Abstract

Using the Kantorovitch method in combination with a Gaussian ansatz, we derive the equations of motion for spatial, temporal, and spatiotemporal optical propagation in a dispersive Kerr medium with a general transverse and spectral gain profile. By rewriting the variational equations as differential equations for the temporal and spatial Gaussian q parameters, optical ABCD matrices for self-focusing and self-phase modulation, a general transverse gain profile, and nonparabolic spectral gain filtering are obtained. Further effects can easily be taken into account by adding the corresponding ABCD matrices. Applications include the temporal pulse dynamics in gain fibers and the beam propagation or spatiotemporal pulse evolution in bulk gain media. As an example, the steady-state spatiotemporal Gaussian pulse dynamics in a Kerr-lens mode-locked laser resonator is studied.

© 2006 Optical Society of America

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  1. M. Manousakis, S. Droulias, P. Papagiannis, and K. Hizanidis, "Propagation of chirped solitary pulses in optical transmission lines: perturbed variational approach," Opt. Commun. 213, 293-299 (2002).
    [CrossRef]
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    [CrossRef]
  3. 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]
  4. C. Jirauschek, F. X. Kärtner, and U. Morgner, "Spatiotemporal Gaussian pulse dynamics in Kerr-lens mode-locked lasers," J. Opt. Soc. Am. B 20, 1356-1368 (2003).
    [CrossRef]
  5. D. Anderson and M. Bonnedal, "Variational approach to nonlinear self-focusing of Gaussian laser beams," Phys. Fluids 22, 105-109 (1979).
    [CrossRef]
  6. D. Anderson, M. Bonnedal, and M. Lisak, "Self-trapped cylindrical laser beams," Phys. Fluids 22, 1838-1840 (1979).
    [CrossRef]
  7. D. Anderson, "Variational approach to nonlinear pulse propagation in optical fibers," Phys. Rev. A 27, 3135-3145 (1983).
    [CrossRef]
  8. M. Desaix, D. Anderson, and M. Lisak, "Variational approach to collapse of optical pulses," J. Opt. Soc. Am. B 8, 2082-2086 (1991).
    [CrossRef]
  9. C. Jirauschek, U. Morgner, and F. X. Kärtner, "Variational analysis of spatio-temporal pulse dynamics in dispersive Kerr media," J. Opt. Soc. Am. B 19, 1716-1721 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. S. C. Cerda, S. B. Cavalcanti, and J. M. Hickmann, "A variational approach of nonlinear dissipative pulse propagation," Eur. Phys. J. D 1, 313-316 (1998).
    [CrossRef]
  13. D. Anderson, F. Cattani, and M. Lisak, "On the Pereira-Stenflo solitons," Phys. Scr. T82, 32-35 (1999).
    [CrossRef]
  14. N. Aközbek, C. M. Bowden, A. Talebpour, and S. L. Chin, "Femtosecond pulse propagation in air: variational analysis," Phys. Rev. E 61, 4540-4549 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. M. A. Larotonda and A. A. Hnilo, "Short laser pulse parameters in a nonlinear medium: different approximations of the ray-pulse matrix," Opt. Commun. 183, 207-213 (2000).
    [CrossRef]
  22. A. G. Kostenbauder, "Ray-pulse matrices: a rational treatment for dispersive optical systems," IEEE J. Quantum Electron. 26, 1148-1157 (1990).
    [CrossRef]
  23. J. L. A. Chilla and O. E. Martínez, "Spatial-temporal analysis of the self-mode-locked Ti:sapphire laser," J. Opt. Soc. Am. B 10, 638-643 (1993).
    [CrossRef]
  24. M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, "Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission," IEEE J. Quantum Electron. 34, 1075-1081 (1998).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]

2003 (1)

2002 (2)

M. Manousakis, S. Droulias, P. Papagiannis, and K. Hizanidis, "Propagation of chirped solitary pulses in optical transmission lines: perturbed variational approach," Opt. Commun. 213, 293-299 (2002).
[CrossRef]

