Abstract

A three dimensional analysis of a special class of anisotropic materials is presented. We introduce an extension of the Scattering Matrix Method (SMM) to investigate the behavior of anisotropic Photonic Crystal Slabs (PhCS) subject to external radiation. We show how the Fano effect can play a fundamental role in the realization of tunable optical devices. Moreover, we show how to utilize electron injection, electric field and temperature as parameters to control the Fano resonance shift in both isotropic and anisotropic materials as Si and Potassium Titanium Oxide Phosphate (KTP). We will see that because Fano modes are sensitive and controllable, a broad range of applications can be considered.

© 2006 Optical Society of America

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    [CrossRef]

2006

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, "Air-bridged photonic crystal slabs at visible and near-infrared wavelength," Phys. Rev. B 73, 115126 (2006).
[CrossRef]

E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, "Extraordinary optical transmission without plasmons: the s-polarization case," J. Opt. A: Pure Appl. Opt. 8, S94-S97 (2006).
[CrossRef]

R. Gomez-Medina, M. Laroche, and J. J. Saenz, "Extraordinary optical reflection from sub-wavelength cylinder arrays," Opt. Express 14, 3730-3737 (2006).
[CrossRef] [PubMed]

2005

2004

J. S. Jensen and O. Sigmund, "Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends," Appl. Phys. Lett. 84, 2021-2023 (2004).
[CrossRef]

L. H. Frandsen, P. I. Borel, Y. X. Zhuang, A. Harpøth, M. Thorhauge, M. Kristensen, W. Bogaerts, P. Dumon, R. Baets, V. Wiaux, J. Wouters, and S. Beckx, "Ultralow-loss 3-dB photonic crystal waveguide splitter," Opt. Lett. 29, 1623-1625 (2004).
[CrossRef] [PubMed]

X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, A. Talneau, "Distributed feedback regime of photonic crystal waveguide lasers at 1.5 mm," Appl. Phys. Lett. 85, 5502-5504 (2004).
[CrossRef]

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, "Surface Plasmon Polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film," Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

W. V. Lousse and J. P. Vigneron, "Use of Fano resonances for bistable optical transfer through photonic crystal film," Phys. Rev. B 69, 155106 (2004).
[CrossRef]

H. J. Lezec and T. Thio, "Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays," Opt. Express 12, 3629-3651 (2004).
[CrossRef] [PubMed]

2003

M. Sarrazin, J. P. Vigneron, and J. M. Vigoureux, "Role of Wood anomalies in optical properties of thin metallic films with bidimensional array of subwavelength holes," Phys. Rev. B 67,085415 (2003).
[CrossRef]

J. M. Steele, C. E. Moran, A. Lee, C. M. Aguirre, and N. J. Halas, "Metallodielectric gratings with subwavelength slots: Optical properties," Phys. Rev. B 68, 205103 (2003).
[CrossRef]

A. Irace, G. Breglio, and A. Cutolo, "All-silicon optoelectronic modulator with 1 GHz switching capability," Electron. Lett. 39, 232-233 (2003).
[CrossRef]

2002

S. Fan and J. D. Joannopoulos, "Analysis of guided resonances in photonic crystal slabs," Phys. Rev. B 65, 235112 (2002).
[CrossRef]

S. G. Tikhodeew, A. L. Yablonskii, E. A. Muljarow, N. A. Gippius, and T. Ishihara, "Quasiguide modes and optical properties of photonic crystal slabs," Phys. Rev. B 66, 045102 (2002).
[CrossRef]

M. G. Banaee, A. R. Cowan, and J. F. Young, "Third-order nonlinear influence on specular reflectivity of two-dimensional waveguide-based photonic crystals," J. Opt. Soc. Am. B 19, 2224 (2002).
[CrossRef]

2000

B. Boulanger, J. P. Feve, and Y. Guillien, "Thermo-optical effect and saturation of nonlinear absorption induced by gray tracking in a 532-nm pumped KTP optical parametric oscillator," Opt. Lett. 25, 484-486 (2000).
[CrossRef]

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

V. N. Astratov, R. M. Stevenson, I. S. Culshaw, D. M. Whittaker, M. S. Skolnick, T. F. Krauss and R. M. De La Rue, "Heavy photon dispersions in photonic crystal waveguides," Appl. Phys. Lett. 77, 178-180 (2000).
[CrossRef]

1999

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

V. N. Astratov, I. S. Culshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, "Resonant coupling of near-infrared radiation to photonic band structure waveguide," J. Lightwave Technol. 17, 2050-2057 (1999).
[CrossRef]

