Abstract

A theoretical analysis of the nonlinear quadratic interaction between two monochromatic plane waves at frequencies ω and 2ω in a Fabry–Perot cavity shows that all-optical control of reflectivity and transmissivity of the fundamental beam is possible under suitable conditions. We report all-optical switching of the fundamental beam for cavity lengths that vary from a few micrometers to a few millimeters. Specifically, we show that 100% all-optical switching can occur for a 6-µm cavity when the nonlinear coefficient is not greater than 20 pm/V and the input intensity does not exceed 10 GW/cm2. Our analytical results are obtained without use of the slowly varying envelope approximation and without resort to the undepleted-pump approximation.

© 2000 Optical Society of America

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1999 (1)

C. Cojocaru, J. Martorell, R. Vilaseca, E. Fazio, and J. Trull, “Active reflection via phase-insensitive quadratic nonlinear interaction within a microcavity,” Appl. Phys. Lett. 74, 504–506 (1999).
[CrossRef]

1998 (3)

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2382 (1998).
[CrossRef]

M. J. Bloemer and M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1998).
[CrossRef]

G. D’Aguanno, C. Sibilia, E. Fazio, E. Ferrari, and M. Bertolotti, “Field phase modulation and input phase and intensity dependence in a nonlinear second-order interaction,” J. Mod. Opt. 45, 1049–1066 (1998).
[CrossRef]

1997 (5)

1996 (4)

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, 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, 1078–1081 (1996).
[CrossRef]

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

E. Fazio, C. Sibilia, F. Senesi, and M. Bertolotti, “All-optical switching during quasi-collinear second-harmonic generation,” Opt. Commun. 127, 62–68 (1996).
[CrossRef]

A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: an analytical study,” Phys. Rev. A 54, 3455–3471 (1996).
[CrossRef] [PubMed]

1995 (2)

Z. Y. Ou, “Observation of nonlinear phase shift in cw harmonic generation,” Opt. Commun. 124, 430–437 (1995).
[CrossRef]

A. Re, C. Sibilia, E. Fazio, and M. Bertolotti, “Field dependent effects in a quadratic nonlinear medium,” J. Mod. Opt. 42, 823–839 (1995).
[CrossRef]

1994 (2)

R. Paschotta, M. Collet, P. Kurz, K. Fiedler, H. A. Bachor, and J. Mlynek, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
[CrossRef] [PubMed]

A. L. Belostotsky, A. S. Leonov, and A. V. Maleshko, “Nonlinear phase change in a type II second-harmonic generation under exact phase-matched conditions,” Opt. Lett. 19, 856–858 (1994).
[CrossRef] [PubMed]

1993 (3)

1992 (1)

1991 (1)

D. G. Hall, “A comment on the coupled-mode equations used in guided-wave optics,” Opt. Commun. 82, 453–455 (1991).
[CrossRef]

1990 (1)

B. Crosignani, P. Di Porto, and A. Yariv, “Coupled-mode theory and slowly-varying approximation in guided-wave optics,” Opt. Commun. 78, 237–239 (1990).
[CrossRef]

1989 (1)

N. R. Belashenkov, S. V. Gagarskij, and M. V. Inochkin, “Nonlinear refraction of light on second harmonic generation,” Opt. Spectrosc. (USSR) 66, 806–808 (1989).

1987 (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

1983 (1)

1979 (1)

1974 (2)

D. N. Klyshko and B. F. Polkovnikov, “Phase modulation and self-modulation of light in three-photon processes,” Sov. J. Quantum Electron. 3, 324–326 (1974).
[CrossRef]

S. A. Akhmanov, A. N. Dubovik, S. M. Saltiel, I. V. Tomov, and V. G. Tunkin, “Nonlinear optical effects of fourth order in the field in a lithium formiate crystal,” JETP Lett. 20, 117–118 (1974).

1972 (2)

E. Yablonovitch, C. Flytzanis, and N. Bloembergen, “Anisotropic interference of three-wave and double-two wave frequency mixing in GaAs,” Phys. Rev. Lett. 29, 865–868 (1972).
[CrossRef]

J. M. R. Thomas and J. P. E. Taran, “Pulse distortion in mismatched second harmonic generation,” Opt. Commun. 4, 329–334 (1972).
[CrossRef]

1971 (1)

J. M. Jarborough and O. Amman, “Simultaneous optical parametric oscillation, second harmonic generation and difference-frequency generation,” Appl. Phys. Lett. 18, 145–147 (1971).
[CrossRef]

1970 (1)

T. K. Gustafson, J. P. E. Taran, P. L. Kelley, and R. Y. Chiao, “Self-modulation of picosecond pulses in electro-optics crystals,” Opt. Commun. 2, 17–21 (1970).
[CrossRef]

1969 (1)

J. P. Coffinet and F. De Martini, “Coherent excitation of polaritons in gallium phosphide,” Phys. Rev. Lett. 22, 60–63 (1969).
[CrossRef]

1967 (1)

L. A. Ostrovskij, “Self-action of light in crystals,” JETP Lett. 5, 272–275 (1967).

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, A. N. Dubovik, S. M. Saltiel, I. V. Tomov, and V. G. Tunkin, “Nonlinear optical effects of fourth order in the field in a lithium formiate crystal,” JETP Lett. 20, 117–118 (1974).

