## Abstract

Geometry dependence of surface plasmon resonance of 2D metallic photonic crystals (PCs) was assessed using rigorous 3D finite difference time domain analysis. PCs of noble metallic rectangular and cylindrical nanopillars in square and triangular lattices on thick noble metal film were simulated for maximum field enhancement. It was found that the period, size and thickness of the nanopillars can be tuned to excite of surface plasmons at desired wavelengths in visible and near-infrared ranges. Maximum electric field enhancement near the nanopillars was found to be greater than 10X. The detail analysis of PCs tuned for 750 nm wavelength showed that thickness of nanopillars was the most sensitive parameter for field enhancement, and triangular lattice PCs had the wider enhancement bandwidth than square lattice PCs. Results showed that these PCs are sensitive with incident angle (θ) but not with polarization angle (ϕ).

©2009 Optical Society of America

## 1. Introduction

The coherent fluctuation of electrons density at a metal surface by electromagnetic radiation is called surface plasmon oscillation [1]. These electron oscillations excite evanescently confined electromagnetic modes at the metal-dielectric interface called surface plasmon polaritons (SPPs) that propagate along the metal/dielectric interface and make the coupling of light with electronic components possible [2]. Strong enhancement of electromagnetic fields at the metal-dielectric interface has applications such as photonic circuit miniaturization, optical signal processing, optical signal switching, optical data storage, biosensors and absorption enhancement in solar cells [3]. Therefore, a detailed analysis of field enhancement by metallic nanostructures is very important for exploitation of these phenomena and coupling of light with electronic devices. Over the last decade, surface plasmon resonance and electromagnetic field enhancement by metal particles and nano-wires have been investigated experimentally and theoretically. Noble-metal nano-particles on dielectric surface were experimentally investigated for enhancement of photocurrent [4,5]. Islands of nano-disks on dielectric materials were investigated for enhancement of absorption [6], transmission [7] and photocurrent [8]. Nano-wires on dielectric materials were investigated for absorption enhancement and fluorescence emission enhancement [9]. Photonic band gap of surface modes in two dimensional arrays of metal nanostrucutes on metal surface has been investigated theoretically [10] and demonstrated experimentally [11,12]. Investigation of scattering and propagation of SPPs along textured metal surface by triangular lattice nanodisks has been demonstrated from both experimental [12] and numerical [13,14] aspects. Electric field enhancement by metallic nanostructure for exciting surface plamson resonance is becoming attractive for nonlinear optical phenomenons which need high intensity of field. Therefore, the investigation of field enhancement by two-dimensional noble-metallic photonic crystals (PCs) of different sizes, shapes, and spacing on thick metal films, for tuning surface plasmon resonance has the importance. These PCs have the potential to be used as SPP based photonic devices and for radiation absorption in solar cells. In this paper, the arrays of cylindrical and rectangular (square prism) nanopillars of silver (and gold) on silver (and gold) surface arranged in square and triangular lattices (2D metallic PCs) with varying lattice periods and sizes were investigated.

Numerical methods are usually employed for analysis of electric field enhancement by metallic nanostructures. Most of the analytical analysis support only one dimensional or two dimensional nanostructures (gratings) in dielectric mediums in the absence of a metallic sheet. Mathematical treatment of gratings has been done using Kirchhoff’s scalar diffraction theory but this faced serious problems when periodicity was smaller than the wavelength of light [15]. Rayleigh’s expansion method solved the problem of size, however, this method also failed in deep gratings [16]. Subsequently, various numerical techniques such as: integral method [17], rigorous vector diffraction method [18], modal expansion method [19–21] and coupled wave method [22,23] were developed for one-dimensional lamellar gratings. Green’s tensor method was used for analysis SPPs scattering [13,24,25] and photonic band gap [13]. Analytical methods for field calculations by approximating the nanostructures as a spherical or spheroid arrays in dielectric medium or on dielectric substrates were developed by dipole-dipole interaction method [26], couple-dipole approximation method [27] and truncated spheroid method [28]. These methods were developed from approximations which were applicable for nanostructures of specific shapes and sizes. They have further restrictions including nanostructures in dielectric mediums and specific nanostructure-substrate distances. There does not appear to be an analytical method for solving fields in practical metallic structures on metallic thick films, which have many potential applications. Due to the recent advances in computational power, simulation of those nano-structures using rigorous numerical methods is becoming practical.

