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Fig
. 9. Degree of coherence at the source plane, for a truncated sZegö CSD with N = 2, 4, 20( from top to bottom), 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.
respectively. The plots of S T and l T are shown in Figures 8 and 9, respectively. It can be seen that, as the number of modes increases, the spectral density becomes more and more similar to that of the sZegö source but never diverges, as expected. On the other hand, the DOC is highly dependent on the number of modes considered.
Two aspects deserve to be noticed. First, the coherence area decreases as N increases, as expected. The second aspect concerns the phase structure of l T. It can be highlighted in a simpler way if we focus our attention on the second fraction in equation( 41), provided the other terms are positive. Using again the variable f = q 1 q 2 exp [ i( u 1 � u 2)], this term can be written as
1 � f N 1 � f ¼�
Q N n¼1
where f n( n = 1,..., N) are the solutions of the equation f N = 1, namely, f n = exp( i2pn / N). Therefore, since f N = 1, we have
�
Q N n¼1 ðf � f n Þ 1 � f
¼ YN�1 n¼1 ðf � f n Þ;
which for fixed values of q 2 and u 2 can be written as
ð43Þ
YN�1
n¼1 ðf � f n Þ¼q ðN�1Þ
2 e �iðN�1Þu 2
YN�1 n¼1
q 1 e iu 1
� 1
e iðu 2þ2pn = NÞ
; ð44Þ q 2
representing the product of N � 1 vortices, each having charge + 1, centered on N � 1 vertices of a regular enneagon, at a distance 1 / q 2 from the center.
To better understand this point, let us take u 2 = 0. Then, the nth term of the product takes the form
q 1 e iu 1
� 1 e i2pn = N ¼ n 1 � 1
2pn cos q 2 q 2 N
þ i g 1 � 1
sin q 2
2pn N
; ð45Þ
which actually represents a vortex with charge + 1 and center at
q ðnÞ
1 ¼ 1
2pn cos; 1 sin 2pn: ð46Þ q 2 N q 2 N
The phase structure of the DOC is shown in Figure 10 for q 2 =( 0.9, 0) and several values of N.
The OAM degree of coherence for this source at z = 0 can be derived using finite geometric series. This quantity and the degree of orbitalization take form