The shot noise level on a spectrum analyzer is calculated using the following formulas:
1. DC Photocurrent:
\[ I_{\text{dc}} = P_{\text{optical}} \times R \]
2. Shot Noise Current Density:
\[ i_n = \sqrt{2eI_{\text{dc}}} \quad \text{[A/}\sqrt{\text{Hz}}\text{]} \]
3. Noise Level on Spectrum Analyzer:
\[ P_{\text{dBm}} = 10 \log_{10}\left(\frac{(i_n \cdot Z_{\text{TIA}})^2 \cdot \text{RBW}}{4 \times 50\,\Omega \times 1\,\text{mW}}\right) \]
where \(e = 1.602 \times 10^{-19}\) C (elementary charge), \(Z_{\text{TIA}}\) is the transimpedance gain, and RBW is the resolution bandwidth of the spectrum analyzer.
Note: The factor of 4 in the denominator accounts for the voltage divider formed by the 50Ω output impedance of the TIA and the 50Ω input impedance of the spectrum analyzer. This divider reduces the voltage by a factor of 2, which reduces the power (proportional to voltage squared) by a factor of 4.
Responsivity Conversion (Optional): If you don't know your photodiode's responsivity directly but have its quantum efficiency and operating wavelength, you can use the optional calculator above. It converts using:
\[ R = \frac{\eta \lambda}{1240} \quad \text{[A/W]} \]
where \(\eta\) is quantum efficiency (as a fraction, 0-1) and \(\lambda\) is wavelength in nanometers.