A 3D-FDTD method was used for simulation of 2D metallic PCs on thick metal films. The initial PC period for surface plasmon resonance was calculated from the analytical grating equation and the surface plasmon dispersion equation. Then rigorous 3D-FDTD method was employed for the optimization of geometrical parameters so that maximum enhancement of the electromagnetic field above the metal surface was achieved through coupling of nano-structure localized plasmons with the surface plasmons. The best photonic crystals were characterized for resonance bandwidth, angle of incidence (θ) and angle of polarization (ϕ) of plane wave.

## 2. Design methodology

Surface plasmon polaritons are excited when a grating on the surface of a metal changes the momentum of the wave such that the wave vector parallel to the surface is equal to the SPPs wave vector. Each nanopillar on the surface is associated with its own localized plasmons which depend on the shape and size of the structure. The coupling of localized plasmons with surface plasmons enhances the perpendicular component of the electric field at the metal surface. At the plasmon resonance condition the electromagnetic radiation is absorbed and stored in the oscillating surface plasmon field at the metal-dielectric interface and localized plasmon field around the nanopillars of photonic crystals, which increases the near-field near the metal surface and decreases the reflectance from the metal-dielectric interface. A fraction of plasmon energy is dissipated into the metal while some is scattered back by the nanopillars as radiation. Therefore, at maximum plasmon resonance, reflection (zero order diffraction) is minimum, and is proportional to the field enhancement. The initial period of the PCs for simulation is determined from the SPPs dispersion and grating equations that follow:

The wave vector β of SPPs at metal dielectric interface is given by Eq. (1) [2].

*k*is the wave vector in a vacuum,

_{o}*ε*is relative permittivity of the dielectric and

_{d}*ε*is the dielectric function of the metal which is a nonlinear function of the frequency of radiation. At

_{m}*ω*,

_{sp}*ε*is equal to -

_{m}*ε*and the SPPs wave vector (

_{d}*β*) becomes nearly infinity; hence the group velocity of SPPs propagation becomes nearly zero, and SPPs waves behave as stationary oscillating waves. The incident power of radiation is stored in an electromagnetic field at the metal-dielectric interface, which is enhanced by several orders of magnitude, but is evanescent normal to the boundary.

The dispersion Eq. (1) indicates that the wave vector of SPPs is always greater than the wave vector of incident radiation in the dielectric. Therefore, the parallel component of the wave vector in the dielectric cannot excite surface plasmons. Electromagnetic radiation incident on a metallic grating of lattice period *λ _{g}* couples with SPPs when the reciprocal lattice vector fulfills the mismatch between the SPP wave vector

*β*and parallel component of wave vector in the dielectric

*k*[29]. The surface plasmon excitation condition of SPPs wave vector is

_{||}*p*-polarized [29]. A bidirectional grating (2D-PC) can excite both

*p*and

*s*polarized radiation [30]. The momentum transfer caused by the grating changes the direction of the wave vector, but total momentum is conserved. Therefore, wave vectors normal to the boundary surface become imaginary and radiation in direction of normal becomes evanescent. In addition to the propagating surface plasmons which are strongly confined to the metal dielectric interface, these structures excite non-propagating modes known as localized surface plasmons.

Localized surface plasmons are the oscillation of conduction electrons on metallic sub-wavelength nanostructures directly excited by electromagnetic radiation. The localized field near the nanostructure depends on the polarizability of the structure which depends on shape, size and dielectric permittivity of the medium and the nanostructure. The intensity of the field at the metal surface is modified by interactions of dipole fields, image dipole fields and the SPPs field. The polarizability of a nanostructure increases with size. However, the total field enhancement per unit surface area does not always increase with an increase in polarizability.

