172 J. Eur. Opt. Society-Rapid Publ. 22, 19( 2026)
W lm ðr 1; r 2; zÞ ¼ 1 Z 2p Z 2p
Wðr 4p 2 1; r 2; zÞe �i ð lu 1�mu 2 Þ du 1 du 2;
0 0 �1 < l; m < 1:
The COAM matrix is the counterpart of the CSD matrix defined in polarization space to the OAM space and carries similar properties( quasi-Hermiticity, non-negative definiteness, etc.). However, it can have any size, say L L, where L is the largest by magnitude OAM index involved. Also, for theoretical models, infinitely dimensional matrices are possible with certain convergence constraints. The uni-variable beam model is of this type.
Two derivatives of the COAM matrix are important and will be considered here. First, the OAM degree of coherence is the analog of the EM degree of coherence in the OAM space [ 58 ]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Tr½W! y ðr 1; r 2; zÞ W! ðr 1; r 2; zÞŠ oðr 1; r 2; zÞ ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: ð10Þ
Q Tr W! ðr j; r j; zÞ j¼1; 2 ð9Þ
It provides the cumulative information about the normalized correlations at a pair of radii for all pairs of the OAM modes, regardless of the OAM phase information contained in each mode.
Second, it was found that a harmonic beam carrying OAM in L modes traces an orbitalization ellipse in LD OAM space( polar Fourier space) at a fixed radius [ 59 ], analogously to the two / three component electric field tracing a polarization ellipse in real 2D / 3D space. To extend this geometric description to partially coherent beams, the COAM matrix can be considered at a single radius r 1 = r 2 = r, and regarded as the L L orbitalization matrix [ 60 ]:
O! ðr; zÞ ¼W! ðr; r; zÞ; ð11Þ
the term steamming from its formal similarity to a polarization matrix. In LD the orbitalization matrix is shown to decompose into three parts: completely orbitalized, unorbitalized, and partially orbitalized, in which mixing occurs for mode couples, triples,..., uptoL � 1. The completely orbitalized matrix is shown to factorize, representing a beam indistinguishable from harmonic, and, hence, described by an orbitalization ellipse. The contribution of the intensity of the completely orbitalized portion of the beam in the total beam, at a fixed radius, was introduced as the degree of orbitalization [ 60 ]:
Oðr; zÞ ¼ r 0ðr; zÞ�r 1 ðr; zÞ
P 1
; ð12Þ r n ðr; zÞ n¼0
where r 0( r, z) r 1( r, z)... r L( r, z)... is the
ordered sequence of the eigenvalues of O!, all being real and nonnegative due to its Hermiticity and non-negative-definiteness. This quantity is analogous in nature to the degree of polarization and reduces to it in form for L = 2 and 3. In this work, we will thoroughly examine its behavior for uni-variable beams.
We note that for a univariable CSD the COAM matrix is diagonal and hence the calculations of both derivatives are relatively simple. For example, the OAM DOC becomes rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P 1 c 2 n ðr 1r 2 = r 2
0Þ 2n n¼0 oðr 1; r 2; 0Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q r P: ð13Þ 1 c n ðr j = r 0 Þ n j¼1; 2
As we show below, for uni-variable beams r 0 and r 1 may depend on different c n at different radii and, hence, it appears impossible to write down an explicit single formula for the degree of orbitalization. Instead, a piecewise-continuous function will be used, with its interval endpoints determined numerically. n¼0
3 Beams radiated by uni-variable CSDs
The propagation from sources with uni-variable CSD, both in the near and in the far field, can be dealt with starting from their expansion in terms of coherent modes. In such a way, from equation( 6) we have, at any plane z = const.,
W ðr 1; r 2; zÞ ¼ X1 n¼0 k n U n ðr 1; zÞU n ðr 2; zÞ; ð14Þ
where the expression of the propagated modes U n( r, z) depends on the approximation we use.
Using the expression above, the spectral density and the degree of coherence of the radiated field can be evaluated as [ 55 ] and
respectively.
Sðr; zÞ W ðr; r; zÞ ¼ X1 n¼0 k n jU n ðr; zÞj 2; ð15Þ
W ðr 1; r 2; zÞ lðr 1; r 2; zÞ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; ð16Þ
Sðr 1; zÞ Sðr 2; zÞ
3.1 Coherent modes in paraxial regime
Within the paraxial approximation, the propagation of the modes can be studied through the Fresnel integral, which, for a typical field V( r, u, z), in polar coordinates reads [ 55 ]
V ðr; u; zÞ ¼� ikeikz 2pz
Z 1 Z 2p
V ðr 0; u 0; 0Þe i 2z k ½ r02 þr 2 �2r 0 r cosðu 0 �uÞŠ r 0 dr 0 du 0: ð17Þ
0 0