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Theory of Ellipsoidal Waves and Seidel Aberrations of Gaussian Beams

Hermite-Gaussian beam type solutions of the Helmholtz' equation are obtained by the method of separation of variables using the ellipsoidal coordinates. Reflection and refraction of ellipsoidal electromagnetic waves are investigated as vector boundary value problems, and first and third order theories are established in accordance with first and third order geometrical optics. the first order theory is formally identical to the conventional theory of mode transformation of Gaussian beams. In the third order theory spherical aberration is interpreted as a superposition of two Gaussian beams. The third order (Seidel) aberrations obtained in this paper include those of geometrical optics (Schwarzschild's perturbation eikonal) as a special limiting case.