Abstract

A rigorous analytical approach for investigating a stratified medium with an arbitrary finite number of homogeneous isotropic layers in a period is developed. The approach is based on the translation matrix method. It is well known that the translation matrix for a period must be found as the product of the layer matrices. It is proved that this matrix can be represented as a finite sum of trigonometric matrices, and thus the dispersion relation of a stratified medium is written in an analytical form. All final expressions are obtained in terms of the constitutive parameters. To this author’s knowledge, this is the first time that the new sign function that allows us to develop the presented analytical results has been described. The condition of the existence of a wave with an arbitrary period divisible by a structure period is found in analytical form. It is proved that changing the layer arrangement within the period does not affect the structure of the transmission and absorption bands.

© 2005 Optical Society of America

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References

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  1. C. Elachi, “Wave in active and passive periodic structures: a review,” Proc. IEEE 64, 1666–1698 (1976).
    [CrossRef]
  2. P. Yeh, A. Yariv, C. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977).
    [CrossRef]
  3. P. Yeh, A. Yariv, C. Hong, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am. 67, 438–448 (1977).
    [CrossRef]
  4. M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965).
  5. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  6. T. Tamir, “Scattering of electromagnetic waves by a sinusoidally stratified half-space: part II,” Can. J. Phys. 44, 2461–2494 (1966).
    [CrossRef]
  7. G. A. Gevorkyan, “On the theory of propagation of electromagnetic waves in a waveguide with a multiperiodically modulated dielectric filling,” Physica A 241, 236–239 (1997).
    [CrossRef]
  8. P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
    [CrossRef]
  9. S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
    [CrossRef]
  10. A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from nonplanar periodic surfaces using a strip current model,” IEEE Trans. Antennas Propag. 37, 1437–1451 (1989).
    [CrossRef]
  11. J. J. Pesque, D. P. Bouche, R. Mittra, “Optimization of multilayer antireflection coatings using an optimal control method,” IEEE Trans. Microwave Theory Tech. 40, 1789–1796 (1992).
    [CrossRef]
  12. G. A. Ybara, S. M. Wu, G. L. Bilbro, S. H. Ardalan, C. P. Hearn, R. T. Neece, “Optimal signal prossing of frequency-stepped SW radar data,” IEEE Trans. Microwave Theory Tech. 43, 94–105 (1995).
    [CrossRef]
  13. L. Brillouin, M. Parodi, Propagation des Ondes dans les Milieux Périodiques (Masson et Cie, Editeurs, Paris, 1956).
  14. N. Blombergen, A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–485 (1970).
    [CrossRef]
  15. A. A. Bulgakov, S. A. Bulgakov, M. Nieto-Vesperinas, “Inhomogeneous waves and energy localization in dielectric superlattices,” Phys. Rev. B 58, 4438–4448 (1998).
    [CrossRef]
  16. F. Abelès, “Recherches sur la propagation des ondes electromagnétiques sinusoidales dans les milieux stratifiés. Application aux conches minces,” Ann. Phys. (Paris) 5, 596–640, 706–782 (1950).
  17. G. Gambill, “Criteria for parametric instability for linear differential systems with periodic coefficients,” Riv. Mat. Univ. Parma 6, 37–43 (1955).
  18. J. K. Hale, Oscillations in Nonlinear Systems (McGraw-Hill, New York, 1963).
  19. A. H. Nayfer, Introduction to Pertrubation Techniques (Wiley, New York, 1981).
  20. L. R. Lewis, A. Hessel, “Propagation characteristics of a periodic array of dielectric slabs,” IEEE Trans. Microwave Theory Tech. 19, 276–286 (1971).
    [CrossRef]
  21. J. C. W. A. Costa, A. J. Giarola, “Electromagnetic wave propagation in multilayered dielectric periodic structures,” IEEE Trans. Antennas Propag. 41, 1432–1438 (1993).
    [CrossRef]
  22. J. W. Strutt, Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philos. Mag. 24, 145–159 (1887).
    [CrossRef]
  23. F. Bloch, “Quantenmechanik der elektronen in kristallgittern,” Z. Phys. 52, 555–500 (1928).
    [CrossRef]
  24. J. R. Pierce, “Coupling of modes of propagation,” J. Appl. Phys. 25, 179–183 (1954).
    [CrossRef]
  25. D. L. Jaggard, C. Elachi, “Floquet and coupled-waves analysis of higher-order Bragg coupling in a periodic medium,” J. Opt. Soc. Am. 66, 674–682 (1976).
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    [CrossRef]
  27. S. Teitler, B. W. Henvis, “Refraction in stratified, anisotropic media,” J. Opt. Soc. Am. 60, 830–834 (1970).
    [CrossRef]
  28. D. Ager, H. P. Hughes, “Optical properties of stratified systems including lamellar gratings,” Phys. Rev. B 44, 13452–13465 (1991).
    [CrossRef]
  29. W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990), 136–140.
  30. K. Vytovtov, “A model of a two-dimensional linear parametric system with a step pumping,” in Proceedings XXXII Sympozjon PTMTS Modelowanie w Mechanice (Gliwice, Poland, 1998), pp. 377–380.
  31. Yu. M. Terent’ev, K. A. Vytovtov, “Investigation of wave behavior in periodical magnetized ferrite,” Electromagn. Waves Electron. Syst. 4, 37–42 (1999).
  32. Yu. M. Terent’ev, K. A. Vytovtov, “Transformation matrix of N-layer periodic medium with anisotropic layers,” J. Commun. Technol. Electron. 45, 255–257 (2000).
  33. K. A. Vytovtov, “The analytical method of investigation of periodic layered media with uniaxial bianisotropy,” J. Commun. Technol. Electron. 46, 144–150 (2001).
  34. K. A. Vytovtov, “Propagation conditions of a harmonic waves within bianisotropic periodic layered media,” in Proceedings of 30th European Microwave Conference (Paris, 2000), Vol. 2, pp. 238–241.

