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Fig. 17. Degree of orbitalization O T( m) for truncated sZegö far fields with( a) N = 2;( b) N = 3;( c) combination of N = 2, N = 3, and non-truncated model.
eigenvalues resulting in the same numerator in equation( 17), while the denominator has an increasing value with increasing N. Thisalwaysoccursatsmallm as the 0th mode always dominates close to the axis.
5 Further examples of uni-variable sources
Different uni-variable CSDs can be defined by choosing different sets of c n coefficients, and in many cases they are given in closed form, as happened for the case in the previous section. Actually, using handbooks of mathematical formulas, dozens of other CSDs can be defined from formulas involving power series. Some of them lead to closed forms even when the series is truncated. This is particularly useful because, as we saw for sZegö and truncated sZegö sources, in such a case the effects of truncation can be evaluated quite simply. Furthermore, in any practical realization of partially coherent sources based on the superimposition of perfectly coherent and mutually uncorrelated fields, the number of modes must be necessarily truncated.
Among others, we quote the following( formula 5.2.2.4 of [ 64 ]): gðfÞ ¼I 0
X 1
n¼1 nf n ¼ I 0 f ð1 � fÞ 2; ð51Þ
together with its truncated version( formula 4.1.7.2 of [ 64 ])
g T ðfÞ ¼I 0
X N
n¼1 nf n ¼ I 0 f þðNf � N � 1Þ f Nþ1 ð1 � fÞ 2; ð52Þ with integer N.
Another interesting example of a source obtained with a finite number of modes is that of the binomial source, for which( formula 4.2.3.1 of [ 64 ])
X N N gðfÞ ¼I 0 f n ¼ I 0 ð1 þ fÞ N: ð53Þ n n¼0
As a last example, we quote the following, which will be discussed in a little more detail. If we take( formula 5.2.7.2 of [ 64 ]) gðfÞ ¼I 0
X 1
n¼0 ðc 2 f n!
Þ n
¼ I 0 e c2f; ð54Þ
with c 2 a positive dimensionless parameter, the corresponding uni-variable CSD turns out to be
W G ðq 1; q 2 Þ¼I 0 exp c 2 q 1 q 2 exp ½ iðu 1 � u 2 ÞŠ
circðq 1 Þcircðq 2 Þ; ð55Þ and the corresponding spectral density has the form
S G ðqÞ ¼I 0 e c2 q 2 circðqÞ: ð56Þ
The interest in such sources comes from what follows. If we express W G through the vectors q 1 and q 2, sinceq j =( n j, g j) and q j exp( iu j)= n j + ig j,( j = 1, 2), equation( 55) can also be written as
W G ðq 1; q 2 Þ ¼ I 0 e c2 ðn 1 n 2 þg 1 g 2 Þ e �ic2 ðn 1 g 2 �n 2 g 1 Þ circðq 1 Þcircðq 2 Þ ¼ I 0 e c2 q 1 q 2 e
�ic 2 ðq 1 q 2 Þ z circ ð q1 Þcircðq 2 Þ ð57Þ or, after some manipulations, W G ðq 1; q 2 Þ
¼ I 0 e c2 ðq 2
1 þq2 2 Þ = 2 e �c2 jq 1 �q 2 j 2 = 2 e �ic2 ðq 1 q 2 Þ z circ ð q1 Þcircðq 2 Þ; ð58Þ
which apparently has the form of a twisted Gaussian Schell-model source with saturated twist [ 40 ], the only difference being the sign in the argument of the Gaussian amplitude term. In particular, the degree of coherence turns out to be
l G ðq 1; q 2 Þ¼e �c2 jq 1 �q 2 j 2 = 2 e �ic2 ðq 1 q 2 Þ z circ ð q1 Þcircðq 2 Þ; ð59Þ whose modulus is jl G ðq 1; q 2 Þj ¼ e �c2 jq 1 �q 2 j 2 = 2 circðq 1 Þcircðq 2 Þ: ð60Þ
The eigenvalues of the Mercer expansion of W G turn out to be k n ¼ I 0 c 2n p r 2
0 ðn þ 1Þ!; ð61Þ