Abstract

We apply the multiple-scales formalism to derive a complete set of equations for a finite medium with periodic linear and nonlinear Kerr optical coefficients. The equations for a single-frequency field reveal three new, nonlinear terms that are related to the difference in the Kerr nonlinearity in two-component media. The nonlinear evolution of coupled forward and backward fields in a multilayered film is numerically simulated by a spectral method. We examine the linear stability of the steady-state solution for an infinite medium and extend previous discussions of modulational instabilities to the new set of equations. We find that the inhomogeneous coefficient can selectively suppress modulational instability in the longitudinal or transverse direction.

© 2002 Optical Society of America

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  1. M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–R1081 (1996).
    [CrossRef]
  2. A. V. Andreev, A. V. Balakin, I. A. Ozheredov, A. P. Shkurinov, P. Maselin, G. Mouret, and D. Boucher, “Compression of femtosecond laser pulses in thin one-dimensional photonic crystals,” Phys. Rev. E 63, 016602 (2001).
    [CrossRef]
  3. A. V. Balakin, V. A. Bushuev, N. I. Koroteev, B. I. Mantsyzov, I. A. Ozheredov, A. P. Shkurinov, D. Boucher, and P. Maselin, “Enhancement of second-harmonic generation with femtosecond laser pulses near the photonic band edge for different polarizations of incident light,” Opt. Lett. 24, 793–795 (1999).
    [CrossRef]
  4. M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
    [CrossRef] [PubMed]
  5. L. Brzozowski and E. H. Sargent, “Nonlinear distributed-feedback structures as passive optical limiters,” J. Opt. Soc. Am. B 17, 1360–1365 (2000).
    [CrossRef]
  6. D. E. Pelinovsky, L. Brzozowski, and E. H. Sargent, “Transmission regimes of periodic nonlinear optical structures,” Phys. Rev. E 62, R4536–R4539 (2000).
    [CrossRef]
  7. M. Scalora, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
    [CrossRef]
  8. B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on a nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
    [CrossRef]
  9. B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
    [CrossRef]
  10. G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 16609 (2001).
    [CrossRef]
  11. A. V. Tarasishin, S. A. Magnitskii, and A. M. Zheltikov, “Matching phase and group velocities in second-harmonic generation in finite one-dimensional photonic band-gap structures,” Laser Phys. 11, 31–8 (2001).
  12. A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, “Enhancement of sum frequency generation near the photonic band gap edge under quasiphase matching conditions,” Phys. Rev. E 63, 046609 (2001).
    [CrossRef]
  13. D. Pezzetta, C. Sibilia, M. Bertolotti, J. W. Haus, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Photonic bandgap structures in planar nonlinear waveguides: application to second-harmonic generation,” J. Opt. Soc. Am. B 18, 1326–1333 (2001).
    [CrossRef]
  14. M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Efficient nonlinear infrared parametric generation on one-dimensional photonic band gap structures,” Opt. Commun. 189, 135–142 (2001).
    [CrossRef]
  15. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
    [CrossRef] [PubMed]
  16. C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
    [CrossRef]
  17. S. Dutta-Gupta, “Nonlinear optics of stratified media,” Prog. Opt. 38, 1–84 (1998).
    [CrossRef]
  18. G. Stegeman and R. H. Stolen, “Waveguides and fibers for nonlinear optics,” J. Opt. Soc. Am. B 6, 652–662 (1989).
    [CrossRef]
  19. D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Prog. Opt. 27, 227–313 (1989).
    [CrossRef]
  20. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, 2001).
  21. A. G. Kalocsai and J. W. Haus, “Asymptotic wave-wave processes beyond cascading in quadratic nonlinear optical materials,” Phys. Rev. E 52, 3166–3183 (1995).
    [CrossRef]
  22. J. W. Haus, R. Viswanathan, A. Kalocsai, J. Cole, M. Scalora, and J. Theimer, “Enhanced second-harmonic generation in weakly periodic media,” Phys. Rev. A 57, 2120–2132 (1998).
    [CrossRef]
  23. M. Scalora and M. Crenshaw, “Beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
    [CrossRef]
  24. J. W. Haus, B. Y. Soon, M. Scalora, M. J. Bloemer, C. M. Bowden, C. Sibilia, and A. Zheltikov, “Spatio-temporal instabilities for counter-propagating waves in periodic media,” submitted to Opt. Express.
  25. N. M. Lichinitser, C. J. McKinstrie, C. M. de Sterke, and G. P. Agrawal, “Spatiotemporal instabilities in nonlinear bulk media with Bragg gratings,” J. Opt. Soc. Am. B 18, 45–54 (2001).
    [CrossRef]

2001

A. V. Andreev, A. V. Balakin, I. A. Ozheredov, A. P. Shkurinov, P. Maselin, G. Mouret, and D. Boucher, “Compression of femtosecond laser pulses in thin one-dimensional photonic crystals,” Phys. Rev. E 63, 016602 (2001).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 16609 (2001).
[CrossRef]

A. V. Tarasishin, S. A. Magnitskii, and A. M. Zheltikov, “Matching phase and group velocities in second-harmonic generation in finite one-dimensional photonic band-gap structures,” Laser Phys. 11, 31–8 (2001).

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, “Enhancement of sum frequency generation near the photonic band gap edge under quasiphase matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

D. Pezzetta, C. Sibilia, M. Bertolotti, J. W. Haus, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Photonic bandgap structures in planar nonlinear waveguides: application to second-harmonic generation,” J. Opt. Soc. Am. B 18, 1326–1333 (2001).
[CrossRef]

M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Efficient nonlinear infrared parametric generation on one-dimensional photonic band gap structures,” Opt. Commun. 189, 135–142 (2001).
[CrossRef]

N. M. Lichinitser, C. J. McKinstrie, C. M. de Sterke, and G. P. Agrawal, “Spatiotemporal instabilities in nonlinear bulk media with Bragg gratings,” J. Opt. Soc. Am. B 18, 45–54 (2001).
[CrossRef]

2000

L. Brzozowski and E. H. Sargent, “Nonlinear distributed-feedback structures as passive optical limiters,” J. Opt. Soc. Am. B 17, 1360–1365 (2000).
[CrossRef]

D. E. Pelinovsky, L. Brzozowski, and E. H. Sargent, “Transmission regimes of periodic nonlinear optical structures,” Phys. Rev. E 62, R4536–R4539 (2000).
[CrossRef]

1999

1998

S. Dutta-Gupta, “Nonlinear optics of stratified media,” Prog. Opt. 38, 1–84 (1998).
[CrossRef]

J. W. Haus, R. Viswanathan, A. Kalocsai, J. Cole, M. Scalora, and J. Theimer, “Enhanced second-harmonic generation in weakly periodic media,” Phys. Rev. A 57, 2120–2132 (1998).
[CrossRef]

1997

1996

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–R1081 (1996).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on a nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
[CrossRef]

1995

A. G. Kalocsai and J. W. Haus, “Asymptotic wave-wave processes beyond cascading in quadratic nonlinear optical materials,” Phys. Rev. E 52, 3166–3183 (1995).
[CrossRef]

1994

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
[CrossRef]

M. Scalora and M. Crenshaw, “Beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

M. Scalora, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

1989

G. Stegeman and R. H. Stolen, “Waveguides and fibers for nonlinear optics,” J. Opt. Soc. Am. B 6, 652–662 (1989).
[CrossRef]

D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Prog. Opt. 27, 227–313 (1989).
[CrossRef]

1987

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Agrawal, G. P.

Andreev, A. V.

A. V. Andreev, A. V. Balakin, I. A. Ozheredov, A. P. Shkurinov, P. Maselin, G. Mouret, and D. Boucher, “Compression of femtosecond laser pulses in thin one-dimensional photonic crystals,” Phys. Rev. E 63, 016602 (2001).
[CrossRef]

Balakin, A. V.

A. V. Andreev, A. V. Balakin, I. A. Ozheredov, A. P. Shkurinov, P. Maselin, G. Mouret, and D. Boucher, “Compression of femtosecond laser pulses in thin one-dimensional photonic crystals,” Phys. Rev. E 63, 016602 (2001).
[CrossRef]

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, “Enhancement of sum frequency generation near the photonic band gap edge under quasiphase matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

A. V. Balakin, V. A. Bushuev, N. I. Koroteev, B. I. Mantsyzov, I. A. Ozheredov, A. P. Shkurinov, D. Boucher, and P. Maselin, “Enhancement of second-harmonic generation with femtosecond laser pulses near the photonic band edge for different polarizations of incident light,” Opt. Lett. 24, 793–795 (1999).
[CrossRef]

Bendickson, J. M.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–R1081 (1996).
[CrossRef]

Bertolotti, M.

