## Complex Bloch-modes calculation of plasmonic crystal slabs by means of finite elements method |

Optics Express, Vol. 20, Issue 15, pp. 16690-16703 (2012)

http://dx.doi.org/10.1364/OE.20.016690

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### Abstract

We present a Finite Element Method (FEM) to calculate the complex valued **k**(ω) dispersion curves of a photonic crystal slab in presence of both dispersive and lossy materials. In particular the method can be exploited to study plasmonic crystal slabs. We adopt Perfectly Matched Layers (PMLs) in order to truncate the open boundaries of the model, including their related anisotropic permittivity and permeability tensors in the weak form of Helmholtz's eigenvalue equation. Results of the model are presented in the interesting case of a holey metal film enabling to study the observed extraordinary optical transmission properties in term of the plasmonic Bloch modes of the structure.

© 2012 OSA

## 1. Introduction

2. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. **7**(6), 442–453 (2008). [CrossRef] [PubMed]

3. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. **9**(3), 205–213 (2010). [CrossRef] [PubMed]

4. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. **408**(3-4), 131–314 (2005). [CrossRef]

4. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. **408**(3-4), 131–314 (2005). [CrossRef]

6. F. J. Garcia de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. **79**(4), 1267–1290 (2007). [CrossRef]

10. J. Bravo-Abad, L. Martín-Moreno, F. J. García-Vidal, E. Hendry, and J. Gómez Rivas, “Transmission of light through periodic arrays of square holes: From a metallic wire mesh to an array of tiny holes,” Phys. Rev. B **76**(24), 241102 (2007). [CrossRef]

4. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. **408**(3-4), 131–314 (2005). [CrossRef]

**408**(3-4), 131–314 (2005). [CrossRef]

**k**), is then obtained as a function of the crystal momentum

**k**. However, when material dispersion plays a crucial role, as in the case of plasmonic crystals, the resulting eigenvalue equation is nonlinear [13]. Besides time consuming methods involving non-linear iterative solvers [14

14. A. Ruhe, “Algorithms for the nonlinear eigenvalue problem,” SIAM J. Numer. Anal. **10**(4), 674–689 (1973). [CrossRef]

**k**(ω)) [15

15. S. Shi, C. Chen, and D. Prather, “Revised plane wave method for dispersive material and its application to band structure calculation of photonic crystal slabs,” Appl. Phys. Lett. **86**(4), 043104 (2005). [CrossRef]

19. G. Shvets and Y. A. Urzhumov, “Electric and magnetic properties of sub-wavelength plasmonic crystals,” J. Opt. A, Pure Appl. Opt. **7**(2), S23–S31 (2005). [CrossRef]

**k**, which can be solved by proper linearization procedures [17

17. M. Davanco, Y. Urzhumov, and G. Shvets, “The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction,” Opt. Express **15**(15), 9681–9691 (2007). [CrossRef] [PubMed]

**k**as well as the real one is naturally retrieved. In a recent work Fietz et al. [18

18. C. Fietz, Y. Urzhumov, and G. Shvets, “Complex k band diagrams of 3D metamaterial/photonic crystals,” Opt. Express **19**(20), 19027–19041 (2011). [CrossRef] [PubMed]

**k**. The weak formulation finds a natural solution in the frame of the Finite Elements Method, which inherently handles weak forms of partial differential equations. This modal method, originally developed by Hiett et al. [16

16. B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett, and K. S. Thomas, “Application of finite element methods to photonic crystal modelling,” IEE Proc. Sci. Meas. Technol. **149**(5), 293–296 (2002). [CrossRef]

*Slabs*(PCS), however, requires proper handling not only of truly bound modes but also of