D. M. Whittaker and I. S. Culshaw, "Scattering-matrix treatment of patterned multilayer photonic structures," Phys. Rev. B 60, 2610-2618 (1999).
[CrossRef]

1998

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

1988

D. Y. K. Ko and J. C. Inkson, "Matrix method for tunneling in heterostructures: Resonant tunneling in multilayer systems," Phys. Rev. B 38, 9945-9951 (1988).
[CrossRef]

1987

R. A. Soref, and B. R. Bennett "Electro-optical Effects in Silicon," IEEE J. Quantum Electron. 23, 123-129 (1987).
[CrossRef]

1965

1961

U. Fano, "Effect of configuration interaction on intensity and phase shifts," Phys. Rev. 124, 1866-1878 (1961).
[CrossRef]

1941

1935

R. W. Wood, "Anomalous diffraction gratings," Phys. Rev. 48, 928-936 (1935).
[CrossRef]

Aguirre, C. M.

J. M. Steele, C. E. Moran, A. Lee, C. M. Aguirre, and N. J. Halas, "Metallodielectric gratings with subwavelength slots: Optical properties," Phys. Rev. B 68, 205103 (2003).
[CrossRef]

Astratov, V. N.

V. N. Astratov, R. M. Stevenson, I. S. Culshaw, D. M. Whittaker, M. S. Skolnick, T. F. Krauss and R. M. De La Rue, "Heavy photon dispersions in photonic crystal waveguides," Appl. Phys. Lett. 77, 178-180 (2000).
[CrossRef]

V. N. Astratov, I. S. Culshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, "Resonant coupling of near-infrared radiation to photonic band structure waveguide," J. Lightwave Technol. 17, 2050-2057 (1999).
[CrossRef]

Baets, R.

Banaee, M. G.

Barnes, W. L.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, "Surface Plasmon Polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film," Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

Beckx, S.

Bennett, B. R.

R. A. Soref, and B. R. Bennett "Electro-optical Effects in Silicon," IEEE J. Quantum Electron. 23, 123-129 (1987).
[CrossRef]

Bogaerts, W.

Bonnefont, S.

X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, A. Talneau, "Distributed feedback regime of photonic crystal waveguide lasers at 1.5 mm," Appl. Phys. Lett. 85, 5502-5504 (2004).
[CrossRef]

Borel, P. I.

Boucaud, P.

X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, A. Talneau, "Distributed feedback regime of photonic crystal waveguide lasers at 1.5 mm," Appl. Phys. Lett. 85, 5502-5504 (2004).
[CrossRef]

Boulanger, B.

Breglio, G.

A. Irace, G. Breglio, and A. Cutolo, "All-silicon optoelectronic modulator with 1 GHz switching capability," Electron. Lett. 39, 232-233 (2003).
[CrossRef]

Carter, M. W.

Casey, J. A.

Checoury, X.

X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, A. Talneau, "Distributed feedback regime of photonic crystal waveguide lasers at 1.5 mm," Appl. Phys. Lett. 85, 5502-5504 (2004).
[CrossRef]

Chen, R. T.

Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chen, "80-micron interaction length silicon photonic crystal waveguide modulator," Appl. Phys. Lett. 87, 221105-221107 (2005).
[CrossRef]

Chen, X.

Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chen, "80-micron interaction length silicon photonic crystal waveguide modulator," Appl. Phys. Lett. 87, 221105-221107 (2005).
[CrossRef]

Cowan, A. R.

Crozier, K. B.

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, "Air-bridged photonic crystal slabs at visible and near-infrared wavelength," Phys. Rev. B 73, 115126 (2006).
[CrossRef]

Cuisin, C.

X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, A. Talneau, "Distributed feedback regime of photonic crystal waveguide lasers at 1.5 mm," Appl. Phys. Lett. 85, 5502-5504 (2004).
[CrossRef]

Culshaw, I. S.

V. N. Astratov, R. M. Stevenson, I. S. Culshaw, D. M. Whittaker, M. S. Skolnick, T. F. Krauss and R. M. De La Rue, "Heavy photon dispersions in photonic crystal waveguides," Appl. Phys. Lett. 77, 178-180 (2000).
[CrossRef]

V. N. Astratov, I. S. Culshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, "Resonant coupling of near-infrared radiation to photonic band structure waveguide," J. Lightwave Technol. 17, 2050-2057 (1999).
[CrossRef]

D. M. Whittaker and I. S. Culshaw, "Scattering-matrix treatment of patterned multilayer photonic structures," Phys. Rev. B 60, 2610-2618 (1999).
[CrossRef]

Cutolo, A.

