in the structured PEC and its dispersion relation are very important in that it can give designers a chance of controlling the dispersion properties.

It is well-known that the self-collimated propagation and the negative refraction of light beam can be achieved in photonic crystals (PhCs) due to the dispersive properties of PhCs [12, 13]. Light modes in the flat regions in the equi-frequncy contours (EFCs) can propagate without diffraction, and some of these modes can be used for the negative refraction phenomena when the light is incident from the air. Recently, a centimeter-scale collimation, referred to as “supercollimation”, in a dielectric SOI PhC slab operating at infrared wavelengths was achieved experimentally [14] and the negative refraction of light in a structured PEC surface is demonstrated using the FDTD method [15].

However, a study on the self-collimation phenomena for surface waves in a structured PEC and metal surfaces has never been reported. Generally, there is no self-collimation phenomena for the conventional SPPs in the metal-air interface. In the case of the isotropically approximated dispersion of the sturctured PEC in the long wavelength range [6], the self-collimation phenomena is not expected. However, because a stuctured PEC or a structured metal has anisotropic dispersion relations for the light of wavelengths comparable to the period of holes as shown by Qiu [11], it is possible to observe self-collimated propagation of the surface waves unlike the conventional SPPs. This self-collimation phenomena for the surface waves is a unique property which can be seen in a structured PEC and a structured metal and it can be applied as a new type of surface wave waveguide and as highly sensitive sensors as well.

In this study, we show that the self-collimation phenomena occurs in PECs with a periodic array of holes by analyzing their band structures and EFCs. Moreover, we present that the self-collimation of light also occurs in a structured metal surface using the FDTD method to which the Drude model is applied.

## 2. Calculations and discussion

First, we consider a PEC structure of square array of square holes with finite depths which was previously studied by Pendry and Qiu [6, 11]. As shown in Fig. 1(b), the PEC structure has a period of *a*, the size of square *d*, and the hole depth *h*. According to Ref. [11], this structure has surface modes which are confined in the interface between the air and the PEC, i.e. on the upper surface of the structure.

To calculate the dispersion relation and the EFCs for the structured PEC, we performed the 3D FDTD simulations using a freely available software package MEEP [16]. Bloch periodic boundary conditions are used in the *x*- and *y*-directions, to consider the propagation of surface waves in a two-dimensional lattice, and the perfectly matched layers(PMLs) absorbing boundary conditions in the *z*-direction to remove unwanted reflections. The grid size (Δ*x*) is (1/30)*a* for the band and the EFC calculations and the Γ*M* direction calculation. All units used in the FDTD calculations are specified in the unit of the period *a*, for example, lengths in *a*, times in *a*, frequencies in 1/*a* and wavevectors in 2*π*/*a*. In this unit system, for example, the time step of 100*a* means the time for which the light propagates 100*a* in free space and corresponds to approximately 333.3*ps* when *a* = 1 *μm*.

Using the FDTD simulation for the unit cell [Fig. 1(b)], we obtain the frequency at which the self-collimated propagation is allowed in the structured PEC surface. To search for this frequency, the band structure and the EFCs are drawn for *d* = 0.875*a* and *h* = *a* (see Fig. 2). By launching a Gaussian source in the unit cell, resonance frequencies are obtained for every 231 k-points in the first irreducible Brillouin zone. The grid size in the k-space (Δ*k _{x}* = Δ

*k*) is 0.025 × 2

_{y}*π*/

*a*. As can be seen in Fig. 2(b), one can find the flat EFCs at the frequencies in the vicinity of

*f*= 0.52

*c*/

*a*where the light propagates to the same direction, i.e. the Γ

*M*direction. This implies that the self-collimation of light beam takes place at this frequency.

To demonstrate the self-collimation phenomena at the frequency of 0.52*c*/*a*, we performed another simulation for a 8√2*a* × √2*a* × 8*a* unit cell with a 5*a*-wide continuous-wave(CW) source [Fig. 1(c)]. In this case, we launch the CW source of the frequency 0.52*c*/*a* and set the boudary condition using the wavevector (*k _{x}*,

*k*) = (0.39 × 2

_{y}*π*/

*a*, 0.39 × 2

*π*/

*a*) which is the center point of the EFC for the frequency of

*f*= 0.52

*c*/

*a*in the Γ

*M*direction. Because of the 45°-rotation of the lattice the wavevector transforms to (

*k*́

_{x},

*k*́

_{y}) = (0,0.11 × 2

*π*/√2

*a*) and (

*k*́

_{x},

*k*́

_{y}) = (0,0.39 × 2

*π*/√2

*a*) in the new

*k*-space. The total simulation time is 1050

*a*which corresponds to 3500

*ps*when

*a*= 1

*μm*and the source is turned off after the time of 50

*a*. The field patterns of

*E*at the time of

_{z}*t*= 1048

*a*are shown in Fig. 3(a–c), which shows that the fields are confined very well at the surface and the surface wave beam propagates collimated without spreading. Even after propagating for the time of 1050

