Bandwidth vs Noise Trade-off
Photodiode on a load resistor: noise grows as √Δf until the R_L·C_j roll-off, then levels off at √(kT/C).
RC 3dB Bandwidth
1.592e+7 Hz
1/(2π·R_L·C_j)
Johnson Noise (V rms)
4.068e-6
Shot Noise (V rms)
5.657e-7
Total Noise (V rms)
4.107e-6
Johnson Limit √(kT/C)
2.035e-5 V
Δf → ∞, independent of R_L
Photodiode on a load resistor R_L; the output voltage across R_L ∥ C_j is low-passed at f_c = 1/(2π·R_L·C_j)
v²_Johnson = 4·k·T·R_L · f_c·atan(Δf/f_c) → k·T/C_j for Δf ≫ f_c
v²_Shot = 2·q·I·R_L² · f_c·atan(Δf/f_c)
Below f_c the noise grows as √Δf; above it, widening the measurement band adds no more noise (or signal).
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
A photodiode on a load resistor R_L: Johnson noise of R_L and photocurrent shot noise pass the single pole 1/(2π·R_L·C_j) and are integrated to the measurement bandwidth. Amplifier noise, dark current and transit time are left out.
- Nyquist H. (1928). Thermal agitation of electric charge in conductors. Phys. Rev. 32, 110. Derivation of 4k_B·T·R per unit bandwidth. doi:10.1103/PhysRev.32.110
- Saleh B. E. A., Teich M. C. (1991). Fundamentals of Photonics. Wiley. §17.5 Noise in photodetectors: shot noise 2qīB and circuit noise. doi:10.1002/0471213748
- Sze S. M., Ng K. K. (2007). Physics of Semiconductor Devices, 3rd ed. Wiley. p–n junction depletion capacitance (abrupt m = 1/2, linearly graded m = 1/3). doi:10.1002/0470068329