Abstract

A new formulation for the solution of wave propagation in inhomogeneous optical systems, based on the extension of conventional differential-transfer matrices into modified differential-transfer matrices, is given. In justification of our proposed method, several examples are presented, and the greater accuracy of our modified differential transfer matrices compared with that of conventional differential-transfer matrices is observed in several cases. It is also shown that the modified differential-transfer-matrix method is accurate enough even in those cases that the conventional differential-transfer-matrix method fails to yield acceptable results.

© 2005 Optical Society of America

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  1. J. H. Simmons and K. S. Potter, Optical Materials (Academic, 2000).
  2. P. K. Kelly and M. Piket-May, "Propagation characteristics for a one-dimensional grounded finite height finite length electromagnetic crystal," J. Lightwave Technol. 17, 2008-2012 (1999).
    [CrossRef]
  3. M. Eguchi and S. Horinouchi, "Finite-element modal analysis of large-core multimode optical fibers," Appl. Opt. 43, 2163-2167 (2004).
    [CrossRef] [PubMed]
  4. J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite Element Method for Electromagnetics (IEEE Press, 1998).
    [CrossRef]
  5. K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis (Wiley, 2001).
    [CrossRef]
  6. L. Adamovics and V. Q. Nguyen, "Electromagnetic field in a slab of photonic crystal by BPM," Opt. Lasers Eng. 35, 67-78 (2001).
    [CrossRef]
  7. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).
  8. P. Yeh, Optical Waves in Layered Media (Wiley, 1988).
  9. S. Khorasani and B. Rashidian, "Modified transfer matrices for conducting interfaces," J. Opt. A 4, 251-256 (2002).
    [CrossRef]
  10. M. Mansuripur, "Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2×2 matrices," J. Appl. Phys. 67, 6466-6475 (1990).
    [CrossRef]
  11. D. J. De Smet, "An application of the 4×4 matrix formalism to the polar magnetic Kerr effect," J. Mod. Opt. 39, 1055-1065 (1992).
    [CrossRef]
  12. M. Schubert, T. E. Tiwald, and J. A. Woollam, "Explicit solutions for the optical properties of arbitrary magneto-optic materials in generalized ellipsometry," Appl. Opt. 38, 177-187 (1999).
    [CrossRef]
  13. J. Zak, E. R. Moog, C. Liu, and S. D. Bader, "Universal approach to magneto-optics," J. Magn. Magn. Mater. 89, 107-123 (1990).
    [CrossRef]
  14. I. Abdulhalim, "Analytic propagation matrix method for linear optics of arbitrary biaxial layered media," J. Opt. A Pure Appl. Opt. 1, 646-653 (1999).
    [CrossRef]
  15. I. Abdulhalim, "2×2 matrix summation method for multiple reflections and transmissions in a biaxial slab between two anisotropic media," Opt. Commun. 163, 9-14 (1999).
    [CrossRef]
  16. I. Abdulhalim, "Exact 2×2 matrix method for the transmission and reflection at the interface between two arbitrarily oriented biaxial crystals," J. Opt. A 1, 655-661 (1999).
    [CrossRef]
