J. Eur. Opt. Society-Rapid Publ. 22, 19( 2026) 181
Fig
. 15. Degree of coherence relative to a point located at q 2 =( 0, 0.5) for a truncated sZegö source with N = 4 at several propagation distances( a) N F = 3;( b) N F = 2;( c) N F = 1;( d) N F = 0.5. The absolute value is represented on the vertical axis, and the phase is coded in a color scale.
Fig. 16. OAM degree of coherence o T( m 1, m 2) for truncated sZegö far fields with( a) N = 2;( b) N = 3;( c) N = 4.
Figure 16 shows this quantity for several values of the summation index:( a) N = 2,( b) N = 3, and( c) N = 4. As N increases, the distributions start resembling that for the non-truncated model, if compared to Figure 7a. This practically occurs already for N = 4, as higher-order modes deliver much smaller contributions to the sums.
For truncated sZegö far fields the degree of orbitalization becomes
O T ðmÞ ¼ J 2 p ð2pr 0mÞ�J 2 q ð2pr 0mÞ; ð50Þ
PN�1
J 2 nþ1 ð2pr 0mÞ n¼0
where, as above, p and q are the indices of the largest and the second largest eigenvalues. Analogously to the nontruncated model, the determination of the largest and the second largest eigenvalues for different radii was carried out numerically.
Figure 17 shows formation of the degree of orbitalization for( a) N = 2and( b) N = 3. ForN = 2 the degree exhibits strictly oscillatory nature with smooth maxima and non-smooth minima, all at zeroes. This is due to the fact that only two modes compete with each other for being maximum and second maximum and can only alternate. This behavior is similar to that of the degree of polarization, also involving competition of two electric field components. Starting from N = 3 both maxima and minima are not smooth, since the switches are determined by three pairs of modes. Figure 17c summarizes the behavior for cases N = 2, N = 3 and N? 1( non-truncated model), same as in Figure 7b. In parts where the curves are almost the same, the same modes act as the largest and second largest