Abstract

A Legendre polynomial expansion of electromagnetic fields for analysis of layers with an inhomogeneous refractive index profile is reported. The solution of Maxwell's equations subject to boundary conditions is sought in a complete space spanned by Legendre polynomials. Also, the permittivity profile is interpolated by polynomials. Different cases including computation of reflection–transmission coefficients of inhomogeneous layers, band-structure extraction of one-dimensional photonic crystals whose unit-cell refractive index profiles are inhomogeneous, and inhomogeneous planar waveguide analysis are investigated. The presented approach can be used to obtain the transfer matrix of an arbitrary inhomogeneous monolayer holistically, and approximation of the refractive index or permittivity profile by dividing into homogeneous sublayers is not needed. Comparisons with other well-known methods such as the transfer-matrix method, WKB, and effective index method are made. The presented approach, based on a nonharmonic expansion, is efficient, shows fast convergence, is versatile, and can be easily and systematically employed to analyze different inhomogeneous structures.

© 2006 Optical Society of America

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  7. L. Brillouin, Wave Propagation in Periodic Structures (McGraw-Hill, 1946).
  8. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2001).
  9. J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772-2775 (1992).
    [CrossRef]
  10. K. Kunz and R. Luebbers, eds., The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).
  11. J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics (IEEE Press, 1998).
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    [CrossRef]
  13. J. Janta and J. Ctyroky, "On the accuracy of WKB analysis of TE and TM modes in planar graded-index waveguides," Opt. Commun. 25, 49-52 (1978).
    [CrossRef]
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  15. F. Xiang and G. L. Yip, "A modified WKB method for the improved phase shift at a turning point," J. Lightwave Technol. 12, 443-452 (1994).
    [CrossRef]
  16. J. Wang and L. Qiao, "A refined WKB method for symmetric planar waveguides with truncated-index profiles and graded-index profiles," IEEE J. Quantum Electron. 27, 878-883 (1991).
  17. T. Hyouguchi, S. Adachi, and M. Ueda, "Divergence-free WKB method," Phys. Rev. Lett. 88, 170404-1-170404-4 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  29. M. H. Eghlidi, K. Mehrany, and B. Rashidian, "Modified differential-transfer-matrix method for solution of one-dimensional linear inhomogeneous optical structures," J. Opt. Soc. Am. B 22, 1521-1528 (2005).
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    [CrossRef]
  32. K. Mehrany and B. Rashidian, "Polynomial expansion of electromagnetic eigenmodes in layered structures," J. Opt. Soc. Am. B 20, 2434-2441 (2003).
  33. M. Chamanzar, K. Mehrany, and B. Rashidian, "Polynomial expansion of electromagnetic fields for grating diffraction analysis," in Proceedings of the 2004 International Symposium on Antennas and Propagation (Institute of Electronics, Information and Communication Engineers, 2004), pp. 161-164.
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2005 (1)

2003 (3)

2002 (5)

2000 (1)

1997 (1)

1996 (1)

C. Thompson and B. L. Weiss, "Modal characteristics of graded multilayer optical waveguides," J. Lightwave Technol. 14, 894-900 (1996).

1994 (1)

F. Xiang and G. L. Yip, "A modified WKB method for the improved phase shift at a turning point," J. Lightwave Technol. 12, 443-452 (1994).
[CrossRef]

1992 (1)

J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772-2775 (1992).
[CrossRef]

1991 (1)

J. Wang and L. Qiao, "A refined WKB method for symmetric planar waveguides with truncated-index profiles and graded-index profiles," IEEE J. Quantum Electron. 27, 878-883 (1991).

1986 (1)

P. K. Mishra and A. Sharma, "Analysis of single mode inhomogeneous planar waveguides," J. Lightwave Technol. 4, 204-212 (1986).

1983 (3)

V. Ramaswamy and R. K. Lagu, "Numerical field solutions for an arbitrary asymmetrical graded-index planar waveguide," J. Lightwave Technol. LT-1, 408-416 (1983).

J. P. Memuer, J. Pigeon, and J. N. Massot, "A numerical technique for determination of propagation characteristics of inhomogeneous planar optical waveguides," Opt. Quantum Electron. 15, 77-85 (1983).
[CrossRef]

L. C. Botten and M. S. Craig, "Complex zeros of analytic functions," Comput. Phys. Commun. 29, 245-259 (1983).
[CrossRef]

1978 (1)

J. Janta and J. Ctyroky, "On the accuracy of WKB analysis of TE and TM modes in planar graded-index waveguides," Opt. Commun. 25, 49-52 (1978).
[CrossRef]

1974 (1)

A. Kumar, K. ThyagaraJan, and A. K. Ghatak, "Modes in inhomogeneous slab waveguides," IEEE J. Quantum Electron. 10, 902-904 (1974).

