CALCULATING A CORRECTING SURFACE

CALCULATING A CORRECTING SURFACE

A laser beam is refracted as it passes through a curved output mirror. If the mirror has a flat second surface, the waist of the refracted beam moves closer to the mirror, and the divergence is increased. To counteract this, laser manufacturers often put a radius on the output coupler’s second surface to collimate the beam by mak- ing a waist at the output coupler. This is illustrated by the case of a typical helium neon laser cavity consisting of a flat high reflector 

and an output mirror with a radius of curvature of 20 cm sepa- rated by 15 cm. If the laser is operating at 633 nm, the beam waist radius, beam radius at the output coupler, and beam half-angle divergence are

respectively; however, with a flat second surface, the divergence nearly doubles to 2.8 mrad. Geometrical optics would give the focal length of the lens formed by the correcting output coupler as 15 cm; a rigorous calculation using Gaussian beam optics shows it should be 15.1 cm. Using the lens-makers formula

  

with the appropriate sign convention and assuming that n = 1.5, we get a convex correcting curvature of approximately 5.5 cm. At this point, the beam waist has been transferred to the output cou- pler, with a radius of 0.26 mm, and the far-field half-angle divergence is reduced to 0.76 mrad, a factor of nearly 4.

Correcting surfaces are used primarily on output couplers whose radius of curvature is a meter or less. For longer radius output cou- plers, the refraction effects are less dramatic, and a correcting sec- ond surface radius is unnecessary.