The transformation of laser beams passing through optical systems is analyzed using a TEM modal decomposition. The laser beam is decomposed and the optical system is described by a matrix that relates the modal composition of the input and output beams. The hard aperture and the parabolic and Gaussian soft apertures are given a typical examples. The matrices that characterize these apertures allow analysis of the losses of laser resonators, including finite and/or variable reflectivity mirrors. The modal decomposition is used for defining modified ABCD matrices that include any kind of apertures.
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