M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Efficient nonlinear infrared parametric generation on one-dimensional photonic band gap structures,” Opt. Commun. 189, 135–142 (2001).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 16609 (2001).
[CrossRef]

D. Pezzetta, C. Sibilia, M. Bertolotti, J. W. Haus, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Photonic bandgap structures in planar nonlinear waveguides: application to second-harmonic generation,” J. Opt. Soc. Am. B 18, 1326–1333 (2001).
[CrossRef]

D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Prog. Opt. 27, 227–313 (1989).
[CrossRef]

Bloemer, M. J.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 16609 (2001).
[CrossRef]

M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Efficient nonlinear infrared parametric generation on one-dimensional photonic band gap structures,” Opt. Commun. 189, 135–142 (2001).
[CrossRef]

D. Pezzetta, C. Sibilia, M. Bertolotti, J. W. Haus, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Photonic bandgap structures in planar nonlinear waveguides: application to second-harmonic generation,” J. Opt. Soc. Am. B 18, 1326–1333 (2001).
[CrossRef]

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–R1081 (1996).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

M. Scalora, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

Boucher, D.

A. V. Andreev, A. V. Balakin, I. A. Ozheredov, A. P. Shkurinov, P. Maselin, G. Mouret, and D. Boucher, “Compression of femtosecond laser pulses in thin one-dimensional photonic crystals,” Phys. Rev. E 63, 016602 (2001).
[CrossRef]

A. V. Balakin, V. A. Bushuev, N. I. Koroteev, B. I. Mantsyzov, I. A. Ozheredov, A. P. Shkurinov, D. Boucher, and P. Maselin, “Enhancement of second-harmonic generation with femtosecond laser pulses near the photonic band edge for different polarizations of incident light,” Opt. Lett. 24, 793–795 (1999).
[CrossRef]

Bowden, C. M.

D. Pezzetta, C. Sibilia, M. Bertolotti, J. W. Haus, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Photonic bandgap structures in planar nonlinear waveguides: application to second-harmonic generation,” J. Opt. Soc. Am. B 18, 1326–1333 (2001).
[CrossRef]

M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Efficient nonlinear infrared parametric generation on one-dimensional photonic band gap structures,” Opt. Commun. 189, 135–142 (2001).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 16609 (2001).
[CrossRef]

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–R1081 (1996).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

M. Scalora, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

Brzozowski, L.

L. Brzozowski and E. H. Sargent, “Nonlinear distributed-feedback structures as passive optical limiters,” J. Opt. Soc. Am. B 17, 1360–1365 (2000).
[CrossRef]

D. E. Pelinovsky, L. Brzozowski, and E. H. Sargent, “Transmission regimes of periodic nonlinear optical structures,” Phys. Rev. E 62, R4536–R4539 (2000).
[CrossRef]

Bushuev, V. A.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, “Enhancement of sum frequency generation near the photonic band gap edge under quasiphase matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

A. V. Balakin, V. A. Bushuev, N. I. Koroteev, B. I. Mantsyzov, I. A. Ozheredov, A. P. Shkurinov, D. Boucher, and P. Maselin, “Enhancement of second-harmonic generation with femtosecond laser pulses near the photonic band edge for different polarizations of incident light,” Opt. Lett. 24, 793–795 (1999).
[CrossRef]

Centini, M.

M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Efficient nonlinear infrared parametric generation on one-dimensional photonic band gap structures,” Opt. Commun. 189, 135–142 (2001).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 16609 (2001).
[CrossRef]

Chen, W.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Cole, J.

J. W. Haus, R. Viswanathan, A. Kalocsai, J. Cole, M. Scalora, and J. Theimer, “Enhanced second-harmonic generation in weakly periodic media,” Phys. Rev. A 57, 2120–2132 (1998).
[CrossRef]

Crenshaw, M.