*leaky*modes. This term addresses eigenmodes whose crystal momentum lies inside the light cone. In these cases the modes can couple to waves propagating in the semi-infinite half spaces surrounding the slab, resulting in radiative losses [12,13]. Simulation of open space boundaries is a tricky task both for methods based on discretization of real space (like Finite Elements and Finite Differences) as well as for methods which discretize the wave vectors space (like Plane Wave Expansion method). A common approach introduces a fictitious periodicity in the direction normal to the slab (Super-cell approach [13]). The period is chosen sufficiently large in order to decouple the bound modes of the slabs. However, the artificial periodicity produces many spurious modes and moreover it perturbs the physical leaky eigenmodes profile. Plane Wave Expansion Method has been extended by Shi et al. [15

15. S. Shi, C. Chen, and D. Prather, “Revised plane wave method for dispersive material and its application to band structure calculation of photonic crystal slabs,” Appl. Phys. Lett. **86**(4), 043104 (2005). [CrossRef]

15. S. Shi, C. Chen, and D. Prather, “Revised plane wave method for dispersive material and its application to band structure calculation of photonic crystal slabs,” Appl. Phys. Lett. **86**(4), 043104 (2005). [CrossRef]

18. C. Fietz, Y. Urzhumov, and G. Shvets, “Complex k band diagrams of 3D metamaterial/photonic crystals,” Opt. Express **19**(20), 19027–19041 (2011). [CrossRef] [PubMed]

7. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. **82**(1), 729–787 (2010). [CrossRef]

9. P. Lalanne, J. C. Rodier, and J. P. Hugonin, “Surface plasmons of metallic surfaces perforated by nano-hole arrays,” J. Opt. A, Pure Appl. Opt. **7**(8), 422–426 (2005). [CrossRef]

## 2. The FEM simulation

17. M. Davanco, Y. Urzhumov, and G. Shvets, “The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction,” Opt. Express **15**(15), 9681–9691 (2007). [CrossRef] [PubMed]

18. C. Fietz, Y. Urzhumov, and G. Shvets, “Complex k band diagrams of 3D metamaterial/photonic crystals,” Opt. Express **19**(20), 19027–19041 (2011). [CrossRef] [PubMed]

**k**is the Bloch-vector. The Finite Elements method relies on the so-called weak formulation of Eq. (1), which consists in annulling the following residueswhere

**v**is a weight function and the integration is over the unitary cell volume [20]. After inserting expression (2) in Eq. (3) and some straightforward algebra, Eq. (3) yields to

**k**must be reduced to one; i.e by fixing a particular

**k**-vector direction,

**k**-vector projection along the chosen direction, i.e.

17. M. Davanco, Y. Urzhumov, and G. Shvets, “The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction,” Opt. Express **15**(15), 9681–9691 (2007). [CrossRef] [PubMed]

**19**(20), 19027–19041 (2011). [CrossRef] [PubMed]

21. F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Rev. **43**(2), 235–286 (2001). [CrossRef]

## 3. The perfect matched layer (PML) implementation

22. Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antenn. Propag. **43**(12), 1460–1463 (1995). [CrossRef]

23. S. D. Gedney, “An Anisotropic Perfectly Matched Layer-Absorbing Medium for the truncation of FDTD Lattices,” IEEE Trans. Antenn. Propag. **44**(12), 1630–1639 (1996). [CrossRef]

*c = α - iβ*, with

*α*equal to the real part of the adjacent medium relative permittivity and

*β*> 0. This condition, without any further restriction on

*β*, can assure perfect absorption of any plane wave incident upon the boundary of an infinitely thick PML, independently of frequency or incidence angle [22

22. Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antenn. Propag. **43**(12), 1460–1463 (1995). [CrossRef]

23. S. D. Gedney, “An Anisotropic Perfectly Matched Layer-Absorbing Medium for the truncation of FDTD Lattices,” IEEE Trans. Antenn. Propag. **44**(12), 1630–1639 (1996). [CrossRef]

*β*. In other words, after discretization, PMLs are still absorbing materials and waves that propagate across them are still attenuated, but the boundary between PML and regular medium is no longer reflectionless. This mismatch problem in the discretized space can be tempered by using spatially varying material parameters [23

23. S. D. Gedney, “An Anisotropic Perfectly Matched Layer-Absorbing Medium for the truncation of FDTD Lattices,” IEEE Trans. Antenn. Propag. **44**(12), 1630–1639 (1996). [CrossRef]

*L*and

*n*are constant parameters.

