Abstract

A band structure of one-dimensional periodic media having an arbitrary inhomogeneous refractive index profile is extracted based on the modification committed on the differential-transfer-matrix method (DTMM). Furthermore, the most recently modified differential transfer matrix is improved by reshaping the formulation in terms of which the electromagnetic fields are expanded and closed form formulas, providing the allowed values of Bloch wavenumbers, which were not available before. Although the frequency gaps in previously published results that we derived by using the conventional DTMM were not in agreement with the well-known Bragg law at the edge of each Brillouin zone, the new results obtained by the proposed method are now matched with the Bragg condition. The final results are also justified by either employing conventional transfer-matrix method or comparing it with exact analytical solutions, wherever such exact solutions were available.

© 2006 Optical Society of America

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  1. P. Yeh, Optical Waves in Layered Media (Wiley,1988).
  2. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).
  3. D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, "All-dielectric one-dimensional periodic structures for total omnidirectional reflection and partial spontaneous emission control," J. Lumin. 17, 2018-2024 (1999).
  4. P. St. J. Russell, T. A. Birks, F. D. Llyold-Lucas, "Photonic Bloch waves and photonic band gaps," in Confined Electron and Photon: New Physics and Applications, E.Burstein and C.Weisbuch, eds. (Plenum, 1995).
    [CrossRef]
  5. 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]
  6. I. Abdulhalim, "Analytic propagation matrix method for anisotropic magneto-optic layered media," J. Opt. A Pure Appl. Opt. 2, 557-564 (2000).
    [CrossRef]
  7. I. Abdulhalim, "Omnidirectional reflection from anisotropic periodic dielectric stack," Opt. Commun. 174, 43-50 (2000).
    [CrossRef]
  8. D. Lusk, I. Abdulhalim, and F. Placido, "Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal," Opt. Commun. 198, 273-279 (2001).
    [CrossRef]
  9. D.-Y. Jeong, Y. H. Ye, and Q. M. Zhang, "Effective optical properties associated with wave propagation in photonic crystals of finite length along the propagation direction," J. Appl. Phys. 92, 4194-4200 (2002).
    [CrossRef]
  10. D. Felbacq, B. Guizal, and F. Zolla, "Wave propagation in one-dimensional photonic crystals," Opt. Commun. 152, 119-126 (1998).
    [CrossRef]
  11. P. K. Kelly and M. Piket-May, "Propagation characteristics for a one-dimensional grounded finite height finite length electromagnetic crystal," J. Lumin. 17, 2008-2012 (1999).
  12. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2000).
  13. S. Khorasani and B. Rashidian, "Modified transfer matrix method for conducting interfaces," J. Opt. A Pure Appl. Opt. 4, 251-256 (2002).
    [CrossRef]
  14. K. Mehrany, B. Momeni, S. Khorasani, and B. Rashidian, "Interface electromagnetic waves between Kronig-Penney photonic crystals," Proc. SPIE 4833, pp. 535-541 (2002).
    [CrossRef]
  15. K. Mehrany and B. Rashidian, "Band structures of coupled electromagnetic slow waves," J. Opt. A Pure Appl. Opt. 6, 937-942 (2004).
    [CrossRef]
  16. E.Wolf, ed., Progress in Optics, Volume XXXIV (Elsevier, 1995).
  17. M. R. Chamanzar, K. Mehrany, and B. Rashidian, "Legendre polynomial expansion for analysis of linear one-dimensional inhomogeneous optical structures and photonic crystals," J. Opt. Soc. Am. B (to be published).
  18. 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]
  19. K. Mehrany and S. Khorasani, "Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrix method," J. Opt. A Pure Appl. Opt. 4, 624-635 (2002).
    [CrossRef]
  20. S. Khorasani and A. Adibi, "New analytical approach for computation of band structures in one-dimensional periodic media," Opt. Commun. 216, 439-451 (2003).
    [CrossRef]
  21. J. N. Mait, "Understanding diffractive optic design in the scalar domain," J. Opt. Soc. Am. A 12, 2145-2158 (1995).
    [CrossRef]
  22. S. Martellucci and A. N. Chester, Diffractive Optics and Optical Microsystems (Springer, 1997).
  23. 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).
  24. 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).
    [CrossRef]
  25. 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]
  26. J. C. Butcher, "Numerical methods for ordinary differential equations in the 20th century," J. Comput. Appl. Math. 125, 1-29 (2000).
    [CrossRef]
  27. W. Kahan, R. Li, "Composition constants for rising the orders of unconventional schemes for ordinary differential equations," Math. Comput. 66, 1089-1099 (1997).
    [CrossRef]
  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. F. J. Dyson, "The S matrix in quantum electrodynamics," Phase Transitions 75, 1736-1755 (1949).
  30. P. Roman, Advanced Quantum Theory (Addison-Wesley, 1965), p. 310.
  31. J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, 1965), p. 177.
  32. K. 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]
  33. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

