Abstract

We consider a laser that is incident normally upon a planar waveguide with a Kerr nonlinearity. In a previous paper [ J. Opt. Soc. Am. 8, 786 ( 1991)] we demonstrated that the waveguide fields can participate in a transverse instability of the incident beam. We examine the evolution of this instability, show that the system evolves to a steady-state pattern, and develop a general instability criterion.

© 1992 Optical Society of America

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References

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  1. See, e.g., feature on transverse effects in nonlinear-optical systems, N. B. Abraham, W. J. Firth, eds., J. Opt. Soc. Am. B 7, 947–1157, 1259–1373 (1990).
  2. E.g., K. Dworschak, J. E. Sipe, H. M. van Driel, “Solid–melt patterns induced on silicon by a continuous laser beam at nonnormal incidence,” J. Opt. Soc. Am. B 7, 981–989 (1990); J. V. Moloney, H. Adachihara, R. Indik, C. Lizarraga, R. Northcutt, D. W. McLaughlin, A. C. Newell, “Modulation-induced optical pattern formation in a passive optical-feedback system,” J. Opt. Soc. Am. B 7, 1039–1044 (1990).
    [CrossRef]
  3. E.g., F. Hollinger, Chr. Jung, H. Weber, “Simple mathematical model describing multitransversal solid-state lasers,” J. Opt. Soc. Am. B 7, 1013–1018 (1990).
    [CrossRef]
  4. E.g., M. Dagenais, H. G. Winful, “Low transverse optical bistability near bound excitons in cadmium sulfide,” Appl. Phys. Lett. 44, 574–576 (1984); Chr. Tamm, C. O. Weiss, “Bistability and optical switching of spatial patterns in a laser,” J. Opt. Soc. Am. B 7, 1034–1038 (1990).
    [CrossRef]
  5. B. D. Robert, J. E. Sipe, “Transverse instability in a nonlinear waveguide. I. Linear analysis,” J. Opt. Soc. Am. B 8, 786–796 (1991). We take this opportunity to correct an error in that paper: In all equations from Eq. (3.4) onward any occurrences of ω, the detuning between the waveguide field and the incident beam, should be replaced by ω/vg, where vg= 1/(dκ/dω) is the group velocity of the waveguide mode. This holds for both the real and the imaginary parts ω.
    [CrossRef]
  6. H. Goldstein, Classical Mechanics (Addison-Wesley, New York, 1981).
  7. J. H. Marburger, J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979).
    [CrossRef]
  8. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
  9. E. Hairer, S. P. Nørsett, G. Wanner, Solving Ordinary Differential Equations I (Springer-Verlag, Berlin, 1987).
  10. K. Dekker, J. G. Verwer, Stability of Runge–Kutte Methods for Nonstiff Nonlinear Equations (North-Holland, Amsterdam, 1984).
  11. C. M. de Sterke, K. R. Jackson, B. D. Robert, “Nonlinear coupled-mode equations on a finite interval: a numerical procedure,” J. Opt. Soc. Am. B 8, 403–412, (1991).
    [CrossRef]
  12. R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983); B. T. Zel’dovich, N. F. Pipiletsky, V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, New York, 1983).
  13. C. M. de Sterke, J. E. Sipe, “Switching behavior of finite periodic nonlinear media,” Phys. Rev. A 42, 2858–2869 (1990).
    [CrossRef] [PubMed]
  14. T. J. Karr, J. R. Morris, D. H. Chambers, J. A. Viecelli, P. G. Cramer, “Perturbation growth by thermal blooming in turbulence,” J. Opt. Soc. Am. B 7, 1103–1124 (1990); M. D. Fiet, J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
    [CrossRef]
  15. A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, P. W. Milonni, “Instabilities and chaos in the polarization of counterpropagating light fields,” Phys. Rev. Lett. 58, 2432–2435 (1987).
    [CrossRef] [PubMed]
  16. W. J. Firth, C. Paré, “Transverse modulation instabilities for counterpropagating beams in Kerr media,” Opt. Lett. 13, 1096–1098 (1988).
    [CrossRef] [PubMed]
  17. Y. Silberberg, I. Bar Joseph, “Instabilities, self-oscillation, and chaos in a simple nonlinear interaction,” Phys. Rev. Lett. 48, 1541–1543 (1982); I. Bar-Joseph, Y. Silberberg, “The mechanism of instabilities in an optical cavity,” Opt. Commun. 48, 53–56 (1983).
    [CrossRef]
  18. C. T. Law, A. E. Kaplan, “Dispersion-related multimode instabilities and self-sustained oscillations in nonlinear counterpropagating waves,” Opt. Lett. 14, 734–736 (1989).
    [CrossRef] [PubMed]

