How many megapixels? A worked-out example

Recently, my faithful Nexus 5X "died" of a disease that eventually affects everyone of its kind: the dreadful boot loop. The replacement was a Google Pixel 3a XL, and I was curious to know the true resolution of its camera.

The rationale of the following calculations can be found in the articles Depth of field vs sensor size and How many megapixels does your equipment have? The relevant specs of the Pixel camera are: aperture f/1.8, sensor size 5.76mm x 4.29mm.

First we calculate the "Airy disk" or diffaction disk. This is the sharpest spot or smaller circle of confusion the lens can project upon the sensor, with perfect focus.

coc = 2.44 . λ . f λ = average light wavelength coc = 2.44 . 0.5µm . 1.8 coc ~= 2.2µm

The result above hinges on two factors: the lens aperture (the wider, the better) and the constant 2.44, which is different accordingly to the version of the formula. The smaller the constant is, the more "optimistic" is the estimate. For example, in Wikipedia the adopted constant is 1.22.

Dividing sensor's width and height by the diffraction disk, we find a resolution of

h = 5760 / 2.2 = 2618px w = 4290 / 2.2 = 1950px mp = 2618 x 1950 = 5105100

In a first approximation, the effective resolution is 5.1 megapixels, not bad for a smartphone. But, as the abovementioned articles explain, we can consider that 1 circle of confusion = 1.5 pixel, so

mp = 2618 x 1.5 x 1950 x 1.5 = 11486475

We find the optical resolution (11.4MP) is almost the same as the sensor resolution (12.2MP). This is great; it means the sensor is well-matched with the optical system. We can shoot at maximum resolution and actually get pictures with such resolution, so we have room for cropping and other adjustments.