Rami Arieli: "The Laser Adventure" Chapter 8 Section 1, page 11a
Mathematical formulas of beam expander:
The relation between the beam diameters and the beam divergence angles is:
f1 = Focal length [m] of the input lens - ocular.
f2 = Focal length [m] of the output lens - objective.

d1 = Diameter of the input beam [m].

d2 = Diameter of the output beam [m].

q1 = Divergence angle (Rad) of the beam at the input to the beam expander.

q2 = Divergence angle (Rad) of the beam at the output to the beam expander.

From the equation it is clear that the ratio between the beam diameters is directly related to the ratio of the focal lengths of the lenses