Abstract

The spatiotemporal dynamics of Kerr-lens mode-locked lasers is described by means of equations of motion obtained from a variational ansatz. Included are dispersion and self-phase modulation as temporal, and diffraction and self-focusing as spatial, effects. The system has steady-state solutions considering only these energy-conserving effects. The Kerr-lens mode-locking (KLM) action and gain filtering can be considered a perturbation to this dynamics. By imposing suitable boundary conditions at the end mirrors, steady-state solutions can be obtained directly from the equations of motion. The approach is used for studying the steady-state dynamics of a KLM laser system. The variational results are compared with spatiotemporal simulation.

© 2003 Optical Society of America

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References

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  1. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992).
    [CrossRef]
  2. G. Cerullo, S. De Silvestri, and V. Magni, “Self-starting Kerr-lens mode locking of a Ti:sapphire laser,” Opt. Lett. 19, 1040–1042 (1994).
    [CrossRef] [PubMed]
  3. A. Penzkofer, M. Wittmann, M. Lorenz, E. Siegert, and S. MacNamara, “Kerr lens effects in a folded-cavity four-mirror linear resonator,” Opt. Quantum Electron. 28, 423–442 (1996).
    [CrossRef]
  4. V. P. Kalosha, M. Müller, J. Herrmann, and S. Gatz, “Spatiotemporal model of femtosecond pulse generation in Kerr-lens mode-locked solid-state lasers,” J. Opt. Soc. Am. B 15, 535–550 (1998).
    [CrossRef]
  5. I. P. Christov, H. C. Kapteyn, M. M. Murnane, C.-P. Huang, and J. Zhou, “Space-time focusing of femtosecond pulses in a Ti:sapphire laser,” Opt. Lett. 20, 309–311 (1995).
    [CrossRef] [PubMed]
  6. I. P. Christov and V. D. Stoev, “Kerr-lens mode-locked laser model: role of space-time effects,” J. Opt. Soc. Am. B 15, 1960–1966 (1998).
    [CrossRef]
  7. Y. Chen and H. A. Haus, “Dispersion-managed solitons in the net positive dispersion regime,” J. Opt. Soc. Am. B 16, 24–30 (1999).
    [CrossRef]
  8. Y. Chen, F. X. Kärtner, U. Morgner, S. H. Cho, H. A. Haus, E. P. Ippen, and J. G. Fujimoto, “Dispersion-managed mode locking,” J. Opt. Soc. Am. B 16, 1999–2004 (1999).
    [CrossRef]
  9. U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser,” Opt. Lett. 24, 411–413 (1999).
    [CrossRef]
  10. R. Ell, U. Morgner, F. X. Kärtner, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, T. Tschudi, M. J. Lederer, A. Boiko, and B. Luther-Davies, “Generation of 5-fs pulses and octave-spanning spectra directly from a Ti:sapphire laser,” Opt. Lett. 26, 373–375 (2001).
    [CrossRef]
  11. D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22, 105–109 (1979).
    [CrossRef]
  12. D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).
    [CrossRef]
  13. D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
    [CrossRef]
  14. M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of optical pulses,” J. Opt. Soc. Am. B 8, 2082–2086 (1991).
    [CrossRef]
  15. C. Jirauschek, U. Morgner, and F. X. Kärtner, “Variational analysis of spatiotemporal pulse dynamics in dispersive Kerr media,” J. Opt. Soc. Am. B 19, 1716–1721 (2002).
    [CrossRef]
  16. D. J. Kaup and B. A. Malomed, “The variational principle for nonlinear waves in dissipative systems,” Physica D 87, 155–159 (1995).
    [CrossRef]
  17. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  18. L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch, “Route to phase control of ultrashort light pulses,” Opt. Lett. 21, 2008–2010 (1996).
    [CrossRef] [PubMed]
  19. A. M. Dunlop, W. J. Firth, and E. M. Wright, “Master equation for spatiotemporal beam propagation and Kerr lens mode-locking,” Opt. Commun. 138, 211–226 (1997).
    [CrossRef]
  20. V. Magni, G. Cerullo, and S. De Silvestri, “Closed form Gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers,” Opt. Commun. 101, 365–370 (1993).
    [CrossRef]
  21. S. K. Turitsyn, J. H. B. Nijhof, V. K. Mezentsev, and N. J. Doran, “Symmetries, chirp-free points, and bistability in dispersion-managed fiber lines,” Opt. Lett. 24, 1871–1873 (1999).
    [CrossRef]
  22. V. Magni, G. Cerullo, S. De Silvestri, and A. Monguzzi, “Astigmatism in Gaussian-beam self-focusing and in resonators for Kerr-lens mode locking,” J. Opt. Soc. Am. B 12, 476–485 (1995).
    [CrossRef]
  23. L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, “Quasi-discrete Hankel transform,” Opt. Lett. 23, 409–411 (1998).
    [CrossRef]
  24. T. Brabec, C. Spielmann, and F. Krausz, “Limits of pulse shortening in solitary lasers,” Opt. Lett. 17, 748–750 (1992).
    [CrossRef] [PubMed]
  25. I. P. Christov, M. M. Murnane, H. C. Kapteyn, J. Zhou, and C.-P. Huang, “Fourth-order dispersion-limited solitary pulses,” Opt. Lett. 19, 1465–1467 (1994).
    [CrossRef] [PubMed]
  26. J. Herrmann, V. P. Kalosha, and M. Müller, “Higher-order phase dispersion in femtosecond Kerr-lens mode-locked solid-state lasers: sideband generation and pulse splitting,” Opt. Lett. 22, 236–238 (1997).
    [CrossRef] [PubMed]
  27. D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5-fs pulses from a Ti:sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
    [CrossRef]
  28. P. Chernev and V. Petrov, “Self-focusing of light pulses in the presence of normal group-velocity dispersion,” Opt. Lett. 17, 172–174 (1992).
    [CrossRef] [PubMed]

