Do full frame sensors gather more light than crop sensors?

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.

Answer

EV is a measure of illuminance, which is defined in the link you provided as "luminous flux incident on a surface, per unit area". You are correct in stating that when if you keep field of view, depth of field and subject brightness constant:

Ev_crop = Ev_ff x c²

however since:

Area_crop = Area_ff / c²

and

Light(total) = EV x Area

we arrive at

Light_crop = Light_ff

In other words your APS-C system will collect more light per unit area of the sensor, however by virtue of a larger sensor a FF system will collect the same amount of light in total.

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However, when comparing systems in any practical sense you have to take lens availability into account. For a given full frame lens there may not exist a lens for APS-C with focal length c times shorter and f number c times lower.

From 135mm and up you can generally achieve equality in light gathering, let c = 1.6:

135mm f/2.0 -> 135/1.6 = 84.3, 2.0/1.6 = 1.25 -> 85mm f/1.2
500mm f/4.5 -> 500/1.6 = 312.5, 4.5/1.6 = 2.8 -> 300mm f/2.8

In the normal to short tele range the best you can hope for is to maintain the same f-stop, which means projecting the same amount per unit area onto the sensor, meaning the larger sensor gathers more light total.

     FF         APS-C
85mm f/1.2 -> 50mm f/1.2
50mm f/1.4 -> 30mm f/1.4

At the wide end lenses for full frame can be significantly faster, giving the full frame system more light per unit area and more area for a significantly greater light gathering ability:

     FF         APS-C

24mm f/1.4 -> 14mm f/2.8