does the size of projected image circle matters in the size and weight of two lenses with the same aperture/DOF (different f-number) and field of view (different mm)?
So one lens is (say) a 20mm, f/2, 20 degree field of view and the other is a 40mm, f/4 20 degree field of view? The two lenses have the same field of view and same size entrance pupil (roughly, aperture). The 40mm lens must be longer if the designs are the same, but scaled.
From the perspective of optical design, the two are not equivalent. A 20 degree f/4 lens is much easier to design than a 20 degree f/2 lens, and the increase in aberrations from f/4 to f/2 (for some aberrations, this increase is 4^8, or 65,000 times) is much larger than the reduction in geometrical aberrations (exactly 2x) by scaling the focal length.
If the lenses are not allowed to touch the sensor, and must be shorter than some length, then the designs are in some way restricted. If, for example, the image clearance must be greater than 38mm, as for EF mount, then the 20mm design is far more difficult than the 40mm design because it is both a faster aperture, and also much more inverse-telephoto.
All other things equal, will a 40mm F4 full-frame lens be identical in dimensions and weight to a 20mm F2 Micro 4/3 lens? Or bigger? Or smaller? Why?
There is no single, general answer to this question. Allow me to add some qualifiers.
- The M4/3 lens has an image clearance > 16mm.
- The FF lens is for Sony E or Leica M mount, with image clearance > 18mm.
- The two lenses must have the same pictoral resolution (e.g. 20MP, though MP is an inappropriate unit for a lens).
Under these conditions, the M4/3 lens must be better corrected than the FF lens and thus should be a bit longer, but probably lighter. If you changed the apertures to, f/2.8 and f/5.6, the FF lens would probably be doable as a triplet or 4-element thing. The M4/3 lens might require 5 elements.
If you push things so that both are far from a diffraction limited regime, e.g. f/1 for M4/3 and f/2 for FF, the FF lens may become larger due to the clearance constraint. Or, more specifically, the ratio of the clearance constraint to the focal length.
As a side question, the design of both lenses has to be totally different?
They naturally want to be different, but they could just be scaled versions of each other with suboptimal performance for one or both designs.
Or is it just a matter of changing one or two glass elements at the base of the lens to adjust the projected image size?
You can do that, Angenieux offers it for some cine lenses, e.g. this one. I leave it to you to find the price of this lens.
In general, the "rear convert" or "speedbooster" design space is more restricted than complete freedom to let the APS-C and FF lenses be the distinct designs they want to be. For that to work, the booster and lens must be independently corrected, and you have to hope that the sign of their aberrations are opposite, and magnitude about equal. Or, the booster must be diffraction limited. This is quite difficult to do while maintaining a compact overall design.
Edit to address a new question
In the first part I understood that lenses for smaller sensors will be shorter because the smaller focal range but maybe heavier because more glass is needed to correct the aberrations.
This is correct; scaling the focal design by 1/2 will make it half the length, adding a lens or two to clean up the performance won't double it.
But then in the second part you said the M4/3 lens should be a bit longer and probably lighter? Is it longer because it must employ a retrofocal design that the 40mm didn't needed?
Correct again -- consider the double gauss lens, which has historically been used for most 50mm SLR lenses, and quite a few of the 40mm f/2 and 40mm f/2.8 ones as well. The focal length is 50mm and the lens' rear element sits (about) 38mm from the image1 - the principal plane is somewhere inside the lens.
In a telephoto lens, the prinncipal plane is generally in front of and outside of the lens. In an inverse-telephoto, it is generally outside of and behind the lens. In a "mild" retrofocus design it may still be inside the lens, but close to the back.
If you start with a double gauss-like design, the first step to improving its correction is to split (turn into 2) or compound (cement into a doublet) the rear group. Then the front, then do the rear again, repeat ad infinitum. This stretches the lens out, and you will run into the image clearance constraint and have to change to a retrofocus form. The retrofocus form is naturally longer than the very compact double gauss.
I said the M4/3 lens would probably be lighter because the manufactures tend to use lots of lighter materials in the mechanics of the lenses. Those are responsible for the bulk of the weight in a lens like this example.
1 EF mount has a flange distance of 44mm but this is measured to the larger flat surface, not the tip of the bayonette. Lenses such as the 85mm f/1.2 place elements as close as the tip of the bayonette, and thus the clearance requirement can be said to be 38mm.