Partially Coherent Beams in Free Space and in Lenslike Media

The behaviour of partially coherent Gaussian beams under free propagation and under passage through lenslike media or first order systems (FOS) is governed by the smplectic group of real linear canonical transformations repesented by the ray-transfer matrix of the system under consideration. And this, coupled with the Wigner representation of the cross-spectral density of the beam, reduces the propagation problem to one of 2×2 or 4×4 matrices depending on whether the beam under consideration is rotationally symmetric or asymmetric about its axis, thus often resulting in complete answers to interesting questions, without too much effort. These considerations hold good not only for Gaussian beams, but also for non-Gaussian partially coherent beams as long as one's interest is restricted (as often happens to be the case) to the first and second moments of the beam. The combined power offered by the Wigner representation on the one hand and the waveoptic representation of canonical transformations on the other will be briefly explored in this sequence of three talks. No prior knowledge or experience with group structures is reqired to follow the arguments and to appreciate the results.

Lecture 1: The first obvious question in respect of partially coherent Gaussian beams is this: Given a hermitian Gaussian two-point function, how to test if it is 'nonnegative' so that it qualifies to be a physical cross-spectral density in some transvese plane? This nontrivial issue gets resolved relatively easily by the 'combined power' referred to above. The evolution of the beam parameters as the beam propagates is obtained as a bye-product. In particular, the invariants or quality parameters associated with a beam present themselves with no additional effort. And so are generalizations of the ABCD-law as well as the presence of the twist phase.

Lecture 2: Gori [Opt. Commun. vol. 34, 301 (1980)] was probably the first to exploit symmetry consideratons to achieve coherent-mode decomposition of a class of partially coherent beams. Reinterpretation of this approach in the light of the 'combined power' noted above, shows that this approach leads to coherent-mode decomposition of the much broader ten-parameter family of general anisotropic Gaussian Schell-model beams and, as a paticular case, to that of the twisted Gaussian beams. What may be more interesting, perhaps, is the fact that this approach leads quite simply to the first nontrivial twodimensional generalization of Factional Fourier Transform.

Lecture 3: This 'combined power' becomes also the natural setting to generalize the notion of shape-invariant propagation studied by Gori and collaborators [Opt. Commun. vo1.48, 7 (1893); Opt. Lett. vol.21, 1205 (1996)]. There are two types of shape-invariant propagatons: one in which the intensity ellipse representing the beam cross-section simply undergoes a scaling as a function of the propagation distance, retaining invariant its eccentricity as well as its orientation; and another in which only the eccentricity, and not its orientation, is maitained invariant. The 'combined power' leads to omplete characterization in respect of both types of shape-invariant propagations.

Note from Publisher: This article contains the abstract only.