C. Jirauschek, U. Morgner, and F. X. Kärtner, "Variational analysis of spatio-temporal pulse dynamics in dispersive Kerr media," J. Opt. Soc. Am. B 19, 1716-1721 (2002).
[CrossRef]

2001 (1)

2000 (2)

M. A. Larotonda and A. A. Hnilo, "Short laser pulse parameters in a nonlinear medium: different approximations of the ray-pulse matrix," Opt. Commun. 183, 207-213 (2000).
[CrossRef]

N. Aközbek, C. M. Bowden, A. Talebpour, and S. L. Chin, "Femtosecond pulse propagation in air: variational analysis," Phys. Rev. E 61, 4540-4549 (2000).
[CrossRef]

1999 (3)

1998 (4)

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, "Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission," IEEE J. Quantum Electron. 34, 1075-1081 (1998).
[CrossRef]

S. C. Cerda, S. B. Cavalcanti, and J. M. Hickmann, "A variational approach of nonlinear dissipative pulse propagation," Eur. Phys. J. D 1, 313-316 (1998).
[CrossRef]

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]

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]

1996 (2)

F. Riewe, "Nonconservative Lagrangian and Hamiltonian mechanics," Phys. Rev. E 53, 1890-1899 (1996).
[CrossRef]

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 (1)

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

1993 (2)

V. Magni, G. Cerullo, and S. De Silvestri, "ABCD matrix analysis of propagation of Gaussian beams through Kerr media," Opt. Commun. 96, 348-355 (1993).
[CrossRef]

J. L. A. Chilla and O. E. Martínez, "Spatial-temporal analysis of the self-mode-locked Ti:sapphire laser," J. Opt. Soc. Am. B 10, 638-643 (1993).
[CrossRef]

1991 (1)

1990 (2)

S. P. Dijaili, A. Dienes, and J. S. Smith, "ABCD matrices for dispersive pulse propagation," IEEE J. Quantum Electron. 26, 1158-1164 (1990).
[CrossRef]

A. G. Kostenbauder, "Ray-pulse matrices: a rational treatment for dispersive optical systems," IEEE J. Quantum Electron. 26, 1148-1157 (1990).
[CrossRef]

1989 (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]

Aközbek, N.

N. Aközbek, C. M. Bowden, A. Talebpour, and S. L. Chin, "Femtosecond pulse propagation in air: variational analysis," Phys. Rev. E 61, 4540-4549 (2000).
[CrossRef]

Anderson, D.

D. Anderson, F. Cattani, and M. Lisak, "On the Pereira-Stenflo solitons," Phys. Scr. T82, 32-35 (1999).
[CrossRef]

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]

Bonnedal, M.

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

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

Bowden, C. M.

N. Aközbek, C. M. Bowden, A. Talebpour, and S. L. Chin, "Femtosecond pulse propagation in air: variational analysis," Phys. Rev. E 61, 4540-4549 (2000).
[CrossRef]

Cattani, F.

D. Anderson, F. Cattani, and M. Lisak, "On the Pereira-Stenflo solitons," Phys. Scr. T82, 32-35 (1999).
[CrossRef]

Cavalcanti, S. B.

S. C. Cerda, S. B. Cavalcanti, and J. M. Hickmann, "A variational approach of nonlinear dissipative pulse propagation," Eur. Phys. J. D 1, 313-316 (1998).
[CrossRef]

Cerda, S. C.

S. C. Cerda, S. B. Cavalcanti, and J. M. Hickmann, "A variational approach of nonlinear dissipative pulse propagation," Eur. Phys. J. D 1, 313-316 (1998).
[CrossRef]

Cerullo, G.

V. Magni, G. Cerullo, and S. De Silvestri, "ABCD matrix analysis of propagation of Gaussian beams through Kerr media," Opt. Commun. 96, 348-355 (1993).
[CrossRef]

Chen, Y.

Chilla, J. L. A.

Chin, S. L.

