Abstract

We present a comprehensive numerical study of (2+1)D counter-propagating incoherent vortices in photorefractive crystals, in both space and time. We consider a local isotropic dynamical model with Kerr-type saturable nonlinearity, and identify the corresponding conserved quantities. We show, both analytically and numerically, that stable beam structures conserve angular momentum, as long as their stability is preserved. As soon as the beams loose stability, owing to radiation or non-elastic collisions, their angular momentum becomes non-conserved. We discover novel types of rotating beam structures that have no counterparts in the copropagating geometry. We consider the counterpropagation of more complex beam arrangements, such as regular arrays of vortices. We follow the transition from a few beam propagation behavior to the transverse pattern formation dynamics.

© 2005 Optical Society of America

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References

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J. Opt. B: Quantum Semiclass. Opt.

M. Beli�?, Ph. Jander, K. Motzek, A. Desyatnikov, D. Jovi�?, A. Strini�?, M. Petrovi�?, C. Denz, and F. Kaiser, �??Counterpropagating self-trapped beams in photorefractive crystals,�?? J. Opt. B: Quantum Semiclass. Opt. 6, S190�??S196 (2004).
[CrossRef]

A. S. Desyatnikov and Y. Kivshar, �??Spatial optical solitons and soliton clusters carrying an angular momentum,�?? J. Opt. B: Quantum Semiclass. Opt. 4, S58 (2002).
[CrossRef]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, �??Propagation of a mutually incoherent optical vortex pair in anisotropic nonlinear media,�?? J. Opt. B: Quantum Semiclass. Opt. 6, S318�??S322 (2004).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

M. Haelterman, A. P. Sheppard, and A. W. Snyder, �??Bimodal counterpropagating spatial solitary-waves,�?? Opt. Commun. 103, 145 (1993).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Phot. News

Special Issue on solitons, ed. M. Segev, Opt. Phot. News 13, No. 2 (2002).

Phy. Rev. E

K. Motzek, M. Beli�?, T. Richter, C. Denz, A. Desyatnikov, Ph. Jander, and F. Kaiser, �??Counterpropagating beams in biased photorefractive crystals: Anisotropic theory,�?? Phy. Rev. E 71, 016610 (2005).
[CrossRef]

Phy. Rev. Lett

Alexander V. Buryak, Yuri S. Kivshar, Ming-feng Shih, and Mordchai Segev, �??Induced Coherence and Stable Soliton Spiraling,�?? Phy. Rev. Lett 82, 81 (1999).
[CrossRef]

Phys. Lett. A

V. I. Kruglov, and R. A. Vlasov, �??Spiral self-trapping propagation of optical beams in media with cubic nonlinearity,�?? Phys. Lett. A 111, 401 (1985).
[CrossRef]

Phys. Rev. A

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, �??Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,�?? Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, �??Solitary attractors and low-order filamentation in anisotropic self-focusing media,�?? Phys. Rev. A 57, 522 (1998).
[CrossRef]

Phys. Rev. E

N. Fressengeas, J. Maufoy, and G. Kugel, �??Temporal behavior of bidimensional photorefractive bright spatial solitons,�?? Phys. Rev. E 54, 6866 (1996).
[CrossRef]

M. Beli�?, Ph. Jander, A. Strini�? A. Desyatnikov, and C. Denz, �??Self-trapped bidirectional waveguides in a saturable photorefractive medium,�?? Phys. Rev. E 68, 025601 (2003).
[CrossRef]

K. Motzek, Ph. Jander, A. Desyatnikov, M. Beli�?, C. Denz, and F. Kaiser, �??Dynamic counterpropagating vector solitons in saturable self-focusing media,�?? Phys. Rev. E 68, 066611 (2003).
[CrossRef]

D. V. Skryabin, and W. J. Firth, �??Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media,�?? Phys. Rev. E 58, 3916 (1998).
[CrossRef]

M. Beli�?, D. Vuji�?, A. Stepken, F. Kaiser, G. F. Calvo, F. Agullo-Lopez, and M. Carrascossa, �??Isotropic vs. anisotropic modeling of photorefractive solitons,�?? Phys. Rev. E 65, 066610 (2002).
[CrossRef]

Phys. Rev. Lett.

C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar, �??Partially incoherent optical vortices in self-focusing nonlinear media,�?? Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odulov, �??Collisions between optical spatial solitons propagating in opposite directions,�?? Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

M. Shih, M. Segev, and G. Salamo, �??Three-dimensional spiraling of interacting spatial solitons,�?? Phys. Rev. Lett. 78, 2551 (1997).
[CrossRef]

Phys. Today

M. Padgett, J. Courtial, and L. Allen, �??Light�??s Orbital Angular Momentum,�?? Phys. Today, May Issue, 35 (2004).

Other

S. Trillo, and W. Torruellas, eds., Spatial Solitons (Springer, New York, 2001).

Y. S. Kivshar, and G. P. Agrawal, Optical Solitons, Academic Press, London (2003).

C. Denz, M. Schwab, and C. Weilnau, Transverse pattern formation in photorefractive optics (Springer, Berlin, 2003).
[CrossRef]

Supplementary Material (38)

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Figures (11)

Fig. 1.
Fig. 1.

