Abstract

Gap-acoustic solitons (GASs) are stable pulses that exist in nonlinear Bragg waveguides. They are a mathematical generalization of gap solitons, in which the model includes the dependence of the refractive index on the material density. We derive unified dynamical equations for gap solitons along with Brillouin scattering, which also results from the dependence of the refractive index on the material density. We find accurate values of the coefficients for fused silica. The analysis of the GAS conserved quantities—Hamiltonian, momentum, photon energy (or number of photons), and material mass—shows dramatic differences compared to the model neglecting the dependence of the refractive index on the material density. In particular, subsonic GASs in fused silica have far more momentum at low velocities than at high velocities. The dependence of the GAS momentum on velocity due to acoustic effects is dramatic up to approximately 1% of the speed of light. These momentum-connected effects mean that instability of a slow GAS may make it suddenly accelerate to high speeds, and also that an unstable high-speed GAS can abruptly decelerate to close to zero velocity. The predictions are confirmed by a direct numerical simulation.

© 2010 Optical Society of America

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2009 (1)

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[Crossref] [PubMed]

2008 (1)

Y. Zhang, W. Qui, J. Ye, N. Wang, J. Wang, H. Tian, and P. Yuan, “Controllable ultraslow light propagation in highly-doped erbium fiber,” Opt. Commun. 281, 2633–2637 (2008).
[Crossref]

2007 (1)

R. S. Tasgal, Y. B. Band, and B. A. Malomed, “Optoacoustic solitons in Bragg gratings,” Phys. Rev. Lett. 98, 243902 (2007).
[Crossref] [PubMed]

2006 (2)

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
[Crossref]

P. Maák, G. Kurdi, A. Barócsi, K. Osvay, A. P. Kovács, L. Jakab, and P. Richter, “Shaping of ultrashort pulses using bulk acousto-optic filter,” Appl. Phys. B 82, 283–287 (2006).
[Crossref]

2005 (5)

2004 (1)

A. A. Zabolotskii, “Generation of pulses upon nonresonant acousto-electromagnetic interaction,” Opt. Spectrosc. 97, 936–944 (2004).
[Crossref]

2003 (1)

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[Crossref] [PubMed]

2002 (1)

R. W. Boyd and D. J. Gauthier, “‘Slow’ and ‘fast’ light,” Prog. Opt. 43, 497–530 (2002).
[Crossref]

2001 (2)

2000 (2)

I. V. Barashenkov and E. V. Zemlyanaya, “Oscillatory instabilities of gap solitons: a numerical study,” Comput. Phys. Commun. 126, 22–27 (2000).
[Crossref]

E. M. Dianov, M. E. Sukharev, and A. S. Biriukov, “Electrostrictive response in single-mode ring-index-profile fibers,” Opt. Lett. 25, 390–392 (2000).
[Crossref]

1999 (3)

1998 (4)

A. Melloni, M. Frasca, A. Garavaglia, A. Tonini, and M. Martinelli, “Direct measurement of electrostriction in optical fibers,” Opt. Lett. 23, 691–693 (1998).
[Crossref]

D. Milam, “Review and assessment of measured values of the nonlinear refractive-index coefficient of fused silica,” Appl. Opt. 37, 546–550 (1998).
[Crossref]

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[Crossref]

A. De Rossi, C. Conti, and S. Trillo, “Stability, multistability, and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
[Crossref]

1997 (2)

1996 (2)

1994 (1)

B. A. Malomed and R. S. Tasgal, “Vibration modes of a gap soliton in a nonlinear optical medium,” Phys. Rev. E 49, 5787–5796 (1994).
[Crossref]

1993 (1)

J. Feng and F. K. Kneubuhl, “Solitons in a periodic structure with Kerr nonlinearity,” IEEE J. Quantum Electron. 29, 590–597 (1993).
[Crossref]

1992 (2)

E. M. Dianov, A. V. Luchnikov, A. N. Pilipetskii, and A. M. Prokhorov, “Long-range interaction of picosecond solitons through excitation of acoustic waves in optical fibers,” Appl. Phys. B 54, 175–180 (1992).
[Crossref]

G. A. Maugin, H. Hadouaj, and B. A. Malomed, “Nonlinear coupling between shear horizontal surface solitons and Rayleigh waves on elastic structures,” Phys. Rev. B 45, 9688–9694 (1992).
[Crossref]

1991 (2)

H. Hadouaj, B. A. Malomed, and G. A. Maugin, “Dynamics of a soliton in a generalized Zakharov system with dissipation,” Phys. Rev. A 44, 3925–3931 (1991).
[Crossref] [PubMed]

H. Hadouaj, B. A. Malomed, and G. A. Maugin, “Soliton–soliton collisions in a generalized Zakharov system,” Phys. Rev. A 44, 3932–3940 (1991).
[Crossref] [PubMed]

1990 (1)

1989 (3)

K. Smith and L. F. Mollenauer, “Experimental observation of soliton interaction over long fiber paths: discovery of a long-range interaction,” Opt. Lett. 14, 1284–1286 (1989).
[Crossref] [PubMed]

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[Crossref] [PubMed]

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37–42 (1989).
[Crossref]

1987 (1)

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

1986 (1)

L. Stenflo, “Nonlinear equations for acoustic gravity waves,” Phys. Scr. 33, 156–158 (1986).
[Crossref]

1985 (1)

R. M. Shelby, M. D. Levenson, and P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B 31, 5244–5252 (1985).
[Crossref]

1980 (1)

1979 (2)

A. S. Davydov, “Solitons in molecular systems,” Phys. Scr. 20, 387–394 (1979).
[Crossref]

P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fibers,” Phys. Rev. B 19, 4986–4998 (1979).
[Crossref]

1977 (2)

E. A. Kuznetsov and A. V. Mikhailov, “On the complete integrability of the two-dimensional classical Thirring model,” Theor. Math. Phys. 30, 193–200 (1977).
[Crossref]

D. J. Kaup and A. C. Newell, “On the Coleman correspondence and the soliton of the massive Thirring model,” Lett. Nuovo Cimento 20, 325–331 (1977).
[Crossref]

1972 (1)

V. E. Zakharov, “Collapse of Langmuir waves,” Zh. Eksp. Teor. Fiz. 62, 1745–1751 (1972) V. E. Zakharov, “[Sov. Phys. JETP 35, 908–914 (1972)].

V. E. Zakharov, “Collapse of Langmuir waves,” Zh. Eksp. Teor. Fiz. 62, 1745–1751 (1972) V. E. Zakharov, “[Sov. Phys. JETP 35, 908–914 (1972)].

1958 (1)

W. E. Thirring, “A soluble relativistic field theory,” Ann. Phys. (N.Y.) 3, 91–112 (1958).
[Crossref]

Aceves, A. B.

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37–42 (1989).
[Crossref]

Afshar V., S.

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2006).

Arai, M.

E. Rat, M. Foret, G. Massiera, R. Vialla, M. Arai, R. Vacher, and E. Courtens, “Anharmonic versus relaxational sound damping in glasses. I. Brillouin scattering in densified silica,” Phys. Rev. B 72, 214204 (2005).
[Crossref]

Bahloul, D.

Band, Y. B.

R. S. Tasgal, Y. B. Band, and B. A. Malomed, “Optoacoustic solitons in Bragg gratings,” Phys. Rev. Lett. 98, 243902 (2007).
[Crossref] [PubMed]

Bao, X.