N. Aközbek, C. M. Bowden, A. Talebpour, and S. L. Chin, "Femtosecond pulse propagation in air: variational analysis," Phys. Rev. E 61, 4540-4549 (2000).
[CrossRef]

Cho, S. H.

Christov, I. P.

De Silvestri, S.

V. Magni, G. Cerullo, and S. De Silvestri, "ABCD matrix analysis of propagation of Gaussian beams through Kerr media," Opt. Commun. 96, 348-355 (1993).
[CrossRef]

Desaix, M.

Dienes, A.

S. P. Dijaili, A. Dienes, and J. S. Smith, "ABCD matrices for dispersive pulse propagation," IEEE J. Quantum Electron. 26, 1158-1164 (1990).
[CrossRef]

Dijaili, S. P.

S. P. Dijaili, A. Dienes, and J. S. Smith, "ABCD matrices for dispersive pulse propagation," IEEE J. Quantum Electron. 26, 1158-1164 (1990).
[CrossRef]

Droulias, S.

M. Manousakis, S. Droulias, P. Papagiannis, and K. Hizanidis, "Propagation of chirped solitary pulses in optical transmission lines: perturbed variational approach," Opt. Commun. 213, 293-299 (2002).
[CrossRef]

French, P. M. W.

Fujimoto, J. G.

Gatz, S.

Grace, E. J.

Haus, H. A.

Herrmann, J.

Hickmann, J. M.

S. C. Cerda, S. B. Cavalcanti, and J. M. Hickmann, "A variational approach of nonlinear dissipative pulse propagation," Eur. Phys. J. D 1, 313-316 (1998).
[CrossRef]

Hizanidis, K.

M. Manousakis, S. Droulias, P. Papagiannis, and K. Hizanidis, "Propagation of chirped solitary pulses in optical transmission lines: perturbed variational approach," Opt. Commun. 213, 293-299 (2002).
[CrossRef]

Hnilo, A. A.

M. A. Larotonda and A. A. Hnilo, "Short laser pulse parameters in a nonlinear medium: different approximations of the ray-pulse matrix," Opt. Commun. 183, 207-213 (2000).
[CrossRef]

Ippen, E. P.

Jirauschek, C.

Kalosha, V. P.

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]

Kolner, B. H.

Kostenbauder, A. G.

A. G. Kostenbauder, "Ray-pulse matrices: a rational treatment for dispersive optical systems," IEEE J. Quantum Electron. 26, 1148-1157 (1990).
[CrossRef]

Kubota, H.

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, "Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission," IEEE J. Quantum Electron. 34, 1075-1081 (1998).
[CrossRef]

Larotonda, M. A.

M. A. Larotonda and A. A. Hnilo, "Short laser pulse parameters in a nonlinear medium: different approximations of the ray-pulse matrix," Opt. Commun. 183, 207-213 (2000).
[CrossRef]

Lisak, M.

D. Anderson, F. Cattani, and M. Lisak, "On the Pereira-Stenflo solitons," Phys. Scr. T82, 32-35 (1999).
[CrossRef]

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]

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.

V. Magni, G. Cerullo, and S. De Silvestri, "ABCD matrix analysis of propagation of Gaussian beams through Kerr media," Opt. Commun. 96, 348-355 (1993).
[CrossRef]

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]

Manousakis, M.

M. Manousakis, S. Droulias, P. Papagiannis, and K. Hizanidis, "Propagation of chirped solitary pulses in optical transmission lines: perturbed variational approach," Opt. Commun. 213, 293-299 (2002).
[CrossRef]

Martínez, O. E.

Morgner, U.

Müller, M.

Nakazawa, M.

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, "Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission," IEEE J. Quantum Electron. 34, 1075-1081 (1998).
[CrossRef]

Nazarathy, M.

New, G. H. C.

Papagiannis, P.

M. Manousakis, S. Droulias, P. Papagiannis, and K. Hizanidis, "Propagation of chirped solitary pulses in optical transmission lines: perturbed variational approach," Opt. Commun. 213, 293-299 (2002).
[CrossRef]

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]

Riewe, F.

