Abstract

Waveguide and resonant properties of diffractive structures are often explained through the complex poles of their scattering matrices. Numerical methods for calculating poles of the scattering matrix with applications in grating theory are discussed and analyzed. A new iterative method for computing the scattering matrix poles is proposed. The method takes account of the scattering matrix form in the pole vicinity and relies upon solving matrix equations with use of matrix decompositions. Using the same mathematical approach, we also describe a Cauchy-integral-based method that allows all of the poles in a specified domain to be calculated. Calculation of the modes of a metal-dielectric diffraction grating shows that the iterative method proposed has the high rate of convergence and is numerically stable for large-dimension scattering matrices. An important advantage of the proposed method is that it usually converges to the nearest pole.

© 2012 IEEE

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  1. V. Lomakin, E. Michielssen, "Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes," IEEE Trans. Antennas Propag. 54, 970-984 (2006).
  2. E. Anemogiannis, E. Glytsis, T. Gaylord, "Efficient solution of eigenvalue equations of optical waveguiding structures," J. Lightw. Technol. 12, 2080-2084 (1994).
  3. C. Chen, P. Berini, D. Feng, S. Tanev, V. Tzolov, "Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media," Opt. Exp. 7, 260-272 (2000).
  4. S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, T. Ishihara, "Quasiguided modes and optical properties of photonic crystal slabs," Phys. Rev. B 66, 045102 (2002).
  5. M. Nevière, E. Popov, R. Reinisch, "Electromagnetic resonances in linear and nonlinear optics: Phenomenological study of grating behavior through the poles and zeros of the scattering operator," J. Opt. Soc. Amer. A 12, 513-523 (1995).
  6. F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College, 2005).
  7. N. A. Gippius, S. G. Tikhodeev, "The scattering matrix and optical properties of metamaterials," Phys. Usp. 52, 1027-1030 (2009).
  8. N. A. Gippius, T. Weiss, S. G. Tikhodeev, H. Giessen, "Resonant mode coupling of optical resonances in stacked nanostructures," Opt. Exp. 18, 7569-7574 (2010).
  9. D. A. Bykov, L. L. Doskolovich, "Magneto-optical resonances in periodic dielectric structures magnetized in plane," J. Mod. Opt. 57, 1611-1618 (2010).
  10. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, "Channel drop tunneling through localized states," Phys. Rev. Lett. 80, 960-963 (1998).
  11. W. Suh, S. Fan, "Mechanically switchable photonic crystal filter with either all-pass transmission or flat-top reflection characteristics," Opt. Lett. 28, 1763-1765 (2003).
  12. E. Centeno, D. Felbacq, "Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity," Phys. Rev. B 62, R7683-R7686 (2000).
  13. E. Centeno, D. Felbacq, "Rabi oscillations in bidimensional photonic crystals," Phys. Rev. B 62, 10 101-10 108 (2000).
  14. T. Vallius, J. Tervo, P. Vahimaa, J. Turunen, "Electromagnetic field computation in semiconductor laser resonators," J. Opt. Soc. Amer. A 23, 906-911 (2006).
  15. M. Liscidini, D. Gerace, L. C. Andreani, J. E. Sipe, "Scattering-matrix analysis of periodically patterned multilayers with asymmetric unit cells and birefringent media," Phys. Rev. B 77, (2008) Art. ID 035324.
  16. D. Felbacq, "Numerical computation of resonance poles in scattering theory," Phys. Rev. E 64, (2001) Art. ID 047702.
  17. T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, H. Giessen, "Derivation of plasmonic resonances in the Fourier modal method with adaptive spatial resolution and matched coordinates," J. Opt. Soc. Amer. A 28, 238-244 (2011).
  18. M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Amer. A 12, 1068-1076 (1995).
  19. L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Amer. A 13, 1870-1876 (1996).
  20. L. Li, "Field singularities at lossless metal-dielectric arbitrary-angle edges and their ramifications to the numerical modeling of gratings," J. Opt. Soc. Amer. A 29, 593-604 (2012).
  21. T. Weiss, Advanced numerical and semi-analytical scattering matrix calculations for modern nano-optics Ph.D. dissertation Fakult ä t Mathematik und Physik Physikalisches Institut der UniversitätStuttgartGermany (2011).
  22. D. Felbacq, "Finding resonance poles by means of cauchy integrals," Proc. 13th Int. Conf. Transparent Optical Networks (2011).
  23. A. Akimov, N. Gippius, S. Tikhodeev, "Optical Fano resonances in photonic crystal slabs near diffraction threshold anomalies," JETP Lett. 93, 427-430 (2011).
  24. F. R. Gantmacher, The Theory of Matrices (Nauka, 1988).
  25. A. S. Householder, The Numerical Treatment of a Single Nonlinear Equation (McGraw-Hill, 1970).
  26. L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Amer. A 13, 1024-1035 (1996).
  27. A. D. Rakic, A. B. Djurišic, J. M. Elazar, M. L. Majewski, "Optical properties of metallic films for vertical-cavity optoelectronic devices," Appl. Opt. 37, 5271-5283 (1998).
  28. L. Yau, A. Ben-Israel, "The Newton and Halley methods for complex roots," Am. Math. Mon. 105, 806-818 (1998).
  29. Y. Hua, T. K. Sarkar, "Matrix pencil and system poles," Signal Process. 21, 195-198 (1990).
  30. Y. Hua, T. K. Sarkar, "Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise," IEEE Trans. Acoust., Speech, Signal Process. 38, 814-824 (1990).
  31. Y. Hua, T. K. Sarkar, "On SVD for estimating generalized eigenvalues of singular matrix pencil in noise," IEEE Trans. Signal Process. 39, 892-900 (1991).

