Shot noise

Photon noise simulation.

Shot noise is a type of electronic noise that occurs when the finite number of particles that carry energy (such as electrons in an electronic circuit or photons in an optical device) is small enough to give rise to detectable statistical fluctuations in a measurement. It is important in electronics, telecommunications, and fundamental physics.

It also refers to an analogous noise in particle simulations, where due to the small number of particles, the simulation exhibits detectable statistical fluctuations not observed in the real-world system. Magnitude of this noise increases with the average magnitude of the current or intensity of the light. However, since the magnitude of the average signal increases more rapidly than that of the shot noise (its relative strength decreases with increasing signal), shot noise is often only a problem with small currents or light intensities.

The number of photons that are collected by a given detector varies, and follows a Poisson distribution, depicted here for averages of 1, 4, and 10.

The intensity of a source will yield the average number of photons collected, but knowing the average number of photons which will be collected will not give the actual number collected. The actual number collected will be more than, equal to, or less than the average, and their distribution about that average will be a Poisson distribution.

Since the Poisson distribution approaches a normal distribution for large numbers, the photon noise in a signal will approach a normal distribution for large numbers of photons collected. The standard deviation of the photon noise is equal to the square root of the average number of photons. The signal-to-noise ratio is then

${\mbox{SNR}}={\frac {N}{\sqrt {N}}}={\sqrt {N}}$

where N is the average number of photons collected. When N is very large, the signal-to-noise ratio is very large as well. It can be seen that photon noise becomes more important when the number of photons collected is small.

Explanation

Intuitive explanation

It is known to everyone that in a statistical experiment such as tossing a fair coin and counting the occurrences of heads and tails, the numbers of heads and tails after a great many throws will differ by only a tiny percentage, while after only few throws outcomes with a significant excess of heads over tails or vice versa are common; if an experiment with a few throws is repeated over and over, the outcomes will fluctuate a lot. (It can be proved that the relative fluctuations reduce as the square root of the number of throws, a result valid for all statistical fluctuations, including shot noise.)

Shot noise exists because phenomena such as light and electric current consist of the movement of discrete, quantized 'packets'. Consider light—a stream of discrete photons—coming out of a laser pointer and hitting a wall to create a visible spot. The fundamental physical processes that govern light emission are such that these photons are emitted from the laser at random times; but the many billions of photons needed to create a spot are so many that the brightness, the number of photons per unit time, varies only infinitesimally with time. However, if the laser brightness is reduced until only a handful of photons hit the wall every second, the relative fluctuations in number of photons, i.e. brightness, will be significant, just as when tossing a coin a few times. These fluctuations are shot noise.

In electronic devices

Shot noise in electronic devices consists of random fluctuations of the electric current in many electrical conductors, due to the current being carried by discrete charges (electrons) whose number per unit time fluctuates. This is often an issue in p-n junctions. In metal wires this is not an issue, as correlations between individual electrons remove these random fluctuations.

Shot noise is distinct from current fluctuations in equilibrium, which happen without any applied voltage and without any average current flowing. These equilibrium current fluctuations are known as Johnson-Nyquist noise.

Shot noise is a Poisson process and the charge carriers which make up the current will follow a Poisson distribution. The current fluctuations have a standard deviation of

$\sigma _{i}={\sqrt {2\,q\,I\,\Delta f}}$

where q is the elementary charge, Δf is the bandwidth in hertz over which the noise is measured, and I is the average current through the device. All quantities are assumed to be in SI units.

For a current of 100 mA this gives a value of

$\sigma _{i}=0.18\,\mathrm {nA}$

if the noise current is filtered with a filter having a bandwidth of 1 Hz.

If this noise current is fed through a resistor the resulting noise power will be

$P=2\,q\,I\,\Delta fR.$

If the charge is not fully localized in time but has a temporal distribution of q F(t) where the integral of F(t) over t is unity then the power spectral density of the noise current signal will be,

$S_{i}(f)=2\,q\,I\,|\Psi (f)|^{2},$

where Ψ(f) is the Fourier transform of F(t).

Note: Shot noise and Johnson–Nyquist noise are both quantum fluctuations. Some authors treat them as a single unified concept (see discussion).

In quantum optics

In quantum optics, shot noise is caused by the fluctuations of detected photons, again therefore a consequence of discretization (of the energy in the electromagnetic field in this case). Shot noise is a main part of quantum noise (see quantum limit).

Shot noise is measurable not only in measurements at the few-photons level using photomultipliers, but also at stronger light intensities measured by photodiodes when using high temporal resolution oscilloscopes. As the photocurrent is proportional to the light intensity (number of photons), the fluctuations of the electromagnetic field are usually contained in the electric current measured.

In the case of a coherent light source such as a laser, the shot noise scales as the square-root of the average intensity:

$\Delta I^{2}\ {\stackrel {\mathrm {def} }{=}}\ \langle \left(I-\langle I\rangle \right)^{2}\rangle \propto I.$

A similar lower bound of quantum noise occurs in linear quantum amplifiers. The only exception being if a squeezed coherent state can be formed through correlated photon generation. The reduction of uncertainty of the number of photons per mode (and therefore the photocurrent) may take place just due to the saturation of gain; this is intermediate case between a laser with locked phase and amplitude-stabilized laser.

Space charge

Low noise active electronic devices are designed such that shot noise is suppressed by the electrostatic repulsion of the charge carriers. Space charge limiting is not possible in photon devices.

See also

• 1/f noise
• Burst noise
• Image noise
• Quantum efficiency

References

 This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22. (in support of MIL-STD-188).

External links

• Shot noise on arxiv.org