Abstract

Multiple-scale analysis is employed for the study of nonlinear wave propagation in periodic layered media. In a first step, wave propagation in each individual layer is modeled by a corresponding equivalent nonlinear transmission line. The multiple-scale analysis is then employed to establish a system of nonlinear equations for the amplitudes of the forward and backward waves in the transmission lines of the model mentioned. This system of nonlinear equations is solved with the aid of a continuation technique for derivation of transmission characteristics of the periodic structure. To study optical bistability, a parameter-switching algorithm is utilized for obtaining the solutions of the system including both stable and unstable ones. For the sake of verification, we have also utilized a nonlinear finite-difference time-domain (NFDTD) method to analyze the wave propagation in the aforementioned structure. While there is an acceptable agreement between the bistability results obtained using our method and the NFDTD, there is a considerable difference between our results and the ones derived using the conventional coupled-mode theory.

© 2007 Optical Society of America

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2005

J. Poon, E. Istrate, M. Allard, and E. H. Sargent, "Multiple-scale analysis of photonic crystal waveguides," IEEE J. Quantum Electron. 39, 778-786 (2005).
[CrossRef]

K. Zamani Aghaie and M. Shahabadi, "Perturbation analysis of plane-wave transmission through a nonlinear slab with Kerr-type nonlinearity," Opt. Express 13, 6587-6596 (2005).
[CrossRef] [PubMed]

2004

2002

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistability in nonlinear photonic crystals," Phys. Rev. E 66, 055601R (2002).
[CrossRef]

C. Lixue, D. Xiaoxu, D. Weiqiang, Cao Liangcai, and L. Shutian, "Finite-difference time-domain analysis of optical bistability with low threshold in one-dimensional nonlinear photonic crystal with Kerr medium," Opt. Commun. 209, 491-500 (2002).
[CrossRef]

2001

R. H. Goodman, M. I. Weinstein, and P. J. Holmes, "Nonlinear propagation of light in one-dimensional periodic structures," J. Nonlinear Sci. 11, 123-168 (2001).

N. A. R. Bhat and J. E. Sipe, "Optical pulse propagation in nonlinear photonic crystals," Phys. Rev. E 64, 056604 (2001).
[CrossRef]

S. H. Jeong, T. Mizumoto, K. Nakatsuhara, M. Takenaka, and Y. Nakano, "Deep-ridge distributed feedback waveguide for polarization independent all-optical switching," Electron. Lett. 37, 498-499 (2001).
[CrossRef]

2000

A. Melloni, M. Chinello, and M. Martinelli, "All-optical switching in phase-shifted fiber Bragg grating," IEEE Photon. Technol. Lett. 12, 42-44 (2000).
[CrossRef]

I. S. Nefedov, V. N. Gusyatnikov, P. K. Kashkarov, and A. M. Zheltikov, "Low-threshold photonic band gap optical logic gates," Laser Phys. 10, 640-644 (2000).

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

1998

M. Scalora, M. J. Bloemer, A. S. Pethal, J. P. Dowling, C. M. Bowden, and A. S. Manka, "Transparent, metallo-dielectric one-dimensional, photonic band-gap structures," J. Appl. Phys. 83, 2377-2383 (1998).
[CrossRef]

D. Taverner, N. G. R. Broderick, J. J. Richardson, M. Ibsen, and R. I. Laming, "All-optical AND gate based on coupled gap-soliton formation in a fiber Bragg grating," Opt. Lett. 23, 259-261 (1998).
[CrossRef]

Z. L. Wang, X. J. Liu, J. Wu, and N. B. Ming, "An effective Bloch wave theory for light propagation in nonlinear media," Phys. Lett. A 248, 263-266 (1998).
[CrossRef]

1997

1996

J. M. Bendickson, J. P. Dowling, and M. Scalora, "Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures," Phys. Rev. E 53, 4107-4121 (1996).
[CrossRef]

C. M. de Sterke, D. J. Salinas, and J. E. Sipe, "Coupled-mode theory for light propagation through deep nonlinear gratings," Phys. Rev. E 54, 1969-1989 (1996).
[CrossRef]

1994

P. St. J. Russell and J. L. Archambault, "Field microstructure and spatial instability of photonic Bloch waves in nonlinear periodic media," J. Phys. III France 4, 2471-2491 (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]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, "The photonic band edge laser: A new approach to gain enhancement," J. Appl. Phys. 75, 1896-1899 (1994).
[CrossRef]

1992

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M. A. Dupertuis, and J. M. Glinski, "All-optical reflectivity tuning and logic gating in a GaAs/AlAs periodic layered structure," Appl. Phys. Lett. 60, 404-406 (1992).
[CrossRef]

