Depth of field

An example of very shallow depth of field in a macro photograph.

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.

The area within the depth of field appears sharp while the areas in front and behind the depth of field appear blurry.

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.)

A 35mm lens set to f/11. The depth-of-field scale (top) indicates that a subject which is anywhere between 1 and 2 meters in front of the camera will be rendered acceptably sharp. If the aperture were set to f/22 instead, everything from 0.7 meters to infinity would be in focus.

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. Of course, this latter approach may be appropriate for images that do not extend to infinity.

The Object Field Method

Traditional depth-of-field formulae and tables assume equal circles ofconfusion for near and far objects. Some authors, such asMerklinger (1992),^[1]have suggested that distant objects often need to be much sharper to beclearly recognizable, whereas closer objects, being larger on the film, donot need to be so sharp. The loss of detail in distant objects may beparticularly noticeable with extreme enlargements. Achieving this additionalsharpness in distant objects usually requires focusing beyond thehyperfocal distance, sometimes almost at infinity. For example, ifphotographing a cityscape with a traffic bollard in the foreground, thisapproach, termed the Object Field Method by Merklinger, would recommendfocusing very close to infinity, and stopping down to make the bollardsharp enough. With this approach, foreground objects cannot always be madeperfectly sharp, but the loss of sharpness in near objects may beacceptable if recognizability of distant objects is paramount.

Depth of field formulae

Hyperfocal Distance

Let <math>f</math> be the lens focal length,<math>N</math> be the lens f-number, and <math>c</math> be thecircle of confusion for a given image format. Thehyperfocal distance <math>H</math> is given by

<math>H \approx \frac {f^2} {N c}</math>

DOF at moderate-to-large subject distances

Let <math>s</math> be the distance at which the camera is focused (the“subject distance”). When <math>s</math> is large in comparison with thelens focal length, the distance <math>D_{\mathrm N}</math> from thecamera to the near limit of DOF and the distance <math>D_{\mathrm F}</math>from the camera to the far limit of DOF are

<math>D_{\mathrm N} \approx \frac {H s} {H + s}</math>

<math>D_{\mathrm F} \approx \frac {H s} {H - s} \mbox{ for } s < H</math>

When the subject distance is the hyperfocal distance,

<math>D_{\mathrm F} = \infty</math>

<math>D_{\mathrm N} = \frac H 2</math>

The depth of field <math>D_{\mathrm F} - D_{\mathrm N}</math> is

<math>

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

For <math>s \ge H</math>, the far limit of DOF is at infinity and the DOFis infinite; of course, only objects at or beyond the near limit of DOFwill be recorded with acceptable sharpness.

Substituting for <math>H</math> and rearranging, DOF can be expressed as

<math>\mathrm {DOF} \approx \frac {2 N c f^2 s^2} {f^4 - N^2 c^2 s^2}</math>

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

Focus and f-number from DOF limits

Not all images require that sharpness extend to infinity; for given nearand far DOF limits <math>D_{\mathrm N}</math> and <math>D_{\mathrm F}</math>,the required f-number is smallest when focus is set to

<math>s = \frac {2 D_{\mathrm N} D_{\mathrm F} }

{D_{\mathrm N} + D_{\mathrm F} }</math>

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

<math>N \approx \frac {f^2} {c}

\frac {D_{\mathrm F} - D_{\mathrm N} } {2 D_{\mathrm N} D_{\mathrm F} }</math>

In practice, these settings usually are determined on the image side of thelens, using measurements on the bed or rail with a view camera, or usinglens DOF scales on manual-focus lenses for small- and medium-formatcameras.

Close-up DOF

This pen was very close to the camera, giving the image a very shallow depth of field.

When the subject distance <math>s</math> approaches the focal length, usingthe formulae given above can result in significant errors. For close-upwork, the hyperfocal distance has little applicability, and it usually ismore convenient to express DOF in terms of image magnification. Let<math>m</math> be the magnification; when the subject distance is small incomparison with the hyperfocal distance,

<math>\mathrm {DOF} \approx 2 N c \left ( \frac {m + 1} {m^2} \right ),</math>

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

The discussion thus far has assumed a symmetrical lens for which theentrance and exit pupils coincide with the object andimage nodal planes, and for which the pupil magnification(the ratio of exit pupil diameter to that of theentrance pupil)^[2] is unity.Although this assumption usually is reasonable for large-format lenses, itoften is invalid for medium- and small-format lenses.

When <math>s \ll H</math>, the DOF for an asymmetrical lens is

<math>\mathrm {DOF} \approx \frac {2 N c (1 + m/P)}{m^2},</math>

where <math>P</math> is the pupil magnification. When thepupil magnification is unity, this equation reduces to that for asymmetrical lens.

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

Complications in practical application of the DOF formulae

The distance scales on most medium- and small-format lenses indicatedistance from the camera's image plane. Most DOFformulae, including those in this article, use the object distance<math>s</math> from the lens's object nodal plane, which often is not easy tolocate. Moreover, for many zoom lenses and internal-focusing non-zoomlenses, the location of the object nodal plane, as well as focal length,changes with subject distance. When the subject distance is large incomparison with the lens focal length, the exact location of the objectnodal plane is not critical; the distance is essentially the same whethermeasured from the front of the lens, the image plane, or the actual nodalplane. The same is not true for close-up photography; at unitymagnification, a slight error in the location of the object nodal plane canresult in a DOF error greater than the errors from any approximations inthe DOF equations.

The asymmetrical lens formulae require knowledge of thepupil magnification, which usually is not specified for medium- andsmall-format lenses. The pupil magnification can be estimated by lookinginto the front and rear of the lens and measuring the diameters of theapparent apertures, and computing the ratio (rear diameter divided by frontdiameter).^[3]However, for many zoom lenses and internal-focusing non-zoom lenses, thepupil magnification changes with subject distance, and several measurementsmay be required.

Limitations of DOF formulae

Most DOF formulae, including those discussed in this article, employseveral simplifications:

1. Paraxial (Gaussian) optics is assumed, and technically, the formulae are valid only for rays that are infinitessimally close to the lens axis. However, Gaussian optics usually is more than adequate for determining DOF, and non-paraxial formulae 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 formulae.
3. Diffraction is ignored. DOF formulae 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) and in 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, 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.

A Cowslip flower, showing how limited depth of field is a great advantage in separating the flower from the background.

Artistic effect of using shallow depth of field to blur foreground and background in a portrait

The lens designer cannot restrict analysis to Gaussian optics and cannotignore lens aberrations. However, the requirements of practicalphotography are less demanding than those of lens design, and despite thesimplifications employed in development of most DOF formulae, theseformulae have proven useful in determining camera settings that result inacceptably sharp pictures. It should be recognized that DOF limits are nothard boundaries between sharp and unsharp, and that there is little pointin determining DOF limits to a precision of many significant figures.

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.

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.

Depth of field versus format size

As the equations above show, depth of field is related to the circle of confusion criterion, which is typically chosen as a fraction, such as 1/1000 or 1/1500, of the image format size. Larger imaging devices (such as 8×10 cameras) 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 the same field of view and f-number, DOF is, to a first approximation, inversely proportional to the format size. Strictly speaking, this relationship 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.

f/32

f/5

f/2.8

f/22

f/8

f/4

f/2.8

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

At a given f-number and field of view, a smaller camera has greater DOF than a larger camera. The depth of field on an 8×10 camera using a normal lens at f/22 is one half that on a 4×5 with a normal lens at f/22. Similarly, a 35 mm camera with a normal lens at f/8 has the same depth of field as a 6×7 cm camera with a normal lens at f/16. This can be an advantage or disadvantage, depending on the desired effect. For the same amount of foreground and background blur, a small-format camera requires a smaller f-number than a large-format camera. 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″ sensor (7.18 mm × 5.32 mm) at a normal focal length and f/2.8 has the same DOF as a 35 mm camera with a normal lens at f/13.

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 f/11, a 4×5 camera would require f/45 to give the same DOF. For the same ISO speed, the exposure time on the 4×5 would be sixteen times as long; if the 35 camera required 1/250 second, the 4×5 camera would require 1/15 second. In windy conditions, the exposure time with the larger camera might allow motion blur.

For cameras of different formats to achieve the same depth of field when shooting from the same position, with focal lengths that capture the same field of view, it is necessary to use the same absolute aperture diameter with each, not the same f-number. Consider formats that differ approximately by factors of two: 35 mm, 6×7 cm, 4×5 inch, 8×10 inch. For a chosen camera position and field of view, to keep the same depth of field, double the f-number each time the film is stepped up to the next size. For example: f/5.6 on 35 mm, f/11 on 6×7, f/22 on 4×5, f/45 on 8×10. This doubling is not exact but is a very good rule of thumb. Also adjust exposure, ISO speed, or both by two stops (factor of four) each time: 1/60, 1/15, 1/4, 1 sec.

In some cases, movements (tilt or swing) can be used with view cameras to better fit the DOF to the scene, and achieve the required sharpness at a smaller f-number. A few small-format cameras can employ the same principle by using tilt/shift lenses.

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 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.

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.

Basis of the DOF formulae

DOF limits and hyperfocal distance

Let <math>s</math> be the distance at which the camera is focused (the“subject distance”), <math>f</math> be the lens focal length,<math>N</math> be the lens f-number, and <math>c</math> be thecircle of confusion for a given image format. Thedistance <math>D_{\mathrm N}</math> from the camera to the near limit ofdepth of field and the distance <math>D_{\mathrm F}</math> from the camerato the far limit of depth of field then are given by^[4]

<math>D_{\mathrm N} = \frac {s f^2} {f^2 + N c ( s - f ) }</math>

<math>D_{\mathrm F} = \frac {s f^2} {f^2 - N c ( s - f ) }</math>

Setting the far limit of DOF <math>D_{\mathrm F}</math> to infinity andsolving for the focus distance <math>s</math> gives

<math>s = H = \frac {f^2} {N c} + f,</math>

where <math>H</math> is the hyperfocal distance. Setting the subjectdistance to the hyperfocal distance and solving for the near limit of DOFgives

<math>D_{\mathrm N} = \frac {f^2 / ( N c ) + f} {2} = \frac {H}{2}</math>

For any practical value of <math>H</math>, the focal length is negligiblein comparison, so that

<math>H \approx \frac {f^2} {N c}</math>

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

<math>D_{\mathrm N} = \frac {H s}{H + ( s - f )}</math>

<math>D_{\mathrm F} = \frac {H s}{H - ( s - f )}</math>

Combining, the depth of field <math>D_{\mathrm F} - D_{\mathrm N}</math> is

<math>

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

DOF at moderate-to-large subject distances

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

<math>D_{\mathrm N} \approx \frac {H s} {H + s}</math>

<math>D_{\mathrm F} \approx \frac {H s} {H - s} \mbox{ for } s < H</math>

<math>

\mathrm {DOF} \approx \frac {2 H s^2}{H^2 - s^2} \mbox{ for } s < H</math>

For <math>s \ge H</math>, the far limit of DOF is at infinity and the DOFis infinite; of course, only objects at or beyond the near limit of DOFwill be recorded with acceptable sharpness.

Focus and f-number from DOF limits

Not all images require that sharpness extend to infinity; the equations forthe DOF limits can be combined to eliminate <math>Nc</math> and solve forthe subject distance. For given near and far DOF limits<math>D_{\mathrm N}</math> and <math>D_{\mathrm F}</math>, thesubject distance is

<math>s = \frac {2 D_{\mathrm N} D_{\mathrm F} }

{D_{\mathrm N} + D_{\mathrm F} }</math>

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

<math>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 ) }</math>

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

<math>N \approx \frac {f^2} {c}

\frac {D_{\mathrm F} - D_{\mathrm N} } {2 D_{\mathrm N} D_{\mathrm F} }</math>

Most discussions of DOF concentrate on the object side of the lens, but theformulae are simpler and the measurements usually easier to make on theimage side. If <math>v_{\mathrm N}</math> and <math>v_{\mathrm F}</math>are the image distances that correspond to the near and far limits of DOF,the optimum image distance <math>v</math> is

<math>v = \frac {2 v_{\mathrm N} v_{\mathrm F} }

{v_{\mathrm N} + v_{\mathrm F} }</math>

The required f-number is

<math>N = \frac {f^2} {c}

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

{v_{\mathrm N} + v_{\mathrm F} }</math>

The image distances are measured from the camera's image plane to thelens's image nodal plane, which is not always easy to locate. In mostcases, focus and f-number can be determined with sufficient accuracy usingthe approximate formulae

<math>v \approx \frac { v_{\mathrm N} + v_{\mathrm F} } {2}

= v_{\mathrm F} + \frac { v_{\mathrm N} - v_{\mathrm F} } {2}</math>

<math>N \approx \frac { v_{\mathrm N} - v_{\mathrm F} } { 2 c },</math>

which require only the difference between the near and far image distances;view camera users often refer to the difference<math>v_{\mathrm N} - v_{\mathrm F}</math> as the focus spread.With a view camera, the focus spread usually is measured on the bed orfocusing rail. On manual-focus small- and medium-format lenses, the focusand f-number usually are determined using the lens DOF scales, whichoften are based on the two equations above.

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

<math>N \approx \frac {1} { 1 + m } \frac { v_{\mathrm N} - v_{\mathrm F} } { 2 c },</math>

where <math>m</math> is the magnification.

Close-up DOF

When the subject distance <math>s</math> approaches the lens focal length,the focal length no longer is negligible, and the approximate formulaeabove cannot be used without introducing significant error. At closedistances, the hyperfocal distance has little applicability, and it usuallyis more convenient to express DOF in terms of magnification. Substituting

<math>s = \frac {m + 1} {m} f</math>

and

<math>s - f = \frac {f} {m}</math>

into the formula for DOF and rearranging gives

<math>

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

At the hyperfocal distance, the terms in the denominator are equal, andthe DOF is infinite. As the subject distance decreases, so does the secondterm in the denominator; when <math>s \ll H</math>, the second term becomessmall in comparison with the first, and

