How do you measure the true f-number of a lens when the aperture is behind lens elements?

Measuring the f-number of a lens

The f-number or f-stop of a lens is defined as: $$\textit{f-number}=\frac{\textit{focal length}}{\textit{diameter of the entrance pupil}}$$ The entrance pupil is defined to be the optical image of the physical aperture, as seen by looking into the front of the lens. (The exit pupil is the image seen by looking into the back of the lens.) The entrance pupil is not necessarily the same size as the physical aperture: it may be larger or smaller than the physical aperture and it may be located in a different position (sometimes even outside the lens, either in front of or behind the lens).

Optically, the entrance pupil determines the amount of light that passes through the lens to form the image. Light rays that enter the lens, but do not go through the entrance pupil, play no part in image formation (they may contribute to lens flare, however).

You can see the entrance pupil by looking into the front of the lens when it is off the camera and there is a brightly lit background behind the lens. It is the brightly lit disc that you can see against the much darker surrounding parts of the lens.

The entrance pupil is the image of the aperture diaphragm (or other parts of the lens barrel that limit the passage of light through the lens). This image is usually a virtual image inside the lens, so it is not possible to put a ruler up against it to measure its diameter.

One way of measuring the diameter of the entrance pupil is to take a photo of the entrance pupil using a camera with a macro lens set to manual focus. Then, without changing the focus, take another photo of a ruler. You can then compare the photo of the entrance pupil with the photo of the ruler to determine the diameter of the entrance pupil. It is important that both the entrance pupil and the ruler are in sharp focus when these photos are taken.

Additional Note:

Another way of thinking about the entrance pupil is that it is the maximum size of a bundle of parallel rays entering the lens that can all pass through the lens and be brought to a focus on the other side of the lens. These rays must all be parallel to the optical axis (through the centre of the lens), hence from a point at infinity (on the optical axis).

This bundle of rays must all pass through the aperture diaphragm somewhere inside the lens. The lens elements in front of the diaphragm bend the light rays and typically reduce the diameter of this bundle of rays so that the aperture diaphragm may be much smaller than the diameter of the bundle before it entered the lens. Lens designers may do this so that the diaphragm can be made smaller and not occupy so much space in the lens. It also allows a smaller shutter mechanism if the shutter is inside the lens.

For working out the light-gathering power of the lens and depth of field, etc., it is the diameter of this bundle of rays before they enter the lens that really matters, not what happens to them inside the lens.

In principle, it is possible to generate this bundle of rays (but going in the opposite direction, i.e. into the back of the lens and out the front). If a point source of light is placed at the focal point of the lens, so that the light from this source enters the back of the lens, then the light will exit the front of the lens as a parallel beam whose cross-section is the exact size (and shape) of the entrance pupil.

In practice, it may be difficult to obtain a small enough source of light and to position it sufficiently accurately at the focal point of the lens. However, even if done approximately, it will give an indication of the size of the entrance pupil if the beam is projected onto a sheet of ground glass (or tissue paper) placed as close as possible to the front of the lens.