## Saturday, August 25, 2007

### Shot noise

Shot noise
Photon noise simulation.
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.

The strength of this noise increases with the average magnitude of the current or intensity of the light. Often, however, as the signal increases more rapidly as the average signal becomes stronger, shot noise often is only a problem with small currents and 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

S\!N\!R = \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.
Contents
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* 1 Explanation
o 1.1 In electronic devices
o 1.2 In quantum optics
* 2 Space charge
* 3 References

 Explanation

 In electronic devices

Shot noise in electronic devices consists of random fluctuations of the electric current in an electrical conductor, which are caused by the fact that the current is carried by discrete charges (electrons). This occurs not only in p-n junctions but also in any conductor, and even in the case where the charge is not well localized.

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 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 = \frac{0.18\,nA}{\sqrt{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 in 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 [1] (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{\mathrm{def}}{=}\ \langle\left(I-\langle I\rangle \right)^2\rangle \propto I.

Similar lower bound of quantum noise takes place for linear quantum amplifier. 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.

 References

1. ^ R. Sarpeshkar, T. Delbruck, and C. A. Mead, "White noise in MOS transistors and resistors", IEEE Circuits Devices Mag., pp. 23–29, Nov. 1993.

This article contains material from the Federal Standard 1037C (in support of MIL-STD-188), which, as a work of the United States Government, is in the public domain.

* Shot noise on arxiv.org

Categories: Noise | Electrical parameters | Quantum optics
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