Abstract

We consider propagation of coherent light through a nonlinear periodic optical structure consisting of two alternating layers with different linear and nonlinear refractive indices. A coupled-mode system is derived from the Maxwell equations and analyzed for the stationary-transmission regimes and linear time-dependent dynamics. We find the domain for existence of true all-optical limiting when the input–output transmission characteristic is monotonic and clamped below a limiting value for output intensity. True all-optical limiting can be managed by compensating the Kerr nonlinearities in the alternating layers, when the net-average nonlinearity is much smaller than the nonlinearity variance. The periodic optical structures can be used as uniform switches between lower-transmissive and higher-transmissive states if the structures are sufficiently long and out-of-phase, i.e., when the linear grating compensates the nonlinearity variations at each optical layer. We prove analytically that true all-optical limiting for zero net-average nonlinearity is asymptotically stable in time-dependent dynamics. We also show that weakly unbalanced out-of-phase gratings with small net-average nonlinearity exhibit local multistability, whereas strongly unbalanced gratings with large net-average nonlinearity display global multistability.

© 2002 Optical Society of America

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    [CrossRef]
  15. S. Dubovitsky and W. H. Steier, “Analysis of optical bistability in a nonlinear coupled resonator,” IEEE J. Quantum Electron. 28, 585–589 (1992).
    [CrossRef]
  16. J. He and M. Cada, “Optical bistability in semiconductor periodic structures,” IEEE J. Quantum Electron. 27, 1182–1188 (1991).
    [CrossRef]
  17. H. G. Winful and G. D. Cooperman, “Self-pulsing and chaos in distributed feedback bistable optical devices,” Appl. Phys. Lett. 40, 298–300 (1982).
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    [CrossRef]
  19. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
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  24. E. Kumacheva, O. Kalinina, and L. Lige, “Three-dimensional arrays in polymer nanocomposites,” Adv. Mater. 11, 231–234 (1999).
    [CrossRef]
  25. C. M. de Sterke, “Stability analysis of nonlinear periodic media,” Phys. Rev. A 45, 8252–8258 (1992).
    [CrossRef] [PubMed]
  26. C. M. de Sterke and J. E. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
    [CrossRef] [PubMed]
  27. M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform—Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).
  28. D. E. Pelinovsky and R. H. J. Grimshaw, “Structural transformation of eigenvalues for a perturbed algebraic soliton potential,” Phys. Lett. A 229, 165–172 (1997).
    [CrossRef]

2000 (4)

L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (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]

S. Trillo, C. Conti, G. Assanto, and A. V. Buryak, “From parametric gap solitons to chaos by means of second-harmonic generation in Bragg gratings,” Chaos 10, 590–599 (2000).
[CrossRef]

F. E. Hernandex, S. Yang, E. W. Van Stryland, and D. J. Hagan, “High-dynamic-range cascaded-focus optical limiter,” Opt. Lett. 25, 1180–1182 (2000).
[CrossRef]

1999 (3)

P. Tran, “All-optical switching with a nonlinear chiral photonic bandgap structure,” J. Opt. Soc. Am. B 16, 70–73 (1999).
[CrossRef]

P. W. E. Smith and L. Qian, “Switching to optical for a faster tomorrow,” IEEE Circuits Devices Mag. 15(11), 28–33 (1999).
[CrossRef]

E. Kumacheva, O. Kalinina, and L. Lige, “Three-dimensional arrays in polymer nanocomposites,” Adv. Mater. 11, 231–234 (1999).
[CrossRef]

1998 (3)

1997 (2)

D. E. Pelinovsky and R. H. J. Grimshaw, “Structural transformation of eigenvalues for a perturbed algebraic soliton potential,” Phys. Lett. A 229, 165–172 (1997).
[CrossRef]

T. Xia, D. J. Hagan, A. Dogariu, A. A. Said, and E. W. Van Stryland, “Optimization of optical limiting devices based on excited-state absorption,” Appl. Opt. 36, 4110–4122 (1997).
[CrossRef] [PubMed]

1995 (1)