  17. I. Abdulhalim, "Omnidirectional reflection from anisotropic periodic dielectric stack," Opt. Commun. 174, 43-50 (2000).
    [CrossRef]
  18. I. Abdulhalim, "Analytic propagation matrix method for anisotropic magneto-optic layered media," J. Opt. A 2, 557-564 (2000).
    [CrossRef]
  19. I. Abdulhalim, "Analytic formulae for the refractive indices and the propagation angles in biaxial and gyrotropic media," Opt. Commun. 157, 265-272 (1998).
    [CrossRef]
  20. D. M. Whittaker and I. S. Culshaw, "Scattering-matrix treatment of patterned multiplayer photonic structures," Phys. Rev. B 60, 2610-2618 (1999).
    [CrossRef]
  21. S. F. Mingaleev and K. Busch, "Scattering matrix approach to large-scale photonic crystal circuits," Opt. Lett. 28, 619-621 (2003).
    [CrossRef] [PubMed]
  22. Z. Menachem and E. Jerby, "Matrix transfer function (MTF) for wave propagation in dielectric waveguides with arbitrary transverse profiles," IEEE Trans. Microwave Theory Tech. 46, 975-982 (1998).
    [CrossRef]
  23. T. F. Jablonski, "Complex modes in open lossless dielectric waveguides," J. Opt. Soc. Am. A 11, 1272-1282 (1994).
    [CrossRef]
  24. J. S. Bagby, D. Nyquist, and B. C. Drachman, "Integral formulation for analysis of integrated dielectric waveguides," IEEE Trans. Microwave Theory Tech. 33, 906-915 (1985).
    [CrossRef]
  25. 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).
    [CrossRef]
  26. K. Mehrany and S. Khorasani, "Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrix method," J. Opt. A 4, 624-635 (2002).
    [CrossRef]
  27. 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).
  28. A. Lakhtakia, "Comment on 'Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrices'," J. Opt. A 5, 432-433 (2003).
    [CrossRef]
  29. S. Khorasani, "Reply to Comment on 'Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrices'," J. Opt. A 5, 434-435 (2003).
    [CrossRef]
  30. D. Sarafyan, "Approximate solution of ordinary differential equations and their systems through discrete and continuous embedded Runge-Kutta formulae and upgrading of their order," Comput. Math. Appl. 28, 353-384 (1992).
    [CrossRef]
  31. W. Kahan and R. Li, "Composition constants for rising the orders of unconventional schemes for ordinary differential equations," Math. Comput. 66, 1089-1099 (1997).
    [CrossRef]
  32. J. C. Butcher, "Numerical methods for ordinary differential equations in the 20th century," J. Comput. Appl. Math. 125, 1-29 (2000).
    [CrossRef]
  33. F. J. Dyson, "The S matrix in quantum electrodynamics," Phys. Rev. 75, 1736-1755 (1949).
    [CrossRef]
  34. P. Roman, Advanced Quantum Theory (Addison-Wesley, 1965), p. 310.
  35. J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, 1965), p. 177.
  36. J. Cronin, Differential Equations, 2nd ed. (Dekker, 1994), p. 73.
  37. S. Khorasani and A. Adibi, "New analytical approach for computation of band structure in one-dimensional periodic media," Opt. Commun. 216, 439-451 (2003).
    [CrossRef]
  38. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