2002 (1)

2001 (1)

1999 (4)

1998 (4)

1997 (2)

A. M. Dunlop, W. J. Firth, and E. M. Wright, “Master equation for spatiotemporal beam propagation and Kerr lens mode-locking,” Opt. Commun. 138, 211–226 (1997).
[CrossRef]

J. Herrmann, V. P. Kalosha, and M. Müller, “Higher-order phase dispersion in femtosecond Kerr-lens mode-locked solid-state lasers: sideband generation and pulse splitting,” Opt. Lett. 22, 236–238 (1997).
[CrossRef] [PubMed]

1996 (2)

L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch, “Route to phase control of ultrashort light pulses,” Opt. Lett. 21, 2008–2010 (1996).
[CrossRef] [PubMed]

A. Penzkofer, M. Wittmann, M. Lorenz, E. Siegert, and S. MacNamara, “Kerr lens effects in a folded-cavity four-mirror linear resonator,” Opt. Quantum Electron. 28, 423–442 (1996).
[CrossRef]

1995 (3)

1994 (2)

1993 (1)

V. Magni, G. Cerullo, and S. De Silvestri, “Closed form Gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers,” Opt. Commun. 101, 365–370 (1993).
[CrossRef]

1992 (3)

1991 (1)

1983 (1)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

1979 (2)

D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22, 105–109 (1979).
[CrossRef]

D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).
[CrossRef]

Anderson, D.

M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of optical pulses,” J. Opt. Soc. Am. B 8, 2082–2086 (1991).
[CrossRef]

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22, 105–109 (1979).
[CrossRef]

D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).
[CrossRef]

Angelow, G.

Boiko, A.

Bonnedal, M.

D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22, 105–109 (1979).
[CrossRef]

D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).
[CrossRef]

Brabec, T.

Cerullo, G.

Chen, M.

Chen, W.

Chen, Y.

Chernev, P.

Cho, S. H.