*L*is the PML thickness while

*L*and

*n*. The problem will be addressed in next section. We notice that, in general, the

*β*parameter depends on ω and therefore the PML is a dispersive material. This does not represent a problem since, as mentioned before, in the present modal method frequency is fixed.

## 4. 2D sinusoidal grating and PML test

*d*) in the x-direction is set to 600nm while the peak-valley amplitude of the sinusoid (

*a*) is increased from 0 to 150nm. We adopt silver as metal, its permittivity being taken from Palik [24]. Periodic boundary conditions are set in both x and z directions. In z direction this condition is not mandatory because of the presence of the PML. This choice is made in order to reduce to zero the last term in Eq. (4). Since the PMLs are expected to absorb all radiation impinging on them, the particular boundary condition set at the end of the PML layer actually does not influence the resulting field distribution.

**k**-vector is parallel to the grating vector,

*λ = k*,

_{x}*ε*and

_{ii}*μ*are the diagonal components of the relative permittivity and permeability tensors of the medium involved (both physical domains and PMLs).

_{ii}*n*parameter, it was shown for example in [23

**44**(12), 1630–1639 (1996). [CrossRef]

*ε*are respectively the metal and dielectric relative permittivities, while

_{m,d}*k*is the vacuum wave vector. When a shallow periodic perturbation is introduced the mode dispersion remains close to the flat case. The only remarkable difference is observed at the crossing point of the flat SPP dispersions at the edge of first Brillouin zone (

_{0}*k*), where an energy gap appears with increasing grating amplitude [28

_{x}= π/d28. W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter **54**(9), 6227–6244 (1996). [CrossRef] [PubMed]

30. P. Lalanne, J. P. Hugonin, and P. Chavel, “Optical properties of deep lamellar grating: a coupled Bloch-mode insight,” J. Lightwave Technol. **24**(6), 2442–2449 (2006). [CrossRef]

*gaplike*region [31

31. K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, “Nature of lossy Bloch states in polaritonic photonic crystals,” Phys. Rev. B **69**(19), 195111 (2004). [CrossRef]

32. A. Kolomenskii, S. Peng, J. Hembd, A. Kolomenski, J. Noel, J. Strohaber, W. Teizer, and H. Schuessler, “Interaction and spectral gaps of surface plasmon modes in gold nano-structures,” Opt. Express **19**(7), 6587–6598 (2011). [CrossRef] [PubMed]

33. Y. Ding and R. Magnusson, “Band gaps and leaky-wave effects in resonant photonic-crystal waveguides,” Opt. Express **15**(2), 680–694 (2007). [CrossRef] [PubMed]

^{15}Hz) the imaginary part of the Bloch-wave-vector has values much greater than its corresponding values at frequencies below the gap. This is because the folded plasmonic band enters the light cone and the radiative coupling to propagating waves in the upper air half space becomes possible.

*d*= 600nm and amplitude

*a*= 50nm. Results are reported in Fig. 5(b). We see that there is a good agreement between the direct modal calculation and the indirect method based on SPP excitation for a wide spectral range of our interest (up to ω ≈4∙10

^{15}Hz). For frequencies above ω ≈4∙10

^{15}Hz a growing mismatch is observed since the considered PML setup is not optimal any more for too short wavelengths. The deviation may be reduced by tuning again the PML parameters.

## 5. 2-D periodic array

*a*milled in a thin silver slab. Period and silver thickness are fixed to

_{x}= a_{y}= a*d*= 940nm and

*h*= 200nm respectively. Two PMLs are introduced in the unit cell in order to absorb the leakage radiation propagating toward open space. They are set at distance

*z*= ± 1470nm from the slab and their parameters are set according to previous section.

