Shot noise

Shot noise is the unavoidable fluctuation in any quantity formed from discrete particle counts (photons, electrons, etc.) arriving randomly. In photodetection, shot noise arises because photons arrive at the detector as discrete quanta with Poisson statistics. The mean and the variance of the photon count are equal. This produces a fundamental noise floor that no detector engineering can eliminate.

Photon shot noise. The number of photons $N$ arriving in a time interval has Poisson statistics:

$$P(N)=\frac{{\bar{N}}^N{e}^{-\bar{N}}}{N!},{\sigma }_N=\sqrt{\bar{N}}\mathrm{.}$$

The fractional uncertainty in photon count decreases with average count: ${\sigma }_N\mathrm{/}\bar{N}=1\mathrm{/}\sqrt{\bar{N}}$.

For a 1 mW optical beam at 1550 nm:

• Photon energy: $h\nu =1.28\times {10}^{-19}$ J
• Photon rate: $P\mathrm{/}h\nu =7.8\times {10}^{15}$ photons/s
• In 1 ns: $\bar{N}=7.8\times {10}^6$ photons; ${\sigma }_N\mathrm{/}\bar{N}=0.036\mathrm{\%}$
• In 1 ms: $\bar{N}=7.8\times {10}^{12}$ photons; ${\sigma }_N\mathrm{/}\bar{N}=0.00004\mathrm{\%}$

Photocurrent shot noise. In a photodetector, the photocurrent $I=RP$ (where $R$ is responsivity, $P$ is optical power) has shot-noise current spectral density:

$$S_I(f)=2qI{\text{[A}}^2\text{/Hz]}\mathrm{.}$$

The RMS shot noise current in bandwidth $\mathrm{\Delta }f$:

$$i_{\text{shot}}^{\text{RMS}}=\sqrt{2qI\mathrm{\Delta }f}\text{[A]}\mathrm{.}$$

For 1 mA photocurrent and 1 MHz bandwidth: $i_{\text{shot}}^{\text{RMS}}=17.9$ nA; in 1 GHz it is 566 nA.

Signal-to-shot-noise ratio (SNR). The fundamental ratio for a shot-noise-limited measurement:

$$\text{SNR}=\frac{I^2}{i_{\text{shot}}^2}=\frac{I}{2q\mathrm{\Delta }f}\mathrm{.}$$

In photon-number form:

$${\text{SNR}}_{\text{shot}}=\frac{{\bar{N}}^2}{\bar{N}}=\bar{N}\mathrm{.}$$

The signal-to-noise ratio of a shot-noise-limited measurement equals the photon number. To achieve 60 dB (${10}^6$) SNR, you need ${10}^6$ photons.

Where shot noise dominates.

Shot-noise-limited operation is the gold standard for sensitive optical detection. Any detector with measurable thermal noise above the shot-noise level is not optimally designed.

Shot noise vs photon number squeezing. Classical and "semiclassical" light fields exhibit shot-noise-limited intensity fluctuations. Specially-prepared quantum states of light (number-squeezed states) can exhibit sub-shot-noise fluctuations:

$${\sigma }_N^2<\bar{N}\text{(squeezed)},$$

at the cost of larger phase fluctuations (uncertainty principle requires conjugate-variable conservation). Squeezed light enables sub-shot-noise interferometric measurements, used in LIGO since 2019 to improve gravitational-wave detection sensitivity by ~50%.

Practical shot-noise observations.

• Direct detection at 1 mW with 100 MHz bandwidth: 0.8 mA of signal (0.8 A/W responsivity) carries 160 nA rms of shot noise, an SNR of 74 dB.
• Coherent detection: a strong LO lifts the signal beat above the receiver's thermal noise, so the noise is dominated by the LO's own shot noise and the SNR approaches the shot-noise limit set by the signal photons; SNR depends on the signal-LO beat amplitude.
• Single-photon detection: the "shot noise" reduces to discrete-time photon counts; the amplitude SNR, mean over standard deviation, is $\sqrt{N}$ for $N$ photons counted (the power SNR defined above is $N$).

Relative intensity noise (RIN) and shot noise. RIN measures intensity fluctuations relative to mean power:

$$\text{RIN}(f)=\frac{S_I(f)}{I^2}$$

with $S_I$ the one-sided current noise spectral density, so RIN is in 1/Hz and usually quoted in dB/Hz.

For shot-noise-limited light: ${\text{RIN}}_{\text{shot}}=2q\mathrm{/}I=2h\nu \mathrm{/}\eta P$. For 1 mW at 1550 nm and $\eta =1$: ${\text{RIN}}_{\text{shot}}=-156$ dB/Hz.

Most lasers have RIN levels well above the shot-noise limit (typically $-130$ to $-160$ dB/Hz), so "shot-noise-limited" operation of communication systems is the exception rather than the rule.

Beam-splitter behavior. Splitting a shot-noise-limited beam into two: each beam has the appropriate fraction of the mean photon flux, and the shot noise of the two beams is uncorrelated. A 50:50 beamsplitter on a coherent state gives two outputs each with the shot-noise-limited fluctuation of half the mean. Classical intensity noise on the input appears in both outputs at once, positively correlated, and that common part is what balanced detection cancels.

Why Poisson statistics? Laser light (a coherent state) has Poisson photon-number statistics in any time bin or aperture, because the detection events are statistically independent, the same logic that produces the Poisson law in radioactive decay or in counting rare events. Thermal light does not: in a single mode its count variance is $\bar{n}+{\bar{n}}^2$, and its counts approach Poisson statistics only when each mode, one coherence time by one coherence area, holds much less than one photon on average, as it does for ordinary thermal sources in the visible and near infrared.

Shot noise and quantum efficiency. A detector with quantum efficiency $\eta <1$ has lower signal but also proportionally lower shot noise:

$${\text{SNR}}_{\text{shot, real}}=\eta {\bar{N}}_{\text{incident}}\mathrm{.}$$

So a detector with $\eta =0.7$ achieves 70% of the maximum shot-noise-limited SNR: quantum efficiency translates directly to SNR.

Distinguishing shot noise from other noises in measurement. Standard test: vary the optical power and measure noise. Shot noise scales as $\sqrt{P}$; thermal noise is power-independent. Plotting noise vs $\sqrt{P}$ should give a straight line with slope corresponding to the shot-noise expectation; deviations indicate other noise sources. The Photodiode Noise Calculator plots this comparison for a chosen detector and front end and reports the optical power at which the signal shot noise overtakes the fixed noise floor.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 18 (photodetection noise); Yariv & Yeh, Photonics (6th ed., 2007), Ch. 11; Loudon, The Quantum Theory of Light (3rd ed., 2000) for the quantum-mechanical treatment.