Abstract

A model and numerical solutions of Maxwell’s equations describing the propagation of short, solitonlike pulses in nonlinear dispersive optical media are presented. The model includes linear dispersion expressed in the time domain, a Kerr nonlinearity, and a coordinate system moving with the group velocity of the pulse. Numerical solutions of Maxwell’s equations are presented for circularly polarized and linearly polarized electromagnetic fields. When the electromagnetic fields are assumed to be circularly polarized, numerical solutions are compared directly with solutions of the nonlinear Schrödinger (NLS) equation. These comparisons show good agreement and indicate that the NLS equation provides an excellent model for short-pulse propagation. When the electromagnetic fields are assumed to be linearly polarized, the propagation of daughter pulses, small-amplitude pulses at three times the frequency of the solitonlike pulse, are observed in the numerical solution. These daughter pulses are shown to be the direct result of third harmonics generated by the main, solitonlike, pulse.

© 1996 Optical Society of America

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References

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  1. A. Hasegawa, Optical Solitons in Fibers (Springer-Verlag, New York, 1989).
    [CrossRef]
  2. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).
  3. R. Lubbers, F. P. Hunsberger, K. S. Kunz, R. S. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
    [CrossRef]
  4. R. M. Joseph, S. C. Hagness, and A. Taflove, “Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses,” Opt. Lett. 16, 1412–1414 (1991).
    [CrossRef] [PubMed]
  5. R. Luebbers, F. Hunsberger, and K. Kunz, “Finite difference time domain recursive convolution for second order dispersive materials,” in 1992 IEEE Conference Proceedings in Electromagnetics (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 1972–1975.
  6. P. M. Goorjian and A. Taflove, “Direct time integration of Maxwell’s equations in nonlinear dispersive media for propagation and scattering of femtosecond electromagnetic solitons,” Opt. Lett. 17, 180–182 (1992); P. M. Goorjian, A. Taflove, R. M. Joseph, and S. C. Hagness, “Computational modeling of femtosecond optical solitons from Maxwell’s equations,” IEEE J. Quantum Electron. 28, 2416–2422 (1992).
    [CrossRef]
  7. R. W. Ziolkowski and J. B. Judkins, “Full-wave vector Maxwell equation modeling of the self-focusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time,” J. Opt. Soc. Am. B 10, 186–198 (1993).
    [CrossRef]
  8. P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1990).
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    [CrossRef]
  11. A. Hasegawa and Y. Kodama, “Signal transmission by optical solitons in monomode fiber,” Proc. IEEE 69, 1145–1150 (1981).
    [CrossRef]
  12. C. Hile, “Numerical studies of nonlinear optical pulse propagation,” Ph.D. dissertation (Northwestern University, Evanton, Illinois, 1993).
  13. W. F. Ames, Numerical Methods for Partial Differential Equations (Academic, New York, 1977).
  14. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Prop. AP-14, 302–307 (1966).
  15. P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–466 (1986).
    [CrossRef] [PubMed]
  16. J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
    [CrossRef]

1993 (1)

1992 (1)

1991 (1)

1990 (1)

R. Lubbers, F. P. Hunsberger, K. S. Kunz, R. S. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
[CrossRef]

1989 (1)

1986 (1)

1981 (1)

A. Hasegawa and Y. Kodama, “Signal transmission by optical solitons in monomode fiber,” Proc. IEEE 69, 1145–1150 (1981).
[CrossRef]

1974 (1)

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
[CrossRef]

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Prop. AP-14, 302–307 (1966).

1965 (1)

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).

Ames, W. F.

W. F. Ames, Numerical Methods for Partial Differential Equations (Academic, New York, 1977).

Butcher, P. N.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1990).

Chen, H. H.

Cotter, D.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1990).

Goorjian, P. M.

Gordon, J. P.

Hagness, S. C.

Hasegawa, A.

A. Hasegawa and Y. Kodama, “Signal transmission by optical solitons in monomode fiber,” Proc. IEEE 69, 1145–1150 (1981).
[CrossRef]

A. Hasegawa, Optical Solitons in Fibers (Springer-Verlag, New York, 1989).
[CrossRef]

Haus, H. A.

Hile, C.

C. Hile, “Numerical studies of nonlinear optical pulse propagation,” Ph.D. dissertation (Northwestern University, Evanton, Illinois, 1993).

