A general class of orthonormal sets of Hermite-Gaussian type modes is introduced, by generalizing the quadratic form that arises in the generating function of the common Hermite-Gaussian modes. We study how these modes propagate through first-order optical systems and express the generating function of the set of output modes in terms of the generating function of the set of input modes.
The requirement of orthonormality yields some additional conditions for the quadratic form of the generating function. As a result of that, we will be able to express the elements of this quadratic form in terms of four matrices that can be combined into a symplectic matrix.
The main result of the paper is that this symplectic matrix propagates through a first-order optical system by a mere multiplication with the system's ray transformation matrix. From this simple propagation law we can easily derive how different members from the class of the Hermite-Gaussian-type modes (such as the common Hermite-Gaussian and Laguerre-Gaussian modes) can be converted into each other.
The propagation law reduces to the well-known bilinear ABCD law in the case of Hermite-to-Hermite conversion (by means of a separable first-order system) and in the case of Laguerre-to-Laguerre conversion (by means of an isotropic first-order system). Knowledge of the generating function and in particular its propagation law may be valuable in the design of more general mode converters.
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