C.-X. Shi, “Optical bistability in reflective fiber grating,” IEEE J. Quantum Electron. 31, 2037–2043 (1995).
[CrossRef]

1994 (1)

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

1992 (4)

C. M. de Sterke, “Stability analysis of nonlinear periodic media,” Phys. Rev. A 45, 8252–8258 (1992).
[CrossRef] [PubMed]

S. Dubovitsky and W. H. Steier, “Analysis of optical bistability in a nonlinear coupled resonator,” IEEE J. Quantum Electron. 28, 585–589 (1992).
[CrossRef]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

M. W. Chbat, B. Hong, M. N. Islam, C. E. Soccoloich, and P. R. Prucnal, “Ultrafast soliton-trapping AND gate,” J. Lightwave Technol. 10, 2011–2016 (1992).
[CrossRef]

1991 (2)

M. N. Islam, C. E. Soccolich, C.-J. Chen, K. S. Kim, J. R. Simpson, and U. C. Paek, “All-optical inverter with one picojoule switching energy,” Electron. Lett. 27, 130–132 (1991).
[CrossRef]

J. He and M. Cada, “Optical bistability in semiconductor periodic structures,” IEEE J. Quantum Electron. 27, 1182–1188 (1991).
[CrossRef]

1990 (1)

C. M. de Sterke and J. E. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
[CrossRef] [PubMed]

1989 (1)

G. L. Wood, W. W. Clark III, M. J. Miller, G. J. Salamo, and E. J. Sharp, “Evaluation of passive optical limiters and switches,” Proc. SPIE 1105, 154–166 (1989).
[CrossRef]

1982 (1)

H. G. Winful and G. D. Cooperman, “Self-pulsing and chaos in distributed feedback bistable optical devices,” Appl. Phys. Lett. 40, 298–300 (1982).
[CrossRef]

1981 (1)

P. W. Smith, “Bistable optical devices promise subpicosecond switching,” IEEE Spectrum 8(6), 26–30 (1981).
[CrossRef]

1979 (1)

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structure,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

1974 (1)

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform—Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

Ablowitz, M. J.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform—Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

Assanto, G.

S. Trillo, C. Conti, G. Assanto, and A. V. Buryak, “From parametric gap solitons to chaos by means of second-harmonic generation in Bragg gratings,” Chaos 10, 590–599 (2000).
[CrossRef]

Bang, O.

Brzozowski, L.

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, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
[CrossRef]

Buryak, A. V.

S. Trillo, C. Conti, G. Assanto, and A. V. Buryak, “From parametric gap solitons to chaos by means of second-harmonic generation in Bragg gratings,” Chaos 10, 590–599 (2000).
[CrossRef]

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Cada, M.

J. He and M. Cada, “Optical bistability in semiconductor periodic structures,” IEEE J. Quantum Electron. 27, 1182–1188 (1991).
[CrossRef]

Chbat, M. W.

M. W. Chbat, B. Hong, M. N. Islam, C. E. Soccoloich, and P. R. Prucnal, “Ultrafast soliton-trapping AND gate,” J. Lightwave Technol. 10, 2011–2016 (1992).
[CrossRef]

Chen, C.-J.

M. N. Islam, C. E. Soccolich, C.-J. Chen, K. S. Kim, J. R. Simpson, and U. C. Paek, “All-optical inverter with one picojoule switching energy,” Electron. Lett. 27, 130–132 (1991).
[CrossRef]

Clark III, W. W.

G. L. Wood, W. W. Clark III, M. J. Miller, G. J. Salamo, and E. J. Sharp, “Evaluation of passive optical limiters and switches,” Proc. SPIE 1105, 154–166 (1989).
[CrossRef]

Conti, C.

S. Trillo, C. Conti, G. Assanto, and A. V. Buryak, “From parametric gap solitons to chaos by means of second-harmonic generation in Bragg gratings,” Chaos 10, 590–599 (2000).
[CrossRef]

Cooperman, G. D.

H. G. Winful and G. D. Cooperman, “Self-pulsing and chaos in distributed feedback bistable optical devices,” Appl. Phys. Lett. 40, 298–300 (1982).
[CrossRef]

de Sterke, C. M.

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

C. M. de Sterke, “Stability analysis of nonlinear periodic media,” Phys. Rev. A 45, 8252–8258 (1992).
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
[CrossRef] [PubMed]

Dogariu, A.

Dubovitsky, S.

S. Dubovitsky and W. H. Steier, “Analysis of optical bistability in a nonlinear coupled resonator,” IEEE J. Quantum Electron. 28, 585–589 (1992).
[CrossRef]

Fejer, M. M.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Garmire, E.

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structure,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

Grimshaw, R. H. J.

D. E. Pelinovsky and R. H. J. Grimshaw, “Structural transformation of eigenvalues for a perturbed algebraic soliton potential,” Phys. Lett. A 229, 165–172 (1997).
[CrossRef]

Hagan, D. J.

Hall, K. L.

He, J.

J. He and M. Cada, “Optical bistability in semiconductor periodic structures,” IEEE J. Quantum Electron. 27, 1182–1188 (1991).
[CrossRef]

Hernandex, F. E.

Hong, B.

M. W. Chbat, B. Hong, M. N. Islam, C. E. Soccoloich, and P. R. Prucnal, “Ultrafast soliton-trapping AND gate,” J. Lightwave Technol. 10, 2011–2016 (1992).
[CrossRef]

Islam, M. N.

M. W. Chbat, B. Hong, M. N. Islam, C. E. Soccoloich, and P. R. Prucnal, “Ultrafast soliton-trapping AND gate,” J. Lightwave Technol. 10, 2011–2016 (1992).
[CrossRef]

M. N. Islam, C. E. Soccolich, C.-J. Chen, K. S. Kim, J. R. Simpson, and U. C. Paek, “All-optical inverter with one picojoule switching energy,” Electron. Lett. 27, 130–132 (1991).
[CrossRef]

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Kalinina, O.

E. Kumacheva, O. Kalinina, and L. Lige, “Three-dimensional arrays in polymer nanocomposites,” Adv. Mater. 11, 231–234 (1999).
[CrossRef]

Kaup, D. J.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform—Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

Kim, K. S.

M. N. Islam, C. E. Soccolich, C.-J. Chen, K. S. Kim, J. R. Simpson, and U. C. Paek, “All-optical inverter with one picojoule switching energy,” Electron. Lett. 27, 130–132 (1991).
[CrossRef]

Kivshar, Yu. S.

Kobyakov, A.

Koshiba, M.

Kumacheva, E.

E. Kumacheva, O. Kalinina, and L. Lige, “Three-dimensional arrays in polymer nanocomposites,” Adv. Mater. 11, 231–234 (1999).
[CrossRef]

Lederer, F.

Lige, L.

E. Kumacheva, O. Kalinina, and L. Lige, “Three-dimensional arrays in polymer nanocomposites,” Adv. Mater. 11, 231–234 (1999).
[CrossRef]

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Marburger, J. H.

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structure,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

Miller, M. J.

G. L. Wood, W. W. Clark III, M. J. Miller, G. J. Salamo, and E. J. Sharp, “Evaluation of passive optical limiters and switches,” Proc. SPIE 1105, 154–166 (1989).
[CrossRef]

Newell, A. C.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform—Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

Niiyama, A.

Paek, U. C.

M. N. Islam, C. E. Soccolich, C.-J. Chen, K. S. Kim, J. R. Simpson, and U. C. Paek, “All-optical inverter with one picojoule switching energy,” Electron. Lett. 27, 130–132 (1991).
[CrossRef]

Patel, N. S.

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]

D. E. Pelinovsky and R. H. J. Grimshaw, “Structural transformation of eigenvalues for a perturbed algebraic soliton potential,” Phys. Lett. A 229, 165–172 (1997).
[CrossRef]

Prucnal, P. R.

