J. Eur. Opt. Society-Rapid Publ. 22, 19( 2026) 183
from which the propagation features of the radiated beams could be evaluated.
In this case, too, the truncated version of the series can be expressed in closed form because( formula 4.1.7.10 of [ 64 ])
X N ðc 2 fÞ n
CðN þ 1; c 2 fÞ g T ðfÞ ¼I 0 ¼ I 0 e c2f; ð62Þ n! N! n¼0 where C(,) is the incomplete Gamma function [ 63 ].
6 Conclusions
Uni-variable CSDs can be derived from any function of a single complex argument whose Taylor series involves only non-negative coefficients. The convergence range of the series determines the spatial extent of the source. The Taylor series can be directly related to the Mercer expansion of the CSD of the source in such a way that the coherent modes turn out to be optical vortices, restricted to a circle at the source plane, and the corresponding eigenvalues are proportional to the Taylor coefficients.
Although the coherence features of light across the source plane can be directly derived from the analytical form of the CSD, the analogous properties for the radiated beam can hardly be evaluated in closed form. Nevertheless, from the knowledge of modes and eigenvalues of the CSD across the source, the coherence features of the radiated beam can be evaluated both in the Fresnel regime and in the Fraunhofer regime.
In this paper, the tools for studying the properties of the propagated field have been provided and applied to a test case, that is, the sZegö source. Furthermore, the effects, on the coherence properties of the source and the radiated field, of truncating the Taylor series have also been investigated. This is crucial from an experimental point of view, because limiting the Taylor series( i. e., the coherent-mode expansion of the CSD) to a finite number of terms becomes necessary whenever sources of this kind have to be synthesized through the superposition of mutually uncorrelated perfectly coherent fields.
Since the coherent modes of uni-variable CSDs possess vortex-like structures at various OAM indices, the overall fields can be regarded as carrying OAM in multiple states. Hence, besides the spectral density and the degree of coherence, which carry information about correlations in the physical space, some recently introduced measures, namely, the OAM degree of coherence and the degree of orbitalization, have also been invoked to characterize the radial correlations of the investigated sources in the OAM space, i. e., the polar Fourier space. Although the technique presented here has been applied to the sZegö CSD and to its truncated version( the truncated sZegö CSD), it can be applied to any source of the uni-variable type. The number of uni-variable CSD that can be envisaged starting from the Taylor expansion of a single-argument function is actually huge, and this work lays the foundation for the design and tayloring of sources with required coherence characteristics, both on the source plane and in propagation.
Funding
This work has been supported by the Spanish Ministerio de Economía y Competitividad under project PID2023-148021NB-I00 and Project FASLIGTH( RED2022-134391-T).
Conflicts of interest This work has no financial or non-financial competing interests.
Data availability statement Data will be made available on request.
Author contribution statement
All authors have contributed equally to all aspects of this work, including conceptualization, methodology, formal analysis, and the writing of the original draft and final manuscript.
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