1991

1990

1989

1988

1987

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, P. W. Milonni, “Instabilities and chaos in the polarization of counterpropagating light fields,” Phys. Rev. Lett. 58, 2432–2435 (1987).
[CrossRef] [PubMed]

1984

E.g., M. Dagenais, H. G. Winful, “Low transverse optical bistability near bound excitons in cadmium sulfide,” Appl. Phys. Lett. 44, 574–576 (1984); Chr. Tamm, C. O. Weiss, “Bistability and optical switching of spatial patterns in a laser,” J. Opt. Soc. Am. B 7, 1034–1038 (1990).
[CrossRef]

1982

Y. Silberberg, I. Bar Joseph, “Instabilities, self-oscillation, and chaos in a simple nonlinear interaction,” Phys. Rev. Lett. 48, 1541–1543 (1982); I. Bar-Joseph, Y. Silberberg, “The mechanism of instabilities in an optical cavity,” Opt. Commun. 48, 53–56 (1983).
[CrossRef]

1979

J. H. Marburger, J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Ackerhalt, J. R.

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, P. W. Milonni, “Instabilities and chaos in the polarization of counterpropagating light fields,” Phys. Rev. Lett. 58, 2432–2435 (1987).
[CrossRef] [PubMed]

Bar Joseph, I.

Y. Silberberg, I. Bar Joseph, “Instabilities, self-oscillation, and chaos in a simple nonlinear interaction,” Phys. Rev. Lett. 48, 1541–1543 (1982); I. Bar-Joseph, Y. Silberberg, “The mechanism of instabilities in an optical cavity,” Opt. Commun. 48, 53–56 (1983).
[CrossRef]

Boyd, R. W.

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, P. W. Milonni, “Instabilities and chaos in the polarization of counterpropagating light fields,” Phys. Rev. Lett. 58, 2432–2435 (1987).
[CrossRef] [PubMed]

Chambers, D. H.

Cramer, P. G.

Dagenais, M.

E.g., M. Dagenais, H. G. Winful, “Low transverse optical bistability near bound excitons in cadmium sulfide,” Appl. Phys. Lett. 44, 574–576 (1984); Chr. Tamm, C. O. Weiss, “Bistability and optical switching of spatial patterns in a laser,” J. Opt. Soc. Am. B 7, 1034–1038 (1990).
[CrossRef]

de Sterke, C. M.

Dekker, K.

K. Dekker, J. G. Verwer, Stability of Runge–Kutte Methods for Nonstiff Nonlinear Equations (North-Holland, Amsterdam, 1984).

Dworschak, K.

Firth, W. J.

Gaeta, A. L.

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, P. W. Milonni, “Instabilities and chaos in the polarization of counterpropagating light fields,” Phys. Rev. Lett. 58, 2432–2435 (1987).
[CrossRef] [PubMed]

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, New York, 1981).

Hairer, E.

E. Hairer, S. P. Nørsett, G. Wanner, Solving Ordinary Differential Equations I (Springer-Verlag, Berlin, 1987).

Hollinger, F.

Jackson, K. R.

Jung, Chr.

Kaplan, A. E.

Karr, T. J.

Lam, J. F.

J. H. Marburger, J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979).
[CrossRef]

Law, C. T.

Marburger, J. H.