Christov, I. P.

De Silvestri, S.

Desaix, M.

Doran, N. J.

Dunlop, A. M.

A. M. Dunlop, W. J. Firth, and E. M. Wright, “Master equation for spatiotemporal beam propagation and Kerr lens mode-locking,” Opt. Commun. 138, 211–226 (1997).
[CrossRef]

Ell, R.

Firth, W. J.

A. M. Dunlop, W. J. Firth, and E. M. Wright, “Master equation for spatiotemporal beam propagation and Kerr lens mode-locking,” Opt. Commun. 138, 211–226 (1997).
[CrossRef]

Fujimoto, J. G.

Gatz, S.

Hänsch, T. W.

Haus, H. A.

Herrmann, J.

Huang, C.-P.

Huang, M.

Huang, W.

Ippen, E. P.

Jirauschek, C.

Jung, I. D.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5-fs pulses from a Ti:sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Kalosha, V. P.

Kapteyn, H. C.

Kärtner, F. X.

Kaup, D. J.

D. J. Kaup and B. A. Malomed, “The variational principle for nonlinear waves in dissipative systems,” Physica D 87, 155–159 (1995).
[CrossRef]

Keller, U.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5-fs pulses from a Ti:sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Krausz, F.

Lederer, M. J.

Lisak, M.

M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of optical pulses,” J. Opt. Soc. Am. B 8, 2082–2086 (1991).
[CrossRef]

D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).
[CrossRef]

Lorenz, M.

A. Penzkofer, M. Wittmann, M. Lorenz, E. Siegert, and S. MacNamara, “Kerr lens effects in a folded-cavity four-mirror linear resonator,” Opt. Quantum Electron. 28, 423–442 (1996).
[CrossRef]

Luther-Davies, B.

MacNamara, S.

A. Penzkofer, M. Wittmann, M. Lorenz, E. Siegert, and S. MacNamara, “Kerr lens effects in a folded-cavity four-mirror linear resonator,” Opt. Quantum Electron. 28, 423–442 (1996).
[CrossRef]

Magni, V.

Malomed, B. A.

D. J. Kaup and B. A. Malomed, “The variational principle for nonlinear waves in dissipative systems,” Physica D 87, 155–159 (1995).
[CrossRef]

Matuschek, N.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5-fs pulses from a Ti:sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Mezentsev, V. K.

Monguzzi, A.

Morgner, U.

Morier-Genoud, F.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5-fs pulses from a Ti:sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Müller, M.

Murnane, M. M.

Nijhof, J. H. B.

Penzkofer, A.

A. Penzkofer, M. Wittmann, M. Lorenz, E. Siegert, and S. MacNamara, “Kerr lens effects in a folded-cavity four-mirror linear resonator,” Opt. Quantum Electron. 28, 423–442 (1996).
[CrossRef]

Petrov, V.

Poppe, A.

Scheuer, V.

Siegert, E.

A. Penzkofer, M. Wittmann, M. Lorenz, E. Siegert, and S. MacNamara, “Kerr lens effects in a folded-cavity four-mirror linear resonator,” Opt. Quantum Electron. 28, 423–442 (1996).
[CrossRef]

Spielmann, C.

Stoev, V. D.

Sutter, D. H.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5-fs pulses from a Ti:sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Tilsch, M.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5-fs pulses from a Ti:sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Tschudi, T.

Turitsyn, S. K.

Wittmann, M.

A. Penzkofer, M. Wittmann, M. Lorenz, E. Siegert, and S. MacNamara, “Kerr lens effects in a folded-cavity four-mirror linear resonator,” Opt. Quantum Electron. 28, 423–442 (1996).
[CrossRef]

Wright, E. M.

A. M. Dunlop, W. J. Firth, and E. M. Wright, “Master equation for spatiotemporal beam propagation and Kerr lens mode-locking,” Opt. Commun. 138, 211–226 (1997).
[CrossRef]

Xu, L.

Yu, L.

