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

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2011

2010

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

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]

2008

H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452(7188), 728–731 (2008).
[CrossRef] [PubMed]

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. 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]

2007

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

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]

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]

2006

2005

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]

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]

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]

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]

2004

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]

2002

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]

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. 66(6), 066608 (2002).
[CrossRef] [PubMed]

2001

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

1996

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]

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]

1995

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]

1979

P. Vincent and M. Neviere, “Corrugated dielectric waveguides: A numerical Study of the Second-Order Stop bands,” Appl. Phys. (Berl.) 20(4), 345–351 (1979).
[CrossRef]

1973

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

Anker, J. N.

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]

Atwater, H. A.

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

Barnes, W. L.

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]

Beckett, D. H.

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]

Bienstman, P.

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]

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. 66(6), 066608 (2002).
[CrossRef] [PubMed]

Bravo-Abad, J.

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]

Chavel, P.

Chen, C.

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]

Cox, S. J.

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]

Davanco, M.

Ding, Y.

Ebbesen, T. W.

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]

Fan, S.

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]

Fietz, C.

Garcia de Abajo, F. J.

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

Garcia-Vidal, F. J.

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]

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).

García-Vidal, F. J.

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]

Gedney, S. D.

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]

Generowicz, J. M.

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]

Gómez Rivas, J.

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]

Hall, W. P.

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]

Hembd, J.

Hendry, E.

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]

Hiett, B. P.

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]

Huang, K. C.

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]

Hugonin, J. P.

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]

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]

Ibanescu, M.

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. 66(6), 066608 (2002).
[CrossRef] [PubMed]

Jiang, X.

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]

Joannopoulos, J. D.

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]

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. 66(6), 066608 (2002).
[CrossRef] [PubMed]

Johnson, S. G.

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. 66(6), 066608 (2002).
[CrossRef] [PubMed]

Kingsland, D. M.

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]

Kitson, S. C.

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]

Kolomenski, A.

Kolomenskii, A.

Kuipers, L.

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]

Lalanne, P.

H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452(7188), 728–731 (2008).
[CrossRef] [PubMed]

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]

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]

Lee, J. F.

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]

Lee, R.

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]

Lidorikis, E.

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]

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. 66(6), 066608 (2002).
[CrossRef] [PubMed]

Liu, H.

H. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452(7188), 728–731 (2008).
[CrossRef] [PubMed]

Lyandres, O.

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]

Magnusson, R.

Maradudin, A. A.

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]

Martin-Moreno, L.

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]

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).

Martín-Moreno, L.

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]

Mary, A.

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).

Meerbergen, K.

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

Molinari, M.

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]

Nelson, K. A.

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]

Neviere, M.

P. Vincent and M. Neviere, “Corrugated dielectric waveguides: A numerical Study of the Second-Order Stop bands,” Appl. Phys. (Berl.) 20(4), 345–351 (1979).
[CrossRef]

Noel, J.

Peng, S.

Polman, A.

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

Prather, D.

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]

Preist, T. W.

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]

Rodier, J. C.

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]

Rodrigo, S. G.

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[CrossRef]

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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).
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[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

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Figures (8)

Fig. 1
Fig. 1

Scheme of photonic crystal slab (green) with PMLs (violet) truncating the cladding domains.

Fig. 2
Fig. 2

(a) Scheme of unit cell for the sinusoidal plasmonic grating; (b) Variation of a sample eigenvalue (λL = d = 0.0905356 at ω = 1.7∙1015Hz) as a function of PML thickness L and σ, ∆λ is defined as ∆λm = λL = (m + 1)d - λL = md (m>0), scale is in decibels.

Fig. 3
Fig. 3

(a) Reflectance map compared with the calculated dispersion curves (black lines), green dashed line is the light line; (b) Comparison between bands obtained with (black) and without (red) PML domains (in both cases the tolerance parameter κ was set to 0.1, and periodicity in z direction is 5μm).

Fig. 4
Fig. 4

Real part (a) and imaginary part (b) of the modes varying the sinusoidal grating amplitude a: a = 0 (flat case, black), 30nm (cyan), 50nm (magenta), 75nm (green), 100nm (red), 150nm (blue). The dashed black curve in (a) is the light line.

Fig. 5
Fig. 5

(a) z-component of the magnetic field obtained by exciting the proper mode by mean of excitation boundary conditions. Frequency is ω = 3.5∙1015Hz, d = 600nm and a = 50nm. (b) Comparison between the imaginary part of the modes calculated with illumination (blue curve) and by modal analysis (magenta curve) respectively at fixed amplitude a = 50nm.

Fig. 6
Fig. 6

Maps of TM and TE transmittance through arrays of square nano-holes (period d = 940nm) in a 200nm thick Silver film in air compared with the calculated Bloch-modes dispersion curves (black lines, labeled as (|m|,|n|) ± ); (a) and (b) refer to holes with size a = 250 nm, (c) and (d) to a = 500nm; (e) and (f) are zooms of the regions marked with 1 and 2 respectively in (a). Black dashed line marks the light line, white and red solid lines mark the flat SPP dispersions, ( ± 1,0) and (0, ± 1) respectively. Figure 6(f) contains also a comparison between the x-component of the magnetic field profile in the z-y plane obtained with modal analysis and illumination respectively.

Fig. 7
Fig. 7

(a) Hy and (b) |Ex| fields of the TM modes found at frequency of 1.6∙1015Hz for a = 250nm (I,II) and a = 500nm (III,IV). The calculated eigenvalues in the four cases are respectively kx/(π/d) = 0.3828 + 0.0019i, 0.3878 + 0.0011i, 0.2792 + 0.1251i, 0.3829 + 0.0025i. Modes are classified as antisymmetric (I,III) and symmetric (II,IV) according to the symmetry of Hy field with respect to the z = 0 plane. PML size is fixed at 2d for convenience.

Fig. 8
Fig. 8

Real and imaginary parts of the modes for a = 250nm (a,b) and a = 500nm (c,d). Blue and red dots represent the TM (1,0)- and TM (1,0)+ modes respectively, black dots represent the TE and TM (0,1) ± modes. Insets in (a) report the dominant magnetic field component profile (colorscale) and the (H) field (arrows) respectively in a x-y plane laying 10 nm above the slab at frequency of 2.05x1015Hz for the (|m|,|n|)- modes.

Equations (9)

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×( p ^ ×V ) ω 2 q ^ V=0,
V(r;k)= e ikr u(r)= e i( k x x+ k y y+ k z z ) u(r),
R k ( v,u )= Ω v[ ×( p ^ ×V( r;k ) ) ω 2 q ^ V( r;k ) ] d 3 r,
Ω d 3 r( v[ k× p ^ ( k×u ) ]iv[ k× p ^ ( ×u ) ]i( ×v ) p ^ ( k×u ) ) + + Ω d 3 r( ( ×v ) p ^ ( ×u ) ω 2 c 2 v q ^ u ) + Ω dAv[ n ^ × p ^ ( ik×u+×u ) ]=0 ,
ε ^ =ε Λ ^ , μ ^ =μ Λ ^ ,
Λ ^ =( c 0 0 0 c 0 0 0 c 1 ),
Ω d 2 r[ λ 2 uv ε 33 +λ i ε 33 ( u x vu v x )+ u z v z 1 ε 11 + u x v x 1 ε 33 ( ω c ) 2 uv μ 22 ] =0.
κ= | H | 2 (1) | H | 2 (2)
ω ( k x , k y ) m,n = 1 n eff ( Re( k x )+ 2πm d ) 2 + ( Re( k y )+ 2πn d ) 2 ,

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