Abstract

Optical-resonator modes and optical-beam-propagation problems have been conventionally analyzed using as the basis set the hermite-gaussian eigenfunctions ψ<sub><i>n</i></sub> (<i>x, z</i>)consisting of a hermite polynomial of real argument <i>H<sub>n</sub></i> [√2<i>x</i>/<i>w</i> (<i>z</i>)] times the complex gaussian function exp[-<i>jkx</i><sup>2</sup>/2<i>q</i> (<i>z</i>)], in which <i>q</i> (<i>z</i>) is a complex quantity. This note shows that an alternative and in some ways more-elegant set of eigensolutions to the same basic wave equation is a hermite-gaussian set ψ⌃<sub><i>n</i></sub>(<i>x, z</i>) of the form <i>H<sub>n</sub></i> [√<i>cx</i>]exp [-<i>c x</i><sup>2</sup>], in which the hermite polynomial and the gaussian function now have the same complex argument √<i>cx</i> ≡ (<i>jk</i>/2<i>q</i>)<sup>½</sup><i>x</i>. The conventional functions ψ<sub><i>n</i></sub> are orthogonal in <i>x</i> in the usual fashion. The new eigenfunctions ψ⌃<sub><i>n</i></sub>, however, are not solutions of a hermitian operator in <i>x</i> and hence form a biorthogonal set with a conjugate set of functions φ⌃<sub><i>n</i></sub> (√<i>cx</i>). The new eigenfunctions ψ⌃<sub><i>n</i></sub> are not by themselves eigenfunctions of conventional spherical-mirror optical resonators, because the wave fronts of the ψ⌃<sub><i>n</i></sub> functions are not spherical for <i>n</i> > 1. However, they resonator and beam-propagation problems.

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References

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  1. H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).

Kogelnik, H.

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).

Li, T.

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).

Other (1)

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).

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