Lots of people say that full frame sensors receive more light than cropped sensors. I have never found a proof of this claim so I tried to do the computation by myself, and proved the contrary! Could you tell me if I am wrong ?
We want to compare the same frame with the same depth of field. I'm not concerned by the quantity of photons by photosite, which is unrelated to sensor size but to the density of pixels. I have neglected effects like vignetting or angle effect on the edge of the micro-lens array. Here is my simple reasoning :
If you want the same angle of view α~2arctan(size/2f) with a full-frame sensor and a crop sensor with crop ratio c, you have to multiply the focal length by approximately c. Now, in order to maintain the same depth of field, the f-number N have to be divided by c. If we measure the "amount of light" with the well defined Illuminance Ev provided by the same frame of the same scene (so the luminance is fixed), we have Ev ~ f/N.
Putting all together, Ev_crop = Ev_ff x c², so the cropped sensor receive more light than the full frame sensor !
For those who are interested in the price of two equivalent systems, one with a FF+50mm+135mm and the other with Crop+35mm+85mm, see this example.