Amman, O.

J. M. Jarborough and O. Amman, “Simultaneous optical parametric oscillation, second harmonic generation and difference-frequency generation,” Appl. Phys. Lett. 18, 145–147 (1971).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Assanto, G.

Bachor, H. A.

R. Paschotta, M. Collet, P. Kurz, K. Fiedler, H. A. Bachor, and J. Mlynek, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
[CrossRef] [PubMed]

Banfi, G. P.

Belashenkov, N. R.

N. R. Belashenkov, S. V. Gagarskij, and M. V. Inochkin, “Nonlinear refraction of light on second harmonic generation,” Opt. Spectrosc. (USSR) 66, 806–808 (1989).

Belostotsky, A. L.

Bendickson, J. M.

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

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, 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, 1078–1081 (1996).
[CrossRef]

Berger, V.

Bertolotti, M.

G. D’Aguanno, C. Sibilia, E. Fazio, E. Ferrari, and M. Bertolotti, “Field phase modulation and input phase and intensity dependence in a nonlinear second-order interaction,” J. Mod. Opt. 45, 1049–1066 (1998).
[CrossRef]

G. D’Aguanno, C. Sibilia, E. Fazio, and M. Bertolotti, “Three-wave mixing in a quadratic material under perfect phase-matching,” Opt. Commun. 142, 75–78 (1997).
[CrossRef]

E. Fazio, C. Sibilia, F. Senesi, and M. Bertolotti, “All-optical switching during quasi-collinear second-harmonic generation,” Opt. Commun. 127, 62–68 (1996).
[CrossRef]

A. Re, C. Sibilia, E. Fazio, and M. Bertolotti, “Field dependent effects in a quadratic nonlinear medium,” J. Mod. Opt. 42, 823–839 (1995).
[CrossRef]

Bloembergen, N.

E. Yablonovitch, C. Flytzanis, and N. Bloembergen, “Anisotropic interference of three-wave and double-two wave frequency mixing in GaAs,” Phys. Rev. Lett. 29, 865–868 (1972).
[CrossRef]

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bloemer, M. J.

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2382 (1998).
[CrossRef]

M. J. Bloemer and M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1998).
[CrossRef]

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, 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, 1078–1081 (1996).
[CrossRef]

Bowden, C. M.

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2382 (1998).
[CrossRef]

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, 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, 1078–1081 (1996).
[CrossRef]

Chiao, R. Y.

T. K. Gustafson, J. P. E. Taran, P. L. Kelley, and R. Y. Chiao, “Self-modulation of picosecond pulses in electro-optics crystals,” Opt. Commun. 2, 17–21 (1970).
[CrossRef]

Coffinet, J. P.

J. P. Coffinet and F. De Martini, “Coherent excitation of polaritons in gallium phosphide,” Phys. Rev. Lett. 22, 60–63 (1969).
[CrossRef]

Cojocaru, C.

C. Cojocaru, J. Martorell, R. Vilaseca, E. Fazio, and J. Trull, “Active reflection via phase-insensitive quadratic nonlinear interaction within a microcavity,” Appl. Phys. Lett. 74, 504–506 (1999).
[CrossRef]

Collet, M.

R. Paschotta, M. Collet, P. Kurz, K. Fiedler, H. A. Bachor, and J. Mlynek, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
[CrossRef] [PubMed]

Crosignani, B.

B. Crosignani, P. Di Porto, and A. Yariv, “Coupled-mode theory and slowly-varying approximation in guided-wave optics,” Opt. Commun. 78, 237–239 (1990).
[CrossRef]

D’Aguanno, G.

G. D’Aguanno, C. Sibilia, E. Fazio, E. Ferrari, and M. Bertolotti, “Field phase modulation and input phase and intensity dependence in a nonlinear second-order interaction,” J. Mod. Opt. 45, 1049–1066 (1998).
[CrossRef]

G. D’Aguanno, C. Sibilia, E. Fazio, and M. Bertolotti, “Three-wave mixing in a quadratic material under perfect phase-matching,” Opt. Commun. 142, 75–78 (1997).
[CrossRef]

Danielius, R.

De Martini, F.

L. A. Lugiato, G. Strini, and F. De Martini, “Squeezed states in second-harmonic generation,” Opt. Lett. 8, 256–258 (1983).
[CrossRef] [PubMed]

J. P. Coffinet and F. De Martini, “Coherent excitation of polaritons in gallium phosphide,” Phys. Rev. Lett. 22, 60–63 (1969).
[CrossRef]

Debray, J. P.

DeSalvo, R.

Di Porto, P.

