Modes in unstable optical resonators and lens waveguides

Abstract

Optical resonators and/or lens waveguides are "unstable" when they have divergent focusing properties such that they fall in the unstable region of the Fox and Li mode chart. Although such resonators have large diffraction losses, their large mode volume and good transverse-mode discrimination may nonetheless make them useful for high-gain diffraction-coupled laser oscillators. A purely geometrical mode analysis (valid for Fresnel number <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N = \infty</tex> ) shows that the geometrical eigenmodes of an unstable system are spherical waves diverging from unique virtual centers. As Burch has noted, the higher-order transverse modes in the geometrical limit have the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u_{n}(x) = x^{n}</tex> with eigenvalues <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\gamma_{n} = 1/M^{n+1/2}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> is the linear magnification of the spherical wave per period. The higher-order modes have nodes on-axis only, and there is substantial transverse-mode discrimination. More exact computer results for finite <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> show that the spherical-wave phase approximation remains very good even at very low <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> , but the exact mode amplitudes become more complicated than the geometrical results. The exact mode loss versus <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> exhibits an interesting quasi-periodicity, with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n = 0</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n = 2</tex> mode degeneracy occurring at the loss peaks. Defining a new equivalent Fresnel number based on the actual spherical waves rather than plane waves shows that the loss peaks occur at integer values of N <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">eq</inf> for all values of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> .