2001

K. A. Vytovtov, “The analytical method of investigation of periodic layered media with uniaxial bianisotropy,” J. Commun. Technol. Electron. 46, 144–150 (2001).

2000

Yu. M. Terent’ev, K. A. Vytovtov, “Transformation matrix of N-layer periodic medium with anisotropic layers,” J. Commun. Technol. Electron. 45, 255–257 (2000).

1999

Yu. M. Terent’ev, K. A. Vytovtov, “Investigation of wave behavior in periodical magnetized ferrite,” Electromagn. Waves Electron. Syst. 4, 37–42 (1999).

1998

A. A. Bulgakov, S. A. Bulgakov, M. Nieto-Vesperinas, “Inhomogeneous waves and energy localization in dielectric superlattices,” Phys. Rev. B 58, 4438–4448 (1998).
[CrossRef]

1997

G. A. Gevorkyan, “On the theory of propagation of electromagnetic waves in a waveguide with a multiperiodically modulated dielectric filling,” Physica A 241, 236–239 (1997).
[CrossRef]

1995

G. A. Ybara, S. M. Wu, G. L. Bilbro, S. H. Ardalan, C. P. Hearn, R. T. Neece, “Optimal signal prossing of frequency-stepped SW radar data,” IEEE Trans. Microwave Theory Tech. 43, 94–105 (1995).
[CrossRef]

1993

J. C. W. A. Costa, A. J. Giarola, “Electromagnetic wave propagation in multilayered dielectric periodic structures,” IEEE Trans. Antennas Propag. 41, 1432–1438 (1993).
[CrossRef]

1992

J. J. Pesque, D. P. Bouche, R. Mittra, “Optimization of multilayer antireflection coatings using an optimal control method,” IEEE Trans. Microwave Theory Tech. 40, 1789–1796 (1992).
[CrossRef]

1991

D. Ager, H. P. Hughes, “Optical properties of stratified systems including lamellar gratings,” Phys. Rev. B 44, 13452–13465 (1991).
[CrossRef]

1989

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from nonplanar periodic surfaces using a strip current model,” IEEE Trans. Antennas Propag. 37, 1437–1451 (1989).
[CrossRef]

1982

S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

1977

1976

1975

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[CrossRef]

1972

1971

L. R. Lewis, A. Hessel, “Propagation characteristics of a periodic array of dielectric slabs,” IEEE Trans. Microwave Theory Tech. 19, 276–286 (1971).
[CrossRef]

1970

N. Blombergen, A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–485 (1970).
[CrossRef]

S. Teitler, B. W. Henvis, “Refraction in stratified, anisotropic media,” J. Opt. Soc. Am. 60, 830–834 (1970).
[CrossRef]

1966

T. Tamir, “Scattering of electromagnetic waves by a sinusoidally stratified half-space: part II,” Can. J. Phys. 44, 2461–2494 (1966).
[CrossRef]

1955

G. Gambill, “Criteria for parametric instability for linear differential systems with periodic coefficients,” Riv. Mat. Univ. Parma 6, 37–43 (1955).

