shot noise

Author: the photonics expert (RP)

Definition: quantum-limited intensity noise

More general term: quantum noise

Categories: quantum optics, fluctuations and noise

DOI: 10.61835/iwy   Cite the article: BibTex plain textHTML   Link to this page!   LinkedIn

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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. For measuring the shot noise of some light, one requires detection with a high quantum efficiency.

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 <$I$> (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. If the current is a photocurrent of a photodiode, for example, we have <$I = \eta \: e \: P / h\nu$>, involving the detector's quantum efficiency <$\eta$>. The 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. Of course, the formula holds only for a shot-noise-limited current. The noise may be lower if a photodetector with high quantum efficiency is illuminated with amplitude-squeezed light – or higher e.g. due to thermal noise.

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

Bibliography

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