Shape invariance and a universal form for the Gouy phase

It is shown that Hermite-Gaussian beams, Laguerre-Gaussian beams, and certain linear combinations thereof are the only finite-energy coherent beams that propagate, on free propagation, in a shape-invariant manner. All shape-invariant beams have Gouy phase of the universal c arctan(Z/Z(R)) form, with quantized values for the prefactor c. It is also shown that, as limiting cases, even two- and three-dimensional non-diffracting beams belong to this class when the Rayleigh distance goes to infinity. The results are deduced from the transport-of-intensity equations, by elementary means as well as by use of the Iwasawa decomposition. A pivotal role in the analysis is the finding that the only possible change in the phase front of a shape-invariant beam from one transverse plane to another is quadratic.