Hunsberger, F.

R. Luebbers, F. Hunsberger, and K. Kunz, “Finite difference time domain recursive convolution for second order dispersive materials,” in 1992 IEEE Conference Proceedings in Electromagnetics (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 1972–1975.

Hunsberger, F. P.

R. Lubbers, F. P. Hunsberger, K. S. Kunz, R. S. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
[CrossRef]

Joseph, R. M.

Judkins, J. B.

Kodama, Y.

A. Hasegawa and Y. Kodama, “Signal transmission by optical solitons in monomode fiber,” Proc. IEEE 69, 1145–1150 (1981).
[CrossRef]

Kunz, K.

R. Luebbers, F. Hunsberger, and K. Kunz, “Finite difference time domain recursive convolution for second order dispersive materials,” in 1992 IEEE Conference Proceedings in Electromagnetics (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 1972–1975.

Kunz, K. S.

R. Lubbers, F. P. Hunsberger, K. S. Kunz, R. S. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
[CrossRef]

Lee, Y. C.

Lubbers, R.

R. Lubbers, F. P. Hunsberger, K. S. Kunz, R. S. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
[CrossRef]

Luebbers, R.

R. Luebbers, F. Hunsberger, and K. Kunz, “Finite difference time domain recursive convolution for second order dispersive materials,” in 1992 IEEE Conference Proceedings in Electromagnetics (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 1972–1975.

Malitson, I. H.

Menyuk, C. R.

Satsuma, J.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
[CrossRef]

Schneider, M.

R. Lubbers, F. P. Hunsberger, K. S. Kunz, R. S. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
[CrossRef]

Standler, R. S.

R. Lubbers, F. P. Hunsberger, K. S. Kunz, R. S. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
[CrossRef]

Stolen, R. H.

Taflove, A.

Tomlinson, W. J.

Wai, P. K. A.

Yajima, N.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
[CrossRef]

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Prop. AP-14, 302–307 (1966).

Ziolkowski, R. W.

IEEE Trans. Antenn. Prop. (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Prop. AP-14, 302–307 (1966).

IEEE Trans. Electromagn. Compat. (1)

R. Lubbers, F. P. Hunsberger, K. S. Kunz, R. S. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

Opt. Lett. (3)

Proc. IEEE (1)

A. Hasegawa and Y. Kodama, “Signal transmission by optical solitons in monomode fiber,” Proc. IEEE 69, 1145–1150 (1981).
[CrossRef]

Prog. Theor. Phys. Suppl. (1)

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
[CrossRef]

Other (6)

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1990).

C. Hile, “Numerical studies of nonlinear optical pulse propagation,” Ph.D. dissertation (Northwestern University, Evanton, Illinois, 1993).

W. F. Ames, Numerical Methods for Partial Differential Equations (Academic, New York, 1977).

R. Luebbers, F. Hunsberger, and K. Kunz, “Finite difference time domain recursive convolution for second order dispersive materials,” in 1992 IEEE Conference Proceedings in Electromagnetics (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 1972–1975.

A. Hasegawa, Optical Solitons in Fibers (Springer-Verlag, New York, 1989).
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).

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Figures (8)

Fig. 1
Fig. 1

Field magnitude | E ˜ | of a first-order solitonlike pulse propagating for one soliton period.

Fig. 2
Fig. 2

Field magnitude | E ˜ | of a second-order solitonlike pulse propagating for one soliton period.

Fig. 3
Fig. 3

Initial and final (after one soliton period) electric field magnitudes ( | E ˜ |) of a second-order solitonlike pulse, showing the asymmetry that is due to higher-order dispersion.

Fig. 4
Fig. 4

Electric field magnitude ( | E ˜ |) of a solitonlike pulse that radiates energy.

Fig. 5
Fig. 5

Maximum electric field magnitudes ( | E ˜ |) of a solitonlike pulse that radiates energy. The dashed curve in each figure corresponds to the pulse produced by the NLS equation, and the solid curve corresponds to the pulse produced by the numerical solution of Eqs. (15). The size of the computational region for (b) is double the size of the computational region for (a).