1991

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

1986

Y. H. Lee, A. Chavez-Pirson, S. W. Koch, H. M. Gibbs, S. H. Park, J. Morhange, A. Jeffery, N. Peyghambarian, L. Banyai, A. C. Gossard, and W. Wiegmann, "Room-temperature optical nonlinearities in GaAs," Phys. Rev. Lett. 57, 2446-2449 (1986).
[CrossRef] [PubMed]

1985

S. Adachi, "GaAs, AlAs, and AlxGa1−xAs; Material parameters for use in research and device applications," J. Appl. Phys. 58, R1-R29 (1985).
[CrossRef]

1977

1976

L. O. Chua, "A switching-parameter algorithm for finding multiple solutions of nonlinear resistive circuits," Int. J. Circuit Theory Appl. 4, 215-239 (1976).
[CrossRef]

Acklin, B.

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M. A. Dupertuis, and J. M. Glinski, "All-optical reflectivity tuning and logic gating in a GaAs/AlAs periodic layered structure," Appl. Phys. Lett. 60, 404-406 (1992).
[CrossRef]

Adachi, S.

S. Adachi, "GaAs, AlAs, and AlxGa1−xAs; Material parameters for use in research and device applications," J. Appl. Phys. 58, R1-R29 (1985).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic,1995).

Allard, M.

J. Poon, E. Istrate, M. Allard, and E. H. Sargent, "Multiple-scale analysis of photonic crystal waveguides," IEEE J. Quantum Electron. 39, 778-786 (2005).
[CrossRef]

Archambault, J. L.

P. St. J. Russell and J. L. Archambault, "Field microstructure and spatial instability of photonic Bloch waves in nonlinear periodic media," J. Phys. III France 4, 2471-2491 (1994).
[CrossRef]

Arregui, F. J.

Banyai, L.

Y. H. Lee, A. Chavez-Pirson, S. W. Koch, H. M. Gibbs, S. H. Park, J. Morhange, A. Jeffery, N. Peyghambarian, L. Banyai, A. C. Gossard, and W. Wiegmann, "Room-temperature optical nonlinearities in GaAs," Phys. Rev. Lett. 57, 2446-2449 (1986).
[CrossRef] [PubMed]

Bassi, P.

A. Parini, G. Bellanca, S. Trillo, L. Saccomandi, and P. Bassi, "Transfer matrix and full Maxwell time domain analysis of nonlinear gratings," Opt. Quantum Electron. 36, 189-199 (2004).
[CrossRef]

Bellanca, G.

A. Parini, G. Bellanca, S. Trillo, L. Saccomandi, and P. Bassi, "Transfer matrix and full Maxwell time domain analysis of nonlinear gratings," Opt. Quantum Electron. 36, 189-199 (2004).
[CrossRef]

Bendickson, J. M.

J. M. Bendickson, J. P. Dowling, and M. Scalora, "Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures," Phys. Rev. E 53, 4107-4121 (1996).
[CrossRef]

Bhat, N. A. R.

N. A. R. Bhat and J. E. Sipe, "Optical pulse propagation in nonlinear photonic crystals," Phys. Rev. E 64, 056604 (2001).
[CrossRef]

Bloemer, M. J.

M. Scalora, M. J. Bloemer, A. S. Pethal, J. P. Dowling, C. M. Bowden, and A. S. Manka, "Transparent, metallo-dielectric one-dimensional, photonic band-gap structures," J. Appl. Phys. 83, 2377-2383 (1998).
[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]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, "The photonic band edge laser: A new approach to gain enhancement," J. Appl. Phys. 75, 1896-1899 (1994).
[CrossRef]

Bowden, C. M.

M. Scalora, M. J. Bloemer, A. S. Pethal, J. P. Dowling, C. M. Bowden, and A. S. Manka, "Transparent, metallo-dielectric one-dimensional, photonic band-gap structures," J. Appl. Phys. 83, 2377-2383 (1998).
[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]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, "The photonic band edge laser: A new approach to gain enhancement," J. Appl. Phys. 75, 1896-1899 (1994).
[CrossRef]

Brillouin, L.

L. Brillouin, Wave Propagation in Periodic Structures (McGraw-Hill, 1946).

Broderick, N. G. R.

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]

Cada, M.

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M. A. Dupertuis, and J. M. Glinski, "All-optical reflectivity tuning and logic gating in a GaAs/AlAs periodic layered structure," Appl. Phys. Lett. 60, 404-406 (1992).
[CrossRef]

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

Chavez-Pirson, A.

Y. H. Lee, A. Chavez-Pirson, S. W. Koch, H. M. Gibbs, S. H. Park, J. Morhange, A. Jeffery, N. Peyghambarian, L. Banyai, A. C. Gossard, and W. Wiegmann, "Room-temperature optical nonlinearities in GaAs," Phys. Rev. Lett. 57, 2446-2449 (1986).
[CrossRef] [PubMed]

Chinello, M.

