Collision-induced shape transformations of partially coherent solitons Wieslaw Kro ´ likowski Australian Photonics CRC, Laser Physics Centre, Research School of Physical Sciences and Engineering, The Australian National University, Canberra 0200, Australian Capital Territory, Australia Nail Akhmediev Australian Photonics CRC, Optical Sciences Centre, Research School of Physical Sciences and Engineering, Australian National University, Canberra 0200, Australian Capital Territory, Australia Barry Luther-Davies Australian Photonics CRC, Laser Physics Centre, Research School of Physical Sciences and Engineering, The Australian National University, Canberra 0200, Australian Capital Territory, Australia ~Received 22 October 1998! We present experimental results related to collisional properties of partially coherent solitons ~PCSs!. In experiments with a photorefractive crystal we show that collisions in nonlinear media cause the PCS comprised of few mutually incoherent soliton components to change its shape. @S1063-651X~99!08804-2# PACS number~s!: 42.65.Tg, 42.65.Jx, 42.65.Hw Partially coherent solitons are objects which have recently attracted a great deal of attention @1–10#. Focusing of the spatially partially coherent beams has been suggested as early as in 1967 @11# and studied subsequently in @12,13#. On the other hand, temporally incoherent solitons have been considered in the original works of Hasegawa @14–16#, both for plasma waves and for nonlinear pulses in multimode fi- bers. However, the creation of incoherent solitons in optical fibers requires unrealistically high pulse energies. In contrast, photorefractive materials allow experimental studies at very low optical powers as they generally exhibit strong nonlinear self-action effects @17–21#. In fact, the first experiments with the partially coherent soliton have been conducted with pho- torefractive nonlinearity @1,2,6#. The interaction of partially coherent solitons ~PCS! is an interesting area of research and it has only been addressed in the recent works @8,22#. In the present paper, we experimen- tally demonstrate new collisional properties of PCS. Namely, we have discovered that the PCS may have variable shape @23# and that the collision of the PCS leads to a transforma- tion of the soliton profile. In order to explain these phenom- ena we will first briefly discuss the main features of partially coherent solitons considering the solvable model of a Kerr- like medium. However, the nonlinear susceptibility of a gen- eral nonlinear medium, such as a photorefractive crystal, is usually well approximated by a saturable nonlinearity. Therefore, we will comment on PCS in a saturable nonlin- earity discussing experimental results. It has been shown that incoherent solitons can be repre- sented as N self-trapped mutually incoherent wave packets which represent modes of the self-induced waveguide @5,6#. From this point of view, incoherent solitons are analogous to higher-order vector solitons which can also be considered as multimoded self-induced waveguides @24#. One of the physi- cal reasons which makes various modes phase independent ~incoherent! is a slow response of the nonlinear medium. Usually in photorefractive media the response time is much slower than the random fluctuation of the phases of the par- tially coherent beam so its refractive index change is a func- tion of the time-averaged light intensity. On the other hand, the light beam components can be initially incoherent in time leading to qualitatively the same phenomena. In either case, the evolution of the N self-trapped mutually incoherent wave packets in a medium with Kerr-like nonlinearity can be rep- resented by a set of N coupled dimensionless equations i ] c i ] z 1 1 2 ] 2 c i ] x 2 1d n ~ c i ! c i 50, ~1! c i denotes the i th component of the beam, x is the transverse coordinate, z is the co-ordinate along the direction of propa- gation, and d n ~ c i ! 5 f S ( i 51 N u c i u 2 D 5 f ~ I ! ~2! is the change in refractive index profile created by the par- tially coherent beam, which, because of the lack of temporal correlation between the modes, is a nonlinear function of just a sum of modal intensities. For the sake of simplicity, we take here the function f ( I ) 5I . As a result, the set of equations ~1! is an integrable set of coupled nonlinear Schro ¨ dinger equations ~NLSE!. This simplifies the analysis in the sense that all solutions can be written in analytical form. Examples of the lowest-order symmetric solutions and their interactions have been pre- sented in recent works @7,8#. However, those solutions con- tain only one free parameter. They are symmetric solutions of a given ~sech! profile and have variable amplitude ~and width! for any particular N. More general analysis has been done in @22#. In particular, it has been shown that the actual PCS profiles are multipa- rameter families of asymmetric solutions and that their shape and amplitude may vary. Moreover, it has been shown @22# that PCS can be best understood using the principle of complementarity. Namely, we can think of PCS both as a PHYSICAL REVIEW E APRIL 1999 VOLUME 59, NUMBER 4 PRE 59 1063-651X/99/59~4!/4654~5!/$15.00 4654 ©1999 The American Physical Society