Abstract

The spatiotemporal dynamics of two counterpropagating beams in a photorefractive crystal with nonlocal and sluggish response is investigated in the longitudinal and one transverse dimension. A static external electric field is applied to the crystal to control the coupling strength of the two-wave mixing process. A nonautonomous linear stability analysis is performed that takes a nonconstant modulation depth into account. The onset of pattern formation for arbitrary coupling constants and pump ratios and the influence of linear absorption are discussed. Above the threshold predicted by stability analysis, running transverse waves appear in the optical near field and wandering spots appear in the corresponding far field. A nonlinear eigenmode analysis reveals the running transverse waves as secondary instabilities.

© 1998 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  26. G. K. Harkness, W. J. Firth, J. B. Geddes, J. V. Moloney, and E. M. Wright, “Boundary effects in large-aspect-ratio lasers,” Phys. Rev. A 50, 4310–4317 (1994).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

1997 (2)

1996 (3)

1995 (4)

1994 (6)

J. B. Geddes, R. A. Indik, J. V. Moloney, and W. J. Firth, “Hexagons and squares in a passive nonlinear optical system,” Phys. Rev. A 50, 3471–3485 (1994).
[Crossref] [PubMed]

M. Saffman, A. A. Zozulya, and D. Z. Anderson, “Transverse instability of energy exchanging counterpropagating two-wave mixing,” J. Opt. Soc. Am. B 11, 1409–1417 (1994).
[Crossref]

M. R. Belić, J. Leonardy, D. Timotijević, and F. Kaiser, “Transverse effects in double phase conjugation,” Opt. Commun. 111, 99–104 (1994).
[Crossref]

O. Hess and E. Schöll, “Eigenmodes of the dynamically coupled twin-stripe semiconductor lasers,” Phys. Rev. A 50, 787–792 (1994).
[Crossref] [PubMed]

G. K. Harkness, W. J. Firth, J. B. Geddes, J. V. Moloney, and E. M. Wright, “Boundary effects in large-aspect-ratio lasers,” Phys. Rev. A 50, 4310–4317 (1994).
[Crossref] [PubMed]

Ch. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E.-H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
[Crossref]

1993 (3)

G. A. Brost, “Numerical analysis of photorefractive grating formation dynamics at large modulation in BSO,” Opt. Commun. 96, 113–116 (1993).
[Crossref]

M. Saffman, D. Montgomery, A. A. Zozulya, K. Kuroda, and D. Z. Anderson, “Transverse instability of counterpropagating waves in photorefractive media,” Phys. Rev. A 48, 3209–3215 (1993).
[Crossref] [PubMed]

T. Honda, “Hexagonal pattern formation due to counterpropagation in KNbO3,” Opt. Lett. 18, 598–600 (1993).
[Crossref]

1992 (5)

G. G. Luther and C. J. McKinstrie, “Transverse modulational instability of counterpropagating light waves,” J. Opt. Soc. Am. B 9, 1047–1060 (1992).
[Crossref]

G. Grynberg, “Transverse-pattern formation for counterpropagating beams in rubidium vapor,” Europhys. Lett. 18, 689–695 (1992).
[Crossref]

J. B. Geddes, J. V. Moloney, and R. Indik, “Spontaneous transverse spatial pattern formation due to Brillouin scattering of counterpropagating light fields,” Opt. Commun. 90, 117–122 (1992).
[Crossref]

J. E. Millerd, E. M. Garmire, M. B. Klein, M. B. Klein, B. A. Wechsler, F. P. Strohkendl, and G. A. Brost, “Photorefractive response at high modulation depths in Bi12TiO20,” J. Opt. Soc. Am. B 9, 1449–1453 (1992).
[Crossref]

M. Kirby, “Minimal dynamical systems from PDEs using Sobolev eigenfunctions,” Physica D 57, 466–475 (1992).
[Crossref]

1990 (3)

1988 (1)

1984 (1)

M. R. Belić, “Comment on using the shooting method to solve boundary-value problems involving coupled-wave equations,” Opt. Quantum Electron. 16, 551–557 (1984).
[Crossref]

1983 (1)

P. Yeh, “Contra-directional two-wave mixing in photorefractive media,” Opt. Commun. 45, 323–326 (1983).
[Crossref]

1979 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[Crossref]

Anderson, D. Z.

M. Saffman, A. A. Zozulya, and D. Z. Anderson, “Transverse instability of energy exchanging counterpropagating two-wave mixing,” J. Opt. Soc. Am. B 11, 1409–1417 (1994).
[Crossref]

M. Saffman, D. Montgomery, A. A. Zozulya, K. Kuroda, and D. Z. Anderson, “Transverse instability of counterpropagating waves in photorefractive media,” Phys. Rev. A 48, 3209–3215 (1993).
[Crossref] [PubMed]

Au, L. B.

