Depth of field

In optics, particularly film and photography, the depth of field (DOF) is the distance in front of and behind the subject which appears to be in focus. For any given lens setting, there is only one distance at which a subject is precisely in focus, but focus falls off gradually on either side of that distance, so there is a region in which the blurring is tolerable. This region is greater behind the point of focus than it is in front, as the angle of the light rays change more rapidly; they approach being parallel with increasing distance.

Definition of "focus"

Several factors determine whether the objective error in focus becomes noticeable. Subject matter, movement, the distance of the subject from the camera, and the way in which the image is displayed all have an influence. However, the most important factor is the actual degree of error in relation to the area of film exposed.

Light from a point source at the correct distance will produce the image of a point on the film. A point farther away or nearer will produce an image in the form of a disk known as a "circle of confusion." The diameter of these circles increases with distance from the point of focus and so can be used as the measure of error or blurring of the image.

For a 35 mm motion picture, the image area on the camera negative is roughly 0.87 by 0.63 in (22 by 16 mm). The limit of tolerable error is usually set at 0.002 in (0.05 mm) diameter. For 16 mm film, where the image area is smaller, the tolerance is stricter, .001 in (0.025 mm). Standard depth of field tables are constructed on this basis, although generally 35 mm productions set it at 0.001 in (0.025 mm). Note that the acceptable circle of confusion values for these formats are different because of the relative amount of magnification each format will need in order to be projected on a full-sized movie screen.

(A table for 35 mm still photography would be somewhat different since more of the film is used for each image and the amount of enlargement is usually much less.)

Another factor to be considered is that the film format's size will affect the relative depth of field. The larger the area of the film is, the longer a lens will need to be to capture the same framing as a smaller film format. In motion pictures, for example, a frame with a 12 degree horizontal field of view will require a 50 mm lens on 16 mm film, a 100 mm lens on 35 mm film, and a 250 mm lens on 65 mm film. Conversely, using the same focal length lens with each of these formats will yield a progressively wider image as the film format gets larger: a 50 mm lens has a horizontal field of view of 12 degrees on 16 mm film, 23.6 degrees on 35 mm film, and 55.6 degrees on 65 mm film. What this all means is that as the larger formats require longer lenses than the smaller ones, they will accordingly have a smaller depth of field. Therefore, compensations in exposure, framing, or subject distance need to be made in order to make one format look like it was filmed like another.

Hyperfocal distance

The hyperfocal distance is the nearest distance at which the far end of the depth of field stretches to infinity. Focusing the camera at the hyperfocal distance results in the largest possible depth of field. Focusing beyond the hyperfocal distance does not add depth of field to the far end (which is already at infinity), but it does subtract from the focus area in front of the hyperfocal point. Therefore there is less total depth of field. Likewise, focusing ahead of the hyperfocal distance results in a gain of focus area ahead of the focus point but loses some of the focus area beyond the focus point including the subjects near infinity.

Depth of field formula

Let H be the hyperfocal distance (calculated below from N, the f-number, and c, the circle of confusion for a given film format), let s be the distance at which the camera is focused, let f be the focal length, let D_F be the distance from the camera to the far limit of depth of field, and let D_N be the distance from the camera to the near limit of depth of field. Then depth of field (DOF) is given by

DOF = \Big(D_F - D_N \Big) \mbox{, where }

H = \frac {f^2}{N \cdot c}

D_F = \frac {H \cdot s}{H - s} \mbox{ for } s < H

D_N = \frac {H \cdot s}{H + s}

Combining, the depth of field formula becomes

DOF = \frac {2\cdot H \cdot s^2}{H^2 - s^2} \mbox{ for } s < H

At the hyperfocal distance, the subject distance s that produces maximum depth of field, the limits are:

D_F = \Big(\infty \Big)

D_N = \frac {H}{2}

\mbox{ for } s=H

Thus for a given film format, depth of field is calculated from three factors: the focal length of the lens, the f-number of the lens opening (the aperture), and the camera-to-subject distance. While it is commonly said that lenses of short focal length have greater depth of field than long lenses, this rule of thumb is not strictly true because it takes into account only one of the three factors. In fact, for a given subject framing and aperture, lenses of all focal lengths have approximately the same depth of field. This is because subject framing is dependent on two of the factors (focal length and subject distance), while aperture is the third. Once the three factors are set in a fixed proportion, the depth of field will be almost the same.