Figure 1
depicts the photonic crystals of cylindrical and rectangular nanopillars of diameter *d* (*l* in rectangular) and thickness *t* arranged in square lattice of period λ_{g.} The initial size and thickness of the nanopillars used for the FDTD simulation were 1/2 and 1/8 of the period respectively. Based on initial values, the cylindrical and rectangular nanopillars in the square and triangular lattice were designed in the 3D-FDTD simulation tool, EM Explorer [31]. The dispersive dielectric functions of gold and silver were taken from Palik [32]. The refractive index of the dielectric medium was 1.49 which is equivalent to the dielectric constant of Polymethyl Methacrylate (PMMA). The unit cell of the simulation for the square lattice PC consists of a single structure due to its periodicity in x and y directions. Similarly, the unit cell of isosceles triangular lattice consisted of a pattern of four structures which periodically repeated along the x and y axes. Therefore, the Bloch periodic boundary condition was applied along the horizontal axes of unit cell and absorbing boundary condition was applied in z-axis.

The smallest wavelength in the material should be represented by more than 20 grid points to provide acceptable accuracy in FDTD simulation [33]. Therefore, mesh size in this simulation was 10 nm. Unless otherwise stated the PCs were illuminated perpendicularly with an arbitrarily chosen wavelength *λ _{o}* = 750 nm and x-polarized plane wave. The PC was tuned for maximum field enhancement by varying the lattice period (

*λ*), diameter (

_{g}*d*) or side length (

*l*in rectangular lattice) and thickness (

*t*) for cylindrical and rectangular nanopillars in square and triangular lattices. The total numbers of simulations would be 4000 for the silver PCs and 4000 for the gold PCs for 10 different diameters (or side-lengths) and 10 different thicknesses with 10 different periods. So instead of point by point simulations, initial values were obtained from analytical equations and successive simulations were chosen in the neighborhood of the minimum reflectance by increasing or decreasing the design parameters (

*d*,

*t*, and

*λ*) by 10 nm. We continued the simulation until all new reflectance were greater the previous reflectance. The PCs that had minimum reflectance and maximum enhancement were analyzed in detail for its response with incident angle (θ) and polarization angle (ϕ).

_{g}## 3. Results and Analysis

The initial period (*λ _{g}*) of the nanopillars calculated from analytical grating formula (2) and dispersion Eq. (1) was 480 nm for silver PC illuminated perpendicularly at

*λ*= 750 nm. Therefore, the initial values of the design parameters were

_{0}*λ*= 480 nm,

_{g}*d*= 240 nm and

*t*= 60 nm. Figure 2 shows the reflectance pattern of PCs with cylindrical nanopillars in square lattice. The color and size of dots indicates the value of reflectance. The minimum reflectance had diagonal characteristics in a 2D space of periods and diameters. For smaller periods the minimum reflectance shifted towards the larger diameters and vice-versa. For smaller thickness the minimum reflectance shifted towards the larger diameters. Thus, the minimum reflectance had a narrow domain of period, diameter and thickness. For example, decreasing

*λ*from 470 nm to 450 nm for

_{g}*d*= 210 nm and

*t*= 50 nm decreased the reflectance from 0.402 to 0.108, where minimum reflectance took place. By decreasing

*λ*to 440 nm the reflectance increased to 0.257. Similarly, at

_{g}*λ*= 450 nm and thickness

_{g}*t*= 50 nm, decreasing in the diameter,

*d*, from 220 nm to 210 nm decreased the reflectance from 0.214 to 0.108, while a further decrease in diameter to 200 nm increased the reflectance to 0.365. Thus, there was an optimum period for minimum reflectance and by increasing or decreasing

*λ*from the optimal point, the reflectance increased. Moreover, by increasing or decreasing the thickness (

_{g}*t*) to 40 nm and 60 nm, the reflectance increased for a given values of

*d*and

*λ*.

