Catadioptric Mirror Lenses
Whilst normally found in amateur telescopes the catadioptric (mirror) system is also used in some photographic lenses. Whilst the system never replaced the more traditional glass element based lenses they did generate their own cult following, which still lives to this day.
To understand why mirror lenses are less common than their element based counterparts we need to look at the design difference between the two.
The mirror lens has a very simple construction, using mirrors to bend light onto a smaller point and focusing it. The design isn’t perfect however, as you are losing a large portion of light by blocking the middle of the front of the lens. This leads to lower contrast, darker images and less light to work with at the camera end. So why would you use a mirror lens? The main benefits are size and weight, with mirror lenses being half the size and even less on the weight front. Digital manipulation can help to add contrast and brighten darker images.
The vast majority of lenses use glass elements to bend light, without blocking part of the lens with mirrors. The drawbacks? Weight and aberrations caused by the glass. The weight only really becomes an issue with larger lenses, and with the constant improvement in digital manipulation of images, the aberrations are easily rectified.
Mirror lenses are a very niche product, where size and weight are paramount. If you see a photo with donut bokeh (the out of focus area) then it was likely taken with a mirror lens.