Banerjee, P. P.

Belic, M. R.

Bledowski, A.

Brost, G. A.

Caulfeld, H. J.

Chernykh, A.

Chernykh, A. I.

Cronin-Golomb, M.

Firth, W. J.

G. K. Harkness, W. J. Firth, J. B. Geddes, J. V. Moloney, and E. M. Wright, “Boundary effects in large-aspect-ratio lasers,” Phys. Rev. A 50, 4310–4317 (1994).
[Crossref] [PubMed]

J. B. Geddes, R. A. Indik, J. V. Moloney, and W. J. Firth, “Hexagons and squares in a passive nonlinear optical system,” Phys. Rev. A 50, 3471–3485 (1994).
[Crossref] [PubMed]

W. J. Firth, A. Fitzgerald, and C. Pare, “Transverse instabilities due to counterpropagation in Kerr media,” J. Opt. Soc. Am. B 7, 1087–1097 (1990).
[Crossref]

W. J. Firth and C. Pare, “Transverse modulational instabilities for counterpropagating beams in Kerr media,” Opt. Lett. 13, 1096–1098 (1988).
[Crossref] [PubMed]

Fitzgerald, A.

Garmire, E. M.

Geddes, J. B.

J. B. Geddes, R. A. Indik, J. V. Moloney, and W. J. Firth, “Hexagons and squares in a passive nonlinear optical system,” Phys. Rev. A 50, 3471–3485 (1994).
[Crossref] [PubMed]

G. K. Harkness, W. J. Firth, J. B. Geddes, J. V. Moloney, and E. M. Wright, “Boundary effects in large-aspect-ratio lasers,” Phys. Rev. A 50, 4310–4317 (1994).
[Crossref] [PubMed]

J. B. Geddes, J. V. Moloney, and R. Indik, “Spontaneous transverse spatial pattern formation due to Brillouin scattering of counterpropagating light fields,” Opt. Commun. 90, 117–122 (1992).
[Crossref]

Gregory, D. A.

Grynberg, G.

G. Grynberg, “Transverse-pattern formation for counterpropagating beams in rubidium vapor,” Europhys. Lett. 18, 689–695 (1992).
[Crossref]

Harkness, G. K.

G. K. Harkness, W. J. Firth, J. B. Geddes, J. V. Moloney, and E. M. Wright, “Boundary effects in large-aspect-ratio lasers,” Phys. Rev. A 50, 4310–4317 (1994).
[Crossref] [PubMed]

Hess, O.

J. Leonardy, F. Kaiser, M. R. Belić, and O. Hess, “Running transverse waves in optical phase conjugation,” Phys. Rev. A 53, 4519–4527 (1996).
[Crossref] [PubMed]

O. Hess and E. Schöll, “Eigenmodes of the dynamically coupled twin-stripe semiconductor lasers,” Phys. Rev. A 50, 787–792 (1994).
[Crossref] [PubMed]

Honda, T.

Indik, R.

J. B. Geddes, J. V. Moloney, and R. Indik, “Spontaneous transverse spatial pattern formation due to Brillouin scattering of counterpropagating light fields,” Opt. Commun. 90, 117–122 (1992).
[Crossref]

Indik, R. A.

J. B. Geddes, R. A. Indik, J. V. Moloney, and W. J. Firth, “Hexagons and squares in a passive nonlinear optical system,” Phys. Rev. A 50, 3471–3485 (1994).
[Crossref] [PubMed]

Jeong, J. S.

Ch. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E.-H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
[Crossref]

Kaiser, F.

O. Sandfuchs, J. Leonardy, F. Kaiser, and M. R. Belić, “Transverse instabilities in photorefractive counterpropagating two-wave mixing,” Opt. Lett. 22, 498–500 (1997).
[Crossref] [PubMed]

J. Leonardy, F. Kaiser, M. R. Belić, and O. Hess, “Running transverse waves in optical phase conjugation,” Phys. Rev. A 53, 4519–4527 (1996).
[Crossref] [PubMed]

M. R. Belić, J. Leonardy, D. Timotijević, and F. Kaiser, “Spatiotemporal effects in double phase conjugation,” J. Opt. Soc. Am. B 12, 1602–1616 (1995).
[Crossref]

M. R. Belić, J. Leonardy, D. Timotijević, and F. Kaiser, “Transverse effects in double phase conjugation,” Opt. Commun. 111, 99–104 (1994).
[Crossref]

Kirby, M.