Zhou, J.

Zhu, Z.

IEEE J. Quantum Electron. (1)

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5-fs pulses from a Ti:sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

J. Opt. Soc. Am. B (7)

Opt. Commun. (2)

A. M. Dunlop, W. J. Firth, and E. M. Wright, “Master equation for spatiotemporal beam propagation and Kerr lens mode-locking,” Opt. Commun. 138, 211–226 (1997).
[CrossRef]

V. Magni, G. Cerullo, and S. De Silvestri, “Closed form Gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers,” Opt. Commun. 101, 365–370 (1993).
[CrossRef]

Opt. Lett. (11)

I. P. Christov, H. C. Kapteyn, M. M. Murnane, C.-P. Huang, and J. Zhou, “Space-time focusing of femtosecond pulses in a Ti:sapphire laser,” Opt. Lett. 20, 309–311 (1995).
[CrossRef] [PubMed]

J. Herrmann, V. P. Kalosha, and M. Müller, “Higher-order phase dispersion in femtosecond Kerr-lens mode-locked solid-state lasers: sideband generation and pulse splitting,” Opt. Lett. 22, 236–238 (1997).
[CrossRef] [PubMed]

L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, “Quasi-discrete Hankel transform,” Opt. Lett. 23, 409–411 (1998).
[CrossRef]

U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser,” Opt. Lett. 24, 411–413 (1999).
[CrossRef]

S. K. Turitsyn, J. H. B. Nijhof, V. K. Mezentsev, and N. J. Doran, “Symmetries, chirp-free points, and bistability in dispersion-managed fiber lines,” Opt. Lett. 24, 1871–1873 (1999).
[CrossRef]

L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch, “Route to phase control of ultrashort light pulses,” Opt. Lett. 21, 2008–2010 (1996).
[CrossRef] [PubMed]

R. Ell, U. Morgner, F. X. Kärtner, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, T. Tschudi, M. J. Lederer, A. Boiko, and B. Luther-Davies, “Generation of 5-fs pulses and octave-spanning spectra directly from a Ti:sapphire laser,” Opt. Lett. 26, 373–375 (2001).
[CrossRef]

P. Chernev and V. Petrov, “Self-focusing of light pulses in the presence of normal group-velocity dispersion,” Opt. Lett. 17, 172–174 (1992).
[CrossRef] [PubMed]

T. Brabec, C. Spielmann, and F. Krausz, “Limits of pulse shortening in solitary lasers,” Opt. Lett. 17, 748–750 (1992).
[CrossRef] [PubMed]

G. Cerullo, S. De Silvestri, and V. Magni, “Self-starting Kerr-lens mode locking of a Ti:sapphire laser,” Opt. Lett. 19, 1040–1042 (1994).
[CrossRef] [PubMed]

I. P. Christov, M. M. Murnane, H. C. Kapteyn, J. Zhou, and C.-P. Huang, “Fourth-order dispersion-limited solitary pulses,” Opt. Lett. 19, 1465–1467 (1994).
[CrossRef] [PubMed]

Opt. Quantum Electron. (1)

A. Penzkofer, M. Wittmann, M. Lorenz, E. Siegert, and S. MacNamara, “Kerr lens effects in a folded-cavity four-mirror linear resonator,” Opt. Quantum Electron. 28, 423–442 (1996).
[CrossRef]

Phys. Fluids (2)

D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22, 105–109 (1979).
[CrossRef]

D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).
[CrossRef]

Phys. Rev. A (1)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

Physica D (1)

D. J. Kaup and B. A. Malomed, “The variational principle for nonlinear waves in dissipative systems,” Physica D 87, 155–159 (1995).
[CrossRef]

Other (1)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

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Figures (11)

Fig. 1
Fig. 1

Simplified spatiotemporal laser model. The end mirrors are represented by dashed lines.

Fig. 2
Fig. 2

Equivalent periodic system. The end mirrors are represented by dashed lines.