A. Irace, G. Breglio, and A. Cutolo, "All-silicon optoelectronic modulator with 1 GHz switching capability," Electron. Lett. 39, 232-233 (2003).
[CrossRef]

De La Rue, R. M.

V. N. Astratov, R. M. Stevenson, I. S. Culshaw, D. M. Whittaker, M. S. Skolnick, T. F. Krauss and R. M. De La Rue, "Heavy photon dispersions in photonic crystal waveguides," Appl. Phys. Lett. 77, 178-180 (2000).
[CrossRef]

V. N. Astratov, I. S. Culshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, "Resonant coupling of near-infrared radiation to photonic band structure waveguide," J. Lightwave Technol. 17, 2050-2057 (1999).
[CrossRef]

Dems, M.

Derouin, E.

X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, A. Talneau, "Distributed feedback regime of photonic crystal waveguide lasers at 1.5 mm," Appl. Phys. Lett. 85, 5502-5504 (2004).
[CrossRef]

Devaux, E.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, "Surface Plasmon Polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film," Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

Dintinger, J.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, "Surface Plasmon Polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film," Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

Drisse, O.

X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, A. Talneau, "Distributed feedback regime of photonic crystal waveguide lasers at 1.5 mm," Appl. Phys. Lett. 85, 5502-5504 (2004).
[CrossRef]

Duan, G. H.

X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, A. Talneau, "Distributed feedback regime of photonic crystal waveguide lasers at 1.5 mm," Appl. Phys. Lett. 85, 5502-5504 (2004).
[CrossRef]

Dumon, P.

Ebbesen, T. W.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, "Surface Plasmon Polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film," Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Eddy, C. R.

Enoch, S.

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

Fan, S.

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, "Air-bridged photonic crystal slabs at visible and near-infrared wavelength," Phys. Rev. B 73, 115126 (2006).
[CrossRef]

S. Fan and J. D. Joannopoulos, "Analysis of guided resonances in photonic crystal slabs," Phys. Rev. B 65, 235112 (2002).
[CrossRef]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

Fano, U.

Feve, J. P.

Frandsen, L. H.

Garcia-Vidal, F. J.

E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, "Extraordinary optical transmission without plasmons: the s-polarization case," J. Opt. A: Pure Appl. Opt. 8, S94-S97 (2006).
[CrossRef]

Gauthier-Lafaye, O.

X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, A. Talneau, "Distributed feedback regime of photonic crystal waveguide lasers at 1.5 mm," Appl. Phys. Lett. 85, 5502-5504 (2004).
[CrossRef]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Gippius, N. A.

S. G. Tikhodeew, A. L. Yablonskii, E. A. Muljarow, N. A. Gippius, and T. Ishihara, "Quasiguide modes and optical properties of photonic crystal slabs," Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Gomez-Medina, R.

Gu, L.

Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chen, "80-micron interaction length silicon photonic crystal waveguide modulator," Appl. Phys. Lett. 87, 221105-221107 (2005).
[CrossRef]

Guillien, Y.

Halas, N. J.

J. M. Steele, C. E. Moran, A. Lee, C. M. Aguirre, and N. J. Halas, "Metallodielectric gratings with subwavelength slots: Optical properties," Phys. Rev. B 68, 205103 (2003).
[CrossRef]

Harpøth, A.

Henry, R. L.

Hessel, A.

Holm, R. T.

Inkson, J. C.

D. Y. K. Ko and J. C. Inkson, "Matrix method for tunneling in heterostructures: Resonant tunneling in multilayer systems," Phys. Rev. B 38, 9945-9951 (1988).
[CrossRef]

Irace, A.

A. Irace, G. Breglio, and A. Cutolo, "All-silicon optoelectronic modulator with 1 GHz switching capability," Electron. Lett. 39, 232-233 (2003).
[CrossRef]

Ishihara, T.

S. G. Tikhodeew, A. L. Yablonskii, E. A. Muljarow, N. A. Gippius, and T. Ishihara, "Quasiguide modes and optical properties of photonic crystal slabs," Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Jensen, J. S.

J. S. Jensen and O. Sigmund, "Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends," Appl. Phys. Lett. 84, 2021-2023 (2004).
[CrossRef]

Jiang, W.

Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chen, "80-micron interaction length silicon photonic crystal waveguide modulator," Appl. Phys. Lett. 87, 221105-221107 (2005).
[CrossRef]

Jiang, Y.

Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chen, "80-micron interaction length silicon photonic crystal waveguide modulator," Appl. Phys. Lett. 87, 221105-221107 (2005).
[CrossRef]

Joannopoulos, J. D.