2004 (1)

2003 (6)

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).
[CrossRef]

S. F. Mingaleev and K. Busch, "Scattering matrix approach to large-scale photonic crystal circuits," Opt. Lett. 28, 619-621 (2003).
[CrossRef] [PubMed]

S. Khorasani and A. Adibi, "New analytical approach for computation of band structure in one-dimensional periodic media," Opt. Commun. 216, 439-451 (2003).
[CrossRef]

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).

A. Lakhtakia, "Comment on 'Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrices'," J. Opt. A 5, 432-433 (2003).
[CrossRef]

S. Khorasani, "Reply to Comment on 'Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrices'," J. Opt. A 5, 434-435 (2003).
[CrossRef]

2002 (2)

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

S. Khorasani and B. Rashidian, "Modified transfer matrices for conducting interfaces," J. Opt. A 4, 251-256 (2002).
[CrossRef]

2001 (1)

L. Adamovics and V. Q. Nguyen, "Electromagnetic field in a slab of photonic crystal by BPM," Opt. Lasers Eng. 35, 67-78 (2001).
[CrossRef]

2000 (3)

I. Abdulhalim, "Omnidirectional reflection from anisotropic periodic dielectric stack," Opt. Commun. 174, 43-50 (2000).
[CrossRef]

I. Abdulhalim, "Analytic propagation matrix method for anisotropic magneto-optic layered media," J. Opt. A 2, 557-564 (2000).
[CrossRef]

J. C. Butcher, "Numerical methods for ordinary differential equations in the 20th century," J. Comput. Appl. Math. 125, 1-29 (2000).
[CrossRef]

1999 (6)

M. Schubert, T. E. Tiwald, and J. A. Woollam, "Explicit solutions for the optical properties of arbitrary magneto-optic materials in generalized ellipsometry," Appl. Opt. 38, 177-187 (1999).
[CrossRef]

P. K. Kelly and M. Piket-May, "Propagation characteristics for a one-dimensional grounded finite height finite length electromagnetic crystal," J. Lightwave Technol. 17, 2008-2012 (1999).
[CrossRef]

I. Abdulhalim, "Analytic propagation matrix method for linear optics of arbitrary biaxial layered media," J. Opt. A Pure Appl. Opt. 1, 646-653 (1999).
[CrossRef]

I. Abdulhalim, "2×2 matrix summation method for multiple reflections and transmissions in a biaxial slab between two anisotropic media," Opt. Commun. 163, 9-14 (1999).
[CrossRef]

I. Abdulhalim, "Exact 2×2 matrix method for the transmission and reflection at the interface between two arbitrarily oriented biaxial crystals," J. Opt. A 1, 655-661 (1999).
[CrossRef]

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

1998 (2)

I. Abdulhalim, "Analytic formulae for the refractive indices and the propagation angles in biaxial and gyrotropic media," Opt. Commun. 157, 265-272 (1998).
[CrossRef]

Z. Menachem and E. Jerby, "Matrix transfer function (MTF) for wave propagation in dielectric waveguides with arbitrary transverse profiles," IEEE Trans. Microwave Theory Tech. 46, 975-982 (1998).
[CrossRef]

1997 (1)

W. Kahan and R. Li, "Composition constants for rising the orders of unconventional schemes for ordinary differential equations," Math. Comput. 66, 1089-1099 (1997).
[CrossRef]

1994 (1)

1992 (2)

D. J. De Smet, "An application of the 4×4 matrix formalism to the polar magnetic Kerr effect," J. Mod. Opt. 39, 1055-1065 (1992).
[CrossRef]

D. Sarafyan, "Approximate solution of ordinary differential equations and their systems through discrete and continuous embedded Runge-Kutta formulae and upgrading of their order," Comput. Math. Appl. 28, 353-384 (1992).
[CrossRef]

1990 (2)

J. Zak, E. R. Moog, C. Liu, and S. D. Bader, "Universal approach to magneto-optics," J. Magn. Magn. Mater. 89, 107-123 (1990).
[CrossRef]

M. Mansuripur, "Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2×2 matrices," J. Appl. Phys. 67, 6466-6475 (1990).
[CrossRef]

1985 (1)

J. S. Bagby, D. Nyquist, and B. C. Drachman, "Integral formulation for analysis of integrated dielectric waveguides," IEEE Trans. Microwave Theory Tech. 33, 906-915 (1985).
[CrossRef]

1949 (1)

F. J. Dyson, "The S matrix in quantum electrodynamics," Phys. Rev. 75, 1736-1755 (1949).
[CrossRef]

Abdulhalim, I.

I. Abdulhalim, "Omnidirectional reflection from anisotropic periodic dielectric stack," Opt. Commun. 174, 43-50 (2000).
[CrossRef]

I. Abdulhalim, "Analytic propagation matrix method for anisotropic magneto-optic layered media," J. Opt. A 2, 557-564 (2000).
[CrossRef]

I. Abdulhalim, "Analytic propagation matrix method for linear optics of arbitrary biaxial layered media," J. Opt. A Pure Appl. Opt. 1, 646-653 (1999).
[CrossRef]

I. Abdulhalim, "2×2 matrix summation method for multiple reflections and transmissions in a biaxial slab between two anisotropic media," Opt. Commun. 163, 9-14 (1999).
[CrossRef]

I. Abdulhalim, "Exact 2×2 matrix method for the transmission and reflection at the interface between two arbitrarily oriented biaxial crystals," J. Opt. A 1, 655-661 (1999).
[CrossRef]

I. Abdulhalim, "Analytic formulae for the refractive indices and the propagation angles in biaxial and gyrotropic media," Opt. Commun. 157, 265-272 (1998).
[CrossRef]

Adamovics, L.