M. Scalora and M. Crenshaw, “Beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
[CrossRef]

D’Aguanno, G.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 16609 (2001).
[CrossRef]

de Sterke, C. M.

N. M. Lichinitser, C. J. McKinstrie, C. M. de Sterke, and G. P. Agrawal, “Spatiotemporal instabilities in nonlinear bulk media with Bragg gratings,” J. Opt. Soc. Am. B 18, 45–54 (2001).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on a nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
[CrossRef]

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
[CrossRef]

Dowling, J. P.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–R1081 (1996).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

M. Scalora, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

Dumeige, Y.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 16609 (2001).
[CrossRef]

Dutta-Gupta, S.

S. Dutta-Gupta, “Nonlinear optics of stratified media,” Prog. Opt. 38, 1–84 (1998).
[CrossRef]

Eggleton, B. J.

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on a nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
[CrossRef]

Flynn, R. J.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–R1081 (1996).
[CrossRef]

Fork, R. L.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–R1081 (1996).
[CrossRef]

Haus, J. W.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 16609 (2001).
[CrossRef]

M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Efficient nonlinear infrared parametric generation on one-dimensional photonic band gap structures,” Opt. Commun. 189, 135–142 (2001).
[CrossRef]

D. Pezzetta, C. Sibilia, M. Bertolotti, J. W. Haus, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Photonic bandgap structures in planar nonlinear waveguides: application to second-harmonic generation,” J. Opt. Soc. Am. B 18, 1326–1333 (2001).
[CrossRef]

J. W. Haus, R. Viswanathan, A. Kalocsai, J. Cole, M. Scalora, and J. Theimer, “Enhanced second-harmonic generation in weakly periodic media,” Phys. Rev. A 57, 2120–2132 (1998).
[CrossRef]

A. G. Kalocsai and J. W. Haus, “Asymptotic wave-wave processes beyond cascading in quadratic nonlinear optical materials,” Phys. Rev. E 52, 3166–3183 (1995).
[CrossRef]

Kalocsai, A.

J. W. Haus, R. Viswanathan, A. Kalocsai, J. Cole, M. Scalora, and J. Theimer, “Enhanced second-harmonic generation in weakly periodic media,” Phys. Rev. A 57, 2120–2132 (1998).
[CrossRef]

Kalocsai, A. G.

A. G. Kalocsai and J. W. Haus, “Asymptotic wave-wave processes beyond cascading in quadratic nonlinear optical materials,” Phys. Rev. E 52, 3166–3183 (1995).
[CrossRef]

Koroteev, N. I.

Leavitt, R. P.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–R1081 (1996).
[CrossRef]

Ledbetter, H. S.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–R1081 (1996).
[CrossRef]

Levenson, J. A.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 16609 (2001).
[CrossRef]

Lichinitser, N. M.

Magnitskii, S. A.

A. V. Tarasishin, S. A. Magnitskii, and A. M. Zheltikov, “Matching phase and group velocities in second-harmonic generation in finite one-dimensional photonic band-gap structures,” Laser Phys. 11, 31–8 (2001).

Mantsyzov, B. I.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, “Enhancement of sum frequency generation near the photonic band gap edge under quasiphase matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

A. V. Balakin, V. A. Bushuev, N. I. Koroteev, B. I. Mantsyzov, I. A. Ozheredov, A. P. Shkurinov, D. Boucher, and P. Maselin, “Enhancement of second-harmonic generation with femtosecond laser pulses near the photonic band edge for different polarizations of incident light,” Opt. Lett. 24, 793–795 (1999).
[CrossRef]

Maselin, P.

A. V. Andreev, A. V. Balakin, I. A. Ozheredov, A. P. Shkurinov, P. Maselin, G. Mouret, and D. Boucher, “Compression of femtosecond laser pulses in thin one-dimensional photonic crystals,” Phys. Rev. E 63, 016602 (2001).
[CrossRef]

A. V. Balakin, V. A. Bushuev, N. I. Koroteev, B. I. Mantsyzov, I. A. Ozheredov, A. P. Shkurinov, D. Boucher, and P. Maselin, “Enhancement of second-harmonic generation with femtosecond laser pulses near the photonic band edge for different polarizations of incident light,” Opt. Lett. 24, 793–795 (1999).
[CrossRef]

McKinstrie, C. J.

