Depth of field

In optics, particularly as it relates to film and photography, the depth of field (DOF) is the distance in front of and beyond the subject that appears to be in focus. Although a lens can precisely focus at only one distance, the decrease in sharpness is gradual on either side of the focused distance, so that within the DOF, the unsharpness is imperceptible under normal viewing conditions.

For some images, such as landscapes, a large DOF may be appropriate, while for others, such as portraits, a small DOF may be more effective.

The DOF is determined by the subject distance, the lens focal length, and the lens f -number (relative aperture). Except at close-up distances, DOF is approximately determined by the subject magnification and the lens f -number. For a given f -number, increasing the magnification, either by moving closer to the subject or using a lens of greater focal length, decreases the DOF; decreasing magnification increases DOF. For a given subject magnification, increasing the f -number (decreasing the aperture diameter) increases the DOF; decreasing f -number decreases DOF.

When focus is set to the hyperfocal distance, the DOF extends from half the hyperfocal distance to infinity, and is the largest DOF possible for a given f -number.

The advent of digital technology in photography has provided additional means of controlling the extent of image sharpness; some methods allow DOF that would be impossible with traditional techniques, and some allow the DOF to be determined after the image is made.



Apparent sharp focus
Precise focus is possible at only one distance; at that distance, a point object will produce a point image. At any other distance, a point object is defocused, and will produce a blur spot shaped like the aperture, which for the purpose of analysis is usually assumed to be circular. When this circular spot is sufficiently small, it is indistiguishable from a point, and appears to be in focus; it is rendered as “acceptably sharp”. The diameter of the circle increases with distance from the point of focus; the largest circle that is indistinguishable from a point is known as the acceptable circle of confusion, or informally, simply as the circle of confusion. The acceptable circle of confusion is influenced by visual acuity, viewing conditions, and the amount by which the image is enlarged. The increase of the circle diameter with defocus is gradual, so the limits of depth of field are not hard boundaries between sharp and unsharp.

Several other factors, such as subject matter, movement, and the distance of the subject from the camera, also influence when a given defocus becomes noticeable.

For a 35 mm motion picture, the image area on the negative is roughly 22 mm by 16 mm (0.87 in by 0.63 in). The limit of tolerable error is usually set at 0.05 mm (0.002 in) diameter. For 16 mm film, where the image area is smaller, the tolerance is stricter, 0.025 mm (0.001 in). Standard depth-of-field tables are constructed on this basis, although generally 35 mm productions set it at 0.025 mm (0.001 in). 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.)



The image format size also will affect the depth of field. The larger the format size, the longer a lens will need to be to capture the same framing as a smaller 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 because 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 in another format.

Effect of f -number
For a given subject framing, the DOF is controlled by the lens f -number. Increasing the f -number (reducing the aperture diameter) increases the DOF; however, it also reduces the amount of light transmitted, and increases diffraction, placing a practical limit on the extent to which the aperture size may be reduced. 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.

Camera movements and DOF
When the lens axis is perpendicular to the image plane, as is normally the case, the plane of focus (POF) is parallel to the image plane, and the DOF extends between parallel planes on either side of the POF. When the lens axis is not perpendicular to the image plane, the POF is no longer parallel to the image plane; the ability to rotate the POF is known as the Scheimpflug principle. Rotation of the POF is accomplished with camera movements (tilt, a rotation of the lens about a horizontal axis, or swing, a rotation about a vertical axis). Tilt and swing are available on most view cameras, and are also available with specific lenses on some small- and medium-format cameras.

When the POF is rotated, the near and far limits of DOF are no longer parallel; the DOF becomes wedge-shaped, with the apex of the wedge nearest the camera. With tilt, the height of the DOF increases with distance from the camera; with swing, the width of the DOF increases with distance.

Rotating the POF with tilt or swing (or both) can be used either to maximize or minimize the part of an image that is within the DOF.

Lens DOF scales
Many lenses for small- and medium-format cameras include scales that indicate the DOF for a given focus distance and f -number; the 35 mm Nikkor lens in the image above is typical. That lens includes distance scales in feet and meters; when a marked distance is set opposite the large white index mark, the focus is set to that distance. The DOF scale below the distance scales includes markings on either side of the index that correspond to f -numbers; when the lens is set to a given f -number, the DOF extends between the distances that align with the f -number markings.

Zone focusing
When the 35 mm lens above is set to f &thinsp;/&thinsp;11 and focused at approximately 1.4 m, the DOF (a “zone” of acceptable sharpness) extends from 1 m to 2 m. Conversely, the required focus and f -number can be determined from the desired DOF limits by locating the near and far DOF limits on the lens distance scale and setting focus so that the index mark is centered between the near and far distances; the required f -number is determined by finding the markings on the DOF scale that are closest to the near and far distances. For the 35 mm lens above, if it were desired for the DOF to extend from 1 m to 2 m, focus would be set to approximately 1.4 m and the aperture set to f &thinsp;/&thinsp;11. The DOF limits can be determined from a scene by focusing on the farthest object to be within the DOF and noting the distance on the lens distance scale, and repeating the process for the nearest object to be within the DOF. If the near and far distances fall outside the largest f -number markings on the DOF scale, the desired DOF cannot be obtained; for example, with the 35 mm lens above, it is not possible to have the DOF extend from 0.7 m to infinity.