L. Adamovics and V. Q. Nguyen, "Electromagnetic field in a slab of photonic crystal by BPM," Opt. Lasers Eng. 35, 67-78 (2001).
[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).

S. Khorasani and A. Adibi, "New analytical approach for computation of band structure in one-dimensional periodic media," Opt. Commun. 216, 439-451 (2003).
[CrossRef]

Bader, S. D.

J. Zak, E. R. Moog, C. Liu, and S. D. Bader, "Universal approach to magneto-optics," J. Magn. Magn. Mater. 89, 107-123 (1990).
[CrossRef]

Bagby, J. S.

J. S. Bagby, D. Nyquist, and B. C. Drachman, "Integral formulation for analysis of integrated dielectric waveguides," IEEE Trans. Microwave Theory Tech. 33, 906-915 (1985).
[CrossRef]

Bjorken, J. D.

J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, 1965), p. 177.

Born, M.

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

Busch, K.

Butcher, J. C.

J. C. Butcher, "Numerical methods for ordinary differential equations in the 20th century," J. Comput. Appl. Math. 125, 1-29 (2000).
[CrossRef]

Chatterjee, A.

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

Cronin, J.

J. Cronin, Differential Equations, 2nd ed. (Dekker, 1994), p. 73.

Culshaw, I. S.

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

De Smet, D. J.

D. J. De Smet, "An application of the 4×4 matrix formalism to the polar magnetic Kerr effect," J. Mod. Opt. 39, 1055-1065 (1992).
[CrossRef]

Drachman, B. C.

J. S. Bagby, D. Nyquist, and B. C. Drachman, "Integral formulation for analysis of integrated dielectric waveguides," IEEE Trans. Microwave Theory Tech. 33, 906-915 (1985).
[CrossRef]

Drell, S. D.

J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, 1965), p. 177.

Dyson, F. J.

F. J. Dyson, "The S matrix in quantum electrodynamics," Phys. Rev. 75, 1736-1755 (1949).
[CrossRef]

Eguchi, M.

Horinouchi, S.

Jablonski, T. F.

Jerby, E.

Z. Menachem and E. Jerby, "Matrix transfer function (MTF) for wave propagation in dielectric waveguides with arbitrary transverse profiles," IEEE Trans. Microwave Theory Tech. 46, 975-982 (1998).
[CrossRef]

Kahan, W.

W. Kahan and R. Li, "Composition constants for rising the orders of unconventional schemes for ordinary differential equations," Math. Comput. 66, 1089-1099 (1997).
[CrossRef]

Kawano, K.

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis (Wiley, 2001).
[CrossRef]

Kelly, P. K.

Kempel, L. C.

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

Khorasani, S.

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).

S. Khorasani and A. Adibi, "New analytical approach for computation of band structure in one-dimensional periodic media," Opt. Commun. 216, 439-451 (2003).
[CrossRef]

S. Khorasani, "Reply to Comment on 'Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrices'," J. Opt. A 5, 434-435 (2003).
[CrossRef]

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).
[CrossRef]

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

S. Khorasani and B. Rashidian, "Modified transfer matrices for conducting interfaces," J. Opt. A 4, 251-256 (2002).
[CrossRef]

Kitoh, T.

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis (Wiley, 2001).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia, "Comment on 'Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrices'," J. Opt. A 5, 432-433 (2003).
[CrossRef]

Li, R.

