Imaging of Optical Modes -- Resonators with Internal Lenses

The theory of Fresnel diffraction is the basis for an understanding of optical resonators1"5 and of optical modes of propagation. 234 Fresnel diffraction explains the mode patterns and diffraction losses of optical resonators, and the beam waist and spreading of the modes of propagation or "Gaussian beams." In this paper we will discuss how these Gaussian beams of light are transformed on their passage through lenses, telescopes, various lens combinations, and lenslike (guiding) media, and how these optical systems affect the properties of optical resonators when inserted between the resonator mirrors. We will assume that no additional aperture diffraction effects are introduced by these optical systems, i.e., that the apertures of the internal lenses can be regarded as infinitely large. The imaging laws of geometrical optics are therefore expected to apply, and we will use them 455 404 TIIE B E L L SYSTEM TECHNICAL JOURNAL, MARCH 1965 wherever possible, as they generally simplify the algebraic derivations and at the same time provide some physical insight. Some of the problems to be investigated here in greater detail have already been treated in the literature. Goubau6 has given some mathematical relations between the parameters of Gaussian beams transformed by a thin lens. The recently published mode matching formulae7 are the result of a computation which will now be presented. Resonators with internal lenses have also been discussed in the literature,8'11 and we have used the concept of an effective distance9,10 in a previous publication.9 In several cases an alternative to our algebraic approach is the graphical method of Collins,11 who introduced the circle diagram11-12 for Gaussian beams.