I like the bokeh and limitations of mirror lenses. I however, also love my 50mm 1.4. Does a 50mm mirror lens exist? I searched but could not find one. If not 50mm what is the closest to 50mm one can get?
The mirror lens design resolves two key optical problems:
All lenses suffer from chromatic aberration. This is color fringing due to the failure of the lens to refract (bend inward) all colors of light accurately. In a conventional lens, this is accomplished by sandwiching two or more lenses, of different powers using different recipes for the glass. The result is an achromatic (without color error) lens. The mirror lens avoids chromatic aberration because it uses a “first surface” mirror. The mirror coat is thus on the surface of the primary lens. This configuration evades chromatic aberration because the image forming light does not transverse the primary lens element. Chromatic aberration gets worse if the lens is a powerful telephoto. Thus mirror lenses are favored for super long focal length lenses.
The mirror lens folds the light path thus a long focal length can be accommodated by a shorter lens barrel.
The 50mm lens is only about 2 inches in focal length. Correcting chromatic aberration in such a short lens is a piece of cake. Lens designers would never attempt to fold a 50mm and make the barrel super short except for a very specialize application. Bottom line, you are not likely to find a find a 50mm mirror lens.
You like the bokeh! It is created by the fact that the mirror lens uses two mirrors. The secondary is up front. It is a small silvered circular mirror centered in the tubes entrance. You can experiment and obtain nearly the same bokeh with your existing 50mm. Mount a UV filter and glue an obstructing circle on it's center. Cut some opaque disks, different sizes and using double stick tape, and affix one to the center of the UV. Now make a shot or two and then try a different size opaque center blocker. You might like to know, some famous portrait lenses of the past used a center opaque obstacle.
I don't know if anyone still cares about this, but you can get very short focal length mirror lenses. Schwarzschild microscope objectives use mirrors. They might have focal lengths ~0.5-5mm. These are highly corrected for microscopic work, not for conventional photography.
As others have mentioned, mirror lenses have a central obstruction, this would reduce the speed of the lens (light input) and also lower contrast. Also, the sharpest part of a lens is near the optical axis (the centre), with a large central obstruction, this region is lost.
Here is a patent by Tamron on a quite wideangle lens for cameraphone. (JP2016256557 JP2018109673A). https://worldwide.espacenet.com/patent/search/family/062721391/publication/CN108254859A?q=JP2016256557%20&called_by=epo.org
Widest example from patent has FOV of 23.6 degrees which corresponds to about 95 mm in FF terms. Of course, such design is impossible to use on a SLR camera due to short backfocus, but possible with MILC camera. If you find something wider, let me know. (except there are even wider speciality mirror lenses but with a blind spot or non-rotationally symmetric)