M. W. Chbat, B. Hong, M. N. Islam, C. E. Soccoloich, and P. R. Prucnal, “Ultrafast soliton-trapping AND gate,” J. Lightwave Technol. 10, 2011–2016 (1992).
[CrossRef]

Qian, L.

P. W. E. Smith and L. Qian, “Switching to optical for a faster tomorrow,” IEEE Circuits Devices Mag. 15(11), 28–33 (1999).
[CrossRef]

Rauschenbach, K. A.

Said, A. A.

Salamo, G. J.

G. L. Wood, W. W. Clark III, M. J. Miller, G. J. Salamo, and E. J. Sharp, “Evaluation of passive optical limiters and switches,” Proc. SPIE 1105, 154–166 (1989).
[CrossRef]

Sargent, E. H.

L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (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]

Segur, H.

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform—Fourier analysis for nonlinear problems,” Stud. Appl. Math. 53, 249–315 (1974).

Sharp, E. J.

G. L. Wood, W. W. Clark III, M. J. Miller, G. J. Salamo, and E. J. Sharp, “Evaluation of passive optical limiters and switches,” Proc. SPIE 1105, 154–166 (1989).
[CrossRef]

Shi, C.-X.

C.-X. Shi, “Optical bistability in reflective fiber grating,” IEEE J. Quantum Electron. 31, 2037–2043 (1995).
[CrossRef]

Simpson, J. R.

M. N. Islam, C. E. Soccolich, C.-J. Chen, K. S. Kim, J. R. Simpson, and U. C. Paek, “All-optical inverter with one picojoule switching energy,” Electron. Lett. 27, 130–132 (1991).
[CrossRef]

Sipe, J. E.

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

C. M. de Sterke and J. E. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
[CrossRef] [PubMed]

Smith, P. W.

P. W. Smith, “Bistable optical devices promise subpicosecond switching,” IEEE Spectrum 8(6), 26–30 (1981).
[CrossRef]

Smith, P. W. E.

P. W. E. Smith and L. Qian, “Switching to optical for a faster tomorrow,” IEEE Circuits Devices Mag. 15(11), 28–33 (1999).
[CrossRef]

Soccolich, C. E.

M. N. Islam, C. E. Soccolich, C.-J. Chen, K. S. Kim, J. R. Simpson, and U. C. Paek, “All-optical inverter with one picojoule switching energy,” Electron. Lett. 27, 130–132 (1991).
[CrossRef]

Soccoloich, C. E.

M. W. Chbat, B. Hong, M. N. Islam, C. E. Soccoloich, and P. R. Prucnal, “Ultrafast soliton-trapping AND gate,” J. Lightwave Technol. 10, 2011–2016 (1992).
[CrossRef]

Steier, W. H.

S. Dubovitsky and W. H. Steier, “Analysis of optical bistability in a nonlinear coupled resonator,” IEEE J. Quantum Electron. 28, 585–589 (1992).
[CrossRef]

Tran, P.

Trillo, S.

S. Trillo, C. Conti, G. Assanto, and A. V. Buryak, “From parametric gap solitons to chaos by means of second-harmonic generation in Bragg gratings,” Chaos 10, 590–599 (2000).
[CrossRef]

Van Stryland, E. W.

Winful, H. G.

H. G. Winful and G. D. Cooperman, “Self-pulsing and chaos in distributed feedback bistable optical devices,” Appl. Phys. Lett. 40, 298–300 (1982).
[CrossRef]

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structure,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

Wood, G. L.

G. L. Wood, W. W. Clark III, M. J. Miller, G. J. Salamo, and E. J. Sharp, “Evaluation of passive optical limiters and switches,” Proc. SPIE 1105, 154–166 (1989).
[CrossRef]

Xia, T.

Yang, S.

