Physica D 184 (2003) 352–375 Numerical integration of Maxwell’s full-vector equations in nonlinear focusing media Paul M. Bennett b , Alejandro Aceves a,∗ a Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA b Computing Sciences Corporation, 2524 South Frontage Road, Suite A Vicksburg, MS 39180, USA Communicated by C.K.R.T. Jones Dedicated to Alan Newell on the occasion of his 60th birthday Abstract In this paper, we present results of parallel numerical simulations on Maxwell’s equations. The parallel code is used to study the effect of the instantaneous focusing nonlinearity upon dispersionless pulse propagation in bulk dielectric. Indications are given of the development of shocks on the optical carrier wave and upon the pulse envelope. We then use the code to study focusing and collapse of optical pulses at anomalously dispersive frequencies. We examine the effect of varying the focusing of the light by varying the intensity as a way to compensate linear dispersion. We demonstrate blow up of sufficiently intense short pulses at finite propagation distances, and we show numerically that the location of blow up depends nontrivially upon the intensity of the light. © 2003 Elsevier B.V. All rights reserved. Keywords: Nonlinear optics; Parallel numerical algorithms; Ultrashort intense pulses; Maxwell’s equations; Anomalous dispersion; Instantaneous nonlinearity 1. Introduction Historically, optical pulse propagation in nonlinear focusing dielectrics has been studied by analyzing the nonlinear Schrödinger equation, or NLSE. It is well known and thoroughly documented that in optical fibers, restriction of the wave-guide profiles leads to stable pulse propagation. This has made it possible to extend the research of optical pulses in fibers into different regimes by application of higher order perturbations. Much less is known about optical pulses propagating in slab wave-guides or bulk dielectric. It is known [1] that the NLSE in two and three space dimensions does not have stable pulse solutions [2–5]. In particular, we know that pulse solutions to the NLSE in critical dimension 2 that exceed a certain power threshold will undergo collapse and blow up at a point [6,7]. However, the analysis is difficult and time consuming. Given the difficulties and the breakup of the validity of the envelope model, it is natural to turn to the underlying Maxwell’s equations governing the propagation of light in nonlinear focusing media. ∗ Corresponding author. E-mail address: aceves@math.unm.edu (A. Aceves). 0167-2789/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0167-2789(03)00240-9