_{g}Reflection patterns showed that the smaller reflectance were found from period *λ _{g}* = 470 nm to 430 nm for thicknesses at t = 60 nm and 50 nm. Periods for minimum reflectance were 10 nm less than the initially calculated lattice vector needed for surface plasmon resonance. The resonance for lattice periods lower than

*λ*= 470 was due to both localized plasmons and surface plasmons which will be further described with Fig. 6 .

_{g}Figure 3
shows the reflection pattern from silver PCs of cylindrical nanopillars in a triangular lattice for various periods and diameters when thickness (*t*) were equal to 50 nm, 60 nm and 70 nm. The triangular lattice also had the diagonal characteristic but to a lesser extent than the square lattice. It had a wider resonance domain in period (*p*) and diameter (d) space than the square lattice. And it also had a wider frequency response as shown in Fig. 7(a)
and further explained later. In contrast with square lattice, the triangular lattice had resonance at periods greater than *λ _{g}* = 480 nm. This may be due to the higher packing fraction of triangular lattice versus the square lattice.

To compare cylindrical PCs with rectangular PCs, the reflectance of rectangular PCs in a square lattice with side-length (*l*) and period (*λ _{g}*) were plotted in the neighborhood of minimum reflectance as shown in Fig. 4
which shows that reflectance from rectangular PCs is higher than cylindrical PCs. The rectangular nanopillars have sharp corners and edges which prevent homogeneous polarization in contrast with that of the cylindrical nanopillars [34]. There are six distinct polarization modes, each having its own resonance frequency. The resulting polarization is the summation of all modes [34,35]. Therefore, the localized plasmon resonance of rectangular structure was smaller than that of the cylindrical structure and the reflectance on periods other than period

*λ*= 480 nm were much higher. Thus, cylindrical nanopillars in a square lattice exhibited better performance in terms of minimum reflectance than that of rectangular nanopillars in a square lattice. Rectangular PCs have lower reflectance at periods 480 nm and 470 nm, beyond that the reflectance increased sharply.

_{g}Figure 5
compares the z-component of the electric field (E_{z}) of rectangular PCs which had similar reflectance at different periods. The PC in Fig. 5(a) had *λ _{g}* = 480 nm,

*l*= 160,

*t*= 40 nm and reflectance of 0.38 while the PC in Fig. 5(b) had

*λ*= 460 nm,

_{g}*l*= 200,

*t*= 40 nm and reflectance of 0.312. The E

_{z}field around the edge of the nanopillar was the same in both PCs, however E

_{z}field between the nanopillars was much higher in the PC having a period

*λ*of 480 nm. Therefore, the minimum reflectance near the period

_{g}*λ*= 480 nm (equal to calculated lattice period for surface plasmon resonance) was due to better coupling of surface plasmons with localized plasmons. Simulations of rectangular nanopillars in triangular lattice had lower enhancement than that of cylindrical of nanopillars in a triangular lattice, so is not presented here. Cylindrical nanopillars in square and triangular lattices had lower reflectance and hence better field enhancement so were used for further analysis of angles of incidence (θ) and polarization (ϕ).

_{g}Figures 6(a) and (b) are the z-component and x-component of normalized electric field of cylindrical silver nanopillar in square lattice of *λ _{g}* = 450 nm,

*d*= 210 nm and

*t*= 50 nm excited by x-polarized plane wave. The electric field at the metal-dielectric interface in PC has been enhanced by an order of magnitude; nevertheless, sharp peak enhancement at the edges of nanopillar was about 50 times.

The normalized electric field of surface plasmons on a metal surface in term of reflectance from surface can be approximated by

*θ*,

*R, ε’*and

_{m}*ε”*are angle of incidence, reflectance and real and imaginary components of dielectric function of metal respectively. The calculated enhancement is 10 times at reflectance 0.108, which is approximately equal to the field enhancement as shown in Fig. 6(a). Figure 6(c) shows the x-component of electric field when nanocylinders were imbedded in dielectric material without metal layer with the same geometrical size, period and lattice arrangement. It can be concluded that the enhanced electric field in islands of nanocylinders is due to localized plasmons and coupling of field among nanocylinders. These islands of nanocylinders were not optimized for maximum field enhancement. Nonetheless, they indicate that the metal layer at the base of nanocylinders enhances the x-component of electric field by more than two times due to surface plasmon resonance. For further support of this argument the thin cylinder was approximated by oblate spheroid and near-field of isolated oblate spheroid was calculated analytically as described in following paragraph.