2012

L. Li, "Field singularities at lossless metal-dielectric arbitrary-angle edges and their ramifications to the numerical modeling of gratings," J. Opt. Soc. Amer. A 29, 593-604 (2012).

2011

A. Akimov, N. Gippius, S. Tikhodeev, "Optical Fano resonances in photonic crystal slabs near diffraction threshold anomalies," JETP Lett. 93, 427-430 (2011).

T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, H. Giessen, "Derivation of plasmonic resonances in the Fourier modal method with adaptive spatial resolution and matched coordinates," J. Opt. Soc. Amer. A 28, 238-244 (2011).

2010

N. A. Gippius, T. Weiss, S. G. Tikhodeev, H. Giessen, "Resonant mode coupling of optical resonances in stacked nanostructures," Opt. Exp. 18, 7569-7574 (2010).

D. A. Bykov, L. L. Doskolovich, "Magneto-optical resonances in periodic dielectric structures magnetized in plane," J. Mod. Opt. 57, 1611-1618 (2010).

2009

N. A. Gippius, S. G. Tikhodeev, "The scattering matrix and optical properties of metamaterials," Phys. Usp. 52, 1027-1030 (2009).

2008

M. Liscidini, D. Gerace, L. C. Andreani, J. E. Sipe, "Scattering-matrix analysis of periodically patterned multilayers with asymmetric unit cells and birefringent media," Phys. Rev. B 77, (2008) Art. ID 035324.

2006

T. Vallius, J. Tervo, P. Vahimaa, J. Turunen, "Electromagnetic field computation in semiconductor laser resonators," J. Opt. Soc. Amer. A 23, 906-911 (2006).

V. Lomakin, E. Michielssen, "Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes," IEEE Trans. Antennas Propag. 54, 970-984 (2006).

2003

2002

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, T. Ishihara, "Quasiguided modes and optical properties of photonic crystal slabs," Phys. Rev. B 66, 045102 (2002).

2001

D. Felbacq, "Numerical computation of resonance poles in scattering theory," Phys. Rev. E 64, (2001) Art. ID 047702.

2000

E. Centeno, D. Felbacq, "Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity," Phys. Rev. B 62, R7683-R7686 (2000).

E. Centeno, D. Felbacq, "Rabi oscillations in bidimensional photonic crystals," Phys. Rev. B 62, 10 101-10 108 (2000).

C. Chen, P. Berini, D. Feng, S. Tanev, V. Tzolov, "Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media," Opt. Exp. 7, 260-272 (2000).

1998

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, "Channel drop tunneling through localized states," Phys. Rev. Lett. 80, 960-963 (1998).

A. D. Rakic, A. B. Djurišic, J. M. Elazar, M. L. Majewski, "Optical properties of metallic films for vertical-cavity optoelectronic devices," Appl. Opt. 37, 5271-5283 (1998).

L. Yau, A. Ben-Israel, "The Newton and Halley methods for complex roots," Am. Math. Mon. 105, 806-818 (1998).

1996

L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Amer. A 13, 1024-1035 (1996).

L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Amer. A 13, 1870-1876 (1996).

1995

M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Amer. A 12, 1068-1076 (1995).

M. Nevière, E. Popov, R. Reinisch, "Electromagnetic resonances in linear and nonlinear optics: Phenomenological study of grating behavior through the poles and zeros of the scattering operator," J. Opt. Soc. Amer. A 12, 513-523 (1995).

1994

E. Anemogiannis, E. Glytsis, T. Gaylord, "Efficient solution of eigenvalue equations of optical waveguiding structures," J. Lightw. Technol. 12, 2080-2084 (1994).

1991

Y. Hua, T. K. Sarkar, "On SVD for estimating generalized eigenvalues of singular matrix pencil in noise," IEEE Trans. Signal Process. 39, 892-900 (1991).

1990

Y. Hua, T. K. Sarkar, "Matrix pencil and system poles," Signal Process. 21, 195-198 (1990).

Y. Hua, T. K. Sarkar, "Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise," IEEE Trans. Acoust., Speech, Signal Process. 38, 814-824 (1990).

Am. Math. Mon.

L. Yau, A. Ben-Israel, "The Newton and Halley methods for complex roots," Am. Math. Mon. 105, 806-818 (1998).

Appl. Opt.

