Abstract
We develop a general representation for ensembles of non-stationary random pulses in terms of statistically uncorrelated, time-delayed, frequency-shifted Gaussian pulses which are classical counterparts of coherent states of a quantum harmonic oscillator. We show that the two-time correlation function describing second-order statistics of the pulses can be expanded in terms of the complex Gaussian pulses. We also demonstrate how the novel formalism can be applied to describe recently introduced Gaussian Schell-model pulses and pulse trains generated by typical mode-locked lasers.
Highlights
We represent each statistical pulse as a linear superposition of uncorrelated, time-delayed, frequency-shifted Gaussian pulses–which can be routinely produced in the laboratory by standard lasers–with a statistical distribution of emission times and carrier frequency shifts
The cross-correlation function of the pulse can be expanded into a Mercer-type series as
We examine the P-representation of a recently introduced [8, 9] nonstationary Gaussian Schell-model (GSM) source with the cross-correlation function
Summary
Relentless recent progress in ultrafast optics [1] has motivated the quest for a better insight into statistical features of non-stationary sources generating ultrashort optical pulses. The purpose of this work is to formulate a statistical theory of random pulses in the language that is sufficiently flexible to describe a variety of partially coherent pulse models on the one hand, and on the other hand, establishes a clear link with experimentally realizable ultrashort pulses To this end, we represent each statistical pulse as a linear superposition of uncorrelated, time-delayed, frequency-shifted Gaussian pulses–which can be routinely produced in the laboratory by standard lasers–with a statistical distribution of emission times and carrier frequency shifts. By analogy with the Glauber-Sudarshan P-representation in quantum optics [3], we can express the second-order two-time correlation function of any statistical pulse as an integral over an over-complete non-orthogonal set of complex Gaussian pulses. Received 8 Jul 2011; revised 29 Jul 2011; accepted 29 Jul 2011; published 16 Aug 2011 29 August 2011 / Vol 19, No 18 / OPTICS EXPRESS 17087 It follows from Eq (5) that the coordinate-representation wave function of the coherent state is given by.
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