If you rely on experience and experiment, then pupil ratio is not very important and can be ignored for most purposes. You can safely skip the remainder of this discussion.

If you use calculations to predict effective aperture, depth of field, exposure factor, or diffraction effects, then ignoring the pupil ratio can result in significant errors. This posting may help avoid to those.

I’ll start with a quick summary of results, then back those up with some discussion that complements what can be found in books and web pages.

**Quick summary...**

The f-number system for rating camera lenses is based on focal length divided by diameter of the entrance pupil. By itself, this gives an accurate indication of effective aperture only when the lens is focused at infinity.

When you extend a lens to focus closer, there is a standard simple formula to predict what happens: f_effective = f_nominal*(magnification+1).

However, this simple formula is in fact too simple. With many lenses, adding extension causes the effective aperture to change either faster or slower than the simple formula predicts. Because diffraction, DOF, and exposure correction all depend on the effective aperture, this means that as you extend such lenses to get higher magnifications, they act as if they had larger or smaller apertures than indicated by their rated f-number.

This sounds like the situation might get arbitrarily complicated, but it is a curious fact that one additional number provides everything you need to know. That number is the PMF -- "pupillary magnification factor" or just "pupil ratio".

The details of measuring and calculating with PMF are described in gory detail below. But the essential intuition is simple: when you extend a non-reversed lens to get higher magnification, its effective aperture depends more and more on diameter of its exit pupil rather than on diameter of the entrance pupil that is the basis for its rated f-number.

**Now, gory details...**

The notation that I’m using here follows Lefkowitz in “The Manual of Close-Up Photography” (1979), page 258.

**Symbol definitions**

**P**is the pupil ratio, computed as rear pupil diameter divided by front pupil diameter.**f_r**is the rated f-number of a lens. This is the value that’s marked on the aperture ring of a manual lens. It’s computed by the manufacturer as focal length divided by the diameter of the entrance pupil.**f_e**is the effective (working) f-number, as seen by the camera.**m**is magnification, measured as image size divided by object size.**TDF**is total depth of field, including both in front and behind the focus plane.**C**is the maximum tolerable circle of confusion, measured in the image plane.

**Formulas for effective (working) f-number**

**f_e = f_r*((m/P)+1)**gives the effective (working) f-number for an extended lens used in its non-reversed orientation, that is, in the same orientation that the manufacturer used to determine f_r. When other calculations such as DOF, exposure factor, and diffraction are expressed in terms of f_e, then incorporating P into this one equation is all that’s needed to account for its effects.**f_e = f_r*(1/P)*(1+(Pm))**gives the effective (working) f-number for an extended lens used in its reversed orientation. Note that f_r in this equation is still the number determined for the non-reversed orientation. The number f_r*(1/P) can be recognized as the effective f-number of the lens at infinity focus but reversed, with the usual “rear” pupil then acting as entrance pupil.

Notice also that when m=1, both equations give the same value. In other words, at m=1 the working aperture is not changed by reversing the lens.

**Formulas for depth of field**

**TDF = 2*C*f_r*((m+P)/(P*m*m))**gives the total depth of field for an extended lens used in its non-reversed orientation.**TDF = 2*C*f_r*((1+Pm)/(P*m*m))**gives the total depth of field for an extended lens used in its reversed orientation. Again, f_r is still the number determined for the non-reversed orientation.

Are these formulas exact?

Are these formulas exact?

Maybe yes, maybe no -- it depends on what you mean by “exact”. Here’s a more precise way of asking the same question: Remember that the classic equation f_e=f_r*(m+1) is significantly wrong for lens designs that have unequal pupils. Does f_e= f_r*((m/P)+1) completely account for differences in lens design, or does it omit other important features also?

The answer is that f_e= f_r*((m/P)+1) completely and exactly accounts for lenses that follow the “thick lens model”. This means it’s pretty good for most lenses other than fisheyes.

To see why this is true, let’s briefly review how apertures work.

Somewhere in every lens, there is a physical "limiting aperture" that selects which light rays get through the lens. Viewed from in front of the lens, the limiting aperture is seen as the “entrance pupil”; viewed from behind the lens, the limiting aperture is seen as the “exit pupil”. Due to refracting elements in the light path, the entrance and exit pupils usually do not have the same size and location as the physical limiting aperture. Nonetheless, they have some size and location, and with most lenses, the size and location of each pupil is fixed with respect to the lens as long as the limiting aperture does not change. (Fisheye lenses are an exception to this rule. Their entrance pupils move significantly depending on viewpoint.)

The effective (working) aperture of a lens really depends on the relationship between the entrance pupil and the object, or equivalently, between the exit pupil and the image.

In the simple “thin lens” model often used for illustrations, it happens that both pupils coincide with the aperture, and all of them occur at the single reference point from which focus is determined. This situation leads to the usual formula that f_e=f_r*(m+1).

Real lenses are more complicated, but most real lenses behave very much like a simple “thick lens” in which the thin lens’s single reference point for focus is just replaced by two points, one for measuring distances from the front and one for measuring from the rear. Using the “thick lens model” (see HERE), it turns out that the locations of pupils with respect to the focus points are completely determined by the ratio of pupil diameters. As a result, formulas that incorporate the pupil ratio can completely describe how a lens behaves, at least to the extent that the real lens conforms to the thick lens model in the first place.

To see how this works, consider what happens when we cast two carefully chosen rays past a single point on the edge of the lens’s limiting aperture. Each ray is parallel to the optical axis on one side of the lens, and thus (courtesy the thick lens model) appears to bend just once at a “principal plane” so as to pass through the focal point on the other side of the lens. Here is the diagram for a lens with exit pupil bigger than entrance pupil:

Each ray (one red, one green) represents the apparent position of the limiting aperture as seen from a different viewpoint. On the front, both rays pass by the edge of the entrance pupil, and the location and size of the entrance pupil can be determined by triangulation using those rays. Similarly the location and size of the exit pupil can be determined by triangulation using rays on the rear.

Changing some of the labels (to match those of http://toothwalker.org/optics/dofderivation.html), and slogging through the geometry of similar triangles, we get this diagram:

I’ll spare you the tedious details, but continuing forward from here, we can eventually reach the formulas listed earlier, as well as the ones shown at toothwalker.org. Tying this into another good page at http://photo.net/learn/optics/lensTutorial, note that the distance from H to E is f-f/P = f*(1-1/P). On that page, this distance is called “zE”, and as they say, “It can be shown that zE = f*(1-1/p)”. Yep, just did that.

Again, the main point of this discussion is only that the pupil ratio really does exactly capture everything that matters about asymmetry in a lens, at least to the extent that the lens obeys the thick-lens model.

Personally, I find this surprising. In fact I was far from convinced that the equations involving P were exact and complete, until I drew the diagram and wrote up the analysis given here. It’s a nice feeling when things finally “click”.

I hope this helps somebody else too!

Please let me know if you see any errors in this posting. It's a lot of math to get right.

--Rik

PS. As a dedicated fan of telecentric lenses, I have to warn you that the whole concept of pupil ratio falls apart with them. But that’s a topic for another day.

Edit, March 14, 2011: added quick summary.