J. Eur. Opt. Society-Rapid Publ. 22, 19( 2026) 175
Fig. 3. Degree of coherence for a sZegö source at the source plane relative to a point located at( a) q 2 =( 0, 0);( b) q 2 =( 0.3, 0);( c) q 2 =( 0.6, 0); and( d) q 2 =( 0.9, 0). Absolute value is represented on the vertical axis, and the phase is coded in a color scale.
gðfÞ ¼ I 0 1 � f ðjfj < 1Þ; ð29Þ
and the corresponding CSD turns out to be W S ðq 1; q 2; 0Þ ¼
I 0
1 � q 1 q 2 e circ ð q i ð u 1�u 2 Þ 1Þ circðq 2 Þ: ð30Þ
Spectral density and degree of coherence across the source plane turn out to be, from equations( 15) and( 16),
S S ðq; 0Þ ¼ I 0 circðqÞ; ð31Þ 1 � q2 and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 � q 2
1Þð1 � q 2 2Þ l S ðq 1; q 2; 0Þ ¼ circðq 1 � q 1 q 2 e iðu 1�u 2 Þ 1 Þ circðq 2 Þ; ð32Þ
respectively. It can be seen that the spectral density grows to infinity as q approaches one. This somewhat anomalous behavior has to be ascribed to the fact that an infinite number of modes contribute to the CSD of the source. It will be solved in the next subsection, where the series that gives rise to sZegö source will be truncated. Plots of l S( absolute value and phase) are shown in Figure 3 as a function of q 1 for fixed q 2. The absolute value of l S is seen to vary from 1( when q 1 = q 2) to 0( along the circle q 1 = 1). Furthermore, when q 2 is in the center of the source, the phase is constant and equals to 0, while the range of variation of the phase increases with increasing q 2. The presence of a coherence vortex can also be observed, which has charge �1, centered at a point outside the source at a distance 1 / q 2 from the center [ 38 ].
The OAM degree of coherence at z = 0 can be derived using infinite geometric series. Together with the degree of orbitalization they take forms: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 � q 2 o S ðq 1; q 2; 0Þ ¼
1Þð1 � q 2 2Þ
; 1 � q 2
1q 2 2
� O S ðq; 0Þ ¼ 1 � q 2 2
: ð33Þ
Note that O S( q, 0) has a particularly simple form since for this source k n( q, 0) k n + 1( q, 0) for all n and q, whichis not generally true for other source classes. The color density plot of the OAM degree of coherence for this beam is shown in Figure 4a. As expected, this quantity takes the unity value at q 1 = q 2 = 0 but gradually decreases and reaches zero for q 1 = 1orq 2 = 1. We also note that it is not generally unity at the coinciding radii q 1 = q 2 6¼ 0, unlike the classic degree of coherence that would be at the coinciding points.
An explicit expression of the CSD of the field radiated in the far zone, W S, 1, can hardly be derived directly from the CSD in z = 0, but from its modal expansion, with the modes of equation( 26) and the eigenvalues given by pr 2 0 k n ¼ I 0; ðn ¼ 0; 1;::: Þ; ð34Þ n þ 1 takes the form
W S; 1 ðm 1; m 2 Þ / X1 n¼0
J nþ1 ð2pr 0 m 1 Þ 2pr 0 m 1
J nþ1ð2pr 0 m 2 Þ 2pr 0 m 2 e inð # 1� # 2 Þ; ð35Þ