Note that M T = − f u − f {\textstyle M_{T}=-{\frac {f}{u-f}}} is the transverse magnification which is the ratio of the lateral image size to the lateral subject size.[5]

Paraxial raysmirror

Figure 4 shows how the beam radius, calculated with wave optics, evolves. Here, one can see that the beam radius in the focal point has a finite value related to diffraction.

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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, so the ratio is 1:∞; as the subject distance decreases, near:far DOF ratio increases, approaching unity at high magnification. For large apertures at typical portrait distances, the ratio is still close to 1:1.

The blur disk diameter b of a detail at distance xd from the subject can be expressed as a function of the subject magnification ms, focal length f, f-number N, or alternatively the aperture d, according to

Hansma and Peterson have discussed determining the combined effects of defocus and diffraction using a root-square combination of the individual blur spots.[30][31] Hansma's approach determines the f-number that will give the maximum possible sharpness; Peterson's approach determines the minimum f-number that will give the desired sharpness in the final image and yields a maximum depth of field for which the desired sharpness can be achieved.[d] In combination, the two methods can be regarded as giving a maximum and minimum f-number for a given situation, with the photographer free to choose any value within the range, as conditions (e.g., potential motion blur) permit. Gibson gives a similar discussion, additionally considering blurring effects of camera lens aberrations, enlarging lens diffraction and aberrations, the negative emulsion, and the printing paper.[27][e] Couzin gave a formula essentially the same as Hansma's for optimal f-number, but did not discuss its derivation.[32]

s = 2 D N D F D N + D F , {\displaystyle s={\frac {2D_{\mathrm {N} }D_{\mathrm {F} }}{D_{\mathrm {N} }+D_{\mathrm {F} }}},}

For a given subject framing and camera position, the DOF is controlled by the lens aperture diameter, which is usually specified as the f-number (the ratio of lens focal length to aperture diameter). Reducing the aperture diameter (increasing the f-number) increases the DOF because only the light travelling at shallower angles passes through the aperture so only cones of rays with shallower angles reach the image plane. In other words, the circles of confusion are reduced or increasing the DOF.[10]

Geometrical optics is a widely used concept in optics, where the propagation of light is described with geometric light rays. An equivalent term is ray optics.

Some methods and equipment allow altering the apparent DOF, and some even allow the DOF to be determined after the image is made. These are based or supported by computational imaging processes. For example, focus stacking combines multiple images focused on different planes, resulting in an image with a greater (or less, if so desired) apparent depth of field than any of the individual source images. Similarly, in order to reconstruct the 3-dimensional shape of an object, a depth map can be generated from multiple photographs with different depths of field. Xiong and Shafer concluded, in part, "... the improvements on precisions of focus ranging and defocus ranging can lead to efficient shape recovery methods."[21]

Image sensor size affects DOF in counterintuitive ways. Because the circle of confusion is directly tied to the sensor size, decreasing the size of the sensor while holding focal length and aperture constant will decrease the depth of field (by the crop factor). The resulting image however will have a different field of view. If the focal length is altered to maintain the field of view, while holding the f-number constant, the change in focal length will counter the decrease of DOF from the smaller sensor and increase the depth of field (also by the crop factor). However, if the focal length is altered to maintain the field of view, while holding the aperture diameter constant, the DOF will remain constant. [6][7][8][9]

Nonparaxial rays

For a given size of the subject's image in the focal plane, the same f-number on any focal length lens will give the same depth of field.[11] This is evident from the above DOF equation by noting that the ratio u/f is constant for constant image size. For example, if the focal length is doubled, the subject distance is also doubled to keep the subject image size the same. This observation contrasts with the common notion that "focal length is twice as important to defocus as f/stop",[12] which applies to a constant subject distance, as opposed to constant image size.

Precise focus is only possible at an exact distance from a lens;[a] at that distance, a point object will produce a small spot image. Otherwise, a point object will produce a larger or blur spot image that is typically and approximately a circle. When this circular spot is sufficiently small, it is visually indistinguishable from a point, and appears to be in focus. The diameter of 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.