2005 (1)

2004 (1)

K. Mehrany and B. Rashidian, "Band structures of coupled electromagnetic slow waves," J. Opt. A Pure Appl. Opt. 6, 937-942 (2004).
[CrossRef]

2003 (5)

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

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

K. 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 (4)

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

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

K. Mehrany, B. Momeni, S. Khorasani, and B. Rashidian, "Interface electromagnetic waves between Kronig-Penney photonic crystals," Proc. SPIE 4833, pp. 535-541 (2002).
[CrossRef]

D.-Y. Jeong, Y. H. Ye, and Q. M. Zhang, "Effective optical properties associated with wave propagation in photonic crystals of finite length along the propagation direction," J. Appl. Phys. 92, 4194-4200 (2002).
[CrossRef]

2001 (1)

D. Lusk, I. Abdulhalim, and F. Placido, "Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal," Opt. Commun. 198, 273-279 (2001).
[CrossRef]

2000 (3)

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

I. Abdulhalim, "Omnidirectional reflection from anisotropic periodic dielectric stack," Opt. Commun. 174, 43-50 (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 (2)

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, "All-dielectric one-dimensional periodic structures for total omnidirectional reflection and partial spontaneous emission control," J. Lumin. 17, 2018-2024 (1999).

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

1998 (1)

D. Felbacq, B. Guizal, and F. Zolla, "Wave propagation in one-dimensional photonic crystals," Opt. Commun. 152, 119-126 (1998).
[CrossRef]

1997 (1)

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

1995 (1)

1992 (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]

1990 (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]

1949 (1)

F. J. Dyson, "The S matrix in quantum electrodynamics," Phase Transitions 75, 1736-1755 (1949).

Abdulhalim, I.

D. Lusk, I. Abdulhalim, and F. Placido, "Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal," Opt. Commun. 198, 273-279 (2001).
[CrossRef]

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

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

Adibi, A.

S. Khorasani and A. Adibi, "New analytical approach for computation of band structures 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).

Birks, T. A.

P. St. J. Russell, T. A. Birks, F. D. Llyold-Lucas, "Photonic Bloch waves and photonic band gaps," in Confined Electron and Photon: New Physics and Applications, E.Burstein and C.Weisbuch, eds. (Plenum, 1995).
[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).

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]

Chamanzar, M. R.

M. R. Chamanzar, K. Mehrany, and B. Rashidian, "Legendre polynomial expansion for analysis of linear one-dimensional inhomogeneous optical structures and photonic crystals," J. Opt. Soc. Am. B (to be published).

Chester, A. N.

S. Martellucci and A. N. Chester, Diffractive Optics and Optical Microsystems (Springer, 1997).

Chigrin, D. N.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, "All-dielectric one-dimensional periodic structures for total omnidirectional reflection and partial spontaneous emission control," J. Lumin. 17, 2018-2024 (1999).

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," Phase Transitions 75, 1736-1755 (1949).

Eghlidi, M. H.

Felbacq, D.

D. Felbacq, B. Guizal, and F. Zolla, "Wave propagation in one-dimensional photonic crystals," Opt. Commun. 152, 119-126 (1998).
[CrossRef]

Gaponenko, S. V.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, "All-dielectric one-dimensional periodic structures for total omnidirectional reflection and partial spontaneous emission control," J. Lumin. 17, 2018-2024 (1999).

