Abstract

Two optical beams, copropagating in a Kerr medium, interact with each other through cross-phase modulation. Such nonlinear beam coupling leads to a transverse modulation instability that is evident as spatial modulation of the beam profiles. A linear-stability analysis in the plane-wave approximation predicts the range of spatial frequencies over which modulation can occur. The case of self-defocusing media is particularly interesting, since modulation instability occurs only when both beams are present simultaneously. Numerical simulations are used to study how modulation instability can occur for finite-size beams. In particular, the mutual coupling of two copropagating Gaussian beams is studied in detail.

© 1990 Optical Society of America

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References

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  1. For a recent review, see G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989), Sec. 5.1.
  2. K. Tai, A. Hasegawa, A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
    [CrossRef] [PubMed]
  3. S. A. Akhmanov, V. A. Vysloukh, A. S. Chirkin, Sov. Phys. Usp. 29, 642 (1986).
    [CrossRef]
  4. V. I. Bespalov, V. I. Talanov, JETP Lett. 3, 307 (1966).
  5. R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
    [CrossRef]
  6. E. Garmire, R. Y. Chiao, C. H. Townes, Phys. Rev. Lett. 16, 347 (1966).
    [CrossRef]
  7. G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989), Chap. 7.
  8. P. L. Baldeck, P. P. Ho, R. R. Alfano, in The Supercontinuum Laser Source, R. R. Alfano, ed. (Springer-Verlag, New York, 1989), Chap. 4.
  9. W. J. Firth, C. Paré, Opt. Lett. 13, 1096 (1988).
    [CrossRef] [PubMed]
  10. G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
    [CrossRef] [PubMed]
  11. G. P. Agrawal, P. L. Baldeck, R. R. Alfano, Phys. Rev. A 39, 3406 (1989).
    [CrossRef] [PubMed]
  12. C. J. McKinstrie, R. Bingham, Phys. Fluids B 1, 230 (1989).
    [CrossRef]
  13. See, for example, Sec. 2.4 of Ref. 7.
  14. C. J. McKinstrie, D. A. Russell, Phys. Rev. Lett. 61, 2929 (1988).
    [CrossRef] [PubMed]
  15. P. L. Baldeck, F. Raccah, R. R. Alfano, Opt. Lett. 12, 588 (1987).
    [CrossRef] [PubMed]

1989

G. P. Agrawal, P. L. Baldeck, R. R. Alfano, Phys. Rev. A 39, 3406 (1989).
[CrossRef] [PubMed]

C. J. McKinstrie, R. Bingham, Phys. Fluids B 1, 230 (1989).
[CrossRef]

1988

C. J. McKinstrie, D. A. Russell, Phys. Rev. Lett. 61, 2929 (1988).
[CrossRef] [PubMed]

W. J. Firth, C. Paré, Opt. Lett. 13, 1096 (1988).
[CrossRef] [PubMed]

1987

1986

K. Tai, A. Hasegawa, A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

S. A. Akhmanov, V. A. Vysloukh, A. S. Chirkin, Sov. Phys. Usp. 29, 642 (1986).
[CrossRef]

1966

V. I. Bespalov, V. I. Talanov, JETP Lett. 3, 307 (1966).

E. Garmire, R. Y. Chiao, C. H. Townes, Phys. Rev. Lett. 16, 347 (1966).
[CrossRef]

1964

R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, P. L. Baldeck, R. R. Alfano, Phys. Rev. A 39, 3406 (1989).
[CrossRef] [PubMed]

G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[CrossRef] [PubMed]

For a recent review, see G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989), Sec. 5.1.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989), Chap. 7.

Akhmanov, S. A.

S. A. Akhmanov, V. A. Vysloukh, A. S. Chirkin, Sov. Phys. Usp. 29, 642 (1986).
[CrossRef]

Alfano, R. R.

G. P. Agrawal, P. L. Baldeck, R. R. Alfano, Phys. Rev. A 39, 3406 (1989).
[CrossRef] [PubMed]

P. L. Baldeck, F. Raccah, R. R. Alfano, Opt. Lett. 12, 588 (1987).
[CrossRef] [PubMed]

P. L. Baldeck, P. P. Ho, R. R. Alfano, in The Supercontinuum Laser Source, R. R. Alfano, ed. (Springer-Verlag, New York, 1989), Chap. 4.

Baldeck, P. L.

G. P. Agrawal, P. L. Baldeck, R. R. Alfano, Phys. Rev. A 39, 3406 (1989).
[CrossRef] [PubMed]

P. L. Baldeck, F. Raccah, R. R. Alfano, Opt. Lett. 12, 588 (1987).
[CrossRef] [PubMed]

P. L. Baldeck, P. P. Ho, R. R. Alfano, in The Supercontinuum Laser Source, R. R. Alfano, ed. (Springer-Verlag, New York, 1989), Chap. 4.

Bespalov, V. I.

V. I. Bespalov, V. I. Talanov, JETP Lett. 3, 307 (1966).

Bingham, R.

C. J. McKinstrie, R. Bingham, Phys. Fluids B 1, 230 (1989).
[CrossRef]

Chiao, R. Y.

E. Garmire, R. Y. Chiao, C. H. Townes, Phys. Rev. Lett. 16, 347 (1966).
[CrossRef]

R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Chirkin, A. S.

S. A. Akhmanov, V. A. Vysloukh, A. S. Chirkin, Sov. Phys. Usp. 29, 642 (1986).
[CrossRef]

Firth, W. J.

Garmire, E.

E. Garmire, R. Y. Chiao, C. H. Townes, Phys. Rev. Lett. 16, 347 (1966).
[CrossRef]

R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Hasegawa, A.

K. Tai, A. Hasegawa, A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

Ho, P. P.

P. L. Baldeck, P. P. Ho, R. R. Alfano, in The Supercontinuum Laser Source, R. R. Alfano, ed. (Springer-Verlag, New York, 1989), Chap. 4.

McKinstrie, C. J.

C. J. McKinstrie, R. Bingham, Phys. Fluids B 1, 230 (1989).
[CrossRef]

C. J. McKinstrie, D. A. Russell, Phys. Rev. Lett. 61, 2929 (1988).
[CrossRef] [PubMed]

Paré, C.

Raccah, F.

Russell, D. A.

C. J. McKinstrie, D. A. Russell, Phys. Rev. Lett. 61, 2929 (1988).
[CrossRef] [PubMed]

Tai, K.

K. Tai, A. Hasegawa, A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

Talanov, V. I.

V. I. Bespalov, V. I. Talanov, JETP Lett. 3, 307 (1966).

Tomita, A.

K. Tai, A. Hasegawa, A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

Townes, C. H.

E. Garmire, R. Y. Chiao, C. H. Townes, Phys. Rev. Lett. 16, 347 (1966).
[CrossRef]

R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Vysloukh, V. A.

S. A. Akhmanov, V. A. Vysloukh, A. S. Chirkin, Sov. Phys. Usp. 29, 642 (1986).
[CrossRef]

JETP Lett.

V. I. Bespalov, V. I. Talanov, JETP Lett. 3, 307 (1966).

Opt. Lett.

Phys. Fluids B

C. J. McKinstrie, R. Bingham, Phys. Fluids B 1, 230 (1989).
[CrossRef]

Phys. Rev. A

G. P. Agrawal, P. L. Baldeck, R. R. Alfano, Phys. Rev. A 39, 3406 (1989).
[CrossRef] [PubMed]

Phys. Rev. Lett.

C. J. McKinstrie, D. A. Russell, Phys. Rev. Lett. 61, 2929 (1988).
[CrossRef] [PubMed]

G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[CrossRef] [PubMed]

K. Tai, A. Hasegawa, A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

E. Garmire, R. Y. Chiao, C. H. Townes, Phys. Rev. Lett. 16, 347 (1966).
[CrossRef]

Sov. Phys. Usp.

S. A. Akhmanov, V. A. Vysloukh, A. S. Chirkin, Sov. Phys. Usp. 29, 642 (1986).
[CrossRef]

Other

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989), Chap. 7.

P. L. Baldeck, P. P. Ho, R. R. Alfano, in The Supercontinuum Laser Source, R. R. Alfano, ed. (Springer-Verlag, New York, 1989), Chap. 4.

For a recent review, see G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989), Sec. 5.1.

See, for example, Sec. 2.4 of Ref. 7.

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Figures (9)

Fig. 1
Fig. 1

Modulation instability gain versus spatial frequency S for the self-focusing (n2 > 0) and self-defocusing (n2 < 0) cases when two plane waves of equal intensities such that Δn = n2I = 10−6 copropagate inside the nonlinear medium.

Fig. 2
Fig. 2

Evolution of the pump-beam profile in a self-defocusing medium (n2 < 0) over a propagation distance LD/2 (ξ = z/LD). The input beam is Gaussian with a peak intensity such that N = 10.

