# Shot Noise

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: quantum-limited intensity noise

More general term: quantum noise

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, but a feature of the detection process: 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.

Note, however, that noise measurements at high optical power levels often require optical attenuation, which raises the shot noise level of the relative intensity (→ *relative intensity noise*). In such situations, the detector setup (including the attenuator) is substantially responsible for increased shot 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 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

If a photocurrent is measured with a photodetector, e.g. a photodiode, the photocurrent will be influenced by various shortcomings:

- A photodetector usually has a non-perfect quantum efficiency, which leads to a reduced photocurrent. Still, the photocurrent noise is given by the equation given below (at least for noise frequencies well within the bandwidth) if the optical source has noise at the level of shot noise. For a sub-shot noise source (→ squeezed states of light), the non-perfect quantum efficiency brings the noise closer to the shot noise level.
- The limited detection bandwidth can lead to reduced noise at high frequencies.
- There is also some detector noise added, which occurs even without any optical input (see below).

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 single-mode 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

$$S(f) = 2\;h\nu \;\bar P$$which is proportional to the average power and the photon energy <$h\nu$>, 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 (which is compatible with the equation above for the PSD on the optical side) is

$$\left\langle {{\delta I^2}} \right\rangle = 2e\;I\;\Delta f$$where <$e$> is the elementary charge. This formula indicates the variance of the current for an average current <$I$> and a measurement bandwidth <$\Delta f$>. The equation corresponds to a one-sided power spectral density

$${S_i}(f) = 2e\;I$$of the photocurrent.

The distribution function of the fluctuating current is a normal distribution, having a Gaussian shape:

$$f(\delta I) = \frac{e^{-\frac{1}{2}(\frac{\delta I}{\sigma})^2}}{\sigma \sqrt{2\pi}}$$where <$\sigma$> is the standard variation of the current, i.e., the square root of its variance as given above. This distribution multiplied with the (small) width of a current interval gives the probability of finding a particular value within that interval around <$\delta I$>.

## More to Learn

Encyclopedia articles:

Blog articles:

- The Photonics Spotlight 2007-04-16: “Questions and Answers on Shot Noise”
- The Photonics Spotlight 2009-11-13: “Photodetection: Optical and Electrical Powers”

### 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); https://doi.org/10.1002/andp.19183622304 |

[3] | E. N. Gilbert and H. O. Pollak, “Amplitude distribution of shot noise”, Bell Syst. Tech. J. 39, 333 (1960); https://doi.org/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); https://doi.org/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); https://doi.org/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); https://doi.org/10.1103/PhysRevLett.66.2867 |

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

## 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 = <$10 \lg(2 h\nu / P)$> 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 = <$10 \: \log(2 h\nu )$>, 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?

The author's answer:

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 <$2 h\nu / P$>. 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.

The author's answer:

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?

The author's answer:

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.

The author's answer:

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.

2021-06-20

What is the difference between shot noise and intensity noise? Are these different sources of noise? Talking about laser sources, as I have understood the shot noise is the lowest boundary of intensity noise.

If we have an electronic system, should we consider them as separate noises and add them up to perform the total noise of the system?

The author's answer:

Short noise is intensity noise resulting from the discreteness of randomly arriving photons. Intensity noise can also have all sorts of other origins.

Indeed, the output of a laser usually exhibits intensity noise at least on the shot noise level.

If you detect noise with a photodetector and electronics, you can assume that they are noise contributions are not correlated with the laser noise. Therefore, you can add up the noise powers.

2021-12-08

The formula for shot noise contains the bandwidth. Is this the inverse of the measurement time or the true bandwidth of the electronic system?

The author's answer:

It is the bandwidth for which you consider the shot noise. That might be limited by the inverse of the measurement time or by other factors, e.g. some electronic bandwidth.

2022-06-17

What is the difference between the photon shot noise and electronic shot noise? Can the electronic shot noise be considered to show a quantum effect?

The author's answer:

As is pointed out in the article, electric currents (e.g. in resistors) normally do not tend to exhibit shot noise, despite the discreteness of the electric charge.

2022-09-02

Suppose I have an incoherent optical incident on a typical silicon photodetector. Let N = (average incident energy in time t)/h*nu, i.e. colloquially the # of photons in time t. Let QE = the detector's average quantum efficiency (about 0.6 for visible light on silicon). Is the limiting noise photocurrent then sqrt(N)*QE or sqrt(N*QE)? I can find both in the literature, with surprisingly little discussion for such a basic question.

The author's answer:

First, one should be clear about which quantity we are talking. If you assume a proportionality to sqrt(N), you apparently mean standard deviations, not noise powers.

It must be sqrt(N * QE), since the photocurrent is proportional to both N and QE, and shot noise can be calculated simply based on the photocurrent. Of course, we assume that the light input is really at the shot noise limit.

The power spectral density is then proportional to N * QE.

2022-11-06

How does shot noise depend on the wavelength?

The author's answer:

For a given optical power, shot noise will get larger for shorter wavelengths, since you have fewer photons per second. You also see this from the formulas containing the photon energy <$h \nu$>, which then gets larger.

2023-01-10

For a typical laser system at higher frequencies, the intensity noise reduces and almost equals the shot noise as shown in figure 1 here. Does this mean at higher frequencies, there is not much noise affecting a typical laser system besides the shot noise? And ultimately, the shot noise is the limiting noise from a laser system?

The author's answer:

Basically yes, but I would word it somewhat differently, and try to explain it a little more. At high noise frequencies, there is hardly any mechanical noise (mirror vibrations or so) and thermal noise. Quantum noise influences are then dominating. The resulting laser noise is then close to the shot noise limit, as can be confirmed with a sufficiently good photodetector.

2023-08-14

What about shot noise for optical pulses? For the same average optical power, do the pulses of 1% duty cycle induce the same shot noise as the continuous wave?

The author's answer:

During the pulses, you have 100 times the average power, and correspondingly get a 100 times higher power spectral density of the photocurrent. Therefore, on average you indeed get the same PSD.

2024-05-04

I see that shot noise is a kind of laser intensity variation that follows a Poisson distribution. And there is also the shot noise at the detector due to the fact that detection event also has a Poisson distribution sqrt(2 q I). Are these two noise sources (laser shot noise and detector shot noise) really just one noise source? Or these are two separate noise sources and need to be added up to give the total shot noise?

The author's answer:

It is just one noise process. The idea of a random detection probably for a constant optical intensity was an early one which should be abandoned. We now see all the noise to be contained in the light field.

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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?The author's answer:

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?