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Optics Letters

Cross-spectral purity of electromagnetic fields

Timo Hassinen, Jani Tervo, Ari T. Friberg

Published Dec 9, 2009• 24 citations
Publisher

Abstract

We extend Mandel's scalar-wave concept of cross-spectral purity to electromagnetic fields. We show that in the electromagnetic case, assumptions similar to the scalar cross-spectral purity lead to a reduction formula, analogous with the one introduced by Mandel. We also derive a condition that shows that the absolute value of the normalized zeroth two-point Stokes parameter of two cross-spectrally pure electromagnetic fields is the same for every frequency component of the field. In analogy with the scalar theory we further introduce a measure of the cross-spectral purity of two electromagnetic fields, namely, the degree of electromagnetic cross-spectral purity.

DOI
10.1364/ol.34.003866

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Physical Review Letters•Aug 2, 2010•15 citations
We extend Mandel's scalar-wave concept of cross-spectral purity to electromagnetic fields. We show that in the electromagnetic case, assumptions similar to the scalar cross-spectral purity lead to a reduction formula, analogous with the one introduced by Mandel. We also derive a condition that shows that the absolute value of the normalized zeroth two-point Stokes parameter of two cross-spectrally pure electromagnetic fields is the same for every frequency component of the field. In analogy with the scalar theory we further introduce a measure of the cross-spectral purity of two electromagnetic fields, namely, the degree of electromagnetic cross-spectral purity.
Cross-spectral purity, a concept introduced by Mandel in 1961, refers to an optical field having identical spectra at two points whose superposition yields the same spectrum. Here, we generalize this concept after discretizing the modal structure of the field, so that cross-purity can then apply to any optical degree-of-freedom (DoF). Making use of two binary DoFs (polarization and two spatial modes), the separability of the associated 4×4 coherence matrix is the condition for cross-purity. We find that the rank of the coherence matrix (the number of non-zero eigenvalues) plays a surprising yet crucial role in determining optical cross-purity: whereas rank-1 and rank-2 fields are always cross-pure, rank-3 fields in contrast are never cross-pure, and only a subset of rank-4 fields are cross-pure.
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.
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.