J. H. Marburger, J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979).
[CrossRef]

Milonni, P. W.

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, P. W. Milonni, “Instabilities and chaos in the polarization of counterpropagating light fields,” Phys. Rev. Lett. 58, 2432–2435 (1987).
[CrossRef] [PubMed]

Morris, J. R.

Nørsett, S. P.

E. Hairer, S. P. Nørsett, G. Wanner, Solving Ordinary Differential Equations I (Springer-Verlag, Berlin, 1987).

Paré, C.

Robert, B. D.

Silberberg, Y.

Y. Silberberg, I. Bar Joseph, “Instabilities, self-oscillation, and chaos in a simple nonlinear interaction,” Phys. Rev. Lett. 48, 1541–1543 (1982); I. Bar-Joseph, Y. Silberberg, “The mechanism of instabilities in an optical cavity,” Opt. Commun. 48, 53–56 (1983).
[CrossRef]

Sipe, J. E.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

van Driel, H. M.

Verwer, J. G.

K. Dekker, J. G. Verwer, Stability of Runge–Kutte Methods for Nonstiff Nonlinear Equations (North-Holland, Amsterdam, 1984).

Viecelli, J. A.

Wanner, G.

E. Hairer, S. P. Nørsett, G. Wanner, Solving Ordinary Differential Equations I (Springer-Verlag, Berlin, 1987).

Weber, H.

Winful, H. G.

E.g., M. Dagenais, H. G. Winful, “Low transverse optical bistability near bound excitons in cadmium sulfide,” Appl. Phys. Lett. 44, 574–576 (1984); Chr. Tamm, C. O. Weiss, “Bistability and optical switching of spatial patterns in a laser,” J. Opt. Soc. Am. B 7, 1034–1038 (1990).
[CrossRef]

Appl. Phys. Lett.

E.g., M. Dagenais, H. G. Winful, “Low transverse optical bistability near bound excitons in cadmium sulfide,” Appl. Phys. Lett. 44, 574–576 (1984); Chr. Tamm, C. O. Weiss, “Bistability and optical switching of spatial patterns in a laser,” J. Opt. Soc. Am. B 7, 1034–1038 (1990).
[CrossRef]

J. H. Marburger, J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979).
[CrossRef]

J. Opt. Soc. Am. B

B. D. Robert, J. E. Sipe, “Transverse instability in a nonlinear waveguide. I. Linear analysis,” J. Opt. Soc. Am. B 8, 786–796 (1991). We take this opportunity to correct an error in that paper: In all equations from Eq. (3.4) onward any occurrences of ω, the detuning between the waveguide field and the incident beam, should be replaced by ω/vg, where vg= 1/(dκ/dω) is the group velocity of the waveguide mode. This holds for both the real and the imaginary parts ω.
[CrossRef]

See, e.g., feature on transverse effects in nonlinear-optical systems, N. B. Abraham, W. J. Firth, eds., J. Opt. Soc. Am. B 7, 947–1157, 1259–1373 (1990).

E.g., K. Dworschak, J. E. Sipe, H. M. van Driel, “Solid–melt patterns induced on silicon by a continuous laser beam at nonnormal incidence,” J. Opt. Soc. Am. B 7, 981–989 (1990); J. V. Moloney, H. Adachihara, R. Indik, C. Lizarraga, R. Northcutt, D. W. McLaughlin, A. C. Newell, “Modulation-induced optical pattern formation in a passive optical-feedback system,” J. Opt. Soc. Am. B 7, 1039–1044 (1990).
[CrossRef]

E.g., F. Hollinger, Chr. Jung, H. Weber, “Simple mathematical model describing multitransversal solid-state lasers,” J. Opt. Soc. Am. B 7, 1013–1018 (1990).
[CrossRef]

C. M. de Sterke, K. R. Jackson, B. D. Robert, “Nonlinear coupled-mode equations on a finite interval: a numerical procedure,” J. Opt. Soc. Am. B 8, 403–412, (1991).
[CrossRef]

T. J. Karr, J. R. Morris, D. H. Chambers, J. A. Viecelli, P. G. Cramer, “Perturbation growth by thermal blooming in turbulence,” J. Opt. Soc. Am. B 7, 1103–1124 (1990); M. D. Fiet, J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
[CrossRef]

Opt. Lett.

