Excess noise factor (F)

The excess noise factor $F$ describes the extra noise added by a detector's internal gain when that gain is random. In an avalanche photodiode each photocarrier is multiplied by impact ionization into a number of carriers whose mean is the gain $M$ but which fluctuates from one primary carrier to the next. The shot noise of the primary photocurrent is therefore multiplied by more than $M^2$:

$$i_\text{n}^2 = 2e\,M^2 F(M)\,I_p\,B,$$

with $I_p$ the primary photocurrent and $B$ the bandwidth. Formally $F = \langle m^2\rangle/\langle m\rangle^2$, the mean square gain over the square of the mean gain, so $F = 1$ for a perfectly deterministic gain and $F > 1$ for any real one. At $M = 10$ it is about 2.1 in silicon APDs and about 6 in InGaAs APDs, 3.2 dB and 7.8 dB of extra noise power.

McIntyre's formula

For a uniform multiplication region with the more strongly ionizing carrier injected, McIntyre derived

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

where $k$ is the ratio of the ionization coefficient of the weaker ionizing carrier to that of the stronger one. Two limits frame it: when only one carrier ionizes ($k = 0$), $F$ rises toward 2 at high gain; when both ionize equally ($k = 1$), $F = M$ and the noise power grows as $M^3$.

Silicon keeps a low $F$ to high gain, 3.95 at $M = 100$, which is why silicon APDs run at gains of 100 or more. InGaAs APDs, whose InP multiplication layer has $k$ of about 0.5–0.7, are run at gains of 10 to 40.

Datasheets sometimes give the excess noise index $x$ instead, defined by $F \approx M^x$. Over gains of 10 to 100 the McIntyre formula corresponds to $x \approx 0.3$ for silicon with $k = 0.02$ and about 0.8–0.95 for an InP multiplier with $k$ = 0.5–0.7.

Optimum gain

Gain multiplies the signal and its shot noise but leaves the thermal noise of the load and the amplifier noise unchanged, so the signal-to-noise ratio first rises with $M$ and then falls once $F$ grows. The optimum is reached when the multiplied shot noise somewhat exceeds the circuit noise; for $kM \gg 1$ the ratio at the optimum tends to 2. As a worked case, an InGaAs APD receiving −30 dBm at 1550 nm with a responsivity of 1.00 A/W gives $I_p$ = 1.00 µA; with a 10 pA/√Hz amplifier in 10 GHz and $k = 0.5$, the optimum gain is 10.7, where $F = 6.3$, the multiplied shot noise is 2.3 times the amplifier noise and the SNR is 15.4 dB, against −0.01 dB at unity gain. The excess noise is what keeps the APD's receiver sensitivity short of the shot-noise limit, which at this power is 24.9 dB.

Measurement

$F$ is measured by recording the noise spectral density of the APD photocurrent with an electrical spectrum analyzer or a calibrated noise meter at several gains and illumination levels. After the amplifier noise and dark-current contribution are subtracted, $F = S_i/(2e M^2 I_p)$, where $S_i$ is the photocurrent noise spectral density and $M$ is taken from the ratio of photocurrent to its unity-gain value at low bias.

Other gain processes

A photomultiplier tube has an excess noise factor close to $\delta/(\delta-1)$, where $\delta$ is the secondary-emission yield of the first dynode: 1.33 for $\delta = 4$ and 1.20 for $\delta = 6$. Electron-multiplying CCDs approach a noise-power factor of 2 at high gain, often written as an amplitude factor of √2, equivalent to halving the quantum efficiency. HgCdTe APDs in which only electrons multiply reach $F$ near 1. In a silicon photomultiplier the gain of each cell is nearly fixed, and the excess noise comes instead from crosstalk and afterpulsing.

The noise figure of an optical amplifier is a different quantity with a similar name: it is the SNR degradation through the amplifier, expressed as a noise factor (linear) or a noise figure (dB), and it has a quantum limit of 3 dB at high gain.

Common questions

What is a good excess noise factor?

Close to 2 is the best a conventional APD reaches, and only with $k$ near zero, as in silicon. Values of 5–8 at $M = 10$ are normal for InGaAs APDs.

Is the excess noise factor the same as the noise figure?

No. $F(M)$ describes the statistics of a detector's internal gain; the noise figure of an amplifier describes the loss of SNR between its input and output.

References: R. J. McIntyre, "Multiplication noise in uniform avalanche diodes," IEEE Transactions on Electron Devices ED-13, 164 (1966); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); G. P. Agrawal, Fiber-Optic Communication Systems, 4th ed. (Wiley, 2010); S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (Wiley, 2007).