W. Kahan and R. Li, "Composition constants for rising the orders of unconventional schemes for ordinary differential equations," Math. Comput. 66, 1089-1099 (1997).
[CrossRef]

Liu, C.

J. Zak, E. R. Moog, C. Liu, and S. D. Bader, "Universal approach to magneto-optics," J. Magn. Magn. Mater. 89, 107-123 (1990).
[CrossRef]

Mansuripur, M.

M. Mansuripur, "Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2×2 matrices," J. Appl. Phys. 67, 6466-6475 (1990).
[CrossRef]

Mehrany, K.

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).
[CrossRef]

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

Menachem, Z.

Z. Menachem and E. Jerby, "Matrix transfer function (MTF) for wave propagation in dielectric waveguides with arbitrary transverse profiles," IEEE Trans. Microwave Theory Tech. 46, 975-982 (1998).
[CrossRef]

Mingaleev, S. F.

Moog, E. R.

J. Zak, E. R. Moog, C. Liu, and S. D. Bader, "Universal approach to magneto-optics," J. Magn. Magn. Mater. 89, 107-123 (1990).
[CrossRef]

Nguyen, V. Q.

L. Adamovics and V. Q. Nguyen, "Electromagnetic field in a slab of photonic crystal by BPM," Opt. Lasers Eng. 35, 67-78 (2001).
[CrossRef]

Nyquist, D.

J. S. Bagby, D. Nyquist, and B. C. Drachman, "Integral formulation for analysis of integrated dielectric waveguides," IEEE Trans. Microwave Theory Tech. 33, 906-915 (1985).
[CrossRef]

Piket-May, M.

Potter, K. S.

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

Rashidian, B.

S. Khorasani and B. Rashidian, "Modified transfer matrices for conducting interfaces," J. Opt. A 4, 251-256 (2002).
[CrossRef]

Roman, P.

P. Roman, Advanced Quantum Theory (Addison-Wesley, 1965), p. 310.

Sarafyan, D.

D. Sarafyan, "Approximate solution of ordinary differential equations and their systems through discrete and continuous embedded Runge-Kutta formulae and upgrading of their order," Comput. Math. Appl. 28, 353-384 (1992).
[CrossRef]

Schubert, M.

Simmons, J. H.

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

Tiwald, T. E.

Volakis, J. L.

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

Whittaker, D. M.

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

Wolf, E.

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

Woollam, J. A.

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).

Zak, J.

J. Zak, E. R. Moog, C. Liu, and S. D. Bader, "Universal approach to magneto-optics," J. Magn. Magn. Mater. 89, 107-123 (1990).
[CrossRef]

Appl. Opt. (2)

Comput. Math. Appl. (1)

D. Sarafyan, "Approximate solution of ordinary differential equations and their systems through discrete and continuous embedded Runge-Kutta formulae and upgrading of their order," Comput. Math. Appl. 28, 353-384 (1992).
[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 Trans. Microwave Theory Tech. (2)

Z. Menachem and E. Jerby, "Matrix transfer function (MTF) for wave propagation in dielectric waveguides with arbitrary transverse profiles," IEEE Trans. Microwave Theory Tech. 46, 975-982 (1998).
[CrossRef]

J. S. Bagby, D. Nyquist, and B. C. Drachman, "Integral formulation for analysis of integrated dielectric waveguides," IEEE Trans. Microwave Theory Tech. 33, 906-915 (1985).
[CrossRef]

J. Appl. Phys. (1)

M. Mansuripur, "Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2×2 matrices," J. Appl. Phys. 67, 6466-6475 (1990).
[CrossRef]

J. Comput. Appl. Math. (1)

J. C. Butcher, "Numerical methods for ordinary differential equations in the 20th century," J. Comput. Appl. Math. 125, 1-29 (2000).
[CrossRef]

