Infinity Focus

In landscape photography, it is often recommended to focus at the hyperfocal distance to maximize the scene in focus. But not everyone agrees.

Harold Merklinger, a keen photographer with a sound background in the sciences, wondered why his images taken with the hyperfocal distance didn't really meet his expectations. In particular, he noticed that the more distant objects often appeared substantially fuzzy. The following is mostly based on his studies of the subject [Merklinger 1992, Merklinger 2002]. The basic problem is illustrated in fig. 7.

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Fig. 7.

Objects not in focus (i.e. located before or behind the object plane) will need a certain size to be visible in the image.

Consider a lens with a focal length f focused at some distance g + h, i.e. object distance g and image distance h are adjusted according to the lens equation. In this case, the rays emitted from a point in the object plane towards the lens form a cone, as illustrated by the purple rays.

If the object is somewhat closer or farther away, it intersects the cone where it has some finite diameter s. If the object is smaller than s, only part of the rays reaching the image plane at the original point will come from the object; the recorded intensity will thus depend on some other contents of the scene as well. In other words, the image of the object gets blurred. The diameter s thus determines the smallest detail the lens can (surely) resolve at that distance.

In the following, the diameter s is called the (minimum) spot size. It can be calculated as follows. For an object located at a distance gn closer to the lens and a lens aperture a, we get

s / a = (g - gn) / g (I1)

Solving for s gives

s = (1 - gn / g) a (I2)

For an object located near the focusing distance (gng), the object can thus be very small and still be resolved. If the object is somewhere near the camera (gn ≈ 0), its size must be about the aperture a of the lens.

If the object is located at a distance gf farther away from the lens, we are facing a more serious problem. In this case, we get

s / a = (gf - g) / g (I3)

Solving for s results in

s = (gf / g - 1) a (I4)

The minimum spot size is thus not limited, but will grow proportional to the distance gf. Objects may thus completely disappear, depending on their size and distance.

Merklinger's approach

Fortunately, this problem can be avoided if we focus at infinity. In this case, all objects in the scene are closer than the focusing distance, such that equation (I2) applies. Furthermore, the fraction disappears, and we get a surprisingly simple description of the spot size

s = a (I5)

Note that the aperture a of the lens is a distance which can be measured in mm, inches, or any other unit of length. With equation (A1), we can express a with the commonly used f-stop A and get

s = f / A (I6)

For a camera focused at infinity, the minimum size of an object to be resolved thus depends on the focal length of the lens and the f-stop used, but not on the object distance. This formula is used for the resolution spot size calculator.

For example, consider a lens with a focal length of 50 mm and the f-stop set to 8. The circle of confusion may be assumed as 0.033 mm. The hyperfocal distance is thus at about 9.6 m (31 ft). In this case, we get a spot size of just 3 mm (0.1 in) at a distance of 5 m, but a whopping 600 mm (2 ft) at 1000 m. Anything smaller than that will more or less disappear. On the other hand, for the lens focused at infinity, the spot size is about 6 mm (0.2 in), regardless if the object is 5 m or 1000 m away. While the difference at 5 m is rather small, the effect at 1000 m may be quite dramatic.

Practical considerations

The good news about Merklinger's approach is that it does not only provide sharp images, but is also very easy to use. He actually recommends to focus at the most distant object of interest, and adjust the f-stop to the size of nearby objects that you still want to see. From equation (I6), the f-stop can easily be calculated as

A = f / s (I7)

In his publications, Merklinger also discusses other uses of his approach, as in wildlife photography.

Note that equation (I6) describes a lower limit for the resolved object size only, as other limitatons such as the finite resolution of the image sensor and diffraction also apply. However, this focusing strategy ensures that we can get as good as lens and camera permit. Also note that neither the circle of confusion nor the size of the image sensor are used, so results are valid for all kinds of cameras alike.

Hyperfocal distance revisited

Merklinger's approach leaves us with the somewhat disturbing realization that the hyperfocal distance, even though based on a seemingly solid theory about the depth of field, does not lead to optimal results. So what's wrong?

Simply put, it is optimizing the wrong parameter. The aim of the hyperfocal distance is to ensure that the picture of a point is never spread beyond a certain size on the image plane. However, as Merklinger showed, this is not the major problem, as more distant objects will be completely smoothed out. In other words, the point may be limited in size, but this doesn't really matter if it's not visible. Actually, it makes much more sense to ensure that we can see it at all.

So do I use the hyperfocal distance? No, never.