## Geometry dependence of field enhancement in 2D metallic photonic crystals

Optics Express, Vol. 17, Issue 24, pp. 22179-22189 (2009)

http://dx.doi.org/10.1364/OE.17.022179

Acrobat PDF (798 KB)

### 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

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]

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]

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]

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]

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. 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]

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]

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]

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]

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]

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]

## 2. Design methodology

*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.

*λ*couples with SPPs when the reciprocal lattice vector fulfills the mismatch between the SPP wave vector

_{g}*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]

*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]

*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]

*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.

*λ*= 750 nm and x-polarized plane wave. The PC was tuned for maximum field enhancement by varying the lattice period (

_{o}*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

*λ*) of the nanopillars calculated from analytical grating formula (2) and dispersion Eq. (1) was 480 nm for silver PC illuminated perpendicularly at

_{g}*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}*λ*= 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

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

_{g}*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*= 480 nm. This may be due to the higher packing fraction of triangular lattice versus the square lattice.

_{g}*l*) and period (

*λ*) 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

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

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]

*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}_{z}) of rectangular PCs which had similar reflectance at different periods. The PC in Fig. 5(a) had

*λ*= 480 nm,

_{g}*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}*λ*= 450 nm,

_{g}*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.

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]

*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}*p*is defined in term of polarizability, α, as

*p = αε*. The polarizability of spheroid is given by [2,37]:Where,

_{o}ε_{d}E_{o}*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:

*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

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. 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]

*λ*) 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.

_{g}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. 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]

*λ*= 480 nm,

_{g}*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.

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. 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]

*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.

*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.

*λ*= 980 nm were done using both EM Explorer and Meep, another FDTD simulation tool [39

_{o}39. MEEP, FDTD package, http://ab-initio.mit.edu/wiki/index.php/Meep

*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

## References and links

1. | H. Raether, |

2. | S. A. Maier, |

3. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature |

4. | D. M. Schaadt, B. Feng, and E. T. Yu, “Enhanced semiconductor optical absorption via surface Plasmon excitation in metal nanoparticles,” Appl. Phys. Lett. |

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

6. | C. Langhammer, M. Schwind, B. Kasemo, and I. Zorić, “Localized surface plasmon resonances in aluminum nanodisks,” Nano Lett. |

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 |

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. |

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 |

10. | M. Kretschmann, “Phase diagrams of surface plasmon polaritons crystals,” Phys. Rev. B |

11. | S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full Photonic Band Gap for Surface Modes in the Visible,” Phys. Rev. Lett. |

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. |

13. | T. Søndergaard and S. I. Bozhevolnyi, “Theoretical analysis of finite-size surface plasmon polaritons band-gap structures,” Phys. Rev. B |

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 |

15. | J. D. Jackson, |

16. | L. O. M. Rayleigh, “On Dynamical theory of gratings,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character |

17. | D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. A |

18. | D. Maystre, “Rigorous vector theories of diffraction gratings,” in |

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 |

20. | K. Yusuura and H. Ikuno, “Improved point matching method with application to scattering from periodic surface,” IEEE Trans. Antennas Propag. AP |

21. | T. Matsuda, D. Zhou, and Y. Okuno, “Numerical analysis of plasmon-resonance absorption in bisinusoidal metal gratings,” J. Opt. Soc. Am. A |

22. | T. K. Gaylord, and M. G. Maharam, “Analysis and Application of Optical Diffraction by Gratings,” in |

23. | A. Benabbas, V. Halté, and J.-Y. Bigot, “Analytical model of the optical response of periodically structured metallic films,” Opt. Express |

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. |

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. |

26. | B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A |

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 |

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 |

29. | Y. Teng and E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. |

30. | P. T. Worthing and W. L. Barnes, “Efficient coupling of surface plasmons polaritons to radiation using a bi-grating,” Appl. Phys. Lett. |

31. | |

32. | D. W. Lynch, and W. R. Hunter, “Comments on the Optical Constants of Metals and an Introduction to the Data for Several metals” |

33. | A. Taflove, and S. C. Hagness, |

34. | R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B |

35. | R. Ruppin, “Plasmon frequencies of cube shaped metal clusters,” Z. Phys. D |

36. | W. H. Weber and G. W. Ford, “Optical electric-field enhancement at a metal surface arising from surface-plasmon excitation,” Opt. Lett. |

37. | C. F. Bohren, and D. R. Huffman, |

38. | J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, |

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

- H. Raether, Surface Plasmon on smooth and rough surface and on grating (Spinger-Verlag, Berlin Heidelberg, 1988).
- S. A. Maier, Plasmonics: Fundamentals and application, (Springer, New York, 2007).
- W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]
- 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]
- S. Phillai, K. R. Catchpole, T. Trupke, and M. A. Green, “Surface Plasmon enhanced silicon solar cells,” J. Appl. Phys. 101, 093104 (2007).
- 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]
- 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]
- 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]
- 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]
- M. Kretschmann, “Phase diagrams of surface plasmon polaritons crystals,” Phys. Rev. B 68(12), 125419 (2003). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- J. D. Jackson, Classical Electrodynamics, (Wiley India, 1999).
- 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]
- D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. A 68(4), 490–495 (1978). [CrossRef]
- D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in optics, Vol. xxi, E. Wolf ed. (1984).
- 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]
- 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]
- 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]
- 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.
- 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]
- 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]
- 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).
- B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1498 (1994). [CrossRef]
- 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]
- 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]
- Y. Teng and E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19(9), 511–514 (1967). [CrossRef]
- 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]
- http://www.emexplorer.net
- 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).
- A. Taflove and S. C. Hagness, Computational Electrodynamics: Finite-Difference Time-Domain Method, (Artech House, 1995).
- R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B 11(4), 1732–1740 (1975). [CrossRef]
- R. Ruppin, “Plasmon frequencies of cube shaped metal clusters,” Z. Phys. D 36(1), 69–71 (1996). [CrossRef]
- 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]
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (John Wiley & Sons, New York, 1983).
- J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals, (Princeton University Press 2008).
- MEEP, FDTD package, http://ab-initio.mit.edu/wiki/index.php/Meep

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