F. Riewe, "Nonconservative Lagrangian and Hamiltonian mechanics," Phys. Rev. E 53, 1890-1899 (1996).
[CrossRef]

Sahara, A.

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, "Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission," IEEE J. Quantum Electron. 34, 1075-1081 (1998).
[CrossRef]

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]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Smith, J. S.

S. P. Dijaili, A. Dienes, and J. S. Smith, "ABCD matrices for dispersive pulse propagation," IEEE J. Quantum Electron. 26, 1158-1164 (1990).
[CrossRef]

Stoev, V. D.

Talebpour, A.

N. Aközbek, C. M. Bowden, A. Talebpour, and S. L. Chin, "Femtosecond pulse propagation in air: variational analysis," Phys. Rev. E 61, 4540-4549 (2000).
[CrossRef]

Tamura, K.

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, "Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission," IEEE J. Quantum Electron. 34, 1075-1081 (1998).
[CrossRef]

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]

Eur. Phys. J. D (1)

S. C. Cerda, S. B. Cavalcanti, and J. M. Hickmann, "A variational approach of nonlinear dissipative pulse propagation," Eur. Phys. J. D 1, 313-316 (1998).
[CrossRef]

IEEE J. Quantum Electron. (3)

S. P. Dijaili, A. Dienes, and J. S. Smith, "ABCD matrices for dispersive pulse propagation," IEEE J. Quantum Electron. 26, 1158-1164 (1990).
[CrossRef]

A. G. Kostenbauder, "Ray-pulse matrices: a rational treatment for dispersive optical systems," IEEE J. Quantum Electron. 26, 1148-1157 (1990).
[CrossRef]

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, "Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission," IEEE J. Quantum Electron. 34, 1075-1081 (1998).
[CrossRef]

J. Opt. Soc. Am. B (8)

Opt. Commun. (3)

M. Manousakis, S. Droulias, P. Papagiannis, and K. Hizanidis, "Propagation of chirped solitary pulses in optical transmission lines: perturbed variational approach," Opt. Commun. 213, 293-299 (2002).
[CrossRef]

V. Magni, G. Cerullo, and S. De Silvestri, "ABCD matrix analysis of propagation of Gaussian beams through Kerr media," Opt. Commun. 96, 348-355 (1993).
[CrossRef]

M. A. Larotonda and A. A. Hnilo, "Short laser pulse parameters in a nonlinear medium: different approximations of the ray-pulse matrix," Opt. Commun. 183, 207-213 (2000).
[CrossRef]

Opt. Lett. (2)

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]

Phys. Rev. E (2)

F. Riewe, "Nonconservative Lagrangian and Hamiltonian mechanics," Phys. Rev. E 53, 1890-1899 (1996).
[CrossRef]

N. Aközbek, C. M. Bowden, A. Talebpour, and S. L. Chin, "Femtosecond pulse propagation in air: variational analysis," Phys. Rev. E 61, 4540-4549 (2000).
[CrossRef]

Phys. Scr. (1)

D. Anderson, F. Cattani, and M. Lisak, "On the Pereira-Stenflo solitons," Phys. Scr. T82, 32-35 (1999).
[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, 1986).

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Figures (2)

Fig. 1
Fig. 1

Simplified model of a KLM laser resonator. The end mirrors are represented by dashed lines.

Fig. 2
Fig. 2

Steady-state pulse solution of the setup shown in Fig. 1 for a fixed intracavity pulse energy of 20 nJ at the right end mirror. Shown is (a) the pulse duration T at the right end mirror and (b) the stability factor Γ as a function of the gain per round trip G. Results for Δ ω ( 2 π ) = 40 , 45, 56 THz and Δ ω are represented by dotted, dashed dotted, dashed, and solid curves, respectively. Transverse gain widths of Δ x = 10 μ m , Δ x = 20 μ m , and Δ x are indicated by light gray, dark gray, and black, respectively.

