I see we're not quite on the same page.The fast axis is the one with the higher divergence. It starts out with a smaller diameter but diverges quicker, forming a line in the far field. The slow axis starts larger but has a lower divergence and thus forms the narrow side of the rectangle in the far field.
The slow axis is smaller in the far field because the divergence is significantly lower.
There's a reason it's called a cylinder pair. Not sure how you'd form a beam expander with one lens.
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Broad area laser diodes
Broad area (or broad stripe) laser diodes are high-power laser diodes with a strongly asymmetric shape of the emitting region.www.rp-photonics.com
The fast axis is vertical....
The reason I suggested slowing the slow axis was to square up the beam only (with no implication to make a beam expander). If not, the rectangular or bar shaped beam profile (spot) is maintained and gets bigger with distance. Slowing the fast axis still leaves a bar shape beam profile (spot). So the OP wants to get rid of the bar shape beam spot and slowing the slow axis is the way to do that.