An example makes this easier to understand. Take a photographer using a 400 mm lens to shoot a subject (for example, a bird) 10 metres away. Assuming an aperture of f/2.8, the depth of field of this shot would be 10 cm. Should the photographer now switch to a 50 mm f/2.8 lens, the depth of field at 10 metres is now 7.62 metres. However, once the photographer has moved to 1.25 metres from the bird, being the distance required such that the bird fills as much of the frame as it did with the 400 mm lens at 10 metres, the depth of field is almost exactly the same as before, 10 cm. In some cases, though, the DOF of one lens may extend to infinity, or nearly so, and the other still finite, so the approximation can break down in such cases.

Complications

There are more complex formulas that are commonly used, but they are not necessarily more accurate. All DOF formulae should be considered approximate. The approximations are excellent for subject distances greater than about 10 focal lengths, and somewhat less accurate in the close-up and macro regime. However, the differences between the various common approximations are usually small compared to the errors from effects not modeled. As P. van Walree points out, the effect of non-symmetric lens designs with non-unity pupil magnification factors, which is usually ignored, is bigger than the difference between the typical approximate expressions. Furthermore, the accuracy of DOF formulae in the macro regime depends on where distances are measured from; some formulae presume the measurement of distances from front principal plane, some from front focal point, and some from the entrance pupil. Such distinctions are not often clear in published derivations.

In the macro regime, it is common to use a formula involving magnification rather than focal length:

DOF = \frac {2 \cdot c \cdot N \cdot (1 + m)}{m^2} \mbox{, where } m =  \mbox{ magnification }

As as example, at 1:1 macro shooting ( $m = 1$ , $s = 2 f$ from the front principal plane), this equation gives $DOF = 4 c N$ . The focal-length-based approximate equation above, treating $s$ as negligible compared to $H$ , gives $DOF = 8 c N$ , which is quite a bit off; the same equations, measuring s from the front focal point (one $f$ in front of the front principal plane) gives $DOF = 2 c N$ , which is off by as big a factor in the other direction. The slightly more complex equations that lead to the right answer at 1:1 macro are:

D_F = \frac {H \cdot s}{(H - f) - (s - f)} \mbox{ for } s < H

D_N = \frac {H \cdot s}{(H - f) + (s - f)}

Sometimes $H$ is used instead of the $H - f$ in the denominators, depending on which formula for $H$ is used.

If pupil magnification is taken into account, the macro equation becomes

DOF = \frac {2 \cdot c \cdot N \cdot (1 + m/P)}{m^2} \mbox{, where } P =  \mbox{ pupil magnification }

which for a telephoto lens of $P = 0.5$ gives $DOF = 3 c N$ and for a retrofocus wideangle lens with $P = 2$ gives $DOF = 6 c N$ . These variations based on the lens asymmetry can be as large as the other approximations involved.

In the Photographic Lenses Tutorial, David Jacobson points out a set of elegant formulae that generalize this macro formula and are said to be exact for near and far DOF for all conditions:

DOF_N = \frac {c \cdot N \cdot (1 + m/P)}{m^2 \cdot (1+(s-f)/H)}

DOF_F = \frac {c \cdot N \cdot (1 + m/P)}{m^2 \cdot (1-(s-f)/H)}

DOF_N = \frac {c \cdot N \cdot (1 + m/P)}{m^2 \cdot (1+(c \cdot N)/(f \cdot m))}

DOF_F = \frac {c \cdot N \cdot (1 + m/P)}{m^2 \cdot (1-(c \cdot N)/(f \cdot m))}

To apply these formulae in practice, one needs to know the pupil magnification of the particular lens, which may not be constant if the lens has internally moving elements for focus or zoom. The pupil magnification factor can be estimated by simply looking into the front and rear of the lens and estimating the ratio of the sizes of the apparent apertures (rear diameter divided by front diameter).

