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Reciprocity (photography)

Reciprocity (photography)

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In photography
Photography
Photography is the art, science and practice of creating durable images by recording light or other electromagnetic radiation, either electronically by means of an image sensor or chemically by means of a light-sensitive material such as photographic film...

 reciprocity refers to the inverse relationship between the intensity and duration of light that determines the reaction of light-sensitive material. Within a normal exposure
Exposure (photography)
In photography, exposure is the total amount of light allowed to fall on the photographic medium during the process of taking a photograph. Exposure is measured in lux seconds, and can be computed from exposure value and scene luminance over a specified area.In photographic jargon, an exposure...

 range for film stock, for example, the reciprocity law states that the film response will be determined by the total exposure, defined as intensity × time. Therefore, the same response (for example, the optical density of the developed film) can result from reducing duration and increasing light intensity, and vice versa.

The reciprocal relationship is assumed in most sensitometry
Sensitometry
Sensitometry is the scientific study of light-sensitive materials, especially photographic film. The study has its origins in the work by Ferdinand Hurter and Vero Charles Driffield with early black-and-white emulsions...

, for example when measuring a Hurter and Driffield
Hurter and Driffield
Ferdinand Hurter and Vero Charles Driffield were nineteenth-century photographic scientists who brought quantitative scientific practice to photography through the methods of sensitometry and densitometry....

 curve (optical density versus logarithm of total exposure) for a photographic emulsion. Total exposure of the film or sensor, the product of focal-plane illuminance
Illuminance
In photometry, illuminance is the total luminous flux incident on a surface, per unit area. It is a measure of the intensity of the incident light, wavelength-weighted by the luminosity function to correlate with human brightness perception. Similarly, luminous emittance is the luminous flux per...

 times exposure time, is measured in lux
Lux
The lux is the SI unit of illuminance and luminous emittance, measuring luminous flux per unit area. It is used in photometry as a measure of the intensity, as perceived by the human eye, of light that hits or passes through a surface...

 second
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....

s.

History


The idea of reciprocity, once known as Bunsen–Roscoe reciprocity, originated from the work of Robert Bunsen
Robert Bunsen
Robert Wilhelm Eberhard Bunsen was a German chemist. He investigated emission spectra of heated elements, and discovered caesium and rubidium with Gustav Kirchhoff. Bunsen developed several gas-analytical methods, was a pioneer in photochemistry, and did early work in the field of organoarsenic...

 and Henry Roscoe in 1862.

Deviations from the reciprocity law were reported by Captain William de Wiveleslie Abney
William de Wiveleslie Abney
William de Wiveleslie Abney FRS was an English astronomer, chemist, and photographer.-Biography:Abney was born in Derby, England, the son of Edward Abney vicar of St Alkmund's Derby, and owner of the Firs Estate...

 in 1893,
and extensively studied by Karl Schwarzschild
Karl Schwarzschild
Karl Schwarzschild was a German physicist. He is also the father of astrophysicist Martin Schwarzschild.He is best known for providing the first exact solution to the Einstein field equations of general relativity, for the limited case of a single spherical non-rotating mass, which he accomplished...

 in 1899. Schwarzschild's model was found wanting by Abney and by Englisch, and better models have been proposed in subsequent decades of the early twentieth century. In 1913, Kron formulated an equation to describe the effect in terms of curves of constant density, which J. Halm adopted and modified, leading to the "Kron–Halm catenary
Catenary
In physics and geometry, the catenary is the curve that an idealised hanging chain or cable assumes when supported at its ends and acted on only by its own weight. The curve is the graph of the hyperbolic cosine function, and has a U-like shape, superficially similar in appearance to a parabola...

 equation"
or "Kron–Halm–Webb formula"
to describe departures from reciprocity.

In chemical photography


In photography
Photography
Photography is the art, science and practice of creating durable images by recording light or other electromagnetic radiation, either electronically by means of an image sensor or chemically by means of a light-sensitive material such as photographic film...

, reciprocity refers to the relationship whereby the total light energy – proportional to the total exposure
Exposure (photography)
In photography, exposure is the total amount of light allowed to fall on the photographic medium during the process of taking a photograph. Exposure is measured in lux seconds, and can be computed from exposure value and scene luminance over a specified area.In photographic jargon, an exposure...