B. Crosignani, P. Di Porto, and A. Yariv, “Coupled-mode theory and slowly-varying approximation in guided-wave optics,” Opt. Commun. 78, 237–239 (1990).
[CrossRef]

Di Trapani, P.

Dowling, J. P.

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2382 (1998).
[CrossRef]

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, 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, 1078–1081 (1996).
[CrossRef]

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

Dubietis, A.

Dubovik, A. N.

S. A. Akhmanov, A. N. Dubovik, S. M. Saltiel, I. V. Tomov, and V. G. Tunkin, “Nonlinear optical effects of fourth order in the field in a lithium formiate crystal,” JETP Lett. 20, 117–118 (1974).

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Fazio, E.

C. Cojocaru, J. Martorell, R. Vilaseca, E. Fazio, and J. Trull, “Active reflection via phase-insensitive quadratic nonlinear interaction within a microcavity,” Appl. Phys. Lett. 74, 504–506 (1999).
[CrossRef]

G. D’Aguanno, C. Sibilia, E. Fazio, E. Ferrari, and M. Bertolotti, “Field phase modulation and input phase and intensity dependence in a nonlinear second-order interaction,” J. Mod. Opt. 45, 1049–1066 (1998).
[CrossRef]

G. D’Aguanno, C. Sibilia, E. Fazio, and M. Bertolotti, “Three-wave mixing in a quadratic material under perfect phase-matching,” Opt. Commun. 142, 75–78 (1997).
[CrossRef]

E. Fazio, C. Sibilia, F. Senesi, and M. Bertolotti, “All-optical switching during quasi-collinear second-harmonic generation,” Opt. Commun. 127, 62–68 (1996).
[CrossRef]

A. Re, C. Sibilia, E. Fazio, and M. Bertolotti, “Field dependent effects in a quadratic nonlinear medium,” J. Mod. Opt. 42, 823–839 (1995).
[CrossRef]

Ferrari, E.

G. D’Aguanno, C. Sibilia, E. Fazio, E. Ferrari, and M. Bertolotti, “Field phase modulation and input phase and intensity dependence in a nonlinear second-order interaction,” J. Mod. Opt. 45, 1049–1066 (1998).
[CrossRef]

Fiedler, K.

R. Paschotta, M. Collet, P. Kurz, K. Fiedler, H. A. Bachor, and J. Mlynek, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
[CrossRef] [PubMed]

Flynn, R. J.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, 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, 1078–1081 (1996).
[CrossRef]

Flytzanis, C.

E. Yablonovitch, C. Flytzanis, and N. Bloembergen, “Anisotropic interference of three-wave and double-two wave frequency mixing in GaAs,” Phys. Rev. Lett. 29, 865–868 (1972).
[CrossRef]

Fork, R. L.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, 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, 1078–1081 (1996).
[CrossRef]

Gagarskij, S. V.

N. R. Belashenkov, S. V. Gagarskij, and M. V. Inochkin, “Nonlinear refraction of light on second harmonic generation,” Opt. Spectrosc. (USSR) 66, 806–808 (1989).

Gustafson, T. K.

T. K. Gustafson, J. P. E. Taran, P. L. Kelley, and R. Y. Chiao, “Self-modulation of picosecond pulses in electro-optics crystals,” Opt. Commun. 2, 17–21 (1970).
[CrossRef]

Hagan, D. J.

Hall, D. G.

D. G. Hall, “A comment on the coupled-mode equations used in guided-wave optics,” Opt. Commun. 82, 453–455 (1991).
[CrossRef]

Harmand, J. C.

Inochkin, M. V.

N. R. Belashenkov, S. V. Gagarskij, and M. V. Inochkin, “Nonlinear refraction of light on second harmonic generation,” Opt. Spectrosc. (USSR) 66, 806–808 (1989).

Jarborough, J. M.

J. M. Jarborough and O. Amman, “Simultaneous optical parametric oscillation, second harmonic generation and difference-frequency generation,” Appl. Phys. Lett. 18, 145–147 (1971).
[CrossRef]

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Kaplan, A. E.

Kelley, P. L.

T. K. Gustafson, J. P. E. Taran, P. L. Kelley, and R. Y. Chiao, “Self-modulation of picosecond pulses in electro-optics crystals,” Opt. Commun. 2, 17–21 (1970).
[CrossRef]

Klyshko, D. N.

D. N. Klyshko and B. F. Polkovnikov, “Phase modulation and self-modulation of light in three-photon processes,” Sov. J. Quantum Electron. 3, 324–326 (1974).
[CrossRef]

Kobyakov, A.

A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: an analytical study,” Phys. Rev. A 54, 3455–3471 (1996).
[CrossRef] [PubMed]

Kurz, P.

R. Paschotta, M. Collet, P. Kurz, K. Fiedler, H. A. Bachor, and J. Mlynek, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
[CrossRef] [PubMed]

Leavitt, R. P.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, 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, 1078–1081 (1996).
[CrossRef]

Ledbetter, H. S.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, 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, 1078–1081 (1996).
[CrossRef]

Lederer, F.

