How would I turn a 60 degree beam pattern into as wide as possible?

Let's say I have a light source with a 60 degree light pattern 50mm wide and I wanted to spread that light pattern as wide as possible, like 120 or even 180 degrees. How do I learn about the lenses required to do that? I don't need to be very precise.

How would I source such a lens? What search terms do I use? Can you recommend a supplier? Shameless commercial links happily accepted.

I could cut custom polycarbonate lenses on my lathe but that is plan C.