1954

J. R. Pierce, “Coupling of modes of propagation,” J. Appl. Phys. 25, 179–183 (1954).
[CrossRef]

1950

F. Abelès, “Recherches sur la propagation des ondes electromagnétiques sinusoidales dans les milieux stratifiés. Application aux conches minces,” Ann. Phys. (Paris) 5, 596–640, 706–782 (1950).

1928

F. Bloch, “Quantenmechanik der elektronen in kristallgittern,” Z. Phys. 52, 555–500 (1928).
[CrossRef]

1887

J. W. Strutt, Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philos. Mag. 24, 145–159 (1887).
[CrossRef]

Abelès, F.

F. Abelès, “Recherches sur la propagation des ondes electromagnétiques sinusoidales dans les milieux stratifiés. Application aux conches minces,” Ann. Phys. (Paris) 5, 596–640, 706–782 (1950).

Ager, D.

D. Ager, H. P. Hughes, “Optical properties of stratified systems including lamellar gratings,” Phys. Rev. B 44, 13452–13465 (1991).
[CrossRef]

Ardalan, S. H.

G. A. Ybara, S. M. Wu, G. L. Bilbro, S. H. Ardalan, C. P. Hearn, R. T. Neece, “Optimal signal prossing of frequency-stepped SW radar data,” IEEE Trans. Microwave Theory Tech. 43, 94–105 (1995).
[CrossRef]

Berreman, D. W.

Bilbro, G. L.

G. A. Ybara, S. M. Wu, G. L. Bilbro, S. H. Ardalan, C. P. Hearn, R. T. Neece, “Optimal signal prossing of frequency-stepped SW radar data,” IEEE Trans. Microwave Theory Tech. 43, 94–105 (1995).
[CrossRef]

Bloch, F.

F. Bloch, “Quantenmechanik der elektronen in kristallgittern,” Z. Phys. 52, 555–500 (1928).
[CrossRef]

Blombergen, N.

N. Blombergen, A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–485 (1970).
[CrossRef]

Boag, A.

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from nonplanar periodic surfaces using a strip current model,” IEEE Trans. Antennas Propag. 37, 1437–1451 (1989).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from nonplanar periodic surfaces using a strip current model,” IEEE Trans. Antennas Propag. 37, 1437–1451 (1989).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965).

Bouche, D. P.

J. J. Pesque, D. P. Bouche, R. Mittra, “Optimization of multilayer antireflection coatings using an optimal control method,” IEEE Trans. Microwave Theory Tech. 40, 1789–1796 (1992).
[CrossRef]

Brillouin, L.

L. Brillouin, M. Parodi, Propagation des Ondes dans les Milieux Périodiques (Masson et Cie, Editeurs, Paris, 1956).

Bulgakov, A. A.

A. A. Bulgakov, S. A. Bulgakov, M. Nieto-Vesperinas, “Inhomogeneous waves and energy localization in dielectric superlattices,” Phys. Rev. B 58, 4438–4448 (1998).
[CrossRef]

Bulgakov, S. A.

A. A. Bulgakov, S. A. Bulgakov, M. Nieto-Vesperinas, “Inhomogeneous waves and energy localization in dielectric superlattices,” Phys. Rev. B 58, 4438–4448 (1998).
[CrossRef]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990), 136–140.

Chuang, S. L.

S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Costa, J. C. W. A.

J. C. W. A. Costa, A. J. Giarola, “Electromagnetic wave propagation in multilayered dielectric periodic structures,” IEEE Trans. Antennas Propag. 41, 1432–1438 (1993).
[CrossRef]

Elachi, C.

Gambill, G.

G. Gambill, “Criteria for parametric instability for linear differential systems with periodic coefficients,” Riv. Mat. Univ. Parma 6, 37–43 (1955).

Gevorkyan, G. A.

G. A. Gevorkyan, “On the theory of propagation of electromagnetic waves in a waveguide with a multiperiodically modulated dielectric filling,” Physica A 241, 236–239 (1997).
[CrossRef]

Giarola, A. J.

J. C. W. A. Costa, A. J. Giarola, “Electromagnetic wave propagation in multilayered dielectric periodic structures,” IEEE Trans. Antennas Propag. 41, 1432–1438 (1993).
[CrossRef]

Hale, J. K.

J. K. Hale, Oscillations in Nonlinear Systems (McGraw-Hill, New York, 1963).

Hearn, C. P.

G. A. Ybara, S. M. Wu, G. L. Bilbro, S. H. Ardalan, C. P. Hearn, R. T. Neece, “Optimal signal prossing of frequency-stepped SW radar data,” IEEE Trans. Microwave Theory Tech. 43, 94–105 (1995).
[CrossRef]

Henvis, B. W.

Hessel, A.

L. R. Lewis, A. Hessel, “Propagation characteristics of a periodic array of dielectric slabs,” IEEE Trans. Microwave Theory Tech. 19, 276–286 (1971).
[CrossRef]

Hong, C.

Hughes, H. P.

D. Ager, H. P. Hughes, “Optical properties of stratified systems including lamellar gratings,” Phys. Rev. B 44, 13452–13465 (1991).
[CrossRef]

Jaggard, D. L.

Kong, J. A.

S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Leviatan, Y.

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from nonplanar periodic surfaces using a strip current model,” IEEE Trans. Antennas Propag. 37, 1437–1451 (1989).
[CrossRef]

Lewis, L. R.

L. R. Lewis, A. Hessel, “Propagation characteristics of a periodic array of dielectric slabs,” IEEE Trans. Microwave Theory Tech. 19, 276–286 (1971).
[CrossRef]

Mittra, R.

J. J. Pesque, D. P. Bouche, R. Mittra, “Optimization of multilayer antireflection coatings using an optimal control method,” IEEE Trans. Microwave Theory Tech. 40, 1789–1796 (1992).
[CrossRef]

Nayfer, A. H.

A. H. Nayfer, Introduction to Pertrubation Techniques (Wiley, New York, 1981).

Neece, R. T.

G. A. Ybara, S. M. Wu, G. L. Bilbro, S. H. Ardalan, C. P. Hearn, R. T. Neece, “Optimal signal prossing of frequency-stepped SW radar data,” IEEE Trans. Microwave Theory Tech. 43, 94–105 (1995).
[CrossRef]

Nieto-Vesperinas, M.

A. A. Bulgakov, S. A. Bulgakov, M. Nieto-Vesperinas, “Inhomogeneous waves and energy localization in dielectric superlattices,” Phys. Rev. B 58, 4438–4448 (1998).
[CrossRef]

Parodi, M.

L. Brillouin, M. Parodi, Propagation des Ondes dans les Milieux Périodiques (Masson et Cie, Editeurs, Paris, 1956).

Pesque, J. J.

J. J. Pesque, D. P. Bouche, R. Mittra, “Optimization of multilayer antireflection coatings using an optimal control method,” IEEE Trans. Microwave Theory Tech. 40, 1789–1796 (1992).
[CrossRef]

Pierce, J. R.

J. R. Pierce, “Coupling of modes of propagation,” J. Appl. Phys. 25, 179–183 (1954).
[CrossRef]

Rayleigh,

J. W. Strutt, Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philos. Mag. 24, 145–159 (1887).
[CrossRef]

Sievers, A. J.

N. Blombergen, A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–485 (1970).
[CrossRef]

Strutt, J. W.

J. W. Strutt, Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philos. Mag. 24, 145–159 (1887).
[CrossRef]

Tamir, T.

T. Tamir, “Scattering of electromagnetic waves by a sinusoidally stratified half-space: part II,” Can. J. Phys. 44, 2461–2494 (1966).
[CrossRef]

Teitler, S.

Terent’ev, Yu. M.

Yu. M. Terent’ev, K. A. Vytovtov, “Transformation matrix of N-layer periodic medium with anisotropic layers,” J. Commun. Technol. Electron. 45, 255–257 (2000).

Yu. M. Terent’ev, K. A. Vytovtov, “Investigation of wave behavior in periodical magnetized ferrite,” Electromagn. Waves Electron. Syst. 4, 37–42 (1999).

Vytovtov, K.

K. Vytovtov, “A model of a two-dimensional linear parametric system with a step pumping,” in Proceedings XXXII Sympozjon PTMTS Modelowanie w Mechanice (Gliwice, Poland, 1998), pp. 377–380.

Vytovtov, K. A.

K. A. Vytovtov, “The analytical method of investigation of periodic layered media with uniaxial bianisotropy,” J. Commun. Technol. Electron. 46, 144–150 (2001).

