How it is calculated
A photon of vacuum wavelength \(\lambda\) carries energy \(E_\text{ph} = hc/\lambda\). In convenient units that is \(E_\text{ph}\,[\text{eV}] = 1239.842/\lambda\,[\text{nm}]\). The photon rate in a beam of average power \(P\), and the number of photons in a pulse of energy \(E_p\), are
For a Gaussian beam of 1/e² radius \(w\), the on-axis photon flux density is twice the average over the area \(\pi w^2\): \(\phi_0 = 2\Phi/(\pi w^2)\). The same factor of two applies to the peak photon fluence per pulse.
Detector responsivity and quantum efficiency
A detector with external quantum efficiency \(\eta\) produces \(\eta\) electrons per incident photon. Its current responsivity is
The relation also runs backwards: a measured responsivity gives \(\eta = \mathcal{R}\,hc/(e\lambda)\). Even an ideal detector (\(\eta = 1\)) has a responsivity that rises linearly with wavelength, because a watt of red light contains more photons than a watt of blue. Silicon photodiodes have a responsivity peak in the near-IR for this reason, even though their quantum efficiency is broadly flat across the visible.
Worked example
A 1 mW, 532 nm beam: \(E_\text{ph} = 2.3305\) eV \(= 3.7339\times10^{-19}\) J, so \(\Phi = 2.678\times10^{15}\) photons/s. At 10 kHz that is \(2.678\times10^{11}\) photons per pulse. Over a 1 mm (1/e²) beam the average flux density is \(3.41\times10^{17}\) photons s⁻¹ cm⁻² and the on-axis value is \(6.82\times10^{17}\). A photodiode with \(\eta = 80\,\%\) has \(\mathcal{R} = 0.343\) A/W and delivers 343 µA.
Assumptions and limits
- Vacuum wavelength. The photon energy depends on frequency, so use the vacuum wavelength. Air wavelengths differ by about 0.03 %, which is negligible here.
- Monochromatic light. For broadband sources, integrate \(P_\lambda \lambda/hc\) over the spectrum. Using the center wavelength is a good approximation when the bandwidth is small.
- Gaussian spatial profile. The peak-to-average factor of 2 holds for a TEM₀₀ beam. Flat-top beams have a factor near 1.
- Linear detector. Responsivity assumes no saturation, gain, or multiphoton response. Avalanche and photomultiplier detectors multiply the current by their internal gain.
References
- E. Tiesinga, P. J. Mohr, D. B. Newell, B. N. Taylor, “CODATA recommended values of the fundamental physical constants: 2018,” Rev. Mod. Phys. 93, 025010 (2021).
- B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed., Wiley (2019) — photodetector chapter.