A. Melloni, M. Chinello, and M. Martinelli, "All-optical switching in phase-shifted fiber Bragg grating," IEEE Photon. Technol. Lett. 12, 42-44 (2000).
[CrossRef]

Chua, L. O.

L. O. Chua, "A switching-parameter algorithm for finding multiple solutions of nonlinear resistive circuits," Int. J. Circuit Theory Appl. 4, 215-239 (1976).
[CrossRef]

Chung, Y.

de Sterke, C. M.

C. M. de Sterke, D. J. Salinas, and J. E. Sipe, "Coupled-mode theory for light propagation through deep nonlinear gratings," Phys. Rev. E 54, 1969-1989 (1996).
[CrossRef]

Del Villar, I.

Dowling, J. P.

M. Scalora, M. J. Bloemer, A. S. Pethal, J. P. Dowling, C. M. Bowden, and A. S. Manka, "Transparent, metallo-dielectric one-dimensional, photonic band-gap structures," J. Appl. Phys. 83, 2377-2383 (1998).
[CrossRef]

J. M. Bendickson, J. P. Dowling, and M. Scalora, "Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures," Phys. Rev. E 53, 4107-4121 (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]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, "The photonic band edge laser: A new approach to gain enhancement," J. Appl. Phys. 75, 1896-1899 (1994).
[CrossRef]

Dupertuis, M. A.

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M. A. Dupertuis, and J. M. Glinski, "All-optical reflectivity tuning and logic gating in a GaAs/AlAs periodic layered structure," Appl. Phys. Lett. 60, 404-406 (1992).
[CrossRef]

Eggleton, B. J.

Fink, Y.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistability in nonlinear photonic crystals," Phys. Rev. E 66, 055601R (2002).
[CrossRef]

Gibbs, H. M.

Y. H. Lee, A. Chavez-Pirson, S. W. Koch, H. M. Gibbs, S. H. Park, J. Morhange, A. Jeffery, N. Peyghambarian, L. Banyai, A. C. Gossard, and W. Wiegmann, "Room-temperature optical nonlinearities in GaAs," Phys. Rev. Lett. 57, 2446-2449 (1986).
[CrossRef] [PubMed]

Glinski, J. M.

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M. A. Dupertuis, and J. M. Glinski, "All-optical reflectivity tuning and logic gating in a GaAs/AlAs periodic layered structure," Appl. Phys. Lett. 60, 404-406 (1992).
[CrossRef]

Goodman, R. H.

R. H. Goodman, M. I. Weinstein, and P. J. Holmes, "Nonlinear propagation of light in one-dimensional periodic structures," J. Nonlinear Sci. 11, 123-168 (2001).

Gossard, A. C.

Y. H. Lee, A. Chavez-Pirson, S. W. Koch, H. M. Gibbs, S. H. Park, J. Morhange, A. Jeffery, N. Peyghambarian, L. Banyai, A. C. Gossard, and W. Wiegmann, "Room-temperature optical nonlinearities in GaAs," Phys. Rev. Lett. 57, 2446-2449 (1986).
[CrossRef] [PubMed]

Gusyatnikov, V. N.

I. S. Nefedov, V. N. Gusyatnikov, P. K. Kashkarov, and A. M. Zheltikov, "Low-threshold photonic band gap optical logic gates," Laser Phys. 10, 640-644 (2000).

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2001).

Han, W. T.

He, J.

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M. A. Dupertuis, and J. M. Glinski, "All-optical reflectivity tuning and logic gating in a GaAs/AlAs periodic layered structure," Appl. Phys. Lett. 60, 404-406 (1992).
[CrossRef]

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

Holmes, P. J.

R. H. Goodman, M. I. Weinstein, and P. J. Holmes, "Nonlinear propagation of light in one-dimensional periodic structures," J. Nonlinear Sci. 11, 123-168 (2001).

Hong, C.

Ibanescu, M.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistability in nonlinear photonic crystals," Phys. Rev. E 66, 055601R (2002).
[CrossRef]

Ibsen, M.

Istrate, E.

J. Poon, E. Istrate, M. Allard, and E. H. Sargent, "Multiple-scale analysis of photonic crystal waveguides," IEEE J. Quantum Electron. 39, 778-786 (2005).
[CrossRef]

Jeffery, A.

Y. H. Lee, A. Chavez-Pirson, S. W. Koch, H. M. Gibbs, S. H. Park, J. Morhange, A. Jeffery, N. Peyghambarian, L. Banyai, A. C. Gossard, and W. Wiegmann, "Room-temperature optical nonlinearities in GaAs," Phys. Rev. Lett. 57, 2446-2449 (1986).
[CrossRef] [PubMed]

Jeong, S. H.