A. D. Rakic, A. B. Djurišic, J. M. Elazar, M. L. Majewski, "Optical properties of metallic films for vertical-cavity optoelectronic devices," Appl. Opt. 37, 5271-5283 (1998).

IEEE Trans. Signal Process.

Y. Hua, T. K. Sarkar, "On SVD for estimating generalized eigenvalues of singular matrix pencil in noise," IEEE Trans. Signal Process. 39, 892-900 (1991).

IEEE Trans. Antennas Propag.

V. Lomakin, E. Michielssen, "Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes," IEEE Trans. Antennas Propag. 54, 970-984 (2006).

IEEE Trans. Acoust., Speech, Signal Process.

Y. Hua, T. K. Sarkar, "Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise," IEEE Trans. Acoust., Speech, Signal Process. 38, 814-824 (1990).

J. Lightw. Technol.

E. Anemogiannis, E. Glytsis, T. Gaylord, "Efficient solution of eigenvalue equations of optical waveguiding structures," J. Lightw. Technol. 12, 2080-2084 (1994).

J. Mod. Opt.

D. A. Bykov, L. L. Doskolovich, "Magneto-optical resonances in periodic dielectric structures magnetized in plane," J. Mod. Opt. 57, 1611-1618 (2010).

J. Opt. Soc. Amer. A

L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Amer. A 13, 1870-1876 (1996).

J. Opt. Soc. Amer. A

M. Nevière, E. Popov, R. Reinisch, "Electromagnetic resonances in linear and nonlinear optics: Phenomenological study of grating behavior through the poles and zeros of the scattering operator," J. Opt. Soc. Amer. A 12, 513-523 (1995).

J. Opt. Soc. Amer. A

T. Vallius, J. Tervo, P. Vahimaa, J. Turunen, "Electromagnetic field computation in semiconductor laser resonators," J. Opt. Soc. Amer. A 23, 906-911 (2006).

T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, H. Giessen, "Derivation of plasmonic resonances in the Fourier modal method with adaptive spatial resolution and matched coordinates," J. Opt. Soc. Amer. A 28, 238-244 (2011).

M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Amer. A 12, 1068-1076 (1995).

L. Li, "Field singularities at lossless metal-dielectric arbitrary-angle edges and their ramifications to the numerical modeling of gratings," J. Opt. Soc. Amer. A 29, 593-604 (2012).

L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Amer. A 13, 1024-1035 (1996).

JETP Lett.

A. Akimov, N. Gippius, S. Tikhodeev, "Optical Fano resonances in photonic crystal slabs near diffraction threshold anomalies," JETP Lett. 93, 427-430 (2011).

Opt. Exp.

C. Chen, P. Berini, D. Feng, S. Tanev, V. Tzolov, "Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media," Opt. Exp. 7, 260-272 (2000).

Opt. Exp.

N. A. Gippius, T. Weiss, S. G. Tikhodeev, H. Giessen, "Resonant mode coupling of optical resonances in stacked nanostructures," Opt. Exp. 18, 7569-7574 (2010).

Opt. Lett.

Phys. Rev. B

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, T. Ishihara, "Quasiguided modes and optical properties of photonic crystal slabs," Phys. Rev. B 66, 045102 (2002).

Phys. Rev. B

E. Centeno, D. Felbacq, "Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity," Phys. Rev. B 62, R7683-R7686 (2000).

E. Centeno, D. Felbacq, "Rabi oscillations in bidimensional photonic crystals," Phys. Rev. B 62, 10 101-10 108 (2000).

M. Liscidini, D. Gerace, L. C. Andreani, J. E. Sipe, "Scattering-matrix analysis of periodically patterned multilayers with asymmetric unit cells and birefringent media," Phys. Rev. B 77, (2008) Art. ID 035324.

Phys. Rev. E

D. Felbacq, "Numerical computation of resonance poles in scattering theory," Phys. Rev. E 64, (2001) Art. ID 047702.

Phys. Rev. Lett.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, "Channel drop tunneling through localized states," Phys. Rev. Lett. 80, 960-963 (1998).

Phys. Usp.

N. A. Gippius, S. G. Tikhodeev, "The scattering matrix and optical properties of metamaterials," Phys. Usp. 52, 1027-1030 (2009).

Signal Process.

Y. Hua, T. K. Sarkar, "Matrix pencil and system poles," Signal Process. 21, 195-198 (1990).

Other

F. R. Gantmacher, The Theory of Matrices (Nauka, 1988).

A. S. Householder, The Numerical Treatment of a Single Nonlinear Equation (McGraw-Hill, 1970).

T. Weiss, Advanced numerical and semi-analytical scattering matrix calculations for modern nano-optics Ph.D. dissertation Fakult ä t Mathematik und Physik Physikalisches Institut der UniversitätStuttgartGermany (2011).

D. Felbacq, "Finding resonance poles by means of cauchy integrals," Proc. 13th Int. Conf. Transparent Optical Networks (2011).

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College, 2005).

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