S. Fan and J. D. Joannopoulos, "Analysis of guided resonances in photonic crystal slabs," Phys. Rev. B 65, 235112 (2002).
[CrossRef]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

Johnson, S. G.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

Kilic, O.

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, "Air-bridged photonic crystal slabs at visible and near-infrared wavelength," Phys. Rev. B 73, 115126 (2006).
[CrossRef]

Kim, M.

Kim, S.

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, "Air-bridged photonic crystal slabs at visible and near-infrared wavelength," Phys. Rev. B 73, 115126 (2006).
[CrossRef]

Kivshar, Y. S.

A. E. Miroshnichenko and Y. S. Kivshar, "Sharp bends in photonic crystal waveguides as nonlinear Fano resonators," Opt. Express 13, 3969-3976 (2005).
[CrossRef] [PubMed]

A. E. Miroshnichenko and Y. S. Kivshar, "Engineering Fano resonances in discrete arrays," Phys. Rev. E 72, 056611 (2005).
[CrossRef]

Ko, D. Y. K.

D. Y. K. Ko and J. C. Inkson, "Matrix method for tunneling in heterostructures: Resonant tunneling in multilayer systems," Phys. Rev. B 38, 9945-9951 (1988).
[CrossRef]

Kolodziejski, L. A.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

Kotynski, R.

Krauss, T. F.

V. N. Astratov, R. M. Stevenson, I. S. Culshaw, D. M. Whittaker, M. S. Skolnick, T. F. Krauss and R. M. De La Rue, "Heavy photon dispersions in photonic crystal waveguides," Appl. Phys. Lett. 77, 178-180 (2000).
[CrossRef]

V. N. Astratov, I. S. Culshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, "Resonant coupling of near-infrared radiation to photonic band structure waveguide," J. Lightwave Technol. 17, 2050-2057 (1999).
[CrossRef]

Kristensen, M.

Kuramochi, E.

Laroche, M.

Lee, A.

J. M. Steele, C. E. Moran, A. Lee, C. M. Aguirre, and N. J. Halas, "Metallodielectric gratings with subwavelength slots: Optical properties," Phys. Rev. B 68, 205103 (2003).
[CrossRef]

Legouezigou, L.

X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, A. Talneau, "Distributed feedback regime of photonic crystal waveguide lasers at 1.5 mm," Appl. Phys. Lett. 85, 5502-5504 (2004).
[CrossRef]

Lelarge, F.

X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, A. Talneau, "Distributed feedback regime of photonic crystal waveguide lasers at 1.5 mm," Appl. Phys. Lett. 85, 5502-5504 (2004).
[CrossRef]

Lezec, H. J.

H. J. Lezec and T. Thio, "Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays," Opt. Express 12, 3629-3651 (2004).
[CrossRef] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Lourtioz, J-M.

X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, A. Talneau, "Distributed feedback regime of photonic crystal waveguide lasers at 1.5 mm," Appl. Phys. Lett. 85, 5502-5504 (2004).
[CrossRef]

Lousse, V.

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, "Air-bridged photonic crystal slabs at visible and near-infrared wavelength," Phys. Rev. B 73, 115126 (2006).
[CrossRef]

Lousse, W. V.

W. V. Lousse and J. P. Vigneron, "Use of Fano resonances for bistable optical transfer through photonic crystal film," Phys. Rev. B 69, 155106 (2004).
[CrossRef]

Lozes, F.

X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, A. Talneau, "Distributed feedback regime of photonic crystal waveguide lasers at 1.5 mm," Appl. Phys. Lett. 85, 5502-5504 (2004).
[CrossRef]

Martin-Moreno, L.

E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, "Extraordinary optical transmission without plasmons: the s-polarization case," J. Opt. A: Pure Appl. Opt. 8, S94-S97 (2006).
[CrossRef]

Miroshnichenko, A. E.

A. E. Miroshnichenko and Y. S. Kivshar, "Engineering Fano resonances in discrete arrays," Phys. Rev. E 72, 056611 (2005).
[CrossRef]

A. E. Miroshnichenko and Y. S. Kivshar, "Sharp bends in photonic crystal waveguides as nonlinear Fano resonators," Opt. Express 13, 3969-3976 (2005).
[CrossRef] [PubMed]

Mitsugi, S.

Moran, C. E.

J. M. Steele, C. E. Moran, A. Lee, C. M. Aguirre, and N. J. Halas, "Metallodielectric gratings with subwavelength slots: Optical properties," Phys. Rev. B 68, 205103 (2003).
[CrossRef]

Moreno, E.