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Attempts for physical interpretations of light rays can be successful only to a quite limited extent. For example, rays were interpreted as the paths of some rapidly moving light particles, but this picture is not consistent with various observations. There are some similarities between geometrical light rays and real light beams, in particular with laser beams; for example, a laser beam can at least be relatively narrow and propagate along a straight line in a homogeneous medium. However, real light beams always have a finite transverse extension and exhibit the phenomenon of diffraction. Therefore, geometrical rays are only a rather abstract representation of actual light rays. Their behavior can be derived from wave optics in the limiting case of vanishing optical wavelength.

Moreover, traditional depth-of-field formulas assume equal acceptable circles of confusion for near and far objects. Merklinger[c] 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.[19] 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.

The depth of field (DOF) is the distance between the nearest and the farthest objects that are in acceptably sharp focus in an image captured with a camera. See also the closely related depth of focus.

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The term "camera movements" refers to swivel (swing and tilt, in modern terminology) and shift adjustments of the lens holder and the film holder. These features have been in use since the 1800s and are still in use today on view cameras, technical cameras, cameras with tilt/shift or perspective control lenses, etc. Swiveling the lens or sensor causes the plane of focus (POF) to swivel, and also causes the field of acceptable focus to swivel with the POF; and depending on the DOF criteria, to also change the shape of the field of acceptable focus. While calculations for DOF of cameras with swivel set to zero have been discussed, formulated, and documented since before the 1940s, documenting calculations for cameras with non-zero swivel seem to have begun in 1990.

Rays may be split up into multiple rays, e.g. due to partial reflection and transmission that interfaces, or due to multiple diffraction orders at gratings. In the context of diffuse optical scattering, one may employ stochastic methods for representing the scattered light with some limited number of rays. For multiple diffuse reflections, this may of course result in a very large number of rays to be treated.

Other technologies use a combination of lens design and post-processing: Wavefront coding is a method by which controlled aberrations are added to the optical system so that the focus and depth of field can be improved later in the process.[25]

Paraxial raysformula

Thomas Sutton and George Dawson first wrote about hyperfocal distance (or "focal range") in 1867.[42] Louis Derr in 1906 may have been the first to derive a formula for hyperfocal distance. Rudolf Kingslake wrote in 1951 about the two methods of measuring hyperfocal distance.

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

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One may not only need to know the paths of multiple rays, but also derive various results from them. For example, ray tracing software may locate focal planes, calculate image magnification or estimates resulting optical intensities and colors.

Ray tracing can be used for many purposes, for example for studying the detailed properties of imaging systems including their optical aberrations and effects of misalignment and imperfections from optical fabrication, or for the design of illumination systems.

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Modified laws can be applied in the case of diffraction gratings, where additional diffracted rays emerge at different angles.

Paraxialray tracing

Hopkins,[33] Stokseth,[34] and Williams and Becklund[35] have discussed the combined effects using the modulation transfer function.[36][37]

Many useful relations can be derived based on such paraxial optics, which would otherwise be far more complicated or not analytically solvable at all.

The main limitation of geometrical optics is that it ignores the wave properties of light, as described in wave optics. In particular, that means that the phenomena of diffraction, interference and polarization are not taken into account. This is not a substantial problem in many practical cases, where such effects may be negligible or can be taken into account separately. For example, one can study the optical aberrations of an imaging system with geometrical optics, being aware that even for perfect compensation of aberrations one will not obtain perfectly sharp images due to the diffraction limit. Anyway, aberrations often remain a more severe limitation than diffraction, which can thus often be safely ignored.

Motion pictures make limited use of aperture control; to produce a consistent image quality from shot to shot, cinematographers usually choose a single aperture setting for interiors (e.g., scenes inside a building) and another for exteriors (e.g., scenes in an area outside a building), 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.

the harmonic mean of the near and far distances. In practice, this is equivalent to the arithmetic mean for shallow depths of field.[44] Sometimes, view camera users refer to the difference vN − vF as the focus spread.[45]

Paraxial raysexamples

When the POF is rotated, the near and far limits of DOF may be thought of as wedge-shaped, with the apex of the wedge nearest the camera; or they may be thought of as parallel to the POF.[17][18]

The hyperfocal distance has a property called "consecutive depths of field", where a lens focused at an object whose distance from the lens is at the hyperfocal distance H will hold a depth of field from H/2 to infinity, if the lens is focused to H/2, the depth of field will be from H/3 to H; if the lens is then focused to H/3, the depth of field will be from H/4 to H/2, etc.

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In optically inhomogeneous media, light beams may propagate along curves instead of straight lines. In geometrical optics, one may correspondingly assume curved ray paths. An example is shown in Figure 3, showing the focusing of light in a gradient-index lens. The rays get deflected in the lens and may exactly meet in a focal point if the lens is optimized.

Photographers can use the lens scales to work backwards from the desired depth of field to find the necessary focus distance and aperture.[38] For the 35 mm lens shown, if it were desired for the DOF to extend from 1 m to 2 m, focus would be set so that index mark was centered between the marks for those distances, and the aperture would be set to f/11.[f]

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Similarly, light propagation in multilayer coatings cannot be realistically analyzed with ray optics because interference effects are essential.

The cylinder value is required for the production of contact lenses for astigmatism. It is a negative value that corrects the corneal curvature. Axis (A/ACH/ ...

The blur increases with the distance from the subject; when b is less than the circle of confusion, the detail is within the depth of field.

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Ignoring diffraction becomes a serious problem when treating the propagation of light under conditions where it experiences tight confinement. For example, light propagation in single-mode fibers can not be realistically described at all with geometrical optics. One may still define the numerical aperture of a fiber, for example, based on geometrical optics, but such a quantity then has only a limited meaning for the actual propagation of light in the fiber. Even for multimode fibers with many modes, geometrical optics is only of quite limited utility. It can be completely misleading, for example, concerning optical phase delays of fiber modes [1].

When a ray hits an interface between two different transparent media, a portion is reflected, and another portion is transmitted; for the latter one, which is refracted, the propagation direction is generally modified according to the Snellius law of refraction. Figure 2 shows an example case for a ball lens, where only the refracted rays (which are usually stronger) have been drawn.

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DOF ≈ 2 N c ( u f ) 2 = 2 N c ( 1 − 1 M T ) 2 {\displaystyle {\text{DOF}}\approx 2Nc\left({\frac {u}{f}}\right)^{2}=2Nc\left(1-{\frac {1}{M_{T}}}\right)^{2}}

Traditional depth-of-field formulas can be hard to use in practice. As an alternative, the same effective calculation can be done without regard to the focal length and f-number.[b] Moritz von Rohr and later Merklinger observe that the effective absolute aperture diameter can be used for similar formula in certain circumstances.[19]

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Light Scanning Photomacrography (LSP) is another technique used to overcome depth of field limitations in macro and micro photography. This method allows for high-magnification imaging with exceptional depth of field. LSP involves scanning a thin light plane across the subject that is mounted on a moving stage perpendicular to the light plane. This ensures the entire subject remains in sharp focus from the nearest to the farthest details, providing comprehensive depth of field in a single image. Initially developed in the 1960s and further refined in the 1980s and 1990s, LSP was particularly valuable in scientific and biomedical photography before digital focus stacking became prevalent.[23][24]

The lens design can be changed even more: in colour apodization the lens is modified such that each colour channel has a different lens aperture. For example, the red channel may be f/2.4, green may be f/2.4, whilst the blue channel may be f/5.6. Therefore, the blue channel will have a greater depth of field than the other colours. The image processing identifies blurred regions in the red and green channels and in these regions copies the sharper edge data from the blue channel. The result is an image that combines the best features from the different f-numbers.[26]

At the extreme, a plenoptic camera captures 4D light field information about a scene, so the focus and depth of field can be altered after the photo is taken.

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To some extent, the deficits of geometrical optics can be amended by adding additional properties to rays. For example, one may attribute some optical power to each ray in a ray tracing simulation, taking into account power losses by absorption, incomplete reflection, etc. Similarly, one may add polarization properties and optical phases, for example for calculations on an interferometer setup. A simpler example is the calculation of different ray paths for different polarization directions, for example when analyzing a polarizing prism.

On a view camera, the focus and f-number can be obtained by measuring the depth of field and performing simple calculations. Some view cameras include DOF calculators that indicate focus and f-number without the need for any calculations by the photographer.[39][40]

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On the surface of a flat mirror, a light ray is assumed to be reflected such that the output angle equals the input angle (both measured against the normal direction). For a curved mirror, one does that calculation based on a tangential flat plane. Figure 1 shows an example with reflection on a curved mirror.