1973 (1)

E. M. Conwell, "Modes in optical waveguides formed by diffusion," Appl. Phys. Lett. 23, 328-329 (1973).
[CrossRef]

1967 (1)

L. M. Delves and J. N. Lyness, "A numerical method for locating the zeros of an analytic function," Math. Comput. 21, 543-560 (1967).

Adachi, S.

T. Hyouguchi, S. Adachi, and M. Ueda, "Divergence-free WKB method," Phys. Rev. Lett. 88, 170404-1-170404-4 (2002).
[CrossRef]

Adibi, A.

S. Khorasani and A. Adibi, "Analytical solution of linear ordinary differential equations by differential transfer matrix method," Electron. J. Differ. Equations 2003, 1-18 (2003).

Assanto, G.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Botten, L. C.

L. C. Botten and M. S. Craig, "Complex zeros of analytic functions," Comput. Phys. Commun. 29, 245-259 (1983).
[CrossRef]

Brillouin, L.

L. Brillouin, Wave Propagation in Periodic Structures (McGraw-Hill, 1946).

Chamanzar, M.

M. Chamanzar, K. Mehrany, and B. Rashidian, "Polynomial expansion of electromagnetic fields for grating diffraction analysis," in Proceedings of the 2004 International Symposium on Antennas and Propagation (Institute of Electronics, Information and Communication Engineers, 2004), pp. 161-164.

Chatterjee, A.

J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics (IEEE Press, 1998).

Conwell, E. M.

E. M. Conwell, "Modes in optical waveguides formed by diffusion," Appl. Phys. Lett. 23, 328-329 (1973).
[CrossRef]

Craig, M. S.

L. C. Botten and M. S. Craig, "Complex zeros of analytic functions," Comput. Phys. Commun. 29, 245-259 (1983).
[CrossRef]

Ctyroky, J.

J. Janta and J. Ctyroky, "On the accuracy of WKB analysis of TE and TM modes in planar graded-index waveguides," Opt. Commun. 25, 49-52 (1978).
[CrossRef]

de Sande, J. C.

Delves, L. M.

L. M. Delves and J. N. Lyness, "A numerical method for locating the zeros of an analytic function," Math. Comput. 21, 543-560 (1967).

Dre'villon, B.

Eghlidi, M. H.

El-Mikati, H. A.

Froman, N.

N. Froman and P. O. Froman, JWKB Approximation: Contributions to the Theory (North-Holland, 1965).

Froman, P. O.

N. Froman and P. O. Froman, JWKB Approximation: Contributions to the Theory (North-Holland, 1965).

Ghatak, A. K.

A. Kumar, K. ThyagaraJan, and A. K. Ghatak, "Modes in inhomogeneous slab waveguides," IEEE J. Quantum Electron. 10, 902-904 (1974).

Grattan, K. T. V.

Hunderi, O.

Hyouguchi, T.

T. Hyouguchi, S. Adachi, and M. Ueda, "Divergence-free WKB method," Phys. Rev. Lett. 88, 170404-1-170404-4 (2002).
[CrossRef]

Ishigure, T.

Janta, J.

J. Janta and J. Ctyroky, "On the accuracy of WKB analysis of TE and TM modes in planar graded-index waveguides," Opt. Commun. 25, 49-52 (1978).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Kempel, L. C.

J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics (IEEE Press, 1998).

Khorasani, S.

S. Khorasani and K. Mehrany, "Differential transfer-matrix method for solution of one-dimensional linear nonhomogeneous optical structures," J. Opt. Soc. Am. B 20, 91-96 (2003).

S. Khorasani and A. Adibi, "Analytical solution of linear ordinary differential equations by differential transfer matrix method," Electron. J. Differ. Equations 2003, 1-18 (2003).

K. Mehrany and S. Khorasani, "Analytical solution of nonhomogeneous anisotropic wave equations based on differential transfer matrix method," J. Opt. A Pure Appl. Opt. 4, 624-635 (2002).
[CrossRef]

Kildemo, L.

Kobayashi, E.

Koike, Y.

Koshiba, M.

Kumar, A.

A. Kumar, K. ThyagaraJan, and A. K. Ghatak, "Modes in inhomogeneous slab waveguides," IEEE J. Quantum Electron. 10, 902-904 (1974).

Kunz, K.

K. Kunz and R. Luebbers, eds., The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).

Lagu, R. K.

V. Ramaswamy and R. K. Lagu, "Numerical field solutions for an arbitrary asymmetrical graded-index planar waveguide," J. Lightwave Technol. LT-1, 408-416 (1983).

Leo, G.

Luebbers, R.

K. Kunz and R. Luebbers, eds., The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).

Lyness, J. N.

L. M. Delves and J. N. Lyness, "A numerical method for locating the zeros of an analytic function," Math. Comput. 21, 543-560 (1967).

MacKinnon, A.

J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772-2775 (1992).
[CrossRef]

Massot, J. N.

J. P. Memuer, J. Pigeon, and J. N. Massot, "A numerical technique for determination of propagation characteristics of inhomogeneous planar optical waveguides," Opt. Quantum Electron. 15, 77-85 (1983).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Mehrany, K.

M. H. Eghlidi, K. Mehrany, and B. Rashidian, "Modified differential-transfer-matrix method for solution of one-dimensional linear inhomogeneous optical structures," J. Opt. Soc. Am. B 22, 1521-1528 (2005).

S. Khorasani and K. Mehrany, "Differential transfer-matrix method for solution of one-dimensional linear nonhomogeneous optical structures," J. Opt. Soc. Am. B 20, 91-96 (2003).

K. Mehrany and B. Rashidian, "Polynomial expansion of electromagnetic eigenmodes in layered structures," J. Opt. Soc. Am. B 20, 2434-2441 (2003).

K. Mehrany and S. Khorasani, "Analytical solution of nonhomogeneous anisotropic wave equations based on differential transfer matrix method," J. Opt. A Pure Appl. Opt. 4, 624-635 (2002).
[CrossRef]

M. Chamanzar, K. Mehrany, and B. Rashidian, "Polynomial expansion of electromagnetic fields for grating diffraction analysis," in Proceedings of the 2004 International Symposium on Antennas and Propagation (Institute of Electronics, Information and Communication Engineers, 2004), pp. 161-164.

Memuer, J. P.

J. P. Memuer, J. Pigeon, and J. N. Massot, "A numerical technique for determination of propagation characteristics of inhomogeneous planar optical waveguides," Opt. Quantum Electron. 15, 77-85 (1983).
[CrossRef]

Mishra, P. K.

P. K. Mishra and A. Sharma, "Analysis of single mode inhomogeneous planar waveguides," J. Lightwave Technol. 4, 204-212 (1986).

Obayya, S. S. A.

Pendry, J. B.

J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772-2775 (1992).
[CrossRef]

Pigeon, J.

J. P. Memuer, J. Pigeon, and J. N. Massot, "A numerical technique for determination of propagation characteristics of inhomogeneous planar optical waveguides," Opt. Quantum Electron. 15, 77-85 (1983).
[CrossRef]

Potter, K. S.

J. H. Simmons and K. S. Potter, Optical Materials (Academic, 2000).

Qiao, L.

J. Wang and L. Qiao, "A refined WKB method for symmetric planar waveguides with truncated-index profiles and graded-index profiles," IEEE J. Quantum Electron. 27, 878-883 (1991).

Rahman, B. M. A.

Ramaswamy, V.

V. Ramaswamy and R. K. Lagu, "Numerical field solutions for an arbitrary asymmetrical graded-index planar waveguide," J. Lightwave Technol. LT-1, 408-416 (1983).

Rashidian, B.

M. H. Eghlidi, K. Mehrany, and B. Rashidian, "Modified differential-transfer-matrix method for solution of one-dimensional linear inhomogeneous optical structures," J. Opt. Soc. Am. B 22, 1521-1528 (2005).

K. Mehrany and B. Rashidian, "Polynomial expansion of electromagnetic eigenmodes in layered structures," J. Opt. Soc. Am. B 20, 2434-2441 (2003).

M. Chamanzar, K. Mehrany, and B. Rashidian, "Polynomial expansion of electromagnetic fields for grating diffraction analysis," in Proceedings of the 2004 International Symposium on Antennas and Propagation (Institute of Electronics, Information and Communication Engineers, 2004), pp. 161-164.

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2001).

Sharma, A.

P. K. Mishra and A. Sharma, "Analysis of single mode inhomogeneous planar waveguides," J. Lightwave Technol. 4, 204-212 (1986).

Simmons, J. H.

J. H. Simmons and K. S. Potter, Optical Materials (Academic, 2000).

Tanaka, S.