Mihalache, D.

D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Prog. Opt. 27, 227–313 (1989).
[CrossRef]

Mills, D. L.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Mouret, G.

A. V. Andreev, A. V. Balakin, I. A. Ozheredov, A. P. Shkurinov, P. Maselin, G. Mouret, and D. Boucher, “Compression of femtosecond laser pulses in thin one-dimensional photonic crystals,” Phys. Rev. E 63, 016602 (2001).
[CrossRef]

Ozheredov, I. A.

A. V. Andreev, A. V. Balakin, I. A. Ozheredov, A. P. Shkurinov, P. Maselin, G. Mouret, and D. Boucher, “Compression of femtosecond laser pulses in thin one-dimensional photonic crystals,” Phys. Rev. E 63, 016602 (2001).
[CrossRef]

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, “Enhancement of sum frequency generation near the photonic band gap edge under quasiphase matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

A. V. Balakin, V. A. Bushuev, N. I. Koroteev, B. I. Mantsyzov, I. A. Ozheredov, A. P. Shkurinov, D. Boucher, and P. Maselin, “Enhancement of second-harmonic generation with femtosecond laser pulses near the photonic band edge for different polarizations of incident light,” Opt. Lett. 24, 793–795 (1999).
[CrossRef]

Pelinovsky, D. E.

D. E. Pelinovsky, L. Brzozowski, and E. H. Sargent, “Transmission regimes of periodic nonlinear optical structures,” Phys. Rev. E 62, R4536–R4539 (2000).
[CrossRef]

Petrov, E. V.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, “Enhancement of sum frequency generation near the photonic band gap edge under quasiphase matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

Pezzetta, D.

Reinhardt, S. B.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–R1081 (1996).
[CrossRef]

Sargent, E. H.

D. E. Pelinovsky, L. Brzozowski, and E. H. Sargent, “Transmission regimes of periodic nonlinear optical structures,” Phys. Rev. E 62, R4536–R4539 (2000).
[CrossRef]

L. Brzozowski and E. H. Sargent, “Nonlinear distributed-feedback structures as passive optical limiters,” J. Opt. Soc. Am. B 17, 1360–1365 (2000).
[CrossRef]

Scalora, M.

M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Efficient nonlinear infrared parametric generation on one-dimensional photonic band gap structures,” Opt. Commun. 189, 135–142 (2001).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 16609 (2001).
[CrossRef]

D. Pezzetta, C. Sibilia, M. Bertolotti, J. W. Haus, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Photonic bandgap structures in planar nonlinear waveguides: application to second-harmonic generation,” J. Opt. Soc. Am. B 18, 1326–1333 (2001).
[CrossRef]

J. W. Haus, R. Viswanathan, A. Kalocsai, J. Cole, M. Scalora, and J. Theimer, “Enhanced second-harmonic generation in weakly periodic media,” Phys. Rev. A 57, 2120–2132 (1998).
[CrossRef]

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–R1081 (1996).
[CrossRef]

M. Scalora, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

M. Scalora and M. Crenshaw, “Beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
[CrossRef]

Shkurinov, A. P.

A. V. Andreev, A. V. Balakin, I. A. Ozheredov, A. P. Shkurinov, P. Maselin, G. Mouret, and D. Boucher, “Compression of femtosecond laser pulses in thin one-dimensional photonic crystals,” Phys. Rev. E 63, 016602 (2001).
[CrossRef]

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, “Enhancement of sum frequency generation near the photonic band gap edge under quasiphase matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

A. V. Balakin, V. A. Bushuev, N. I. Koroteev, B. I. Mantsyzov, I. A. Ozheredov, A. P. Shkurinov, D. Boucher, and P. Maselin, “Enhancement of second-harmonic generation with femtosecond laser pulses near the photonic band edge for different polarizations of incident light,” Opt. Lett. 24, 793–795 (1999).
[CrossRef]

Sibilia, C.