Adv. Mater. (1)

E. Kumacheva, O. Kalinina, and L. Lige, “Three-dimensional arrays in polymer nanocomposites,” Adv. Mater. 11, 231–234 (1999).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

H. G. Winful and G. D. Cooperman, “Self-pulsing and chaos in distributed feedback bistable optical devices,” Appl. Phys. Lett. 40, 298–300 (1982).
[CrossRef]

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structure,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

Chaos (1)

S. Trillo, C. Conti, G. Assanto, and A. V. Buryak, “From parametric gap solitons to chaos by means of second-harmonic generation in Bragg gratings,” Chaos 10, 590–599 (2000).
[CrossRef]

Electron. Lett. (1)

M. N. Islam, C. E. Soccolich, C.-J. Chen, K. S. Kim, J. R. Simpson, and U. C. Paek, “All-optical inverter with one picojoule switching energy,” Electron. Lett. 27, 130–132 (1991).
[CrossRef]

IEEE Circuits Devices Mag. (1)

P. W. E. Smith and L. Qian, “Switching to optical for a faster tomorrow,” IEEE Circuits Devices Mag. 15(11), 28–33 (1999).
[CrossRef]

IEEE J. Quantum Electron. (5)

L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
[CrossRef]

C.-X. Shi, “Optical bistability in reflective fiber grating,” IEEE J. Quantum Electron. 31, 2037–2043 (1995).
[CrossRef]

S. Dubovitsky and W. H. Steier, “Analysis of optical bistability in a nonlinear coupled resonator,” IEEE J. Quantum Electron. 28, 585–589 (1992).
[CrossRef]

J. He and M. Cada, “Optical bistability in semiconductor periodic structures,” IEEE J. Quantum Electron. 27, 1182–1188 (1991).
[CrossRef]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

IEEE Spectrum (1)

P. W. Smith, “Bistable optical devices promise subpicosecond switching,” IEEE Spectrum 8(6), 26–30 (1981).
[CrossRef]

J. Lightwave Technol. (2)

M. W. Chbat, B. Hong, M. N. Islam, C. E. Soccoloich, and P. R. Prucnal, “Ultrafast soliton-trapping AND gate,” J. Lightwave Technol. 10, 2011–2016 (1992).
[CrossRef]

A. Niiyama and M. Koshiba, “Three-dimensional beam propagation analysis of nonlinear optical fibers and optical logic gates,” J. Lightwave Technol. 16, 162–168 (1998).
[CrossRef]

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

Opt. Lett. (2)

Phys. Lett. A (1)

D. E. Pelinovsky and R. H. J. Grimshaw, “Structural transformation of eigenvalues for a perturbed algebraic soliton potential,” Phys. Lett. A 229, 165–172 (1997).
[CrossRef]

Phys. Rev. A (2)

C. M. de Sterke, “Stability analysis of nonlinear periodic media,” Phys. Rev. A 45, 8252–8258 (1992).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

Phys. Rev. E (1)

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

Proc. SPIE (1)

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[CrossRef]

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[CrossRef]

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Other (2)

I. C. Khoo, M. Wood, B. D. Guenther, “Nonlinear liquid crystal optical fiber array for all-optical switching/limiting,” in Proceedings of the Nineth Annual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE, New York, 1996), Vol. 2, pp. 211–212.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

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Figures (7)

Fig. 1
Fig. 1

Periodic nonlinear structure consisting of alternating layers with different linear refractive indices and different Kerr nonlinearities.

Fig. 2
Fig. 2

Balanced nonlinearity management with a linear grating, where nnl=0, n2k=1. Horizontal lines show the limiting intensity Ilim, and the dashed line displays the regime of complete transparence: Iout=Iin. An out-of-phase (n0k=-0.02) grating increases Ilim, whereas an in-phase grating (n0k=0.02) decreases it.

Fig. 3
Fig. 3

Balanced nonlinearity management with an out-of-phase grating, where nnl=0, n0k=-0.02, and n2k=1. The figure emphasizes the sharpening of the S-shaped curves with an increase in the number of layers. Each curve has been normalized to its limiting intensity.

Fig. 4
Fig. 4

Unbalanced nonlinearity management with no linear grating, where n0k=0, n2k=1. The threshold between the limiting regime and multistability is nnl=1.33. Note that the multilevel oscillations become tighter for larger nnl.

