J. Eur. Opt. Society-Rapid Publ. 22, 19( 2026) 171
can be expressed as the superposition of coherent modes [ 55 ] carrying optical vortices. The analytical form of the vortex modes is quite simple, both across the source plane and in the far zone, enabling the study of the coherence features of the source, as well as those of the field radiated under Fraunhofer conditions [ 38 ]. One of the important features of univariable CSDs is their capability to customize at will the contributions from various coherent modes, serving, at the same rate, as the OAM modes. This renders them ideal for high-capacity information-transfer applications employing structured light.
The main aim of the present paper is, on one hand, to extend to the Fresnel regime the study of the propagation of beams radiated from sources with uni-variable CSD; on the other hand, to study the effects of truncating the number of the modes involved in the expansion. Examples will be presented concerning the so-called sZegö source, introduced in [ 38 ], which is obtained with an infinite number of Taylor terms, all with the same coefficient. Taking only a finite number of terms gives rise to the truncated sZegö source, introduced here, which represents a more realistic model for a partially coherent source, suitable to be physically realized. Finally, hints for devising other types of uni-variable CSDs are given for some cases in which the CSD takes a simple closed form.
2 Preliminaries
2.1 Uni-variable CSDs
A uni-variable CSD [ 38 ] isdefined from any function g of a single complex variable f such that:
where gðÞ¼ f X1 n¼0 c n f n;
c n 0 ð8nÞ; ð1Þ
ð2Þ
and f belongs to the convergence domain of the series. By suitably scaling the argument of g it is always possible to set such a domain as jj f < 1: The CSD associated with g is obtained by letting f ¼ r 1r 2 e iðu 1�u 2 Þ; ð4Þ r 2 0
where( r 1, u 1) and( r 2, u 2) are the polar coordinates of two points across the source plane( r 1 and r 2) and r 0 is the radius of the circular region where the source is defined. It should be noted that although the CSD depends on two points, for this class of sources, the dependence enters only through the single complex parameter f, following the terminology introduced previously in reference [ 38 ]. Therefore, using equations( 1) and( 4) we define, across the plane z = 0, ð3Þ
W ðr 1; r 2; 0Þ ¼ g r 1r 2
ðjrj < r 0 Þ:
¼ X1 n¼0 r 2 0 e i ð u 1�u 2 Þ c n r 1 r 2 r 2 0
n e in u 1�u 2
ð Þ; ð5Þ
The sum in equation( 5) can be read as the Mercer expansion of W 0, thatis,[ 55 ] W ðr 1; r 2; 0Þ ¼ X1 n¼0 k n U n ðr 1; 0ÞU n ðr 2; 0Þ; ð6Þ
with coherent modes( taking into account their normalization) given by sffiffiffiffiffiffiffiffiffiffiffi
n n þ 1 r r
U n ðr; 0Þ ¼ circ e inu; pr 2 0 r 0 r 0 ð7Þ ðn ¼ 0; 1;::: Þ; and eigenvalues pr 2 0 k n ¼ c n; ðn ¼ 0; 1;::: Þ; ð8Þ n þ 1
where circ( r) is the characteristic function of a unit disk centered at the origin of the axes. The second argument of the modes U n refers to the z-coordinate of the transverse plane where they are evaluated( in this case, across the source, that is, z = 0). The condition in equation( 2), together with equation( 8), guarantees that k n > 0(" n), so that the CSD turns out to be well defined [ 13, 14 ].
The modes in equation( 7) are recognized at once as optical vortices [ 49 ] with charge n. Theyhavethesame structure as Laguerre – Gaussian beams [ 56 ] in which the Laguerre polynomial has order zero and where the Gaussian function has been replaced by a circ.
The number of genuine CSDs that can be obtained following the present approach is practically unlimited, the only constraint that the function g has to fulfill being the fact that the coefficients of its Taylor expansion are all non-negative. Examples will be given in the following, where simple forms of CSDs will be presented, together with the mean features of the beams they radiate.
2.2 Characterization of uni-variable CSDs in OAM space
Uni-variable CSD represents one of a few known model optical fields that carry OAM in multiple modes, either finite or infinite. For the characterization of radial-only correlations among the OAM modes in such fields, the coherence – orbital angular momentum( COAM) matrix
W! ðq 1; q 2; zÞ and its derivatives can be used [ 57 ]. Each element of the COAM matrix is a scalar radial field correlation at a pair of OAM indices, given by the expression: