Abstract

We report the coherent propagation of optical pulses through a three-level atomic system embedded in a nonlinear dispersive waveguide. We treat the case where an initially chirped soliton in resonance with the pump transition propagates together with a seed without chirp in resonance with the Stokes transition. It is found that the initial chirp is suppressed during the soliton frequency conversion, demonstrating the principle of a frequency converter that produces chirp-free soliton pulses. Numerical solutions of the complete equation set describing our system corroborate our analytical result.

© 2006 Optical Society of America

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  1. A. Hasegawa and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. 1. Anomalous dispersion," Appl. Phys. Lett. 23, 142-144 (1973).
    [CrossRef]
  2. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
    [CrossRef]
  3. S. L. McCall and E. L. Hahn, "Self-induced transparency by pulsed coherent light," Phys. Rev. Lett. 18, 908-911 (1967).
    [CrossRef]
  4. S. L. McCall and E. L. Hahn, "Self-induced transparency," Phys. Rev. 183, 457-485 (1969).
    [CrossRef]
  5. H. M. Gibbs and R. E. Slusher, "Peak amplification and breakup of a coherent optical pulse in a simple atomic absorber," Phys. Rev. Lett. 24, 638-641 (1970).
    [CrossRef]
  6. A. I. Maimistov and E. A. Manykin, "Propagation of ultrashort optical pulses in resonant non-linear light guides," Zh. Eksp. Teor. Fiz. 85, 1177-1181 (1983).
  7. M. Nakazawa, F. Yamada, and H. Kubota, "Coexistence of a self-induced-transparency soliton and a nonlinear Schrödinger soliton in an erbium-doped fiber," Phys. Rev. A 44, 5973-5987 (1991).
    [CrossRef] [PubMed]
  8. D. P. Caetano, S. B. Cavalcanti, J. M. Hickmann, A. M. Kamchatnov, R. A. Kraenkel, and E. A. Makarova, "Soliton propagation in a medium with Kerr nonlinearity and resonant impurities: a variational approach," Phys. Rev. E 67, 046615 (2003).
    [CrossRef]
  9. G. Vemuri, G. S. Agarwal, and K. V. Vasavada, "Cloning, dragging, and parametric amplification of solitons in a coherently driven, nonabsorbing system," Phys. Rev. Lett. 79, 3889-3892 (1997).
    [CrossRef]
  10. D. P. Caetano, S. B. Cavalcanti, and J. M. Hickmann, "Coherent interaction effects in pulses propagating through a doped nonlinear dispersive medium," Phys. Rev. E 65, 036617 (2002).
    [CrossRef]
  11. L. Matulic and J. H. Eberly, "Analytic study of pulse chirping in self-induced transparency," Phys. Rev. A 6, 822-836 (1972).
    [CrossRef]
  12. J. H. Eberly, "Area theorem rederived," Opt. Express 2, 173-176 (1998).
    [CrossRef] [PubMed]
  13. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995).
  14. V. B. Gorfinkel and S. Luryi, "High-frequency modulation and suppression of chirp in semiconductor lasers," Appl. Phys. Lett. 62, 2923-2925 (1993).
    [CrossRef]
  15. J. Li, P. A. Andrekson, and B. Bakhshi, "Direct generation of subpicosecond chirp-free pulses at 10 GHz from a nonpolarization maintaining actively mode-locked fiber ring laser," IEEE Photon. Technol. Lett. 12, 1150-1152 (2000).
    [CrossRef]
  16. N. Dokhane and G. L. Lippi, "Chirp reduction in semiconductor lasers through injection current patterning," Appl. Phys. Lett. 78, 3938-3940 (2001).
    [CrossRef]
  17. L. V. Hmurcik and D. J. Kaup, "Solitons created by chirped initial profiles in coherent pulse-propagation," J. Opt. Soc. Am. 69, 597-604 (1979).
    [CrossRef]
  18. Z. V. Lewis, "Semiclassical solutions of the Zakharov-Shabat scattering problem for phase modulated potentials," Phys. Lett. 112A, 99-103 (1985).
  19. D. J. Kaup, J. Elreedy, and B. A. Malomed, "Effect of a chirp on soliton production," Phys. Rev. E 50, 1635-1637 (1994).
    [CrossRef]
  20. M. Desaix, L. Helczynski, D. Anderson, and M. Lisak, "Propagation properties of chirped soliton pulses in optical nonlinear Kerr media," Phys. Rev. E 65, 056602 (2002).
    [CrossRef]

2003

D. P. Caetano, S. B. Cavalcanti, J. M. Hickmann, A. M. Kamchatnov, R. A. Kraenkel, and E. A. Makarova, "Soliton propagation in a medium with Kerr nonlinearity and resonant impurities: a variational approach," Phys. Rev. E 67, 046615 (2003).
[CrossRef]

2002

D. P. Caetano, S. B. Cavalcanti, and J. M. Hickmann, "Coherent interaction effects in pulses propagating through a doped nonlinear dispersive medium," Phys. Rev. E 65, 036617 (2002).
[CrossRef]

M. Desaix, L. Helczynski, D. Anderson, and M. Lisak, "Propagation properties of chirped soliton pulses in optical nonlinear Kerr media," Phys. Rev. E 65, 056602 (2002).
[CrossRef]

2001

N. Dokhane and G. L. Lippi, "Chirp reduction in semiconductor lasers through injection current patterning," Appl. Phys. Lett. 78, 3938-3940 (2001).
[CrossRef]

2000

J. Li, P. A. Andrekson, and B. Bakhshi, "Direct generation of subpicosecond chirp-free pulses at 10 GHz from a nonpolarization maintaining actively mode-locked fiber ring laser," IEEE Photon. Technol. Lett. 12, 1150-1152 (2000).
[CrossRef]

1998

1997

G. Vemuri, G. S. Agarwal, and K. V. Vasavada, "Cloning, dragging, and parametric amplification of solitons in a coherently driven, nonabsorbing system," Phys. Rev. Lett. 79, 3889-3892 (1997).
[CrossRef]

1994

D. J. Kaup, J. Elreedy, and B. A. Malomed, "Effect of a chirp on soliton production," Phys. Rev. E 50, 1635-1637 (1994).
[CrossRef]

1993

V. B. Gorfinkel and S. Luryi, "High-frequency modulation and suppression of chirp in semiconductor lasers," Appl. Phys. Lett. 62, 2923-2925 (1993).
[CrossRef]

1991

M. Nakazawa, F. Yamada, and H. Kubota, "Coexistence of a self-induced-transparency soliton and a nonlinear Schrödinger soliton in an erbium-doped fiber," Phys. Rev. A 44, 5973-5987 (1991).
[CrossRef] [PubMed]

1985

Z. V. Lewis, "Semiclassical solutions of the Zakharov-Shabat scattering problem for phase modulated potentials," Phys. Lett. 112A, 99-103 (1985).

1983

A. I. Maimistov and E. A. Manykin, "Propagation of ultrashort optical pulses in resonant non-linear light guides," Zh. Eksp. Teor. Fiz. 85, 1177-1181 (1983).

1980

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
[CrossRef]

1979

1973

A. Hasegawa and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. 1. Anomalous dispersion," Appl. Phys. Lett. 23, 142-144 (1973).
[CrossRef]

1972

L. Matulic and J. H. Eberly, "Analytic study of pulse chirping in self-induced transparency," Phys. Rev. A 6, 822-836 (1972).
[CrossRef]

1970

H. M. Gibbs and R. E. Slusher, "Peak amplification and breakup of a coherent optical pulse in a simple atomic absorber," Phys. Rev. Lett. 24, 638-641 (1970).
[CrossRef]

1969

S. L. McCall and E. L. Hahn, "Self-induced transparency," Phys. Rev. 183, 457-485 (1969).
[CrossRef]

1967

S. L. McCall and E. L. Hahn, "Self-induced transparency by pulsed coherent light," Phys. Rev. Lett. 18, 908-911 (1967).
[CrossRef]

Agarwal, G. S.

G. Vemuri, G. S. Agarwal, and K. V. Vasavada, "Cloning, dragging, and parametric amplification of solitons in a coherently driven, nonabsorbing system," Phys. Rev. Lett. 79, 3889-3892 (1997).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995).

Anderson, D.

M. Desaix, L. Helczynski, D. Anderson, and M. Lisak, "Propagation properties of chirped soliton pulses in optical nonlinear Kerr media," Phys. Rev. E 65, 056602 (2002).
[CrossRef]

Andrekson, P. A.

J. Li, P. A. Andrekson, and B. Bakhshi, "Direct generation of subpicosecond chirp-free pulses at 10 GHz from a nonpolarization maintaining actively mode-locked fiber ring laser," IEEE Photon. Technol. Lett. 12, 1150-1152 (2000).
[CrossRef]

Bakhshi, B.

J. Li, P. A. Andrekson, and B. Bakhshi, "Direct generation of subpicosecond chirp-free pulses at 10 GHz from a nonpolarization maintaining actively mode-locked fiber ring laser," IEEE Photon. Technol. Lett. 12, 1150-1152 (2000).
[CrossRef]

Caetano, D. P.

D. P. Caetano, S. B. Cavalcanti, J. M. Hickmann, A. M. Kamchatnov, R. A. Kraenkel, and E. A. Makarova, "Soliton propagation in a medium with Kerr nonlinearity and resonant impurities: a variational approach," Phys. Rev. E 67, 046615 (2003).
[CrossRef]

D. P. Caetano, S. B. Cavalcanti, and J. M. Hickmann, "Coherent interaction effects in pulses propagating through a doped nonlinear dispersive medium," Phys. Rev. E 65, 036617 (2002).
[CrossRef]

Cavalcanti, S. B.

D. P. Caetano, S. B. Cavalcanti, J. M. Hickmann, A. M. Kamchatnov, R. A. Kraenkel, and E. A. Makarova, "Soliton propagation in a medium with Kerr nonlinearity and resonant impurities: a variational approach," Phys. Rev. E 67, 046615 (2003).
[CrossRef]

D. P. Caetano, S. B. Cavalcanti, and J. M. Hickmann, "Coherent interaction effects in pulses propagating through a doped nonlinear dispersive medium," Phys. Rev. E 65, 036617 (2002).
[CrossRef]

Desaix, M.

M. Desaix, L. Helczynski, D. Anderson, and M. Lisak, "Propagation properties of chirped soliton pulses in optical nonlinear Kerr media," Phys. Rev. E 65, 056602 (2002).
[CrossRef]

Dokhane, N.

N. Dokhane and G. L. Lippi, "Chirp reduction in semiconductor lasers through injection current patterning," Appl. Phys. Lett. 78, 3938-3940 (2001).
[CrossRef]

Eberly, J. H.

J. H. Eberly, "Area theorem rederived," Opt. Express 2, 173-176 (1998).
[CrossRef] [PubMed]

L. Matulic and J. H. Eberly, "Analytic study of pulse chirping in self-induced transparency," Phys. Rev. A 6, 822-836 (1972).
[CrossRef]

Elreedy, J.

D. J. Kaup, J. Elreedy, and B. A. Malomed, "Effect of a chirp on soliton production," Phys. Rev. E 50, 1635-1637 (1994).
[CrossRef]

Gibbs, H. M.

H. M. Gibbs and R. E. Slusher, "Peak amplification and breakup of a coherent optical pulse in a simple atomic absorber," Phys. Rev. Lett. 24, 638-641 (1970).
[CrossRef]

Gordon, J. P.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
[CrossRef]

Gorfinkel, V. B.

V. B. Gorfinkel and S. Luryi, "High-frequency modulation and suppression of chirp in semiconductor lasers," Appl. Phys. Lett. 62, 2923-2925 (1993).
[CrossRef]

Hahn, E. L.

S. L. McCall and E. L. Hahn, "Self-induced transparency," Phys. Rev. 183, 457-485 (1969).
[CrossRef]

S. L. McCall and E. L. Hahn, "Self-induced transparency by pulsed coherent light," Phys. Rev. Lett. 18, 908-911 (1967).
[CrossRef]

Hasegawa, A.

A. Hasegawa and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. 1. Anomalous dispersion," Appl. Phys. Lett. 23, 142-144 (1973).
[CrossRef]

Helczynski, L.

M. Desaix, L. Helczynski, D. Anderson, and M. Lisak, "Propagation properties of chirped soliton pulses in optical nonlinear Kerr media," Phys. Rev. E 65, 056602 (2002).
[CrossRef]

Hickmann, J. M.