Artistic considerations

Depth of field can be anywhere from a fraction of an inch to virtually infinite. For instance a shot of a woman's face in closeup may have shallow depth of field (with someone just behind her visible but out of focus—common, for instance, in melodramas and horror films); a shot of rolling hills would be likely to have great depth of field, with the objects both in the foreground and in the background in focus.

Example of how the F-number affects
depth of field. Above, from top to bottom:
f/22, f/8, f/4, f/2.8.

Aperture effects

The aperture controls the effective diameter of the lens opening. Reducing the aperture size (by increasing the f-number) increases the depth of field; however, it also reduces the amount of light transmitted, placing a practical limit on the extent to which the aperture size may be reduced. Photography lenses almost invariably work best at medium apertures. Motion pictures make only limited use of this control. To produce a consistent image quality from shot to shot, cinematographers usually choose a single aperture setting for interiors and another for exteriors and adjust exposure through the use of camera filters or light levels. Aperture settings are adjusted more frequently in still photography, where variations in depth of field are used to produce a variety of special effects.

Depth of field versus film format size

As the equations above show, depth of field is also related to the circle of confusion criterion, which is typically chosen as a fraction such as 1/1000 or 1/1500 of the film format size. Larger imaging devices (such as 8x10 inch photographic plates) can tolerate a larger circle of confusion, while smaller imaging devices such as point-and-shoot digital cameras need a smaller circle of confusion. For equal field of view and f-number, depth of field is inversely proportional to the film format size.

In practical terms this means that smaller cameras have deeper depth of field than larger cameras. This can be an advantage or disadvantage, depending on the desired effect. A large format camera is better for photographs where the foreground and background are blurred (cf. bokeh), while a small camera maximizes depth of field, so that objects behind or in front of the focus plane are still in good focus. This difference between formats goes away if the cameras are compared with equal aperture diameters rather than equal f-numbers; but the smaller camera can not usually use a large aperture diameter, so can not achieve a very limited depth of field.

Depth of field in photolithography

In semiconductor photolithography applications, depth of field is extremely important as integrated circuit layout features must be printed with high accuracy at extremely small size. The difficulty is that the wafer surface is not perfectly flat, but may vary by several micrometres. Even this small variation causes some distortion in the projected image, and results in unwanted variations in the resulting pattern. Thus photolithography engineers take extreme measures to maximize the optical depth of field of the photolithography equipment. To mimimize this distortion further, chip makers like IBM are forced to use chemical mechanical polishing machines to make the wafer surface even flatter before lithographic patterning.

In ophthalmology and optometry

A person may sometimes experience better vision in daylight than at night because of an increased depth of field due to constriction of the pupil (i.e. miosis).

Digital editing of depth of field

Digital image processing can increase the depth of field of a photograph by combining images from multiple shots at different focus depths, or by using techniques such as Wavefront coding. Available programs for multi-shot DOF enhancement include Helicon Focus and CombineZ5. See the linked online article by Rik Littlefield.

References

Hummel, Rob (editor). American Cinematographer Manual, 8th edition. Hollywood: ASC Press, 2001.

External links

Depth of field calculator
Demonstration that all focal lengths have identical depth of field
Depth of Field: illustrations and terminology for photographers
Explanation of why "... all focal lengths have identical depth of field" is true only in some circumstances.
Depth of Field explanation and comparison photographs
Depth of Field - the Third Dimension
P. van Walree's DOF with Pupil Magnification
D. Jacobson's Photographic Lenses Tutorial
Littlefield's "An Introduction to Extended Depth of Field Digital Photography"

Geometrical optics Photographic terms

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