Guizal, B.

D. Felbacq, B. Guizal, and F. Zolla, "Wave propagation in one-dimensional photonic crystals," Opt. Commun. 152, 119-126 (1998).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2000).

Jeong, D.-Y.

D.-Y. Jeong, Y. H. Ye, and Q. M. Zhang, "Effective optical properties associated with wave propagation in photonic crystals of finite length along the propagation direction," J. Appl. Phys. 92, 4194-4200 (2002).
[CrossRef]

Kahan, W.

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

Kelly, P. K.

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

Khorasani, K.

K. 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]

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

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

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

K. Mehrany, B. Momeni, S. Khorasani, and B. Rashidian, "Interface electromagnetic waves between Kronig-Penney photonic crystals," Proc. SPIE 4833, pp. 535-541 (2002).
[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]

Lavrinenko, A. V.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, "All-dielectric one-dimensional periodic structures for total omnidirectional reflection and partial spontaneous emission control," J. Lumin. 17, 2018-2024 (1999).

Li, R.

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

Llyold-Lucas, F. D.

P. St. J. Russell, T. A. Birks, F. D. Llyold-Lucas, "Photonic Bloch waves and photonic band gaps," in Confined Electron and Photon: New Physics and Applications, E.Burstein and C.Weisbuch, eds. (Plenum, 1995).
[CrossRef]

Lusk, D.

D. Lusk, I. Abdulhalim, and F. Placido, "Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal," Opt. Commun. 198, 273-279 (2001).
[CrossRef]

Mait, J. N.

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]

Martellucci, S.

S. Martellucci and A. N. Chester, Diffractive Optics and Optical Microsystems (Springer, 1997).

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).
[CrossRef]

K. Mehrany and B. Rashidian, "Band structures of coupled electromagnetic slow waves," J. Opt. A Pure Appl. Opt. 6, 937-942 (2004).
[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 Pure Appl. Opt. 4, 624-635 (2002).
[CrossRef]

K. Mehrany, B. Momeni, S. Khorasani, and B. Rashidian, "Interface electromagnetic waves between Kronig-Penney photonic crystals," Proc. SPIE 4833, pp. 535-541 (2002).
[CrossRef]

M. R. Chamanzar, K. Mehrany, and B. Rashidian, "Legendre polynomial expansion for analysis of linear one-dimensional inhomogeneous optical structures and photonic crystals," J. Opt. Soc. Am. B (to be published).

Momeni, B.

K. Mehrany, B. Momeni, S. Khorasani, and B. Rashidian, "Interface electromagnetic waves between Kronig-Penney photonic crystals," Proc. SPIE 4833, pp. 535-541 (2002).
[CrossRef]

Piket-May, M.

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

Placido, F.

D. Lusk, I. Abdulhalim, and F. Placido, "Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal," Opt. Commun. 198, 273-279 (2001).
[CrossRef]

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).
[CrossRef]

K. Mehrany and B. Rashidian, "Band structures of coupled electromagnetic slow waves," J. Opt. A Pure Appl. Opt. 6, 937-942 (2004).
[CrossRef]

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

K. Mehrany, B. Momeni, S. Khorasani, and B. Rashidian, "Interface electromagnetic waves between Kronig-Penney photonic crystals," Proc. SPIE 4833, pp. 535-541 (2002).
[CrossRef]

M. R. Chamanzar, K. Mehrany, and B. Rashidian, "Legendre polynomial expansion for analysis of linear one-dimensional inhomogeneous optical structures and photonic crystals," J. Opt. Soc. Am. B (to be published).

Roman, P.

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

Russell, P. St. J.

P. St. J. Russell, T. A. Birks, F. D. Llyold-Lucas, "Photonic Bloch waves and photonic band gaps," in Confined Electron and Photon: New Physics and Applications, E.Burstein and C.Weisbuch, eds. (Plenum, 1995).
[CrossRef]

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]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2000).

Wolf, E.

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

Yariv, A.

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

Yarotsky, D. A.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, "All-dielectric one-dimensional periodic structures for total omnidirectional reflection and partial spontaneous emission control," J. Lumin. 17, 2018-2024 (1999).

