How do you maximize an electric engine's thrust?
Generally, if you want to optimize for area/diameter (without the complexity and weight of the many/multi stage option mentioned in comments to the question), you'll want to use a ducted fan (roughly similar to the fan on a turbofan).  Not only do you avoid tip losses due to the end plate effect of the duct (if the clearance isn't excessive), you can run the fan at arbitrarily high speed, even with supersonic tips, without the extreme drag and noise production of an open propeller in such a regime.

Look at the design of EDF (Electric Ducted Fan) engines used to simulate jets in small radio control models; some of these can product thrust comparable to fans driven by high performance methanol/nitromethane glow engines (15-25 kW/L displacement, with 10 cc, 1.5 kW engines driving fans 15 cm diameter), with 10-20 minute flight duration on a charge with light enough batteries to fly well.  Scaling these up to 500 mm diameter will generally require shortening the blade chord relative to diameter, and you'll need to optimize pitch for the combination of rotation speed and torque available.