Fig. 5
Fig. 5

Unbalanced nonlinearity management with linear grating, where nnl=1, n2k=1. Bistability occurs for n0k-0.03, so the curve for n0k=-0.04 shows two local bistability cascades.

Fig. 6
Fig. 6

Unbalanced nonlinearity management with an out-of-phase grating, where n0k=-0.04, nnl=1, n2k=1 and (a) N=20, (b) N=50, (c) N=200, and (d) N=400. The Iout versus Iin curves become multistable as the device gets longer.

Fig. 7
Fig. 7

(a), (b) Functions D+ and (c), (d) D- versus γ as given in Eqs. (46) and (48) for two sets: (a), (c) μ=0 and θ0=0.25π[0.2, 0.4, 0.6, 0.8]; (b), (d) μ=-0.75,-0.25, 0.25, 0.75, and θ0=0.1π(1-μ2)1/2. No zeros of D±(γ) exist for γ0.

Equations (63)

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2Ez2-n2(z, |E|2)c2 2Et2=0,
n(z, |E|2)=nln+2n0k cos kz+nnl|E|2+2n2k|E|2 cos kz,
E(z, t)=A+(z, t)exp[i(k0z-ω0t)]+A-(z, t)exp[-i(k0z+ω0t)]+higher-order terms,
iA+Z+A+T+n0kA-+nnl(|A+|2+2|A-|2)A+
+n2k[(2|A+|2+|A-|2)A-+A+2A¯-]=0,
-iA-Z-A-T+n0kA++nnl(2|A+|2+|A-|2)A-
+n2k[(|A+|2+2|A-|2)A++A-2A¯+]=0,
nln=n01+n022,nnl=nnl1+nnl22,
n0k=n01-n02π,n2k=nnl1-nnl2π.
|A+(0)|2=Iin,|A-(0)|2=Iref,
|A+(L)|2=Iout,|A-(L)|2=0.
|A+(Z)|2-|A-(Z)|2=Iin-Iref=Iout.
A±Z=±i HA¯±,
H=[n0k+n2k(|A+|2+|A-|2)](A¯+A-+A+A¯-)+12nnl(|A+|4+4|A+|2|A-|2+|A-|4).
A+(Z)=Iout+Q exp[i(Φ-Ψ)],
A-(Z)=Q exp(iΦ)
dQdZ=-2Q(Iout+Q) sin Ψ[n0k+n2k(Iout+2Q)],
dΨdZ=-3nnl(Iout+2Q)-cos ΨQ(Iout+Q)×[n0k(Iout+2Q)+n2k(Iout2+8IoutQ+8Q2)].
H=2Q(Iout+Q)cos Ψ[n0k+n2k(Iout+2Q)]+3nnlQ(Iout+Q)+12nnlIout2
cos Ψ=-3nnlQ(Iout+Q)2[n0k+n2k(Iout+2Q)].
Ψ(Z)=π2 for n0k+n2kIout0,
Ψ(Z)=-π2 for n0k+n2kIout<0.
Q(Z)=Iout(n0k+n2kIout)sin2 θn2kIout cos 2θ-n0k,
-1cos(2n2k2Ilim2-n0k2L)=n0kn2kIlim1.
Ilim=π4n2k L 1-8n0k Lπ2+O(n0k)2.
Ilim|n0k|n2k for n0k<0,nnl=0.
Q(Z)=Iout(n0k+n2kIout)sinh2 ϕn0k-n2kIout cosh 2ϕ,
