Cory Ahrens Abstract

PhD, May 2006

The Asymptotic Analysis of Communications

and Wave Collapse Problems in Nonlinear Optics

Advisor: Mark J. Ablowitz

This thesis investigates two problems in nonlinear optics. The first is the calculation of collision induced timing shifts in nonlinear fiber optic communication systems, while the second is the use of dispersion management in preventing wave collapse in (2+1) dimensions.

Optical fiber communication systems are a key technology in the long distance transmission of information. Long distance propagation implies that nonlinear effects are important. The nonlinear effects of interest here are cross-phase modulation (XPM) and four-wave mixing (FWM). In the first part of this thesis, an asymptotic theory to calculate frequency and timing shifts due to XPM and FWM is developed. The theory is based on a perturbed Nonlinear Schrödinger (NLS) equation. From the NLS equation, ordinary differential equations describing pulse temporal position and frequency are derived. Effects of FWM on temporal position and frequency are then shown to be negligible compared to effects from XPM. By neglecting FWM, computation of the timing and frequency shift is greatly simplified. Using asymptotic methods, formulas for timing and frequency shift due to XPM are derived, giving an accurate, computationally efficient method to estimate frequency and timing shifts. The utility of the theory is demonstrated for several realistic systems.

The search for light bullets is an outstanding problem in nonlinear optics. In two or more spatial dimensions, pulses governed by the cubic NLS equation can undergo collapse. Recently, researchers have proposed using dispersion management to prevent pulse collapse. The second part of this thesis investigates the effects of dispersion management in (2+1) dimensions on pulse evolution and development of pulse collapse. A multiple scale analysis is used to derive the (2+1) dimensional dispersion-managed NLS (DMNLS) equation, which describes average pulse dynamics. Local existence of solutions to the DMNLS equation in the Sobolev space H^s(R^2), s>1, is established. With appropriate a priori estimates, global existence is then proved. The asymptotic validity of the (2+1) dimensional DMNLS equation is shown to hold for finite, but long distances. Therefore it is proved that for long distances, pulses evolving under the (2+1) dimensional NLS equation with dispersion management do not collapse.