Tables (2)

Tables Icon

Table 1 Gaussian Test Function; Relevant Equations of Motion; and Coefficients for Spatiotemporal, Spatial, and Temporal Dynamics

Tables Icon

Table 2 Optical Matrix Elements for Spatiotemporal Gaussian Pulse Propagation through Kerr Media with a Spatial and Spectral Gain Profile a

Equations (87)

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

i z U D t r 2 U + B ( x 2 + y 2 ) U + δ U 2 U = Q
Q = i ( g 0 g x x 2 g y y 2 + g ω t r 2 ) U .
U ( z , t r , x , y ) = U ̂ ( z ) exp { [ 1 2 T 2 ( z ) i b ( z ) ] t r 2 [ 1 2 w x 2 ( z ) i a x ( z ) ] x 2 [ 1 2 w y 2 ( z ) i a y ( z ) ] y 2 } ,
U ̂ ( z ) = A ( z ) exp [ i ϕ ( z ) ] .
w p = 4 B a p w p g p w p 3 ,
T = 4 D b T + g ω ( 1 T 4 b 2 T 3 ) ,
a p = B ( 1 w p 4 4 a p 2 ) c a δ A 2 w p 2 ,
b = D ( 1 T 4 4 b 2 ) 4 g ω b T 2 c a δ A 2 T 2 ,
A = A ( g 0 g ω T 2 + 2 D b 2 B a x 2 B a y ) ,
ϕ = 2 g ω b + D 1 T 2 B 1 w x 2 B 1 w y 2 + c ϕ δ A 2 ,
U ( z , x , y ) = U ̂ ( z ) exp { [ 1 2 w x 2 ( z ) i a x ( z ) ] x 2 [ 1 2 w y 2 ( z ) i a y ( z ) ] y 2 } ,
U ( z , t r ) = U ̂ ( z ) exp { [ 1 2 T 2 ( z ) i b ( z ) ] t r 2 } ,
Q = i F t 1 { g F t { U } } ,
F t { U } = U ̃ = d t r U exp ( i ω t r ) .
g ω = 1 2 π 1 Ω 4 E ( Ω 2 2 ω 2 ) g U ̃ 2 d ω d x d y ,
g x = 1 2 π 1 w x 4 E ( w x 2 2 x 2 ) g U ̃ 2 d ω d x d y ,
g y = 1 2 π 1 w y 4 E ( w y 2 2 y 2 ) g U ̃ 2 d ω d x d y ,
g 0 = 1 2 π 1 E g U ̃ 2 d ω d x d y + g x 2 w x 2 + g y 2 w y 2 + g w 2 Ω 2 ,
Ω = 1 T 2 + 4 b 2 T 2 .
U ̃ 2 = A 2 2 π T Ω exp ( x 2 w x 2 y 2 w y 2 ω 2 Ω 2 ) .
g x = 1 w x 4 P ( w x 2 2 x 2 ) g U 2 d x d y ,
g y = 1 w y 4 P ( w y 2 2 y 2 ) g U 2 d x d y ,
g 0 = 1 P g U 2 d x d y + g x 2 w x 2 + g y 2 w y 2 ,
U 2 = A 2 exp ( x 2 w x 2 y 2 w y 2 ) .
g ω = 1 2 π 1 Ω 4 F ( Ω 2 2 ω 2 ) g U ̃ 2 d ω ,
g 0 = 1 2 π 1 F g U ̃ 2 d ω + g ω 2 Ω 2 ,
U ̃ 2 = A 2 2 π T Ω exp ( ω 2 Ω 2 ) .
