Extension of the Jones matrix formalism to higher-order transverse modes

The well-known Jones matrix formalism, which can be directly applied to the propagation of the polarization of fundamental (TEM(00)) laser resonator modes, has to be modified for higher-order transverse modes. It is shown that this can be done in straightforward manner by using N x N matrices instead of the 2 x 2 Jones matrices, where N denotes the number of orthogonal polarization states of the transverse mode under consideration. The most common case of TEM(01) Hermite-Gaussian modes, where N is four, is discussed in detail.