A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: an analytical study,” Phys. Rev. A 54, 3455–3471 (1996).
[CrossRef] [PubMed]

Leonov, A. S.

Levenson, J. A.

Lovering, D. J.

Lugiato, L. A.

Maleshko, A. V.

Manka, A. S.

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2382 (1998).
[CrossRef]

Martorell, J.

C. Cojocaru, J. Martorell, R. Vilaseca, E. Fazio, and J. Trull, “Active reflection via phase-insensitive quadratic nonlinear interaction within a microcavity,” Appl. Phys. Lett. 74, 504–506 (1999).
[CrossRef]

Mlynek, J.

R. Paschotta, M. Collet, P. Kurz, K. Fiedler, H. A. Bachor, and J. Mlynek, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
[CrossRef] [PubMed]

Ostrovskij, L. A.

L. A. Ostrovskij, “Self-action of light in crystals,” JETP Lett. 5, 272–275 (1967).

Ou, Z. Y.

Z. Y. Ou, “Observation of nonlinear phase shift in cw harmonic generation,” Opt. Commun. 124, 430–437 (1995).
[CrossRef]

Paschotta, R.

R. Paschotta, M. Collet, P. Kurz, K. Fiedler, H. A. Bachor, and J. Mlynek, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
[CrossRef] [PubMed]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Pethel, A. S.

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2382 (1998).
[CrossRef]

Piskarskas, A.

Podenas, D.

Polkovnikov, B. F.

D. N. Klyshko and B. F. Polkovnikov, “Phase modulation and self-modulation of light in three-photon processes,” Sov. J. Quantum Electron. 3, 324–326 (1974).
[CrossRef]

Re, A.

A. Re, C. Sibilia, E. Fazio, and M. Bertolotti, “Field dependent effects in a quadratic nonlinear medium,” J. Mod. Opt. 42, 823–839 (1995).
[CrossRef]

Reinhardt, S. B.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, 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, 1078–1081 (1996).
[CrossRef]

Russell, P. St. J.

Saltiel, S. M.

S. A. Akhmanov, A. N. Dubovik, S. M. Saltiel, I. V. Tomov, and V. G. Tunkin, “Nonlinear optical effects of fourth order in the field in a lithium formiate crystal,” JETP Lett. 20, 117–118 (1974).

Scalora, M.

M. J. Bloemer and M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1998).
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2382 (1998).
[CrossRef]

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, 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, 1078–1081 (1996).
[CrossRef]

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

Senesi, F.

E. Fazio, C. Sibilia, F. Senesi, and M. Bertolotti, “All-optical switching during quasi-collinear second-harmonic generation,” Opt. Commun. 127, 62–68 (1996).
[CrossRef]

Sheik-Bahae, M.

Sibilia, C.

G. D’Aguanno, C. Sibilia, E. Fazio, E. Ferrari, and M. Bertolotti, “Field phase modulation and input phase and intensity dependence in a nonlinear second-order interaction,” J. Mod. Opt. 45, 1049–1066 (1998).
[CrossRef]

G. D’Aguanno, C. Sibilia, E. Fazio, and M. Bertolotti, “Three-wave mixing in a quadratic material under perfect phase-matching,” Opt. Commun. 142, 75–78 (1997).
[CrossRef]

E. Fazio, C. Sibilia, F. Senesi, and M. Bertolotti, “All-optical switching during quasi-collinear second-harmonic generation,” Opt. Commun. 127, 62–68 (1996).
[CrossRef]

A. Re, C. Sibilia, E. Fazio, and M. Bertolotti, “Field dependent effects in a quadratic nonlinear medium,” J. Mod. Opt. 42, 823–839 (1995).
[CrossRef]

Simonneu, C.

Sipe, J. E.

Stegeman, G. I.

Strini, G.

Taran, J. P. E.

J. M. R. Thomas and J. P. E. Taran, “Pulse distortion in mismatched second harmonic generation,” Opt. Commun. 4, 329–334 (1972).
[CrossRef]

T. K. Gustafson, J. P. E. Taran, P. L. Kelley, and R. Y. Chiao, “Self-modulation of picosecond pulses in electro-optics crystals,” Opt. Commun. 2, 17–21 (1970).
[CrossRef]

Thomas, J. M. R.

J. M. R. Thomas and J. P. E. Taran, “Pulse distortion in mismatched second harmonic generation,” Opt. Commun. 4, 329–334 (1972).
[CrossRef]

Tocci, M. D.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, 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, 1078–1081 (1996).
[CrossRef]

Tomov, I. V.

S. A. Akhmanov, A. N. Dubovik, S. M. Saltiel, I. V. Tomov, and V. G. Tunkin, “Nonlinear optical effects of fourth order in the field in a lithium formiate crystal,” JETP Lett. 20, 117–118 (1974).