Paraxial rayspdf

The depth of field can be determined by focal length, distance to subject (object to be imaged), the acceptable circle of confusion size, and aperture.[2] Limitations of depth of field can sometimes be overcome with various techniques and equipment. The approximate depth of field can be given by:

The acceptable circle of confusion depends on how the final image will be used. The circle of confusion as 0.25 mm for an image viewed from 25 cm away is generally accepted.[14]

Other authors such as Ansel Adams have taken the opposite position, maintaining that slight unsharpness in foreground objects is usually more disturbing than slight unsharpness in distant parts of a scene.[20]

Many lenses include scales that indicate the DOF for a given focus distance and f-number; the 35 mm lens in the image 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.

More so than in the case of the zero swivel camera, there are various methods to form criteria and set up calculations for DOF when swivel is non-zero. There is a gradual reduction of clarity in objects as they move away from the POF, and at some virtual flat or curved surface the reduced clarity becomes unacceptable. Some photographers do calculations or use tables, some use markings on their equipment, some judge by previewing the image.

These explain quite comprehensively a wide range of aspects, not only physical principles of operation, but also various practical issues.

Diffraction causes images to lose sharpness at high f-numbers (i.e., narrow aperture stop opening sizes), and hence limits the potential depth of field.[27] (This effect is not considered in the above formula giving approximate DOF values.) In general photography this is rarely an issue; because large f-numbers typically require long exposure times to acquire acceptable image brightness, motion blur may cause greater loss of sharpness than the loss from diffraction. However, diffraction is a greater issue in close-up photography, and the overall image sharpness can be degraded as photographers are trying to maximize depth of field with very small apertures.[28][29]

In optics and photography, hyperfocal distance is a distance from a lens beyond which all objects can be brought into an "acceptable" focus. As the hyperfocal distance is the focus distance giving the maximum depth of field, it is the most desirable distance to set the focus of a fixed-focus camera.[41] The hyperfocal distance is entirely dependent upon what level of sharpness is considered to be acceptable.

Paraxialapproximation

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b = f m s N x d s ± x d = d m s x d D . {\displaystyle b={\frac {fm_{\mathrm {s} }}{N}}{\frac {x_{\mathrm {d} }}{s\pm x_{\mathrm {d} }}}=dm_{\mathrm {s} }{\frac {x_{\mathrm {d} }}{D}}.}

In many situations, one can use simplified equations which describe the approximate propagation of rays which stay close to the optical axis in terms of lateral offset and direction. Any terms of second or higher order are ignored; for example, one may consider the deflection of a ray at a curved lens surface as occurring in the plane touching the surface, ignoring a longitudinal position error of second order in the lateral offset.

For 35 mm motion pictures, the image area on the film is roughly 22 mm by 16 mm. The limit of tolerable error was traditionally set at 0.05 mm (0.0020 in) diameter, while for 16 mm film, where the size is about half as large, the tolerance is stricter, 0.025 mm (0.00098 in).[15] More modern practice for 35 mm productions set the circle of confusion limit at 0.025 mm (0.00098 in).[16]

The propagation of light rays, as shown in the figures above, is calculated based on purely geometrical considerations. The used technique is called ray tracing and is usually applied with specialized optics software. The calculations can be geometrically exact, i.e., valid even for large incidence angles. Curved surface may have any geometrical shapes. Depending on the initial direction of a light ray, it may or may not hit a certain optical component.

For cameras that can only focus on one object distance at a time, depth of field is the distance between the nearest and the farthest objects that are in acceptably sharp focus in the image.[1] "Acceptably sharp focus" is defined using a property called the "circle of confusion".

This section covers some additional formula for evaluating depth of field; however they are all subject to significant simplifying assumptions: for example, they assume the paraxial approximation of Gaussian optics. They are suitable for practical photography, lens designers would use significantly more complex ones.

Paraxial raysin physics

Another approach is focus sweep. The focal plane is swept across the entire relevant range during a single exposure. This creates a blurred image, but with a convolution kernel that is nearly independent of object depth, so that the blur is almost entirely removed after computational deconvolution. This has the added benefit of dramatically reducing motion blur.[22]

As distance or the size of the acceptable circle of confusion increases, the depth of field increases; however, increasing the size of the aperture (i.e., reducing f-number) or increasing the focal length reduces the depth of field. Depth of field changes linearly with f-number and circle of confusion, but changes in proportion to the square of the distance to the subject and inversely in proportion to the square of the focal length. As a result, photos taken at extremely close range (i.e., so small u) have a proportionally much smaller depth of field.