Q(Z)=4Ioutn0k2(L-Z)21-4n0k2(L-Z)2 as Ioutn0kn2k>0.
cosh(2n0k2-n2k2 Ilim2L)=n0kn2k Ilim>1.
Q(Z)=Iout(n0k+n2kIout)2(L-Z)2.
IoutIlim=2 IinIlim-1for 12IlimIinIlim.
Iout=Ilim=n0kn2k for Iin>0.
n2k3|nnl|4.
dQdZ2=Q(Iout+Q){4[n0k+n2k(Iout+2Q)]2-9nnl2Q(Iout+Q)}.
Ilim4|n0k|16n2k2-9nnl2 for n0k<0.
 Q(Z)=4(n0k+n2kIout)29nnl2 Iout sin232nnlIout(L-Z).
0>n0k-πn2k3|nnl|L.
A+A¯+(Z, T)=A+A¯+(Z)+a1a2(Z)exp(λT),
A-A¯-(Z, T)=A-A¯-(Z)+b1b2(Z)exp(λT),
Hψ=λJψ,
H=-iσ300-σ3 ddZ-n0kσ100σ1-nnl2(|A+|2+|A-|2)2A+A¯-A+22A+A-2A¯+A-2(|A+|2+|A-|2)2A+A-A-2-A¯+22A¯+A¯-2(|A+|2+|A-|2)2A¯+A-2A¯+A¯-A¯-22A+A¯-2(|A+|2+|A-|2)-n2k2(A+A¯-+A¯+A-)2(|A+|2+|A-|2)2A+A-(A+2+A-2)2(|A+|2+|A-|2)2(A+A¯-+A¯+A-)(A+2+A-2)2A+A-2A¯+A¯-(A¯+2+A¯-2)2(A+A¯-+A¯+A-)2(|A+|2+|A-|2)(A¯+2+A¯-2)2A¯+A¯-2(|A+|2+|A-|2)2(A+A¯-+A¯+A-).
σ1=0110,σ3=100-1.
a1(0)=a2(0)=b1(L)=b2(L)=0.
ψ+(Z)=a1+a2b1-b2(Z),ψ-(Z)=a1-a2b1+b2(Z).
[H1±n2kH2]ψ±=iλψ±,
H1=-iσ3 ddZ-σ1[n0k+2n2k(Iout+2Q)],
H2=2iσ3Q(Iout+Q)+σ1σ3Iout.
ψ-1(Z)=ψ+2(2L-Z),ψ-2(Z)=ψ+1(2L-Z).
θ=n2k2 Iout2-n0k2(L-Z),λ=n2k2 Iout2-n0k2γ,
ψ+1(Z)=-i[(n2kIout-n0k)(n2kIout cos2θ-n0k)]1/2φ1(θ),
ψ+2(Z)=[(n2k Iout+n0k)(n2k Iout cos 2θ-n0k)]1/2φ2(θ).
dφ1dθ-1+2(n0k+n2kIout)n2kIout cos 2θ-n0kφ2=γφ1,
-dφ2dθ-1+2(n0k-n2kIout)n2kIout cos 2θ-n0kφ1=γφ2.
φ(θ)=Φ±(θ, γ)=exp(±ikθ)×±ik+n2kIout sin 2θn2kIout cos 2θ-n0k×-(γ±ik)1+1n2kIout cos 2θ-n0k×n0k+n2kIout(n0k-n2kIout)(γ±ik),
ψ+(Z):φ2(0)=0,φ1(θ0)=0,
D+(γ; u, θ0)=γ+sin 2θ0cos 2θ0-μ-tan(kθ0)k×k2+1+μ-γ sin 2θ0cos 2θ0-μ=0,
ψ-(Z):φ1(0)=0,φ2(-θ0)=0.
D-(γ, μ, θ0)
=2-γ2(1-μ)-(2+γ sin 2θ0)(1-μ)cos 2θ0-μ+tan(kθ0)k (1-μ)γk2+γ(1-μ2)+[2-γ2(1-μ)]sin 2θ0cos 2θ0-μ=0.
Dˆ+(Γ; μ, ϕ0)
=Γ+sinh 2ϕ0μ-cosh 2ϕ0+tanh(κϕ0)κ×κ2-1+μ-Γ sinh 2ϕ0μ-cosh 2ϕ0=0,
Dˆ-(Γ, μ, ϕ0)
=2+Γ2(1-μ)+(2+Γ sinh 2ϕ0)(1-μ)μ-cosh 2ϕ0+tanh(κϕ0)κ (1-μ)Γκ2-Γ(1-μ2)-[2+Γ2(1-μ)]sinh 2ϕ0μ-cosh 2ϕ0=0,

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