g = g ̂ exp ( Δ x 2 x 2 Δ y 2 y 2 Δ ω 2 ω 2 ) ,
g ω = g ̂ Δ ω 2 ( Δ ω 2 Ω 2 + 1 ) 3 2 ( Δ x 2 w x 2 + 1 ) 1 2 ( Δ y 2 w y 2 + 1 ) 1 2 ,
g x = g ̂ Δ x 2 ( Δ ω 2 Ω 2 + 1 ) 1 2 ( Δ x 2 w x 2 + 1 ) 3 2 ( Δ y 2 w y 2 + 1 ) 1 2 ,
g y = g ̂ Δ y 2 ( Δ ω 2 Ω 2 + 1 ) 1 2 ( Δ x 2 w x 2 + 1 ) 1 2 ( Δ y 2 w y 2 + 1 ) 3 2 ,
g 0 = 1 6 [ g ω ( 5 Ω 2 + 2 Δ ω 2 ) + g x ( 5 w x 2 + 2 Δ x 2 ) + g y ( 5 w y 2 + 2 Δ y 2 ) ] .
g = g ̂ exp ( Δ x 2 x 2 Δ y 2 y 2 ) ,
g x = g ̂ Δ x 2 ( Δ x 2 w x 2 + 1 ) 3 2 ( Δ y 2 w y 2 + 1 ) 1 2 ,
g y = g ̂ Δ y 2 ( Δ x 2 w x 2 + 1 ) 1 2 ( Δ y 2 w y 2 + 1 ) 3 2 ,
g 0 = g x ( w x 2 + Δ x 2 2 ) + g y ( w y 2 + Δ y 2 2 ) ;
g = g ̂ exp ( Δ ω 2 ω 2 ) ,
g ω = g ̂ Δ ω 2 ( Δ ω 2 Ω 2 + 1 ) 3 2 ,
g 0 = g ω ( 3 2 Ω 2 + Δ ω 2 ) .
U ( z , t r , x , y ) = U ̂ ( z ) exp [ i k 0 x 2 2 q x ( z ) + i k 0 y 2 2 q y ( z ) i ω 0 t r 2 2 q t ( z ) ] .
q p 1 = 1 k 0 ( i w p 2 + 2 a p ) ,
q t 1 = 1 ω 0 ( i T 2 + 2 b ) .
U = U ̂ exp ( i k 0 x 2 2 q x + i k 0 y 2 2 q y ) ,
U = U ̂ exp ( i ω 0 t r 2 2 q t ) .
z q s = q s 2 C s + B s ,
z U ̂ = U ̂ ( B t 2 q t B x 2 q x B y 2 q y + α ) .
M s = [ 1 B s δ z C s δ z 1 ] ,
B p = 2 B k 0 = 1 n 0 ,
B t = 2 D ω 0 + 2 i ω 0 g ω ,
C p = 2 i g p k 0 2 c a δ k 0 w p 2 U ̂ 2 ,
C t = 2 c a δ ω 0 T 2 U ̂ 2 ,
α = g 0 + i c ϕ δ U ̂ 2 .
L = i 2 ( U * U z U U * z ) + D U t r 2 B U x 2 B U y 2 + δ 2 U 4 ,
L f d d z L f = R f ,
L = L d t r d x d y
R f = 2 R { Q U * f d t r d x d y } .
R f = 1 π R { i g U ̃ U ̃ * f d ω d x d y } .
R ϕ = E [ 2 g 0 g x w x 2 g y w y 2 g ω Ω 2 ] ,
R T = 4 g ω E b T 1 ,
R b = 1 2 E T 2 [ 2 g 0 g x w x 2 g y w y 2 + g ω ( T 2 12 T 2 b 2 ) ] ,
R a x = 1 2 E w x 2 [ 2 g 0 3 g x w x 2 g y w y 2 g ω Ω 2 ] ,
E = π 3 2 A 2 T w x w y .
g ω = R T T ( 4 E b ) ,
g x = ( R ϕ w x 2 2 R a x ) ( E w x 4 ) ,
g y = ( R ϕ w y 2 2 R a y ) ( E w y 4 ) ,
g 0 = ( R ϕ E + g x w x 2 + g y w y 2 + g ω Ω 2 ) 2 .
