Technical shading @ Animal Logic. Special interest in lens simulation. To quote Matt Pharr: 'It seemed worth writing up at the time.'

On accurate computation of the f-stop

the constants

The following table of optical elements, taken from patent GB157040A, is standardized to a focal length of 1 inch ($\approx$ 25.4 mm). The patent describes an aperture of $f$/2.

Unrelated but interesting never the less, the uncoated Cooke Speed Panchro designed by Horace W. Lee in 1920 described below, is incredibly significant in the history of cinema.

radius thickness material ior abbe housing-radius
16.59 2.1234 abbe 1.6118 59 6.5
76.809 0.35306 air - - 6.5
11.3309 2.4765 abbe 1.6118 59 6
100000 1.06172 abbe 1.576 41 6
7.1856 1.7691 air - - 4.65
infinite 1.7691 aperture - - 4.65
-7.2872 1.06172 abbe 1.576 41 4.65

Note that the housing radii are not usually described in optics literature. In this case it was matched visually by overlaying the patent’s lens drawing with my own so small errors are to be expected


So, how do we compute the f-stop ourselves, given the constants above? Using the first equations we find on the internet, it seems straightforward:

where $N$ is the fstop, $f$ the focal length and $D$ the effective aperture radius.

However, quickly placing in the numbers shows that this can’t be the full story:

Note that we’re talking about the effective aperture here. This is the aperture as viewed from the sensor, which might be occluded by other lens elements. So, to take that into account we need to do some raytracing: In this case I start tracing parallel rays with increasing height until the ray is blocked by any of the lens elements.

We’re now interested in the position on the entry pupil of the last ray that was able to pass as this describes the effective aperture:

In this image I scaled the lens to a focal length of 100mm. This makes it a little bit easier to visualise. If you scale the radius of curvature, thickness and housing radius by the same constant, you can adjust the focal length of the lens by that same constant.

Plugging this value into the equation:

Closer.. But clearly still incorrect.

numerical aperture

Instead, the following equation for the numerical aperture should be used, which describes the range of angles over which the system can accept or emit light:

Substituting now, this brings us much closer:

I think the main idea to take away from this, is that there’s an issue regarding the nomenclature for this popular subject. The word f-stop is passed around rather carelessly. In optics literature, usually the numerical aperture is used. In photography, the f-number is used. It is incredibly confusing they have the same f/~ notation.

The same issue goes for the focal length definitions. The effective focal length [distance between intersection(ray, optical axis) and the principal plane] is not the same as the back focal length [distance between intersection(ray, optical axis) and the entry pupil vertex].

For subjects like these, let us please use explicit terminology.