Barashenkov, I. V.

I. V. Barashenkov and E. V. Zemlyanaya, “Oscillatory instabilities of gap solitons: a numerical study,” Comput. Phys. Commun. 126, 22–27 (2000).
[Crossref]

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[Crossref]

Barócsi, A.

P. Maák, G. Kurdi, A. Barócsi, K. Osvay, A. P. Kovács, L. Jakab, and P. Richter, “Shaping of ultrashort pulses using bulk acousto-optic filter,” Appl. Phys. B 82, 283–287 (2006).
[Crossref]

Bayer, P. W.

R. M. Shelby, M. D. Levenson, and P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B 31, 5244–5252 (1985).
[Crossref]

Bigelow, M. S.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[Crossref] [PubMed]

Biriukov, A. S.

Blow, K. J.

Bongrand, I.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Botineau, J.

Boyd, R. W.

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[Crossref] [PubMed]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[Crossref] [PubMed]

R. W. Boyd and D. J. Gauthier, “‘Slow’ and ‘fast’ light,” Prog. Opt. 43, 497–530 (2002).
[Crossref]

E. L. Buckland and R. W. Boyd, “Measurement of the frequency response of the electrostrictive nonlinearity in optical fibers,” Opt. Lett. 22, 676–678 (1997).
[Crossref] [PubMed]

E. L. Buckland and R. W. Boyd, “Electrostrictive contribution to the intensity-dependent refractive index of optical fibers,” Opt. Lett. 21, 1117–1119 (1996).
[Crossref] [PubMed]

R. W. Boyd, Nonlinear Optics (Academic, 2003).

Buckland, E. L.

Chen, L.

Chen, W.

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

Cheval, G.

Christodoulides, D. N.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[Crossref] [PubMed]

Conti, C.

A. De Rossi, C. Conti, and S. Trillo, “Stability, multistability, and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
[Crossref]

Courtens, E.

E. Rat, M. Foret, G. Massiera, R. Vialla, M. Arai, R. Vacher, and E. Courtens, “Anharmonic versus relaxational sound damping in glasses. I. Brillouin scattering in densified silica,” Phys. Rev. B 72, 214204 (2005).
[Crossref]

Dahan, D.

Davydov, A. S.

A. S. Davydov, “Solitons in molecular systems,” Phys. Scr. 20, 387–394 (1979).
[Crossref]

De Rossi, A.

A. De Rossi, C. Conti, and S. Trillo, “Stability, multistability, and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
[Crossref]

de Sterke, C. M.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
[Crossref]

Dianov, E. M.

Eggleton, B. J.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
[Crossref]

B. J. Eggleton, C. Martijn de Sterke, and R. E. Slusher, “Bragg solitons in the nonlinear Schrödinger limit: experiment and theory,” J. Opt. Soc. Am. B 16, 587–599 (1999).
[Crossref]

Eisenstein, G.

Fabelinskii, I. L.

I. L. Fabelinskii, Molecular Scattering of Light (Plenum, 1968).

Fellegara, A.

Feng, J.

J. Feng and F. K. Kneubuhl, “Solitons in a periodic structure with Kerr nonlinearity,” IEEE J. Quantum Electron. 29, 590–597 (1993).
[Crossref]

Foret, M.

E. Rat, M. Foret, G. Massiera, R. Vialla, M. Arai, R. Vacher, and E. Courtens, “Anharmonic versus relaxational sound damping in glasses. I. Brillouin scattering in densified silica,” Phys. Rev. B 72, 214204 (2005).
[Crossref]

Frasca, M.

Garavaglia, A.

Gauthier, D. J.

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[Crossref] [PubMed]

R. W. Boyd and D. J. Gauthier, “‘Slow’ and ‘fast’ light,” Prog. Opt. 43, 497–530 (2002).
[Crossref]

Hadouaj, H.

G. A. Maugin, H. Hadouaj, and B. A. Malomed, “Nonlinear coupling between shear horizontal surface solitons and Rayleigh waves on elastic structures,” Phys. Rev. B 45, 9688–9694 (1992).
[Crossref]

H. Hadouaj, B. A. Malomed, and G. A. Maugin, “Soliton–soliton collisions in a generalized Zakharov system,” Phys. Rev. A 44, 3932–3940 (1991).
[Crossref] [PubMed]

H. Hadouaj, B. A. Malomed, and G. A. Maugin, “Dynamics of a soliton in a generalized Zakharov system with dissipation,” Phys. Rev. A 44, 3925–3931 (1991).
[Crossref] [PubMed]

Hardman, P. J.

Herraez, M. G.

Hon, D. T.

Iizuka, T.

T. Iizuka and Y. S. Kivshar, “Optical gap solitons in nonresonant quadratic media,” Phys. Rev. E 59, 7148–7151 (1999).
[Crossref]

Imamoglu, A.

M. D. Lukin and A. Imamoglu, “Controlling photons using electromagnetically induced transparency,” Nature 413, 273–276 (2001).
[Crossref] [PubMed]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

Jakab, L.

P. Maák, G. Kurdi, A. Barócsi, K. Osvay, A. P. Kovács, L. Jakab, and P. Richter, “Shaping of ultrashort pulses using bulk acousto-optic filter,” Appl. Phys. B 82, 283–287 (2006).
[Crossref]

Joseph, R. I.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[Crossref] [PubMed]

Kalosha, V. P.

Kaup, D. J.

D. J. Kaup and A. C. Newell, “On the Coleman correspondence and the soliton of the massive Thirring model,” Lett. Nuovo Cimento 20, 325–331 (1977).
[Crossref]

Kivshar, Y. S.

T. Iizuka and Y. S. Kivshar, “Optical gap solitons in nonresonant quadratic media,” Phys. Rev. E 59, 7148–7151 (1999).
[Crossref]

Kneubuhl, F. K.

J. Feng and F. K. Kneubuhl, “Solitons in a periodic structure with Kerr nonlinearity,” IEEE J. Quantum Electron. 29, 590–597 (1993).
[Crossref]

Kovács, A. P.

P. Maák, G. Kurdi, A. Barócsi, K. Osvay, A. P. Kovács, L. Jakab, and P. Richter, “Shaping of ultrashort pulses using bulk acousto-optic filter,” Appl. Phys. B 82, 283–287 (2006).
[Crossref]

Kurdi, G.

P. Maák, G. Kurdi, A. Barócsi, K. Osvay, A. P. Kovács, L. Jakab, and P. Richter, “Shaping of ultrashort pulses using bulk acousto-optic filter,” Appl. Phys. B 82, 283–287 (2006).
[Crossref]

Kuznetsov, E. A.

E. A. Kuznetsov and A. V. Mikhailov, “On the complete integrability of the two-dimensional classical Thirring model,” Theor. Math. Phys. 30, 193–200 (1977).
[Crossref]

Lepeshkin, N. N.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[Crossref] [PubMed]

Levenson, M. D.

R. M. Shelby, M. D. Levenson, and P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B 31, 5244–5252 (1985).
[Crossref]

Littler, I. C. M.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
[Crossref]

Luchnikov, A. V.