Fig. 3
Fig. 3

Rotationally symmetric laser setup. The end mirrors are represented by dashed lines.

Fig. 4
Fig. 4

Comparison of the pulse dynamics in the Kerr medium with breathing (solid curves) and without breathing (dashed curves): (a) pulse duration, (b) spectral width, (c) beam width.

Fig. 5
Fig. 5

Energy dependence of the stationary pulse solution: (a) stability factor, (b) minimum (solid curve) and maximum (dashed curve) beam width in the Kerr medium, (c) minimum (solid curve) and maximum (dashed curve) pulse duration in the Kerr medium.

Fig. 6
Fig. 6

Maximum achievable stability factor Γmax. The parameter regions with Γmax<1 are shaded gray.

Fig. 7
Fig. 7

Intracavity pulse energy Emax for which the stability factor reaches its maximum.

Fig. 8
Fig. 8

Energy dependence of the steady-state solution for the case of zero (solid curves) and slightly negative (dashed curves) net dispersion: (a) minimum pulse length in the Kerr medium, (b) pulse length at the right end mirror, (c) stability factor.

Fig. 9
Fig. 9

Comparison between numerical solution (solid curves) and Gaussian approximation (dashed curves) at the left end mirror: (a) temporal pulse shape, (b) spectrum, (c) transverse distribution.

Fig. 10
Fig. 10

Comparison between numerical solution (solid curves) and Gaussian approximation (dashed curves) at the right end mirror: (a) temporal pulse shape, (b) spectrum, (c) transverse distribution.

Fig. 11
Fig. 11

Comparison between numerical solution (solid curves) and Gaussian approximation (dashed curves) at the right end of the Kerr medium: (a) temporal pulse shape, (b) spectrum, (c) transverse distribution.

Tables (2)

Tables Icon

Table 1 Pulse Durations for Gaussian Approximation and Numerical Simulation

Tables Icon

Table 2 Transverse Widths for Gaussian Approximation and Numerical Simulation

Equations (67)