Phys. Rev. A

C. M. de Sterke, J. E. Sipe, “Switching behavior of finite periodic nonlinear media,” Phys. Rev. A 42, 2858–2869 (1990).
[CrossRef] [PubMed]

Phys. Rev. Lett.

Y. Silberberg, I. Bar Joseph, “Instabilities, self-oscillation, and chaos in a simple nonlinear interaction,” Phys. Rev. Lett. 48, 1541–1543 (1982); I. Bar-Joseph, Y. Silberberg, “The mechanism of instabilities in an optical cavity,” Opt. Commun. 48, 53–56 (1983).
[CrossRef]

A. L. Gaeta, R. W. Boyd, J. R. Ackerhalt, P. W. Milonni, “Instabilities and chaos in the polarization of counterpropagating light fields,” Phys. Rev. Lett. 58, 2432–2435 (1987).
[CrossRef] [PubMed]

Other

R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983); B. T. Zel’dovich, N. F. Pipiletsky, V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, New York, 1983).

H. Goldstein, Classical Mechanics (Addison-Wesley, New York, 1981).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

E. Hairer, S. P. Nørsett, G. Wanner, Solving Ordinary Differential Equations I (Springer-Verlag, Berlin, 1987).

K. Dekker, J. G. Verwer, Stability of Runge–Kutte Methods for Nonstiff Nonlinear Equations (North-Holland, Amsterdam, 1984).

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Figures (11)

Fig. 1
Fig. 1

We consider an intense beam that is incident normally upon a waveguide. An instantaneous Kerr nonlinearity is present in the center (film) region but is uniform in the plane of the guide.

Fig. 2
Fig. 2

Intensity of the right-going wave as a function of time for a uniform beam profile and zero attenuation. The profile has a width (area) of 6 times the threshold value, and the intensity is recorded in the center of the pattern.

Fig. 3
Fig. 3

Intensity of the right-going field as a function of distance across the interaction region for the same parameters as in Fig. 2 and time τ = 20. The dashed curve shows the final, steady-state intensity of the right-going field.

Fig. 4
Fig. 4

Plot similar to Fig. 2 but for large attenuations, 0.5 and 0.8. Note that the transients are almost completely damped out for η0 = 0.5 and vanish for η0 = 0.8.

Fig. 5
Fig. 5

Steady-state field for attenuations η0 of 0, 0.5, and 0.8. The incident beam profile is the same as in Fig. 2.

Fig. 6
Fig. 6

Steady-state field for a Gaussian beam profile just beyond threshold, - I ( ζ ) d ζ = 1.1 π / 2and η0 = 0. We compare this result with the linear solution from Eq. (3.25), where we have fitted the amplitude of the field leaving the interaction region.

Fig. 7
Fig. 7

Intensity of the right-going field at the center of a Gaussian beam plotted as a function of time for two different attenuations, η0 = 0 and η0 = 0.5. The field for η0 = 0.5 is highly localized about the center of the beam by the attenuation, and so it appears larger than the field for η0 = 0 (see Fig. 9). The profile of the incident beam is 10 times threshold.

Fig. 8
Fig. 8

Right-going field intensity across the incident beam at time τ = 30 for the system in Fig. 7.

Fig. 9
Fig. 9

Right-going field intensity across the incident beam in steady state for the system in Fig. 7.

Fig. 10
Fig. 10

Instability diagram for a Gaussian beam; the solid curve is the estimate from relation (5.1), and the crosses are from numerical simulations.

Fig. 11
Fig. 11

Dispersion relation for waves traveling in the center of the steady-state pattern. The cross indicates the waves that produce the oscillations in Fig. 2.