E. M. Dianov, A. V. Luchnikov, A. N. Pilipetskii, and A. M. Prokhorov, “Long-range interaction of picosecond solitons through excitation of acoustic waves in optical fibers,” Appl. Phys. B 54, 175–180 (1992).
[Crossref]

E. M. Dianov, A. V. Luchnikov, A. N. Pilipetskii, and A. N. Starodumov, “Electrostriction mechanism of soliton interaction in optical fibers,” Opt. Lett. 15, 314–316 (1990).
[Crossref] [PubMed]

Lukin, M. D.

M. D. Lukin and A. Imamoglu, “Controlling photons using electromagnetically induced transparency,” Nature 413, 273–276 (2001).
[Crossref] [PubMed]

Maák, P.

P. Maák, G. Kurdi, A. Barócsi, K. Osvay, A. P. Kovács, L. Jakab, and P. Richter, “Shaping of ultrashort pulses using bulk acousto-optic filter,” Appl. Phys. B 82, 283–287 (2006).
[Crossref]

Malomed, B. A.

R. S. Tasgal, Y. B. Band, and B. A. Malomed, “Optoacoustic solitons in Bragg gratings,” Phys. Rev. Lett. 98, 243902 (2007).
[Crossref] [PubMed]

B. A. Malomed and R. S. Tasgal, “Vibration modes of a gap soliton in a nonlinear optical medium,” Phys. Rev. E 49, 5787–5796 (1994).
[Crossref]

G. A. Maugin, H. Hadouaj, and B. A. Malomed, “Nonlinear coupling between shear horizontal surface solitons and Rayleigh waves on elastic structures,” Phys. Rev. B 45, 9688–9694 (1992).
[Crossref]

H. Hadouaj, B. A. Malomed, and G. A. Maugin, “Soliton–soliton collisions in a generalized Zakharov system,” Phys. Rev. A 44, 3932–3940 (1991).
[Crossref] [PubMed]

H. Hadouaj, B. A. Malomed, and G. A. Maugin, “Dynamics of a soliton in a generalized Zakharov system with dissipation,” Phys. Rev. A 44, 3925–3931 (1991).
[Crossref] [PubMed]

Martijn de Sterke, C.

Martinelli, M.

Massiera, G.

E. Rat, M. Foret, G. Massiera, R. Vialla, M. Arai, R. Vacher, and E. Courtens, “Anharmonic versus relaxational sound damping in glasses. I. Brillouin scattering in densified silica,” Phys. Rev. B 72, 214204 (2005).
[Crossref]

Maugin, G. A.

G. A. Maugin, H. Hadouaj, and B. A. Malomed, “Nonlinear coupling between shear horizontal surface solitons and Rayleigh waves on elastic structures,” Phys. Rev. B 45, 9688–9694 (1992).
[Crossref]

H. Hadouaj, B. A. Malomed, and G. A. Maugin, “Soliton–soliton collisions in a generalized Zakharov system,” Phys. Rev. A 44, 3932–3940 (1991).
[Crossref] [PubMed]

H. Hadouaj, B. A. Malomed, and G. A. Maugin, “Dynamics of a soliton in a generalized Zakharov system with dissipation,” Phys. Rev. A 44, 3925–3931 (1991).
[Crossref] [PubMed]

Melloni, A.

Mikhailov, A. V.

E. A. Kuznetsov and A. V. Mikhailov, “On the complete integrability of the two-dimensional classical Thirring model,” Theor. Math. Phys. 30, 193–200 (1977).
[Crossref]

Milam, D.

Mills, D. L.

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

Mok, J. T.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
[Crossref]

Mollenauer, L. F.

Montes, C.

Newell, A. C.

D. J. Kaup and A. C. Newell, “On the Coleman correspondence and the soliton of the massive Thirring model,” Lett. Nuovo Cimento 20, 325–331 (1977).
[Crossref]

Osvay, K.

P. Maák, G. Kurdi, A. Barócsi, K. Osvay, A. P. Kovács, L. Jakab, and P. Richter, “Shaping of ultrashort pulses using bulk acousto-optic filter,” Appl. Phys. B 82, 283–287 (2006).
[Crossref]

Pelinovsky, D. E.

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[Crossref]

Picholle, E.

Picozzi, A.

Pilipetskii, A. N.

E. M. Dianov, A. V. Luchnikov, A. N. Pilipetskii, and A. M. Prokhorov, “Long-range interaction of picosecond solitons through excitation of acoustic waves in optical fibers,” Appl. Phys. B 54, 175–180 (1992).
[Crossref]

E. M. Dianov, A. V. Luchnikov, A. N. Pilipetskii, and A. N. Starodumov, “Electrostriction mechanism of soliton interaction in optical fibers,” Opt. Lett. 15, 314–316 (1990).
[Crossref] [PubMed]

Poustie, A. J.

Prokhorov, A. M.

E. M. Dianov, A. V. Luchnikov, A. N. Pilipetskii, and A. M. Prokhorov, “Long-range interaction of picosecond solitons through excitation of acoustic waves in optical fibers,” Appl. Phys. B 54, 175–180 (1992).
[Crossref]

Qui, W.

Y. Zhang, W. Qui, J. Ye, N. Wang, J. Wang, H. Tian, and P. Yuan, “Controllable ultraslow light propagation in highly-doped erbium fiber,” Opt. Commun. 281, 2633–2637 (2008).
[Crossref]

Rat, E.

E. Rat, M. Foret, G. Massiera, R. Vialla, M. Arai, R. Vacher, and E. Courtens, “Anharmonic versus relaxational sound damping in glasses. I. Brillouin scattering in densified silica,” Phys. Rev. B 72, 214204 (2005).
[Crossref]

Richter, P.

P. Maák, G. Kurdi, A. Barócsi, K. Osvay, A. P. Kovács, L. Jakab, and P. Richter, “Shaping of ultrashort pulses using bulk acousto-optic filter,” Appl. Phys. B 82, 283–287 (2006).
[Crossref]

Rowell, N. L.

P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fibers,” Phys. Rev. B 19, 4986–4998 (1979).
[Crossref]

Sazonov, S. V.

S. V. Sazonov, “Optical-acoustic soliton under the conditions of slow light and stimulated Mandelstam–Brillouin scattering,” JETP Lett. 81, 201–204 (2005).
[Crossref]

Shelby, R. M.

R. M. Shelby, M. D. Levenson, and P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B 31, 5244–5252 (1985).
[Crossref]

Slusher, R. E.

Smith, K.

Song, K. Y.

Starodumov, A. N.

Stegeman, G. I.

P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fibers,” Phys. Rev. B 19, 4986–4998 (1979).
[Crossref]

Stenflo, L.

L. Stenflo, “Nonlinear equations for acoustic gravity waves,” Phys. Scr. 33, 156–158 (1986).
[Crossref]

Sukharev, M. E.

Tasgal, R. S.

R. S. Tasgal, Y. B. Band, and B. A. Malomed, “Optoacoustic solitons in Bragg gratings,” Phys. Rev. Lett. 98, 243902 (2007).
[Crossref] [PubMed]

B. A. Malomed and R. S. Tasgal, “Vibration modes of a gap soliton in a nonlinear optical medium,” Phys. Rev. E 49, 5787–5796 (1994).
[Crossref]

Thevenaz, L.

Thirring, W. E.

W. E. Thirring, “A soluble relativistic field theory,” Ann. Phys. (N.Y.) 3, 91–112 (1958).
[Crossref]

Thomas, P. J.

P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fibers,” Phys. Rev. B 19, 4986–4998 (1979).
[Crossref]

Tian, H.

