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:

$$P(N) \;=\; \frac{\bar{N}^N e^{-\bar{N}}}{N!}, \quad \sigma_N \;=\; \sqrt{\bar{N}}.$$

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

For a 1 mW optical beam at 1550 nm:

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

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

$$S_I(f) \;=\; 2 q I \quad \text{[A}^2\text{/Hz]}.$$

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

$$i_\text{shot}^\text{RMS} \;=\; \sqrt{2 q I \, \Delta f} \quad \text{[A]}.$$

For 1 mA photocurrent and 1 GHz bandwidth: $i_\text{shot}^\text{RMS} = 18$ nA.

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

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

In photon-number form:

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

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.

Detection regime	Dominant noise	Typical conditions
Bright (cooled) detector	Shot noise	Photocurrent $>$ 100 nA, low dark current
Dim signal (PMT, APD)	Multiplication noise + dark current	Signal photon rates $\ll 10^9$/s
High-bandwidth receiver	Thermal noise of input stage	Bandwidth $>$ 1 GHz, signal $<$ 1 μA
Very low signal (single photon)	Dark count rate	Single-photon detectors
High-frequency RF detection	Excess RIN + shot noise	Coherent detection systems

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:

$$\sigma_N^2 \;<\; \bar{N} \quad \text{(squeezed)},$$

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: 5.7 nA shot noise on 0.8 mA signal (200 mA/W responsivity). 102 dB SNR.
• Coherent detection: shot-noise floor lowered by LO amplification of signal; SNR depends on signal-LO beat amplitude.
• Single-photon detection: the "shot noise" reduces to discrete-time photon counts; SNR scales as $\sqrt{N}$ for $N$ photons counted.

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

$$\text{RIN}(f) \;=\; \frac{S_I(f) / I^2}{\Delta f} \quad \text{[1/Hz, often expressed in dB/Hz]}.$$

For shot-noise-limited light: $\text{RIN}_\text{shot} = 2q/I = 2 h\nu / P$. For 1 mW at 1550 nm: $\text{RIN}_\text{shot} = -166$ 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 two beams have classically-correlated (not anti-correlated!) shot noise. A 50:50 beamsplitter on a coherent state gives two outputs each with the shot-noise-limited fluctuation of half the mean — uncorrelated photon-number fluctuations between the two output arms.

Why Poisson statistics? Photons emerging from a thermal source or a coherent laser have Poisson photon-number statistics in any given time bin. The Poisson distribution applies because each emission is independent and infrequent — the same logic that produces it in radioactive decay or in counting rare events.

For laser light (coherent state), Poisson statistics applies to any time bin or aperture size — confirmed by quantum-optics measurements at sub-photon levels.

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

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

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

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.