E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, "Extraordinary optical transmission without plasmons: the s-polarization case," J. Opt. A: Pure Appl. Opt. 8, S94-S97 (2006).
[CrossRef]

Mulin, D.

X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, A. Talneau, "Distributed feedback regime of photonic crystal waveguide lasers at 1.5 mm," Appl. Phys. Lett. 85, 5502-5504 (2004).
[CrossRef]

Muljarow, E. A.

S. G. Tikhodeew, A. L. Yablonskii, E. A. Muljarow, N. A. Gippius, and T. Ishihara, "Quasiguide modes and optical properties of photonic crystal slabs," Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Murray, W. A.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, "Surface Plasmon Polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film," Phys. Rev. Lett. 92, 107401 (2004).
[CrossRef] [PubMed]

Nevière, M.

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

Notomi, M.

Oliner, A. A.

Panajotov, K.

Poingt, F.

X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, A. Talneau, "Distributed feedback regime of photonic crystal waveguide lasers at 1.5 mm," Appl. Phys. Lett. 85, 5502-5504 (2004).
[CrossRef]

Pommereau, F.

X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, A. Talneau, "Distributed feedback regime of photonic crystal waveguide lasers at 1.5 mm," Appl. Phys. Lett. 85, 5502-5504 (2004).
[CrossRef]

Popov, E.

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

Reinisch, R.

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

Rosenberg, A.

Saenz, J. J.

Sarrazin, M.

M. Sarrazin, J. P. Vigneron, and J. M. Vigoureux, "Role of Wood anomalies in optical properties of thin metallic films with bidimensional array of subwavelength holes," Phys. Rev. B 67,085415 (2003).
[CrossRef]

Shamamian, V. A.

Shinya, A.

Sigmund, O.

J. S. Jensen and O. Sigmund, "Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends," Appl. Phys. Lett. 84, 2021-2023 (2004).
[CrossRef]

Skolnick, M. S.

V. N. Astratov, R. M. Stevenson, I. S. Culshaw, D. M. Whittaker, M. S. Skolnick, T. F. Krauss and R. M. De La Rue, "Heavy photon dispersions in photonic crystal waveguides," Appl. Phys. Lett. 77, 178-180 (2000).
[CrossRef]

V. N. Astratov, I. S. Culshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, "Resonant coupling of near-infrared radiation to photonic band structure waveguide," J. Lightwave Technol. 17, 2050-2057 (1999).
[CrossRef]

Solgaard, O.

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, "Air-bridged photonic crystal slabs at visible and near-infrared wavelength," Phys. Rev. B 73, 115126 (2006).
[CrossRef]

Soref, R. A.

R. A. Soref, and B. R. Bennett "Electro-optical Effects in Silicon," IEEE J. Quantum Electron. 23, 123-129 (1987).
[CrossRef]

Steele, J. M.

J. M. Steele, C. E. Moran, A. Lee, C. M. Aguirre, and N. J. Halas, "Metallodielectric gratings with subwavelength slots: Optical properties," Phys. Rev. B 68, 205103 (2003).
[CrossRef]

Stevenson, R. M.

V. N. Astratov, R. M. Stevenson, I. S. Culshaw, D. M. Whittaker, M. S. Skolnick, T. F. Krauss and R. M. De La Rue, "Heavy photon dispersions in photonic crystal waveguides," Appl. Phys. Lett. 77, 178-180 (2000).
[CrossRef]

V. N. Astratov, I. S. Culshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, "Resonant coupling of near-infrared radiation to photonic band structure waveguide," J. Lightwave Technol. 17, 2050-2057 (1999).
[CrossRef]

Talneau, A.

X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, A. Talneau, "Distributed feedback regime of photonic crystal waveguide lasers at 1.5 mm," Appl. Phys. Lett. 85, 5502-5504 (2004).
[CrossRef]

Thio, T.

H. J. Lezec and T. Thio, "Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays," Opt. Express 12, 3629-3651 (2004).
[CrossRef] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Thorhauge, M.

Tikhodeew, S. G.

S. G. Tikhodeew, A. L. Yablonskii, E. A. Muljarow, N. A. Gippius, and T. Ishihara, "Quasiguide modes and optical properties of photonic crystal slabs," Phys. Rev. B 66, 045102 (2002).
[CrossRef]

Valentin, J.

X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, A. Talneau, "Distributed feedback regime of photonic crystal waveguide lasers at 1.5 mm," Appl. Phys. Lett. 85, 5502-5504 (2004).
[CrossRef]

Vigneron, J. P.