D. P. Caetano, S. B. Cavalcanti, J. M. Hickmann, A. M. Kamchatnov, R. A. Kraenkel, and E. A. Makarova, "Soliton propagation in a medium with Kerr nonlinearity and resonant impurities: a variational approach," Phys. Rev. E 67, 046615 (2003).
[CrossRef]

D. P. Caetano, S. B. Cavalcanti, and J. M. Hickmann, "Coherent interaction effects in pulses propagating through a doped nonlinear dispersive medium," Phys. Rev. E 65, 036617 (2002).
[CrossRef]

Hmurcik, L. V.

Kamchatnov, A. M.

D. P. Caetano, S. B. Cavalcanti, J. M. Hickmann, A. M. Kamchatnov, R. A. Kraenkel, and E. A. Makarova, "Soliton propagation in a medium with Kerr nonlinearity and resonant impurities: a variational approach," Phys. Rev. E 67, 046615 (2003).
[CrossRef]

Kaup, D. J.

D. J. Kaup, J. Elreedy, and B. A. Malomed, "Effect of a chirp on soliton production," Phys. Rev. E 50, 1635-1637 (1994).
[CrossRef]

L. V. Hmurcik and D. J. Kaup, "Solitons created by chirped initial profiles in coherent pulse-propagation," J. Opt. Soc. Am. 69, 597-604 (1979).
[CrossRef]

Kraenkel, R. A.

D. P. Caetano, S. B. Cavalcanti, J. M. Hickmann, A. M. Kamchatnov, R. A. Kraenkel, and E. A. Makarova, "Soliton propagation in a medium with Kerr nonlinearity and resonant impurities: a variational approach," Phys. Rev. E 67, 046615 (2003).
[CrossRef]

Kubota, H.

M. Nakazawa, F. Yamada, and H. Kubota, "Coexistence of a self-induced-transparency soliton and a nonlinear Schrödinger soliton in an erbium-doped fiber," Phys. Rev. A 44, 5973-5987 (1991).
[CrossRef] [PubMed]

Lewis, Z. V.

Z. V. Lewis, "Semiclassical solutions of the Zakharov-Shabat scattering problem for phase modulated potentials," Phys. Lett. 112A, 99-103 (1985).

Li, J.

J. Li, P. A. Andrekson, and B. Bakhshi, "Direct generation of subpicosecond chirp-free pulses at 10 GHz from a nonpolarization maintaining actively mode-locked fiber ring laser," IEEE Photon. Technol. Lett. 12, 1150-1152 (2000).
[CrossRef]

Lippi, G. L.

N. Dokhane and G. L. Lippi, "Chirp reduction in semiconductor lasers through injection current patterning," Appl. Phys. Lett. 78, 3938-3940 (2001).
[CrossRef]

Lisak, M.

M. Desaix, L. Helczynski, D. Anderson, and M. Lisak, "Propagation properties of chirped soliton pulses in optical nonlinear Kerr media," Phys. Rev. E 65, 056602 (2002).
[CrossRef]

Luryi, S.

V. B. Gorfinkel and S. Luryi, "High-frequency modulation and suppression of chirp in semiconductor lasers," Appl. Phys. Lett. 62, 2923-2925 (1993).
[CrossRef]

Maimistov, A. I.

A. I. Maimistov and E. A. Manykin, "Propagation of ultrashort optical pulses in resonant non-linear light guides," Zh. Eksp. Teor. Fiz. 85, 1177-1181 (1983).

Makarova, E. A.

D. P. Caetano, S. B. Cavalcanti, J. M. Hickmann, A. M. Kamchatnov, R. A. Kraenkel, and E. A. Makarova, "Soliton propagation in a medium with Kerr nonlinearity and resonant impurities: a variational approach," Phys. Rev. E 67, 046615 (2003).
[CrossRef]

Malomed, B. A.

D. J. Kaup, J. Elreedy, and B. A. Malomed, "Effect of a chirp on soliton production," Phys. Rev. E 50, 1635-1637 (1994).
[CrossRef]

Manykin, E. A.

A. I. Maimistov and E. A. Manykin, "Propagation of ultrashort optical pulses in resonant non-linear light guides," Zh. Eksp. Teor. Fiz. 85, 1177-1181 (1983).

Matulic, L.

L. Matulic and J. H. Eberly, "Analytic study of pulse chirping in self-induced transparency," Phys. Rev. A 6, 822-836 (1972).
[CrossRef]

McCall, S. L.

S. L. McCall and E. L. Hahn, "Self-induced transparency," Phys. Rev. 183, 457-485 (1969).
[CrossRef]

S. L. McCall and E. L. Hahn, "Self-induced transparency by pulsed coherent light," Phys. Rev. Lett. 18, 908-911 (1967).
[CrossRef]

Mollenauer, L. F.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
[CrossRef]

Nakazawa, M.

M. Nakazawa, F. Yamada, and H. Kubota, "Coexistence of a self-induced-transparency soliton and a nonlinear Schrödinger soliton in an erbium-doped fiber," Phys. Rev. A 44, 5973-5987 (1991).
[CrossRef] [PubMed]

Slusher, R. E.

H. M. Gibbs and R. E. Slusher, "Peak amplification and breakup of a coherent optical pulse in a simple atomic absorber," Phys. Rev. Lett. 24, 638-641 (1970).
[CrossRef]

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
[CrossRef]

Tappert, F.

A. Hasegawa and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. 1. Anomalous dispersion," Appl. Phys. Lett. 23, 142-144 (1973).
[CrossRef]

Vasavada, K. V.

G. Vemuri, G. S. Agarwal, and K. V. Vasavada, "Cloning, dragging, and parametric amplification of solitons in a coherently driven, nonabsorbing system," Phys. Rev. Lett. 79, 3889-3892 (1997).
[CrossRef]

Vemuri, G.

G. Vemuri, G. S. Agarwal, and K. V. Vasavada, "Cloning, dragging, and parametric amplification of solitons in a coherently driven, nonabsorbing system," Phys. Rev. Lett. 79, 3889-3892 (1997).
[CrossRef]

Yamada, F.

M. Nakazawa, F. Yamada, and H. Kubota, "Coexistence of a self-induced-transparency soliton and a nonlinear Schrödinger soliton in an erbium-doped fiber," Phys. Rev. A 44, 5973-5987 (1991).
[CrossRef] [PubMed]

Appl. Phys. Lett.

A. Hasegawa and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. 1. Anomalous dispersion," Appl. Phys. Lett. 23, 142-144 (1973).
[CrossRef]

N. Dokhane and G. L. Lippi, "Chirp reduction in semiconductor lasers through injection current patterning," Appl. Phys. Lett. 78, 3938-3940 (2001).
[CrossRef]

V. B. Gorfinkel and S. Luryi, "High-frequency modulation and suppression of chirp in semiconductor lasers," Appl. Phys. Lett. 62, 2923-2925 (1993).
[CrossRef]

IEEE Photon. Technol. Lett.

J. Li, P. A. Andrekson, and B. Bakhshi, "Direct generation of subpicosecond chirp-free pulses at 10 GHz from a nonpolarization maintaining actively mode-locked fiber ring laser," IEEE Photon. Technol. Lett. 12, 1150-1152 (2000).
[CrossRef]

J. Opt. Soc. Am.

Opt. Express

Phys. Lett.

Z. V. Lewis, "Semiclassical solutions of the Zakharov-Shabat scattering problem for phase modulated potentials," Phys. Lett. 112A, 99-103 (1985).

Phys. Rev.

S. L. McCall and E. L. Hahn, "Self-induced transparency," Phys. Rev. 183, 457-485 (1969).
[CrossRef]

Phys. Rev. A

M. Nakazawa, F. Yamada, and H. Kubota, "Coexistence of a self-induced-transparency soliton and a nonlinear Schrödinger soliton in an erbium-doped fiber," Phys. Rev. A 44, 5973-5987 (1991).
[CrossRef] [PubMed]

L. Matulic and J. H. Eberly, "Analytic study of pulse chirping in self-induced transparency," Phys. Rev. A 6, 822-836 (1972).
[CrossRef]

Phys. Rev. E

D. P. Caetano, S. B. Cavalcanti, J. M. Hickmann, A. M. Kamchatnov, R. A. Kraenkel, and E. A. Makarova, "Soliton propagation in a medium with Kerr nonlinearity and resonant impurities: a variational approach," Phys. Rev. E 67, 046615 (2003).
[CrossRef]

D. J. Kaup, J. Elreedy, and B. A. Malomed, "Effect of a chirp on soliton production," Phys. Rev. E 50, 1635-1637 (1994).
[CrossRef]

M. Desaix, L. Helczynski, D. Anderson, and M. Lisak, "Propagation properties of chirped soliton pulses in optical nonlinear Kerr media," Phys. Rev. E 65, 056602 (2002).
[CrossRef]

D. P. Caetano, S. B. Cavalcanti, and J. M. Hickmann, "Coherent interaction effects in pulses propagating through a doped nonlinear dispersive medium," Phys. Rev. E 65, 036617 (2002).
[CrossRef]

Phys. Rev. Lett.