Ye, Y. H.

D.-Y. Jeong, Y. H. Ye, and Q. M. Zhang, "Effective optical properties associated with wave propagation in photonic crystals of finite length along the propagation direction," J. Appl. Phys. 92, 4194-4200 (2002).
[CrossRef]

Yeh, P.

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

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

Zhang, Q. M.

D.-Y. Jeong, Y. H. Ye, and Q. M. Zhang, "Effective optical properties associated with wave propagation in photonic crystals of finite length along the propagation direction," J. Appl. Phys. 92, 4194-4200 (2002).
[CrossRef]

Zolla, F.

D. Felbacq, B. Guizal, and F. Zolla, "Wave propagation in one-dimensional photonic crystals," Opt. Commun. 152, 119-126 (1998).
[CrossRef]

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).

J. Appl. Phys. (2)

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]

D.-Y. Jeong, Y. H. Ye, and Q. M. Zhang, "Effective optical properties associated with wave propagation in photonic crystals of finite length along the propagation direction," J. Appl. Phys. 92, 4194-4200 (2002).
[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. Lumin. (2)

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

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, "All-dielectric one-dimensional periodic structures for total omnidirectional reflection and partial spontaneous emission control," J. Lumin. 17, 2018-2024 (1999).

J. Opt. A (2)

K. 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]

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]

J. Opt. A Pure Appl. Opt. (4)

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

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

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

K. Mehrany and B. Rashidian, "Band structures of coupled electromagnetic slow waves," J. Opt. A Pure Appl. Opt. 6, 937-942 (2004).
[CrossRef]

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

J. Opt. Soc. Am. B (2)

Math. Comput. (1)

W. Kahan, 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 structures in one-dimensional periodic media," Opt. Commun. 216, 439-451 (2003).
[CrossRef]

D. Felbacq, B. Guizal, and F. Zolla, "Wave propagation in one-dimensional photonic crystals," Opt. Commun. 152, 119-126 (1998).
[CrossRef]

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

D. Lusk, I. Abdulhalim, and F. Placido, "Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal," Opt. Commun. 198, 273-279 (2001).
[CrossRef]

Phase Transitions (1)

F. J. Dyson, "The S matrix in quantum electrodynamics," Phase Transitions 75, 1736-1755 (1949).

Proc. SPIE (1)

K. Mehrany, B. Momeni, S. Khorasani, and B. Rashidian, "Interface electromagnetic waves between Kronig-Penney photonic crystals," Proc. SPIE 4833, pp. 535-541 (2002).
[CrossRef]

Other (10)

E.Wolf, ed., Progress in Optics, Volume XXXIV (Elsevier, 1995).

M. R. Chamanzar, K. Mehrany, and B. Rashidian, "Legendre polynomial expansion for analysis of linear one-dimensional inhomogeneous optical structures and photonic crystals," J. Opt. Soc. Am. B (to be published).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2000).

P. St. J. Russell, T. A. Birks, F. D. Llyold-Lucas, "Photonic Bloch waves and photonic band gaps," in Confined Electron and Photon: New Physics and Applications, E.Burstein and C.Weisbuch, eds. (Plenum, 1995).
[CrossRef]

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

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

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.

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

S. Martellucci and A. N. Chester, Diffractive Optics and Optical Microsystems (Springer, 1997).

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Figures (9)

Fig. 1
Fig. 1

Illustration of a one-dimensional, lossless, and nonmagnetic inhomogeneous medium.

Fig. 2
Fig. 2

Band structure of the photonic crystal, with the unit cell shown in the inset. The angle of incidence is zero. Inset, illustration of the sawtooth profile of refractive index used in this example.

Fig. 3
Fig. 3

Illustration of the exponential profile of refractive index used in examples.

Fig. 4
Fig. 4

Band structure of the exponential grating obtained by reproduction of the unit cell in Fig. 3: (a) TE, n 0 = 2.8 , n s = 3.5 ; (b) TM, n 0 = 2.8 , n s = 3.5 ; (c) TE, n 0 = 1 , n s = 4 ; (d) TM, n 0 = 1 , n s = 4 . The angle of incidence is π 4 , and m = 1 .