_{0}6. F. J. Garcia de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. **79**(4), 1267–1290 (2007). [CrossRef]

*a*= 250nm, the second for a wider hole,

*a*= 500nm. We focus on modes along x-direction,

6. F. J. Garcia de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. **79**(4), 1267–1290 (2007). [CrossRef]

9. P. Lalanne, J. C. Rodier, and J. P. Hugonin, “Surface plasmons of metallic surfaces perforated by nano-hole arrays,” J. Opt. A, Pure Appl. Opt. **7**(8), 422–426 (2005). [CrossRef]

*n*is the real part of effective index of the SPP mode propagating on a flat silver-air interface [1] and

_{eff}*m, n*are integers. We consider dispersions along the x-direction by setting Re(

*k*) = 0. The white and red lines in Fig. 6(a,c,e) denote the (

_{y}*m,n*) = (

*m|*,|

*n|*)

^{-}) and symmetric modes (higher frequency, labeled as (|

*m|*,|

*n|*)

^{+}), depending on the distribution of the dominant magnetic field component with respect to the z = 0. This is a consequence of the mirror symmetry of the system with respect to the z = 0 plane [12]. The comparison between modal analysis and direct illumination gives important information about the modes which can be excited with the two types of illumination. We see that the (1,0)

^{±}modes can be excited only in presence of TM illumination and both correspond to transmittance peaks (see Fig. 6(e)). On the other hand, different (0,1)

^{±}modes can be excited both in TM and TE illumination. In particular, in Fig. 6(f), only one transmittance peak is clearly visible corresponding to the antisymmetric TM (0,1)

^{-}mode (see the comparison between

*H*in the y-z plane obtained with modal analysis and illumination at frequency of 2.022∙10

_{x}^{15}Hz).

*H*and |

_{y}*E*| in the x-z plane, for the two TM modes observed at frequency ω = 1.6∙10

_{x}^{15}Hz with

*a*= 250nm (I-II) and a = 500nm (III-IV). As can be expected, for small hole sizes, the modes are well confined close to the metal slab. The mode confinement decreases in case of

*a*= 500nm and a stronger leakage radiation is observed in the cladding regions. The presence of the PMLs, however, makes it possible to properly absorb this radiation and allows reconstructing the correct mode field profiles. This is clearly seen in Fig. 7(b). In fact, the almost uniform |

*E*| in the region far from the slab indicates that no waves reflected from the PMLs are present. Comparing Fig. 7(b) I and III with 7(b) II and IV, we observe that the electric field intensity is mainly concentrated within the hole in the anti-symmetric modes, while the field has a node at the plane z = 0 in the symmetric case.

_{x}7. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. **82**(1), 729–787 (2010). [CrossRef]

10. J. Bravo-Abad, L. Martín-Moreno, F. J. García-Vidal, E. Hendry, and J. Gómez Rivas, “Transmission of light through periodic arrays of square holes: From a metallic wire mesh to an array of tiny holes,” Phys. Rev. B **76**(24), 241102 (2007). [CrossRef]

7. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. **82**(1), 729–787 (2010). [CrossRef]

^{±}modes. Their imaginary parts present strong increases in correspondence of the edge of the first Brillouin zone around the frequency of 2.2∙10

^{15}Hz (Fig. 8(b) inset). This behavior indicates the presence of gaps similar to the ones observed for the TM modes. In the insets of Fig. 8(a) we report the dominant magnetic field component profiles of the (|

*m|*,|

*n|*)

^{-}modes for

*a*= 250 nm in a x-y plane laying 10 nm above the slab at frequency of 2.05∙10

^{15}Hz. We see that the TM (1,0)

^{-}mode is a transversal mode with the magnetic field polarized in the y-direction, while both the TE and TM (0,1)

^{-}modes are longitudinal modes with the magnetic field mostly polarized in the x-direction of propagation.