Y. Zhang, W. Qui, J. Ye, N. Wang, J. Wang, H. Tian, and P. Yuan, “Controllable ultraslow light propagation in highly-doped erbium fiber,” Opt. Commun. 281, 2633–2637 (2008).
[Crossref]

Tonini, A.

Townsend, P. D.

Trillo, S.

A. De Rossi, C. Conti, and S. Trillo, “Stability, multistability, and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
[Crossref]

Vacher, R.

E. Rat, M. Foret, G. Massiera, R. Vialla, M. Arai, R. Vacher, and E. Courtens, “Anharmonic versus relaxational sound damping in glasses. I. Brillouin scattering in densified silica,” Phys. Rev. B 72, 214204 (2005).
[Crossref]

van Driel, H. M.

P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fibers,” Phys. Rev. B 19, 4986–4998 (1979).
[Crossref]

Vialla, R.

E. Rat, M. Foret, G. Massiera, R. Vialla, M. Arai, R. Vacher, and E. Courtens, “Anharmonic versus relaxational sound damping in glasses. I. Brillouin scattering in densified silica,” Phys. Rev. B 72, 214204 (2005).
[Crossref]

Wabnitz, S.

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37–42 (1989).
[Crossref]

Wang, J.

Y. Zhang, W. Qui, J. Ye, N. Wang, J. Wang, H. Tian, and P. Yuan, “Controllable ultraslow light propagation in highly-doped erbium fiber,” Opt. Commun. 281, 2633–2637 (2008).
[Crossref]

Wang, N.

Y. Zhang, W. Qui, J. Ye, N. Wang, J. Wang, H. Tian, and P. Yuan, “Controllable ultraslow light propagation in highly-doped erbium fiber,” Opt. Commun. 281, 2633–2637 (2008).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Ye, J.

Y. Zhang, W. Qui, J. Ye, N. Wang, J. Wang, H. Tian, and P. Yuan, “Controllable ultraslow light propagation in highly-doped erbium fiber,” Opt. Commun. 281, 2633–2637 (2008).
[Crossref]

Yuan, P.

Y. Zhang, W. Qui, J. Ye, N. Wang, J. Wang, H. Tian, and P. Yuan, “Controllable ultraslow light propagation in highly-doped erbium fiber,” Opt. Commun. 281, 2633–2637 (2008).
[Crossref]

Zabolotskii, A. A.

A. A. Zabolotskii, “Generation of pulses upon nonresonant acousto-electromagnetic interaction,” Opt. Spectrosc. 97, 936–944 (2004).
[Crossref]

Zakharov, V. E.

V. E. Zakharov, “Collapse of Langmuir waves,” Zh. Eksp. Teor. Fiz. 62, 1745–1751 (1972) V. E. Zakharov, “[Sov. Phys. JETP 35, 908–914 (1972)].

V. E. Zakharov, “Collapse of Langmuir waves,” Zh. Eksp. Teor. Fiz. 62, 1745–1751 (1972) V. E. Zakharov, “[Sov. Phys. JETP 35, 908–914 (1972)].

Zemlyanaya, E. V.

I. V. Barashenkov and E. V. Zemlyanaya, “Oscillatory instabilities of gap solitons: a numerical study,” Comput. Phys. Commun. 126, 22–27 (2000).
[Crossref]

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[Crossref]

Zhang, Y.

Y. Zhang, W. Qui, J. Ye, N. Wang, J. Wang, H. Tian, and P. Yuan, “Controllable ultraslow light propagation in highly-doped erbium fiber,” Opt. Commun. 281, 2633–2637 (2008).
[Crossref]

Ann. Phys. (N.Y.) (1)

W. E. Thirring, “A soluble relativistic field theory,” Ann. Phys. (N.Y.) 3, 91–112 (1958).
[Crossref]

Appl. Opt. (1)

Appl. Phys. B (2)

P. Maák, G. Kurdi, A. Barócsi, K. Osvay, A. P. Kovács, L. Jakab, and P. Richter, “Shaping of ultrashort pulses using bulk acousto-optic filter,” Appl. Phys. B 82, 283–287 (2006).
[Crossref]

E. M. Dianov, A. V. Luchnikov, A. N. Pilipetskii, and A. M. Prokhorov, “Long-range interaction of picosecond solitons through excitation of acoustic waves in optical fibers,” Appl. Phys. B 54, 175–180 (1992).
[Crossref]

Comput. Phys. Commun. (1)

I. V. Barashenkov and E. V. Zemlyanaya, “Oscillatory instabilities of gap solitons: a numerical study,” Comput. Phys. Commun. 126, 22–27 (2000).
[Crossref]

IEEE J. Quantum Electron. (1)

J. Feng and F. K. Kneubuhl, “Solitons in a periodic structure with Kerr nonlinearity,” IEEE J. Quantum Electron. 29, 590–597 (1993).
[Crossref]

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

JETP Lett. (1)

S. V. Sazonov, “Optical-acoustic soliton under the conditions of slow light and stimulated Mandelstam–Brillouin scattering,” JETP Lett. 81, 201–204 (2005).
[Crossref]

Lett. Nuovo Cimento (1)

D. J. Kaup and A. C. Newell, “On the Coleman correspondence and the soliton of the massive Thirring model,” Lett. Nuovo Cimento 20, 325–331 (1977).
[Crossref]

Nat. Phys. (1)

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
[Crossref]

Nature (1)

M. D. Lukin and A. Imamoglu, “Controlling photons using electromagnetically induced transparency,” Nature 413, 273–276 (2001).
[Crossref] [PubMed]

Opt. Commun. (1)

Y. Zhang, W. Qui, J. Ye, N. Wang, J. Wang, H. Tian, and P. Yuan, “Controllable ultraslow light propagation in highly-doped erbium fiber,” Opt. Commun. 281, 2633–2637 (2008).
[Crossref]

Opt. Express (2)

Opt. Lett. (12)

E. M. Dianov, A. V. Luchnikov, A. N. Pilipetskii, and A. N. Starodumov, “Electrostriction mechanism of soliton interaction in optical fibers,” Opt. Lett. 15, 314–316 (1990).
[Crossref] [PubMed]

P. J. Hardman, P. D. Townsend, A. J. Poustie, and K. J. Blow, “Experimental investigation of resonant enhancement of the acoustic interaction of optical pulses in an optical fiber,” Opt. Lett. 21, 393–395 (1996).
[Crossref] [PubMed]

E. L. Buckland and R. W. Boyd, “Electrostrictive contribution to the intensity-dependent refractive index of optical fibers,” Opt. Lett. 21, 1117–1119 (1996).
[Crossref] [PubMed]

E. L. Buckland and R. W. Boyd, “Measurement of the frequency response of the electrostrictive nonlinearity in optical fibers,” Opt. Lett. 22, 676–678 (1997).
[Crossref] [PubMed]

A. Fellegara, A. Melloni, and M. Martinelli, “Measurement of the frequency response induced by electrostriction in optical fibers,” Opt. Lett. 22, 1615–1617 (1997).
[Crossref]

A. Melloni, M. Frasca, A. Garavaglia, A. Tonini, and M. Martinelli, “Direct measurement of electrostriction in optical fibers,” Opt. Lett. 23, 691–693 (1998).
[Crossref]

E. L. Buckland, “Mode-profile dependence of the electrostrictive response in fibers,” Opt. Lett. 24, 872–874 (1999).
[Crossref]