D. Pezzetta, C. Sibilia, M. Bertolotti, J. W. Haus, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Photonic bandgap structures in planar nonlinear waveguides: application to second-harmonic generation,” J. Opt. Soc. Am. B 18, 1326–1333 (2001).
[CrossRef]

M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Efficient nonlinear infrared parametric generation on one-dimensional photonic band gap structures,” Opt. Commun. 189, 135–142 (2001).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 16609 (2001).
[CrossRef]

D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Prog. Opt. 27, 227–313 (1989).
[CrossRef]

Sipe, J. E.

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on a nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
[CrossRef]

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
[CrossRef]

Slusher, R. E.

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on a nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
[CrossRef]

Stegeman, G.

Stolen, R. H.

Tarasishin, A. V.

A. V. Tarasishin, S. A. Magnitskii, and A. M. Zheltikov, “Matching phase and group velocities in second-harmonic generation in finite one-dimensional photonic band-gap structures,” Laser Phys. 11, 31–8 (2001).

Theimer, J.

J. W. Haus, R. Viswanathan, A. Kalocsai, J. Cole, M. Scalora, and J. Theimer, “Enhanced second-harmonic generation in weakly periodic media,” Phys. Rev. A 57, 2120–2132 (1998).
[CrossRef]

Tocci, M. D.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–R1081 (1996).
[CrossRef]

Vidakovic, P.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 16609 (2001).
[CrossRef]

Viswanathan, R.

J. W. Haus, R. Viswanathan, A. Kalocsai, J. Cole, M. Scalora, and J. Theimer, “Enhanced second-harmonic generation in weakly periodic media,” Phys. Rev. A 57, 2120–2132 (1998).
[CrossRef]

Zheltikov, A. M.

A. V. Tarasishin, S. A. Magnitskii, and A. M. Zheltikov, “Matching phase and group velocities in second-harmonic generation in finite one-dimensional photonic band-gap structures,” Laser Phys. 11, 31–8 (2001).

Electron. Lett.

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on a nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
[CrossRef]

J. Appl. Phys.

M. Scalora, J. P. Dowling, M. J. Bloemer, and C. M. Bowden, “The photonic band edge optical diode,” J. Appl. Phys. 76, 2023–2026 (1994).
[CrossRef]

J. Opt. Soc. Am. B

Laser Phys.

A. V. Tarasishin, S. A. Magnitskii, and A. M. Zheltikov, “Matching phase and group velocities in second-harmonic generation in finite one-dimensional photonic band-gap structures,” Laser Phys. 11, 31–8 (2001).

Opt. Commun.

M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Efficient nonlinear infrared parametric generation on one-dimensional photonic band gap structures,” Opt. Commun. 189, 135–142 (2001).
[CrossRef]

M. Scalora and M. Crenshaw, “Beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
[CrossRef]

Opt. Lett.

Phys. Rev. A

J. W. Haus, R. Viswanathan, A. Kalocsai, J. Cole, M. Scalora, and J. Theimer, “Enhanced second-harmonic generation in weakly periodic media,” Phys. Rev. A 57, 2120–2132 (1998).
[CrossRef]

Phys. Rev. E

A. G. Kalocsai and J. W. Haus, “Asymptotic wave-wave processes beyond cascading in quadratic nonlinear optical materials,” Phys. Rev. E 52, 3166–3183 (1995).
[CrossRef]

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. D. Tocci, M. J. Bloemer, C. M. Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrashort pulse propagation at the photonic band edge: large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–R1081 (1996).
[CrossRef]

A. V. Andreev, A. V. Balakin, I. A. Ozheredov, A. P. Shkurinov, P. Maselin, G. Mouret, and D. Boucher, “Compression of femtosecond laser pulses in thin one-dimensional photonic crystals,” Phys. Rev. E 63, 016602 (2001).
[CrossRef]

D. E. Pelinovsky, L. Brzozowski, and E. H. Sargent, “Transmission regimes of periodic nonlinear optical structures,” Phys. Rev. E 62, R4536–R4539 (2000).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 16609 (2001).
[CrossRef]

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, and A. P. Shkurinov, “Enhancement of sum frequency generation near the photonic band gap edge under quasiphase matching conditions,” Phys. Rev. E 63, 046609 (2001).
[CrossRef]

Phys. Rev. Lett.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368–1371 (1994).
[CrossRef] [PubMed]

Prog. Opt.

