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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 24 — Nov. 23, 2009
  • pp: 22179–22189
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Geometry dependence of field enhancement in 2D metallic photonic crystals

Hari P. Paudel, Khadijeh Bayat, Mahdi Farrokh Baroughi, Stanley May, and David W. Galipeau  »View Author Affiliations


Optics Express, Vol. 17, Issue 24, pp. 22179-22189 (2009)
http://dx.doi.org/10.1364/OE.17.022179


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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 OSA

1. Introduction

The coherent fluctuation of electrons density at a metal surface by electromagnetic radiation is called surface plasmon oscillation [1

1. H. Raether, Surface Plasmon on smooth and rough surface and on grating (Spinger-Verlag, Berlin Heidelberg, 1988).

]. 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

2. S. A. Maier, Plasmonics: Fundamentals and application, (Springer, New York, 2007).

]. 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

3. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 ( 2003). [CrossRef] [PubMed]

]. 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

4. D. M. Schaadt, B. Feng, and E. T. Yu, “Enhanced semiconductor optical absorption via surface Plasmon excitation in metal nanoparticles,” Appl. Phys. Lett. 86(6), 063106 ( 2005). [CrossRef]

,5

5. S. Phillai, K. R. Catchpole, T. Trupke, and M. A. Green, “Surface Plasmon enhanced silicon solar cells,” J. Appl. Phys. 101, 093104 ( 2007).

]. Islands of nano-disks on dielectric materials were investigated for enhancement of absorption [6

6. C. Langhammer, M. Schwind, B. Kasemo, and I. Zorić, “Localized surface plasmon resonances in aluminum nanodisks,” Nano Lett. 8(5), 1461–1471 ( 2008). [CrossRef] [PubMed]

], transmission [7

7. C. H. Liu, M. H. Hong, H. W. Cheung, F. Zhang, Z. Q. Huang, L. S. Tan, and T. S. A. Hor, “Bimetallic structure fabricated by laser interference lithography for tuning surface plasmon resonance,” Opt. Express 16(14), 10701–10709 ( 2008). [CrossRef] [PubMed]

] and photocurrent [8

8. C. Hägglund, M. Zäch, G. Petersson, and B. Kasemo, “Electromagnetic coupling of light into a silicon solar cell by nanodisk plasmon,” Appl. Phys. Lett. 92, 153110 ( 2008). [CrossRef]

]. Nano-wires on dielectric materials were investigated for absorption enhancement and fluorescence emission enhancement [9

9. K. Tawa, H. Hori, K. Kintaka, K. Kiyosue, Y. Tatsu, and J. Nishii, “Optical microscopic observation of fluorescence enhanced by grating-coupled surface plasmon resonance,” Opt. Express 16(13), 9781–9790 ( 2008). [CrossRef] [PubMed]

]. Photonic band gap of surface modes in two dimensional arrays of metal nanostrucutes on metal surface has been investigated theoretically [10

10. M. Kretschmann, “Phase diagrams of surface plasmon polaritons crystals,” Phys. Rev. B 68(12), 125419 ( 2003). [CrossRef]

] and demonstrated experimentally [11

11. S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full Photonic Band Gap for Surface Modes in the Visible,” Phys. Rev. Lett. 77(13), 2670–2673 ( 1996). [CrossRef] [PubMed]

,12

12. S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86(14), 3008–3011 ( 2001). [CrossRef] [PubMed]

]. Investigation of scattering and propagation of SPPs along textured metal surface by triangular lattice nanodisks has been demonstrated from both experimental [12

12. S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86(14), 3008–3011 ( 2001). [CrossRef] [PubMed]

] and numerical [13

13. T. Søndergaard and S. I. Bozhevolnyi, “Theoretical analysis of finite-size surface plasmon polaritons band-gap structures,” Phys. Rev. B 71(12), 125429 ( 2005). [CrossRef]

,14

14. A. Boltasseva, T. Søndergaard, T. Nikolajsen, K. Leosson, S. I. Bozhevolnyi, and J. M. Hvam, “Propagation of long-range surface plasmon polaritons in photonic crystals,” J. Opt. Soc. Am. B 22(9), 2027 ( 2005). [CrossRef]

] 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.

