Tool · Detectors and noise

Photodiode Noise Calculator

Which noise source limits a photodiode receiver at a given optical power, and what is the weakest signal it can see? The calculator adds the signal shot, dark-current shot, thermal and amplifier noise of a PIN or avalanche photodiode over the bandwidth and reports the signal-to-noise ratio, the noise-equivalent power, the minimum detectable power, the power at which the receiver becomes shot-noise limited and the optimum APD gain. Background: responsivity, shot noise, and noise-equivalent power.

Detector

Type

Wavelength λ (nm)

Responsivity set by

Quantum efficiency η (%)

Responsivity R₀ (A/W), unity gain

Avalanche gain M

slider logarithmic from 1 to 1000

Ionization ratio k

Dark current I_d primary (unmultiplied) for an APD

Receiver

Load or feedback resistance R_L

Temperature (K)

Amplifier input noise (pA/√Hz)

Bandwidth B

Optical power (dBm)

Presets

Noise densities are summed as uncorrelated white sources over a brick-wall bandwidth B. R_L is the load, or the feedback resistor of a transimpedance amplifier, and the amplifier term covers the rest of the front end. The APD uses McIntyre's excess noise factor F = kM + (1 − k)(2 − 1/M), with the whole dark current multiplied; surface dark current that bypasses the gain region is not modelled.

Learn with it

Three short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Readouts

The minimum detectable power is the power whose signal equals the signal-independent noise in the bandwidth B (SNR ≈ 1, exactly 1 when the signal shot noise is negligible); it is not a link sensitivity. A digital receiver needs a Q factor near 7 for a bit-error ratio of 10⁻¹², far above SNR = 1; see receiver sensitivity.

Noise current densities versus optical power

signal shotdark shotthermalamplifiertotal

Signal-to-noise ratio versus optical power

this receivershot-noise limit at the set η, ηP/2hνB

Signal-to-noise ratio versus avalanche gain

Checked against

These checks run in your browser on every load. Responsivity, shot and thermal noise are compared with textbook values, the excess-noise factor with McIntyre's limits, the general model with the closed-form shot-noise limit and the optimum-gain condition, and the unit paths with hand-computed numbers.

Check	Expected	Computed	Tolerance

Expected values come from the 2019 SI values of e, h, c and k_B, from the formulas in Agrawal, Fiber-Optic Communication Systems, section 4.4, and from McIntyre (1966) for the excess noise factor; the optimum-gain rows compare the root of the stationarity cubic with a brute-force search for the largest SNR. The tolerance is the largest difference from Expected, in the same units, that still passes; rows whose expected value is itself rounded to three figures carry a correspondingly wider band.

The model

A photodiode of quantum efficiency $\eta$ at wavelength $\lambda$ has the unity-gain responsivity

R_0 = \frac{\eta e \lambda}{hc}

so an optical power $P$ gives the primary photocurrent $I_p = R_0 P$ . An avalanche photodiode multiplies it by the gain $M$ , and the randomness of the multiplication raises the shot noise by McIntyre’s excess noise factor

F(M) = kM + (1-k)\left(2 - \frac{1}{M}\right)

where $k$ is the ratio of the ionization coefficient of the weaker ionizing carrier to that of the stronger one ( $F = 1$ for a PIN). The four noise sources are treated as white and uncorrelated, so their current spectral densities add:

S_n = 2e(I_p + I_d)M^2F + S_e

S_e = \frac{4k_BT}{R_L} + i_n^2

with $I_d$ the dark current, $R_L$ the load or feedback resistance and $i_n$ the input noise of the amplifier. Over a brick-wall bandwidth $B$ the electrical signal-to-noise ratio is

\mathrm{SNR} = \frac{(M I_p)^2}{S_n B}

With no dark current and no electronic noise a PIN reaches the shot-noise limit at its own quantum efficiency,

\mathrm{SNR}_{\mathrm{shot}} = \frac{\eta P}{2h\nu B}

and a shot-limited APD sits below that line by the factor $F$ , that is by $10\log_{10}F$ dB. The noise-equivalent power uses only the signal-independent part of the noise, $S_0 = 2eI_dM^2F + S_e$ :

\mathrm{NEP} = \frac{\sqrt{S_0}}{R_0 M}

and the minimum detectable power is $P_{\min} = \mathrm{NEP}\sqrt{B}$ , the power whose signal equals that noise in the bandwidth (an SNR of about 1). It is not a receiver sensitivity: a digital link at a bit-error ratio of $10^{-12}$ needs a Q factor near 7, and the power that achieves it is treated under receiver sensitivity. The receiver becomes shot-noise limited above the power at which the signal shot noise equals $S_0$ , $P_x = S_0/(2eR_0M^2F)$ . Setting $d\,\mathrm{SNR}/dM = 0$ gives the gain that maximizes the SNR as the root of

kM^3 + (1-k)M = \frac{S_e}{e(I_p + I_d)}

which the tool solves by bisection between 1 and 1000. The model assumes white noise, a brick-wall bandwidth, McIntyre’s local-field excess noise, and the whole dark current multiplied by the gain. Surface dark current that bypasses the multiplication region, 1/f noise, the frequency dependence of a transimpedance amplifier’s input noise, relative intensity noise of the source and background light are not modelled. In responsivity mode a typed value above the $\eta = 1$ maximum, $\lambda/1239.84$ A/W with $\lambda$ in nm, raises a warning; the results then use the typed value, and the shot-noise limit is computed at the implied $\eta$ , so the receiver never sits above it.

Worked example

The default preset is an InGaAs PIN at 1550 nm with $\eta = 0.8$ , 1 nA of dark current, a 500 Ω feedback resistor at 295 K, 8 pA/√Hz of amplifier noise and a 10 GHz bandwidth, receiving −10 dBm (100 µW). Then $R_0 = 0.8\times1550/1239.84 = 1.000$ A/W and $I_p = 100\ \mu\mathrm{A}$ . The noise densities are 5.66 pA/√Hz of signal shot noise, 5.71 pA/√Hz of thermal noise, 8.00 pA/√Hz from the amplifier and 17.9 fA/√Hz of dark-current shot noise, 11.3 pA/√Hz in total; the amplifier carries 50 % of the variance and the signal shot and thermal noise 25 % each. The SNR is the squared photocurrent, $10^{-8}\ \mathrm{A}^2$ , divided by $1.286\times10^{-22}\ \mathrm{A^2/Hz}$ times 10 GHz, which is $7.78\times10^3$ or 38.9 dB, 6.0 dB below the 44.9 dB shot-noise limit. The signal-independent noise, 9.83 pA/√Hz, gives an NEP of 9.83 pW/√Hz and a minimum detectable power of 983 nW (−30.1 dBm); the receiver would become shot-noise limited above 301 µW (−5.2 dBm).

References: G. P. Agrawal, Fiber-Optic Communication Systems, 4th ed., Wiley (2010), sections 4.4 and 4.5. R. J. McIntyre, “Multiplication noise in uniform avalanche diodes,” IEEE Transactions on Electron Devices 13, 164–168 (1966). H. Nyquist, “Thermal agitation of electric charge in conductors,” Physical Review 32, 110–113 (1928).