Analyze detector multiplication noise across realistic operating gains. Choose known k, M, or F values. Download tables as CSV and print to PDF easily.
This calculator uses a common avalanche photodiode (APD) noise model (McIntyre) for electron-initiated multiplication:
If you select a different mode, the same relationship is rearranged to solve for the missing variable (including a quadratic solve for M).
Sample values illustrate typical trends: higher gain and higher k often increase F.
| Case | M | k | F | Iₚ (A) | B (Hz) | iᵣₘₛ (A RMS) |
|---|---|---|---|---|---|---|
| A | 1 | 0.02 | 2.062 | 2.0e-6 | 1.0e+7 | 3.635e-8 |
| B | 3 | 0.02 | 2.527333 | 2.0e-6 | 1.0e+7 | 1.207e-7 |
| C | 5 | 0.05 | 4.381 | 2.0e-6 | 1.0e+7 | 2.649e-7 |
| D | 8 | 0.1 | 9.78875 | 5.0e-6 | 2.0e+7 | 1.417e-6 |
| E | 12 | 0.2 | 25.593333 | 5.0e-6 | 2.0e+7 | 3.436e-6 |
In avalanche photodiodes, multiplication boosts signal current, but it also amplifies stochastic gain fluctuations. The excess noise factor, F, summarizes how much additional noise the multiplication process adds beyond ideal noiseless gain. For the same average gain M, a lower F directly improves signal-to-noise ratio, detection margin, and measurement repeatability.
This tool uses the McIntyre electron-initiated relationship, which is widely applied for first-order APD noise estimates. It connects F to gain M and the ionization-coefficient ratio k = β/α. The expression is compact, differentiable, and practical for fitting lab data when you measure gain curves and want a consistent noise parameterization.
The ratio k indicates how strongly the “secondary” carrier ionizes relative to the primary carrier. Smaller k generally means a more one-sided multiplication process and therefore less noise. In many device discussions, k values like 0.01–0.05 indicate favorable low-noise behavior, while k near 0.1–0.3 can produce visibly larger F at moderate-to-high gains.
When M increases, F typically increases as well, but the rate depends on k. For small k, F grows more slowly and can remain close to 2 at moderate gains. For larger k, the linear kM term dominates and F rises quickly. This matters because pushing gain higher can stop improving overall sensitivity once multiplication noise dominates.
Shot-noise-limited current scales with the product M²F, so small differences in F become large differences in RMS noise at high gain. For example, at fixed Iₚ and bandwidth B, doubling M approximately quadruples M², and any increase in F further increases noise. This is why reporting M without F can hide major performance differences.
The optional estimate uses iᵣₘₛ = √(2 q Iₚ M² F B). Use average primary photocurrent (before multiplication) and a realistic electrical noise bandwidth. If you only know a measurement’s sampling rate, convert to an equivalent bandwidth carefully. For narrowband lock-in style measurements, B can be orders of magnitude smaller than the raw ADC rate.
F is dimensionless and should remain positive. In the electron-initiated model, F approaches 2 − 1/M as k → 0 and approaches M as k → 1. If your computed k lands outside 0–1, it often signals inconsistent inputs (for example, an F value too small for the stated gain) or a device regime not captured by this simplified model.
For clean reporting, export a CSV row set that includes mode, M, k, and F, plus Iₚ and B if you used the noise estimate. When comparing devices or operating points, keep the same bandwidth definition, temperature, and bias conditions. A short table of (M, F) points across bias is often more informative than a single operating number.
F represents noise added by random multiplication. A perfectly noiseless gain would give F = 1, but real avalanche gain fluctuates, so F is typically above 1 and often near or above 2 at practical gains.
Small k means the secondary carrier ionizes much less than the primary carrier. The multiplication becomes more one-sided, reducing gain randomness. This usually yields lower F at the same gain, improving sensitivity and stability.
The math may return a value, but k outside 0–1 is usually non-physical. It often indicates inconsistent inputs, measurement error, or conditions where the simplified model is not appropriate.
Excess noise factor is defined for avalanche multiplication. At M = 1 there is no multiplication, and the model’s terms like 1/M become edge cases. For photodiode operation without avalanche, set M close to 1 and interpret results cautiously.
Use the electrical noise bandwidth that matches your measurement chain. It may be set by a filter, amplifier, or lock-in time constant. Using an overly large B will overestimate iᵣₘₛ and can mislead comparisons.
Yes. Both k and multiplication behavior depend on electric field and carrier scattering, which vary with bias and temperature. Treat k and F as operating-point parameters, not fixed device constants.
Export M, k, and F, and include Iₚ and B if you computed noise current. Add measurement conditions such as bias voltage, temperature, wavelength, and the bandwidth definition so results remain comparable later.