Propagation of polychromatic Gaussian beams through thin lenses

The transformation by a lens of a polychromatic laser beam composed of on-axis superposed monochromatic TEM00 Gaussian modes in the paraxial approximation is studied. The chromatic aberrations are described by allowing the waist position on the z axis and the Rayleigh range to depend on wavelength. The beam radius, the far-field divergence, the Rayleigh range, the beam product, the beam propagation factor, and the kurtosis parameter are calculated. The relationship between the fourth-order and the second-order moments of Hermite-Gaussian and Laguerre-Gaussian modes is obtained and is used for calculating kurtosis parameter. The results are generalized to polychromatic modes of higher orders. It is shown that the on-axis superposition of monochromatic TEM00 modes with no chromatic aberration is leptokurtic.