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
[CrossRef]

S. Dutta-Gupta, “Nonlinear optics of stratified media,” Prog. Opt. 38, 1–84 (1998).
[CrossRef]

D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Prog. Opt. 27, 227–313 (1989).
[CrossRef]

Other

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

J. W. Haus, B. Y. Soon, M. Scalora, M. J. Bloemer, C. M. Bowden, C. Sibilia, and A. Zheltikov, “Spatio-temporal instabilities for counter-propagating waves in periodic media,” submitted to Opt. Express.

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Figures (8)

Fig. 1
Fig. 1

Transmission spectra for four cases; in all curves the scaled length is L=5. (a) No absorption, solid curve; homogeneous absorption, α0=0.1, dashed–dotted curve. (b) Inhomogeneous absorption: α0=0.1 and α1=0.1, solid curve; α0=0.1 and α1=-0.1, dashed curve.

Fig. 2
Fig. 2

Reflection spectra for the four cases illustrated in Fig. 1. The parameter values are found in the caption of Fig. 1.

Fig. 3
Fig. 3

Gain profile for f=-1, Γ0=1, Γ1=0, Γ2=0. The curves are labeled with the values of the scaled transverse wave number, i.e., k2/F, which starts at zero and increases in increments of Δ(k2/F)=0.1.

Fig. 4
Fig. 4

Gain profile (f=-1): (a) Γ0=1, Γ1=0.1, Γ2=0; (b) Γ0=1, Γ1=0.4, Γ2=0.

Fig. 5
Fig. 5

Gain profile (f=-1): (a) Γ0=1, Γ1=0.6, Γ2=0; (b) Γ0=1, Γ1=1, Γ2=0.

Fig. 6
Fig. 6

Forward and backward field profiles for A=0.6, δ=1.12, η0=1, η1=0, η2=0, and L=6.1 Shown is the absolute amplitude of the fields after a time lapse of T=40. Bars at the top and the bottom of each photograph indicate the PBG’s position; the bar extends from z=0 to z=6.1 in each picture.

Fig. 7
Fig. 7

Forward and backward propagation for A=0.6, δ=1.12, η0=1, η1=0.4, η2=0, and L=6.1. See the caption to Fig. 6 for further details.

Fig. 8
Fig. 8

Forward and backward propagation for A=0.6, δ=1.12, η0=1, η1=1, η2=0, and L=6.1. See the caption to Fig. 6 for further details.

Equations (58)