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

2. S. A. Maier, Plasmonics: Fundamentals and application, (Springer, New York, 2007).

].
β=koεdεmεd+εm
(1)
Where, ko is the wave vector in a vacuum, εd is relative permittivity of the dielectric and εm is the dielectric function of the metal which is a nonlinear function of the frequency of radiation. At ωsp, εm is equal to -εd and the SPPs wave vector (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 λg 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

29. Y. Teng and E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19(9), 511–514 ( 1967). [CrossRef]

]. The surface plasmon excitation condition of SPPs wave vector is
β=k||+2πn λgx^+2πmλgy^
(2)
A unidirectional grating excites surface plasmons only if the excitation radiation is p-polarized [29

29. Y. Teng and E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19(9), 511–514 ( 1967). [CrossRef]

]. A bidirectional grating (2D-PC) can excite both p and s polarized radiation [30

30. P. T. Worthing and W. L. Barnes, “Efficient coupling of surface plasmons polaritons to radiation using a bi-grating,” Appl. Phys. Lett. 79(19), 3035–3037 ( 2001). [CrossRef]

]. 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
Fig. 1 2D photonic crystals in square lattice with period λg, diameter d (side-length l in rectangular) and thickness t. (a) cylindrical nanopillars, (b) rectangular nanopillars
depicts the photonic crystals of cylindrical and rectangular nanopillars of diameter d (l in rectangular) and thickness t arranged in square lattice of period λg. 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

32. D. W. Lynch, and W. R. Hunter, “Comments on the Optical Constants of Metals and an Introduction to the Data for Several metals” Handbook of Optical constant of Solid, E. D. Palik ed. (Academic press, New York 1985).

]. 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.

3. Results and Analysis

The initial period (λg) of the nanopillars calculated from analytical grating formula (2) and dispersion Eq. (1) was 480 nm for silver PC illuminated perpendicularly at λ0 = 750 nm. Therefore, the initial values of the design parameters were λg = 480 nm, d = 240 nm and t = 60 nm. Figure 2
Fig. 2 Reflectance from silver PCs of cylindrical nanopillars in square lattice at λ o = 750 nm illuminated perpendicularly by x-polarized plane wave.
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 λg from 470 nm to 450 nm for d = 210 nm and t = 50 nm decreased the reflectance from 0.402 to 0.108, where minimum reflectance took place. By decreasing λg to 440 nm the reflectance increased to 0.257. Similarly, at λg = 450 nm and thickness 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 λg from the optimal point, the reflectance increased. Moreover, by increasing or decreasing the thickness (t) to 40 nm and 60 nm, the reflectance increased for a given values of d and λg.

Reflection patterns showed that the smaller reflectance were found from period λg = 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 λg = 470 was due to both localized plasmons and surface plasmons which will be further described with Fig. 6
Fig. 6 Electric field components in silver PC of cylindrical nanopillars in square lattice of λg = 450 nm, d = 210 nm and t = 50 nm (a) Ez component in XY plane, (b) Ex component in XY plane and (c) Ex component when the nanocylinder are imbedded only in dielectric without metal sheet.
.

Figure 3
Fig. 3 Reflectance from silver PCs of cylindrical nanopillars in triangular lattice at λo = 750 nm illuminated perpendicularly by x-polarized of plane wave.
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)
Fig. 7 (a) Reflectance versus wavelength of silver PCs. A: cylindrical nanopillar of λg = 450 nm, d = 210 nm and t = 50 nm in square lattice, B: cylindrical nanopillar of λg = 480 nm, d = 210 nm and t = 60 nm in triangular lattice and C: rectangular nanopillar of λg = 480 nm, l = 150 nm and t = 50 nm in square lattice. (b) Reflectance versus dimensions of PC A.
and further explained later. In contrast with square lattice, the triangular lattice had resonance at periods greater than λg = 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 (λg) were plotted in the neighborhood of minimum reflectance as shown in Fig. 4
Fig. 4 Reflectance from rectangular-shaped silver PC in square lattice at λo = 750 nm illuminated perpendicularly by x-polarized plane wave.
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