Equations (59)

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l 2 ( x ) = I ( x ) exp [ i ϕ ( x ) ]
NL ( x , z , t ) = S ( z ) E ( x , z , t ) 2 2 ,
S ( z ) = { 1 z in the film 0 otherwise
β 0 = ω 0 2 I e h e c 2 κ 0 E 0 2 2 ,
I i j k = d z S ( z ) h ( z ) i ( z ) j * ( z ) k ( z ) ,
ζ = x β 0 ,             τ = t v g β 0 ,
- i ( ζ + τ + η 0 ) f = [ α 0 I ( ζ ) + f 2 + 2 b 2 ] f + I ( ζ ) exp [ i ϕ ( ζ ) ] b * ,
- i ( ζ - τ - η 0 ) b * = [ α 0 I ( ζ ) + b 2 + 2 f 2 ] b * + I ( ζ ) exp [ - i ϕ ( ζ ) ] f ,
( f , b * ) ( f , b * ) exp [ i α 0 0 ζ I ( x ) d x ] .
- i ( ζ + τ + η 0 ) f = ( f 2 + 2 b 2 ) f + I ( ζ ) exp [ i ϕ ( ζ ) ] b * ,
- i ( ζ - τ - η 0 ) b * = ( b 2 + 2 f 2 ) b * + I ( ζ ) exp [ - i ϕ ( ζ ) ] f .
E wg ( x , z , t ) = Re [ y ^ E 0 ( ( I e h e I h h h ) 1 / 2 × h ( z ) exp { i [ α 0 0 x I ( x ) d x - ω 0 t ] } × [ f ( x , t ) exp ( i κ 0 x ) + b ( x , t ) exp ( - i κ 0 x ) ] ) ] .
- i f = ( - ω + f 2 + 2 b 2 ) f + I ( ζ ) exp [ i ϕ ( ζ ) ] b * ,
- i ( b * ) = ( + ω + b 2 + 2 f 2 ) b * + I ( ζ ) exp [ - i ϕ ( ζ ) ] f ,
L = ( i / 2 ) [ [ f f * + ( b * ) b ] - c . c . ] + ω ( b 2 - f 2 ) + ½ ( f 4 + b 4 + 4 f 2 b 2 ) + I ( ζ ) ( exp [ i ϕ ( ζ ) ] b * f * + c . c . )
d d ζ ( L q n ) = L q n ,
ν = f 2 + b 2 4 ,             p = f 2 - b 2 f 2 + b 2 , θ = δ f - δ b ,             δ = δ f + δ b ,
L = - 2 ν p θ - 2 ν δ - 4 ω ν p + 8 ν 2 + 4 ν 2 ( 1 - p 2 ) + 4 ν I ( ζ ) ( 1 - p 2 ) 1 / 2 cos [ θ - ϕ ( ζ ) ] .
L ¯ = 1 - 2 ν [ L - ( - 2 ν ) δ ] - 1 - 2 ν d d ζ 8 ν 2 ζ ,
L ¯ = p θ + 2 ω p - 2 ν ( 1 - p 2 ) - 2 I ( ζ ) × ( 1 - p 2 ) 1 / 2 cos [ θ - ϕ ( ζ ) ] .
H = - 2 p ω + 2 ν ( 1 - p 2 ) + 2 I ( ζ ) ( 1 - p 2 ) 1 / 2 cos [ θ - ϕ ( ζ ) ] ,
q = θ - ϕ ( ζ ) ,
H = - 2 p [ ω + ϕ ( ζ ) / 2 ] + 2 ν ( 1 - p 2 ) + 2 I ( ζ ) ( 1 - p 2 ) 1 / 2 cos q ,
½ p = I ( ζ ) ( 1 - p 2 ) 1 / 2 sin q ,
1 2 q = - [ ω + ϕ ( ζ ) 2 ] - 2 ν p - I ( ζ ) p ( 1 - p 2 ) 1 / 2 cos q .
