# Propagation law for Hermite- and Laguerre-Gaussian beams in first-order optical systems

## Martin J. Bastiaans and Tatiana Alieva

We consider the propagation of (one-dimensional) Hermite-Gaussian beams and (rotationally symmetric) Laguerre-Gaussian beams through first-order optical systems. Extensive use is made of the Wigner distribution, in terms of which the systems's input-output relationship takes the form of a mere coordinate transformation in phase space.

We start with a one-dimensional Hermite-Gaussian beam, described by its complex curvature (i.e. its curvature and its width, combined into one complex parameter). We determine the Wigner distribution of the beam and consider the propagation through a first-order optical system expressed in terms of its ray transformation (ABCD) matrix. From the particular dependence of the Wigner distribution on the phase space coordinates, we conclude that a Hermite-Gaussian beam preserves its Hermite-Gaussian character and that its complex curvature propagates according to the bilinear ABCD-law.

We then convert a Hermite-Gaussian beam into a Laguerre-Gaussian beam, using any appropriate Hermite-to-Laguerre mode converter. Since the mode converter is a first-order optical system, the Wigner distributions of the input and output beams are related through a mere coordinate transformation and have roughly the same form, which demonstrates once more the usefulness of the Wigner distribution.

Finally, a (rotationally symmetric) Laguerre-Gaussian beam, completely described again by a complex curvature, is considered with respect to propagation through isotropic first-order optical systems. Much like in the Hermite-Gaussian case, we observe that Laguerre-Gaussian beams keep their Laguerre-Gaussian character and that the bilinear ABCD-law remains valid.

To: Papers by Martin J. Bastiaans