Toruellas, W. E.

Trull, J.

C. Cojocaru, J. Martorell, R. Vilaseca, E. Fazio, and J. Trull, “Active reflection via phase-insensitive quadratic nonlinear interaction within a microcavity,” Appl. Phys. Lett. 74, 504–506 (1999).
[CrossRef]

Tunkin, V. G.

S. A. Akhmanov, A. N. Dubovik, S. M. Saltiel, I. V. Tomov, and V. G. Tunkin, “Nonlinear optical effects of fourth order in the field in a lithium formiate crystal,” JETP Lett. 20, 117–118 (1974).

Van Stryland, E. W.

Vanherzeele, H.

Vidakovic, P.

Vilaseca, R.

C. Cojocaru, J. Martorell, R. Vilaseca, E. Fazio, and J. Trull, “Active reflection via phase-insensitive quadratic nonlinear interaction within a microcavity,” Appl. Phys. Lett. 74, 504–506 (1999).
[CrossRef]

Wang, Z.

Webjorn, J.

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

E. Yablonovitch, C. Flytzanis, and N. Bloembergen, “Anisotropic interference of three-wave and double-two wave frequency mixing in GaAs,” Phys. Rev. Lett. 29, 865–868 (1972).
[CrossRef]

Yariv, A.

B. Crosignani, P. Di Porto, and A. Yariv, “Coupled-mode theory and slowly-varying approximation in guided-wave optics,” Opt. Commun. 78, 237–239 (1990).
[CrossRef]

Zyss, J.

Appl. Phys. Lett. (3)

J. M. Jarborough and O. Amman, “Simultaneous optical parametric oscillation, second harmonic generation and difference-frequency generation,” Appl. Phys. Lett. 18, 145–147 (1971).
[CrossRef]

C. Cojocaru, J. Martorell, R. Vilaseca, E. Fazio, and J. Trull, “Active reflection via phase-insensitive quadratic nonlinear interaction within a microcavity,” Appl. Phys. Lett. 74, 504–506 (1999).
[CrossRef]

M. J. Bloemer and M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1998).
[CrossRef]

J. Appl. Phys. (1)

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2382 (1998).
[CrossRef]

J. Mod. Opt. (2)

A. Re, C. Sibilia, E. Fazio, and M. Bertolotti, “Field dependent effects in a quadratic nonlinear medium,” J. Mod. Opt. 42, 823–839 (1995).
[CrossRef]

G. D’Aguanno, C. Sibilia, E. Fazio, E. Ferrari, and M. Bertolotti, “Field phase modulation and input phase and intensity dependence in a nonlinear second-order interaction,” J. Mod. Opt. 45, 1049–1066 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

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

JETP Lett. (2)

S. A. Akhmanov, A. N. Dubovik, S. M. Saltiel, I. V. Tomov, and V. G. Tunkin, “Nonlinear optical effects of fourth order in the field in a lithium formiate crystal,” JETP Lett. 20, 117–118 (1974).

L. A. Ostrovskij, “Self-action of light in crystals,” JETP Lett. 5, 272–275 (1967).

Opt. Commun. (7)

T. K. Gustafson, J. P. E. Taran, P. L. Kelley, and R. Y. Chiao, “Self-modulation of picosecond pulses in electro-optics crystals,” Opt. Commun. 2, 17–21 (1970).
[CrossRef]

J. M. R. Thomas and J. P. E. Taran, “Pulse distortion in mismatched second harmonic generation,” Opt. Commun. 4, 329–334 (1972).
[CrossRef]

G. D’Aguanno, C. Sibilia, E. Fazio, and M. Bertolotti, “Three-wave mixing in a quadratic material under perfect phase-matching,” Opt. Commun. 142, 75–78 (1997).
[CrossRef]

Z. Y. Ou, “Observation of nonlinear phase shift in cw harmonic generation,” Opt. Commun. 124, 430–437 (1995).
[CrossRef]

E. Fazio, C. Sibilia, F. Senesi, and M. Bertolotti, “All-optical switching during quasi-collinear second-harmonic generation,” Opt. Commun. 127, 62–68 (1996).
[CrossRef]

B. Crosignani, P. Di Porto, and A. Yariv, “Coupled-mode theory and slowly-varying approximation in guided-wave optics,” Opt. Commun. 78, 237–239 (1990).
[CrossRef]

D. G. Hall, “A comment on the coupled-mode equations used in guided-wave optics,” Opt. Commun. 82, 453–455 (1991).
[CrossRef]

Opt. Lett. (8)

P. Vidakovic, D. J. Lovering, J. A. Levenson, J. Webjorn, and P. St. J. Russell, “Large nonlinear phase shift owing to cascaded χ(2) in quasi-phase matched bulk LiNbO3,” Opt. Lett. 22, 277–279 (1997).
[CrossRef]

C. Simonneu, J. P. Debray, J. C. Harmand, P. Vidakovic, D. J. Lovering, and J. A. Levenson, “Second harmonic generation in a doubly resonant semiconductor microcavity,” Opt. Lett. 23, 1775–1777 (1997).
[CrossRef]

L. A. Lugiato, G. Strini, and F. De Martini, “Squeezed states in second-harmonic generation,” Opt. Lett. 8, 256–258 (1983).
[CrossRef] [PubMed]

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. I. Stegeman, E. W. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–30 (1992).
[CrossRef] [PubMed]

G. I. Stegeman, M. Sheik-Bahae, E. W. Van Stryland, and G. Assanto, “Large nonlinear phase shifts in second-order nonlinear optical processes,” Opt. Lett. 18, 13–15 (1993).
[CrossRef] [PubMed]

R. Danielius, P. Di Trapani, A. Dubietis, A. Piskarskas, D. Podenas, and G. P. Banfi, “Self-diffraction through cascaded second-order frequency-mixing effects in β-barium borate,” Opt. Lett. 18, 574–576 (1993).
[CrossRef]

A. E. Kaplan, “Eigenmodes of χ(2) wave mixings: cross-induced second-order nonlinear refraction,” Opt. Lett. 18, 1223–1225 (1993).
[CrossRef] [PubMed]

A. L. Belostotsky, A. S. Leonov, and A. V. Maleshko, “Nonlinear phase change in a type II second-harmonic generation under exact phase-matched conditions,” Opt. Lett. 19, 856–858 (1994).
[CrossRef] [PubMed]

Opt. Spectrosc. (USSR) (1)

N. R. Belashenkov, S. V. Gagarskij, and M. V. Inochkin, “Nonlinear refraction of light on second harmonic generation,” Opt. Spectrosc. (USSR) 66, 806–808 (1989).

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Phys. Rev. A (1)

A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: an analytical study,” Phys. Rev. A 54, 3455–3471 (1996).
[CrossRef] [PubMed]

Phys. Rev. E (2)

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, 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, 1078–1081 (1996).
[CrossRef]

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

Phys. Rev. Lett. (5)

E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

R. Paschotta, M. Collet, P. Kurz, K. Fiedler, H. A. Bachor, and J. Mlynek, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
[CrossRef] [PubMed]

E. Yablonovitch, C. Flytzanis, and N. Bloembergen, “Anisotropic interference of three-wave and double-two wave frequency mixing in GaAs,” Phys. Rev. Lett. 29, 865–868 (1972).
[CrossRef]

J. P. Coffinet and F. De Martini, “Coherent excitation of polaritons in gallium phosphide,” Phys. Rev. Lett. 22, 60–63 (1969).
[CrossRef]

Sov. J. Quantum Electron. (1)

D. N. Klyshko and B. F. Polkovnikov, “Phase modulation and self-modulation of light in three-photon processes,” Sov. J. Quantum Electron. 3, 324–326 (1974).
[CrossRef]

Other (8)

G. I. Stegeman and P. L. Wa, in Nonlinear Optical Materials and Devices for Applications in Information Technology (Kluwer Academic, Dordrecht, The Netherlands, 1995), pp. 285–317.

G. Assanto, “Quadratic cascading: effects and applications,” in Beam Shaping and Control with Nonlinear Optics, F. Kajzar and R. Reinisch, eds. (Plenum, New York, 1998), pp. 341–374.

R. L. Byer and A. Piskarskas, eds., Feature on optical parametric oscillation and amplification, J. Opt. Soc. Am. B 10, 1656–1791; 2148–2243 (1993).

R. Reinisch, E. Popov, and M. Nevière, “Second harmonic generation-induced optical bistability in prism or grating couplers,” Opt. Lett. 20, 854–856 (1995); R. Reinisch, M. Nevière, and E. Popov, “Phase-matched guided wave optical bistability in χ(2) optical resonators,” Opt. Lett. 20, 2472–2474 (1995).
[CrossRef] [PubMed]

L. Caleo, C. Sibilia, P. Masciulli, and M. Bertolotti, “Nonlinear optical filter based on cascading second order effect,” J. Opt. Soc. Am. B 14, 2315–2324 (1997); C. Sibilia, A. Re, E. Fazio, and M. Bertolotti, “Cascading effect on second-harmonic generation in a ring cavity,” J. Opt. Soc. Am. B 13, 1151–1159 (1996).
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997); J. W. Haus, R. Viswanathan, M. Scalora, A. G. Kalocsai, J. D. Cole, and J. Theimer, “Enhanced second-harmonic generation in media with a weak periodicity,” Phys. Rev. A 57, 2120–2128 (1998).
[CrossRef]

G. D’Aguanno, M. Centini, C. Sibilia, M. Bertolotti, M. Scalora, M. J. Bloemer, and C. M. Bowden, “Enhancement of χ(2) cascading processes in one-dimensional photonic bandgap structures,” Opt. Lett. 24, 1663–1665 (1999); M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), pp. 87–88.

