Definitions

# Shot noise

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.

The strength 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, shot noise is often only a problem with small currents or light intensities.

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

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

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

Shot noise exists because things like light and electrical charge come in little packets — that is, they are quantized. Imagine light coming out of a laser pointer and hitting a wall. That light comes in small packets, or photons. When the spot is bright enough to see, there are many billions of light photons that hit the wall per second. Now, imagine turning down the laser brightness until the laser is almost off. Then, only a few photons hit the wall every second. But the fundamental physical processes that govern light emission say that these photons are emitted from the laser at random times. So if the average number of photons that hit the wall each second is 5, some seconds when you measure you will find 2 photons, and some 10 photons. These fluctuations are shot noise.

An analogy is to think of what happens when you pour sugar out of a cup. Start off by tilting the cup a lot so that a lot of sugar runs out. Now, decrease the tilt so less and less sugar pours out each second, until only 5 sugar grains are dropping out each second. Even if you hold the cup perfectly still, you won't get exactly 5 grains to come out every second -- sometimes you will see 2 grains and sometimes you will see 10. The physics that governs how sugar grains fall off a pile causes the number to be random, just like the physics that governs light emission from a light source. And for large flows, shot noise is not important -- when you tip the cup a lot, the small fluctuations due to random individual grains become small relative to the large continuous flow.

### In electronic devices

Shot noise in electronic devices consists of random fluctuations of the electric current in many electrical conductors, which are caused by the fact that the current is carried by discrete charges (electrons). 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 to be distinguished 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 100mA 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 f R.

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.

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:

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