J. Lightwave Technol. (1)

J. Magn. Magn. Mater. (1)

J. Zak, E. R. Moog, C. Liu, and S. D. Bader, "Universal approach to magneto-optics," J. Magn. Magn. Mater. 89, 107-123 (1990).
[CrossRef]

J. Mod. Opt. (1)

D. J. De Smet, "An application of the 4×4 matrix formalism to the polar magnetic Kerr effect," J. Mod. Opt. 39, 1055-1065 (1992).
[CrossRef]

J. Opt. A (6)

A. Lakhtakia, "Comment on 'Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrices'," J. Opt. A 5, 432-433 (2003).
[CrossRef]

S. Khorasani, "Reply to Comment on 'Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrices'," J. Opt. A 5, 434-435 (2003).
[CrossRef]

S. Khorasani and B. Rashidian, "Modified transfer matrices for conducting interfaces," J. Opt. A 4, 251-256 (2002).
[CrossRef]

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

I. Abdulhalim, "Exact 2×2 matrix method for the transmission and reflection at the interface between two arbitrarily oriented biaxial crystals," J. Opt. A 1, 655-661 (1999).
[CrossRef]

I. Abdulhalim, "Analytic propagation matrix method for anisotropic magneto-optic layered media," J. Opt. A 2, 557-564 (2000).
[CrossRef]

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

I. Abdulhalim, "Analytic propagation matrix method for linear optics of arbitrary biaxial layered media," J. Opt. A Pure Appl. Opt. 1, 646-653 (1999).
[CrossRef]

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

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

Math. Comput. (1)

W. Kahan and R. Li, "Composition constants for rising the orders of unconventional schemes for ordinary differential equations," Math. Comput. 66, 1089-1099 (1997).
[CrossRef]

Opt. Commun. (4)

S. Khorasani and A. Adibi, "New analytical approach for computation of band structure in one-dimensional periodic media," Opt. Commun. 216, 439-451 (2003).
[CrossRef]

I. Abdulhalim, "2×2 matrix summation method for multiple reflections and transmissions in a biaxial slab between two anisotropic media," Opt. Commun. 163, 9-14 (1999).
[CrossRef]

I. Abdulhalim, "Analytic formulae for the refractive indices and the propagation angles in biaxial and gyrotropic media," Opt. Commun. 157, 265-272 (1998).
[CrossRef]

I. Abdulhalim, "Omnidirectional reflection from anisotropic periodic dielectric stack," Opt. Commun. 174, 43-50 (2000).
[CrossRef]

Opt. Lasers Eng. (1)

L. Adamovics and V. Q. Nguyen, "Electromagnetic field in a slab of photonic crystal by BPM," Opt. Lasers Eng. 35, 67-78 (2001).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

F. J. Dyson, "The S matrix in quantum electrodynamics," Phys. Rev. 75, 1736-1755 (1949).
[CrossRef]

Phys. Rev. B (1)

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

Other (9)

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

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

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

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis (Wiley, 2001).
[CrossRef]

P. Roman, Advanced Quantum Theory (Addison-Wesley, 1965), p. 310.

J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, 1965), p. 177.

J. Cronin, Differential Equations, 2nd ed. (Dekker, 1994), p. 73.

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

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

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Figures (6)

Fig. 1
Fig. 1

Illustration of the inhomogeneous refractive index profile used in examples.

Fig. 2
Fig. 2

Reflection coefficient from the single inhomogeneous layer having the refractive index profile of (38) with n 0 , n s , and m being equal to 2.8, 3.5, and 1, respectively.

Fig. 3
Fig. 3

Reflection coefficient from the single inhomogeneous layer having the refractive index profile of (38) with n 0 , n s , and m being equal to 2.8, 3.5, and 4, respectively.

Fig. 4
Fig. 4

Reflection coefficient from the grating obtained by reproduction of an inhomogeneous unit cell having the refractive index of (38) with n 0 = 2.8 , n s = 3.5 , and m = 1 , to 50 times.