ω ¯ = ω + ϕ 1 / 2.
I ( ζ ) = { 1 0 < ζ < L 0 otherwise
½ p = ± { 1 - p 2 - [ H / 2 + ω ¯ p - ν ( 1 - p 2 ) ] 2 } 1 / 2 ,
H = 0 ,             ω ¯ = 0.
p = cn [ 2 ( ζ - ζ 0 ) ν 2 ] .
cn ( 2 L ν 2 ) = - 1.
p = - cn ( 2 ζ ν 2 ) ,             K ( ν 2 ) = L / ( 2 n + 1 ) ,             n = 0 , 1 , 2 , .
L ( π / 2 ) ( 2 n + 1 ) .
½ p = I ( ζ ) ( 1 - p 2 ) 1 / 2 sin q ,
½ q = - I ( ζ ) p ( 1 - p 2 ) 1 / 2 cos q .
X = 2 0 ζ I ( x ) d x ,
d p d X = ( 1 - p 2 ) 1 / 2 sin q ,
d q d X = - p ( 1 - p 2 ) 1 / 2 cos q ,
H ¯ = ( 1 - p 2 ) 1 / 2 cos q .
p = ( 1 - H ¯ 2 ) 1 / 2 sin ( X - X 0 ) .
p ( ζ ) = ( 1 - H ¯ 2 ) 1 / 2 sin [ 2 ζ 0 ζ I ( x ) d x ] ,
p ( ζ - ) = - 1 = sin [ - 2 - ζ 0 I ( x ) d x ] , p ( ζ + ) = 1 = sin [ 2 ζ 0 + I ( x ) d x ] ,
- I ( ζ ) d ζ = ( π / 2 ) ( 2 n + 1 ) ,             n = 0 , 1 , 2 , .
- I ( ζ ) d ζ π / 2.
δ ( ζ ) - δ ( ζ 0 ) = ζ 0 ζ { - p [ q + ϕ ( x ) ] + 8 ν + 4 ν ( 1 - p 2 ) + 2 I ( x ) ( 1 - p 2 ) 1 / 2 cos q } d x .
X = ( ζ + τ ) / 2 ,             T = ( ζ - τ ) / 2 ,
f X = [ - η 0 + i ( f 2 + 2 b 2 ) ] f + i I ( ζ ) exp [ i ϕ ( ζ ) ] b * ,
b * T = [ η 0 + i ( b 2 + 2 f 2 ) ] b * + i I ( ζ ) exp [ - i ϕ ( ζ ) ] f .
L π / 2 + tan - 1 [ n 0 / ( 1 - η 0 2 ) 1 / 2 ] ( 1 - η 0 2 ) 1 / 2 ,             η 0 1 ,
p = 0 + O [ ( 1 - ν 2 ) 1 / 2 ] .
q q 0 = ( 0 or π ) + O [ ( 1 - ν 2 ) 1 / 2 ] ,
δ = 0 ζ ( 2 α 0 + 12 + 2 cos q ) d x + O ( 1 - ν ) .
f ¯ = 2 exp ( i δ 0 ζ ) exp [ i ( ϕ 0 - q ) / 2 ] ,
b ¯ = 2 exp ( i δ 0 ζ ) exp [ - i ( ϕ 0 - q ) / 2 ] ,
f ( ζ , τ ) = f ¯ ( ζ ) [ 1 + f ˜ ( ζ , τ ) ] ,             b ( ζ , τ ) = b ¯ ( ζ ) [ 1 + b ˜ ( ζ , τ ) ] .
- i ( ζ + τ ) f ˜ = 3 f ˜ + 2 f ˜ * + 3 b ˜ + 4 b ˜ * ,
- i ( ζ - τ ) b ˜ = 3 b ˜ + 2 b ˜ * + 3 f ˜ + 4 f ˜ * ,
( f ˜ , b ˜ ) = ( f ˜ 0 , b ˜ 0 ) exp [ - i ( k ζ - ω τ ) ] .
k ( ω ) = ± { ( ω 2 - 2 ) ± [ 20 ω 2 + ( 2 ) 2 ] 1 / 2 } 1 / 2 .

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