LOCATING THE BEAM WAIST

LOCATING THE BEAM WAIST

For a Gaussian laser beam, the location (and radius) of the beam waist is determined uniquely by the radius of curvature and optical spacing of the laser cavity mirrors because, at the reflect- ing surfaces of the cavity mirrors, the radius of curvature of the propagating beam is exactly the same as that of the mirrors. Con- sequently, for the flat/curved cavity shown in figure 36.7 (a), the beam waist is located at the surface of the flat mirror. For a sym- metric cavity (b), the beam waist is halfway between the mirrors; for non-symmetric cavities (c and d), the beam waist is located by using the equation

 

and

 

where L is the effective mirror spacing, R1 and R2 are the radii of curvature of the cavity mirrors, and z1 and z2 are the distances from the beam waist of mirrors 1 and 2, respectively. (Note that dis- tances are measured from the beam waist, and that, by conven- tion, mirror curvatures that are concave when viewed from the waist

are considered positive, while those that are convex are considered negative.)

In any case but that of a flat output mirror, the beam waist is refracted as it passes through the mirror substrate. If the output coupler’s second surface is flat, the effective waist of the refracted beam is moved toward the output coupler and is reduced in diam- eter. However, by applying a spherical correction to the second sur- face of the output coupler, the location of the beam waist can be moved to the output coupler itself, increasing the beam waist diam- eter and reducing far-field divergence. (See Calculating a Correct- ing Surface.)

It is useful, particularly when designing laser cavities, to under- stand the effect that mirror spacing has on the beam radius, both at the waist and at the curved mirror. Figure 36.8 plots equations 36.7 and 36.8 as a function of R/z (curved mirror radius divided by the mirror spacing). As the mirror spacing approaches the radius of curvature of the mirror (R/z = 1), the beam waist decreases dra- matically, and the beam radius at the curved mirror becomes very large. On the other hand, as R/z becomes large, the beam radius at the waist and at the curved mirror are approximately the same.