M. Kirby, “Minimal dynamical systems from PDEs using Sobolev eigenfunctions,” Physica D 57, 466–475 (1992).
[Crossref]

Klein, M. B.

Krolikowski, W.

Kukhtarev, N.

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[Crossref]

Kuroda, K.

M. Saffman, D. Montgomery, A. A. Zozulya, K. Kuroda, and D. Z. Anderson, “Transverse instability of counterpropagating waves in photorefractive media,” Phys. Rev. A 48, 3209–3215 (1993).
[Crossref] [PubMed]

Kwak, Ch. H.

Ch. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E.-H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
[Crossref]

Lee, E.-H.

Ch. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E.-H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
[Crossref]

Leonardy, J.

O. Sandfuchs, J. Leonardy, F. Kaiser, and M. R. Belić, “Transverse instabilities in photorefractive counterpropagating two-wave mixing,” Opt. Lett. 22, 498–500 (1997).
[Crossref] [PubMed]

J. Leonardy, F. Kaiser, M. R. Belić, and O. Hess, “Running transverse waves in optical phase conjugation,” Phys. Rev. A 53, 4519–4527 (1996).
[Crossref] [PubMed]

M. R. Belić, J. Leonardy, D. Timotijević, and F. Kaiser, “Spatiotemporal effects in double phase conjugation,” J. Opt. Soc. Am. B 12, 1602–1616 (1995).
[Crossref]

M. R. Belić, J. Leonardy, D. Timotijević, and F. Kaiser, “Transverse effects in double phase conjugation,” Opt. Commun. 111, 99–104 (1994).
[Crossref]

Luther, G. G.

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[Crossref]

McKinstrie, C. J.

Millerd, J. E.

Moloney, J. V.

J. B. Geddes, R. A. Indik, J. V. Moloney, and W. J. Firth, “Hexagons and squares in a passive nonlinear optical system,” Phys. Rev. A 50, 3471–3485 (1994).
[Crossref] [PubMed]

G. K. Harkness, W. J. Firth, J. B. Geddes, J. V. Moloney, and E. M. Wright, “Boundary effects in large-aspect-ratio lasers,” Phys. Rev. A 50, 4310–4317 (1994).
[Crossref] [PubMed]

J. B. Geddes, J. V. Moloney, and R. Indik, “Spontaneous transverse spatial pattern formation due to Brillouin scattering of counterpropagating light fields,” Opt. Commun. 90, 117–122 (1992).
[Crossref]

Montgomery, D.

M. Saffman, D. Montgomery, A. A. Zozulya, K. Kuroda, and D. Z. Anderson, “Transverse instability of counterpropagating waves in photorefractive media,” Phys. Rev. A 48, 3209–3215 (1993).
[Crossref] [PubMed]

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[Crossref]

Pare, C.

Park, S. Y.

Ch. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E.-H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
[Crossref]

Saffman, M.

M. Saffman, A. A. Zozulya, and D. Z. Anderson, “Transverse instability of energy exchanging counterpropagating two-wave mixing,” J. Opt. Soc. Am. B 11, 1409–1417 (1994).
[Crossref]

M. Saffman, D. Montgomery, A. A. Zozulya, K. Kuroda, and D. Z. Anderson, “Transverse instability of counterpropagating waves in photorefractive media,” Phys. Rev. A 48, 3209–3215 (1993).
[Crossref] [PubMed]

Sandfuchs, O.

Schöll, E.

O. Hess and E. Schöll, “Eigenmodes of the dynamically coupled twin-stripe semiconductor lasers,” Phys. Rev. A 50, 787–792 (1994).
[Crossref] [PubMed]

Solymar, L.

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[Crossref]

Strohkendl, F. P.

Sturman, B.

Sturman, B. I.

Suh, H. H.

Ch. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E.-H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
[Crossref]

Timotijevic, D.

M. R. Belić, J. Leonardy, D. Timotijević, and F. Kaiser, “Spatiotemporal effects in double phase conjugation,” J. Opt. Soc. Am. B 12, 1602–1616 (1995).
[Crossref]

M. R. Belić, J. Leonardy, D. Timotijević, and F. Kaiser, “Transverse effects in double phase conjugation,” Opt. Commun. 111, 99–104 (1994).
[Crossref]

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[Crossref]

Wechsler, B. A.

Wright, E. M.

G. K. Harkness, W. J. Firth, J. B. Geddes, J. V. Moloney, and E. M. Wright, “Boundary effects in large-aspect-ratio lasers,” Phys. Rev. A 50, 4310–4317 (1994).
[Crossref] [PubMed]

Yeh, P.

