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Fig. 2. Plots of the spectral density of the modes in the far field, equation( 26) for n = 5( a), 10( b), 15( c), with r 0 = 1.
( say, V 1) is obtained from the Fourier transform, suitably scaled, of the field across the source [ 61 ]. For simplicity, the coordinate across the Fourier plane( say m) will be taken as the spatial coordinate across a transverse plane in the far zone. Disregarding unessential proportionality factors and curvature terms, we simply write
ZZ
V 1 ðmÞ ¼ V ðr; 0Þe �2pimr dr; ð22Þ
the integral extending over the whole z = 0 plane. Using polar coordinates, the above integral reads
V 1 ðmÞ ¼
Z 1 Z 2p
0 0
V ðr; u; 0Þe �2pimr cosð # �uÞ r dr du; ð23Þ
with m and # being the polar coordinates of m. For the case of the modes in equation( 7), the above integral gives
U n; 1 ðmÞ ¼ 2ð�iÞn pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pðn þ 1Þe in # r 0
Z r0
nþ1 r
J n ð2pmrÞdr; ð24Þ
0
J n being the Bessel function of the first kind and order n [ 62 ]. Exploiting the following relation obeyed by the Bessel functions [ 63 ]: r 0
d
½ dg gnþ1 J nþ1 ðgÞŠ ¼ g nþ1 J n ðgÞ; ð25Þ
the expression of the modes propagated in the far zone takes the simple form pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U n; 1 ðmÞ ¼2r 0 ð�iÞ n pðn þ 1Þe in # J nþ1ð2pr 0 mÞ
: ð26Þ
2pr 0 m
Note that for n = 0 the latter equation gives rise to the wellknown Fraunhofer diffraction pattern from a circular hole [ 61 ]. Indeed, for n = 0 the vortex field coincides with that of an ordinary plane wave. For n 6¼ 0, we can read equation( 26) as the diffraction pattern produced by a vortex field that impinges on a circular aperture [ 48 ].
Plots of the spectral density of the propagated modes of equation( 26) are given in Figure 2 for n = 5, 10, 15. It appears that, in general, the function is very small until its argument reaches values roughly equal to n, and then slowly goes to zero in an oscillating way.
In light of the above result, it is useful to express the CSD of a beam radiated by a uni-variable source through its Mercer expansion, which reads
X 1
W 1 ðm 1; m 2 Þ¼4pr 2
0 k n ðn þ 1Þe inð # 1� # 2 Þ n¼0
J nþ1ð2pr 0 m 1 Þ J nþ1 ð2pr 0 m 2 Þ: ð27Þ
2pr 0 m 1 2pr 0 m 2
In some cases, this expression even leads to closed forms for the degree of coherence in the far field( l 1). This happens, for example, if k n /( n + 1) and we limit ourselves to point along radial directions( i. e., # 1 = # 2), in which case it can be shown that [ 64 ]
l 1 ðm 1;#; m 2;# Þ¼2 J 1ð2pr 0 ðm 1 � m 2 ÞÞ: ð28Þ
2pr 0 ðm 1 � m 2 Þ
4 sZegö and truncated sZegö sources
The first example of a uni-variable CSD we are going to show is that of the so-called sZegö source. The features of such a source across the source plane and in the far zone were studied and commented on in detail in a previous paper [ 38 ]. Here, we limit ourselves to recalling some of the main results obtained there. The effects of paraxial propagation of the radiated field will be studied here on the basis of the expression of the propagated modes obtained in Section 3.1. However, we find it convenient to perform such an analysis starting from a different uni-variable CSD, namely, the truncated sZegö CSD. Unlike sZegö CSD, in fact, the latter requires a finite number of modes and reduces to sZegö CSD when the number of modes goes to infinity. In such a way, the propagation features of the beams they radiate can be evaluated exactly through finite sums. However, we want to stress that a CSD of this type is interesting in itself because it is suitable to be synthesized in the laboratory starting from the superposition of a finite number of coherent fields [ 20, 28, 32, 37 ].
4.1 The sZegö source
To obtain a sZegö CSD we let c n = I 0(" n) in equation( 1), with I 0 a positive constant having dimensions of irradiance, so that