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Numerical simulations show that, because of the spatiotemporal coupling implied by the multidimensional nonlinear Schrödinger equation, self-focusing of ultrashort optical pulses can lead to pulse compression even in the normal-dispersion regime of a nonlinear Kerr medium. We show how this coupling can be further exploited to control the compression by use of spatial phase modulation. Both the compression factor and the position at which the minimum pulse width is realized change with the amplitude of the phase modulation.

A simple scheme to steer optical beams is proposed. The basic idea is to impose a sinusoidal phase modulation on the optical beam and then propagate it in a nonlinear Kerr medium. Spatial phase modulation splits the input beam into multiple subbeams, while the nonlinear medium shapes a particular subbeam into a spatial soliton in such a way that most of the beam power appears in a narrow beam whose direction can be controlled by changing the modulation parameters. We present numerical results showing how spatial phase modulation can be used to alter the path of an optical beam propagating in a nonlinear Kerr medium.In both the temporal and spatial domains the multidimensional nonlinear Schr6dinger equation (NLSE) has long been a useful tool for describing the behavior of optical fields in nonlinear dispersive media. 1' 2 It has proved valuable in the description of such diverse phenomena as pulse compression, dark soliton formation, and self-focusing of ultrashort pulses. In recent years it has been useful in describing some of the new innovations in beam steering. 3 -6 One technique uses area modulation of a second beam to induce a temporal prism in the nonlinear medium, which then deflects the beam. 3 Another two-beam technique uses cross-phase modulation from a pump beam to alter the phase profile of a probe beam and so induce a deflection. 4 Others have employed single beams with asymmetric power profiles, which resulted in self-bending on propagations Another technique uses the properties of dark solitons for beam steering. 6 There are also a variety of techniques for steering beams in linear media. 7 ' 1 The technique that we propose here employs spatial phase modulation of a beam entering a nonlinear medium and shows that high-efficiency beam steering is possible. Spatial phase modulation splits the input beam into many subbeams, while the nonlinear medium shapes a particular subbeam into a spatial soliton in such a way that most of the beam power appears in a narrow beam whose direction can be controlled by changes in the modulation parameters.We model beam propagation with the NLSE in the dispersionless (cw or quasi-cw) approximation by using the well-known split-step Fourier method.' Spatial transverse coordinates t and 7 are normalized to the input beam width o-, and the propagation distance ; is measured in units of the diffraction length, Ld = (27/A)u 2 , where A is the optical wavelength. The normalized NLSE then takes the form where the parameter N = (2iro-/A)\ non 2 Io represents the strength of Kerr nonlinearity. The quantity n 2 1 0 represents the maximum nonlinear index change for an input beam of peak intensity Io in a medium of linear refractive index no and nonlinear index parameter n 2 . We have performed simulations for both one and two transverse dimensions.We focus first on the case of one transverse dimension so that the results are applicable mainly to planar waveguides. The NLSE is solved for a phasemodulated input beam having a spatial profile(2)For the case of sinusoidal phase modu...