^{15}Hz near

*k*= 0. Correspondingly, a divergence in the imaginary parts is observed. Similar deviations are typical of real-frequency (and complex propagation constant) eigenvalue methods. They are not physical and were already reported elsewhere [18

_{x}**19**(20), 19027–19041 (2011). [CrossRef] [PubMed]

31. K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, “Nature of lossy Bloch states in polaritonic photonic crystals,” Phys. Rev. B **69**(19), 195111 (2004). [CrossRef]

33. Y. Ding and R. Magnusson, “Band gaps and leaky-wave effects in resonant photonic-crystal waveguides,” Opt. Express **15**(2), 680–694 (2007). [CrossRef] [PubMed]

## 6. Conclusions

## Acknowledgments

## References and links

1. | H. Rather, |

2. | J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. |

3. | H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. |

4. | A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. |

5. | M. L. Brongersma and P. G. Kik, |

6. | F. J. Garcia de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. |

7. | F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. |

8. | H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature |

9. | P. Lalanne, J. C. Rodier, and J. P. Hugonin, “Surface plasmons of metallic surfaces perforated by nano-hole arrays,” J. Opt. A, Pure Appl. Opt. |

10. | J. Bravo-Abad, L. Martín-Moreno, F. J. García-Vidal, E. Hendry, and J. Gómez Rivas, “Transmission of light through periodic arrays of square holes: From a metallic wire mesh to an array of tiny holes,” Phys. Rev. B |

11. | A. Mary, S. G. Rodrigo, L. Martin-Moreno, and F. J. Garcia-Vidal, “Plasmonic metamaterials based on holey metallic films,” J. Phys. Condens. Matter |

12. | J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and D. Robert, Meade, |

13. | K. Sakoda, |

14. | A. Ruhe, “Algorithms for the nonlinear eigenvalue problem,” SIAM J. Numer. Anal. |

15. | S. Shi, C. Chen, and D. Prather, “Revised plane wave method for dispersive material and its application to band structure calculation of photonic crystal slabs,” Appl. Phys. Lett. |

16. | B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett, and K. S. Thomas, “Application of finite element methods to photonic crystal modelling,” IEE Proc. Sci. Meas. Technol. |

17. | M. Davanco, Y. Urzhumov, and G. Shvets, “The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction,” Opt. Express |

18. | C. Fietz, Y. Urzhumov, and G. Shvets, “Complex k band diagrams of 3D metamaterial/photonic crystals,” Opt. Express |

19. | G. Shvets and Y. A. Urzhumov, “Electric and magnetic properties of sub-wavelength plasmonic crystals,” J. Opt. A, Pure Appl. Opt. |

20. | J. Jin, |

21. | F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Rev. |

22. | Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antenn. Propag. |

23. | S. D. Gedney, “An Anisotropic Perfectly Matched Layer-Absorbing Medium for the truncation of FDTD Lattices,” IEEE Trans. Antenn. Propag. |

24. | D. Edward, Palik, |

25. | G. Steven, Johnson, “Notes on Perfectly Matched Layers (PMLs),” Lecture notes Massachusetts Institute of Technology Massachusetts (2008), 1–16 (2007). |

26. | S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

27. | M. Imran and G. Andrew, Kirk, “Accurate determination of quality factor of axisymmetric resonators by use of perfectly mathed layer,” arXiv:1101.1984v2 [physics.optics], 1–7(2011). |

28. | W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter |

29. | P. Vincent and M. Neviere, “Corrugated dielectric waveguides: A numerical Study of the Second-Order Stop bands,” Appl. Phys. (Berl.) |

30. | P. Lalanne, J. P. Hugonin, and P. Chavel, “Optical properties of deep lamellar grating: a coupled Bloch-mode insight,” J. Lightwave Technol. |

