Characterizing frequency chirps and phase fluctuations during noisy laser transients

Characterizing frequency chirps and phase fluctuations during noisy laser transientsNeal B. AbrahamDepartment of Physics, Bryn Mawr College, Bryn Mawr, PA 19010-2899 USAABSTRACTMethods of determining the frequency of a laser during its transient evolution, such as during theswitch-on of a laser, are considered. This study is motivated by several illustrations of steady state lasersystems which change their operating frequencies as their parameters are varied adiabatically. Forevolving systems one would like an instantaneous measure of the frequency, but for stochastic processesit is not possible. We discuss several different ways of characterizing the evolution of characteristicfrequencies and phase fluctuations during transients that come as close as possible to instantaneousmeasures and which give evidence of the evolution that is taking place.1. INTRODUCTIONChirping of the optical frequency is widely viewed as a detriment in coherent signalcommunication and equally widely used for optical pulse compression. Most treatments consider theprocesses to be deterministic with the hope that the effects of noise can be reduced to acceptable levels ofpulse height fluctuations or timing jitter.We consider some aspects of the general problem of chirping (variation of the optical frequency)during switch-on transients in noisy lasers. Often chirping occurs when a parameter is switchedbetween two values for which the laser operation has different optical frequenices in steady state. Suchshifting frequencies have been observed in semiconductor lasers, in inhomogneously broadened gaslasers, and in lasers with external resonators. Similar phenomena occur in noisy systems, with theevidence of frequency evolution found in changes in the frequency of the peak in the spectrum of thefluctuations. This has been observed in models of the subthreshold operation of detuned lasers withtwo-level media, whose frequency is intensity independent except for below threshold operation, and insubthreshold and above threshold semiconductor lasers in which the peak frequency depends directly onthe carrier number which in turn depends on the operating point of the laser.However, several significant additional problems for determination of a frequency arise when theinevitable and intrinsic noise is included in the models. While the problems are always present in theexperiments, it is often the case that discussions of experimental measurements of frequency and spectralvariations ignore these problems as if deterministic, noise-free models were appropriate. In this paperwe wish to highlight some of the problems and to discuss their resolution. In particular the nature ofdynamical frequency variations in several laser systems will be reviewed together with some indicationsof how they might be characterized in the presence of noise.The basic problem is that the sources of noise we are considering are typically represented bywhite noise in the equations for the phase of the electric field. For example if there is additive whitenoise of constant strength in the equation for the complex electric field, the dynamical equation for thephase of the electric field often can be reduced to the form: