ABSTRACT
G. Fibich, Department of Applied Mathematics, School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel, fibich@math.tau.ac.il
S. Tsynkov, Department of Mathematics and Center for Research in Scientific Computation, North Carolina State University, Box 8205, Raleigh, NC 27695, USA, tsynkov@math.ncsu.edu
The objective of this chapter is to describe a new numerical algorithm for studying nonlinear self-focusing of time-harmonic electromagnetic waves. The physical mechanism that leads to self-focusing is known as the Kerr effect. At the microscopic level, the Kerr effect may originate from electrostriction, nonresonant electrons, or from molecular orientation. At the macroscopic level the Kerr effect is manifested through an increase in the index of refraction, which is proportional to the intensity of the electric field |E|2. Since light rays bend toward regions with higher index of refraction, an impinging laser beam would become narrower as it propagates, a phenomenon known as selffocusing. For more information on self-focusing, see, e.g., [3, 8, 11].
