Diffractive nonlinear geometric optics for short pulses.

Abstract

The propagation of laser light has been studied through shortwave asymptotics. For lasers which produce pulses as short as several picoseconds, the light is assumed to have the form of a modulated wavetrain with slowly varying envelope and rapidly oscillating phase. This slowly varying envelope approximation leads to the nonlinear Schrodinger equation when applied to nonlinear Maxwell's equations. The slowly varying envelope assumption does not hold for ultrafast lasers. In this thesis, an alternative description, valid for ultrafast laser pulses, is derived and analyzed. For medium time scales, in the regime of geometric optics, the approximate short pulse solutions and wavetrain solutions satisfy the same nonlinear profile equation. On longer, or diffractive time scales, the short pulse approximation satisfies a different profile equation than the wavetrain solution. The short pulse approximation is proved valid for each time scale, with the relative error between exact and approximate solution going to zero in both <italic>L</italic><super>2 </super> and <italic>L</italic><super>infinity</super> in the short wavelength limit.