Yu. M. Terent’ev, K. A. Vytovtov, “Transformation matrix of N-layer periodic medium with anisotropic layers,” J. Commun. Technol. Electron. 45, 255–257 (2000).

Yu. M. Terent’ev, K. A. Vytovtov, “Investigation of wave behavior in periodical magnetized ferrite,” Electromagn. Waves Electron. Syst. 4, 37–42 (1999).

K. A. Vytovtov, “Propagation conditions of a harmonic waves within bianisotropic periodic layered media,” in Proceedings of 30th European Microwave Conference (Paris, 2000), Vol. 2, pp. 238–241.

Waterman, P. C.

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965).

Wu, S. M.

G. A. Ybara, S. M. Wu, G. L. Bilbro, S. H. Ardalan, C. P. Hearn, R. T. Neece, “Optimal signal prossing of frequency-stepped SW radar data,” IEEE Trans. Microwave Theory Tech. 43, 94–105 (1995).
[CrossRef]

Yariv, A.

Ybara, G. A.

G. A. Ybara, S. M. Wu, G. L. Bilbro, S. H. Ardalan, C. P. Hearn, R. T. Neece, “Optimal signal prossing of frequency-stepped SW radar data,” IEEE Trans. Microwave Theory Tech. 43, 94–105 (1995).
[CrossRef]

Yeh, P.

Ann. Phys. (Paris)

F. Abelès, “Recherches sur la propagation des ondes electromagnétiques sinusoidales dans les milieux stratifiés. Application aux conches minces,” Ann. Phys. (Paris) 5, 596–640, 706–782 (1950).

Appl. Phys. Lett.

N. Blombergen, A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–485 (1970).
[CrossRef]

Can. J. Phys.

T. Tamir, “Scattering of electromagnetic waves by a sinusoidally stratified half-space: part II,” Can. J. Phys. 44, 2461–2494 (1966).
[CrossRef]

Electromagn. Waves Electron. Syst.

Yu. M. Terent’ev, K. A. Vytovtov, “Investigation of wave behavior in periodical magnetized ferrite,” Electromagn. Waves Electron. Syst. 4, 37–42 (1999).

IEEE Trans. Antennas Propag.

J. C. W. A. Costa, A. J. Giarola, “Electromagnetic wave propagation in multilayered dielectric periodic structures,” IEEE Trans. Antennas Propag. 41, 1432–1438 (1993).
[CrossRef]

A. Boag, Y. Leviatan, A. Boag, “Analysis of two-dimensional electromagnetic scattering from nonplanar periodic surfaces using a strip current model,” IEEE Trans. Antennas Propag. 37, 1437–1451 (1989).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

J. J. Pesque, D. P. Bouche, R. Mittra, “Optimization of multilayer antireflection coatings using an optimal control method,” IEEE Trans. Microwave Theory Tech. 40, 1789–1796 (1992).
[CrossRef]

G. A. Ybara, S. M. Wu, G. L. Bilbro, S. H. Ardalan, C. P. Hearn, R. T. Neece, “Optimal signal prossing of frequency-stepped SW radar data,” IEEE Trans. Microwave Theory Tech. 43, 94–105 (1995).
[CrossRef]

L. R. Lewis, A. Hessel, “Propagation characteristics of a periodic array of dielectric slabs,” IEEE Trans. Microwave Theory Tech. 19, 276–286 (1971).
[CrossRef]

J. Acoust. Soc. Am.

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[CrossRef]

J. Appl. Phys.

J. R. Pierce, “Coupling of modes of propagation,” J. Appl. Phys. 25, 179–183 (1954).
[CrossRef]

J. Commun. Technol. Electron.

Yu. M. Terent’ev, K. A. Vytovtov, “Transformation matrix of N-layer periodic medium with anisotropic layers,” J. Commun. Technol. Electron. 45, 255–257 (2000).

K. A. Vytovtov, “The analytical method of investigation of periodic layered media with uniaxial bianisotropy,” J. Commun. Technol. Electron. 46, 144–150 (2001).

J. Opt. Soc. Am.

Philos. Mag.