Fig. 6
Fig. 6

(a) Electric field ( E ˜ z) solution of Eqs. (30) after propagating 15 nondimensional units in time and (b) a daughter pulse of the electric field ( E ˜ z) solution of Eqs. (30) after propagating 15 nondimensional units in time.

Fig. 7
Fig. 7

Comparison between the third-harmonic components of the linearly polarized case (solid lines) and the model third-harmonic problem (dashed lines) for (a) the region −17 < x < −11 and (b) the region 8 < x < 17.

Fig. 8
Fig. 8

(a) Left-moving and (b) right-moving daughter pulses of the electric field ( E ˜ z) solution of Eqs. (30), shown by the lighter lines, and the daughter pulse of the electric field ( E ˜ z) solution of Eqs. (30), with initial conditions (33), shown by the darker line.

Equations (77)

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P L = 0 χ ( 1 ) ( t t 1 ) E ( t 1 ) d t 1 ,
P NL = 0 χ ( 3 ) ( t t 1 , t t 2 , t t 3 ) × { E ( t 1 ) · E ( t 2 ) } E ( t 3 ) d t 1 d t 2 d t 3 ,
χ ˆ ( 1 ) ( ω ) = n 0 2 ( ω ) 1 .
n 0 2 ( ω ) = 1 + j = 1 m β j ω j 2 ω j 2 ω 2 ,
χ ˆ ( 1 ) ( ω ) = β 1 ω 1 2 ω 1 2 2 i γ ω ω 2 ,
χ ( 1 ) ( t ) = β 1 ω 1 2 ω 1 2 γ 2 exp ( γ t ) sin ( ω 1 2 γ 2 t ) H ( t ) ,
E = E 1 exp ( i ω t ) + E 3 exp ( 3 i ω t ) + c.c. ,
P NL = 0 a { 3 ( E 1 · E 1 * ) E 1 + E 1 × ( E 1 * × E 1 ) } + 0 b ( E 1 · E 1 ) E 1 + c.c.
P NL = 0 a { 3 ( E 1 · E 1 * ) E 1 + E 1 × ( E 1 * × E 1 ) } + 0 a ( E 1 · E 1 ) E 1 + c.c.
P = 0 β 1 ω 1 2 ω 1 2 γ 2 t exp [ γ ( t t 1 ) ] × sin [ ω 1 2 γ 2 ( t t 1 ) ] E ( t 1 ) d t 1 + 0 a ( E · E ) E .
Φ ( x , t ) = β 1 ω 1 2 ω 1 2 γ 2 t exp [ γ ( t t 1 ) ] × sin [ ω 1 2 γ 2 ( t t 1 ) ] E ( x , t 1 ) d t 1 .
H y t = 1 μ 0 E z x ,
D z t = H y x ,
Φ z t = U z ,
U z t = 2 γ U z ω 1 2 Φ z + β 1 ω 1 2 E z ,
D z = 0 E z + 0 Φ z + 0 a ( E y 2 + E z 2 ) E z ;
H z t = 1 μ 0 E y x ,
D y t = H z x ,
Φ y t = U y ,
U y t = 2 γ U y ω 1 2 Φ y + β 1 ω 1 2 E y ,
D z = 0 E y + 0 Φ y + 0 a ( E y 2 + E z 2 ) E y .