S. H. Jeong, T. Mizumoto, K. Nakatsuhara, M. Takenaka, and Y. Nakano, "Deep-ridge distributed feedback waveguide for polarization independent all-optical switching," Electron. Lett. 37, 498-499 (2001).
[CrossRef]

Joannopoulos, J. D.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistability in nonlinear photonic crystals," Phys. Rev. E 66, 055601R (2002).
[CrossRef]

Johnson, S. G.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistability in nonlinear photonic crystals," Phys. Rev. E 66, 055601R (2002).
[CrossRef]

Kashkarov, P. K.

I. S. Nefedov, V. N. Gusyatnikov, P. K. Kashkarov, and A. M. Zheltikov, "Low-threshold photonic band gap optical logic gates," Laser Phys. 10, 640-644 (2000).

Kim, N. S.

Kim, Y. H.

Koch, S. W.

Y. H. Lee, A. Chavez-Pirson, S. W. Koch, H. M. Gibbs, S. H. Park, J. Morhange, A. Jeffery, N. Peyghambarian, L. Banyai, A. C. Gossard, and W. Wiegmann, "Room-temperature optical nonlinearities in GaAs," Phys. Rev. Lett. 57, 2446-2449 (1986).
[CrossRef] [PubMed]

Laming, R. I.

Lee, Y. H.

Y. H. Lee, A. Chavez-Pirson, S. W. Koch, H. M. Gibbs, S. H. Park, J. Morhange, A. Jeffery, N. Peyghambarian, L. Banyai, A. C. Gossard, and W. Wiegmann, "Room-temperature optical nonlinearities in GaAs," Phys. Rev. Lett. 57, 2446-2449 (1986).
[CrossRef] [PubMed]

Liangcai, Cao

C. Lixue, D. Xiaoxu, D. Weiqiang, Cao Liangcai, and L. Shutian, "Finite-difference time-domain analysis of optical bistability with low threshold in one-dimensional nonlinear photonic crystal with Kerr medium," Opt. Commun. 209, 491-500 (2002).
[CrossRef]

Liu, X. J.

Z. L. Wang, X. J. Liu, J. Wu, and N. B. Ming, "An effective Bloch wave theory for light propagation in nonlinear media," Phys. Lett. A 248, 263-266 (1998).
[CrossRef]

Lixue, C.

C. Lixue, D. Xiaoxu, D. Weiqiang, Cao Liangcai, and L. Shutian, "Finite-difference time-domain analysis of optical bistability with low threshold in one-dimensional nonlinear photonic crystal with Kerr medium," Opt. Commun. 209, 491-500 (2002).
[CrossRef]

Manka, A. S.

M. Scalora, M. J. Bloemer, A. S. Pethal, J. P. Dowling, C. M. Bowden, and A. S. Manka, "Transparent, metallo-dielectric one-dimensional, photonic band-gap structures," J. Appl. Phys. 83, 2377-2383 (1998).
[CrossRef]

Martin, D.

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M. A. Dupertuis, and J. M. Glinski, "All-optical reflectivity tuning and logic gating in a GaAs/AlAs periodic layered structure," Appl. Phys. Lett. 60, 404-406 (1992).
[CrossRef]

Martinelli, M.

A. Melloni, M. Chinello, and M. Martinelli, "All-optical switching in phase-shifted fiber Bragg grating," IEEE Photon. Technol. Lett. 12, 42-44 (2000).
[CrossRef]

Matías, I. R.

Melloni, A.

A. Melloni, M. Chinello, and M. Martinelli, "All-optical switching in phase-shifted fiber Bragg grating," IEEE Photon. Technol. Lett. 12, 42-44 (2000).
[CrossRef]

Ming, N. B.

Z. L. Wang, X. J. Liu, J. Wu, and N. B. Ming, "An effective Bloch wave theory for light propagation in nonlinear media," Phys. Lett. A 248, 263-266 (1998).
[CrossRef]

Mizumoto, T.

S. H. Jeong, T. Mizumoto, K. Nakatsuhara, M. Takenaka, and Y. Nakano, "Deep-ridge distributed feedback waveguide for polarization independent all-optical switching," Electron. Lett. 37, 498-499 (2001).
[CrossRef]

Morhange, J.

Y. H. Lee, A. Chavez-Pirson, S. W. Koch, H. M. Gibbs, S. H. Park, J. Morhange, A. Jeffery, N. Peyghambarian, L. Banyai, A. C. Gossard, and W. Wiegmann, "Room-temperature optical nonlinearities in GaAs," Phys. Rev. Lett. 57, 2446-2449 (1986).
[CrossRef] [PubMed]

Morier-Genoud, F.

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M. A. Dupertuis, and J. M. Glinski, "All-optical reflectivity tuning and logic gating in a GaAs/AlAs periodic layered structure," Appl. Phys. Lett. 60, 404-406 (1992).
[CrossRef]

Nakano, Y.

S. H. Jeong, T. Mizumoto, K. Nakatsuhara, M. Takenaka, and Y. Nakano, "Deep-ridge distributed feedback waveguide for polarization independent all-optical switching," Electron. Lett. 37, 498-499 (2001).
[CrossRef]

Nakatsuhara, K.

S. H. Jeong, T. Mizumoto, K. Nakatsuhara, M. Takenaka, and Y. Nakano, "Deep-ridge distributed feedback waveguide for polarization independent all-optical switching," Electron. Lett. 37, 498-499 (2001).
[CrossRef]

Nefedov, I. S.

I. S. Nefedov, V. N. Gusyatnikov, P. K. Kashkarov, and A. M. Zheltikov, "Low-threshold photonic band gap optical logic gates," Laser Phys. 10, 640-644 (2000).

Paek, U. C.

Parini, A.

A. Parini, G. Bellanca, S. Trillo, L. Saccomandi, and P. Bassi, "Transfer matrix and full Maxwell time domain analysis of nonlinear gratings," Opt. Quantum Electron. 36, 189-199 (2004).
[CrossRef]

Park, S. H.

Y. H. Lee, A. Chavez-Pirson, S. W. Koch, H. M. Gibbs, S. H. Park, J. Morhange, A. Jeffery, N. Peyghambarian, L. Banyai, A. C. Gossard, and W. Wiegmann, "Room-temperature optical nonlinearities in GaAs," Phys. Rev. Lett. 57, 2446-2449 (1986).
[CrossRef] [PubMed]

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]

Pethal, A. S.

M. Scalora, M. J. Bloemer, A. S. Pethal, J. P. Dowling, C. M. Bowden, and A. S. Manka, "Transparent, metallo-dielectric one-dimensional, photonic band-gap structures," J. Appl. Phys. 83, 2377-2383 (1998).
[CrossRef]

Peyghambarian, N.

Y. H. Lee, A. Chavez-Pirson, S. W. Koch, H. M. Gibbs, S. H. Park, J. Morhange, A. Jeffery, N. Peyghambarian, L. Banyai, A. C. Gossard, and W. Wiegmann, "Room-temperature optical nonlinearities in GaAs," Phys. Rev. Lett. 57, 2446-2449 (1986).
[CrossRef] [PubMed]

Poon, J.

J. Poon, E. Istrate, M. Allard, and E. H. Sargent, "Multiple-scale analysis of photonic crystal waveguides," IEEE J. Quantum Electron. 39, 778-786 (2005).
[CrossRef]

Proctor, M.

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M. A. Dupertuis, and J. M. Glinski, "All-optical reflectivity tuning and logic gating in a GaAs/AlAs periodic layered structure," Appl. Phys. Lett. 60, 404-406 (1992).
[CrossRef]

Richardson, J. J.

Russell, P. St. J.

P. St. J. Russell and J. L. Archambault, "Field microstructure and spatial instability of photonic Bloch waves in nonlinear periodic media," J. Phys. III France 4, 2471-2491 (1994).
[CrossRef]

Saccomandi, L.

A. Parini, G. Bellanca, S. Trillo, L. Saccomandi, and P. Bassi, "Transfer matrix and full Maxwell time domain analysis of nonlinear gratings," Opt. Quantum Electron. 36, 189-199 (2004).
[CrossRef]

Salinas, D. J.

C. M. de Sterke, D. J. Salinas, and J. E. Sipe, "Coupled-mode theory for light propagation through deep nonlinear gratings," Phys. Rev. E 54, 1969-1989 (1996).
[CrossRef]

Sargent, E. H.

J. Poon, E. Istrate, M. Allard, and E. H. Sargent, "Multiple-scale analysis of photonic crystal waveguides," IEEE J. Quantum Electron. 39, 778-786 (2005).
[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]

Scalora, M.

M. Scalora, M. J. Bloemer, A. S. Pethal, J. P. Dowling, C. M. Bowden, and A. S. Manka, "Transparent, metallo-dielectric one-dimensional, photonic band-gap structures," J. Appl. Phys. 83, 2377-2383 (1998).
[CrossRef]

J. M. Bendickson, J. P. Dowling, and M. Scalora, "Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures," Phys. Rev. E 53, 4107-4121 (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]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, "The photonic band edge laser: A new approach to gain enhancement," J. Appl. Phys. 75, 1896-1899 (1994).
[CrossRef]

Shahabadi, M.

Shutian, L.

C. Lixue, D. Xiaoxu, D. Weiqiang, Cao Liangcai, and L. Shutian, "Finite-difference time-domain analysis of optical bistability with low threshold in one-dimensional nonlinear photonic crystal with Kerr medium," Opt. Commun. 209, 491-500 (2002).
[CrossRef]

Sipe, J. E.

N. A. R. Bhat and J. E. Sipe, "Optical pulse propagation in nonlinear photonic crystals," Phys. Rev. E 64, 056604 (2001).
[CrossRef]

C. M. de Sterke, D. J. Salinas, and J. E. Sipe, "Coupled-mode theory for light propagation through deep nonlinear gratings," Phys. Rev. E 54, 1969-1989 (1996).
[CrossRef]

Slusher, R. E.

Soljacic, M.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistability in nonlinear photonic crystals," Phys. Rev. E 66, 055601R (2002).
[CrossRef]

Stark, J. B.

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2001).

Takenaka, M.

S. H. Jeong, T. Mizumoto, K. Nakatsuhara, M. Takenaka, and Y. Nakano, "Deep-ridge distributed feedback waveguide for polarization independent all-optical switching," Electron. Lett. 37, 498-499 (2001).
[CrossRef]

Taverner, D.

Tran, P.

Trillo, S.

A. Parini, G. Bellanca, S. Trillo, L. Saccomandi, and P. Bassi, "Transfer matrix and full Maxwell time domain analysis of nonlinear gratings," Opt. Quantum Electron. 36, 189-199 (2004).
[CrossRef]

Vengsarkar, A. M.

Wang, Z. L.

Z. L. Wang, X. J. Liu, J. Wu, and N. B. Ming, "An effective Bloch wave theory for light propagation in nonlinear media," Phys. Lett. A 248, 263-266 (1998).
[CrossRef]

Weinstein, M. I.

R. H. Goodman, M. I. Weinstein, and P. J. Holmes, "Nonlinear propagation of light in one-dimensional periodic structures," J. Nonlinear Sci. 11, 123-168 (2001).

Weiqiang, D.

C. Lixue, D. Xiaoxu, D. Weiqiang, Cao Liangcai, and L. Shutian, "Finite-difference time-domain analysis of optical bistability with low threshold in one-dimensional nonlinear photonic crystal with Kerr medium," Opt. Commun. 209, 491-500 (2002).
[CrossRef]

Wiegmann, W.

Y. H. Lee, A. Chavez-Pirson, S. W. Koch, H. M. Gibbs, S. H. Park, J. Morhange, A. Jeffery, N. Peyghambarian, L. Banyai, A. C. Gossard, and W. Wiegmann, "Room-temperature optical nonlinearities in GaAs," Phys. Rev. Lett. 57, 2446-2449 (1986).
[CrossRef] [PubMed]

Wu, J.

Z. L. Wang, X. J. Liu, J. Wu, and N. B. Ming, "An effective Bloch wave theory for light propagation in nonlinear media," Phys. Lett. A 248, 263-266 (1998).
[CrossRef]

Xiaoxu, D.

C. Lixue, D. Xiaoxu, D. Weiqiang, Cao Liangcai, and L. Shutian, "Finite-difference time-domain analysis of optical bistability with low threshold in one-dimensional nonlinear photonic crystal with Kerr medium," Opt. Commun. 209, 491-500 (2002).
[CrossRef]

Yariv, A.

Yeh, P.

Zamani Aghaie, K.

Zheltikov, A. M.

I. S. Nefedov, V. N. Gusyatnikov, P. K. Kashkarov, and A. M. Zheltikov, "Low-threshold photonic band gap optical logic gates," Laser Phys. 10, 640-644 (2000).

Appl. Phys. Lett.

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M. A. Dupertuis, and J. M. Glinski, "All-optical reflectivity tuning and logic gating in a GaAs/AlAs periodic layered structure," Appl. Phys. Lett. 60, 404-406 (1992).
[CrossRef]

Electron. Lett.

S. H. Jeong, T. Mizumoto, K. Nakatsuhara, M. Takenaka, and Y. Nakano, "Deep-ridge distributed feedback waveguide for polarization independent all-optical switching," Electron. Lett. 37, 498-499 (2001).
[CrossRef]

IEEE J. Quantum Electron.

J. Poon, E. Istrate, M. Allard, and E. H. Sargent, "Multiple-scale analysis of photonic crystal waveguides," IEEE J. Quantum Electron. 39, 778-786 (2005).
[CrossRef]

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

IEEE Photon. Technol. Lett.

A. Melloni, M. Chinello, and M. Martinelli, "All-optical switching in phase-shifted fiber Bragg grating," IEEE Photon. Technol. Lett. 12, 42-44 (2000).
[CrossRef]

Int. J. Circuit Theory Appl.