J. W. Strutt, Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philos. Mag. 24, 145–159 (1887).
[CrossRef]

Phys. Rev. B

D. Ager, H. P. Hughes, “Optical properties of stratified systems including lamellar gratings,” Phys. Rev. B 44, 13452–13465 (1991).
[CrossRef]

A. A. Bulgakov, S. A. Bulgakov, M. Nieto-Vesperinas, “Inhomogeneous waves and energy localization in dielectric superlattices,” Phys. Rev. B 58, 4438–4448 (1998).
[CrossRef]

Physica A

G. A. Gevorkyan, “On the theory of propagation of electromagnetic waves in a waveguide with a multiperiodically modulated dielectric filling,” Physica A 241, 236–239 (1997).
[CrossRef]

Proc. IEEE

C. Elachi, “Wave in active and passive periodic structures: a review,” Proc. IEEE 64, 1666–1698 (1976).
[CrossRef]

Radio Sci.

S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Riv. Mat. Univ. Parma

G. Gambill, “Criteria for parametric instability for linear differential systems with periodic coefficients,” Riv. Mat. Univ. Parma 6, 37–43 (1955).

Z. Phys.

F. Bloch, “Quantenmechanik der elektronen in kristallgittern,” Z. Phys. 52, 555–500 (1928).
[CrossRef]

Other

K. A. Vytovtov, “Propagation conditions of a harmonic waves within bianisotropic periodic layered media,” in Proceedings of 30th European Microwave Conference (Paris, 2000), Vol. 2, pp. 238–241.

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990), 136–140.

K. Vytovtov, “A model of a two-dimensional linear parametric system with a step pumping,” in Proceedings XXXII Sympozjon PTMTS Modelowanie w Mechanice (Gliwice, Poland, 1998), pp. 377–380.

J. K. Hale, Oscillations in Nonlinear Systems (McGraw-Hill, New York, 1963).

A. H. Nayfer, Introduction to Pertrubation Techniques (Wiley, New York, 1981).

L. Brillouin, M. Parodi, Propagation des Ondes dans les Milieux Périodiques (Masson et Cie, Editeurs, Paris, 1956).

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965).

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

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Figures (4)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Illustration of the sign function f q , i for a triple-layered structure.

Fig. 3
Fig. 3

Illustration of change in the origin of coordinates.

Fig. 4
Fig. 4

Illustration of long periodic waves.

Tables (1)

Tables Icon

Table 1 Magnitudes of the Sign Function for a Triple-Layered Period

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

L i = cos   ϕ i - j p i sin   ϕ i - jp i sin   ϕ i cos   ϕ i ,
L = i = N 1 L i .
f q , i = sign sin π 2 N + 1 - i   ( 2 q - 1 )
L ( Λ ) = q = 1 N - 1 1 2 N - 1 p 1 / p N × exp i = 1 N - 1 ln 1 + p i + 1 f q , i + 1 p i f q , i L q ,
L q = p N / p 1 cos   ψ q - ( j / f q , N p 1 p N ) sin   ψ q - j p 1 p N sin   ψ q f q , N p 1 / p N cos   ψ q ,
ln 1 + p i + 1 f q , i + 1 p i f q , i
= ln 1 + p i + 1 f q , i + 1 p i f q , i + j arg 1 + p i + 1 f q , i + 1 p i f q , i + 2 n π ,
ψ q = i = 1 N ( ϕ i f q , i ) .
ξ q = 1 2 N - 1 p 1 / p N exp i = 1 N - 1 ln 1 + p q , i + 1 f q , i + 1 p i f i
q = 1 2 N - 1 1 2 N - 1 1 + p 1 p N f q , N × exp i = 1 N - 1 ln 1 + p i + 1 f q , i + 1 p i f q , i cos ( ψ q ) = 2 ,
q = 1 2 N - 1 i = 1 N - 1 m = i + 1 N 2 + p m f q , m p i f q , i + p i f q , i p m f q , m × cos i = 1 N ( ϕ i f q , i ) = 2 .
ψ q = 1 = ϕ 1 + ϕ 2 + ϕ 3 ,
ψ q = 2 = ϕ 1 + ϕ 2 - ϕ 3 ,
ψ q = 3 = ϕ 1 - ϕ 2 + ϕ 3 ,
ψ q = 4 = ϕ 1 - ϕ 2 - ϕ 3 .
ψ q = ϕ 1 f q , 1 + ϕ 2 f q , 2 + ϕ 3 f q , 3
q = 1 : ψ 1 = ϕ 1 + ϕ 2 + ϕ 3 + ϕ 4 + ϕ 5 ÷ a 1 = 00000 , q = 2 : ψ 2 = ϕ 1 + ϕ 2 + ϕ 3 + ϕ 4 - ϕ 5 ÷ a 2 = 00001 , q = 16 : ψ 16 = ϕ 1 - ϕ 2 - ϕ 3 - ϕ 4 - ϕ 5 ÷ a 16 = 01111 .
 