31. | K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, “Nature of lossy Bloch states in polaritonic photonic crystals,” Phys. Rev. B |

32. | A. Kolomenskii, S. Peng, J. Hembd, A. Kolomenski, J. Noel, J. Strohaber, W. Teizer, and H. Schuessler, “Interaction and spectral gaps of surface plasmon modes in gold nano-structures,” Opt. Express |

33. | Y. Ding and R. Magnusson, “Band gaps and leaky-wave effects in resonant photonic-crystal waveguides,” Opt. Express |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(240.6680) Optics at surfaces : Surface plasmons

(260.2030) Physical optics : Dispersion

(050.5298) Diffraction and gratings : Photonic crystals

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: May 9, 2012

Revised Manuscript: May 29, 2012

Manuscript Accepted: June 13, 2012

Published: July 9, 2012

**Citation**

Giuseppe Parisi, Pierfrancesco Zilio, and Filippo Romanato, "Complex Bloch-modes calculation of plasmonic crystal slabs by means of finite elements method," Opt. Express **20**, 16690-16703 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-16690

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### References

- H. Rather, Surface Plasmons (Springer-Verlag, 1988).
- J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008). [CrossRef] [PubMed]
- H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010). [CrossRef] [PubMed]
- A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005). [CrossRef]
- M. L. Brongersma and P. G. Kik, Surface Plasmon Nanophotonics, (Springer, 2007).
- F. J. Garcia de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267–1290 (2007). [CrossRef]
- F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010). [CrossRef]
- H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452(7188), 728–731 (2008). [CrossRef] [PubMed]
- P. Lalanne, J. C. Rodier, and J. P. Hugonin, “Surface plasmons of metallic surfaces perforated by nano-hole arrays,” J. Opt. A, Pure Appl. Opt. 7(8), 422–426 (2005). [CrossRef]
- J. Bravo-Abad, L. Martín-Moreno, F. J. García-Vidal, E. Hendry, and J. Gómez Rivas, “Transmission of light through periodic arrays of square holes: From a metallic wire mesh to an array of tiny holes,” Phys. Rev. B 76(24), 241102 (2007). [CrossRef]
- A. Mary, S. G. Rodrigo, L. Martin-Moreno, and F. J. Garcia-Vidal, “Plasmonic metamaterials based on holey metallic films,” J. Phys. Condens. Matter 20(30), 304215 (2008).
- J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and D. Robert, Meade, Photonic Crystals - Molding the Flow of Light, 2nd ed. (Princeton University Press, 2007).
- K. Sakoda, Optical Properties of Photonic Crystal, ser. Optical Sciences. (Springer, New York, 2001).
- A. Ruhe, “Algorithms for the nonlinear eigenvalue problem,” SIAM J. Numer. Anal. 10(4), 674–689 (1973). [CrossRef]
- S. Shi, C. Chen, and D. Prather, “Revised plane wave method for dispersive material and its application to band structure calculation of photonic crystal slabs,” Appl. Phys. Lett. 86(4), 043104 (2005). [CrossRef]
- B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett, and K. S. Thomas, “Application of finite element methods to photonic crystal modelling,” IEE Proc. Sci. Meas. Technol. 149(5), 293–296 (2002). [CrossRef]
- M. Davanco, Y. Urzhumov, and G. Shvets, “The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction,” Opt. Express 15(15), 9681–9691 (2007). [CrossRef] [PubMed]
- C. Fietz, Y. Urzhumov, and G. Shvets, “Complex k band diagrams of 3D metamaterial/photonic crystals,” Opt. Express 19(20), 19027–19041 (2011). [CrossRef] [PubMed]
- G. Shvets and Y. A. Urzhumov, “Electric and magnetic properties of sub-wavelength plasmonic crystals,” J. Opt. A, Pure Appl. Opt. 7(2), S23–S31 (2005). [CrossRef]
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