J. Eur. Opt. Society-Rapid Publ. 22, 19( 2026) 179
Fig
. 10. Phase of the degree of coherence at the source plane, for a truncated sZegö CSD relative to the point q 2 =( 0.9, 0) for several values of N.
Fig
. 11. OAM degree of coherence o T( q 1, q 2) for truncated sZegö sources with( a) N = 4;( b) N = 8;( c) N = 16.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 � q 2 o T ðq 1; q 2; 0Þ ¼
1Þð1 � q 2 2Þ
1 � q 2
1q 2 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � q 2N
1 q 2N 2
; ð1 � q 2N
1 Þð1 � q 2N
2 Þ
O T ðq; 0Þ ¼ ð1 � q2 Þ 2
: ð47Þ 1 � q2N
The plots of the OAM degree of coherence for this source are shown in Figures 11a – 11c, for several values of N. Unlike for the sZegö sources the distributions do not decrease to zero for q 1 = 1orq 2 = 1butrathertakeon values in the interval( 0, 1) decreasing with increasing N. The plots of the degree of orbitalization are shown in Figure 12 for several values of N. ThecaseN = 1 is also shown corresponding to the un-truncated sZegö source. While all the curves have monotonic behavior taking on unit value at q = 0anddecreasingtozeroatq = 1only the curve for N = 1 is convex on the whole interval, all the others change convexity once.
Fig. 12. Degree of orbitalization O T( q) for truncated sZegö sources with different values of N. The case N = 1 corresponds to the non-truncated sZegö source.
To evaluate the propagated CSD in the near and in the far field equations( 14) and( 19), or( 26) are used. Equations( 38) and( 8) give, for the eigenvalues,