E. M. Dianov, M. E. Sukharev, and A. S. Biriukov, “Electrostrictive response in single-mode ring-index-profile fibers,” Opt. Lett. 25, 390–392 (2000).
[Crossref]

S. Afshar V., V. P. Kalosha, X. Bao, and L. Chen, “Enhancement of stimulated Brillouin scattering of higher-order acoustic modes in single-mode optical fiber,” Opt. Lett. 30, 2685–2687 (2005).
[Crossref]

D. T. Hon, “Pulse compression by stimulated Brillouin scattering,” Opt. Lett. 5, 516–518 (1980).
[Crossref] [PubMed]

I. Bongrand, C. Montes, E. Picholle, J. Botineau, A. Picozzi, G. Cheval, and D. Bahloul, “Soliton compression in Brillouin fiber lasers,” Opt. Lett. 26, 1475–1477 (2001).
[Crossref]

K. Smith and L. F. Mollenauer, “Experimental observation of soliton interaction over long fiber paths: discovery of a long-range interaction,” Opt. Lett. 14, 1284–1286 (1989).
[Crossref] [PubMed]

Opt. Spectrosc. (1)

A. A. Zabolotskii, “Generation of pulses upon nonresonant acousto-electromagnetic interaction,” Opt. Spectrosc. 97, 936–944 (2004).
[Crossref]

Phys. Lett. A (1)

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37–42 (1989).
[Crossref]

Phys. Rev. A (2)

H. Hadouaj, B. A. Malomed, and G. A. Maugin, “Dynamics of a soliton in a generalized Zakharov system with dissipation,” Phys. Rev. A 44, 3925–3931 (1991).
[Crossref] [PubMed]

H. Hadouaj, B. A. Malomed, and G. A. Maugin, “Soliton–soliton collisions in a generalized Zakharov system,” Phys. Rev. A 44, 3932–3940 (1991).
[Crossref] [PubMed]

Phys. Rev. B (4)

G. A. Maugin, H. Hadouaj, and B. A. Malomed, “Nonlinear coupling between shear horizontal surface solitons and Rayleigh waves on elastic structures,” Phys. Rev. B 45, 9688–9694 (1992).
[Crossref]

E. Rat, M. Foret, G. Massiera, R. Vialla, M. Arai, R. Vacher, and E. Courtens, “Anharmonic versus relaxational sound damping in glasses. I. Brillouin scattering in densified silica,” Phys. Rev. B 72, 214204 (2005).
[Crossref]

P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fibers,” Phys. Rev. B 19, 4986–4998 (1979).
[Crossref]

R. M. Shelby, M. D. Levenson, and P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B 31, 5244–5252 (1985).
[Crossref]

Phys. Rev. E (2)

B. A. Malomed and R. S. Tasgal, “Vibration modes of a gap soliton in a nonlinear optical medium,” Phys. Rev. E 49, 5787–5796 (1994).
[Crossref]

T. Iizuka and Y. S. Kivshar, “Optical gap solitons in nonresonant quadratic media,” Phys. Rev. E 59, 7148–7151 (1999).
[Crossref]

Phys. Rev. Lett. (5)

A. De Rossi, C. Conti, and S. Trillo, “Stability, multistability, and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
[Crossref]

R. S. Tasgal, Y. B. Band, and B. A. Malomed, “Optoacoustic solitons in Bragg gratings,” Phys. Rev. Lett. 98, 243902 (2007).
[Crossref] [PubMed]

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[Crossref]

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

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[Crossref] [PubMed]

Phys. Scr. (2)

A. S. Davydov, “Solitons in molecular systems,” Phys. Scr. 20, 387–394 (1979).
[Crossref]

L. Stenflo, “Nonlinear equations for acoustic gravity waves,” Phys. Scr. 33, 156–158 (1986).
[Crossref]

Prog. Opt. (1)

R. W. Boyd and D. J. Gauthier, “‘Slow’ and ‘fast’ light,” Prog. Opt. 43, 497–530 (2002).
[Crossref]

Science (2)

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003).
[Crossref] [PubMed]

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009).
[Crossref] [PubMed]

Theor. Math. Phys. (1)

E. A. Kuznetsov and A. V. Mikhailov, “On the complete integrability of the two-dimensional classical Thirring model,” Theor. Math. Phys. 30, 193–200 (1977).
[Crossref]

Zh. Eksp. Teor. Fiz. (1)

V. E. Zakharov, “Collapse of Langmuir waves,” Zh. Eksp. Teor. Fiz. 62, 1745–1751 (1972) V. E. Zakharov, “[Sov. Phys. JETP 35, 908–914 (1972)].

Other (5)

I. L. Fabelinskii, Molecular Scattering of Light (Plenum, 1968).

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

R. W. Boyd, Nonlinear Optics (Academic, 2003).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2006).

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Figures (14)

Fig. 1
Fig. 1

Schematic illustration of a fiber with a periodically varying refractive index. Light and sound waves propagate in the fiber. Photons are shown as wavy lines with arrows indicating the direction of motion; the low-frequency phonons are shown as a solid line with double-sided arrows, and high-frequency phonons are shown as dashed lines with arrows.

Fig. 2
Fig. 2

Quiescent (zero velocity, β = 0 ) GAS, with frequency in the middle of the bandgap, Q = π / 2 . The physical parameters are typical of bulk fused silica, and the Bragg coefficient is κ = 90 cm 1 . The top part of the figure shows the amplitude of the envelope u of the forward-moving electromagnetic wave, the middle part shows the envelope v of the backward-moving wave, and the bottom part shows the acoustic field (material density). Solid lines are for the magnitudes of the amplitudes, dashed lines for the real parts, and dotted lines are for the imaginary parts.

Fig. 3
Fig. 3

GAS with velocity ten times the speed of sound β = 10 β s = 59   km / s , and frequency in the middle of the bandgap, Q = π / 2 . The physical parameters are typical of bulk fused silica, and the Bragg coefficient is κ = 90 cm 1 . The first part of the figure shows the amplitude of the envelope u of the forward-moving electromagnetic wave, the middle part shows the envelope v of the backward-moving wave, and the bottom part shows the acoustic field (material density). Solid lines are for the magnitudes of the amplitudes, dashed lines for the real parts, and dotted lines are for the imaginary parts.

Fig. 4
Fig. 4

GAS with velocity half the group velocity of light in the medium β = 0.5 / k 0 = 1.03 × 10 8   m / s , and frequency in the middle of the bandgap, Q = π / 2 . The physical parameters are typical of bulk fused silica, and the Bragg coefficient is κ = 90 cm 1 . The top part of the figure shows the amplitude of the envelope u of the forward-moving electromagnetic wave, the middle part shows the envelope v of the backward-moving wave, and the bottom part shows the acoustic field (material density).

Fig. 5
Fig. 5

Mass, or integrated material density variation, per cross-sectional area ( M ) of GASs in bulk silica with physical parameters as given in the text. The velocities ( β ) range from zero up to the group velocity of light, and the frequencies are in the middle of the bandgap, Q = π / 2 . (a) shows what the mass would be, were there no dependence of the refractive index on the material density. (b) Soliton mass with physical values of the electrostrictive constants.

Fig. 6
Fig. 6

Photon energy (or number of photons) per cross-sectional area ( N ) of GASs in bulk silica at light wavelength of 0.8 μ m . The velocities ( β ) range from zero up to the group velocity of light, and the frequencies are in the middle of the bandgap, Q = π / 2 . (a) Photon energy N, if there were no dependence of the refractive index on the material density. (b) The soliton’s photon energy N, with n / W = 0.2 cm 3 / g .

Fig. 7
Fig. 7

Momentum per cross-sectional area ( P ) of GASs in bulk silica at wavelength of 0.8 μ m . The velocities ( β ) range from zero up to the group velocity of light, and the frequencies are in the middle of the bandgap, Q = π / 2 . (a) Momentum, if there were there no dependence of the refractive index on the material density. (b) Soliton momentum with physical values of the electrostrictive constants. (c) Momentum in the light (solid line) and sound (dashed line) separately.

Fig. 8
Fig. 8

Hamiltonian per cross-sectional area ( H ) of GASs in bulk silica with the typical physical parameters as given in the text. The velocities ( β ) range from zero up to the group velocity of light, and the frequencies are in the middle of the bandgap, Q = π / 2 . (a) Hamiltonian, if there were no dependence of the refractive index on the material density. (b) The soliton’s Hamiltonian energy with nonzero values of the electrostrictive constants, χ e s and λ e s .

Fig. 9
Fig. 9

GAS in fused silica with initial velocity ten times the speed of sound, β = 10 β s , and Q = π / 3 . Following realization of the supersonic instability, a much faster GAS ( β = 1600 β s ) is produced, and a slowly decaying density variation remains behind.

Fig. 10
Fig. 10

High-frequency acoustic (Brillouin) waves interacting with light in the run depicted in Fig. 9. The Brillouin waves are initially zero and grow due to excitation by the light. When the GAS suddenly speeds up, some Brillouin waves are left behind, and the now faster-moving soliton produces its own wake.

Fig. 11
Fig. 11

GAS in fused silica with initial velocity ten times the speed of sound, β = 10 β s , and Q = π / 2 . After the supersonic instability, about half of the light escapes as a dispersive (non-soliton) radiation, and the remaining light reforms a much faster GAS (velocity β = 2000 β s ) with Q = 0.25 π . A slowly decaying low-frequency density variation is left behind.

Fig. 12
Fig. 12

The high-frequency acoustic (Brillouin) waves interacting with the light in the run depicted in Fig. 11. As in the prior simulation, the Brillouin waves are initially zero and grow due to excitation by the light. When the GAS suddenly speeds up, some Brillouin waves are left behind, and the now faster-moving soliton produces its own wake.

Fig. 13
Fig. 13

GAS in fused silica with initial velocity of β = 6.9 × 10 5   m / s = v g / 300 and Q = π / 3 . Following realization of the supersonic instability, the GAS comes to a stop while emitting acoustic waves to the left and to the right.

Fig. 14
Fig. 14

High-frequency acoustic (Brillouin) waves interacting with light in the run depicted in Fig. 13.

Equations (88)