q = 5 : ψ 5 = ϕ 1 + ϕ 2 - ϕ 3 + ϕ 4 + ϕ 5 ÷ a 5 = 00100 , q = 9 : ψ 9 = ϕ 1 - ϕ 2 + ϕ 3 + ϕ 4 + ϕ 5 ÷ a 9 = 01000 .
q = 1 2 N - 1 i = 1 N - 1 m = i + 1 N 2 + p m f q , m p i f q , i + p i f q , i p m f q , m × cos i = 1 N ( n ϕ i f q , i ) = 2 .
L ( 1 ) = cos   ϕ 1 - ( j / p 1 ) sin   ϕ 1 - jp 1 sin   ϕ 1 cos   ϕ 1 .
L ( 2 ) = cos   ϕ 1 - ( j / p 1 ) sin   ϕ 1 - jp 1 sin   ϕ 1 cos   ϕ 1 × cos   ϕ 2 - ( j / p 2 ) sin   ϕ 2 - jp 2 sin   ϕ 2 cos   ϕ 2 .
L 11 2 = 1 2 1 + p 2 p 1 cos ( ϕ 1 + ϕ 2 ) + 1 2 1 - p 2 p 1 × cos ( ϕ 1 - ϕ 2 ) ,
L 12 2 = - j 2 1 p 1 + 1 p 2 sin ( ϕ 1 + ϕ 2 ) - j 2 1 p 1 - 1 p 2 × sin ( ϕ 1 - ϕ 2 ) ,
L 21 2 = - j 2   ( p 1 + p 2 ) sin ( ϕ 1 + ϕ 2 ) - j 2   ( p 1 - p 2 ) × sin ( ϕ 1 - ϕ 2 ) ,
L 22 2 = 1 2 1 + p 1 p 2 cos ( ϕ 1 + ϕ 2 ) + 1 2 1 - p 1 p 2 × cos ( ϕ 1 - ϕ 2 ) .
L ( 3 ) = cos   ϕ 1 - ( j / p 1 ) sin   ϕ 1 - jp 1 sin   ϕ 1 cos   ϕ 1 × cos   ϕ 2 - ( j / p 2 ) sin   ϕ 2 - jp 2 sin   ϕ 2 cos   ϕ 2 × cos   ϕ 3 - ( j / p 3 ) sin   ϕ 3 - jp 3 sin   ϕ 3 cos   ϕ 3 .
L 11 3 = 1 4 1 + p 2 p 1 + p 3 p 1 + p 3 p 2 cos ( ϕ 1 + ϕ 2 + ϕ 3 ) + 1 4 1 + p 2 p 1 - p 3 p 1 - p 3 p 2 cos ( ϕ 1 + ϕ 2 - ϕ 3 ) + 1 4 1 - p 2 p 1 + p 3 p 1 - p 3 p 2 cos ( ϕ 1 - ϕ 2 + ϕ 3 ) + 1 4 1 - p 2 p 1 - p 3 p 1 + p 3 p 2 cos ( ϕ 1 - ϕ 2 - ϕ 3 ) ,
L 12 3 = - j 4 1 p 1 + 1 p 2 + 1 p 3 + p 2 p 1 p 3 sin ( ϕ 1 + ϕ 2 + ϕ 3 ) - j 4 1 p 1 + 1 p 2 - 1 p 3 - p 2 p 1 p 3 × sin ( ϕ 1 + ϕ 2 - ϕ 3 ) - j 4 1 p 1 - 1 p 2 + 1 p 3 - p 2 p 1 p 3 sin ( ϕ 1 - ϕ 2 + ϕ 3 ) - j 4 1 p 1 - 1 p 2 - 1 p 3 + p 2 p 1 p 3 × sin ( ϕ 1 - ϕ 2 - ϕ 3 ) ,
L 21 3 = - j 4 p 1 + p 2 + p 3 + p 1 p 3 p 2 sin ( ϕ 1 + ϕ 2 + ϕ 3 ) - j 4 p 1 + p 2 - p 3 - p 1 p 3 p 2 sin ( ϕ 1 + ϕ 2 - ϕ 3 ) - j 4 p 1 - p 2 + p 3 - p 1 p 3 p 2 sin ( ϕ 1 - ϕ 2 + ϕ 3 ) - j 4 p 1 - p 2 - p 3 + p 1 p 3 p 2 × sin ( ϕ 1 = ϕ 2 - ϕ 3 ) ,
L 11 3 = 1 4 1 + p 1 p 2 + p 1 p 3 + p 2 p 3 cos ( ϕ 1 + ϕ 2 + ϕ 3 ) + 1 4 1 + p 1 p 2 - p 1 p 3 - p 2 p 3 cos ( ϕ 1 + ϕ 2 - ϕ 3 ) + 1 4 1 - p 1 p 2 + p 1 p 3 - p 2 p 3 cos ( ϕ 1 - ϕ 2 + ϕ 3 ) + 1 4 1 - p 1 p 2 - p 1 p 3 + p 2 p 3 cos ( ϕ 1 - ϕ 2 - ϕ 3 ) .
