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Physical Review A

Cross-spectral purity of nonstationary light

Matias Koivurova, Chaoliang Ding, Jari P. Turunen, Ari T. Friberg

Published Apr 29, 2019• 13 citations
Publisher

Abstract

The concept of cross-spectral purity of light, established by Mandel for stationary optical fields, is extended to nonstationary fields. Separability conditions that ensure the cross-spectral purity of nonstationary light are derived in the space-frequency and space-time domains. It is also shown that the property of cross-spectral purity is not strictly preserved on propagation of fields with any appreciable spectral bandwidth. Further, we introduce a method for generating cross-spectrally pure (stationary and nonstationary) fields from spatially incoherent light sources by means of achromatic Fourier-transform systems.

DOI
10.1103/physreva.99.043842

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The concept of cross-spectral purity of light, established by Mandel for stationary optical fields, is extended to nonstationary fields. Separability conditions that ensure the cross-spectral purity of nonstationary light are derived in the space-frequency and space-time domains. It is also shown that the property of cross-spectral purity is not strictly preserved on propagation of fields with any appreciable spectral bandwidth. Further, we introduce a method for generating cross-spectrally pure (stationary and nonstationary) fields from spatially incoherent light sources by means of achromatic Fourier-transform systems.
An examination of the conditions under which the normalized coherence function γ(x1,x2,τ), at two points x1 and x2 in an optical field, is reducible to the product of a function of x1, x2 and a function of τ leads to the concept of cross spectral purity. Spectrally pure beams of light are characterized by the property that their coherence function is so reducible and, operationally, by the fact that superposition does not affect the spectral distribution. Light beams that do not have this property are called spectrally impure, and it is shown to be characteristic of such beams that the interference fringes exhibit a detailed periodic coloring. A measure of the departure from cross-spectral purity is introduced and evaluated in some special cases. It is shown by an example that spectrally pure and spectrally impure beams of light may be derived from the same source with similar optical components. Moreover, these beams may be identical as regards their intensity, their spectral distribution, and their degree of coherence, and differ only as regards their state of spectral purity.
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.
Cross-spectral purity was conceived in the context of coherence research and has remained largely unknown to the wider optics community. Over the past few decades, coherence researchers have made progress in uncovering the intricate properties of this fundamental concept. Cross-spectral purity has been shown to be intimately connected to the concept of spatiotemporal separability, generalizing the concept of separability from fields to correlation functions. It has an observable effect on a multitude of experiments, and purity is in fact assumed in many well-known experiments. In this tutorial, we discuss the basic definitions related to cross-spectral purity and investigate some special cases. Additionally, we show how to generate and detect this property, and consider its experimental significance in areas such as ultrashort pulses, quantum optics, scattering, and ghost imaging.
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.