G. Vemuri, G. S. Agarwal, and K. V. Vasavada, "Cloning, dragging, and parametric amplification of solitons in a coherently driven, nonabsorbing system," Phys. Rev. Lett. 79, 3889-3892 (1997).
[CrossRef]

H. M. Gibbs and R. E. Slusher, "Peak amplification and breakup of a coherent optical pulse in a simple atomic absorber," Phys. Rev. Lett. 24, 638-641 (1970).
[CrossRef]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
[CrossRef]

S. L. McCall and E. L. Hahn, "Self-induced transparency by pulsed coherent light," Phys. Rev. Lett. 18, 908-911 (1967).
[CrossRef]

Zh. Eksp. Teor. Fiz.

A. I. Maimistov and E. A. Manykin, "Propagation of ultrashort optical pulses in resonant non-linear light guides," Zh. Eksp. Teor. Fiz. 85, 1177-1181 (1983).

Other

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995).

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Figures (6)

Fig. 1
Fig. 1

Energy levels and applied pulses for the soliton frequency conversion process.

Fig. 2
Fig. 2

Pulse intensity (dotted curves) and frequency chirp (solid curves) at z = 0 for (a) A 1 and (b) A 2 . τ is normalized to the FWHM pulse duration.

Fig. 3
Fig. 3

Propagation of the A 2 pulse. τ is normalized to the FWHM pulse duration.

Fig. 4
Fig. 4

Propagation of the A 1 pulse. τ is normalized to the FWHM pulse duration.

Fig. 5
Fig. 5

Frequency chirp evolution of the A 2 pulse. τ is normalized to the FWHM pulse duration.

Fig. 6
Fig. 6

Propagation of the A 2 pulse for a large distance. τ is normalized to the FWHM pulse duration.

Equations (14)

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

A 1 z = i 2 β 21 2 A 1 t 2 + i γ 1 ( A 1 2 + 2 A 2 2 ) A 1 + i ω 1 n a 2 ε 0 c μ 13 c 3 * c 1 ,
A 2 z = i 2 β 22 2 A 2 t 2 + i γ 2 ( A 2 2 + 2 A 1 2 ) A 2 + i ω 2 n a 2 ε 0 c μ 12 c 2 * c 1 ,
d c 1 d t = i ( c 2 μ 12 A 2 + c 3 μ 13 A 1 ) ,
d c 2 d t = i ( c 1 μ 12 * A 2 ) ,
d c 3 d t = i ( c 1 μ 13 A 1 * ) ,
A 1 2 z = i 2 β 21 t ( A 1 * A 1 t A 1 A 1 * t ) + ω 1 n a 2 ε 0 c c 3 2 t ,
A 2 2 z = i 2 β 22 t ( A 2 * A 2 t A 2 A 2 * t ) + ω 2 n a 2 ε 0 c c 2 2 t .
| ϕ 1 t | z = 0 = f ( t ) , | ϕ 2 t | z = 0 = 0 ,
a 1 2 z = β 21 t ( a 1 ϕ 1 t ) + ω 1 n a 2 ϵ 0 c c 3 2 t ,
a 2 2 z = β 22 t ( a 2 ϕ 2 t ) + ω 2 n a 2 ϵ 0 c c 2 2 t .
z ( a 1 2 + a 2 2 ) = t ( β 21 a 1 ϕ 1 t + β 22 a 2 ϕ 2 t ) + n a 2 ϵ 0 c t [ ω 2 c 2 2 ω 1 ( c 1 2 + c 2 2 ) ] .
β 21 a 1 ϕ 1 t + β 22 a 2 ϕ 2 t = z [ ( a 1 2 + a 2 2 ) d t ] = n a 2 ϵ 0 c [ ω 1 c 1 2 + ( ω 1 ω 2 ) c 2 2 ] .
a 2 ϕ 2 t = 0 ,
ϕ 2 t = 0 .

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