E y ( x , t ) = E 0 ( x , t ) cos ( kx ω t ) = Re { E 0 ( x , t ) exp [ i ( kx ω t ) ] } ,
E z ( x , t ) = E 0 ( x , t ) sin ( kx ω t ) = Re { E 0 ( x , t ) exp [ i ( kx ω t π / 2 ) ] } ,
E ( x , t ) = E y + i E z = E 0 ( x , t ) exp [ i ( kx ω t ) ] ,
H t v g H x ˆ = 1 μ 0 E x ˆ ,
D t v g D x ˆ = H x ˆ ,
Φ t v g Φ x ˆ = U ,
U t v g U x ˆ = 2 γ U ω 1 2 Φ + β 1 ω 1 2 E ,
D = 0 E + 0 Φ + 0 a | E | 2 E .
i q x 1 2 k 2 q t 2 + ω v n 2 c | q | 2 q = 0 ,
q 0 2 t 0 2 = λ k π n 2 = 2 ck ω n 2 .
q 0 2 x 0 2 = 2 c ω ω n 2 ω ,
D 0 = 0 n 2 ( ω ) E 0 ,
n 2 ( ω ) = n 0 2 ( ω ) + a | E 0 | 2 .
q 0 2 x 0 2 = 4 c n 0 ω ω a ω = 4 c 2 k ω a ω 2 ω .
[ H ( x ˆ , t ) D ( x ˆ , t ) Φ ( x ˆ , t ) U ( x ˆ , t ) E ( x ˆ , t ) ] = 1 2 π 1 ω ˜ 2 [ ( 1 / μ 0 ) k ω ˜ ( 1 / μ 0 ) k 2 c 2 k 2 ω ˜ 2 i ( ω ˜ 3 c 2 k 2 ω ˜ ) ω ˜ 2 ] × E 0 ( k ) exp [ i ( kx ω t ) ] d k ,
ω ˜ 4 + 2 i γ ω ˜ 3 [ c 2 k 2 + ( 1 + β 1 ) ω 1 2 ] ω ˜ 2 2 i γ c 2 k 2 ω ˜ + c 2 k 2 ω 1 2 ] = 0 .
H ( x , 0 ) 1 μ 0 { [ k ω ˜ ( k ) ] | k 0 f ( x ) i [ k ω ˜ ( k ) ] | k 0 f ( x ) 1 2 [ k ω ˜ ( k ) ] | k 0 f ( x ) + } exp ( i k 0 x ) .
H ˜ = μ 0 / 0 E ¯ H , D ˜ = 1 0 E ¯ D , Φ ˜ = 1 E ¯ Φ , U ˜ = t 0 E ¯ U , E ˜ = 1 E ¯ E ,
ω ˜ 1 = t 0 ω 1 , γ ˜ = t 0 γ , α = v g / c , a ˜ = a E ¯ 2 .
H ˜ t ˜ α H ˜ x ˜ = E ˜ x ˜ ,
D ˜ t ˜ α D ˜ x ˜ = H ˜ x ˜ ,
Φ ˜ t ˜ α Φ ˜ x ˜ = U ˜ ,
U ˜ t ˜ α U ˜ x ˜ = 2 γ ˜ U ˜ ω ˜ 1 2 Φ ˜ + β 1 ω 1 2 E ˜ ,
D ˜ = E ˜ + Φ + a ˜ | E ˜ | 2 E ˜ ,
D ˜ j n + 1 = D ˜ j n 1 + Δ t Δ x ( H ˜ j + 1 n H ˜ j 1 n ) + α Δ t Δ x ( D ˜ j + 1 n D ˜ j 1 n ) ,
H ˜ j n + 1 = H ˜ j n 1 + Δ t Δ x ( E ˜ j + 1 n E ˜ j 1 n ) + α Δ t Δ x ( H ˜ j + 1 n H ˜ j 1 n ) ,
Φ ˜ j n + 1 = Φ ˜ j n 1 + 2 Δ t U ˜ j n + α Δ t Δ x ( Φ ˜ j + 1 n Φ ˜ j 1 n ) ,
U ˜ j n + 1 = U ˜ j n 1 4 γ ˜ Δ t U ˜ j n 2 ω ˜ 1 2 Δ t Φ ˜ j n + 2 β 1 ω ˜ 1 2 Δ t E ˜ j n + α Δ t Δ x ( U ˜ j + 1 n U ˜ j 1 n ) .
E ˜ j n + 1 = D ˜ j n + 1 Φ ˜ j n + 1 1 + a ˜ | E ˜ j n + 1 | 2 ,
E ˜ t α E ˜ x = H ˜ x ,
H ˜ t α H ˜ x = E ˜ x .
E ( x , t ) = E z ( x , t ) k ˆ = { E 0 ( x , t ) exp [ i ( kx ω t ) ] + c.c. } k ˆ ,