L. O. Chua, "A switching-parameter algorithm for finding multiple solutions of nonlinear resistive circuits," Int. J. Circuit Theory Appl. 4, 215-239 (1976).
[CrossRef]

J. Appl. Phys.

S. Adachi, "GaAs, AlAs, and AlxGa1−xAs; Material parameters for use in research and device applications," J. Appl. Phys. 58, R1-R29 (1985).
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Pethal, J. P. Dowling, C. M. Bowden, and A. S. Manka, "Transparent, metallo-dielectric one-dimensional, photonic band-gap structures," J. Appl. Phys. 83, 2377-2383 (1998).
[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]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, "The photonic band edge laser: A new approach to gain enhancement," J. Appl. Phys. 75, 1896-1899 (1994).
[CrossRef]

J. Lightwave Technol.

J. Nonlinear Sci.

R. H. Goodman, M. I. Weinstein, and P. J. Holmes, "Nonlinear propagation of light in one-dimensional periodic structures," J. Nonlinear Sci. 11, 123-168 (2001).

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

J. Phys. III France

P. St. J. Russell and J. L. Archambault, "Field microstructure and spatial instability of photonic Bloch waves in nonlinear periodic media," J. Phys. III France 4, 2471-2491 (1994).
[CrossRef]

Laser Phys.

I. S. Nefedov, V. N. Gusyatnikov, P. K. Kashkarov, and A. M. Zheltikov, "Low-threshold photonic band gap optical logic gates," Laser Phys. 10, 640-644 (2000).

Opt. Commun.

C. Lixue, D. Xiaoxu, D. Weiqiang, Cao Liangcai, and L. Shutian, "Finite-difference time-domain analysis of optical bistability with low threshold in one-dimensional nonlinear photonic crystal with Kerr medium," Opt. Commun. 209, 491-500 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

A. Parini, G. Bellanca, S. Trillo, L. Saccomandi, and P. Bassi, "Transfer matrix and full Maxwell time domain analysis of nonlinear gratings," Opt. Quantum Electron. 36, 189-199 (2004).
[CrossRef]

Phys. Lett. A

Z. L. Wang, X. J. Liu, J. Wu, and N. B. Ming, "An effective Bloch wave theory for light propagation in nonlinear media," Phys. Lett. A 248, 263-266 (1998).
[CrossRef]

Phys. Rev. E

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistability in nonlinear photonic crystals," Phys. Rev. E 66, 055601R (2002).
[CrossRef]

J. M. Bendickson, J. P. Dowling, and M. Scalora, "Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures," Phys. Rev. E 53, 4107-4121 (1996).
[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]

C. M. de Sterke, D. J. Salinas, and J. E. Sipe, "Coupled-mode theory for light propagation through deep nonlinear gratings," Phys. Rev. E 54, 1969-1989 (1996).
[CrossRef]

N. A. R. Bhat and J. E. Sipe, "Optical pulse propagation in nonlinear photonic crystals," Phys. Rev. E 64, 056604 (2001).
[CrossRef]

Phys. Rev. Lett.

Y. H. Lee, A. Chavez-Pirson, S. W. Koch, H. M. Gibbs, S. H. Park, J. Morhange, A. Jeffery, N. Peyghambarian, L. Banyai, A. C. Gossard, and W. Wiegmann, "Room-temperature optical nonlinearities in GaAs," Phys. Rev. Lett. 57, 2446-2449 (1986).
[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]

Other

L. Brillouin, Wave Propagation in Periodic Structures (McGraw-Hill, 1946).

R.E.Slusher and B.J.Eggleton, eds., Nonlinear Photonic Crystals (Springer, 2003).

E.D.Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2001).

G. P. Agrawal, Nonlinear Fiber Optics (Academic,1995).

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Figures (3)

Fig. 1
Fig. 1

One-dimensional periodic structure consisting of two alternating semiconductor layers of widths l 1 and l 2 , respectively, and an overall period of L x = l 1 + l 2 . An equivalent transmission line model of this structure is also seen in the figure.

Fig. 2
Fig. 2

Reflection coefficient of a 30 period GaAs–AlAs superlattice ( d 1 = 53.8 and d 2 = 74.4 nm ) versus wavelength in the linear regime. Solid curve, our results. Circles, FDTD results. Dashed curve, the results obtained using the conventional coupled-mode theory (Ref. [18]).

Fig. 3
Fig. 3

Reflected intensity, (a)–(c) along with transmitted intensity, (d)–(f), versus incident intensity for a GaAs–AlAs superlattice, at three different wavelengths obtained using the proposed method (solid curve), NFDTD (dotted curve), and the conventional coupled-mode theory (dashed curve) (Ref. [18]).