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n ( z ) = n ( ω , W ) + Δ n   cos ( 2 π z / λ Bragg ) ,
E ( z , t ) = u ( z , t ) exp [ i ( k 0 z ω 0 t ) ] + v ( z , t ) exp [ i ( k 0 z + ω 0 t ) ] + u ( z , t ) exp [ i ( k 0 z ω 0 t ) ] + v ( z , t ) exp [ i ( k 0 z + ω 0 t ) ] ,
W ( z , t ) = w 0 ( z , t ) + w u ( z , t ) exp [ 2 i k 0 ( z β s t ) ] + w v ( z , t ) exp [ 2 i k 0 ( z + β s t ) ] + w u ( z , t ) exp [ 2 i k 0 ( z β s t ) ] + w v ( z , t ) exp [ 2 i k 0 ( z + β s t ) ] ,
0 = i k 0 u t + i u z + κ v + 2 π ( ω 0 / c ) 2 k 0 3 χ ( 3 ) ( | u | 2 + 2 | v | 2 ) u + χ e s [ w 0 u + exp ( 2 i k 0 β s t ) w u v + exp ( 2 i k 0 β s t ) w v v ] ,
0 = i k 0 v t i v z + κ u + 2 π ( ω 0 / c ) 2 k 0 3 χ ( 3 ) ( 2 | u | 2 + | v | 2 ) v + χ e s [ w 0 v + exp ( 2 i k 0 β s t ) w u u + exp ( 2 i k 0 β s t ) w v u ] ,
0 = w 0 , t t β s 2 w 0 , z z Γ w 0 , t z z + λ e s ( | u | 2 + | v | 2 ) z z ,
0 = i w u , t + i β s w u , z + i ( 2 k 0 2 Γ ) w u + k 0 λ e s β s exp ( 2 i k 0 β s t ) u v ,
0 = i w v , t i β s w v , z + i ( 2 k 0 2 Γ ) w v + k 0 λ e s β s exp ( 2 i k 0 β s t ) u v ,
n = n ( ω , W ) + Δ n   cos ( 2 k 0 z ) ,
k ( ω , W ) = n ( ω , W ) ω / c ,
k 0 = k ( ω 0 , W 0 ) ,
k 0 = v g 1 = ω k ( ω , W ) ω = ω 0 , W = W 0 ,
χ s = 3 χ ( 3 ) ( ω 0 ; ω 0 , ω 0 , ω 0 ) ,
χ x = 6 χ ( 3 ) ( ω 0 ; ω 0 , ω 0 , ω 0 ) ,
κ = ω 0 c Δ n 2 ,
χ e s = ω 0 c n W ,
λ e s = n ( ω 0 ) 2 π W n W .
κ Brill χ e s [ exp ( 2 i k 0 β s t ) w u + exp ( 2 i k 0 β s t ) w v ] ,
L Brill χ e s [ exp ( 2 i k 0 β s t ) w u exp ( 2 i k 0 β s t ) w v ] .
0 = i k 0 u t + i u z + ( κ + κ Brill ) v + 2 π ( ω 0 / c ) 2 k 0 3 χ ( 3 ) ( | u | 2 + 2 | v | 2 ) u + χ e s w 0 u ,
0 = i k 0 v t i v z + ( κ + κ Brill ) u + 2 π ( ω 0 / c ) 2 k 0 3 χ ( 3 ) ( 2 | u | 2 + | v | 2 ) v + χ e s w 0 v ,
0 = w 0 , t t β s 2 w 0 , z z Γ w 0 , t z z + λ e s ( | u | 2 + | v | 2 ) z z ,
0 = ( t + 2 k 0 2 Γ ) κ Brill + β s ( 2 i k 0 + z ) L Brill ,
0 = ( t + 2 k 0 2 Γ ) L Brill + β s ( 2 i k 0 z ) κ Brill + 2 i k 0 β s ( χ e s λ e s β s 2 ) u v .
P NL ( x , t ) = χ ( 3 ) ( t ; t 1 , t 2 , t 3 ) E ( x , t 1 ) E ( x , t 2 ) E ( x , t 3 ) d t 1 d t 2 d t 3 .
L = i 2 k 0 ( u u t u u t ) + i 2 k 0 ( v v t v v t ) + i 2 ( u u z u u z ) i 2 ( v v z v v z ) + κ u v + κ u v + 2 π ( ω 0 / c ) 2 k 0 [ χ s 2 ( | u | 4 + | v | 4 ) + χ x | u | 2 | v | 2 ] + χ e s 2 λ e s ( r t 2 β s 2 r z 2 ) + χ e s ( | u | 2 + | v | 2 ) r z + χ e s β s k 0 λ e s i 2 [ ( w u w u , t w u w u , t ) + ( w v w v , t w v w v , t ) ] + χ e s β s 2 k 0 λ e s i 2 [ ( w u w u , z w u w u , z ) ( w v w v , z w v w v , z ) ] + χ e s   exp ( 2 i k 0 β s t ) ( u v w u + u v w v ) + χ e s   exp ( 2 i k 0 β s t ) ( u v w v + u v w u ) .
r ( z , t ) z 0 z w 0 ( z , t ) d z ,
M = A w 0 d z ,
N = n ( ω 0 ) 2 4 π A ( | u | 2 + | v | 2 ) d z ,
P = n ( ω 0 ) 2 4 π ω 0 A { i 2 ( u u z u u z + v v z v v z ) + χ e s λ e s k 0 r z r + β s / k 0 2 k 0 χ e s λ e s i 2 ( κ Brill , z κ Brill κ Brill , z κ Brill + L Brill , z L Brill L Brill , z L Brill ) } d z ,
H = n ( ω 0 ) 2 4 π ω 0 k 0 A { i 2 ( u u z u u z v v z + v v z ) ( κ + κ Brill ) u v ( κ + κ Brill ) u v 2 π ( ω 0 / c ) 2 k 0 [ χ s 2 ( | u | 4 + | v | 4 ) + χ x | u | 2 | v | 2 ] + χ e s 2 λ e s ( r t 2 + β s 2 r z 2 ) χ e s ( | u | 2 + | v | 2 ) r z β s 2 2 k 0 χ e s λ e s i 2 ( κ Brill , z L Brill κ Brill , z L Brill κ Brill L Brill , z + κ Brill L Brill , z ) } d z .
u ( z , t ) = κ γ ( 1 + β k 0 ) α   sin   Q   sech ( ζ   sin   Q i 2 Q ) exp [ i θ ( ζ ) i τ   cos   Q ] ,
v ( z , t ) = κ γ ( 1 β k 0 ) α   sin   Q   sech ( ζ   sin   Q + i 2 Q ) exp [ i θ ( ζ ) i τ   cos   Q ] ,
w ( z , t ) = λ e s β s 2 β 2 γ | κ | ( 4 | α | 2 ) sin 2 Q cosh ( 2 ζ   sin   Q ) + cos   Q ,
θ ( ζ ) = β k 0 γ 2 ( 4 | α | 2 ) [ 2 π ( ω 0 / c ) 2 k 0 χ s + χ e s λ e s β s 2 β 2 ] tan 1 [ tanh ( ζ   sin   Q ) tan ( Q / 2 ) ] ,
α = ( 2 π ( ω 0 / c ) 2 k 0 { χ x + χ s γ 2 [ 1 + ( β k 0 ) 2 ] } + 2 γ 2 χ e s λ e s β s 2 β 2 ) 1 / 2 ,
τ γ | κ | ( t / k 0 β k 0 z ) ,
ζ γ | κ | ( z β t ) ,
γ [ 1 ( β k 0 ) 2 ] 1 / 2 ,
β cr 2 = 1 2 ( k 0 ) 2 χ x + χ s χ x χ s + β s 2 2 ( χ x + χ s ( χ x χ s ) 2 ( k 0 ) 2 β s 2 2 ) 2 k 0 2 π ( ω 0 / c ) 2 2 χ e s λ e s / ( k 0 ) 2 χ x χ s .
M GAS = λ e s β s 2 β 2 A 4 | α | 2 Q ,
N GAS = n ( ω 0 ) 2 4 π A 4 | α | 2 Q ,
P GAS = n ( ω 0 ) 2 4 π ω 0 A ( β k 0 ) γ | κ | ( 4 | α | 2 ) [ sin   Q + ( 4 | α | 2 ) γ 2 ( 2 π ( ω 0 / c ) 2 k 0 χ s + λ e s χ e s β s 2 β 2 ) ( sin   Q Q   cos   Q ) + ( 4 | α | 2 ) λ e s χ e s / ( k 0 ) 2 ( β s 2 β 2 ) 2 ( sin   Q Q   cos   Q ) ] ,
H GAS = n ( ω 0 ) 2 / k 0 4 π ω 0 A γ | κ | ( 4 | α | 2 ) { sin   Q + γ 2 ( sin   Q Q   cos   Q ) | α | 2 2 π ( ω 0 / c ) 2 k 0 { χ s γ 2 [ 1 4 ( β k 0 ) 2 ( β k 0 ) 4 ] + χ x γ 2 } ( sin   Q Q   cos   Q ) + | α | 2 χ e s λ e s β s 2 β 2 [ 2 β s 2 + 6 β 2 β s 2 β 2 + 4 ( β k 0 ) 2 γ 2 ] ( sin   Q Q   cos   Q ) } .
( E + 4 π P linear + 4 π P NL ) = 0 ,
B = 0 ,