*a*, the surface wave beam maintains its initial width. By comparing the field pattern for

*t*= 1036

*a*[Fig. 3(d)] and the mode pattern at the

*M*point in the band structure [Fig. 3(e)], we can see that the self-collimated beam has a field distribution similar to that of the extended mode for the

*M*point. Figure 3(f) shows that the surface wave mode persists for a long time with a small decrease in the field intensity. This small amount of intensity decrease is due to the leakage in the

*x*-direction owing to the uncertainty of the wavevector which stems from the finite beam width. Some wavevector components located at slightly bent EFCs are contained in the beam of finite width and give rise to a propagation loss consequently.

Next, consider a structured metal surface which has the same geometry as the one in the PEC case. Before examining the details, let us consider the SPP on a metal surface. Unlike PECs, the dielectric constants of metals are dispersive, i.e. vary with the frequencies of the light and they are complex numbers. For metals the Drude model is commonly used in which the dielectric constant of a metal is given as

where *ω _{p}* is the plasma frequency and

*γ*the damping constant. The dispersion relation of SPPs in a metal-dielectric interface can be derived from Maxwell’s equations using proper boundary conditions [17]. The dispersion relation is

where *k _{SP}* is the SPP wavevector,

*k*

_{0}is the free space wavevector of light, and

*ε*,

_{m}*ε*are the dielectric constants of the metal and the dielectric material, respectively. One important property of SPP is that it has a larger magnitude of wavevector than the light in air [1, 17]. This makes the SPP dispersion line bent and flat near the SPP frequencies. The surface wave modes in a structured metal surface are called “hybrid surface plasmons” because they are different from the conventional SPPs on a metal surface [6]. Since the field penetration into the metal increases the effective refractive index, the bands in the structured metal surface become lower than the ones in the structured PEC surface. One can check this lowering effect by comparing the band structures for the PEC [Fig. 2(a)] and the metal surfaces [Fig. 4(a)]. The lowering effect becomes stronger as the frequencies approach the plasma frequency. We checked the change in the position of bands with respect to the plasma frequency (not shown here).

_{d}Using the FDTD with the Drude model, we calculated the band structure for the metal surface with a square hole array. For simplicity, we set the plasma frequency to be 1 and the damping constant 0. We performed the EFC calculations as in the PEC case, and found the flat regions for the first band. As shown in Fig. 4(b), the EFC becomes nearly flat at the frequency of 0.364*c*/*a*. In this case, to minimize numerical errors, we set the width *d* to be 0.6*a*, the grid size 1/30*a*, and the unit cell size *a* × *a* × 14*a*.

To confirm the self-collimation phenomena, a CW-calculation was perfomed with the frequency of 0.364*c*/*a* and the wavevector (0.35 × 2*π*/*a*,0.35 × 2*π*/*a*,0). The E-field distribution shows the surface wave is collimated very well and the field intensity shows very little decreasement after propagation for *t* = 500*a* (see Fig. 5). But, the field confinement in the *z*-direction looks poorer compared to the PEC case and some higher frequency oscillation is observed in the time signal. This is due to the reflections of light beam between the bottom surface of the metal structure and the PML layers. But this is not a serious problem and the metal structure still collimates the surface wave very well. In the previous case of the PEC, this does not occur as no field penetrates into the PEC structure.

For the purpose of designing actual devices utilizing the self-collimation phenomena in the visible or infrared range, silver, which has the plasma frequency in the ultraviolet range, is chosen with the damping constant *γ* taken into account. The plasma frequency *ω _{p}* is set to be 6.37 and the damping constant

*γ*0.042441, which corresponds to

*ω*= 1.29 × 10

_{p}^{16}/

*s*and

*γ*= 0.8 × 10

^{14}/

*s*, respectively, when

*a*= 1

*μm*. The E-field distributions are obtained after performing the same procedures as described above. One can see in Fig. 6, the surface wave of frequency,

*f*= 0.414

*c*/

*a*is well-collimated in the structured silver surface. But the introduction of the damping term makes the field intensity decrease significantly.

## 3. Conclusion

In conclusion, we have studied the self-collimation phenomena of surface waves in structured PEC and metal surfaces. By performing intenisve FDTD calculations, we have shown that the self-collimation phenomena take place for surface waves in the structured PEC surface, as well as in the structured metal surfaces. As an aid for the realization of actual devices, a structured silver surface is considered and it is shown that self-collimation phenomena can be achieved for the structure considered in this study. We believe that the self-collimated propagation of light can be demonstrated even for the light of frequencies in the visible and infrared region.

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