P. Yeh, “Contra-directional two-wave mixing in photorefractive media,” Opt. Commun. 45, 323–326 (1983).
[Crossref]

Yu, H.-L.

Zozulya, A. A.

M. Saffman, A. A. Zozulya, and D. Z. Anderson, “Transverse instability of energy exchanging counterpropagating two-wave mixing,” J. Opt. Soc. Am. B 11, 1409–1417 (1994).
[Crossref]

M. Saffman, D. Montgomery, A. A. Zozulya, K. Kuroda, and D. Z. Anderson, “Transverse instability of counterpropagating waves in photorefractive media,” Phys. Rev. A 48, 3209–3215 (1993).
[Crossref] [PubMed]

Europhys. Lett. (1)

G. Grynberg, “Transverse-pattern formation for counterpropagating beams in rubidium vapor,” Europhys. Lett. 18, 689–695 (1992).
[Crossref]

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[Crossref]

J. Opt. Soc. Am. A (1)

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

Opt. Commun. (5)

J. B. Geddes, J. V. Moloney, and R. Indik, “Spontaneous transverse spatial pattern formation due to Brillouin scattering of counterpropagating light fields,” Opt. Commun. 90, 117–122 (1992).
[Crossref]

M. R. Belić, J. Leonardy, D. Timotijević, and F. Kaiser, “Transverse effects in double phase conjugation,” Opt. Commun. 111, 99–104 (1994).
[Crossref]

P. Yeh, “Contra-directional two-wave mixing in photorefractive media,” Opt. Commun. 45, 323–326 (1983).
[Crossref]

G. A. Brost, “Numerical analysis of photorefractive grating formation dynamics at large modulation in BSO,” Opt. Commun. 96, 113–116 (1993).
[Crossref]

Ch. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E.-H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. 105, 353–358 (1994).
[Crossref]

Opt. Lett. (7)

Opt. Quantum Electron. (1)

M. R. Belić, “Comment on using the shooting method to solve boundary-value problems involving coupled-wave equations,” Opt. Quantum Electron. 16, 551–557 (1984).
[Crossref]

Phys. Rev. A (5)

M. Saffman, D. Montgomery, A. A. Zozulya, K. Kuroda, and D. Z. Anderson, “Transverse instability of counterpropagating waves in photorefractive media,” Phys. Rev. A 48, 3209–3215 (1993).
[Crossref] [PubMed]

J. B. Geddes, R. A. Indik, J. V. Moloney, and W. J. Firth, “Hexagons and squares in a passive nonlinear optical system,” Phys. Rev. A 50, 3471–3485 (1994).
[Crossref] [PubMed]

J. Leonardy, F. Kaiser, M. R. Belić, and O. Hess, “Running transverse waves in optical phase conjugation,” Phys. Rev. A 53, 4519–4527 (1996).
[Crossref] [PubMed]

O. Hess and E. Schöll, “Eigenmodes of the dynamically coupled twin-stripe semiconductor lasers,” Phys. Rev. A 50, 787–792 (1994).
[Crossref] [PubMed]

G. K. Harkness, W. J. Firth, J. B. Geddes, J. V. Moloney, and E. M. Wright, “Boundary effects in large-aspect-ratio lasers,” Phys. Rev. A 50, 4310–4317 (1994).
[Crossref] [PubMed]

Physica D (1)

M. Kirby, “Minimal dynamical systems from PDEs using Sobolev eigenfunctions,” Physica D 57, 466–475 (1992).
[Crossref]

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Figures (14)

Fig. 1
Fig. 1

Two-wave mixing configuration in reflection geometry with an externally applied voltage V: A1, A2, pump beams; Q, grating amplitude. z indicates the direction of propagation, and x is the transverse dimension.

Fig. 2
Fig. 2

(a) Threshold curves of the applied electric field and (b) threshold frequency of the grating amplitude, as functions of the transverse wave vector K. The coupling constant is Γ0L =2.0, and the pump ratio r0=20.09. sFP, uFP, regions of stable and unstable fixed points, respectively; Hopf, the type of bifurcation as the threshold curve is crossed in the direction of the arrow.

Fig. 3
Fig. 3

(a) Bifurcation diagram of the primary instability threshold (solid curve), displaying the critical values of E0<0 as a function of Γ0L for fixed r0=20.09. Dashed curve, constant values of the intensity coupling strength γL=1.0, 2.0, 3.0, 4.0, 5.0 (left to right); sFP, and uFP+sLC, regions of stable and unstable fixed points and stable limit cycle, respectively. (b) Oscillation frequency, (c) spatial frequency. Diamonds, points where the threshold values can be obtained analytically.

