Shape-Invariant Propagation of the Cross-Spectral Density
Collett-Wolf sources of both quasi-homogeneous and Schellmodel type have been extensively studied in recent years1−17. In particular, some researches have dealt with the paraxial propagation of the field emitted by such sourcesj5−17. It has been found that both the optical intensity and the degree of coherence remain gaussian across any plane parallel to the source plane. Furthermore, in any such plane the ratio between the transverse coherence length and the r.m.s. width of the optical intensity has the same value. We shall characterize this kind of behavior by saying that the cross-spectral density is shape-invariant. A precise definition of shape-invariance will be given shortly. We can ask whether this property is exhibited by other partially coherent fields. In the following, we shall give two classes of such fields. One of them is generated far enough from spatially incoherent sources. The other one encompasses those fields for which the cross-spectral density has a series expansion18 into Hermite-Gauss modes with arbitrary weights. To illustrate this last class two examples will be given. First, the Collett-Wolf source itself and second, the field generated by the laser mode superposition known as the “doughnut” mode.
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