Depth of Field Revisited

by Harold M. Merklinger


Nearly all lenses - especially those for 35 mm and medium format cameras - come equipped with depth-of-field scales. The purpose of these scales is to provide the photographer with guidance on where a subject needs to be in order to be rendered sharply in the image. We can also learn about depth of field in any of a number of good books on photography. We are often treated to copious tables, or to formulae derived for special applications, such as close-ups.

The frequency of questions about depth-of-field in magazines, and my own experience, suggest to me that something is a little bit wrong with the official story. When I trusted depth-of-field scales, I was very frequently dissatisfied with the results. In scenic photography subjects in the distance were almost always substantially fuzzier than I wanted. I am happy to report that there is a solution to the problem, and it does not rely on either depth-of-field scales or tables. You can do it in your head. Then again, you often don't even need to think at all.

Before I describe the solution, let me say what's wrong with the existing philosophy. There's nothing wrong with the logic or the math. It's the basic assumptions that have been surpassed by technology. The starting point of the existing theory was that there is a limit on the resolving power of films and lenses. That limit was said to yield a smallest spot having a diameter equal to one-fifteen-hundredth of a normal lens's focal length. Thus the maximum permissible "circle of confusion" is 1/30 mm in diameter for a 35 mm camera with its 50 mm standard lens.

The problem is that today's films and lenses are capable of achieving a resolution standard at least five times as stringent, and maybe more. But if one enters that revised standard into the formulae, depth of field just about disappears. And that doesn't square with experience either.

The honored way out is to say: well, that's true, but the human eye also has its limits. The average eye can't see anything smaller than one one-hundredth of an inch on an 8 in. by 10 in. photograph, it is said. And that translates into, you guessed it, 1/30 mm on film. I don't pretend to have super eyes, but that 1/30 mm standard doesn't always look sharp to me on a 3.5 by 5 in. print, let alone an 8 by 10. I don't think photographs meeting that standard will win many sharpness competitions today.

What bothers me even more is the old story about maximizing depth-of-field by focusing at the hyperfocal distance. If you follow that advice you will be guaranteed that scenes in the distance will never be resolved any better than mediocre. You will have sealed in that "minimum acceptable standard".

Another Way to Think about Depth of Field

So what do we do? We start by reviewing how that circle of confusion was determined. The cone of light concentrated by a lens at it's focus is intercepted by a film not quite in the right place. The intersection of the cone and the film is a circle: the circle of confusion. And the size of the circle can be determined by simple geometry, as illustrated in Figure 1.

Figure 1: Light from a tiny object is focused some distance behind the lens. If the film is closer to (or farther from) the lens than required for perfect focus, the image will be a small circle. The diameter, c, of that "circle of confusion" depends upon the focus error e, and the focal length, f, of the lens as well as the diameter of the lens opening, d.

This same scheme can equally be applied in front of the lens, as shown in Figure 2. The rays of light from a small object in perfect focus also form a cone, with the lens at the base of the cone. If some other object is not quite in the right place, it intercepts that cone where it has some measurable diameter. That diameter determines the smallest detail the lens can resolve at that location. So simple geometry can tell us what objects can or cannot be resolved.

Figure 2: We can reverse Figure 1 to determine how large an object must be in order to be resolved in the image. The resolution spot size, S, will depend upon the focus distance, D, the focus error, E, and the diameter of the lens opening, d. An object smaller than S will be enlarged, indistinct and reduced in contrast.

The rule for determining what objects are resolved is ultra-simple. We focus our lens exactly at some distance, D, from our lens. An object one-tenth of the way back from D, towards our camera (that is, at a distance of 0.9 D), will be resolved if it is at least one-tenth as big as the opening in our lens diaphragm. If the object is one-quarter of the way from the point of exact focus to the camera lens, it will have to be one-quarter as big as our lens to be resolved. And so on. The same rule will hold on the far side of that point of exact focus also. An object twice as far from the lens as the point of exact focus will have to be as large as the lens aperture to be resolved. For those who would prefer the math formula, it is included with Figure 2. In this formula d stands for the lens diameter, and S stands for the "spot size" - how big an object must be to be resolved. Distance either side of the point of exact focus is given the symbol, E.