Some distance scales have markings for only a few distances; for example, the 35 mm lens above shows only 3 ft and 5 ft on its upper scale. Using other distances for DOF limits requires visual interpolation between marked distances; because the distance scale is nonlinear, accurate interpolation can be difficult. In most cases, English and metric distance markings are not coincident, so using both scales to note focused distances can sometimes lessen the need for interpolation. Many autofocus lenses have smaller distance and DOF scales and fewer markings than do comparable manual-focus lenses, so that determining focus and f -number from the scales on an autofocus lens may be more difficult than with a comparable manual-focus lens. In most cases, using the lens DOF scales on an autofocus lens requires that the lens or camera body be set to manual focus.

On a view camera, the focus and f -number can be obtained by measuring the focus spread and performing simple calculations; the procedure is described in more detail in the section Focus and f -number from DOF limits. Some view cameras include DOF calculators that indicate focus and f -number without the need for any calculations by the photographer.

Hyperfocal distance
The hyperfocal distance is the nearest focus distance at which the DOF extends to infinity; focusing the camera at the hyperfocal distance results in the largest possible depth of field for a given f -number. Focusing beyond the hyperfocal distance does not increase the far DOF (which already extends to infinity), but it does decrease the DOF in front of the subject, decreasing the total DOF. Some photographers refer to this as “wasting DOF”; however, see The object field method below for a rationale for doing so. If the lens includes a DOF scale, the hyperfocal distance can be set by aligning the infinity mark on the distance scale with the mark on the DOF scale corresponding to the f -number to which the lens is set. For example, with the 35 mm lens shown above set to f &thinsp;/&thinsp;11, aligning the infinity mark with the ‘11’ to the left of the index mark on the DOF scale would set the focus to the hyperfocal distance. Focusing on the hyperfocal distance is a special case of zone focusing in which the far limit of DOF is at infinity.

The object field method
Traditional depth-of-field formulas and tables assume equal circles of confusion for near and far objects. Some authors, such as Merklinger (1992), have suggested that distant objects often need to be much sharper to be clearly recognizable, whereas closer objects, being larger on the film, do not need to be so sharp. The loss of detail in distant objects may be particularly noticeable with extreme enlargements. Achieving this additional sharpness in distant objects usually requires focusing beyond the hyperfocal distance, sometimes almost at infinity. For example, if photographing a cityscape with a traffic bollard in the foreground, this approach, termed the object field method by Merklinger, would recommend focusing very close to infinity, and stopping down to make the bollard sharp enough. With this approach, foreground objects cannot always be made perfectly sharp, but the loss of sharpness in near objects may be acceptable if recognizability of distant objects is paramount.

Moritz von Rohr also used an object field method, but unlike Merklinger, he used the conventional criterion of a maximum circle of confusion diameter in the image plane, leading to unequal front and rear depths of field.

Limited DOF: selective focus
Depth of field can be anywhere from a fraction of a millimeter to virtually infinite. In some cases, such as landscapes, it may be desirable to have the entire image in focus, and a large DOF is appropriate. In other cases, artistic considerations may dictate that only a part of the image be in focus, emphasizing the subject while de-emphasizing the background, perhaps giving only a suggestion of the environment (Langford 1973, 81). For example, a common technique in melodramas and horror films is a closeup of a person's face, with someone just behind that person visible but out of focus. A portrait or closeup still photograph might use a small DOF to isolate the subject from a distracting background. The use of limited DOF to emphasize one part of an image is known as selective focus or differential focus.

Although a small DOF implies that other parts of the image will be unsharp, it does not, by itself, determine how unsharp those parts will be. The amount of background (or foreground) blur depends on the distance from the plane of focus, so if a background is close to the subject, it may be difficult to blur sufficiently even with a small DOF. In practice, the lens f -number is usually adjusted until the background or foreground is acceptably blurred, often without direct concern for the DOF.

Sometimes, however, it is desirable to have the entire subject sharp while ensuring that the background is sufficiently unsharp. When the distance between subject and background is fixed, as is the case with many scenes, the DOF and the amount of background blur are not independent. Although it is not always possible to achieve both the desired subject sharpness and the desired background unsharpness, several techniques can be used to increase the separation of subject and background.

For a given scene and subject magnification, the background blur increases with lens focal length. If it is not important that background objects be unrecognizable, background de-emphasis can be increased by using a lens of longer focal length and increasing the subject distance to maintain the same magnification. This technique requires that sufficient space in front of the subject be available; moreover, the perspective of the scene changes because of the different camera position, and this may or may not be acceptable.