34. R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B 11(4), 1732–1740 ( 1975). [CrossRef]

]. There are six distinct polarization modes, each having its own resonance frequency. The resulting polarization is the summation of all modes [34

34. R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B 11(4), 1732–1740 ( 1975). [CrossRef]

,35

35. R. Ruppin, “Plasmon frequencies of cube shaped metal clusters,” Z. Phys. D 36(1), 69–71 ( 1996). [CrossRef]

]. Therefore, the localized plasmon resonance of rectangular structure was smaller than that of the cylindrical structure and the reflectance on periods other than period λg = 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.

Figure 5
Fig. 5 Ez components of rectangular silver PCs in XZ plane crossing through the nanopillar (right image of each pair) and crossing between the nanopillars (left image of each pair), (a) nm λg = 480 nm, l = 160 nm and t = 40 nm, (b) λg = 460 nm, l = 200 and t = 40 nm, illuminated perpendicularly by λo = 750 nm x-polarized plane wave.
compares the z-component of the electric field (Ez) of rectangular PCs which had similar reflectance at different periods. The PC in Fig. 5(a) had λg = 480 nm, l = 160, t = 40 nm and reflectance of 0.38 while the PC in Fig. 5(b) had λg = 460 nm, l = 200, t = 40 nm and reflectance of 0.312. The Ez field around the edge of the nanopillar was the same in both PCs, however Ez field between the nanopillars was much higher in the PC having a period λg of 480 nm. Therefore, the minimum reflectance near the period λg = 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).

Figures 6(a) and (b) are the z-component and x-component of normalized electric field of cylindrical silver nanopillar in square lattice of λg = 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
ESP2=2cosθεm2(1R)εdεm"εm'εd
(3)
which is derived from energy conservation and is equal to the ratio of propagation length of SPPs along the boundary of layers to the spatial extension of SPPs normal to the boundary [36

36. W. H. Weber and G. W. Ford, “Optical electric-field enhancement at a metal surface arising from surface-plasmon excitation,” Opt. Lett. 6(3), 122–124 ( 1981). [CrossRef] [PubMed]

]. Where, θ, R, ε’m and ε”m 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.

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 = αεoεdEo. The polarizability of spheroid is given by [2

2. S. A. Maier, Plasmonics: Fundamentals and application, (Springer, New York, 2007).

,37

37. C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (John Wiley & Sons, New York, 1983).

]:
α=4πa2b3εmεdεd+L[εmεd]
(4)
Where, 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:

L=a2b2{a2π2a2arctan(ba2b2)ba2b2a2(a2b2)3/2}
(5)

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 2a = 210 nm and 2b = 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 Ex 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

4. D. M. Schaadt, B. Feng, and E. T. Yu, “Enhanced semiconductor optical absorption via surface Plasmon excitation in metal nanoparticles,” Appl. Phys. Lett. 86(6), 063106 ( 2005). [CrossRef]

6

6. C. Langhammer, M. Schwind, B. Kasemo, and I. Zorić, “Localized surface plasmon resonances in aluminum nanodisks,” Nano Lett. 8(5), 1461–1471 ( 2008). [CrossRef] [PubMed]

]. 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 λg) 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

7. C. H. Liu, M. H. Hong, H. W. Cheung, F. Zhang, Z. Q. Huang, L. S. Tan, and T. S. A. Hor, “Bimetallic structure fabricated by laser interference lithography for tuning surface plasmon resonance,” Opt. Express 16(14), 10701–10709 ( 2008). [CrossRef] [PubMed]

,23

23. A. Benabbas, V. Halté, and J.-Y. Bigot, “Analytical model of the optical response of periodically structured metallic films,” Opt. Express 13(22), 8730–8745 ( 2005). [CrossRef] [PubMed]