Now it's important here to understand that the "size of the lens" is the diameter of the lens diaphragm at it stopped-down or working position. If we are shooting at f/8 with a 50 mm lens, that diameter is 50 divided by 8 millimeters: 6.25 mm, or about one-quarter of an inch. For a 300 mm lens at f/5.6, the diameter would be 300 divided by 5.6, or 53.6 mm-about two inches.

What this rule does, is give the photographer direct control of the problem. The photographer can decide what must be resolved, and what need not. Or, he can use the rule to ensure something will be intentionally blurred out and unrecognizable. As the spot size increases relative to the size of the object being photographed, the image of that object grows in size and fades in contrast.

Caption, Figure 3: This scene, the village of Placentia in Newfoundland, was taken with a 90 mm lens set at f/8 and focused at infinity. As explained in the text, to have focused at the hyperfocal distance would have seriously degraded the image of the village, while making negligible improvement to the foreground. (Sorry, no GIF version - just too big as a GIF.)

Let me illustrate with an example. Figure 3 shows a photograph of the Newfoundland village of Placentia. In the foreground are a cannon, some grass, some gravel, and some trees. The picture was taken with a 90 mm lens on a Leitz-Minolta CL at f/8. The aperture is thus about 11 mm in diameter. The hyperfocal distance for a 90 mm lens at f/8 is 106 feet. The cannon is at a distance of about 30 feet, and the distance to the village is about a mile: 5280 feet. Had I focused at the hyperfocal distance, the resolution spot size at the cannon would have been 8 mm and at the village it would have been 0.55 meters or about 21.5 in. If, on the other hand, I focus at infinity, the spot size is 11 mm at the cannon and 11 mm at the village. Or 11 mm anywhere else, for that matter. True, diffraction effects won't let me resolve 11 mm at 5000 ft., but at least resolution at the village will be as sharp as nature permits. The change in spot size from 8 mm to 11 mm at the cannon would be almost impossible to detect in the image. The foreground is admittedly not tack-sharp. But I can recognize the cannon, the gravel, the grass and the trees. Had I focused at the hyperfocal distance the telephone poles in the village would have been almost erased, and windows in buildings would have been just blurs.

A corollary to this example is that there is really only one way to make the foreground sharper, and still have a sharp view of the village. That is to use a smaller lens opening. In this example, I could have reduced the spot size to 5.5 mm by using f/16. Resolution of the village, however, would have been slightly degraded by diffraction effects.

The general rule for scenic photographs, where one wishes to maximize the depth of field, is as follows. Set the focus at the distance of the most distant object. Then set the lens opening to the size of the smallest object to be resolved in the foreground. No calculations needed!

The rule for estimating diffraction effects is also easy to express. The diffraction limited spot size is given by D/1600d, where D is the distance to the village in this case, and d is the diameter of the lens opening, in millimeters. The units of measure for the spot size are whatever units we use to express D. So, for D=5280 ft., and d=11 mm, we get the result that the diffraction limited resolution at the village is 0.3 ft or about 4 inches. That is good enough to resolve telephone poles and window frames. Had I used f/16, the spot size at the village would have increased to 0.6 ft., or about 7 in. That would resolve telephone poles, but window frames would have been a bit blurred. All-in-all, this might actually have resulted in a slightly superior photograph. But it was quite windy that day; I wanted to use a higher shutter speed.

As a general rule, lens aperture diameters of 3 to 5 mm are often the optimum for scenic purposes. Blades of grass and small branches are resolved, and resolution of distant objects is about equal to that allowed by the majority of modern films and lenses. This rule applies to all formats.

Another useful application for this approach to depth of field is in telephoto close-ups-as of animals or flowers. With long lenses and large apertures, depth of field scales are just about useless. But with just a little bit of simple math we can figure out how much of that muskrat will be resolved. If we are using an 800 mm f/8 lens to photograph a muskrat at 25 feet, we have the following situation. The lens working diameter is 100 mm. In order to resolve features of the muskrat 1 mm in size, that detail must be within 1% of 25 ft (answer: 3 in) of the point of exact focus. If the muskrat is looking at us head on, and we focus on his eyes, his tail 1 foot behind the eyes will be fuzzy, but will still be resolved. The reasoning is that the resolution one foot past the point of exact focus is one twenty-fifth of 100 mm, or 4 mm. The muskrat's tail is just a bit bigger than this. We won't see every wet hair in his tail, but we will see the tail.

Simple, isn't it!

© Harold M. Merklinger, July 1992.

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