How Does the Area of Sharp Focus Change Expand?

[Scott Rosenberg]

just wondering here... as a lens is stopped down, is the magnitude of the expansion of the area of sharp focus equivalent in both directions - assuming both standards are perfectly normal to the ground?

that is to say, if i painted a series of equidistant numbers, say from 1 - 10, on the road and opened up my lens, focusing on the number 5, as i stopped down the lens would one expect the area of sharpness to expand equally towards 1 and 10?

scott

[Scott Rosenberg]

yeah, i thought it was a 1/3 to 2/3 relationship... thanks.

[Dan Fromm]

Guys, if you do the arithmetic yourselves, you'll find that the distribution of depth of field about the plane of best focus, given focal length and aperture, varies with distance. There's no simple easy-to-remember rule of thumb.

jj, don't you dare flame me.

[Scott Rosenberg]

dan, no doubt that the distribution varies with distance, but does it always approximate a 1/3 : 2/3 ratio?

[Frank Petronio]

How about cutting jj some slack and welcoming him back? His answer is close enough for most working photographers and all I ever needed to know in practice... afterall, the only really true way to know if what you've got is in focus is to shoot a Polaroid 55 neg and loupe it, and even then you'd want to give a little more stopping down or prayer for film flatness variations.

[Dan Fromm]

Scott, it varies quite a lot. Ask me and I'll send you a spreadsheet that shows what's going on. Or save me the trouble and visit an on-line DoF calculator. Google will find them for you.

Frank, I HAVE cut jj considerable slack. I've even praised him as a good contributor -- he's been one -- when the gang piled on him. But I've received more abuse from him than I've wanted and I don't want any more.

And I agree with you about the difficulty of seeing what's in focus, or even how sharp anything is, on the ground glass.

[Michael Gudzinowicz]

No.

focus_point = 2 * near_limit * far_limit / (near_limit + far_limit)

y=focus_point - near_limit

z=far_limit - focus_point

Ratio: y/z = near_limit / far_limit

[Leonard Evens]

The 1/3 2/3 rule, i.e., the rear focus distance being twice the near focus distance, holds for just one distance, one third the hyperfocal distance. It would certainly never hold for reasonably distant subjects. For example, the rear focus distance is often infinite. In the close-up range, usually defined as within 10 times the focal length, the ranges in focus in front and behind the exact focus point are roughly equal. So what Scott proposed would work if the numbers were close enough to the lens.

One possible source of confusion is that the proper focus point on the rail is very close to halfway between the positions of near and far point on the rail. (See below.) That is what the standard optics calculation gives you assuming that you want equal size cocs on both sides of the exact focus point, but some people use other criteria and come up with a different answer. In any case, the relation between image distances and subjects distance in not a simple linear relation, so equal distances on the image side don't translate into equal distances on the subject side.

I don't understand Mike's formulas when discussing subject distances. I believe the proper distance to focus is a function not only of the near distance and the far distance, but also of of the hyperfocal distance. I can give the formulas, but I don't think they would be very useful. His formulas are correct for distances measured on the rail. They give the exact position to place the standard between the near and far focus points. But in almost all practical view camera photography, taking the midpoint is a close enough approximation. See my essay, www.math.northwestern.edu/~len/photos/pages/dof_essay.pdf, for more details.

This web page has an extensive discussion of focusing. As it makes clear, it is much simpler to concentrate on what is happening on the image side than on the subject side.

[Michael Gudzinowicz]

Leonard,

The formulas are not correct for the rail. The rail formula for the plane of focus is the midpoint between the positions for the planes of focus for the near and far depth of field limits.

The formulas (my preceeding post) refer to the subject distance (focus_point), the near limit of the depth of field (near_limit), and the far limit of the depth of field (far_limit). Unfortunately I was rushed when posting, so the formulas are cryptic.

focus_point = 2 * near_limit * far_limit / (near_limit + far_limit)

The subject distance is twice the product of the near and far limits divided by the sum of the near and far limits. The formula is published and easily derived from the dof equations:

y=focus_point - near_limit
z=far_limit - focus_point

"y" is the distance between the subject and near limit.
"z" is the distance between the subject and far limit.

Ratio: y/z = near_limit / far_limit

"y/z" is the ratio of distances in front of and behind the subject that are in focus.

That ratio is equal to the near limit of depth of field / far limit of depth of field.

Example: The near limit is 10 feet, and far limit is 30 feet. The subject distance focused upon will be 15 feet, and the ratio of the distance in focus in front of the subject to the "in focus" distance behind the subject is 5 feet / 15 feet which equals the ratio of the limits of depth of field, 10 feet / 30 feet.

Of course, if you set the far limit at infinity, the subject distance will be equal to the hyperfocal distance which is twice the distance to the near limit of depth of field.

And getting back to the original question, the ratio of distance in focus in front of the subject to that behind the subject is the ratio of the distances, and that ratio decreases as one stops down. In other words, depth of field behind the subject increases more rapidly than that in front of the subject, but the exact ratio depends upon the depth of field limits.

I've ignored diffraction to keep it simple.

Mike

[Leonard Evens]

Mike,

Mea culpa, mea culpa!

You are completely correct. My excuse---lame as it it is---is that I long ago decided concentrating on the subject distances was a distraction, and if I once knew the formula, I forgot it. And I got confused when trying to verify it quickly in my head when looking at your post, and somehow convinced myself it wasn't correct. At your prodding I derived it for myself, (in at least two different ways.) So I definitely put my foot in it that time.

I do NOT believe you are right about the point on the rail at which you should focus. This I did analyze carefully, and you can find the analysis in my aforementioned essay, as well as an explanation of why the difference between the midpoint and the exact point is negligible in almost all practical circumstances, which I find more interesting than the formula itself. (Unfortunately---as I just discovered---there is a typo in my essay. There are two quantities, v_1 and and v_2, and their product should appear in the numerator of the formula, but in the essay, they are both given as v_2. I will have to correct that.) Paul Hansma, in his original paper on choosing the optimal f-stop also notes that the formula is correct for distances on the rail. Moreover, from the lens equation, the formulas for the image position (on the rail) and subject position imply each other; if one is true, so must the other be true. Finally Jeff Conrad's careful analysis "Depth of Field in Depth" in this web page agrees that the same formula works both for subject distance and image distance.

I do have a couple of quibbles about the formula for subject distance

2 far_limit x near_limit/(far_limit + near_limit)

This formula doesn't work if the far limit is infinite, i.e., if you focus at or beyond the hyperfocal distance, because both numerator and denominator become infinite and it doesn't tell you where to focus. On the other hand, if you divide through by the far_limit you obtain obtain instead

2 near_limit/(1 + near_limit/far_limit)

so if you let the far_limit go to infinity, the denominator approaches 1, and the fraction approaches twice the near limit. That is consistent with the fact that as you approach the hyperfocal distance the far limit approaches infinity and the near limit approaches half the hyper focal distance. So you could conclude in that case that if the far limit is infinite, you should focus at twice the near_limit. If you then choose your aperture correctly, you will be at the hyperfocal distance for that aperture.

Secondly, in the close-up range, the proper subject distance at which to focus is very close to halfway between the near and far limits. Of course, in that case, the near and far limits are quite close to each other, so one could interpret the formula as being consistent with that. Thus, if you take the limiting ratio (in the mathematical sense) as the near and far limits approach one another, you get a ratio of one, suggesting the midpoint is correct. But, the formula itself just tells you that you should focus at the limiting value of both, which is not very helpful.

It should be noted that working with distances on the rail avoids both these problems.