Fig. 5
Fig. 5

Band structure of the exponential grating obtained by reproduction of the unit cell in Fig. 3: (a) TE, n 0 = 2.8 , n s = 3.5 ; (b) TM, n 0 = 2.8 , n s = 3.5 ; (c) TE, n 0 = 1 , n s = 4 ; (d) TM, n 0 = 1 , n s = 4 . The angle of incidence is π 4 , and m = 4 .

Fig. 6
Fig. 6

Band structure of the exponential grating at two different regimes obtained by reproduction of the unit cell in Fig. 3: n 0 = 1 and n s = 4 . The angle of incidence is zero, and m = 1 .

Fig. 7
Fig. 7

Band structure of the corresponding grating obtained by reproduction of the unit cell shown in the inset. The angle of incidence is zero. Inset, illustration of the chirped refractive index profile used in this example.

Fig. 8
Fig. 8

Band structure of sinusoidal grating obtained by reproduction of the unit cell in the inset. The angle of incidence is π 6 . Inset, illustration of the even symmetrical sinusoidal refractive index profile used as a numerical example verifying the accuracy of Eq. (31).

Fig. 9
Fig. 9

Normalized group velocity corresponding to the case of Fig. 5d.

Tables (1)

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Table 1 Parameters Needed for Checking the Bragg Condition in Figs. 2, 4, 5

Equations (41)

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d 2 A ( x ) d x 2 + k x 2 ( k ) 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 ] ,
d d x [ A + ( x ) A ( x ) ] = U TE ( x ) [ A + ( x ) A ( x ) ] ,
f TE ( x ) = k x ( x ) [ 2 k x ( 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 ] ,
A ( x ) = A + ( x ) exp [ j 0 x k x ( x ) d x ] k x ( x ) + A ( x ) exp [ j 0 x k x ( x ) d x ] k x ( x ) ,
U TE ( x ) = f TE ( x ) { 0 exp [ + j 2 ϕ ( x ) ] exp [ j 2 ϕ ( x ) ] 0 } .
d 2 A ( x ) d x 2 2 d ln [ n ( x ) ] d x dA ( 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 ) ] n ( x ) n ( 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 } ,
A ( x ) = A + ( x ) n ( x ) exp [ j 0 x k x ( x ) d x ] k x ( x ) + A ( x ) n ( x ) exp [ j 0 x k x ( x ) d x ] k x ( x ) .
U TM ( x ) = f TM ( x ) { 0 exp [ + j 2 ϕ ( x ) ] exp [ j 2 ϕ ( x ) ] 0 } ,
[ 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 0 x = exp { [ 0 m 12 ( x ) m 21 ( x ) 0 ] } = { cosh [ m 12 ( x ) m 21 ( x ) ] m 12 ( x ) sinh [ m 12 ( x ) m 21 ( x ) ] m 21 ( x ) m 12 ( x ) sinh [ m 12 ( x ) m 21 ( x ) ] m 21 ( x ) cosh [ m 12 ( x ) m 21 ( x ) ] } ,
m 12 ( x ) = x 0 x f ( x ) exp [ + j 2 ϕ ( x ) ] d x ,
m 21 ( x ) = x 0 x f ( x ) exp [ j 2 ϕ ( x ) ] d x ,
Q x 1 x 1 = I ,
Q x 2 x 1 = Q x 1 x 2 1 ,
Q x 1 x 2 = c ,
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 .
A ( 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 ) .
cos ( κ L ) = 1 2 ( f 1 f 2 + f 2 f 1 ) cosh [ m 12 ( L ) m 21 ( L ) ] cos [ ϕ ( L ) ] + 1 2 ( f 1 f 2 f 2 f 1 ) sinh [ m 12 ( L ) m 21 ( L ) ] cos [ ϕ ( L ) + ψ ( L ) ] ,
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 = 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 .
cos [ κ L ] = cos { 2 π ( L λ ) [ 1 L L 2 + L 2 n 2 ( x ) N 2 d x ] } cosh [ m ] ,
n ( x ) = n 0 exp [ ( x L ) m ln ( n s n 0 ) ] ,

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