Yet another depth of field question

[Leonard Evens]

Misinformation about depth of field is rife. The problem is that the formulas, while not terribly complex, can't be summarized in simple rubrics using words. Almost all such statements are right in some circumstances and wrong in others.

I encountered this recently, and I did an analysis, which I think is correct, but I hope someone will check my calculations.

It is well known that for a fixed format and a relatively close subject, depth of field doesn't depend much on focal length, but mainly on the scale of reproduction. (Without the qualifications, this statement is definitely false.) But what happens if you fix the subject size and the image size in the final print, but vary the format. I come up with the conclusion that you get less depth of field (again with relatively close subjects) with a larger format, where it is presumed the larger format is enlarged less for a final print.

Here is the formula I came up with, and perhaps someone can check it.

Let N be the f-number, C the coc in the final print, E the degree of enlargement necessary to produce that print, and M_p the ratio of subject size to size of the image in the final print. Then I get

NC(E + M_p)/(M_p)^2

for the front and rear depth of field. C, N, and M_p are assumed fixed. So if E is smaller, you get less depth of field.

[Dan Fromm]

Leonard, you've backed into the well-known result that DOF varies inversely with magnfication, where magnification is the ratio of the size of the object on film to the size of the subject in front of the camera. If, that is, for each format you use a lens that will fill the frame; since your E varies inversely with format given final print size, there you are.

[Chad Jarvis]

This is why I threw away my calculator and bought a loupe. ; )

[Leonard Evens]

Dan,

I didn't quite back into the result. I started with it. The formula, for relatively close objects is

Nc(1 + M)/M^2

Where N is the f-number, c the coc in the film, and M the magnification. As you see, in that circumstance, it would be more accurate to say it varies inversely with the square of the magnification, although even that would be inaccurate for large magnifications as in close-up photography. Also, this formula also leaves out a factor in the denominator (1 +/- Nc/fM) where f is the focal length. So it is inaccurate unless Nc/fM is relatively small. The one case where these come together is the medium near distance as in portraiture.

Chad,

I'm afraid the loupe doesn't always get you off the hook. In the first place, if you are like me, you can't see much of anything if you stop down to f/22 or smaller, so it is hard to judge the DOF at the taking aperture. Also, using the loupe itself changes the apparent depth of field. What you would see in focus at 4 X is different from what you would see in focus at 8 X. What most people do is combine what they see on the ground glass with knowledge of what happened in their final prints or other images to guesstimate from the former what will happen in the latter. This is a perfectly reasonable way to proceed, particularly if your pictures are roughly similar in important characteristics. And the formulas would not give you a completely accurate answer anyway which would allow you to ignore what you see on the gg. But the formulas do play a role when unusual situations arise.

To my mind, the best way to make use of the gg information is to apply the focus spread method described elsewhere on this webpage.

[Emmanuel BIGLER]

Leonard : I agree with semi-DoF = Nc(1 + M)/M^2, but with M being defined as (object)/(image on film). Am I right ? Usually, at least in French textbooks, magnification factors are defined as (image)/(object).
Also : the formula is valid as soon as the DoF is small with respect to the object-lens distance ; which is valid for close-up right ? in other words when DoF limits corresponds to a small variation of the quantity delta(1/p), can be approximated to (delta_p) / p^2 = (semi-dof)/p^2.
And last, Leonard : do you agree that the general formula would be : 1/p1-2 = 1/p +- (Nc / f^2) (1/(1+M)) where p is the object-image distance and p1-2 the near-far DoF limits in the general case ?

[Leonard Evens]

Emanuel,

I agree with pretty much everything you said. I see in my first post that I mistyped the ratio M_p, which should have been the ratio of image size in the print to subject size.

The general formula you give looks right to me (from memory), but I would have to double to check to be sure. I won't bother because I think I can assume you got it right.

The formula I used is in Jacobson's len tutorial. (www.photo.net/learn/optics/lensTutorial). The formula valid at all distances for symmetric lenses (but ignoring diffraction) is

Nc(1+M)/(M^2(1 +/- Nc/fM))

(One needs to replace 1 + M by 1 + M/p in general, where p is the pupil magnification.)

To be completely precise, M is the ratio of the image size in the film plane to the subject size in the plane of exact focus, or alternately the ratio of the lens to film distance to the lens to plane of exact focus distance. For a real physical lens, these are of course measured from certain reference points associated with the lens.

c is the diameter of the circle of confusion in the film plane, N is the f-number, and f is the focal length. In the term 1 +/- Nc/fM, you use + for near DOF and - for far DOF. That term is close to 1 if Nc/fM is small, and the simpler formula Nc(1+M)/M^2 is valid in that case. For the typical lenses and f-numbers used in large format photography, Nc/f is very small, so if M is large enough, the Nc/fM term will be small. This happens in the close-up range, usually defined as a distance closer than 10 times the focal length, but one can still use the simpler formula at some distances beyond that, such as might be used in portraiture without being too far off. Jacobson uses that as a basis of his argument that DOF is more or less independent of focal length, in many circumstances. Of course, that is false as soon as M is small enough, i.e., subject distance is large enough.

[Emmanuel BIGLER]

Thanks Leonard for reminding us the excellent Lens Tutorial by David Jacobson.
Most classical formulae are scattered in old books out of print for a long time, some formulae are copied with errors in more recent books. It is nice to can count on a document like Jacobson's which summarizes everything in a convenient way.

I like very much the idea to define DOF with respect to the final print I have in hands and not with respect to some obscure rules, even if they were cleverly set by some beloved Masters half a century ago ;-);-)
For example if somebody aims at publishing a print @300 dpi on an A4 page, those final requirements should yield a much better DOF criterion and better choice of an effective C-value for the circle of confusion that what is actually engraved on vintage MF lenses (Ooops! sorry : we are LF-only here, fortunately, no DOF scales on LF lenses ;-);-)