, the product of the light intensity and exposure time, controlled by aperture
Aperture
In optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture of an optical system is the opening that determines the cone angle of a bundle of rays that come to a focus in the image plane. The aperture determines how collimated the admitted rays are,...

 and shutter speed
Shutter speed
In photography, shutter speed is a common term used to discuss exposure time, the effective length of time a camera's shutter is open....

, respectively – determines the effect of the light on the film. That is, an increase of brightness by a certain factor is exactly compensated by a decrease of exposure time by the same factor, and vice versa. In other words there is under normal circumstances a reciprocal proportion
Proportionality (mathematics)
In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...

 between aperture area and shutter speed for a given photographic result, with a wider aperture requiring a faster shutter speed for the same effect. For example, an EV
Exposure value
In photography, exposure value denotes all combinations of a camera's shutter speed and relative aperture that give the same exposure. In an attempt to simplify choosing among combinations of equivalent camera settings, the concept was developed by the German shutter manufacturer in the 1950s...

 of 10 may be achieved with an aperture (f-number
F-number
In optics, the f-number of an optical system expresses the diameter of the entrance pupil in terms of the focal length of the lens; in simpler terms, the f-number is the focal length divided by the "effective" aperture diameter...

) of 2.8 and a shutter speed of 1/125 s
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....

. The same exposure is achieved by doubling the aperture area to 2 and halving the exposure time to 1/250 s, or by halving the aperture area to 4 and doubling the exposure time to 1/60 s; in each case the response of the film is expected to be the same.

Reciprocity failure


For most photographic materials, reciprocity is valid with good accuracy over a range of values of exposure duration, but becomes increasingly inaccurate as we depart from this range: reciprocity failure, reciprocity law failure, or Schwarzschild effect. As the light level decreases out of the reciprocity range, the increase in duration, and hence of total exposure, required to produce an equivalent response becomes higher than the formula states; for instance, at half of the light required for a normal exposure, the duration must be more than doubled for the same result. Multipliers used to correct for this effect are called reciprocity factors (see model below).

At very low light levels, film
Photographic film
Photographic film is a sheet of plastic coated with an emulsion containing light-sensitive silver halide salts with variable crystal sizes that determine the sensitivity, contrast and resolution of the film...

 is less responsive. Light can be considered to be a stream of discrete photons
Wave–particle duality
Wave–particle duality postulates that all particles exhibit both wave and particle properties. A central concept of quantum mechanics, this duality addresses the inability of classical concepts like "particle" and "wave" to fully describe the behavior of quantum-scale objects...

, and a light-sensitive emulsion is composed of discrete light-sensitive grain
Film grain
Film grain or granularity is the random optical texture of processed photographic film due to the presence of small particles of a metallic silver, or dye clouds, developed from silver halide that have received enough photons. While film grain is a function of such particles it is not the same...

s, usually silver halide
Silver halide
A silver halide is one of the compounds formed between silver and one of the halogens — silver bromide , chloride , iodide , and three forms of silver fluorides. As a group, they are often referred to as the silver halides, and are often given the pseudo-chemical notation AgX...

 crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...

s. Each grain must absorb a certain number of photons in order for the light-driven reaction to occur and the latent image
Latent image
A latent image on photographic film is an invisible image produced by the exposure of the film to light. When the film is developed, the area that was exposed darkens and forms a visible image...

 to form. In particular, if the surface of the silver halide crystal has a cluster of approximately four or more reduced silver atoms, resulting from absorption of a sufficient number of photons (usually a few dozen photons are required), it is rendered developable. At low light levels, i.e. few photons per unit time, photons impinge upon each grain relatively infrequently; if the four photons required arrive over a long enough interval, the partial change due to the first one or two is not stable enough to survive before enough photons arrive to make a permanent latent image
Latent image
A latent image on photographic film is an invisible image produced by the exposure of the film to light. When the film is developed, the area that was exposed darkens and forms a visible image...

 center.

This breakdown in the usual tradeoff between aperture and shutter speed is known as reciprocity failure. Each different film type has a different response at low light levels. Some films are very susceptible to reciprocity failure, and others much less so. Some films that are very light sensitive at normal illumination levels and normal exposure times lose much of their sensitivity at low light levels, becoming effectively "slow" films for long exposures. Conversely some films that are "slow" under normal exposure duration retain their light sensitivity better at low light levels.

For example, for a given film, if a light meter
Light meter
A light meter is a device used to measure the amount of light. In photography, a light meter is often used to determine the proper exposure for a photograph...

 indicates a required EV
Exposure value
In photography, exposure value denotes all combinations of a camera's shutter speed and relative aperture that give the same exposure. In an attempt to simplify choosing among combinations of equivalent camera settings, the concept was developed by the German shutter manufacturer in the 1950s...

 of 5 and the photographer sets the aperture to f/11, then ordinarily a 4 second exposure would be required; a reciprocity correction factor of 1.5 would require the exposure to be extended to 6 seconds for the same result. Reciprocity failure generally becomes significant at exposures of longer than about 1 sec for film, and above 30 sec for paper.

Reciprocity also breaks down at extremely high levels of illumination with very short exposures. This is concern for scientific and technical
Technology
Technology is the making, usage, and knowledge of tools, machines, techniques, crafts, systems or methods of organization in order to solve a problem or perform a specific function. It can also refer to the collection of such tools, machinery, and procedures. The word technology comes ;...

 photography, but rarely to general photographers, as exposures significantly shorter than a millisecond
Millisecond
A millisecond is a thousandth of a second.10 milliseconds are called a centisecond....

 are only required for subjects such as explosion
Explosion
An explosion is a rapid increase in volume and release of energy in an extreme manner, usually with the generation of high temperatures and the release of gases. An explosion creates a shock wave. If the shock wave is a supersonic detonation, then the source of the blast is called a "high explosive"...

s and particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

 experiment
Experiment
An experiment is a methodical procedure carried out with the goal of verifying, falsifying, or establishing the validity of a hypothesis. Experiments vary greatly in their goal and scale, but always rely on repeatable procedure and logical analysis of the results...

s, or when taking high-speed motion pictures with very high shutter speeds (1/10,000 sec or faster).

Schwarzschild law


In response to astronomical observations of low intensity reciprocity failure, Karl Schwarzschild
Karl Schwarzschild
Karl Schwarzschild was a German physicist. He is also the father of astrophysicist Martin Schwarzschild.He is best known for providing the first exact solution to the Einstein field equations of general relativity, for the limited case of a single spherical non-rotating mass, which he accomplished...

 wrote (circa 1900):
"In determinations of stellar brightness by the photographic method I have recently been able to confirm once more the existence of such deviations, and to follow them up in a quantative way, and to express them in the following rule, which should replace the law of reciprocity: Sources of light of different intensity I cause the same degree of blackening under different exposures t if the products are equal."


Unfortunately, Schwarzschild's empirically determined 0.86 coefficient turned out to be of limited usefulness.
A modern formulation of Schwarzschild's law is given as


where E is a measure of the "effect of the exposure" that leads to changes in the opacity
Opacity (optics)
Opacity is the measure of impenetrability to electromagnetic or other kinds of radiation, especially visible light. In radiative transfer, it describes the absorption and scattering of radiation in a medium, such as a plasma, dielectric, shielding material, glass, etc...

 of the photosensitive material (in the same degree that an equal value of exposure H = It does in the reciprocity region), I is illuminance
Illuminance
In photometry, illuminance is the total luminous flux incident on a surface, per unit area. It is a measure of the intensity of the incident light, wavelength-weighted by the luminosity function to correlate with human brightness perception. Similarly, luminous emittance is the luminous flux per...

, t is exposure duration and p is the Schwarzschild coefficient.

However, a constant value for p remains elusive, and has not replaced the need for more realistic models or empirical sensitometric data in critical applications. When reciprocity holds, Schwarzschild's law uses p = 1.0.

Since the Schwarzschild's law formula gives unreasonable values for times in the region where reciprocity holds, a modified formula has been found that fits better across a wider range of exposure times. The modification is in terms of a factor the multiplies the ISO film speed
Film speed
Film speed is the measure of a photographic film's sensitivity to light, determined by sensitometry and measured on various numerical scales, the most recent being the ISO system....

:
Relative film speed


where the t + 1 term implies a breakpoint near 1 second separating the region where reciprocity holds from the region where it fails.

Simple model for t > 1 second


Some models of microscope use automatic electronic models for reciprocity failure compensation, generally of a form for correct time, Tc, expressible as a power law
Power law
A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...

 of metered time, Tm, that is, Tc=(Tm)p, for times in seconds. Typical values of p are 1.25 to 1.45, but some are low as 1.1 and high as 1.8.

The Kron–Halm catenary equation


Kron's equation as modified by Halm states that the response of the film is a function of , with the factor defined by an catenary
Catenary
In physics and geometry, the catenary is the curve that an idealised hanging chain or cable assumes when supported at its ends and acted on only by its own weight. The curve is the graph of the hyperbolic cosine function, and has a U-like shape, superficially similar in appearance to a parabola...

 (hyperbolic cosine) equation accounting for reciprocity failure at both very high and very low intensities:


where I0 is the photographic material's optimum intensity level and a is a constant that characterizes the material's reciprocity failure.

Quantum reciprocity-failure model


Modern models of reciprocity failure incorporate an exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

, as opposed to power law
Power law
A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...

, dependence on time or intensity at long exposure times or low intensities, based on the distribution of interquantic times (times between photon absorptions in a grain) and the temperature-dependent lifetimes of the intermediate states of the partially-exposed grains.

Baines and Bomback
explain the "low intensity inefficiency" this way:

Astrophotography


Reciprocity failure is an important effect in the field of film-based astrophotography
Astrophotography
Astrophotography is a specialized type of photography that entails recording images of astronomical objects and large areas of the night sky. The first photographs of an astronomical object were taken in the 1840s, but it was not until the late 19th century that advances in technology allowed for...

. Deep-sky objects such as galaxies and nebulae are often so faint that they are not visible to the un-aided eye. To make matters worse, many objects' spectra do not line up with the film emulsion's sensitivity curves. Many of these targets are small and require long focal lengths, which can push the focal ratio far above 5. Combined, these parameters make these targets extremely difficult to capture with film; exposures from 30 minutes to well over an hour are typical. As a typical example, capturing an image of the Andromeda Galaxy
Andromeda Galaxy
The Andromeda Galaxy is a spiral galaxy approximately 2.5 million light-years from Earth in the constellation Andromeda. It is also known as Messier 31, M31, or NGC 224, and is often referred to as the Great Andromeda Nebula in older texts. Andromeda is the nearest spiral galaxy to the...

 at 4 will take about 30 minutes; to get the same density at 8 would require an exposure of about 200 minutes.

When a telescope is tracking an object, every minute is difficult; therefore, reciprocity failure is one of the biggest motivations for astronomers to switch to digital imaging
Digital imaging
Digital imaging or digital image acquisition is the creation of digital images, typically from a physical scene. The term is often assumed to imply or include the processing, compression, storage, printing, and display of such images...

. Electronic image sensor
Image sensor
An image sensor is a device that converts an optical image into an electronic signal. It is used mostly in digital cameras and other imaging devices...

s have their own limitation at long exposure time and low illuminance levels, not usually referred to as reciprocity failure, namely noise from dark current
Dark current (physics)
In physics and in electronic engineering, dark current is the relatively small electric current that flows through photosensitive devices such as a photomultiplier tube, photodiode, or charge-coupled device even when no photons are entering the device. It is referred to as reverse bias leakage...

, but this effect can be controlled by cooling the sensor.

Holography


A similar problem exists in holography
Holography
Holography is a technique that allows the light scattered from an object to be recorded and later reconstructed so that when an imaging system is placed in the reconstructed beam, an image of the object will be seen even when the object is no longer present...

. The total energy required when exposing holographic film using a continuous wave laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

 (i.e. for several seconds) is significantly less than the total energy required when exposing holographic film using a pulsed laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

 (i.e. around 20–40 nanosecond
Nanosecond
A nanosecond is one billionth of a second . One nanosecond is to one second as one second is to 31.7 years.The word nanosecond is formed by the prefix nano and the unit second. Its symbol is ns....

s) due to a reciprocity failure. It can also be caused by very long or very short exposures with a continuous wave laser. To try to offset the reduced brightness of the film due to reciprocity failure, a method called latensification
Latensification
Latensification is the name given to uniformly pre-exposing a photographic emulsion .The benefits of latensification are applicable in astrophotography - capturing images of stars. Without latensification an image would come out with several visible stars and be a perfectly acceptable image...

can be used. This is usually done directly after the holographic exposure and using an incoherent light source (such as a 25-40 W light bulb). Exposing the holographic film to the light for a few seconds can increase the brightness of the hologram by an order of magnitude.

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