Fig. 4
Fig. 4

(a) Bifurcation diagram of the primary instability threshold, displaying the critical values of E0<0 as a function of r0 for fixed Γ0L=2.0. (b) Oscillation frequency, (c) spatial frequency. Diamonds, points where the threshold values can be obtained analytically.

Fig. 5
Fig. 5

Transverse intensity and phase profiles of beam A2 when it leaves the crystal at z=0: (a), (c) below threshold for E0=-1.8Ed; (b), (d) above threshold for E0=-2.0Ed at times t and t+T/2.

Fig. 6
Fig. 6

Spatiotemporal dynamics above the instability threshold at E0=-2.0Ed. Running transverse waves in the near field for (a) I1 and (b) I2, and (c), (d) wandering spots in the far field; the pump beams have been subtracted. Γ0L=2.0, f=0.025, r0=20.09, αL=0.

Fig. 7
Fig. 7

(a) Bifurcation diagram of the primary instability threshold (solid curve) displaying the critical values of E0<0 as a function of αL for fixed Γ0L=2.0 and r0=20.09. (b) Oscillation frequency, (c) spatial frequency. Asterisks, results obtained from numerical simulations. For the spatial frequency one obtains different values for I1 (⋄) and I2 (□). The dashed curve in (b) indicates the correction for the frequency given by stability analysis.

Fig. 8
Fig. 8

Frequency behavior above threshold for different values of αL: (*) 0.0, (△) 0.3, (□) 0.5, obtained from numerical simulations. Solid curves, cubic splines through the data points; dashed curve, onset of pattern formation given by stability analysis.

Fig. 9
Fig. 9

Transverse intensity profiles for E0/Ed= (a) -2.2, (b) -2.3, (c) -2.6 at times t and t+T/2.

Fig. 10
Fig. 10

Spatial distribution of the slowly varying envelope of |Q| within the crystal at one instant of time for E0/Ed= (a) -2.0, (b) -2.6.

Fig. 11
Fig. 11

Spectra of normalized eigenvalues λ(i) (in percent) for the intensity patterns of (a) I1 and (b) I2 with (●) E0/Ed=-2.0 and (□) E0/Ed=-2.6.

Fig. 12
Fig. 12

(a), (d) Spatiotemporal substructures: the two dominating eigenmodes (b) with λ(1)80.22% and (e) with λ(2)19.39% and (c), (f) their time-dependent expansion coefficients of the running transverse wave of Fig. 6(b).

Fig. 13
Fig. 13

Spatial Fourier spectra of the two largest eigenmodes from Fig. 12(b) (solid curve) and from Fig. 12(e) (dashed curve).

Fig. 14
Fig. 14

Behavior of (a) the basic spatial frequency K0 and (b) the frequency gap ΔK as functions of the applied field strength for (⋄) beam I1 and (□) beam I2. Solid curves, cubic splines through the data points.

Equations (20)

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zA1+ifx2A1+αA1=-QA2,
-zA2+ifx2A2+αA2=Q*A1.
τtQ+ηQ=Γ A1A2*|A1|2+|A2|2,
η=Ed+Eq+iE0EM+Ed+iE0,
Γ=Γ01+EqEd Ed+iE0EM+Ed+iE0,
A1(x, z=0)=C1 exp(-x2/w02),
A2(x, z=L)=C2 exp(-x2/w02).
A1(z, x, t)=A10(z)[1+a(z, x, t)],
A2(z, x, t)=A20(z)[1+b(z, x, t)],
Q(z, x, t)=Q0(z)[1+q(z, x, t)].
za=A(z, K, λ)a(z, K, λ).
A=U-1m02γ+(1-m02)g(λ)-fK20-s1-m02h(λ)m02β+(1-m02)h(λ)+fK200s1-m02g(λ)s1-m02g(λ)00-h(λ)-fK2s1-m02h(λ)0fK2g(λ)U.
g(λ)=λ2 Γeτeλτe+1+Γe*τe*λτe*+1,
h(λ)=λ2i Γeτeλτe+1-Γe*τe*λτe*+1.
m0(z)=2[I10(z)I20(z)]1/2I10(z)+I20(z)
F(L)=exp0L Tr A(s)ds×z=0L D(z).
a(zout)=S(K, λ)a(zin),
coshγ-g2+cos(χ1)cos(χ2)+p sinc(χ1)sinc(χ2)=0,
Γ0L=2.0,r0=20.09,f=0.025.
δI(x, t)=ia(i)(t)p(i)(x).

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