Filamentary Tracks Formed in Transparent Optical Glass by Laser Beam Self-Focusing. II. Theoretical Analysis

Abstract

An electrostriction mechanism for laser-beam self-focusing and track formation in transparent optical glass is analyzed theoretically. Electrostrictive self-focusing occurs when a laser pulse of sufficiently high power and rapid rise time passes through a transparent medium. For a pulse duration of 50 nsec, trapping thresholds vary from 20 kW to 2 MW. During self-focusing, the beam collapses to a small radius. In solid dielectrics, self-focusing causes permanent damage in the form of isolated regions of gross fracture, termed "damage stars" and long straight tracks of very fine fractures. Typical tracks have a diameter of a few wave-lengths of light and extend up to several centimeters. Self-focusing occurs because of an interaction between light and sound. The laser beam electrostrictively excites an ultrasonic cylindrical disturbance or sound wave. The sound wave initially increases the refractive index along the beam axis. This focuses the beam into a waveguide channel called a filament. In the filament, the intensity is so high that many nonlinear mechanisms may occur, leading to damage and track formation. This paper analyzes only the self-focusing mechanism, not the various damage mechanisms. We assume the beam always remains Gaussian. The propagation of the beam is described by the quasioptics beam-tracing equation, which includes the effects of diffraction. The sound wave and beam-tracing equations are solved in several approximate models. A trapping threshold is derived for three pulse shapes, covering the steady-state, transitional, and transient regimes of pulse duration and beam size. There is a trapping-power coefficient $K$ for each material, calculable from the density, speed of sound for a compression wave or elastic moduli, and refractive index at the laser wavelength. A formula for computing the power to achieve a given constant maximum intensity $I$ as a function of beam size, and pulse duration, and trapping-power coefficient is derived. Values of this given constant intensity have been selected so the constant-intensity curve closely matches experimental track-formation thresholds for three optical glasses. The results are $K=221$ kW and $I=2.5$ ${\mathrm{G}\mathrm{W}/\mathrm{c}\mathrm{m}}^{2}$ for dense flint glass, $K=937$ kW and $I=60$ ${\mathrm{G}\mathrm{W}/\mathrm{c}\mathrm{m}}^{2}$ for borosilicate crown glass, and $K=1119$ kW and $I=180$ ${\mathrm{G}\mathrm{W}/\mathrm{c}\mathrm{m}}^{2}$ for fused silica, at a fixed pulse duration of 55 nsec and a laser wavelength of 694.3 nm. A computer movie of beam trapping shows the collapse of the beam to a relatively constant small radius, which causes track formation. It also shows the extremely rapid upstream motion of the focal points at speeds greater than 100 times the speed of sound. The long period of time they dwell at the upstream end of their motion explains the appearance of damage stars at the upstream ends of the tracks.