By direct numerical-integration comparisons, it is established that the Fresnel approximation for collimated propagation is quite good (within about 2% in amplitude and 0.02 rad in phase) in every case, including that with the limit of a high Fresnel number. Moreover, the Fresnel approximation begins to break down in phase for spherical-wave propagation for beams faster than about f/12. It has been discovered, however, that if one also invokes the paraxial approximation, that is, replaces the spherical wave by a quadratic phase front, then the Fresnel approximation becomes valid for expanding (or diverging) beams as well. This result is substantiated through the use of stationary-phase arguments.

Validity of the Fresnel approximation in the near field

The linewidth of an unstable laser exceeds the quantum minimum by the Petermann factor K, which depends on the overlap between left and right eigenvectors of the (non-unitary) round-trip wave operator. When K is plotted as a function of the Fresnel number N, strong resonances occur, which are associated with degeneracies N c lying close to the real axis in the complex N plane. For certain values of the magnification, degeneracies can lie on the real axis, and K is infinite. The Horwitz-Southwell asymptotic theory of the spectrum, presented here in a very accurate form and with a streamlined derivation, is used to show that the peaks occur near N = s - 1/8 (s integer), and to give reliable formulae for the resonance widths Im N c and other features of the degeneracies. Low-lying resonances have discontinuous profiles associated with mode switching where the absolute values of the eigenvalues cross (these are not degeneracies).

https://michaelberryphysics.files.wordpress.com/2013/07/berry350.pdf

Mode degeneracies and the petermann excess-noise factor for unstable lasers

This paper looks back at how the basic concepts related to optical resonators and lensguides emerged during the first decade of the laser era.

Laser beams and resonators: the 1960s

We show that the eigenmodes of unstable-cavity lasers have fractal structure, in contrast with the well-known stable-cavity eigenmodes. As with all fractals, the dynamic range over which self-similarity holds is limited; in this case the range is set by diffraction, i.e., by the Fresnel number of the resonator. We determine the fractal dimension of the mode profiles and show that it is related to the aperture shape.

Fractal structure of eigenmodes of unstable-cavity lasers.

Unstable optical resonator with self-imaging aperture

The integral equation that describes the mode structure of unstable resonators of rectangular aperture is solved by a new method that makes use of an asymptotic expansion of the integral for large Fresnel number. This expansion is carried out to terms of order 1/F, where F is the Fresnel number, and includes effects of diffraction at the edges of the feedback mirror, as well as a term corresponding to the geometric-optics approximation. Results obtained by this method are in good agreement with those obtained by gaussian integration, even at an effective Fresnel number of unity. The method has the advantage that it involves finding the roots of a polynomial, rather than the eigenvalues of a matrix, and therefore requires less computer time and storage. The results of the present work are at variance with a class of theories based on geometric optics, as regards the higher-loss modes even for very large Fresnel number, although the lowest-loss symmetric mode is correctly given by geometric optics in this limit.

Paul Horwitz

Papers

Asymptotic theory of unstable resonator modes