The situation is not as simple if it is important that a background object, such as a sign, be unrecognizable. The magnification of background objects also increases with focal length, so with the technique just described, there is little change in the recognizability of background objects. However, a lens of longer focal length may still be of some help; because of the narrower angle of view, a slight change of camera position may suffice to eliminate the distracting object from the field of view.

Although tilt and swing are normally used to maximize the part of the image that is within the DOF, they also can be used, in combination with a small f -number, to give selective focus to a plane that isn't perpendicular to the lens axis. With this technique, it is possible to have objects at greatly different distances from the camera in sharp focus and yet have a very shallow DOF. The effect can be interesting because it differs from what most viewers are accustomed to seeing.

Near:far distribution
The DOF beyond the subject is always greater than the DOF in front of the subject. When the subject is at the hyperfocal distance or beyond, the far DOF is infinite; as the subject distance decreases, near:far DOF ratio increases, approaching unity at high magnification. The oft-cited “rule” that 1/3 of the DOF is in front of the subject and 2/3 is beyond is true only when the subject distance is 1/3 the hyperfocal distance.

Depth of field formulas
The basis of these formulas is given in the section Derivation of the DOF formulas; refer to the diagram in that section for illustration of the quantities discussed below.

Hyperfocal Distance
Let $$f$$ be the lens focal length, $$N$$ be the lens f -number, and $$c$$ be the circle of confusion for a given image format. The hyperfocal distance $$H$$ is given by


 * $$H \approx \frac {f^2} {N c}$$

Moderate-to-large distances
Let $$s$$ be the distance at which the camera is focused (the “subject distance”). When $$s$$ is large in comparison with the lens focal length, the distance $$D_{\mathrm N}$$ from the camera to the near limit of DOF and the distance $$D_{\mathrm F}$$ from the camera to the far limit of DOF are


 * $$D_{\mathrm N} \approx \frac {H s} {H + s}$$


 * $$D_{\mathrm F} \approx \frac {H s} {H - s} \mbox{ for } s < H$$

When the subject distance is the hyperfocal distance,


 * $$D_{\mathrm F} = \infty$$


 * $$D_{\mathrm N} = \frac H 2$$

The depth of field $$D_{\mathrm F} - D_{\mathrm N}$$ is



\mathrm {DOF} \approx \frac {2 Hs^2} {H^2 - s^2} \mbox{ for } s < H $$

For $$s \ge H$$, the far limit of DOF is at infinity and the DOF is infinite; of course, only objects at or beyond the near limit of DOF will be recorded with acceptable sharpness.

Substituting for $$H$$ and rearranging, DOF can be expressed as


 * $$\mathrm {DOF} \approx \frac {2 N c f^2 s^2} {f^4 - N^2 c^2 s^2}$$

Thus, for a given image format, depth of field is determined by three factors: the focal length of the lens, the f -number of the lens opening (the aperture), and the camera-to-subject distance.

Close-up
When the subject distance $$s$$ approaches the focal length, using the formulas given above can result in significant errors. For close-up work, the hyperfocal distance has little applicability, and it usually is more convenient to express DOF in terms of image magnification. Let $$m$$ be the magnification; when the subject distance is small in comparison with the hyperfocal distance,


 * $$\mathrm {DOF} \approx 2 N c \left ( \frac {m + 1} {m^2} \right ),$$

so that for a given magnification, DOF is independent of focal length. Stated otherwise, for the same subject magnification, all focal lengths give approximately the same DOF. This statement is true only when the subject distance is small in comparison with the hyperfocal distance, however.

The discussion thus far has assumed a symmetrical lens for which the entrance and exit pupils coincide with the front and rear nodal planes, and for which the pupil magnification (the ratio of exit pupil diameter to that of the entrance pupil) is unity. Although this assumption usually is reasonable for large-format lenses, it often is invalid for medium- and small-format lenses.

When $$s \ll H$$, the DOF for an asymmetrical lens is


 * $$\mathrm {DOF} \approx \frac {2 N c (1 + m/P)}{m^2},$$

where $$P$$ is the pupil magnification. When the pupil magnification is unity, this equation reduces to that for a symmetrical lens.

Except for close-up and macro photography, the effect of lens asymmetry is minimal. At unity magnification, however, the errors from neglecting the pupil magnification can be significant. Consider a telephoto lens with $$P = 0.5$$ and a retrofocus wide-angle lens with $$P = 2$$, at $$m = 1.0$$. The asymmetrical-lens formula gives $$\mathrm {DOF} = 6 N c$$ and $$\mathrm {DOF} = 3 N c$$, respectively. The symmetrical-lens formula gives $$\mathrm {DOF} = 4 N c$$ in either case. The errors are &minus;33% and 33%, respectively.

