Gouy phase for higher-order Gaussian modes

Abstract

The Iwasawa decomposition of a ray transfer matrix and its unitary metaplectic counterpart is effectively used to solve for the evolution of a Hermite–Gaussian mode through a rotationally symmetric first-order optical system. Gouy phase as picked by an arbitrary Hermite–Gaussian mode with arbitrary width and quadratic phase on passage through a rotationally symmetric first-order optical system is obtained. The effect of width and quadratic phase of the input Hermite–Gaussian mode, position and momentum displacements, and composition of rotationally symmetric first-order optical systems, on the Gouy phase is obtained. It is seen that the Gouy phase acquired by a Hermite–Gaussian mode is intrinsic, and is retained even when there are position and momentum displacements on the mode while propagating through the desired optical system. The position and momentum displacements on the mode are seen to impart a transverse displacement to the mode, a tilt phase, and an overall displacement phase dependent on the system ray transfer matrix, over and above the Gouy phase and a propagation dependent phase. A method to tune the relative Gouy phases between the modes in a superposition by moving a single convex lens through a fixed distance, is outlined. The evolution of an arbitrary paraxial light field through any rotationally symmetric first-order optical system is in principle readily evaluated through the present method.