W. V. Lousse and J. P. Vigneron, "Use of Fano resonances for bistable optical transfer through photonic crystal film," Phys. Rev. B 69, 155106 (2004).
[CrossRef]

M. Sarrazin, J. P. Vigneron, and J. M. Vigoureux, "Role of Wood anomalies in optical properties of thin metallic films with bidimensional array of subwavelength holes," Phys. Rev. B 67,085415 (2003).
[CrossRef]

Vigoureux, J. M.

M. Sarrazin, J. P. Vigneron, and J. M. Vigoureux, "Role of Wood anomalies in optical properties of thin metallic films with bidimensional array of subwavelength holes," Phys. Rev. B 67,085415 (2003).
[CrossRef]

Villeneuve, P. R.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

Whittaker, D. M.

V. N. Astratov, R. M. Stevenson, I. S. Culshaw, D. M. Whittaker, M. S. Skolnick, T. F. Krauss and R. M. De La Rue, "Heavy photon dispersions in photonic crystal waveguides," Appl. Phys. Lett. 77, 178-180 (2000).
[CrossRef]

V. N. Astratov, I. S. Culshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Krauss, R. M. De La Rue, "Resonant coupling of near-infrared radiation to photonic band structure waveguide," J. Lightwave Technol. 17, 2050-2057 (1999).
[CrossRef]

D. M. Whittaker and I. S. Culshaw, "Scattering-matrix treatment of patterned multilayer photonic structures," Phys. Rev. B 60, 2610-2618 (1999).
[CrossRef]

Wiaux, V.

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
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Wood, R. W.

R. W. Wood, "Anomalous diffraction gratings," Phys. Rev. 48, 928-936 (1935).
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Wouters, J.

Yablonskii, A. L.

S. G. Tikhodeew, A. L. Yablonskii, E. A. Muljarow, N. A. Gippius, and T. Ishihara, "Quasiguide modes and optical properties of photonic crystal slabs," Phys. Rev. B 66, 045102 (2002).
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Young, J. F.

Zhuang, Y. X.

Appl. Opt.

Appl. Phys. Lett.

X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, A. Talneau, "Distributed feedback regime of photonic crystal waveguide lasers at 1.5 mm," Appl. Phys. Lett. 85, 5502-5504 (2004).
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Y. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chen, "80-micron interaction length silicon photonic crystal waveguide modulator," Appl. Phys. Lett. 87, 221105-221107 (2005).
[CrossRef]

V. N. Astratov, R. M. Stevenson, I. S. Culshaw, D. M. Whittaker, M. S. Skolnick, T. F. Krauss and R. M. De La Rue, "Heavy photon dispersions in photonic crystal waveguides," Appl. Phys. Lett. 77, 178-180 (2000).
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J. S. Jensen and O. Sigmund, "Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends," Appl. Phys. Lett. 84, 2021-2023 (2004).
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A. Irace, G. Breglio, and A. Cutolo, "All-silicon optoelectronic modulator with 1 GHz switching capability," Electron. Lett. 39, 232-233 (2003).
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R. A. Soref, and B. R. Bennett "Electro-optical Effects in Silicon," IEEE J. Quantum Electron. 23, 123-129 (1987).
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J. Opt. A: Pure Appl. Opt.

E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, "Extraordinary optical transmission without plasmons: the s-polarization case," J. Opt. A: Pure Appl. Opt. 8, S94-S97 (2006).
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J. Opt. Soc. Am. B

Nature

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
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Phys. Rev. B

D. Y. K. Ko and J. C. Inkson, "Matrix method for tunneling in heterostructures: Resonant tunneling in multilayer systems," Phys. Rev. B 38, 9945-9951 (1988).
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D. M. Whittaker and I. S. Culshaw, "Scattering-matrix treatment of patterned multilayer photonic structures," Phys. Rev. B 60, 2610-2618 (1999).
[CrossRef]

S. Fan and J. D. Joannopoulos, "Analysis of guided resonances in photonic crystal slabs," Phys. Rev. B 65, 235112 (2002).
[CrossRef]

S. G. Tikhodeew, A. L. Yablonskii, E. A. Muljarow, N. A. Gippius, and T. Ishihara, "Quasiguide modes and optical properties of photonic crystal slabs," Phys. Rev. B 66, 045102 (2002).
[CrossRef]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

K. B. Crozier, V. Lousse, O. Kilic, S. Kim, S. Fan, and O. Solgaard, "Air-bridged photonic crystal slabs at visible and near-infrared wavelength," Phys. Rev. B 73, 115126 (2006).
[CrossRef]

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[CrossRef]

W. V. Lousse and J. P. Vigneron, "Use of Fano resonances for bistable optical transfer through photonic crystal film," Phys. Rev. B 69, 155106 (2004).
[CrossRef]

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[CrossRef] [PubMed]

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Figures (9)

Fig. 1.
Fig. 1.

