Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams.

We present the Iwasawa decomposition theorem for the group ${\rm Sp}(2, R)$ in a form particularly suited to first-order optics, and we exploit it to develop a uniform description of the shape-invariant propagation of several families of optical beams. Both coherent and partially coherent beams are considered. We analyze the Hermite-Gaussian beam as an example of the fully coherent case. For the partially coherent case, we treat the Gaussian Schell model beams and the recently discovered twisted Gaussian Schell model beams, both of which are axially symmetric, and also the axially nonsymmetric Gori-Guattari beams. The key observation is that by judicious choice of a free-scale parameter available in the Iwasawa decomposition, appropriately in each case, the one potentially nontrivial factor in the decomposition can be made to act trivially. Invariants of the propagation process are discussed. Shape-invariant propagation is shown to be equivalent to invariance under fractional Fourier transformation