<math>\mathrm {DOF} \approx 2 N c \left ( \frac {m + 1} {m^2} \right ),</math>

so that for a given magnification, DOF is independent of focal length.Stated otherwise, for the same subject magnification, all focal lengths fora given image format give approximately the same DOF. Thisstatement is true only when the subject distance is small in comparisonwith the hyperfocal distance, however. Multiplying the numerator anddenominator of the exact formula by

<math>\frac {N c m} {f}</math>

gives

<math>\mathrm {DOF} = \frac

{2 N c \left ( m + 1 \right )}{m^2 - \left ( \frac {N c} {f} \right )^2}</math>

Decreasing the focal length <math>f</math> increases the second term in thedenominator, decreasing the denominator and increasing the value of theright-hand side, so that a shorter focal length gives greater DOF. Theeffect of focal length is greatest near the hyperfocal distance, anddecreases as subject distance is decreased. However, the near/farperspective will differ for different focal lengths, so the difference inDOF may not be readily apparent. When the subject distance is small incomparison with the hyperfocal distance, the effect of focal length isnegligible, and, as noted above, the DOF essentially is independent offocal length.

Asymmetrical lenses

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

For an asymmetrical lens, the DOF ahead of the subject distance and theDOF beyond the subject distance are given by^[5]

<math>\mathrm {DOF_N} = \frac

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

<math>\mathrm {DOF_F} = \frac

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

where <math>P</math> is the pupil magnification.

Combining gives the total DOF:

<math>\mathrm {DOF} = \frac {2 f ( 1/m + 1/P ) }

{ ( f m ) / ( N c ) - ( N c ) / ( f m ) }</math>

When <math>s \ll H</math>, the second term in the denominator becomessmall in comparison with the first, and

<math>\mathrm {DOF} \approx \frac {2 N c (1 + m/P)}{m^2}</math>

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

Effect of lens asymmetry

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

<math>\mathrm {DOF} \approx \frac {2 N c} {m}

\left ( \frac 1 m + \frac 1 P \right )</math>

As magnification decreases, the <math>1/P</math> term becomes smaller incomparison with the <math>1/m</math> term, and eventually the effect ofpupil magnification becomes negligible.

Notes

1. ^ Joe Englander describes a similar approach in his paperApparent Depth of Field: Practical Use in Landscape Photography. (PDF);Jeff Conrad discusses this approach, under Different Circles ofConfusion for Near and Far Limits of Depth of Field, and The Object FieldMethod, in Depth of Field in Depth (PDF)
2. ^ A well-illustrated discussion of pupils and pupilmagnification that assumes minimal knowledge of optics and mathematics isgiven in Shipman (1977).
3. ^ The procedure for estimating pupil magnification isdescribed in detail in Shipman (1977).
4. ^ The derivation isstraightforward and is covered in many texts, including Larmore (1965)and Ray (2002). Complete derivations also are given in Conrad's Depth of Field in Depth (PDF) and van Walree's Derivation of the DOF equations.
5. ^ This is discussed inJacobson's Photographic Lenses Tutorial.and complete derivations are given in Conrad'sDepth of Field in Depth (PDF)and van Walree's Derivation of the DOF quations.

References

• Hansma, Paul K. 1996. “View Camera Focusing in Practice,” Photo Techniques, March/April 1996, 54–57. Available as GIF images on the Large Format page.
• Larmore, Lewis. 1965. Introduction to Photographic Principles, 2nd ed. New York: Dover Publications, Inc.
• Merklinger, Harold M. 1992. The INs and OUTs of FOCUS, v. 1.0.3. Bedford, Nova Scotia: Seaboard Printing Limited. Version 1.03e available in PDF at http://www.trenholm.org/hmmerk/. ISBN 0-9695025-0-8
• Ray, Sidney F. 2002. Applied Photographic Optics, 3rd ed. Oxford: Focal Press. ISBN 0-240-51540-4
• Shipman, Carl. 1977. SLR Photographers Handbook. Tucson: H.P. Books. ISBN 0-912656-59-X

Further reading

• Hummel, Rob (editor). 2001. American Cinematographer Manual, 8th edition. Hollywood: ASC Press. ISBN 0935578153

See also

External links

• Depth of field calculator
• Depth of Field: illustrations and terminology for photographers
• Demonstration that all focal lengths have identical depth of field
• 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
• Jeff Conrad's Depth of Field in Depth (PDF) Includes derivations of most DoF formulae
• Joe Englander's Apparent Depth of Field: Practical Use in Landscape Photography (PDF). Alternative criteria for circle of confusion.
• David Jacobson's Photographic Lenses Tutorial
• Littlefield's "An Introduction to Extended Depth of Field Digital Photography"
• Paul van Walree's DOF with Pupil Magnification
• CombineZ - free software for increasing DoF of digital photos by combining differently focused versions of the same shot

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