2 ϕ ( b T 2 + a x w x 2 + a y w y 2 ) + D ( 1 T 2 + 4 b 2 T 2 ) B ( 1 w x 2 + 4 a x 2 w x 2 ) B ( 1 w y 2 + 4 a y 2 w y 2 ) + δ A 2 2 = 0 ,
E = E [ 2 g 0 g x w x 2 g y w y 2 g ω ( 1 T 2 + 4 b 2 T 2 ) ] .
ϕ + 1 2 [ b T 2 3 a x w x 2 a y w y 2 + D ( 1 T 2 + 4 b 2 T 2 ) B ( 1 w x 2 + 12 a x 2 w x 2 ) B ( 1 w y 2 + 4 a y 2 w y 2 ) ] + δ 4 2 A 2 = 0 ,
ϕ + 1 2 [ b T 2 a x w x 2 3 a y w y 2 + D ( 1 T 2 + 4 b 2 T 2 ) B ( 1 w x 2 + 4 a x 2 w x 2 ) B ( 1 w y 2 + 12 a y 2 w y 2 ) ] + δ 4 2 A 2 = 0 ,
ϕ + 1 2 [ 3 b T 2 a x w x 2 a y w y 2 + D ( 1 T 2 + 12 b 2 T 2 ) B ( 1 w x 2 + 4 a x 2 w x 2 ) B ( 1 w y 2 + 4 a y 2 w y 2 ) ] + δ 4 2 A 2 = 4 g ω b ,
1 2 E + E w x w x 4 B E a x = E [ g 0 3 g x 2 w x 2 g y 2 w y 2 2 g ω ( 1 4 T 2 + b 2 T 2 ) ] ,
1 2 E + E w y w y 4 B E a y = E [ g 0 g x 2 w x 2 3 g y 2 w y 2 2 g ω ( 1 4 T 2 + b 2 T 2 ) ] ,
1 2 E + E T T + 4 D E b = E [ g 0 g x 2 w x 2 g y 2 w y 2 + 2 g ω ( 1 4 T 2 3 T 2 b 2 ) ] .
q p = ( 2 i g p k 0 2 c a δ k 0 w p 2 U ̂ 2 ) q p 2 + 1 n 0 ,
q t = 2 c a δ ω 0 T 2 U ̂ 2 q t 2 + 2 D ω 0 + 2 i g ω ω 0 ,
U ̂ U ̂ = A A + i ϕ = B k 0 q x B k 0 q y D ω 0 q t i ω 0 g ω q t + g 0 + i c ϕ δ U ̂ 2 .
M s = [ A s B s C s D s ] .
q s ( z 2 ) = A s q s ( z 1 ) + B s C s q s ( z 1 ) + D s ,
U ̂ ( z 2 ) = τ U ̂ ( z 1 ) [ A x + B x q x ( z 1 ) ] 1 2 [ A y + B y q y ( z 1 ) ] 1 2 [ A t + B t q t ( z 1 ) ] 1 2
U ̂ ( z 2 ) = τ U ̂ ( z 1 ) [ A x + B x q x ( z 1 ) ] 1 2 [ A y + B y q y ( z 1 ) ] 1 2
U ̂ ( z 2 ) = τ U ̂ ( z 1 ) [ A t + B t q t ( z 1 ) ] 1 2
q s ( z + δ z ) = q s ( z ) + B s δ z q s ( z ) C s δ z + 1 q s 2 ( z ) C s δ z + q s ( z ) + B s δ z ,
U ̂ ( z + δ z ) = U ̂ ( z ) exp ( α δ z ) ( 1 + B x δ z q x ) 1 2 ( 1 + B y δ z q y ) 1 2 ( 1 + B t δ z q t ) 1 2 U ̂ ( z ) ( 1 B x δ z 2 q x B y δ z 2 q y B t δ z 2 q t + α δ z ) .
c a 2 δ k 0 w p 2 U ̂ 2
c a 2 δ ω 0 T 2 U ̂ 2
i c ϕ δ U ̂ 2

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