D ( x , t ) = D z ( x , t ) k ˆ = { D 0 ( x , t ) exp [ i ( kx ω t ) ] + c.c. } k ˆ ,
H ( x , t ) = H y ( x , t ) j ˆ = { H 0 ( x , t ) exp [ i ( kx ω t ) ] + c.c. } j ˆ .
H y t = 1 μ 0 E z x ,
D z t = H y x ,
Φ z t = U z ,
U z t = 2 γ ˜ U z ω 1 2 Φ z + β 1 ω 1 2 E z ,
D z = 0 E z + 0 Φ z + a 0 E z 3 .
D 0 = 0 [ 1 + χ ˆ ( 1 ) ] E 0 + 3 0 a | E 0 | 2 E 0 + higher harmonics .
q 0 2 x 0 2 = 4 c n 0 ω ω a ω = 4 c 2 k ω 3 a ω 2 ω
E ˜ z ( x ˜ , t ˜ ) = E 0 ( k , ω ˜ ) exp [ i ( k x ˜ ω ˜ t ˜ ) ] + E 1 ( k , ω ˜ ) exp [ 3 i ( k x ˜ ω ˜ t ˜ ) ] ,
[ H ˜ y ( x ˜ , 0 ) D ˜ z ( x ˜ , 0 ) Φ ˜ z ( x ˜ , 0 ) U ˜ z ( x ˜ , 0 ) E ˜ ( x ˜ , 0 ) ] = linear terms + [ 0 0 3 a ˜ | f ( x ˜ ) | 2 f ( x ˜ ) 3 i a ˜ ω ˜ | f ( x ˜ ) | 2 f ( x ˜ ) 0 ] exp ( ik x ˜ ) + 1 ω ˜ 2 [ k 2 k ω ˜ k 2 ω ˜ 2 i ( ω ˜ 3 k 2 ω ˜ ) ω ˜ 2 ] g ( x ˜ ) exp ( 3 ik x ˜ ) + [ 0 0 a ˜ f 3 ( x ˜ ) 3 i a ˜ ω ˜ f 3 ( x ˜ ) 0 ] exp ( 3 ik x ˜ ) + c.c. ,
g ( x ˜ ) = a f 3 ( x ˜ ) k 2 ω ˜ 2 1 + β 1 ω ˜ 1 2 9 ω ˜ 2 + 6 i γ ˜ ω ˜ ω ˜ 1 2
ω ¯ n 4 { [ sin ( k ˜ Δ x ) Δ x ] 2 + ( 1 + β 1 ) ω ˜ 1 2 } ω ˜ n 2 + ω ˜ 1 2 [ sin ( k ˜ Δ x ) Δ x ] 2 = 0 ,
ω ¯ n = sin ( ω ˜ n Δ t ) Δ t + α sin ( k ˜ Δ x ) Δ x .
ω ¯ n = 1 2 [ { [ sin ( k ˜ Δ x ) Δ x ] 2 + ( 1 + β 1 ) ω ˜ 1 2 } + ( { [ sin ( k ˜ Δ x ) Δ x ] 2 + ( 1 + β 1 ) ω ˜ 1 2 } 2 4 ω ˜ 1 2 [ sin ( k ˜ Δ x ) Δ x ] 2 ) 1 / 2 ] 1 / 2 .
P e = | ω ˜ ω ˜ n | ω ˜ ,
( ω ˜ + α k ˜ ) 4 [ k ˜ 2 + ( 1 + β 1 ) ω ˜ 1 2 ] ( ω ˜ + α k ˜ ) 2 + ω ˜ 1 2 k ˜ 2 = 0 .
ω ¯ = ω ˜ + α k ˜ ,
ω ¯ 4 [ k ˜ 2 + ( 1 + β 1 ) ω ˜ 1 2 ] ω ¯ 2 + ω ˜ 1 2 k ˜ 2 = 0 .
ω ¯ = 1 2 ( k ˜ 2 + ( 1 + β 1 ) ω ˜ 1 2 + { [ k ˜ 2 + ( 1 + β 1 ) ω ˜ 1 2 ] 2 4 ω ˜ 1 2 k ˜ 2 } 1 / 2 ) .
P e = | ω ¯ α k ˜ 1 Δ t arcsin { [ ω ¯ n α sin ( k ˜ Δ x ) Δ x ] Δ t } | ω ¯ α k ˜ .
sin [ Δ t ( ω ¯ α k ˜ ) ] = [ ω ¯ n α sin ( k ˜ Δ x ) Δ x ] Δ t .
ω ¯ α k ˜ ω ¯ n α sin ( k ˜ Δ x ) Δ x ,
α ω ¯ ω ¯ n k ˜ sin ( k ˜ Δ x ) Δ x .

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