_{m}Electromagnetic field in sub-wavelength structure can be treated as quasi-static field when the dimensions of the structure are much smaller than the wavelength of radiation in dielectric. Dipole moment *p* is defined in term of polarizability, α, as *p = αε _{o}ε_{d}E_{o}*. The polarizability of spheroid is given by [2,37]:

*a = d/2*and

*b = t/2*are major and minor axes of spheroid equivalent to the radius and half of disk thickness.

*L*is the geometrical factor given by:

The quantity of electric field along x-axis at 15 nm distance from the surface of oblate spheroid with major and minor axes diameters of 2*a* = 210 nm and 2*b* = 50 nm calculated using the well-known near-field electric dipole formula from reference 15 (section 9.2) and Eq. (4) is 2.8. This is in good agreement with the quantity of the E_{x} near the surface of nanocylinder obtained from FDTD simulation, Fig. 6(c). This indicated that the field enhancement by coupling the inter-nanopillars fields had much less contribution than coupling with metal surface. Equation (4) shows that polarizability is the cubic function of size of structure; thus, bigger structure should have stronger localized field near the surface than smaller structure. However, there is an optimum size of structure for extinction of radiation and total near-field enhancement at a particular metal-dielectric interface [4–6]. When these nanopillars are on a metal surface (2D metallic PC), the localized plasmons couple with metal surface and excites the surface plasmons whenever the lattice vector satisfy Eqs. (1) and (2). Therefore, in this work the lattice period (period *λ _{g}*) was defined by the lattice vector required to maintain surface plasmon resonance and the shape and dimensions of nanopillars were defined by the localized plasmons which couple with surface plasmons constructively.

Bandwidth of plasmon resonance is important in some application such as absorption enhancement in solar cells. But, plasmon resonance excited by metallic structures and metal particles are very narrow-band [7,23]. To obtain the bandwidth of plasmon resonance, 3D-FDTD simulation was carried out within a wide window of radiation wavelength, 500-1000 nm. Figure 7(a) shows that PCs of cylindrical nanopillars in a triangular lattice (B: *λ _{g}* = 480 nm,

*d*= 210 nm and

*t*= 60 nm) have a distinctly larger bandwidth in comparison to other simulated structures (A: cylindrical nanopillars of

*λ*= 450 nm,

_{g}*d*= 210 nm and

*t*= 50 nm in square lattice and C: rectangular nanopillar of

*λ*= 480 nm,

_{g}*l*= 150 nm and

*t*= 50 nm in square lattice). The resonance bandwidth increases with an increase in the diameter of the nanopillars (not shown in graph), however increasing diameter increases reflectance, so that overall enhancement is decreased. Figure 7(a) illustrates that rectangular nanopillars in a square lattice (curve C) has higher reflectance, and hence lower field enhancement than that of cylindrical nanopillars.

PCs of cylindrical nanopillars in a square lattice (curve A) was used for further analysis for reflectance versus thickness, diameter and period of PC as illustrated in Fig. 7(b). Figure 2, 3, and 7(b) show that the reflectance was more sensitive to the thickness of nanopillars than diameter and period of nanopillars. When the thickness of nanopillar was kept at optimum value, changes in period or diameter by 10 nm did not change the reflectance as much as the change in thickness by 10 nm. As we know from oblate approximation, the polarizability of nanocylinder is proportional to its volume, but thicker nanocylinder has smaller polarizability along the major axis due to geometrical depolarization factor (L). In perpendicular incidence the thinner nanocylinder has lower normal component of electric field which decreases coupling of field with metal surface. Therefore, the thickness of nanocylinder is sensitive to both localized plasmons and coupling with surface plasmons, exhibiting higher sensitivity. As a result, thickness of nanopillar must have higher accuracy than diameter and period during fabrication. Fortunately, current deposition techniques enable control of thickness within a few nanometers.

