Its not always square but in cheaper greens yes but remember the beam originates from the IR pump diode and may be mounted directly before the lasing crystal or its beam may pass through correction and collimating optics first and another lens after the crystal. In any case the optics are more complex in a DPSS than in a regular diode based laser.
Are you serious with this ?
If your bathroom mirror is square why is the reflection of your head still round ?
Squareness of the crystal has nothing to do with shape of the beam. It only changes the frequency of the wavelength.
The final beam shape has nothing to do with the shape of the pump diode, Mr. Livinloud. There are square pump diodes that emit a square beam, and yet the green beam is still round. The light doesn't just pass through and change color, it is absorbed by the gain medium and the gain medium emits its own light. A guasian dot (or other TEM modes) is simply a more efficient and more stable use of the cavity. One reason would be the intensity at the corners isn't high enough to cause stimulated emission.
oh i know, was just joking around since there was so much thought and effort put into the OP. 445nm are point and case, round diode yet rectangular beam