# Shot Noise

Definition: quantum-limited intensity noise

More general term: quantum noise

German: Schrotrauschen

Categories: quantum optics, fluctuations and noise

How to cite the article; suggest additional literature

Author: Dr. Rüdiger Paschotta

A fundamental limit to the optical intensity noise as observed in many situations (e.g. in measurements with a photodiode or a CCD image sensor) is given by *shot noise*.
This is a quantum noise effect, related to the discreteness of photons and electrons.
Originally, it was interpreted as arising from the random occurrence of photon absorption events in a photodetector, i.e. not as noise in the light field itself.
Intensity noise at the shot noise level is obtained when the probability for an absorption event per unit time is constant and not correlated with former events.
However, the existence of amplitude-squeezed light, which exhibits intensity noise below the shot noise level (*sub-Poissonian intensity noise*), proves that shot noise must be interpreted as a property of the light field itself, rather than as an issue of photodetection only – although a photodetector may be blamed for shot noise if it requires optical attenuation, which raises the shot noise level of the relative intensity (→ *relative intensity noise*).

Intensity noise at the shot noise level is obtained e.g. for a so-called *coherent state*, which may be approximated by the output of a laser at high noise frequencies.
At lower noise frequencies, laser noise is normally much higher due to relaxation oscillations, mode hopping, excess pump noise, and other phenomena.
The intensity noise of a simple incandescent lamp is fairly close to the shot noise level.
Noise below the shot noise level is obtained for amplitude-squeezed light, which can be obtained e.g. by transforming an original coherent state with the help of nonlinear interactions.

Linear absorption of light also pulls the noise level closer to the shot noise level. Therefore, the noise registered with a photodetector having a low quantum efficiency may be close to shot noise even if the incident light is well below the shot noise level.

Note also that background light often introduces not only just a constant addition to an actual signal, but also the corresponding shot noise. That makes it more difficult, for example, to detect a weak signal if the detector is at the same time affected by substantially more intense sunlight.

## Measurements at the Shot Noise Level

Photodetectors with high quantum efficiency and appropriate electronic circuitry are required for obtaining sub-shot noise sensitivity of intensity noise measurements.
A common configuration is that of a *balanced homodyne detector* (Figure 1) containing two photodetectors, where a beam splitter sends 50% of the optical power to each detector, and the sum and difference of the photocurrents are obtained electronically.
Whereas the sum of the photocurrents is the same as for using all light on a single detector, the difference signal provides a reference for the shot noise level.
The article on optical heterodyne detection gives more details.

A severe challenge can come from thermal noise in the electronics, particularly when the photocurrent is converted to a voltage in a small resistor, as is often required for achieving a high detection bandwidth.
Also, the *full* optical power needs to be detected, i.e. the measurement cannot be done on an attenuated beam.
Otherwise, the optical attenuation adds additional quantum noise.
(The finite quantum efficiency of the detector has the same kind of effect.)
If the full optical power is too high for a single detector, a possible method is to use beam splitters for distributing the power on several photodetectors, and to combine the photocurrents.

## Sub-shot-noise Electric Currents and Optical Noise

Note that an electric current with noise below the shot noise level can be obtained very easily, e.g. by connecting a quiet voltage source to a resistor. The reason for this is that electrons, being equally charged particles, experience a mutual repulsion, which gives them a natural tendency to “line up”, i.e. to pass a conductor with more regular than just random distances between them.

Efficient laser diodes, operated at low temperatures, can convert sub-shot-noise electric currents into light with intensity noise below the shot noise level (→ *amplitude-squeezed light*).
Surprisingly, the degree of squeezing is not even limited by the quantum efficiency of the laser diode.

Various optical nonlinearities can be used to generate light with quantum noise below the shot noise limit. This can be squeezed light, where one quadrature component is below the shot noise level, or light exhibiting certain quantum correlations.

## Important Equations

The one-sided power spectral density of the optical power in the case of shot noise is

which is proportional to the average power and the photon energy *hν*, and is independent of the noise frequency (i.e., shot noise is “white noise”).
As the power of a modulation signal with a given relative modulation amplitude scales with the *square* of the average power, the *relative* intensity noise decreases with increasing optical power.
In the formula for the power spectral density of the *relative* intensity noise at the shot noise limit, one would divide by the average power, rather than multiplying with it.

An often quoted equation for the shot noise in an electric current is

where *e* is the elementary charge.
This formula indicates the variance of the current for an average current *I* and a measurement bandwidth Δ*f*.
The equation corresponds to a one-sided power spectral density

of the photocurrent.

## Questions and Comments from Users

2020-07-30

Can shot noise be expressed in dBm/Hz, rather than dBc/Hz? If I understand correctly, the shot noise floor has a single value in dBm/Hz for each wavelength.

Let shot noise = 10 log(2 *hν* / *P*) in dBc/Hz (as indicated in your relative intensity noise article).
The 'c' in dBc means relative to the signal, so we multiply by the signal power *P* (or add the signal power in dBm) to get the shot noise power in dBm/Hz.
The *P* cancels, and we are left with shot noise = 10 * log(2 *hν*), or shot noise in dBm/Hz = 10 * log(2 * photon energy in mJ).
A laser beam at 1064 nm has a one-sided shot noise floor at −154 dBm/Hz.
The two-sided floor would be at −157 dBm/Hz.