Equations (27)

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

ε r ( z ) = { n 1 2 j 2 n 1 κ 1 + 2 n 1 n 1 E x 2 in region 1 n 2 2 in region 2 } ,
d E x d z = j ω μ o H y ,
d H y d z = j ω ε o ε r ( z ) E x .
{ d 2 E x d z 2 + k o 2 ( ε r 1 + α E x 2 ) E x = 0 in region 1 d 2 E x d z 2 + k o 2 ε r 2 E x = 0 in region 2 } ,
{ d V d z = j ω μ o I d I d z = j ω ε o ( ε r 1 + α V 2 ) V } ,
d 2 V d z 2 + k o 2 ( ε r 1 + α V 2 ) V = 0 ,
V ( z ) = m = 0 α m V ( m ) = V ( 0 ) ( z , ζ ) + α V ( 1 ) ( z , ζ ) + .
d 2 V d z 2 = 2 V ( 0 ) z 2 + α ( 2 2 V ( 0 ) ζ z + 2 V ( 1 ) z 2 ) + O ( α 2 ) .
2 V ( 0 ) z 2 + β 1 2 V ( 0 ) = 0 ,
2 V ( 1 ) z 2 + β 1 2 V ( 1 ) = k o 2 V ( 0 ) 2 V ( 0 ) 2 2 V ( 0 ) ζ z ,
V ( 0 ) ( z , ζ ) = a ( ζ ) exp j β 1 z + b ( ζ ) exp j β 1 z ,
V n ( z ) = a n exp ( j ( k 1 n + β 1 ) ( z z n ) ) + b n exp ( j ( k 2 n + β 1 ) ( z z n ) ) ,
I n ( z ) = a n Z c 1 [ 1 + n 1 ε r 1 ( a n 2 + 2 b n 2 ) ] exp ( j ( k 1 n + β 1 ) ( z z n ) ) b n Z c 1 [ 1 + n 1 ε r 1 ( 2 a n 2 + b n 2 ) ] exp ( j ( k 2 n + β 1 ) ( z z n ) ) ,
V n ( z ) = c n exp ( j β 2 ( z z n l 1 ) ) + d n exp ( j β 2 ( z z n l 1 ) ) ,
I n ( z ) = c n Z c 2 exp ( j β 2 ( z z n l 1 ) ) d n Z c 2 exp ( j β 2 ( z z n l 1 ) ) ,
c n + d n = a n exp ( j ( k 1 n + β 1 ) l 1 ) + b n exp ( j ( k 2 n + β 1 ) l 1 ) ,
c n d n Z c 2 = a n Z c 1 [ 1 + n 1 ε r 1 ( a n 2 + 2 b n 2 ) ] exp ( j ( k 1 n + β 1 ) l 1 ) b n Z c 1 [ 1 + n 1 ε r 1 ( 2 a n 2 + b n 2 ) ] exp ( j ( k 2 n + β 1 ) l 1 ) .
a n + 1 + b n + 1 = c n exp ( j β 2 l 2 ) + d n exp ( j β 2 l 2 ) ,
a n + 1 b n + 1 Z c 1 = c n Z c 2 exp ( j β 2 l 2 ) d n Z c 2 exp ( j β 2 l 2 ) .
[ a n + 1 ( 1 + Z c 2 Z c 1 ) + b n + 1 ( 1 Z c 2 Z c 1 ) ] exp ( j β 2 l 2 ) = a n { 1 + Z c 2 Z c 1 [ 1 + n 1 ε r 1 ( a n 2 + 2 b n 2 ) ] } exp ( j ( k 1 n + β 1 ) l 1 ) + b n { 1 Z c 2 Z c 1 [ 1 + n 1 ε r 1 ( 2 a n 2 + b n 2 ) ] } exp ( j ( k 2 n + β 1 ) l 1 ) ,
[ a n + 1 ( 1 Z c 2 Z c 1 ) + b n + 1 ( 1 + Z c 2 Z c 1 ) ] exp ( j β 2 l 2 ) = a n { 1 Z c 2 Z c 1 [ 1 + n 1 ε r 1 ( a n 2 + 2 b n 2 ) ] } exp ( j ( k 1 n + β 1 ) l 1 ) + b n { 1 + Z c 2 Z c 1 [ 1 + n 1 ε r 1 ( 2 a n 2 + b n 2 ) ] } exp ( j ( k 2 n + β 1 ) l 1 ) .
d D x d t + J x = d H y d z ,
μ o d H y d t = d E x d z ,
D x = ε o ( ε r + χ x x x x ( 3 ) E x 2 ) E x ,
H y i + ( 1 2 ) n + ( 1 2 ) = H y i + ( 1 2 ) n ( 1 2 ) Δ t μ o Δ z ( E x i + 1 n E x i n ) .
E x i = 0 n = Λ ( n ) E o exp ( ω n Δ t ) ,
Λ ( n ) = { 1 2 n n max 1 1 n n max 0 otherwise } ,

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