Since we have teleconverters that increase the BFD (back focal distance) of a lens and magnify the image, would it be possible to design a teleconverter that decreases the magnification and therefore acts like a speedbooster for a Nikon FX lens to a DX sensor?
The nice thing about teleconverters is that they extend the effective focal length of the lens. That makes it easy to design them with enough thickness in front of the camera's mounting flange to fit all of the needed optics inside.
In the case of speed boosters, though, we are effectively reducing the lens' focal length, which means the lens elements in the booster need to be brought closer to the camera's sensor than the flange mount. Pretty much every speed booster I've seen is used to adapt a lens made for a longer registration distance to a camera with a shorter registration distance, i.e. a Canon EF lens with a 44mm registration distance adapted to a Sony camera with an 18mm registration distance. The speed booster occupies the space in the 26mm difference between the 44mm Canon EF mount and the 18mm Sony E-mount.
To create a speedbooster placed between a lens and camera with the same mount would require it to have an effective optical thickness of zero if infinity focus is desired. While this could be done using some of the same design techniques that go into making lenses with retrofocus designs, which have an effective focal length shorter than the camera's registration distance, it would necessarily be large, heavy, and expensive. You'd probably also have mirror clearance issues with cameras that have reflex mirrors. So far I've never seen anyone market such an adapter.
My proposal is to take a lens at a given focal length that was designed for an FX camera, ie. having an image circle to cover a FX sensor, and reduce the focal distance, ie. shrink the image to DX sensor size, so that the wide angle of the lens is unchanged on the sensor, the light will be compressed into a smaller area of the chip and therefore the lens will be 'faster'.
Your proposal requires the glass for the reducer to be between the lens' flange and the camera's sensor without moving the lens forward from its position against the camera's flange. There's no practical way to do that because:
- There's nothing on the back of the FX lens for a "countersunk" adapter inside the flange ring to attach to.
- That pesky little mirror would need to occupy the same space as the focal reducer inside the camera's light box.
It's not possible to use a focal reducer on a crop sensor camera with lenses whose flange focal distance is close to the native FFD of the camera because there is no space to put the focal reducer optics.
In addition to reducing the size of the imaging circle, focal reducers also decrease the FFD. So even if you could fit the optics in somewhere, the lens would need to be recessed into the mirror box, which won't work.