A general procedure for treating solid-state laser unstable resonators with tapered reflectivity mirrors within the framework of geometrical optics is presented. A suitable choice of the analytical form of the reflectivity profile, which can simulate many experimental devices, makes possible the derivation of simple design equations for the resonator parameters. Experiments performed with a pulsed Nd:YAG laser containing a variable-reflectivity output coupler, implemented with vacuum evaporation techniques, are discussed. >

Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach

We consider four families of functions--the super-Gaussian, flattened Gaussian, Fermi-Dirac, and super-Lorentzian--that have been used to describe flattened irradiance profiles. We determine the shape and width parameters of the different distributions, when each flattened profile has the same radius and slope of the irradiance at its half-height point, and then we evaluate the implicit functional relationship between the shape and width parameters for matched profiles, which provides a quantitative way to compare profiles described by different families of functions. We conclude from an analysis of each profile with matched parameters using Kirchhoff-Fresnel diffraction theory and M2 analysis that the diffraction patterns as they propagate differ by small amounts, which may not be distinguished experimentally. Thus, beam shaping optics is designed to produce either of these four flattened output irradiance distributions with matched parameters will yield similar irradiance distributions as the beam propagates.

Laser beam shaping profiles and propagation.

A systematic theoretical and experimental investigation of unstable resonators with variable reflectivity mirrors of a super-Gaussian profile is presented. The validity of a geometrical optics formalism for an algebraic mode calculation is discussed in comparison to the diffraction theory. A new method for the calculation of the output energy by means of a closed-form relationship that accounts for the effects of the transverse mode profile on gain saturation is presented. Various dielectric super-Gaussian mirrors have been fabricated, characterized, and tested in a pulsed Nd:YAG laser. Super-Gaussian resonators can provide even higher output energies than those of traditional unstable resonators and with better beam-focusing properties. >

Lasers with super-Gaussian mirrors

The multi-Gaussian beam shape is proposed as a model for aperture functions and laser beam profiles that have a nearly flat top but whose sides decrease continuously. Beams and apertures of this type represent a simple, elegant, and intuitive alternative to super-Gaussian beams, which are important in a number of applications such as laser resonator design. Analytical formulas are developed for the propagation of these beams through free space and optical systems representable by ABCD matrices.

Propagation of flat-topped multi-Gaussian laser beams.

The azimuthally symmetric diffractive transverse eigenmodes in an unstable optical resonator with a super-Gaussian reflectivity output mirror are computed in the low-Fresnel-number region to quantify the limits of validity of the geometrical-optics solution. It is found that for low cavity magnifications (M ≤ 2) and with mirrors of super-Gaussian orders n ≤ 8, the diffractive eigenvalues deviate significantly from the geometrical-optics value for equivalent Fresnel numbers Neq 2. These new results indicate that for the geometrical-optics approximation to be valid, resonators with M ≤ 2 and n ≤ 8 should be designed with Neq ≥ 2. This will ensure that the cavity exhibits good azimuthally symmetric transverse-mode discrimination and a super-Gaussian fundamental mode.

Diffractive analysis of unstable optical resonators with super-Gaussian mirrors.

A new class of tapered reflectivity mirrors with a super-Gaussian profile R ∝ exp(−krn) is introduced, and a geometrical-optics approach for analysis and design of unstable resonators made with these mirrors is presented. A super-Gaussian mirror, built by a special evaporation technique, has been tested in an unstable resonator of a pulsed Nd:YAG laser, demonstrating its effectiveness in generating diffraction-limited beams.

B. Majocchi

Papers

Unstable laser resonators with super-Gaussian mirrors.