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Figures (5)

Fig. 1
Fig. 1

Schematic representation of the geometry that we analyze. Two plane waves, Eω(i) and E2ω(i), at frequencies ω and 2ω propagating in a linear medium (air) are incident at the input mirror of a Fabry–Perot interferometer filled with a thickness L of quadratic material. Eω(r) and E2ω(r) are the reflected components; Eω(t) and E2ω(t) are the transmitted components.

Fig. 2
Fig. 2

Band structure (solid curve) of a dielectric mirror composed of 10 alternating layers of SiO2 and TiO2 materials, such that SiO2 is taken to be a quarter-wave layer and TiO2 is taken to be a half-wave layer with respect to a reference wavelength. Ω=ω/ω0, where ω0 corresponds to a reference wavelength of 1 µm. Dotted curve, the phase of the field (in units of π) on reflection from the dielectric stack.

Fig. 3
Fig. 3

Transmittance versus L, the length of the nonlinear cavity, in the linear regime for both the fundamental and the second-harmonic fields. The transmission peaks have widths of (∼0.1 nm) 1 Å for the fundamental and ∼0.6 Å (∼0.06 nm) for the second harmonic.

Fig. 4
Fig. 4

Linear transmittance of the fundamental (continuous line) and the second-harmonic (dotted curve) fields about a length L of 6.1574 µm of nonlinear material. This corresponds to a peak of transmission for the second-harmonic field and to a reflection maximum for the fundamental field.

Fig. 5
Fig. 5

Transmittance for the fundamental beam in the linear regime and in the nonlinear regime about a length L of 6.1574 µm of nonlinear material. The fundamental field switches from total reflection in the linear regime to total transmission in the nonlinear regime for an input intensity of 9.2 GW/cm2 for the fundamental field and 4.6 GW/cm2 for the second-harmonic field. 100% all-optical switching for the fundamental beam is obtained.

Equations (62)

