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J. Eur. Opt. Society-Rapid Publ. 22, 19( 2026)
Fig
. 13. Spectral density versus the inverse of the Fresnel number for several truncated sZegö sources with N modes.
Fig
. 14. Spectral density versus the inverse of the Fresnel number for different truncated sZegö sources. 8 pr
I 2
0 >< 0 ð0 n < NÞ; nþ1 k n ¼ >:
0 ðn NÞ: ð48Þ
Figure 13 shows examples of the spectral density, across the( x, z) plane, of the field propagated from sZegö sources of various orders. It can be observed that the main contribution for large propagation distances is due to the 0th order mode( compare Fig. 13 with Fig. 1 for n = 0).
As commented above, the addition of more and more modes modifies the distribution of the spectral density only for small propagation distances. This is confirmed by the plots in Figure 14 where the spectral density profiles are calculated at several propagation distances for sZegö truncated sources with different numbers of modes. For a Fresnel number N F = 4, the five profiles are different, although those corresponding to N = 30andN = 100are practically the same up to a normalized distance q = 1. For N F = 1theprofiles practically coincide for all truncated sZegö sources with N 12. This profile should be the same as the one for sZegö source.
The same approach can be used to evaluate the degree of coherence across different planes( Fig. 15). The analogous calculations across the source plane and in the far field [ 38 ] showed the presence of coherence vortices( in both cases).
Thefar-zoneOAMdegreeofcoherenceradiatedbya truncated sZegö source takes the form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o T ðm 1; m 2 Þ¼
PN�1 n¼0
J 2 nþ1 ð2pr 0v 1 ÞJ 2 nþ1 ð2pr 0v 2 Þ
Q j¼1; 2 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: ð49Þ
PN�1
n¼0
J 2 nþ1 ð2pr 0v j Þ