Fig. 5
Fig. 5

Band structure of an infinite grating composed of an inhomogeneous unit cell having the refractive index profile of (38) with n 0 , n s , and m being equal to 2.8, 3.5, and 4, respectively.

Fig. 6
Fig. 6

Normalized group velocity of electromagnetic waves propagating in an infinite grating composed of an inhomogeneous unit cell having the refractive index profile of (38) with n 0 , n s , and m being equal to 2.8, 3.5, and 4, respectively.

Equations (48)

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d 2 A ( x ) d x 2 d ln [ μ ( x ) ] d x d A ( x ) d x + k x 2 ( x ) A ( x ) = 0 .
A ( x ) = A + ( x ) exp [ j k x ( x ) x ] + A ( x ) exp [ j k x ( x ) x ] .
A ( x ) = A + ( x ) exp [ j 0 x k x ( x ) d x ] + A ( x ) exp [ j 0 x k x ( x ) d x ] ,
A + ( x ) exp [ j 0 x k x ( x ) d x ] + A ( x ) exp [ + j 0 x k x ( x ) d x ] = A + ( x + Δ x ) exp [ j 0 x k x ( x ) d x ] + A ( x + Δ x ) exp [ + j 0 x k x ( x ) d x ] .
d A + ( x ) d x exp [ j 0 x k x ( x ) d x ] + d A ( x ) d x exp [ + j 0 x k x ( x ) d x ] = 0 .
[ ϵ ( x ) μ ( x ) ] 1 2 k x ( x ) k 0 n ( x ) A + ( x ) exp [ j 0 x k x ( x ) d x ] [ ϵ ( x ) μ ( x ) ] 1 2 k x ( x ) k 0 n ( x ) A ( x ) exp [ + j 0 x k x ( x ) d x ] = [ ϵ ( x + Δ x ) μ ( x + Δ x ) ] 1 2 k x ( x + Δ x ) k 0 n ( x + Δ x ) A + ( x + Δ x ) exp [ j 0 x k x ( x ) d x ] [ ϵ ( x + Δ x ) μ ( x + Δ x ) ] 1 2 k x ( x + Δ x ) k 0 n ( x + Δ x ) A ( x + Δ x ) exp [ + j 0 x k x ( x ) d x ] ,
d [ A + ( x ) k x ( x ) μ ( x ) ] d x exp [ j 0 x k x ( x ) d x ] d [ A ( x ) k x ( x ) μ ( x ) ] d x × exp [ + j 0 x k x ( x ) d x ] = 0 .
d d x [ A + ( x ) A ( x ) ] = U TE ( x ) [ A + ( x ) A ( x ) ] ,
f TE [ x ] = k x ( x ) 2 k x ( x ) μ ( x ) 2 μ ( x ) ,
Φ ( x ) = 0 x k x ( x ) d x ,
U TE ( x ) = f TE ( x ) { 1 exp [ + j 2 Φ ( x ) ] exp [ j 2 Φ ( x ) ] 1 } ,
U TE ( x ) = k x ( x ) 2 k x ( x ) { 1 exp [ + j 2 0 x k x ( x ) d x ] exp [ j 2 0 x k x ( x ) d x ] 1 } .
d 2 A ( x ) d x 2 d ln [ ϵ ( x ) ] d x d A ( x ) d x + k x 2 ( x ) A ( x ) = 0 .
d d x [ A + ( x ) A ( x ) ] = U TM ( x ) [ A + ( x ) A ( x ) ] .
f TM ( x ) = k x ( x ) 2 k x ( x ) ϵ ( x ) 2 ϵ ( x ) ,
Φ ( x ) = 0 x k x ( x ) d x ,
U TM ( x ) = f TM ( x ) { 1 exp [ + j 2 Φ ( x ) ] exp [ j 2 Φ ( x ) ] 1 } ,
U TM ( x ) = [ k x ( x ) 2 k x ( x ) n ( x ) n ( x ) ] { 1 exp [ + j 2 0 x k x ( x ) d x ] exp [ j 2 0 x k x ( x ) d x ] 1 } ,
[ A + ( x 2 ) A ( x 2 ) ] = Q x 1 x 2 [ A + ( x 1 ) A ( x 1 ) ] ,
Q x 1 x 2 = exp [ x 1 x 2 U ( x ) d x ] = exp ( M ) ,
exp ( M ) = I + n = 1 1 n ! M n ,
Q x 1 x 1 = I ,
Q x 1 x 2 = Q x 2 x 1 1 ,
det ( Q x 1 x 2 ) = k x ( x 1 ) k x ( x 2 ) .
Q x 1 x 3 = exp ( M x 2 x 3 + M x 1 x 2 ) = exp ( M x 2 x 3 ) exp ( M x 1 x 2 ) = Q x 2 x 3 Q x 1 x 2 .
k x ( x ) = { k 1 x < L k 2 x > L } .
Q x 1 x 2 = { I x 1 < x 2 < L Q 1 2 x 1 < L < x 2 I L < x 1 < x 2 } ,
M = { k 1 k 2 d k x 2 k x k 1 k 2 exp [ + j 2 0 L k x ( x ) d x ] d k x 2 k x k 1 k 2 exp [ j 2 0 L k x ( x ) d x ] d k x 2 k x k 1 k 2 d k x 2 k x } .
M = 1 2 ln ( k 1 k 2 ) ( 1 1 1 1 ) .
M n = ln n 1 ( k 1 k 2 ) M ,
exp ( M ) = I + 1 2 n = 1 1 n ! ln n ( k 1 k 2 ) ( 1 1 1 1 ) = I + 1 2 { exp [ ln ( k 1 k 2 ) ] 1 } ( 1 1 1 1 ) = ( k 2 + k 1 2 k 2 k 2 k 1 2 k 2 k 2 k 1 2 k 2 k 2 + k 1 2 k 2 ) .
R r = A 1 A 1 + = q 21 q 22 ,
T r = A 2 + A 1 + = Δ q 22 ,
R l = A 2 + A 2 = q 12 q 22 ,
T l = A 1 A 2 = 1 q 22 ,
Ψ ( x ) = Φ κ ( x ) exp ( j κ x ) .
Φ κ ( x ) = Φ κ ( x + L ) .
[ A + ( x + L ) A ( x + L ) ] = exp ( j κ L ) [ A + ( x ) A ( x ) ] .
[ A + ( x + L ) A ( x + L ) ] = Q x x + L [ A + ( x ) A ( x ) ] .
I Q x x + L exp ( j κ L ) = 0 .
q 22 exp ( j κ L ) q 11 exp ( j κ L ) + 1 + ( q 11 q 22 q 21 q 12 ) exp ( j 2 κ L ) = 0 ,
cos ( κ L ) = 1 2 ( q 11 + q 22 ) .
m 11 = m 22 = 0 ,
m 12 = m 21 = 2 j 0 L 2 sin [ 2 0 x k x ( x ) d x ] 2 k x ( x ) k x ( x ) d x ,
Q L 2 L 2 = exp ( M ) = exp [ ( 0 j m j m 0 ) ] = [ cosh ( m ) j sinh ( m ) j sinh ( m ) cosh ( m ) ] .
m j m 12 = j m 21 = 2 0 L 2 sin [ 2 0 x k x ( x ) d x ] k x ( x ) k x ( x ) d x .
m = 0 L 2 [ k x ( x ) k x ( x ) 2 n ( x ) n ( x ) ] sin [ 2 0 x k x ( x ) d x ] d x .
n ( x ) = n 0 exp [ ( x L ) m ln ( n s n 0 ) ] .

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