Modes of arbitrary order in inhomogeneous laser resonators

An analysis is made of modes of arbitrary order in resonators containing elements with square-law dependences of the refractive index on the transverse coordinates. It is shown that a calculation of the field in such resonators reduces to the application of two transformations. One of them represents the wellknown ABCD law and describes the transformation of the beam variance. The other describes the transformation of the complex argument of Hermite polynomials which occur in the expression for the resonator field. This transformation may be omitted if the medium is loss-free but its use is essential in the case of media characterized by inhomogeneous gain. The proposed calculation method makes it possible to determine uniformly all the mode characteristics for a wide range of resonators, including unstable ones.