Photon flux

Photon flux is the rate at which photons pass through a surface or arrive at a detector, in photons per second. For monochromatic light of power $P$ it follows from the photon energy $h\nu$:

$$\Phi = \frac{P}{h\nu} = \frac{P\lambda}{hc}$$

One milliwatt at 1550 nm is $7.80 \times 10^{15}$ photons per second; at 532 nm the same power is $2.68 \times 10^{15}$, 2.9 times fewer, because each green photon carries 2.33 eV against 0.80 eV at 1550 nm. One picowatt at 1550 nm is $7.80 \times 10^{6}$ photons per second. The term is also used for the flux per unit area, more precisely called photon flux density or photon irradiance, in photons/(cm²·s).

Flux density and spectral flux

Photon flux density is the photon counterpart of irradiance: irradiance divided by photon energy. A 1 mW beam at 532 nm spread uniformly over a 10 µm diameter spot has an irradiance of 1.27 kW/cm² and a photon flux density of $3.4 \times 10^{21}$ photons/(cm²·s), the form in which excitation rates are written in photoluminescence and photochemistry, where each absorbed photon drives at most one event. For broadband sources the spectral photon flux, in photons/(s·nm), is integrated over wavelength; since the conversion factor $\lambda/(hc)$ varies across the band, a power spectrum cannot be converted to a photon count with a single wavelength unless the band is narrow. As a scale for spectral data, 1 µW/nm at 1000 nm is $5.03 \times 10^{12}$ photons/(s·nm).

From photon flux to photocurrent

A detector converts a fraction $\eta$ of the incident photons into electrons, its quantum efficiency, so the photocurrent is $I = \eta q \Phi$ and the responsivity is

$$\mathcal{R} = \frac{I}{P} = \frac{\eta q \lambda}{hc}$$

A photodiode with $\eta = 0.8$ at 1550 nm has a responsivity of 1.00 A/W; with 1 pW incident it produces $6.24 \times 10^6$ electrons per second, a current of 1.0 pA. Image sensors, photon-counting modules and astronomical instruments are usually specified in photons or electrons rather than watts, and the conversion between the two descriptions is this equation.

Shot noise and counting statistics

Photons from a laser arrive at random, with Poisson statistics, so a count of $N$ photons on average fluctuates by $\sqrt{N}$ (see photon statistics). One nanowatt at 1550 nm delivers 7800 photons in 1 µs, with a fluctuation of 88, or 1.1%. This is the shot noise of the measurement; it falls in relative terms as the flux or the integration time grows, and at fixed power the relative fluctuation is lower at longer wavelengths, where there are more photons per joule.

The single-photon regime

At picowatt levels the flux is low enough for each photon to be registered as a separate event by a single-photon avalanche diode, a superconducting nanowire detector or a photomultiplier tube. One picowatt at 1550 nm on a detector with 80% efficiency gives $6.24 \times 10^6$ counts per second. Counting detectors are blind for a dead time after each event, and at high rates they miss photons: for a non-paralyzable detector with an assumed 50 ns dead time, typical of a superconducting nanowire detector (InGaAs SPADs usually need microseconds), a true rate $n$ is recorded as

$$m = \frac{n}{1 + n\,\tau_d}$$

which for this flux is $4.76 \times 10^6$ counts per second, a 24% shortfall. Attenuation is therefore chosen so that $n\tau_d$ stays well below one, or the dead-time correction is applied. One photon per nanosecond at 1550 nm corresponds to 0.13 nW.

For pulsed sources the useful number is photons per pulse: a 1 mW average power at 1550 nm and a 100 MHz repetition rate is $7.8 \times 10^7$ photons per pulse. Quantum communication sources attenuate this to a mean photon number below one per pulse.

Pitfalls

Equal powers at different wavelengths are unequal photon fluxes, so comparisons of detector or source performance across wavelengths must state which is held fixed. Photometric quantities such as lux cannot be converted to photon flux without the spectrum. For a beam focused onto a detector smaller than the spot, only the fraction of the flux that falls on the active area counts.

Common questions

How many photons per second are in 1 mW?

$7.80 \times 10^{15}$ at 1550 nm, $4.28 \times 10^{15}$ at 850 nm and $2.68 \times 10^{15}$ at 532 nm. The count is proportional to wavelength at fixed power.

What is the difference between photon flux and irradiance?

Irradiance is power per unit area, in W/m²; photon flux density is photons per second per unit area. They differ by the photon energy, so their ratio depends on wavelength.

What is the unit of photon flux?

Photons per second (s⁻¹) for a whole beam, and photons/(cm²·s) or photons/(m²·s) for a flux density. In photobiology the molar unit µmol/(m²·s) is also used, with 1 µmol equal to $6.022 \times 10^{17}$ photons.

References: B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983); R. H. Hadfield, "Single-photon detectors for optical quantum information applications," Nature Photonics 3, 696–705 (2009).