Photonic crystal slab structure consisting of a square lattice of holes embedded into a uniform material.

Fig. 2.
Fig. 2.

Three Fano’s resonances at 1.9871a, 2.0658a and 2.0841a are shown. In particular, the top graph shows the variation of transmission in the wavelength region from 1.9a to 2.2a. The bottom graph shows the energy P calculated in a unit cell of PhCS. The energy peaks correspond to wavelength values roughly in between T=0 and T=1.

Fig. 3.
Fig. 3.

(a), (b) and (c) show the Sz distribution for each Fano resonance of Fig (2). The patterns cover four unit cells, which have a hole just in the center. (d) Distribution of the Poynting vector at the plane y/a=0.5, z0.

Fig. 4.
Fig. 4.

(a) Transmission spectrum of the incident light for s (top) and p (bottom) polarization. The incoming angle θ is from 0° to 80°, where zero means a beam parallel to the z axis. Narrow Fano resonances form several characteristic lines. (b) Relation between neff and incident light angle. Light is along ΓX direction. The dashed line denotes the X point, beyond which the mirror effect for neff is shown.

Fig. 5.
Fig. 5.

(a) Transmission and reflection spectra calculated at θ=0. The sharp lines show Fano resonances. The reduction of Si refractive index shifts the resonance lines to shorter wavelengths. (b) Transmission and reflection variation spectra. ΔT and ΔR are defined as T(n0+Δn)-T(n0) and R(n0+Δn)-R(n0). At Fano wavelengths strong changes of T and R are possible. Variation range of Si index is from 0 to -0.01. The range of wavelength is from 1.97a to 2.12a.

Fig. 6.
Fig. 6.

(a) Sketch of 0th and 1st order scattered light directions. Incident light, 0th, and 1st order scattered light are denoted by heavy, dot and thin line, respectively. (b) Power flux spectrum of 0th and 1st order transmitted light. Dot and solid lines represent 0th and 1st order transmitted light, respectively. (c) Power flux ratio between 1st and 0th order transmitted light. (d) and (e) describe the reflection case. Figs. (b), (c), (d) and (e) consist of five graphs each, corresponding to variations of Si refractive index in the interval [-0.001, -0.009].

Fig. 7.
Fig. 7.

Transmission (top) and reflection (bottom) spectra for different incident polarizations. ϕ is the angle between electric field angle and x axis.

Fig. 8.
Fig. 8.

(a) and (b) show transmission spectra at varies temperatures for E‖y and E‖x, respectively. The differences are due to the anisotropic nature of KTP. Each graph consists of ten curves. Temperature is increased from 20°C to 200°C with constant steps of 20°C. The dot line shows the relation between Fano resonance wavelengths and temperature.

Fig. 9.
Fig. 9.

Fano resonant wavelength λf versus temperature. The graph shows linear dependence of λf2 with T2, namely linear relation between λf and temperature. In abscissa is plotted T2-400 to be consistent with Eq. 18.

Equations (39)