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

U(z, tr, x, y)=Uˆ(z)exp-12T(z)2+jb(z)tr2+12wx(z)2+jax(z)x2+12wy(z)2+jay(z)y2.
z U-jD 2tr2 U+jB2x2+2y2U+jδ|U|2U=0.
L=j2U U*z-U*Uz+DUtr2-BUx2-BUy2+δ2 |U|4.
T-γTT3-TT2wxwy=0,
wx-γwwx3+wTwx2wy=0,
wy-γwwy3+wTwxwy2=0,
T=DδE0/π3/22,
w=BδE0/π3/22;
γT=4D2,
γw=4B2.
b=-14DTT=-14D (ln T),
ax=14Bwxwx=14B (ln wx),
ay=14Bwywy=14B (ln wy).
|Uˆ(z)|2=E0π3/2T(z)wx(z)wy(z).
dϕ(z)dz=-DT(z)2+Bwx(z)2+Bwy(z)2-782 δ|Uˆ(z)|2.
u(zβ, f)=u(zα, f)exp[-jS(2πf)2].
u(zα, tr)=exp-12T(zα)2+jb(zα)tr2,
u(zα, f)=2πT(zα)21+2jb(zα)T(zα)21/2×exp-2[πfT(zα)]21+2jb(zα)T(zα)2=2πT(zα)21+2jb(zα)T(zα)21/2×exp-12σω2+jχ(2πf)2,
σω=[1/T(zα)2+4b(zα)2T(zα)2]1/2
χ=-b(zα)T(zα)41+4b(zα)2T(zα)4.
u(zβ, f)=u(zα, f)exp[-jS(2πf)2]=2πT(zα)21+2jb(zα)T(zα)21/2×exp-12σω2+j(χ+S)(2πf)2.
u(zβ, tr)=1/T(zβ)2+2jb(zβ)1/T(zα)2+2jb(zα)1/2×exp-12T(zβ)2+jb(zβ)tr2,
T(zβ)=1σω2+4(χ+S)2σω21/2
b(zβ)=-(χ+S)σω41+4(χ+S)2σω4.
Mx=AxBxCxDx,
My=AyByCyDy,
det(M)=AD-BC=1.
v(zα, x, y)=exp-jk0x22qx(zα)-jk0y22qy(zα).
1qp(z)=1k0-jwp(z)2+2ap(z),
Mn1n2,x=cos γ2cos γ100cos γ1cos γ2
Mn1n2,y=1001.
ML=1L/n01,
Mf,p=10-1/fp1,
qp(zβ)=Apqp(zα)+BpCpqp(zα)+Dp.
v(zβ, x, y)=exp(-jk0L0)Ax+Bxqx(zα)1/2Ay+Byqy(zα)1/2×exp-jk0x22qx(zβ)-jk0y22qy(zβ).
L0=zαzβn(z)dz.
wp(zβ)={wp(zα)4[k0Ap+2ap(zα)Bp]2+Bp2}1/2k0wp(zα)
ap(zβ)=k02wp(zα)4[k0Cp+2ap(zα)Dp][k0Ap+2ap(zα)Bp]+BpDpwp(zα)4[k0Ap+2ap(zα)Bp]2+Bp2.
U(zα, tr, x, y)=Uˆ(zα)exp-12T(zα)2+jb(zα)tr2×exp-12wx(zα)2+jax(zα)x2-12wy(zα)2+jay(zα)y2,
U(zβ, tr, x, y)=Uˆ(zβ)exp-12T(zβ)2+jb(zβ)tr2×exp-12wx(zβ)2+jax(zβ)x2-12wy(zβ)2+jay(zβ)y2,
Uˆ(zβ)=1/T(zβ)2+2jb(zβ)1/T(zα)2+2jb(zα)1/2×exp(-jk0L0)Ax+Bxqx(zα)1/2Ay+Byqy(zα)1/2 Uˆ(zα).
Mp=ApBpCpDp,
M˜p=DpBpCpAp.
T(z0)=T(z6),b(z0)=b(z6),
wp(z0)=wp(z6),ap(z0)=ap(z6).
T-γTT3-TT2wxwy=0,
E0=---|U(z, f, x, y)|2dfdxdy.
ΔE=----L/2L/2g(z, f+f0, x, y)|U(z, f, x, y)|2dzdfdxdy.
P=--I(z, x, y)dxdy,
ΔP=---L/2L/2g(z, f0, x, y)I(z, x, y)dzdxdy.
Γ=PΔEEΔP,
g(z, f, x, y)=g0(z)exp-(f-f0)2σf2-x2σx(z)2-y2σy(z)2
σx(z)=σy(z)=z2n02k02σr2+σr21/2.
g0(z)=g0σr2σx(z)σy(z).
z U(z, f, x, y)gain=g(z, f+f0, x, y)2 U(z, f, x, y).
h(tr)
=121-cos5πttmax-1,ttmax-1<0.21,ttmax-10.2,
 
Ψ(zβ, f, x2, y2)
=jfcBxByexp-j 2πfc L0--dx1dy1×Ψ(zα, f, x1, y1)exp-j πfc G(x1, y1, x2, y2),
G(x1, y1, x2, y2)=AxBx x12+AyBy y12+DxBx x22+DyBy y22-2Bx x1x2-2By y1y2.
L0(f)=zαzβn(z, f)dz,
---||Ui(tr, x, y)|2-|Ui-1(tr, x, y)|2|dtrdxdy---|Ui(tr, x, y)|2dtrdxdy<t,
---||Ui(f, kx, ky)|2-|Ui-1(f, kx, ky)|2|dfdkxdky---|Ui(f, kx, ky)|2dfdkxdky<f,
Pm(tr)=--|Um(tr, x, y)|2dxdy.
Sm(f)=--|Um(f-f0, x, y)|2dxdy
Φm(x, y)=-|Um(tr, x, y)|2dtr.

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