To unlock these stages, simply clear each stage with an A rank or better for Stage Omega, or a C rank or worse for Stage Alpha. The best way to get a good ranking is to get all quickens as fast as possible, then make huge combos. The best way to get a bad ranking is to not do anything at all. Simply dodge the enemies and bullets, then defeat the boss. It's still alright to get quickens, but only quickens, no combos.

Effect - Code

Stage Alpha - Get a C rank or worse on each stage in arcade mode.
Stage Omega - Get an A rank or better on each stage in arcade mode.