A 1064-nm beam at a power level of −157 dBm is just 1 photon per second, which makes a shot noise floor at a sample rate of 1 second seem intuitive (ignoring the statistics of the arrival time of the photons, which is the source of the noise, and matters). At a 10 Hz one-sided bandwidth (1/20 second sampling period), one photon per sampling period is 20 photons per second, or −144 dBm for light at 1064 nm.

Is this a correct understanding of shot noise?

Answer from the author:

Using units of dBm are used to indicate powers, and a power spectral density can in principle have units of dBm/Hz – meaning dBm in a bandwidth of 1 Hz.

Your calculations appear to be flawed: you can apply the logarithm function only two dimensionless arguments. Things like log(2 hv / P) are not defined.

2020-08-01

(follow-up to the previous question and answer)

To clarify, I'm using the equation in the “RIN from Shot Noise” section of one of your other article on relative intensity noise.
In that section, you specify shot-noise limited relative intensity noise as 2 *hν* / *P*.
Considering the value in dBc/Hz is for a bandwidth of 1 Hz (multiply by 1 Hz), once you multiply with the bandwidth, the expression is unitless.

Answer from the author:

Once you multiply … but you didn't multiply, so your applied the logarithm to something in units of inverse Hertz! But we can repair that by inserting a 1-Hz bandwidth into your log argument. Sorry for being a bit pedantic, but multiplying with 1 Hz isn't just doing nothing.

In the end, however, I think your thoughts are a reasonable and intuitive interpretation.

2020-10-21

The equation given above for the power spectral density of shot noise has units of W^{2}/Hz. For a power spectral density I would rather expect units of W/Hz (as also mentioned in your article on power spectral density). How can these units be explained?

Answer from the author:

To resolve that, one needs to consider the meaning of “power” in the term power spectral density.
It is *not* the optical power, but rather the power of a signal.
If we consider the signal amplitude to be the optical power, then the signal power is proportional to the square of that power.

For example, you may think of an analog signal transmission apparatus using an intensity-modulated laser beam. The electrical input signal is reflected by corresponding changes of the transmitted optical power, and leads to a photocurrent or detector voltage which is in the end proportional to the input signal. If you feed that signal into an spectrum analyzer, this will deliver the signal power per unit frequency interval. From that, you can calculate the PSD of the optical power.

2020-11-17

In quantum cryptography, everything is normalized to shot-noise-unit, so what is it exactly and how to measure it experimentally.

Answer from the author:

More precisely speaking, one often normalizes noise to the standard quantum noise level. Shot noise is just a special case: intensity noise at the standard quantum noise level. The article already explains how to measure it.

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### Bibliography

[1] | N. Campbell, “The study of discontinuous phenomena”, Proc. Cambr. Phil. Soc. 15, 117 (1909) |

[2] | W. Schottky, “Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern”, Ann. Physik 57, 541 (1918), doi:10.1002/andp.19183622304 |

[3] | E. N. Gilbert and H. O. Pollak, “Amplitude distribution of shot noise”, Bell Syst. Tech. J. 39, 333 (1960), doi:10.1002/j.1538-7305.1960.tb01603.x |

[4] | C. M. Caves, “Quantum limits on noise in linear amplifiers”, Phys. Rev. D 26 (8), 1817 (1982), doi:10.1103/PhysRevD.26.1817 |

[5] | H. P. Yuen and V. W. S. Chan, “Noise in homodyne and heterodyne detection”, Opt. Lett. 8 (3), 177 (1983), doi:10.1364/OL.8.000177 |

[6] | W. H. Richardson et al., “Squeezed photon-number noise and sub-Poissonian electrical partition noise in a semiconductor laser”, Phys. Rev. Lett. 66 (22), 2867 (1991), doi:10.1103/PhysRevLett.66.2867 |

[7] | G. Brida et al., “Experimental realization of sub-shot-noise quantum imaging”, Nature Photon. 4, 227 (2010), doi:10.1038/nphoton.2010.29 |

See also: intensity noise, relative intensity noise, quantum noise, photodetectors, optical heterodyne detection, coherent states, standard quantum limit, photons, squeezed states of light, The Photonics Spotlight 2007-04-16, The Photonics Spotlight 2009-11-13

and other articles in the categories quantum optics, fluctuations and noise

2020-05-25

Is the here mentioned power spectral density the frequency noise power spectral density, and if not (I guess so since the unit here is not Hz

^{2}/Hz) how can this be computed for shot noise?Answer from the author:

By definition, shot noise is noise of the optical power, not of the frequency. Therefore, strictly speaking your question is meaningless.

However, such questions are often meant in a different sense: how large is the phase noise or frequency noise for a coherent state? (Such a state exhibits shot noise of the optical power, and some well defined level of phase noise and frequency noise.) Unfortunately, I do not have the time to dig out that equation – maybe someone else can help?