Thompson, C.

C. Thompson and B. L. Weiss, "Modal characteristics of graded multilayer optical waveguides," J. Lightwave Technol. 14, 894-900 (1996).

ThyagaraJan, K.

A. Kumar, K. ThyagaraJan, and A. K. Ghatak, "Modes in inhomogeneous slab waveguides," IEEE J. Quantum Electron. 10, 902-904 (1974).

Tsuji, Y.

Ueda, M.

T. Hyouguchi, S. Adachi, and M. Ueda, "Divergence-free WKB method," Phys. Rev. Lett. 88, 170404-1-170404-4 (2002).
[CrossRef]

Volakis, J. L.

J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics (IEEE Press, 1998).

Wang, J.

J. Wang and L. Qiao, "A refined WKB method for symmetric planar waveguides with truncated-index profiles and graded-index profiles," IEEE J. Quantum Electron. 27, 878-883 (1991).

Weiss, B. L.

C. Thompson and B. L. Weiss, "Modal characteristics of graded multilayer optical waveguides," J. Lightwave Technol. 14, 894-900 (1996).

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Xiang, F.

F. Xiang and G. L. Yip, "A modified WKB method for the improved phase shift at a turning point," J. Lightwave Technol. 12, 443-452 (1994).
[CrossRef]

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

Yip, G. L.

F. Xiang and G. L. Yip, "A modified WKB method for the improved phase shift at a turning point," J. Lightwave Technol. 12, 443-452 (1994).
[CrossRef]

Appl. Phys. Lett. (1)

E. M. Conwell, "Modes in optical waveguides formed by diffusion," Appl. Phys. Lett. 23, 328-329 (1973).
[CrossRef]

Comput. Phys. Commun. (1)

L. C. Botten and M. S. Craig, "Complex zeros of analytic functions," Comput. Phys. Commun. 29, 245-259 (1983).
[CrossRef]

Electron. J. Differ. Equations (1)

S. Khorasani and A. Adibi, "Analytical solution of linear ordinary differential equations by differential transfer matrix method," Electron. J. Differ. Equations 2003, 1-18 (2003).

IEEE J. Quantum Electron. (2)

J. Wang and L. Qiao, "A refined WKB method for symmetric planar waveguides with truncated-index profiles and graded-index profiles," IEEE J. Quantum Electron. 27, 878-883 (1991).

A. Kumar, K. ThyagaraJan, and A. K. Ghatak, "Modes in inhomogeneous slab waveguides," IEEE J. Quantum Electron. 10, 902-904 (1974).

J. Lightwave Technol. (8)

J. Opt. A Pure Appl. Opt. (1)

K. Mehrany and S. Khorasani, "Analytical solution of nonhomogeneous anisotropic wave equations based on differential transfer matrix method," J. Opt. A Pure Appl. Opt. 4, 624-635 (2002).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (3)

Math. Comput. (1)

L. M. Delves and J. N. Lyness, "A numerical method for locating the zeros of an analytic function," Math. Comput. 21, 543-560 (1967).

Opt. Commun. (1)

J. Janta and J. Ctyroky, "On the accuracy of WKB analysis of TE and TM modes in planar graded-index waveguides," Opt. Commun. 25, 49-52 (1978).
[CrossRef]

Opt. Quantum Electron. (1)

J. P. Memuer, J. Pigeon, and J. N. Massot, "A numerical technique for determination of propagation characteristics of inhomogeneous planar optical waveguides," Opt. Quantum Electron. 15, 77-85 (1983).
[CrossRef]

Phys. Rev. Lett. (2)

J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772-2775 (1992).
[CrossRef]

T. Hyouguchi, S. Adachi, and M. Ueda, "Divergence-free WKB method," Phys. Rev. Lett. 88, 170404-1-170404-4 (2002).
[CrossRef]

Other (11)

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

J. H. Simmons and K. S. Potter, Optical Materials (Academic, 2000).

K. Kunz and R. Luebbers, eds., The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).

J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics (IEEE Press, 1998).

N. Froman and P. O. Froman, JWKB Approximation: Contributions to the Theory (North-Holland, 1965).

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

L. Brillouin, Wave Propagation in Periodic Structures (McGraw-Hill, 1946).

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2001).

M. Chamanzar, K. Mehrany, and B. Rashidian, "Polynomial expansion of electromagnetic fields for grating diffraction analysis," in Proceedings of the 2004 International Symposium on Antennas and Propagation (Institute of Electronics, Information and Communication Engineers, 2004), pp. 161-164.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

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