To shine a logo on clouds in the night, what power lamp and what size should the outline be?

Say, the outline of a big bat. Visible by anyone within some largish urban area who looks up.

Just saw the movie, and the outline more or less was circumscribed by and on the surface of this large spotlight.

I just realized there’s no way that would work. Leaving aside the power law of illumination, simply the imaging wouldn’t focus. I don’t know the optics, but I’m pretty sure.

How would that image be best created? Either on a fog bank (distance? size?) or the base of a cumulus layer (distance? size?)?

you’re correct. you can’t just put a mask on a lamp and have the image of the mask project clearly anywhere. You’d need to have it as part of a lens assembly, and you’d need to know the distance to the surface/cloud/fog whatever to figure out what kind of lens you need.

all depends on how far away it is.

Actually, that might work OK.
The rays leaving a parabolic reflector are more-or-less parallel, so putting a mask in front of them would project an image.

I used a “Sun Gun” movie light and fresnel lens to light up low clouds as a kid - it probably projected to several thousand feet. A green laser is easily visible on clouds.

It’s been done.

Light Prints Time on Clouds

That’s right. 1930’s-style bat signals.

Laser light show gear might be a good choice.

This is just perfect:
Once engaged in the development of a death ray for possible military use, H. Grindell-Matthews, British inventor…

And from Wikipedia:
Harry Grindell Matthews

Harry Grindell Matthews (17 March 1880 – 11 September 1941) was an English inventor who claimed to have invented a death ray in the 1920s.[1]

Yep. That’s the connection. Interesting guy.

Here’s a 1933 story from Popular Science about movie lighting director Otto K. Oleson, who put together a stereopticon projector with a searchlight and sent the image 2,000 feet to light up the side of Hoover Dam with an American flag. The original image was 2.5" square.

I don’t know exactly what he used, but the articles indicate Oleson’s favorite tool was 875 million candlepower portable carbon arc searchlamps.

I’m thinking of the famous 9/11 light pillar annual memorial. Gonna look that up…

When I was in high school and teen dance clubs and the old Batman TV show were the rage (mid 60’s), a friend of mine (RIP, Brent) had what I will call a stencil of the bat symbol which he could project onto the ceiling of the club with a pocket flashlight. Everybody though it was cool, but I don’t think it would have gone much farther than that.