Enhancement of electromagnetic field by non-localized plasmons is a strong function of angle of incidence [19,21]. Increase in incident angle (θ) increases the wave vector parallel to the surface of PC (*k _{||}*), so that wave vector given by Eq. (2) will be greater than the wave vector of SPPs (

*β*) and the condition of phase matching will not be satisfied. Figure 8(a) shows the reflectance versus incident angle (θ) of PCs A and B at wavelength

*λ*= 750 nm and polarization angle ϕ = 0°. Reflectance of

_{o}*p*-polarized electric field (R

_{p}) of PC A initially increased sharply by increasing the angle of incidence. The maximum reflectance takes place at θ = 20° and dropped off fast to another minimum reflectance at θ = 40°. Second minimum of reflectance curve was due to excitation of another resonance mode as illustrated in Fig. 8(c). The FDTD simulation shows that this resonance mode is excited by coupling the electric field between nanopillars and metal surface along y-axis as wave propagates along x-axis. Physically this resonance mode occurs when the nanopillar couples the opposite phases of

*p*-polarized electric field and provides net vertical polarization. The reflectance of s-polarized electric field (R

_{s}) steadily increased with increase in angle of incidence. Reflectance of

*p*-polarized field (R

_{p}) of PC B slowly increased up to θ = 10°, then reached at a maximum value at θ = 30°. Reflectance of

*s*-polarized electric field (R

_{s}) had sharp peak response at 15°, but it followed the trend of the R

_{p}of PC A at other angles of incidence.

Reflectance curves showed that the *p*-polarized light in a triangular lattice had lower reflectance than square lattice up to 25°. However, it did not have a second resonance mode as that in the square lattice. Figure 8(b) shows reflectance versus azimuth angle ϕ (polarization angle) of incident wave for θ = 0° in first irreducible zone of rotational symmetry (45° for square lattice and 30° for triangular lattice [38]). Both PCs A and B had fairly constant values of reflectance with polarization angles. Above results indicated that there would not be major reduction in field enhancement when changing the polarization angle as long as the wave incident perpendicularly. PCs having dimensions other than those of PCs A and B also had patterns similar to Fig. 8.

EM Explorer was preferred for simulations because it has the advantages of using the experimental values of dielectric constant of metals and provides the reflection even from dispersive material. To verify the robustness of this numerical tool, simulations of PC of cylindrical gold nanopillars for enhancement at wavelength *λ _{o}* = 980 nm were done using both EM Explorer and Meep, another FDTD simulation tool [39]. The dielectric function of gold was obtained by fitting experimental values from Palik [32]. The PC of cylindrical nanopillar having

*λ*= 620 nm,

_{g}*d*= 310 nm and

*t*= 70 nm in square lattice had field enhancement greater than 10 times. The results obtained from both simulation tools matched to a great extent as shown in Fig. 9 , which supports the robustness of the simulations.

## 4. Conclusions

The investigation showed that 2D metallic PCs can be tuned for enhancement of electromagnetic field at desired wavelength by tuning the sizes and lattice parameters of the nanopillars. PCs of cylindrical nanopillars of optimum sizes had electric field enhancement of more than 10X in vicinity of nanopillars and up to 50X on the edge of nanopillars. Depending on lattice types, nanopillar shapes and thickness, each PCs had their own domain of minimum reflection in space of period and size. It was found that PCs of cylindrical nanopillars in triangular lattice had the wider bandwidth of enhancement as well as larger minimum reflectance domain. Thickness of nanopillars was more sensitive than period and diameter. The reflection versus incident angle curves showed that the PCs of a lattice type have similar characteristic curves. It was observed that reflectance form PCs, hence field enhancement, was less sensitive with polarization angle (ϕ).

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