]. 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: λg = 480 nm, d = 210 nm and t = 60 nm) have a distinctly larger bandwidth in comparison to other simulated structures (A: cylindrical nanopillars of λg = 450 nm, d = 210 nm and t = 50 nm in square lattice and C: rectangular nanopillar of λg = 480 nm, 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

19. P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunction for square-wave gratings: Application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26(6), 2907–2916 ( 1982). [CrossRef]

,21

21. T. Matsuda, D. Zhou, and Y. Okuno, “Numerical analysis of plasmon-resonance absorption in bisinusoidal metal gratings,” J. Opt. Soc. Am. A 19(4), 695–701 ( 2002). [CrossRef]

]. 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)
Fig. 8 (a) Reflectance versus incident angle curves of PCs A and B for ϕ = 0°. Continuous lines represent the reflectance of p-polarization (Rp) and broken lines represent the reflectance of s-polarization (Rs). (b) Reflectance versus polarization angle (ϕ) curves of PCs A and B for incident angle θ = 0°. (c) Ez field of two resonance modes at time period t = 0 and t = T/2 and reflectance angles (i) θ = 0° and (ii) θ = 40°.
shows the reflectance versus incident angle (θ) of PCs A and B at wavelength λo = 750 nm and polarization angle ϕ = 0°. Reflectance of p-polarized electric field (Rp) 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 (Rs) steadily increased with increase in angle of incidence. Reflectance of p-polarized field (Rp) of PC B slowly increased up to θ = 10°, then reached at a maximum value at θ = 30°. Reflectance of s-polarized electric field (Rs) had sharp peak response at 15°, but it followed the trend of the Rp 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

38. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals, (Princeton University Press 2008).

]). 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 λo = 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

32. D. W. Lynch, and W. R. Hunter, “Comments on the Optical Constants of Metals and an Introduction to the Data for Several metals” Handbook of Optical constant of Solid, E. D. Palik ed. (Academic press, New York 1985).

]. The PC of cylindrical nanopillar having λg = 620 nm, 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
Fig. 9 Enhancement of Ez component of field excited by λo = 980 nm, p-polarized electric field on λg = 620 nm, d = 310 nm and t = 70 nm gold PC of cylindrical nanopillar in square lattice simulated by (a) Meep and (b) EM explorer.
, 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).

References and links

1.

H. Raether, Surface Plasmon on smooth and rough surface and on grating (Spinger-Verlag, Berlin Heidelberg, 1988).

2.

S. A. Maier, Plasmonics: Fundamentals and application, (Springer, New York, 2007).

3.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 ( 2003). [CrossRef] [PubMed]

4.

D. M. Schaadt, B. Feng, and E. T. Yu, “Enhanced semiconductor optical absorption via surface Plasmon excitation in metal nanoparticles,” Appl. Phys. Lett. 86(6), 063106 ( 2005). [CrossRef]

5.

S. Phillai, K. R. Catchpole, T. Trupke, and M. A. Green, “Surface Plasmon enhanced silicon solar cells,” J. Appl. Phys. 101, 093104 ( 2007).

6.

C. Langhammer, M. Schwind, B. Kasemo, and I. Zorić, “Localized surface plasmon resonances in aluminum nanodisks,” Nano Lett. 8(5), 1461–1471 ( 2008). [CrossRef] [PubMed]

7.

C. H. Liu, M. H. Hong, H. W. Cheung, F. Zhang, Z. Q. Huang, L. S. Tan, and T. S. A. Hor, “Bimetallic structure fabricated by laser interference lithography for tuning surface plasmon resonance,” Opt. Express 16(14), 10701–10709 ( 2008). [CrossRef] [PubMed]

8.

C. Hägglund, M. Zäch, G. Petersson, and B. Kasemo, “Electromagnetic coupling of light into a silicon solar cell by nanodisk plasmon,” Appl. Phys. Lett. 92, 153110 ( 2008). [CrossRef]

9.