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

2E-1c2 2Et2=4πc2 2PLt2+4πc2 2PNLt2.
PL=χ˜z, i tE.
˜z, i t=1+4πχ˜z, i t.
˜(z, ω)=˜r(z, ω)+i˜i(z, ω).
PNL=χ(3)(z)E3.
f(z)=m=- fm exp(i2πnz/Λ),
t=t0+μ t1+μ2 t2+,
z=z0+μ z1+μ2 z2+.
E=E0+μE1+μ2E2+.
2E0z02-1c2 2t02 ˜0i t0E0=0.
E0={Ef(z1, t1,)exp[i(kz0-ωt0)]+Eb(z1, t1 ,)exp[i(-kz0-ωt0)]+c.c.}.
k2=˜r(z, ω)(ω2/c2).
1v Eft=-Efz+iF2Ef+iδ-α02Ef+iκ1-α12Eb+iη0(|Ef|2+2|Eb|2)Ef+iη1(|Eb|2+2|Ef|2)Eb+iη-1Ef2Eb*+iη2Eb2Ef*,
1v Ebt=+Ebz+iF2Eb+iδ-α02Eb+iκ-1-α-12Ef+iη0(|Eb|2+2|Ef|2)Eb+iη-1(|Ef|2+2|Eb|2)Ef+iη1Eb2Ef*+iη-2Ef2Eb*,
Ef (x, y, 0, t)=S(x, y, t),
Eb(x, y, L, t)=0.
Ef (q, z, ω˜)=S(q, ω˜){Δ cos [Δ(z-L)]-iΩˆ sin[Δ(z-L)]}/[Δ cos(ΔL)-iΩˆ sin(ΔL)],
Eb(q, z, ω˜)=S(q, ω˜)iK-1 sin[Δ(z-L)]/[Δ cos(ΔL)-iΩˆ sin(ΔL)],
r=iK-1 sin(ΔL)/[Δ cos(ΔL)-iΩˆ sin(ΔL)],
t=Δ/[Δ cos(ΔL)-iΩˆ sin(ΔL)].
Ut=(L+V)U.
U(t+Δt)=exp(ΔtL/2)exp(ΔtV)exp(ΔtL/2)U(t).
Lff=-z+iF2+iδ-α0/2,
Lbb=z+iF2+iδ-α0/2.
Nff=iη0(|Ef|2+2|Eb|2)+2iη1EbEf*+iη-1EfEb*+iη2Eb2Ef*/Ef,
Nbb=iη0(|Eb|2+2|Ef|2)+2iη-1Ef Eb*+iη1EbEf*+iη-2Ef2Eb*/Eb,
Nbf=iη-1|Ef|2,
Nfb=iη1|Eb|2.
Kfb=-iκ1-α1/2,Kbf=-iκ-1-α-1/2.
exp(ΔtV)=exp(ΔtK/2)exp(ΔtN)exp(ΔtK/2).
exp(ΔtN)exp(ΔtNd)+NoΔt.
Af=a(1+f2)1/2,Ab=af(1+f2)1/2.
Ω=-κ2 f+1f-32η0a2-12η1 a21+f2×6f+1f+f3-12η2a2,
K=-δ-κ2 f-1f-12η0 a21+f2(1-f2)-12η1 a21+f2 1f-f3-12η2 a21+f2(1-f2).
Ex=(Ax+x)exp[i(Kz-Ωt)].
1v ft+fz=iκ1b-ifκ1f+iF2f+iη0 a21+f2[f+f*+2 f(b+b*)]+iη1 a21+f2[(2+f2)b+b*-f(3+f2)f+f2(b+b*)+2 f(f+f*)]+iη2 a21+f2(-f2f+f2f*+2 fb),
1v bt-bz=iκ1f-i κ1fb+iF2b+iη0 a21+f2×[f 2(b+b*)+2 f(f+f*)]+iη1 a21+f2 (1+2 f2)f+f2f*-(1+f2)fb+f+f*+2 f(b+b*)+iη2 a21+f2(-b+b*+2 ff).
x=ax+ exp[i(k·x-ω˜t)]+ax- exp[-i(k)·(x)-ω˜t],
kz-ω˜v+κf+k2F-Γ0+Γ1(-f+f3)+Γ2f2af+
-(Γ0+2 fΓ1+Γ2f2)af--[κ+2 fΓ0+Γ1(2+2 f2)+2 fΓ2]ab+-[2 fΓ0+Γ1(1+f2)]ab-
=0,
-(Γ0+2 fΓ1+Γ2f2)af++-kz-ω˜v+κf+k2F  -Γ0+Γ1(-f+f3)+Γ2f2af--[2 fΓ0+Γ1(1+f2)]ab+-[κ+2 fΓ0
+2Γ1(1+f2)+2 fΓ2]ab-=0,
-[κ+2 fΓ0+2Γ1(1+f2)+2 fΓ2]af+-[2 fΓ0
+Γ1(1+f2)]af-+-kz+ω˜v+κf+k2F-Γ0f2+Γ1 (1-f2)f+Γ2ab+
-(Γ0f2+2 fΓ1+Γ2)ab-=0,
-[2 fΓ0+Γ1(1+f2)]af+-[κ+2 fΓ0+2Γ1(1+f2)
+2 fΓ2]af--(Γ0+2 fΓ1+Γ2)ab++kz+ω˜v+κf+k2F-Γ0f2
+Γ1 (1-f2)f+Γ2ab-=0,
ω˜2+2ω˜2(A2+B2-C2-D2-kz2)+kz4+A4+B4
+C4+D4-8ABCD+2kz2(A2+D2-C2-B2)-2[A2(B2+C2+D2)+B2(C2+D2)
+C2D2]=0,
A=±2Γ0+2Γ1,
B=Γ0±2Γ1+Γ2,
C=κ+k2F-Γ0+Γ2,
D=κ±2Γ0+4Γ1±2Γ2.
G=|Im(ω˜)|.
F(x, y, z, 0)=A exp[-(z-z0)2/σz2]exp(-x2/σx2).

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