Fig. 3
Fig. 3

Probe-beam profiles at ξ = 0.1 and ξ = 0.2 in a self-focusing medium (n2 < 0). The input Gaussian profile at ξ = 0 is also shown for comparison. The probe beam is launched together with a Gaussian pump beam whose peak intensity corresponds to N = 10.

Fig. 4
Fig. 4

Same as in Fig. 3 expect that the probe-beam profile at ξ = 0.5 is compared with the input Gaussian profile.

Fig. 5
Fig. 5

Far-field intensity versus diffraction angle θ. The corresponding near field is shown in Fig. 4 at ξ = 0.5.

Fig. 6
Fig. 6

Induced focusing of a probe beam copropagated with an intense pump beam (N = 10) in a self-focusing medium (n2 > 0). The probe-beam profile at ξ = 0.05 is compared with the input Gaussian profile. The probe peak intensity is much below the self-focusing threshold.

Fig. 7
Fig. 7

Probe beams profiles showing pump-induced focusing in a self-defocusing medium (n2 < 0). The input profile at ξ = 0 peaks at x = w0, whereas the pump-beam profile peaks at x = 0. Other parameters are identical to those used for Fig. 6.

Fig. 8
Fig. 8

Near- and far-field profiles of an optical beam copropagating with another equally intense beam in a self-defocusing medium. The top and bottom rows correspond to ξ = 0.1 and ξ = 0.2, respectively. High-frequency modulation of the near-field profile at ξ = 0.2 is due to the onset of the transverse modulation instability.

Fig. 9
Fig. 9

Same as in Fig. 8 except that the near- and far-field profiles of the second beam are shown. The differences in Figs. 8 and 9 are entirely due to the lower wavelength (λ21 = 0.9) of the second beam. The side peaks in the far-field profile at ξ = 0.2 are due to the onset of the transverse modulation instability.

Equations (20)

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n = n 0 + 2 n 2 | E | 2 ,
E ( r , t ) = 1 2 x ˆ j = 1 2 A j ( r , t ) exp [ i ( k j z ω j t ) ] + c . c . ,
k j = n 0 j ω j / c = 2 π n 0 j / λ j .
A 1 z i 2 k 1 ( 2 A 1 x 2 + 2 A 1 y 2 ) = i k 1 n 2 n 01 ( | A 1 | 2 + 2 | A 2 | 2 ) A 1 ,
A 2 z i 2 k 2 ( 2 A 2 x 2 + 2 A 2 y 2 ) = i k 2 n 2 n 02 ( | A 2 | 2 + 2 | A 1 | 2 ) A 2 .
A j ( x , y , 0 ) = I j , j = 1 , 2.
A ¯ j ( x , y , z ) = I j exp [ i k j z n 2 n 0 j ( I j + 2 I 3 j ) ] ,
A j = A ¯ j × { 1 + u j ( z ) exp [ i ( p x + q y ) ] + υ j * ( z ) exp [ i ( p x + q y ) ] } ,
( K 2 h 1 2 ) ( K 2 h 2 2 ) = C 2 ,
h j = ( S / 2 k j ) [ S 2 sgn ( n 2 ) S c j 2 ] 1 / 2 ,
C = S 2 ( S c 1 S c 2 k 1 k 2 ) ,
S c j = 2 k j ( | n 2 | I j / n 0 j ) 1 / 2 , j = 1 , 2
[ S 2 sgn ( n 2 ) S c 1 2 ] [ S 2 sgn ( n 2 ) S c 2 2 ] < 4 S c 1 2 S c 2 2 .
X = x ω 0 , ξ = z L D , U j = A j ( I 1 ) 1 / 2
U 1 ξ i 2 2 U 1 X 2 = sgn ( n 2 ) i N 2 ( | U 1 | 2 + 2 | U 2 | 2 ) U 1 ,
U 2 ξ i 2 2 U 2 X 2 = sgn ( n 2 ) λ 1 λ 2 i N 2 ( | U 2 | 2 + 2 | U 1 | 2 ) U 2 ,
N 2 = L D ( k 1 / n 01 ) n 2 I 1 = k 1 2 w 0 2 ( | n 2 | I 1 / n 01 ) ,
U 1 ( 0 , X ) = exp ( X 2 / 2 ) ,
U 2 ( 0 , X ) = ( I 2 / I 1 ) 1 / 2 exp [ ( w 0 w 0 ) 2 X 2 2 ] .
I j FF ( p ) = | U j ( L , X ) exp ( 2 π i p X ) d X | 2 ,

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