K. Tawa, H. Hori, K. Kintaka, K. Kiyosue, Y. Tatsu, and J. Nishii, “Optical microscopic observation of fluorescence enhanced by grating-coupled surface plasmon resonance,” Opt. Express 16(13), 9781–9790 ( 2008). [CrossRef] [PubMed]

10.

M. Kretschmann, “Phase diagrams of surface plasmon polaritons crystals,” Phys. Rev. B 68(12), 125419 ( 2003). [CrossRef]

11.

S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full Photonic Band Gap for Surface Modes in the Visible,” Phys. Rev. Lett. 77(13), 2670–2673 ( 1996). [CrossRef] [PubMed]

12.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86(14), 3008–3011 ( 2001). [CrossRef] [PubMed]

13.

T. Søndergaard and S. I. Bozhevolnyi, “Theoretical analysis of finite-size surface plasmon polaritons band-gap structures,” Phys. Rev. B 71(12), 125429 ( 2005). [CrossRef]

14.

A. Boltasseva, T. Søndergaard, T. Nikolajsen, K. Leosson, S. I. Bozhevolnyi, and J. M. Hvam, “Propagation of long-range surface plasmon polaritons in photonic crystals,” J. Opt. Soc. Am. B 22(9), 2027 ( 2005). [CrossRef]

15.

J. D. Jackson, Classical Electrodynamics, (Wiley India, 1999).

16.

L. O. M. Rayleigh, “On Dynamical theory of gratings,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character 79(532), 399–416 ( 1907). [CrossRef]

17.

D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. A 68(4), 490–495 ( 1978). [CrossRef]

18.

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in optics, Vol. xxi, E. Wolf ed. (1984).

19.

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunction for square-wave gratings: Application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26(6), 2907–2916 ( 1982). [CrossRef]

20.

K. Yusuura and H. Ikuno, “Improved point matching method with application to scattering from periodic surface,” IEEE Trans. Antennas Propag. AP 21(5), 657–662 ( 1973). [CrossRef]

21.

T. Matsuda, D. Zhou, and Y. Okuno, “Numerical analysis of plasmon-resonance absorption in bisinusoidal metal gratings,” J. Opt. Soc. Am. A 19(4), 695–701 ( 2002). [CrossRef]

22.

T. K. Gaylord, and M. G. Maharam, “Analysis and Application of Optical Diffraction by Gratings,” in Proceedings of IEEE Conference (1985) 73(5), pp. 894–937.

23.

A. Benabbas, V. Halté, and J.-Y. Bigot, “Analytical model of the optical response of periodically structured metallic films,” Opt. Express 13(22), 8730–8745 ( 2005). [CrossRef] [PubMed]

24.

M. Paulus and O. J. Martin, “Green’s tensor technique for scattering in two-dimensional stratified media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(6), 066615 ( 2001). [CrossRef] [PubMed]

25.

T. Søndergaard and S. I. Bozhevolnyi, “Surface plasmon polariton scattering by a small particle placed near a metal surface:An analytical study,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69, 045422 ( 2004).

26.

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1498 ( 1994). [CrossRef]

27.

L. Zhao, K. L. Kelly, and G. C. Schatz, “The extinction spectra of silver nanoparticle arrays: Influence of arrays structure on plasmon resonance wavelength and width,” J. Phys. Chem. B 107(30), 7343–7350 ( 2003). [CrossRef]

28.

R. Lazzari, I. Simonsen, D. Bedeaux, J. Vlieger, and J. Jupille, “Polarizability of truncated spheroidal particles supported by a substrate: model and application,” Eur. Phys. J. B 24(2), 267–284 ( 2001). [CrossRef]

29.

Y. Teng and E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19(9), 511–514 ( 1967). [CrossRef]

30.

P. T. Worthing and W. L. Barnes, “Efficient coupling of surface plasmons polaritons to radiation using a bi-grating,” Appl. Phys. Lett. 79(19), 3035–3037 ( 2001). [CrossRef]

31.

http://www.emexplorer.net

32.