L 11 3 = 1 4 1 + p 2 p 1 1 + p 3 p 2 cos ( ϕ 1 + ϕ 2 + ϕ 3 ) + 1 4 1 + p 2 p 1 1 + ( - p 3 ) p 2 cos ( ϕ 1 + ϕ 2 - ϕ 3 ) + 1 4 1 + ( - p 2 ) p 1 1 + p 3 ( - p 2 ) cos ( ϕ 1 - ϕ 2 + ϕ 3 ) + 1 4 1 + ( - p 2 ) p 1 1 + ( - p 3 ) ( - p 2 ) cos ( ϕ 1 - ϕ 2 - ϕ 3 ) = 1 4 q = 1 4 1 + f q , 2 p 2 f q , 2 p 1 1 + f q , 2 p 3 f q , 2 p 2 cos ( f q , 1 ϕ 1 + f q , 2 ϕ 2 + f q , 3 ϕ 3 ) = 1 4 q = 1 4 i = 1 2 1 + f q , i + 1 p i + 1 f q , i p i cos i = 1 3 f q , i ϕ i = 1 4 i = 1 4 i = 1 2 exp 1 + f q , i + 1 p i + 1 f q , i p i × cos i = 1 3 f q , i ϕ i .
L 11 N = 1 4 i = 1 2 N - 1 1 2 N - 1 i = 1 N - 1 exp 1 + f q , i + 1 p i + 1 f q , i p i × cos i = 1 N f q , i ϕ i .
L N
= q = 1 2 N - 1 ξ q p N / p 1 cos   ψ q - j / ( f q , N p 1 p N )   sin   ψ q - j p 1 p N sin   ψ q ( 1 / f q , N ) p 1 / p N cos   ψ q × cos   ϕ N + 1 ( - j / p N + 1 ) sin   ϕ N + 1 - jp N + 1 sin   ϕ N + 1 cos   ϕ N + 1 .
L 11 N + 1 = q = 1 2 N + 1 ξ q { p N / p 1 cos   ψ q cos   ϕ N + 1 - [ p N + 1 / ( f q , N p 1 p N ) ] sin   ψ q sin   ϕ N + 1 } = q = 1 2 N + 1 1 2   ξ q { p N / p 1 cos ( ψ q + ϕ N + 1 ) + p N / p 1 cos ( ψ q - ϕ N + 1 ) + [ p N + 1 / ( f q , N p 1 p N ) ] cos ( ψ q + ϕ N + 1 ) - [ p N + 1 / ( f q , N p 1 p N ) ] cos ( ψ q - ϕ N + 1 ) } = q = 1 2 N + 1 1 2   ξ q { [ p N / p 1 + p N + 1 / ( f q , N p 1 p N ) ] × cos ( ψ q + ϕ N + 1 ) + [ p N / p 1 - p N + 1 / ( f q , N p 1 p N ) ] × cos ( ψ q - ϕ N + 1 ) } .
L 11 N + 1 = q = 1 2 N - 1 1 2   ξ q p N p N + 1 1 + p N + 1 f q , N + 1 p N f q , N × p N + 1 p 1 cos ( ψ q + ϕ N + 1 f q , N + 1 )
L 11 N + 1 = q = 1 2 ( N - 1 ) + 1 1 2 ( N - 1 ) + 1 i = 1 ( N - 1 ) + 1 1 + p i + 1 f q , i + 1 p i f q , i × p N + 1 p 1 cos i = 1 N + 1 ϕ i f q , i .
L 11 M = q = 1 2 M - 1 1 2 M - 1 i = 1 M - 1 1 + p i + 1 f q , i + 1 p i f q , i × p M p 1 cos i = 1 M ϕ i f q , i .

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