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× E = i ω B , × H = i ω D , · B = 0 , · D = 0
z E + ( y x ) e z = i ω μ 0 ( μ xx μ xy μ yx μ yy ) H + i ω μ 0 ( μ xz μ yz ) h z
( x y ) E = i ω μ 0 ( μ zx μ zy ) H + i ω μ 0 μ zz h z
z H + ( x y ) h z = i ω ε 0 ( ε yy ε yx ε xy ε xx ) H i ω ε 0 ( ε yz ε xz ) e z
( y x ) H = i ω ε 0 ( ε zy ε zx ) E i ω ε 0 ε zz e z
z H + i F ( 1 ) H = ik 0 ω μ 0 [ F ( 0 ) E ]
z E + i T ( 1 ) E = i ω μ 0 k 0 [ T ( 0 ) H ]
F ̂ ( 1 ) = ( x y ) μ zz 1 ( μ zx μ zy ) ( ε yz ε xz ) ε zz 1 ( y x )
F ̂ ( 0 ) = ( ε yy ε yx ε xy ε xx ) ( ε yz ε xz ) ε zz 1 ( ε zy ε zx ) ( x y ) μ zz 1 ( x y )
T ̂ ( 1 ) = ( μ xz μ yz ) μ zz 1 ( x y ) ( y x ) ε zz 1 ( ε zy ε zx )
T ̂ ( 0 ) = ( μ xx μ xy μ yx μ yy ) + ( μ xz μ yz ) μ zz 1 ( μ zx μ zy ) ( y x ) ε zz 1 ( y x )
H ( r ) = G k + G H ( G ) exp ( i β z )
E ( r ) = G k + G E ( G ) exp ( i β z )
ε ( r ) = G V G exp [ i ( k + G ) · r ]
μ ( r ) = G U G exp [ i ( k + G ) · r ]
V G G = k + G ε ( r ) k + G = V G , G
U G G = k + G μ ( r ) k + G = U G , G
k + G PQ k + G = G P G , G · Q G , G
F ( 1 ) = ( diag ( k x + G x ) diag ( k y + G y ) ) U zz 1 ( U zx U zy ) + ( V yz V xz ) V zz 1 ( diag ( k y + G y ) diag ( k x + G x ) )
F ( 0 ) = ( V yy V yx V xy V xx ) ( V yz V xz ) V zz 1 ( V zy V zx )
( diag ( k x + G x ) diag ( k y + G y ) ) U zz 1 ( diag ( k x + G x ) diag ( k y + G y ) )
T ( 1 ) = ( U xz U yz ) U zz 1 ( diag ( k x + G x ) diag ( k y + G y ) ) + ( diag ( k y + G y ) diag ( k x + G x ) ) V zz 1 ( V zy V zx )
T ( 0 ) = ( U xx U xy U yx U yy ) + ( U xz U yz ) U zz 1 ( U zx U zy )
( diag ( k y + G y ) diag ( k x + G x ) ) V zz 1 ( diag ( k y + G y ) diag ( k x + G x ) )
H ( G ) β 2 + [ F ( 1 ) + F ( 0 ) T ( 1 ) ( F ( 0 ) ) 1 ] H ( G ) β + [ F ( 0 ) T ( 1 ) ( F ( 0 ) ) 1 F ( 1 ) F ( 0 ) T ( 0 ) ] H ( G ) = O
E ( G ) β 2 + [ T ( 1 ) + T ( 0 ) F ( 1 ) ( T ( 0 ) ) 1 ] E ( G ) β + [ T ( 0 ) F ( 1 ) ( T ( 0 ) ) 1 T ( 1 ) T ( 0 ) F ( 0 ) ] E ( G ) = O
H ( G ) β 2 F ( 0 ) T ( 0 ) H ( G ) = O
E ( G ) β 2 T ( 0 ) F ( 0 ) E ( G ) = O
F ( 0 ) = ( V O O V ) ( diag ( k x + G x ) diag ( k y + G y ) ) U 1 ( diag ( k x + G x ) diag ( k y + G y ) )
T ( 0 ) = ( U O O U ) ( diag ( k y + G y ) diag ( k x + G x ) ) V 1 ( diag ( k y + G y ) diag ( k x + G x ) )
F ( 0 ) = ( V yy V yx V xy V xx ) ( diag ( k x + G x ) diag ( k y + G y ) ) U zz 1 ( diag ( k x + G x ) diag ( k y + G y ) )
T ( 0 ) = ( U xx U xy U yx U yy ) ( diag ( k y + G y ) diag ( k x + G x ) ) V zz 1 ( diag ( k y + G y ) diag ( k x + G x ) )
H ( r ) = G , j k + G [ H , j ( + ) ( G ) a j exp ( i β j ( + ) z ) + H , j ( ) ( G ) b j exp ( i β j ( ) z ) ]
E ( r ) = G , j k + G [ E , j ( + ) ( G ) a j exp ( i β j ( + ) z ) + E , j ( ) ( G ) b j exp ( i β j ( ) z ) ]
( E ( + ) ( G ) E ( ) ( G ) H ( + ) ( G ) H ( ) ( G ) ) ( a b ) = ( E ( ) ( G ) E ( + ) ( G ) H ( ) ( G ) H ( + ) ( G ) ) ( b a )
S ( a b ) = ( b a ) ,
S = ( E ( ) ( G ) E ( + ) ( G ) H ( ) ( G ) H ( + ) ( G ) ) 1 ( E ( + ) ( G ) E ( ) ( G ) H ( + ) ( G ) H ( ) ( G ) )
P = cell 1 2 ( D * · E + B * · H ) dV
n i 2 ( λ , T ) = a i + β i ( T 2 400 ) + b i + δ i ( T 2 400 ) λ 2 c i + ϕ i ( T 2 400 ) λ 2 [ d i + ρ i ( T 2 400 ) ]

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