D. W. Lynch, and W. R. Hunter, “Comments on the Optical Constants of Metals and an Introduction to the Data for Several metals” Handbook of Optical constant of Solid, E. D. Palik ed. (Academic press, New York 1985).

33.

A. Taflove, and S. C. Hagness, Computational Electrodynamics: Finite-Difference Time-Domain Method, (Artech House, 1995).

34.

R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B 11(4), 1732–1740 ( 1975). [CrossRef]

35.

R. Ruppin, “Plasmon frequencies of cube shaped metal clusters,” Z. Phys. D 36(1), 69–71 ( 1996). [CrossRef]

36.

W. H. Weber and G. W. Ford, “Optical electric-field enhancement at a metal surface arising from surface-plasmon excitation,” Opt. Lett. 6(3), 122–124 ( 1981). [CrossRef] [PubMed]

37.

C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (John Wiley & Sons, New York, 1983).

38.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals, (Princeton University Press 2008).

39.

MEEP, FDTD package, http://ab-initio.mit.edu/wiki/index.php/Meep

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(350.4238) Other areas of optics : Nanophotonics and photonic crystals

ToC Category:
Photonic Crystals

History
Original Manuscript: September 8, 2009
Revised Manuscript: October 12, 2009
Manuscript Accepted: October 13, 2009
Published: November 19, 2009

Citation
Hari P. Paudel, Khadijeh Bayat, Mahdi Farrokh Baroughi, Stanley May, and David W. Galipeau, "Geometry dependence of field enhancement in 2D metallic photonic crystals," Opt. Express 17, 22179-22189 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-22179


