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/zR) form, with quantized values for the prefactor c. It is also shown that, as limiting cases, even two- and three-dimensional nondiffracting 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.
© 2004 Optical Society of America
(030.4070) Coherence and statistical optics : Modes
Riccardo Borghi, Massimo Santarsiero, and Rajiah Simon, "Shape invariance and a universal form for the Gouy phase," J. Opt. Soc. Am. A 21, 572-579 (2004)