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

d2Aωdz2+kω2Aω=-αAω*A2ω,
d2A2ωdz2+k2ω2A2ω=-2αAω2,
gjω(z, z)=exp(ikjω|z-z|)2ikjω,j=1, 2,
Aω(z)=-a-+gω(z, z)Aω*(z)A2ω(z)dz,
A2ω(z)=-2α -+g2ω(z, z)Aω2(z)dz.
Ajω(z)=A+jω(z)exp(ikjωz)+A-jω(z)exp(-ikjωz),
j=1, 2,
A+ω(z)=-α2ikω-z{A+ω*(z)A+2ω(z)exp[i(k2ω-2kω)z]+A-ω*(z)A+2ω(z)exp(ik2ωz)+A+ω(z)A-2ω(z)exp[-i(k2ω+kω)z]+A-ω*(z)A-2ω(z)exp(-ik2ωz)}dz,
A-ω(z)=-α2ikωz-{A+ω*(z)A+2ω(z)exp(ik2ωz)+A-ω*(z)A+2ω(z)exp[i(k2ω+2kω)z]+A+ω(z)A-2ω(z)exp(-ik2ωz)+A-ω*(z)A-2ω(z)exp[-i(k2ω-2kω)z]}dz,
A+2ω(z)=-αik2ω-z{A+ω2(z)exp[-i(k2ω-2kω)z]+A-ω2(z)exp[-i(k2ω+2kω)z]+2A+ω(z)A-ω(z)exp(-ik2ωz)}dz,
A-2ω(z)=-α2ik2ωz-{A+ω2(z)exp[i(k2ω+2kω)z]+A-ω2(z)exp[i(k2ω-2kω)z]+2A+ω(z)A-ω(z)exp(ik2ωz)}dz.
dA+ωdz=-α2ikω{A+ω*A+2ω exp[i(k2ω-2kω)z]+A-ω*A+2ω exp(ik2ωz)+A+ω*A-2ω exp[-i(k2ω+2kω)z]+A-ω*A-2ω exp(-ik2ωz)},
dA-ωdz=α2ikω{A+ω*A+2ω exp(ik2ωz)+A-ω*A+2ω exp[ik2ω+2kω)z]+A+ω*A-2ω exp(-ik2ωz)+A-ω*A-2ω exp[-i(k2ω-2kω)z)},
dA+2ωdz=-αikω{A+ω2 exp[-i(k2ω-2kω)z]+A-ω2 exp[-i(k2ω+2kω)z]+2A+ωA-ω exp(-i(k2ωz)},
dA-2ωdz=αik2ω{A+ω2 exp[i(k2ω+2kω)z]+A-ω2 exp[i(k2ω-2kω)z]+2A+ωA-ω exp(ik2ωz)}.
dA±ωdz=±iα2kωA±2ωA±ω* exp(±iΔkz),
dA±2ωdz=±iαk2ωA±ω2 exp(iΔkz),
rkl(jω)=|rkl(jω)|exp[iφrkl(jω)],
tkl(jω)=|tkl(jω)|exp[iφtkl(jω)],
j=1, 2,k, l=1, 2, 3,k1,
|r12jω|2+njω|t12jω|2=1,
|r32jω|2+njω|t32jω|2=1,
|r21jω|2+1njω|t21jω|2=1,
|r23jω|2+1njω|t23jω|2=1,j=1, 2
A+jω(0)=t12(jω)Ejω(i)+r21(jω)A-jω(0),
Ejω(r)=r12(jω)Ejω(i)+t21(jω)A-jω(0),
j=1, 2.
Ejω(t)=t23(jω)A+jω(L)exp(ikjωL),
A-jω(L)=r23(jω)A+jω(L)exp(2ikjωL),
j=1, 2.
Ajω(z)=C+jω exp(ikjωz)+C-jω exp(-ikjωz),
j=1, 2.
Tjω(linear)=t12(jω)t23(jω) exp)(ikjωL)1+r21(jω)r23(jω) exp(2ikjωL),j=1, 2.
A±ω(z)=(-i)m±2|A±2ω(0)|exp±i(-i)m±Γ|A±2ω(0)|z+iϕ±2ω(0)2,
A±2ω(z)=|A±2ω(0)|exp[±i2(-1)m±Γ|A±2ω(0)|z+iϕ±2ω(0)],
|A±ω(0)|=2|A±2ω(0)|,
φ±2ω(0)-2φ±ω(0)=m±π.
|Tω|2=|t12(ω)|2|t23(ω)|2[1-|r21(ω)r23(ω)|]2+4|r21(ω)r23(ω)|sin2kωL+Γ˜LI2ω(t)2|t23(2ω)|[(-1)m++(-1)m-|r23(2ω)|]+φr23(ω)+φr21(ω)2,
|T2ω|2
=|t12(2ω)|2|t23(2ω)|2[1-|r21(2ω)r23(2ω)|]2+4|r21(2ω)|r23(2ω)|sin22kωL+Γ˜LI2ω(t)|t23(2ω)|[(-1)m++(-1)m-|r23(2ω)|]+φr23(2ω)+φr21(2ω)2
φT(ω)=φt12(ω)+φt23(ω)+kωL+(-1)m+Γ˜LI2ω(t)|t23(2ω)|-arctan|r21(ω)r23(ω)|sinφr21(ω)+φr23(ω)+2kωL+Γ˜LI2ω(t)|t23(2ω)|[(-1)m++(-1)m-|r23(2ω)|]|r21(ω)r23(ω)|cosφr21(ω)+φr23(ω)+2kωL+Γ˜LI2ω(t)|t23(2ω)|[(-1)m++(-1)m-|r23(2ω)|]-1,
φT(2ω)=φt12(2ω)+φt23(2ω)+2kωL+(-1)m+2Γ˜LI2ω(t)|t23(2ω)|-arctan|r21(2ω)r23(2ω)|sinφr21(2ω)+φr23(2ω)+4kωL+2Γ˜LI2ω(t)|t23(2ω)|[(-1)m++(-1)m-|r23(2ω)|]|r21(2ω)r23(2ω)|cosφr21(2ω)+φr23(2ω)+4kωL+Γ˜LI2ω(t)|t23(2ω)|[(-1)m++(-1)m-|r23(2ω)|]-1,
Iω(i)=2|t23(ω)|2|t23 (2ω)|2I2ω(t)|Tω|2.
φ2ω(i)-2φω(t)=m+π+2φT(ω)-φT(2ω)+φt23(2ω)-2φt23(ω).
|r23(2ω)|=|r23(ω)|,
φr23(2ω)-2φr23(ω)=π(m--m+).
δ=jΓ˜LI2ω(t)2|t23(2ω)|[(-1)m++|r23(2ω)|(-1)m-],j=1, 2.
I2ω(t)thr jω=π2|t23(2ω)|2jΓ˜2L2[(-1)m++|r23(2ω)|(-1)m-]2,
j=1, 2,
Iω(i)thr jω=2π2|t23(ω)|2jΓ˜2L2|Tω|2[(-1)m++|r23(2ω)|(-1)m-]2,
j=1, 2.
|r23(2ω)|=|r23ω|(110-4)
φr23(2ω)=2φr23(ω).
|t12(ω)|=|t12(2ω)|=1.851×10-2,
|r21(ω)|=|r212(ω)|=9.996×10-1,
|t23(ω)|=|t23(2ω)|=3.998×10-2,
|r23(ω)|=|r23(2ω)|=9.996×10-1,
I2ω(t)thr ω=π2|t23(2ω)|2Γ˜2L2[1+|r23(2ω)|]2.
I2ω(i)thr ω=I2ω(t)thr ω4.6 GW/cm2.
Iω(i)thr ω=2π2|t23(ω)|2Γ˜2L2[1+|r23(2ω)|]2.
φ2ω(i)-2φω(i)=mπ+2φt12(ω)-φt12(2ω).
I2ω(i)thr ω1L2,Iω(i)thr ω1L2.

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