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References

  1. H. Raether, Surface Plasmon on smooth and rough surface and on grating (Spinger-Verlag, Berlin Heidelberg, 1988).
  2. S. A. Maier, Plasmonics: Fundamentals and application, (Springer, New York, 2007).
  3. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]
  4. D. M. Schaadt, B. Feng, and E. T. Yu, “Enhanced semiconductor optical absorption via surface Plasmon excitation in metal nanoparticles,” Appl. Phys. Lett. 86(6), 063106 (2005). [CrossRef]
  5. S. Phillai, K. R. Catchpole, T. Trupke, and M. A. Green, “Surface Plasmon enhanced silicon solar cells,” J. Appl. Phys. 101, 093104 (2007).
  6. C. Langhammer, M. Schwind, B. Kasemo, and I. Zorić, “Localized surface plasmon resonances in aluminum nanodisks,” Nano Lett. 8(5), 1461–1471 (2008). [CrossRef] [PubMed]
  7. C. H. Liu, M. H. Hong, H. W. Cheung, F. Zhang, Z. Q. Huang, L. S. Tan, and T. S. A. Hor, “Bimetallic structure fabricated by laser interference lithography for tuning surface plasmon resonance,” Opt. Express 16(14), 10701–10709 (2008). [CrossRef] [PubMed]
  8. C. Hägglund, M. Zäch, G. Petersson, and B. Kasemo, “Electromagnetic coupling of light into a silicon solar cell by nanodisk plasmon,” Appl. Phys. Lett. 92, 153110 (2008). [CrossRef]
  9. K. Tawa, H. Hori, K. Kintaka, K. Kiyosue, Y. Tatsu, and J. Nishii, “Optical microscopic observation of fluorescence enhanced by grating-coupled surface plasmon resonance,” Opt. Express 16(13), 9781–9790 (2008). [CrossRef] [PubMed]
  10. M. Kretschmann, “Phase diagrams of surface plasmon polaritons crystals,” Phys. Rev. B 68(12), 125419 (2003). [CrossRef]
  11. S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full Photonic Band Gap for Surface Modes in the Visible,” Phys. Rev. Lett. 77(13), 2670–2673 (1996). [CrossRef] [PubMed]
  12. S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86(14), 3008–3011 (2001). [CrossRef] [PubMed]
  13. T. Søndergaard and S. I. Bozhevolnyi, “Theoretical analysis of finite-size surface plasmon polaritons band-gap structures,” Phys. Rev. B 71(12), 125429 (2005). [CrossRef]
  14. A. Boltasseva, T. Søndergaard, T. Nikolajsen, K. Leosson, S. I. Bozhevolnyi, and J. M. Hvam, “Propagation of long-range surface plasmon polaritons in photonic crystals,” J. Opt. Soc. Am. B 22(9), 2027 (2005). [CrossRef]
  15. J. D. Jackson, Classical Electrodynamics, (Wiley India, 1999).
  16. L. O. M. Rayleigh, “On Dynamical theory of gratings,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character 79(532), 399–416 (1907). [CrossRef]
  17. D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. A 68(4), 490–495 (1978). [CrossRef]
  18. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in optics, Vol. xxi, E. Wolf ed. (1984).
  19. P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunction for square-wave gratings: Application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26(6), 2907–2916 (1982). [CrossRef]
  20. K. Yusuura and H. Ikuno, “Improved point matching method with application to scattering from periodic surface,” IEEE Trans. Antennas Propag. AP 21(5), 657–662 (1973). [CrossRef]
  21. T. Matsuda, D. Zhou, and Y. Okuno, “Numerical analysis of plasmon-resonance absorption in bisinusoidal metal gratings,” J. Opt. Soc. Am. A 19(4), 695–701 (2002). [CrossRef]
  22. T. K. Gaylord, and M. G. Maharam, “Analysis and Application of Optical Diffraction by Gratings,” in Proceedings of IEEE Conference (1985) 73(5), pp. 894–937.
  23. A. Benabbas, V. Halté, and J.-Y. Bigot, “Analytical model of the optical response of periodically structured metallic films,” Opt. Express 13(22), 8730–8745 (2005). [CrossRef] [PubMed]
  24. M. Paulus and O. J. Martin, “Green’s tensor technique for scattering in two-dimensional stratified media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(6), 066615 (2001). [CrossRef] [PubMed]
  25. T. Søndergaard and S. I. Bozhevolnyi, “Surface plasmon polariton scattering by a small particle placed near a metal surface:An analytical study,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69, 045422 (2004).
  26. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1498 (1994). [CrossRef]
  27. L. Zhao, K. L. Kelly, and G. C. Schatz, “The extinction spectra of silver nanoparticle arrays: Influence of arrays structure on plasmon resonance wavelength and width,” J. Phys. Chem. B 107(30), 7343–7350 (2003). [CrossRef]
  28. R. Lazzari, I. Simonsen, D. Bedeaux, J. Vlieger, and J. Jupille, “Polarizability of truncated spheroidal particles supported by a substrate: model and application,” Eur. Phys. J. B 24(2), 267–284 (2001). [CrossRef]
  29. Y. Teng and E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19(9), 511–514 (1967). [CrossRef]
  30. P. T. Worthing and W. L. Barnes, “Efficient coupling of surface plasmons polaritons to radiation using a bi-grating,” Appl. Phys. Lett. 79(19), 3035–3037 (2001). [CrossRef]
  31. http://www.emexplorer.net
  32. D. W. Lynch and W. R. Hunter, “Comments on the Optical Constants of Metals and an Introduction to the Data for Several metals” Handbook of Optical constant of Solid, E. D. Palik ed., (Academic press, New York 1985).
  33. A. Taflove and S. C. Hagness, Computational Electrodynamics: Finite-Difference Time-Domain Method, (Artech House, 1995).
  34. R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B 11(4), 1732–1740 (1975). [CrossRef]
  35. R. Ruppin, “Plasmon frequencies of cube shaped metal clusters,” Z. Phys. D 36(1), 69–71 (1996). [CrossRef]
  36. W. H. Weber and G. W. Ford, “Optical electric-field enhancement at a metal surface arising from surface-plasmon excitation,” Opt. Lett. 6(3), 122–124 (1981). [CrossRef] [PubMed]
  37. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (John Wiley & Sons, New York, 1983).
  38. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals, (Princeton University Press 2008).
  39. MEEP, FDTD package, http://ab-initio.mit.edu/wiki/index.php/Meep

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