Typical behavior of CP vortices in the parameter plane. In the two cases shown the input vortices have the same topological charge +1, but different input intensities. Insets list the possible outcomes from vortex collisions.

Fig. 2.
Fig. 2.

Movies of the stable dipole (first column), the stable tripole (second column), a transformation of an unstable quadrupole into a stable tripole (third column), the stable quadrupole (fourth column) and a transformation of an unstable quadrupole into a stable quadrupole (fifth column). Output face of the backward beam is shown in the direct (a)–(e) (224KB, 320KB, 475KB, 491KB, 545KB), and the inverse space (f)–(j) (341KB, 288KB, 376KB, 366KB, 382KB). The lower row (k)–(o) (615KB, 432KB, 515KB, 575KB, 737KB) presents time evolution of the total angular momentum. Parameters Γ and L are given in the figures. The total input intensity of each beam in all cases is 1.

Fig. 3.
Fig. 3.

Standing waves: Isosurface plots of (a) stable dipole, (b) stable tripole and (c) stable quadrupole. The coresponding parameters are as in Fig. 2(a), (b) and (d), respectively. The isosurfaces at half-maximum intensity are plotted in the direct space, with the transverse plane being vertical and the z axis horizontal.

Fig. 4.
Fig. 4.

Movies of stable rotating structures. (a) (474KB), (d) (415KB), (g) (490KB) Rotating dipole formed by the CP vortices of the same charge, (b) (837KB), (e) (429KB), (h) (715KB) rotating quadrupole formed by the CP vortices of the opposite charge, (c) (501KB), (f) (281KB), (i) (380KB) rotating soliton formed by the CP head-on Gaussian beams. The figure setup is as in Fig. 2.

Fig. 5.
Fig. 5.

Isosurface plots of a rotating dipole from Fig. 4(a), shown at different times.

Fig. 6.
Fig. 6.

Isosurface plots of a rotating soliton from Fig. 4(c), at different times.

Fig. 7.
Fig. 7.

Rotating (4-on-4) vortices, backward field, out-of-phase: (a) Movie of intensity distribution in the real space (2,34 MB), (b) movie of intensity distribution in the inverse space (1,427 KB), (c) movie of phase distribution (1,623 MB), (d) the total angular momentum of the backward beam, (e) movie of time evolution of the angular momentum of the total field F+B (624 KB).

Fig. 8.
Fig. 8.

Fig. 8. Rotating (4-on-4) vortices, backward field, in-phase. Figure setup as in Fig. 7, (2,418 MB), (1,53 KB), (1,674 MB), (721KB).

Fig. 9.
Fig. 9.

Isosurface plots of the rotating (4-on-4) vortices, from Fig. 8.

Fig. 10.
Fig. 10.

Stable rotating 9-by-9 array, with a 50 µm distance between the incident vortices, at t=126 τ. Movies in (a) (2,395 MB) direct space (13,982 MB version), and (b) inverse space (1,446 MB) depict distributions of the backward field. Other parameters are as in Fig. 4(b).

Fig. 11.
Fig. 11.

Unstable, increasingly chaotic, rotating 9-by-9 array, with a 45 µm, distance between vortices, at t=130 τ. Movies in (a) (2,178 MB) direct space (13,851 MB version), and (b) inverse space (1,353 MB) show the distributions of the backward field. Other parameters are as in Fig. 4(b).

Equations (10)

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

i z F = Δ F + Γ E F , i z B = Δ B + Γ E B
τ t E + E = I 1 + I ,
I F z = 0 , I B z = 0 ,
I F = dxdy ( F F * ) , I B = dxdy ( B B * ) ,
z i ( F * F x B * B x ) dxdy = Γ E x ( F * F + B * B ) dxdy = 0 ,
L z tot z = L z F z + L z B z =
= z dxdy ( x ( i F * F y ) y ( i F * F x ) + x ( i B * B y ) y ( i B * B x ) ) ,
L z tot z = 0 ρ d ρ Γ 0 2 π d φ E ( F * F + B * B ) φ .
H = dxdy [ F x F * x + F y F * y + B x B * x + B y B * y ] +
dxdy [ Γ ln ( 1 + F F * + B B * ) Γ ( F F * + B B * ) ] .

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