× E = 1 c t B ,
× B = 1 c t ( E + 4 π P linear + 4 π P NL ) .
E + 4 π P linear D = n 2 ( ω , w ) E ,
P NL = χ ( 3 ) ( E E ) E .
0 = n 2 ( ω ) E ( x , ω ) + 4 π P NL ( x , ω ) ,
0 = × B ( x , ω ) + i ω c [ n 2 ( ω ) E ( x , ω ) + 4 π P NL ( x , ω ) ] .
0 = [ 2 + n 2 ( ω ) ω 2 c 2 ] E ( x , ω ) + 4 π ω 2 c 2 P NL ( x , ω ) + 4 π n 2 ( ω ) [ P NL ( x , ω ) ] .
0 = [ k 2 n 2 ( ω ) ω 2 c 2 ] E ( k , ω ) 4 π ω 2 c 2 { P NL ( k , ω ) c 2 n 2 ( ω ) ω 2 k [ k P NL ( k , ω ) ] } .
0 = [ 2 z 2 + n 2 ( ω ) ω 2 c 2 ] E ( z , ω ) + 4 π ω 2 c 2 P NL ( z , ω ) ,
0 = [ k 2 n 2 ( ω ) ω 2 c 2 ] E ( k , ω ) 4 π ω 2 c 2 P NL ( k , ω ) .
0 = [ ( k 0 + δ k ) 2 n 2 ( ω 0 + δ ω ) ( ω 0 + δ ω ) 2 c 2 4 π ( ω 0 + δ ω ) 2 c 2 P NL ( k 0 + δ k , ω 0 + δ ω ) E ( k 0 + δ k , ω 0 + δ ω ) ] E ( k 0 + δ k , ω 0 + δ ω ) ,
0 = [ ( k 0 + δ k ) + n ( ω 0 + ω ) ( ω 0 + ω ) c 1 + 4 π [ n ( ω 0 + ω ) ] 2 P NL E ] E ( k 0 + δ k , ω 0 + δ ω )
= ( k 0 + δ k ) E ( k 0 + δ k , ω 0 + δ ω ) + n ( ω 0 + ω ) ( ω 0 + ω ) c E + 2 π ( ω 0 + ω ) n ( ω 0 + ω ) c P NL ( k 0 + δ k , ω 0 + δ ω ) +
= δ k E ( k 0 + δ k , ω 0 + δ ω ) + ( n ( ω 0 ) ω 0 c k 0 ) E + ω ( n ( ω ) ω c ) ω 0 ω E + 2 π ω 0 / c n ( ω 0 ) P NL ( k 0 + δ k , ω 0 + δ ω ) + .
0 = i k 0 t E ( z , t ) ± i z E + ( n ( ω 0 , z , W ) ω 0 c k 0 ) E + 2 π ( ω 0 / c ) 2 k 0 P NL ( z , t ) +
= i k 0 t E ( z , t ) ± i z E + ( ω 0 c Δ n ( z ) + ω 0 c n W W ) E + 2 π ( ω 0 / c ) 2 k 0 P NL ( z , t ) + ,
E ( z , t ) = u ( z , t ) exp [ i ( k 0 z ω 0 t ) ] + v ( z , t ) exp [ i ( k 0 z + ω 0 t ) ] + u ( z , t ) exp [ i ( k 0 z ω 0 t ) ] + v ( z , t ) exp [ i ( k 0 z + ω 0 t ) ] ,
W ( z , t ) = w u ( z , t ) exp [ 2 i k 0 ( z β s t ) ] + w v ( z , t ) exp [ 2 i k 0 ( z + β s t ) ] + w u ( z , t ) exp [ 2 i k 0 ( z β s t ) ] + w v ( z , t ) exp [ 2 i k 0 ( z + β s t ) ] + w 0 ( z , t ) ,
Δ n ( z ) = Δ n   cos ( 2 k 0 z ) .
0 = i k 0 u t + i u z + κ v + 2 π ( ω 0 / c ) 2 k 0 3 χ ( 3 ) ( | u | 2 + 2 | v | 2 ) u + χ e s [ w 0 u + exp ( 2 i k 0 β s t ) w u v + exp ( 2 i k 0 β s t ) w v v ] ,
0 = i k 0 v t i v z + κ u + 2 π ( ω 0 / c ) 2 k 0 3 χ ( 3 ) ( 2 | u | 2 + | v | 2 ) v + χ e s [ w 0 v + exp ( 2 i k 0 β s t ) w u u + exp ( 2 i k 0 β s t ) w v u ] ,
κ ω 0 c Δ n 2 ,     χ e s ω 0 c n W .
0 = 2 t 2 W ( x , t ) β s 2 2 W Γ t 2 W + λ e s 2 2 E ( x , t ) 2 ,
0 = 2 t 2 W ( z , t ) β s 2 2 z 2 W Γ 3 t z 2 W + λ e s 2 2 z 2 E ( z , t ) 2 .
0 = W t t β s 2 W z z Γ W t z z + λ e s [ | u | 2 + | v | 2 + u v   exp ( 2 i k 0 z ) + u v   exp ( 2 i k 0 z ) ] z z ,
0 = ω 2 W ( k , ω ) i ω k 2 Γ W + k 2 β s 2 W k 2 λ e s F { | u | 2 + | v | 2 } ( k , ω ) k 2 λ e s F { u v } ( k 2 k 0 , ω ) k 2 λ e s F { u v } ( k + 2 k 0 , ω ) .
0 = ω 2 w 0 ( k , ω ) i ω k 2 Γ w 0 + k 2 β s 2 w 0 k 2 λ e s F { | u | 2 + | v | 2 } ( k , ω ) ,
0 = ( ω ω 0 ) 2 w u ( k , ω ) + i Γ ( ω ω 0 ) ( k 2 k 0 ) 2 w u ( k 2 k 0 ) 2 β s 2 w u + ( k 2 k 0 ) 2 λ e s F { u v } ( k 2 k 0 , ω ω 0 ) ,
0 = ( ω ω 0 ) 2 w v ( k , ω ) + i Γ ( ω ω 0 ) ( k + 2 k 0 ) 2 w v ( k + 2 k 0 ) 2 β s 2 w v + ( k + 2 k 0 ) 2 λ e s F { u v } ( k + 2 k 0 , ω ω 0 ) .
0 = w 0 , t t β s 2 w 0 , z z Γ w 0 , t z z + λ e s ( | u | 2 + | v | 2 ) z z .
0 = [ ω + 2 i ( k 0 + k / 2 ) 2 Γ + ω u ± 2 ( k 0 + k / 2 ) β s { 1 ( 1 / 2 ) [ ( k 0 + k / 2 ) Γ / β s ] 2 } ] w u ( k , ω ) ( k 0 + k / 2 ) λ e s β s 1 F { u v } ( k , ω + ω 0 ) + .
0 = { ω + 2 i k 0 2 Γ ± k β s [ 1 ( 3 / 2 ) ( k 0 Γ / β s ) 2 ] } w u ( k , ω ) k 0 λ e s β s 1 F { u v } ( k , ω 2 k 0 β s [ 1 ( 1 / 2 ) ( k 0 Γ / β s ) 2 ] ) + .
0 = i w u , t + i ( 2 k 0 2 Γ ) w u i β s [ 1 ( 3 / 2 ) ( k 0 Γ / β s ) 2 ] w u , z k 0 λ e s β s exp { 2 i k 0 β s [ 1 ( 1 / 2 ) ( k 0 Γ / β s ) 2 ] t } ( u v ) + .
0 = i w u , t + i β s w u , z + i ( 2 k 0 2 Γ ) w u + k 0 λ e s β s exp ( 2 i k 0 β s t ) u v .
0 = i w v , t β s w v , z + i ( 2 k 0 2 Γ ) w v + k 0 λ e s β s exp ( 2 i k 0 β s t ) u v .
E ( z , t ) = u ( z , t ) exp [ i ( k 0 z ω 0 t ) ] + v ( z , t ) exp [ i ( k 0 z + ω 0 t ) ] + u ( z , t ) exp [ i ( k 0 z ω 0 t ) ] + v ( z , t ) exp [ i ( k 0 z + ω 0 t ) ] ,
W ( z , t ) = w 0 ( z , t ) + w u ( z , t ) exp [ 2 i k 0 ( z β s t ) ] + w v ( z , t ) exp [ 2 i k 0 ( z + β s t ) ] + w u ( z , t ) exp [ 2 i k 0 ( z β s t ) ] + w v ( z , t ) exp [ 2 i k 0 ( z + β s t ) ] ,
0 = i k 0 u t + i u z + κ v + 2 π ( ω 0 / c ) 2 k 0 3 χ ( 3 ) ( | u | 2 + 2 | v | 2 ) u + χ e s [ w 0 u + exp ( 2 i k 0 β s t ) w u v + exp ( 2 i k 0 β s t ) w v v ] ,
0 = i k 0 v t i v z + κ u + 2 π ( ω 0 / c ) 2 k 0 3 χ ( 3 ) ( 2 | u | 2 + | v | 2 ) v + χ e s [ w 0 v + exp ( 2 i k 0 β s t ) w u u + exp ( 2 i k 0 β s t ) w v u ] ,
0 = w 0 , t t β s 2 w 0 , z z Γ w 0 , t z z + λ e s ( | u | 2 + | v | 2 ) z z ,
0 = i w u , t + i β s w u , z + i ( 2 k 0 2 Γ ) w u + k 0 λ e s β s exp ( 2 i k 0 β s t ) u v ,
0 = i w v , t i β s w v , z + i ( 2 k 0 2 Γ ) w v + k 0 λ e s β s exp ( 2 i k 0 β s t ) u v .

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