Exact Analytical Solution for the Mutual Compensation of Astigmatism Using Curved Mirrors in a Folded Resonator Laser

For the first time, to the best of our knowledge, analytical expressions for mutually compensating the curved mirror astigmatism in the two terminal arms of a folded resonator are deduced and experimentally verified. The analytical expressions are derived using the theory of the propagation and transformation of Gaussian beams under few assumptions. Our exact analytical expressions describe the necessary and sufficient conditions for simultaneously compensating astigmatism in two arms of a sequence of two off-axis curved mirrors. The theoretical results indicate that, when the astigmatism compensation expressions are satisfied, the astigmatism introduced by a folded curved mirror can be mutually compensated by another folded curved mirror, even if there is no additional Brewster element inside the cavity. The astigmatism can only be successfully eliminated by using a pair of concave mirrors or convex mirrors. Our analytical expressions can be used to design astigmatically compensated folded resonators without a Brewster element. A typical side-pumped z-shaped cavity Nd:YAG laser is employed to demonstrate the astigmatic compensation. Experimental measurements of the pattern of the laser output beam show that not only the spot intensity profile deformation but, in addition, the phase distortion in the two terminal arms can be simultaneously compensated completely in the cavity, which is in good agreement with the analytical predictions