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Journal of the Optical Society of America A

General interference law for nonstationary, separable optical fields

V. Manea

Published Aug 10, 2009• 4 citations
Publisher

Abstract

An approach to the theory of partial coherence for nonstationary optical fields is presented. Starting with a spectral representation, a favorable decomposition of the optical signals is discussed that supports a natural extension of the mathematical formalism. The coherence functions are redefined, but still as temporal correlation functions, allowing the obtaining of a more general form of the interference law for partially coherent optical signals. The general theory is applied in some relevant particular cases of nonstationary interference, namely, with quasi-monochromatic beams of different frequencies and with phase-modulated quasi-monochromatic beams of similar frequency spectra. All the results of the general treatment are reducible to the ones given in the literature for the case of stationary interference.

DOI
10.1364/josaa.26.001907

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An approach to the theory of partial coherence for nonstationary optical fields is presented. Starting with a spectral representation, a favorable decomposition of the optical signals is discussed that supports a natural extension of the mathematical formalism. The coherence functions are redefined, but still as temporal correlation functions, allowing the obtaining of a more general form of the interference law for partially coherent optical signals. The general theory is applied in some relevant particular cases of nonstationary interference, namely, with quasi-monochromatic beams of different frequencies and with phase-modulated quasi-monochromatic beams of similar frequency spectra. All the results of the general treatment are reducible to the ones given in the literature for the case of stationary interference.
This thesis contains research related to random light fields.In particular, novel constructions of electromagnetic coherence in beam fields are introduced, and certain fundamental notions concerning the spectrum of light are assessed in the context of nonstationary fields.The singular-value decomposition (SVD) of the cross-spectral density matrix (CSDM) is analysed and interpreted for electromagnetic beams.In addition, the Gram matrix of the CSDM is invoked and combined with the SVD formalism.The Gram matrix shares the essential properties of the polarization matrix, thus appearing as a particularly potential descriptor of electromagnetic coherence.The SVD and the Gram matrix yield results analogous to those traditionally met in polarization theory of beam fields, but in the context of two-point spatial coherence analysis.In particular, two-point analogies of the usual decompositions of the polarization matrix are introduced by applying the SVD.In addition, the Poincaré sphere representation of electromagnetic coherence is established, which is analogous to the traditional Poincaré sphere in polarization optics.Cross-spectral purity and spectral invariance of light are two fundamental concepts concerning stationary random light fields, which were recently generalized to nonstationary fields.In this thesis, a reduction formula for time-integrated coherence functions of cross-spectrally pure nonstationary scalar fields is established.This reduction formula is an equivalent condition for the physical definition of crossspectral purity in nonstationary fields, similarly to the reducibility met in the case of cross-spectrally pure stationary light.In addition, optical systems consisting of hybrid refractive-diffractive lens designs which transform partially coherent isodiffracting pulsed beams into cross-spectrally pure or spectrally invariant nonstationary fields are developed.Furthermore, polarimetric measurement of temporal coherence in electromagnetic beam fields is performed.It is shown that the electromagnetic temporal degree of coherence can be deduced from these measurements, thus without the need for visibility detection or optical elements in the arms of the device, in contrast to earlier studies.
We study the connection between cross-spectral purity and spatiotemporal separability of nonstationary (pulsed) scalar fields. It is found that in the case of complete coherence, there is a two-way relation between global cross-spectral purity and spatiotemporal separability of the field. Moreover, we show that cross-spectral purity generalizes the notion of spatiotemporal separability to partially coherent fields, due to the separability of the correlation functions. We also briefly discuss simple self-referencing linear measurement methods that can reveal cross-spectral purity.
We consider temporally integrating interferometric measurements and their relation to the coherence properties of nonstationary light. We find that performing such experiments as a function of time delay is equivalent to spectrally resolving the interference patterns, and time-domain coherence information can be obtained from field autocorrelation only if the source is of the Schell-model type. In an analogy to autocorrelation, we introduce field cross-correlation, which can be used to determine the complete complex field of unknown signal pulses if suitable probe pulses are available. We demonstrate our findings with simulated supercontinuum and free-electron laser ensembles, and discuss the prospect of carrying out experiments.
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We examine the temporal coherence properties of trains of nonidentical short optical pulses in the framework of the second-order coherence theory of nonstationary light. Considering Michelson's interferometric measurement of temporal coherence, we demonstrate that time-resolved interferograms reveal the full two-time temporal coherence function of the partially coherent pulse train. We also show that the result given by the time-integrated Michelson interferogram equals the true degree of temporal coherence only when the pulse train is quasi-stationary, i.e., the coherence time is a small fraction of the pulse duration. True two-time and integrated coherence functions produced by specific models representing perturbed trains of mode-locked pulses and supercontinuum pulse trains produced in nonlinear fibers are illustrated.