Focus and f -number from DOF limits
Not all images require that sharpness extend to infinity; for given near and far DOF limits $$D_{\mathrm N}$$ and $$D_{\mathrm F}$$, the required f -number is smallest when focus is set to


 * $$s = \frac {2 D_{\mathrm N} D_{\mathrm F} }

{D_{\mathrm N} + D_{\mathrm F} } $$

When the subject distance is large in comparison with the lens focal length, the required f -number is


 * $$N \approx \frac {f^2} {c}

\frac {D_{\mathrm F} - D_{\mathrm N} } {2 D_{\mathrm N} D_{\mathrm F} } $$

In practice, these settings usually are determined on the image side of the lens, using measurements on the bed or rail with a view camera, or using lens DOF scales on manual-focus lenses for small- and medium-format cameras. If $$v_{\mathrm N}$$ and $$v_{\mathrm F}$$ are the image distances that correspond to the near and far limits of DOF, the required f -number is minimized when the image distance $$v$$ is


 * $$v \approx \frac { v_{\mathrm N} + v_{\mathrm F} } {2}

= v_{\mathrm F} + \frac { v_{\mathrm N} - v_{\mathrm F} } {2} $$

In practical terms, focus is set to halfway between the near and far image distances. The required f -number is


 * $$N \approx \frac { v_{\mathrm N} - v_{\mathrm F} } { 2 c }$$

The image distances are measured from the camera's image plane to the lens's image nodal plane, which is not always easy to locate. In most cases, focus and f -number can be determined with sufficient accuracy using the approximate formulas above, which require only the difference between the near and far image distances; view camera users often refer to the difference $$v_{\mathrm N} \, - \, v_{\mathrm F}$$ as the focus spread. Most lens DOF scales are based on the same concept.

Foreground and background blur
If a subject is at distance $$s$$ and the foreground or background is at distance $$D$$, let the distance between the subject and the foreground or background be indicated by


 * $$x_\mathrm d = \left | D - s \right |$$

The blur disk diameter $$b$$ of a detail at distance $$x_\mathrm d$$ from the subject can be expressed as a function of the focal length $$f$$, subject magnification $$m_\mathrm{s}$$, and f -number $$N$$ according to


 * $$b = \frac {fm_\mathrm s} N \frac { x_\mathrm d } { s \pm x_\mathrm d}$$

The minus sign applies to a foreground object, and the plus sign applies to a background object.

The blur increases with the distance from the subject; when $$b \le c$$, the detail is within the depth of field, and the blur is imperceptible. If the detail is only slightly outside the DOF, the blur may be only barely perceptible.

For a given subject magnification, f -number, and distance from the subject of the foreground or background detail, the degree of detail blur varies with the lens focal length. For a background detail, the blur increases with focal length; for a foreground detail, the blur decreases with focal length. For a given scene, the positions of the subject, foreground, and background usually are fixed, and the distance between subject and the foreground or background remains constant regardless of the camera position; however, to maintain constant magnification, the subject distance must vary if the focal length is changed. For small distance between the foreground or background detail, the effect of focal length is small; for large distance, the effect can be significant. For a reasonably distant background detail, the blur disk diameter is


 * $$b \approx \frac {fm_\mathrm s} {N} ,$$

depending only on focal length.

The blur diameter of foreground details is very large if the details are close to the lens.

The ratio $$b/c$$ is independent of camera format; the blur then is in terms of circles of confusion.

The magnification of the detail also varies with focal length; for a given detail, the ratio of the blur disk diameter to imaged size of the detail is independent of focal length, depending only on the detail size and its distance from the subject. This ratio can be useful when it is important that the background be recognizable (as usually is the case in evidence or surveillance photography), or unrecognizable (as might be the case for a pictorial photographer using selective focus to isolate the subject from a distracting background). As a general rule, an object is recognizable if the blur disk diameter is one-tenth to one-fifth the size of the object or smaller (Williams 1990, 205), and unrecognizable when the blur disk diameter is the object size or greater.

The effect of focal length on background blur is illustrated in van Walree's article on Depth of field.

Practical complications
The distance scales on most medium- and small-format lenses indicate distance from the camera's image plane. Most DOF formulas, including those in this article, use the object distance $$s$$ from the lens's object nodal plane, which often is not easy to locate. Moreover, for many zoom lenses and internal-focusing non-zoom lenses, the location of the object nodal plane, as well as focal length, changes with subject distance. When the subject distance is large in comparison with the lens focal length, the exact location of the object nodal plane is not critical; the distance is essentially the same whether measured from the front of the lens, the image plane, or the actual nodal plane. The same is not true for close-up photography; at unity magnification, a slight error in the location of the object nodal plane can result in a DOF error greater than the errors from any approximations in the DOF equations.

The asymmetrical lens formulas require knowledge of the pupil magnification, which usually is not specified for medium- and small-format lenses. The pupil magnification can be estimated by looking into the front and rear of the lens and measuring the diameters of the apparent apertures, and computing the ratio (rear diameter divided by front diameter). However, for many zoom lenses and internal-focusing non-zoom lenses, the pupil magnification changes with subject distance, and several measurements may be required.

Limitations
Most DOF formulas, including those discussed in this article, employ several simplifications:


 * 1) Paraxial (Gaussian) optics is assumed, and technically, the formulas are valid only for rays that are infinitesimally close to the lens axis. However, Gaussian optics usually is more than adequate for determining DOF, and non-paraxial formulas are sufficiently complex that requiring their use would make determination of DOF impractical in most cases.
 * 2) Lens aberrations are ignored. Including the effects of aberrations is nearly impossible, because doing so requires knowledge of the specific lens design.  Moreover, in well-designed lenses, most aberrations are well corrected, and at least near the optical axis, often are almost negligible when the lens is stopped down 2–3 steps from maximum aperture.  Because lenses usually are stopped down at least to this point when DOF is of interest, ignoring aberrations usually is reasonable. Not all aberrations are reduced by stopping down, however, so actual sharpness may be slightly less than predicted by DOF formulas.
 * 3) Diffraction is ignored. DOF formulas imply that any arbitrary DOF can be achieved by using a sufficiently large  f -number. Because of diffraction, however, this isn't quite true. Once a lens is stopped down to where most aberrations are well corrected, stopping down further will decrease sharpness in the center of the field.  At the DOF limits, however, further stopping down decreases the size of the defocus blur spot, and the overall sharpness may increase.  Consequently, choosing an f -number sometimes involves a tradeoff between center and edge sharpness, although viewers typically prefer uniform sharpness to slightly greater center sharpness.  The choice, of course, is subjective, and may depend upon the particular image.  Eventually, the defocus blur spot becomes negligibly small, and further stopping down serves only to decrease sharpness even at DOF limits.  Typically, diffraction at DOF limits becomes significant only at fairly large f -numbers; because large f -numbers typically require long exposure times, motion blur often causes greater loss of sharpness than does diffraction. Combined defocus and diffraction is discussed in Hansma (1996), Conrad's Depth of Field in Depth (PDF), and Jacobson's Photographic Lenses Tutorial.
 * 4) Post-capture manipulation of the image is ignored. Sharpening via techniques such as deconvolution or unsharp mask can increase the DOF in the final image, particularly when the original image has a large DOF. Conversely, image noise reduction can reduce the DOF.
 * 5) For digital capture with color filter array sensors, demosaicing is ignored. Demosaicing alone would normally reduce the DOF, but the demosaicing algorithm used might also include sharpening.

The lens designer cannot restrict analysis to Gaussian optics and cannot ignore lens aberrations. However, the requirements of practical photography are less demanding than those of lens design, and despite the simplifications employed in development of most DOF formulas, these formulas have proven useful in determining camera settings that result in acceptably sharp pictures. It should be recognized that DOF limits are not hard boundaries between sharp and unsharp, and that there is little point in determining DOF limits to a precision of many significant figures.

DOF vs. format size
To a first approximation, DOF is inversely proportional to format size. More precisely, if photographs with the same final-image size are taken in two different camera formats at the same subject distance with the same field of view and f -number, the DOF is, to a first approximation, inversely proportional to the format size. Strictly speaking, this is true only when the subject distance is large in comparison with the focal length and small in comparison with the hyperfocal distance, for both formats, but it nonetheless is generally useful for comparing results obtained from different formats

To maintain the same field of view, the lens focal lengths must be in proportion to the format sizes. Assuming, for purposes of comparison, that the 4&times;5 format is four times the size of 35 mm format, if a 4&times;5 camera used a 300 mm lens, a 35 mm camera would need a 75 mm lens for the same field of view. For the same f -number, the image made with the 35 mm camera would have four times the DOF of the image made with the 4&times;5 camera.

In many cases, the DOF is fixed by the requirements of the desired image. For a given DOF and field of view, the required f -number is proportional to the format size. For example, if a 35 mm camera required 11, a 4&times;5 camera would require 45 to give the same DOF. For the same ISO speed, the exposure time on the 4&times;5 would be sixteen times as long; if the 35 camera required 1/250 second, the 4&times;5 camera would require 1/15 second. In windy conditions, the exposure time with the larger camera might allow motion blur. Adjusting the f -number to the camera format is equivalent to maintaining the same absolute aperture diameter.

The greater DOF with the smaller format can be either an advantage or a disadvantage, depending on the desired effect. For the same amount of foreground and background blur, a small-format camera requires a smaller f -number and allows a shorter exposure time than a large-format camera; however, many point-and-shoot digital cameras cannot provide a very shallow DOF. For example, a point-and-shoot digital camera with a 1/1.8&Prime; sensor (7.18 mm &times; 5.32 mm) at a normal focal length and 2.8 has the same DOF as a 35 mm camera with a normal lens at 13.

In some cases, camera movements (tilt or swing) can be used to better fit the DOF to the scene, and achieve the required sharpness at a smaller f -number.

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 minimize 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.

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).

Focus stacking
Focus stacking is a digital image processing technique which combines multiple images taken at different focus distances to give a resulting image with a greater depth of field than any of the individual source images. Available programs for multi-shot DOF enhancement include PhotoAcute Studio, Helicon Focus and CombineZM.

Getting sufficient depth of field can be particularly challenging in macro photography. The images at right illustrate the increase in DOF that can be achieved by combining multiple exposures.

Wavefront coding
Wavefront coding is a method that convolves rays in such a way to provide an image where fields are in focus simultaneously with all planes out of focus by a constant amount.

Plenoptic cameras
A plenoptic camera uses a microlens array to capture 4D light field information about a scene.

DOF limits
A symmetrical lens is illustrated at right. The subject at distance $$s$$ is in focus at image distance $$v$$. Point objects at distances $$D_\mathrm F$$ and $$D_\mathrm N$$ would be in focus at image distances $$v_\mathrm F$$ and $$v_\mathrm N$$, respectively; at image distance $$v$$, they are imaged as blur spots. The depth of field is controlled by the aperture stop diameter $$d$$; when the blur spot diameter is equal to the acceptable circle of confusion $$c$$, the near and far limits of DOF are at $$D_\mathrm N$$ and $$D_\mathrm F$$. From similar triangles,


 * $$\frac {v_\mathrm N - v} {v_\mathrm N} = \frac c d$$


 * $$\frac {v- v_\mathrm F} {v_\mathrm F} = \frac c d$$

It usually is more convenient to work with the lens f -number than the aperture diameter; the f -number $$N$$ is related to the lens focal length $$f$$ and the aperture diameter $$d$$ by


 * $$N = \frac f d\,;$$

substituting into the previous equations and rearranging gives
 * $$v_\mathrm N = \frac {fv} {f - Nc}$$
 * $$v_\mathrm F = \frac {fv} {f + Nc}$$

The image distance $$v$$ is related to an object distance $$s$$ by the thin lens equation


 * $$\frac 1 s + \frac 1 v = \frac 1 f\,;$$

Substituting into the two previous equations and rearranging gives the near and far limits of DOF:


 * $$D_{\mathrm N} = \frac {s f^2} {f^2 + N c ( s - f ) }$$


 * $$D_{\mathrm F} = \frac {s f^2} {f^2 - N c ( s - f ) }$$

Hyperfocal distance
Setting the far limit of DOF $$D_{\mathrm F}$$ to infinity and solving for the focus distance $$s$$ gives


 * $$s = H = \frac {f^2} {N c} + f,$$

where $$H$$ is the hyperfocal distance. Setting the subject distance to the hyperfocal distance and solving for the near limit of DOF gives


 * $$D_{\mathrm N} = \frac {f^2 / ( N c ) + f} {2} = \frac {H}{2}$$

For any practical value of $$H$$, the focal length is negligible in comparison, so that


 * $$H \approx \frac {f^2} {N c}$$

Substituting the approximate expression for hyperfocal distance into the formulas for the near and far limits of DOF gives


 * $$D_{\mathrm N} = \frac {H s}{H + ( s - f )}$$


 * $$D_{\mathrm F} = \frac {H s}{H - ( s - f )}$$

Combining, the depth of field $$D_{\mathrm F} - D_{\mathrm N}$$ is



\mathrm {DOF} = \frac {2 H s (s - f )} {H^2 - ( s - f )^2} \mbox{ for } s < H $$

Moderate-to-large distances
When the subject distance is large in comparison with the lens focal length,


 * $$D_{\mathrm N} \approx \frac {H s} {H + s}$$


 * $$D_{\mathrm F} \approx \frac {H s} {H - s} \mbox{ for } s < H$$



\mathrm {DOF} \approx \frac {2 H s^2} {H^2 - s^2} \mbox{ for } s < H $$

For $$s \ge H$$, the far limit of DOF is at infinity and the DOF is infinite; of course, only objects at or beyond the near limit of DOF will be recorded with acceptable sharpness.

Close-up
When the subject distance $$s$$ approaches the lens focal length, the focal length no longer is negligible, and the approximate formulas above cannot be used without introducing significant error. At close distances, the hyperfocal distance has little applicability, and it usually is more convenient to express DOF in terms of magnification. Substituting


 * $$s = \frac {m + 1} {m} f$$

and


 * $$s - f = \frac {f} {m}$$

into the formula for DOF and rearranging gives



\mathrm {DOF} = \frac {2 f ( m + 1 ) / m } { ( f m ) / ( N c ) - ( N c ) / ( f m ) } $$

At the hyperfocal distance, the terms in the denominator are equal, and the DOF is infinite. As the subject distance decreases, so does the second term in the denominator; when $$s \ll H$$, the second term becomes small in comparison with the first, and


 * $$\mathrm {DOF} \approx 2 N c \left ( \frac {m + 1} {m^2} \right ),$$

so that for a given magnification, DOF is independent of focal length. Stated otherwise, for the same subject magnification, all focal lengths for a given image format give approximately the same DOF. This statement is true only when the subject distance is small in comparison with the hyperfocal distance, however. Multiplying the numerator and denominator of the exact formula by


 * $$\frac {N c m} {f}$$

gives


 * $$\mathrm {DOF} = \frac

{2 N c \left ( m + 1 \right )} {m^2 - \left ( \frac {N c} {f} \right )^2} $$

Decreasing the focal length $$f$$ increases the second term in the denominator, decreasing the denominator and increasing the value of the right-hand side, so that a shorter focal length gives greater DOF. The effect of focal length is greatest near the hyperfocal distance, and decreases as subject distance is decreased. However, the near/far perspective will differ for different focal lengths, so the difference in DOF may not be readily apparent. When the subject distance is small in comparison with the hyperfocal distance, the effect of focal length is negligible, and, as noted above, the DOF essentially is independent of focal length.

Near:far DOF ratio
From the “exact” equations for near and far limits of DOF, the DOF in front of the subject is


 * $$s - D_{\mathrm N} = \frac {Ncs(s - f)} {f^2 + Nc(s - f)}\,,$$

and the DOF beyond the subject is


 * $$D_{\mathrm F} - s = \frac {Ncs(s - f)} {f^2 - Nc(s - f)}$$

The near:far DOF ratio is


 * $$\frac {s - D_{\mathrm N}} {D_{\mathrm F} - s}

= \frac {f^2 - Nc(s - f)} {f^2 + Nc(s - f)}$$

This ratio is always less than unity; at moderate-to-large subject distances, $$f \ll s$$, and


 * $$\frac {s - D_{\mathrm N}} {D_{\mathrm F} - s}

\approx \frac {f^2 - Ncs} {f^2 + Ncs} = \frac {H - s} {H + s}$$

When the subject is at the hyperfocal distance or beyond, the far DOF is infinite, and the near:far ratio is zero. It's commonly stated that approximately 1/3 of the DOF is in front of the subject and approximately 2/3 is beyond; however, this is true only when $$s \approx H/3$$.

At closer subject distances, it's often more convenient to express the DOF ratio in terms of the magnification


 * $$m = \frac f {s - f}$$

Substitution into the “exact” equation for DOF ratio gives


 * $$\frac {s - D_{\mathrm N}} {D_{\mathrm F} - s}

= \frac {m - Nc/f} {m + Nc/f}$$

As magnification increases, the near:far ratio approaches a limiting value of unity.

Focus and f -number from DOF limits
Not all images require that sharpness extend to infinity; the equations for the DOF limits can be combined to eliminate $$Nc$$ and solve for the subject distance. For given near and far DOF limits $$D_{\mathrm N}$$ and $$D_{\mathrm F}$$, the subject distance is


 * $$s = \frac {2 D_{\mathrm N} D_{\mathrm F} }

{D_{\mathrm N} + D_{\mathrm F} } $$

The equations for DOF limits also can be combined to eliminate $$s$$ and solve for the required f -number, giving


 * $$N = \frac {f^2} {c}

\frac {D_{\mathrm F} - D_{\mathrm N} } {D_{\mathrm F} ( D_{\mathrm N} - f ) + D_{\mathrm N} ( D_{\mathrm F} - f ) } $$

When the subject distance is large in comparison with the lens focal length, this simplifies to


 * $$N \approx \frac {f^2} {c}

\frac {D_{\mathrm F} - D_{\mathrm N} } {2 D_{\mathrm N} D_{\mathrm F} } $$

Most discussions of DOF concentrate on the object side of the lens, but the formulas are simpler and the measurements usually easier to make on the image side. If $$v_{\mathrm N}$$ and $$v_{\mathrm F}$$ are the image distances that correspond to the near and far limits of DOF, the required f -number is minimum when the image distance $$v$$ is


 * $$v = \frac {2 v_{\mathrm N} v_{\mathrm F} }

{v_{\mathrm N} + v_{\mathrm F} } $$

The required f -number is


 * $$N = \frac {f} {c}

\frac { v_{\mathrm N} - v_{\mathrm F} } {v_{\mathrm N} + v_{\mathrm F} } $$

The image distances are measured from the camera's image plane to the lens's image nodal plane, which is not always easy to locate. In most cases, focus and f -number can be determined with sufficient accuracy using the approximate formulas


 * $$v \approx \frac { v_{\mathrm N} + v_{\mathrm F} } {2}

= v_{\mathrm F} + \frac { v_{\mathrm N} - v_{\mathrm F} } {2} $$


 * $$N \approx \frac { v_{\mathrm N} - v_{\mathrm F} } { 2 c },$$

which require only the difference between the near and far image distances; focus is simply set to halfway between the near and far distances. View camera users often refer to the difference $$v_{\mathrm N} \, - \, v_{\mathrm F}$$ as the focus spread; it usually is measured on the bed or focusing rail. On manual-focus small- and medium-format lenses, the focus and f -number usually are determined using the lens DOF scales, which often are based on the two equations above.

For close-up photography, the f -number is more accurately determined using


 * $$N \approx \frac {1} { 1 + m } \frac { v_{\mathrm N} - v_{\mathrm F} } { 2 c },$$

where $$m$$ is the magnification.



Foreground and background blur
If the equation for the far limit of DOF is solved for $$c$$, and the far distance replaced by an arbitrary distance $$D$$, the blur disk diameter $$b$$ at that distance is


 * $$b = \frac {fm_\mathrm s} {N} \frac { D - s } { D }$$

When the background is at the far limit of DOF, the blur disk diameter is equal to the circle of confusion $$c$$, and the blur is just imperceptible. The diameter of the background blur disk increases with the distance to the background. A similar relationship holds for the foreground; the general expression for a defocused object at distance $$D$$ is


 * $$b = \frac {fm_\mathrm s} {N} \frac { \left| D - s \right | } { D }$$

For a given scene, the distance between the subject and a foreground or background object is usually fixed; let that distance be represented by


 * $$x_\mathrm d = \left | D - s \right | ;$$

then


 * $$b = \frac {fm_\mathrm s} {N} \frac { x_\mathrm d } { D }$$

or, in terms of subject distance,


 * $$b = \frac {fm_\mathrm s} {N} \frac { x_\mathrm d } { s \pm x_\mathrm d } ,$$

with the minus sign used for foreground objects and the plus sign used for background objects. For a relatively distant background object,


 * $$b \approx \frac {fm_\mathrm s} N $$

In terms of subject magnification, the subject distance is


 * $$s = \frac { m_\mathrm s + 1 } { m_\mathrm s } f ,$$

so that, for a given f -number and subject magnification,


 * $$b = \frac {fm_\mathrm s} {N} \frac { x_\mathrm d } { \frac { m_\mathrm s + 1} {m_\mathrm s} f \pm x_\mathrm d }

= \frac {fm_\mathrm s ^2} {N} \frac { x_\mathrm d } { \left ( m_\mathrm s + 1 \right ) f \pm m_\mathrm s x_\mathrm d } $$

Differentiating $$b$$ with respect to $$f$$ gives


 * $$\frac {\mathrm d b} {\mathrm d f}

= \frac {\pm m_\mathrm s ^3 x_\mathrm d ^2} {N \left [ \left ( m_\mathrm s + 1 \right ) f \pm m_\mathrm s x_\mathrm d \right ]^2 } $$

With the plus sign, the derivative is everywhere positive, so that for a background object, the blur disk size increases with focal length. With the minus sign, the derivative is everywhere negative, so that for a foreground object, the blur disk size decreases with focal length.

The magnification of the defocused object also varies with focal length; the magnification of the defocused object is


 * $$m_\mathrm d = \frac {v_\mathrm s} {D} = \frac { \left ( m_\mathrm s + 1 \right ) f } { D },$$

where $$v_\mathrm s$$ is the image distance of the subject. For a defocused object with some characteristic dimension $$y$$, the imaged size of that object is


 * $$m_\mathrm d y = \frac { \left ( m_\mathrm s + 1 \right ) f y } { D }$$

The ratio of the blur disk size to the imaged size of that object then is


 * $$ \frac b { m_\mathrm d y } = \frac {m_\mathrm s} { m_\mathrm s + 1 } \frac {x_\mathrm d } { Ny },$$

so for a given defocused object, the ratio of the blur disk diameter to object size is independent of focal length, and depends only on the object size and its distance from the subject.

Asymmetrical lenses
The discussion thus far has assumed a symmetrical lens for which the entrance and exit pupils coincide with the object and image nodal planes, and for which the pupil magnification is unity. Although this assumption usually is reasonable for large-format lenses, it often is invalid for medium- and small-format lenses.

For an asymmetrical lens, the DOF ahead of the subject distance and the DOF beyond the subject distance are given by


 * $$\mathrm {DOF_N} = \frac

{N c (1 + m/P)} {m^2 [ 1 + (N c ) / ( f m ) ] } $$


 * $$\mathrm {DOF_F} = \frac

{N c (1 + m/P)} {m^2 [ 1 - (N c )/ ( f m ) ] }, $$

where $$P$$ is the pupil magnification.

Combining gives the total DOF:


 * $$\mathrm {DOF} = \frac {2 f ( 1/m + 1/P ) }

{ ( f m ) / ( N c ) - ( N c ) / ( f m ) } $$

When $$s \ll H$$, the second term in the denominator becomes small in comparison with the first, and


 * $$\mathrm {DOF} \approx \frac {2 N c (1 + m/P)}{m^2}$$

When the pupil magnification is unity, the equations for asymmetrical lenses reduce to those given earlier for symmetrical lenses.

Effect of lens asymmetry
Except for close-up and macro photography, the effect of lens asymmetry is minimal. A slight rearrangement of the last equation gives


 * $$\mathrm {DOF} \approx \frac {2 N c} {m}

\left ( \frac 1 m + \frac 1 P \right ) $$

As magnification decreases, the $$1/P$$ term becomes smaller in comparison with the $$1/m$$ term, and eventually the effect of pupil magnification becomes negligible.