A recent article from Petapixel about the fastest f-number theoretically possible inspired me to make the following observations about the relationship between f-number and light-gathering ability.

Recall that the f-number is defined as the ratio of the distance between the image plane and the exit pupil, to the diameter of the exit pupil. For an idealized thin lens, this is equivalent to the focal length divided by the aperture diameter.

Also familiar to most of us is the idea that light-gathering ability doubles for each full stop increase in speed; i.e., f/1.4 gathers twice as much light as f/2, which in turn gathers twice as much light as f/2.8, etc.

When I read the Petapixel article and the commentary therein, it seems that this doubling/halving notion of full stop increments was overlooked and not fully understood. That is to say, while this relationship between f-number and light-gathering ability is true at relatively slow f-numbers, it is most certainly NOT true at fast f-numbers, because it is an approximation. f/0.5 is not, strictly speaking, twice as fast as f/0.7, nor is f/0.35 twice as fast as f/0.5, even though the area of the pupil is twice as large.

The true light-gathering ability of a lens is measured not by the f-number but by the square of the angle of incidence of the marginal paraxial rays to the optical axis. If we think of a light source at infinity, its rays enter the lens (which is also focused at infinity) parallel to the optical axis. Off-axis rays get refracted to form an image of the light source, and the angle of the cone of light that forms that image point determines the intensity.

In the small-angle case, the tangent of the angle is a good approximation of the angle itself: theta = tan theta for "sufficiently small" theta. But if the angle is large--as in the case of very fast lenses--this approximation breaks down, and we can no longer use the f-number as a reliable indicator of how much light is being gathered. For example, compared to f/1.0, an f-number of f/0.7 is only 1.76 times brighter; f/0.5 is only 1.63 times brighter than f/0.7; and f/0.35 is only 1.48 times brighter than f/0.5. Of course, it is extremely rare to see even the first case, and we certainly don't see the others, but in the theoretical discussion of the fastest possible lens, it is important to point out that even a theoretical f/0 lens (for which the lens diameter is infinitely large, and the marginal rays somehow manage to make a 90-degree refraction and strike the image plane with no loss), the light-gathering ability is finite. f/0.35 does not get us 32 times the light-gathering ability of f/2, but only about a 15x increase, even assuming an ideal lens and sensor. And this is even true at slower f-numbers, though to a much smaller extent: f/1.0 is really only 13.9x as bright as f/4, not 16x brighter as you would expect just by counting full stops.