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J. Eur. Opt. Society-Rapid Publ. 22, 19( 2026)
Fig. 4.( a) OAM degree of coherence o S( q 1, q 2) and( b) degree of orbitalization O S( q) for a sZegö source.
which can be used to evaluate S S, 1 and l S, 1 numerically. The latter quantities, evaluated by adding the contribution of the first 100 modes, are shown in Figures 5 and 6, respectively.
For the case of the spectral density, it was found that there are no significant differences, at least in the range r 0 m < 1, when calculating it using only the first 6 modes or the first 100 modes, so that the contribution of modes of order higher than 6 turns out to be negligible. Basically, the same conclusions hold for the degree of coherence, but in such a case, the first 10 modes have to be considered. The degree of coherence shows cylindrical symmetry when calculated in relation to the beam axis, with its absolute value reaching a maximum at the center of the beam and showing decreasing oscillations, while the phase alternates between 0 and p. When evaluated in relation to a direction outside the axis, the oscillations of the absolute value of the DOC are distorted, and the phase varies gradually. In this case also the presence of coherence vortices can be observed( with charge + 1 and �1), corresponding to the zeros of l S, 1 [ 38 ]. The absolute value is represented on the vertical axis, and the phase is coded in a color scale. The first 100 modes were considered in the calculation.
The far-zone OAM degree of coherence of a field radiated by a sZegö source takes form rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P
2 1
J 2 nþ1 ð2pr 0m 1 ÞJ 2 nþ1 ð2pr 0m 2 Þ n¼0
o S ðm 1; m 2 Þ¼qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; ð36Þ 1 � J 2
0 ð2pr 0m 1 Þ 1 � J 2 0 ð2pr 0m 2 Þ
after using a summation formula for J 2 nþ1 ð2pr 0m j Þ, j = 1,2 in the denominator. Figure 7a shows this quantity as a function of m 1, m 2 with r 0 = 1, where we restrict ourselves to a region close to the axis. This distribution is structurally different from that in Figure 4a, since the coherence no longer decreases monotonically with the increasing radii but oscillates between wider and narrower distributions. It should also be noted that for m 1 = m 2 = m the OAM coherence remains fairly high within the plotted region around the axis.
The degree of orbitalization becomes
O S ðmÞ ¼2 J 2 p ð2pr 0mÞ�J 2 q ð2pr 0mÞ 1 � J 2
0 ð2pr; ð37Þ 0mÞ
Fig
. 5. Spectral density for a sZegö source in the far field( with r 0 = 1), calculated by adding the first 2, 5, and 100 modes, repectively.
where p and q are the indices of the largest and the second largest eigenvalues r n for a given radius m. We note that unlike in the source plane, where the zeroth and the first eigenvalues are the largest and second largest for all radii, in the far field such eigenvalues must be determined numerically. Figure 7b shows the degree of orbitalization as a function of m, withr 0 = 1, form 1. Unlike in the source plane, where this degree monotonically decreases from unity to zero, see Figure 4b, it shows a piecewise-continuous profile without smoothness at junctions. This is due to the fact that many OAM modes are competing for being the maximum and the second maximum. For example, close to the axis, modes n = 0 and n = 1 constitute this pair, but at m 0.419, modes n = 1 and n = 2 become such. Hence, most of the oscillations are due to switching to higher indexes, with one exception: in the region 0.817 m 0.894, the n = 0 mode has a crest becoming the second largest eigenvalue again.
4.2
Truncated sZegö sources
In this case, we let c n = I 0 if 0 n < N and c n = 0ifn N, so that equation( 1) gives
g T ðfÞ ¼I 0
X N�1
n¼0 f n ¼ I 0
1 � f N 1 � f
and the corresponding CDS turns out to be
ðjfj < 1Þ; ð38Þ