The Mathematical Basis for the Calculation of PPFD

If the spectral power distribution (SPD) of a light source is known across the relevant wavelengths (400-700 nm), then the amount of photosynthetic energy available to plants can be determined. Based on its SPD, a light source will have a conversion factor that can be used to translate luminous flux density (illuminance) received by the plant into photosynthetic photon flux density (PPFD), in μmol/s-m2.


One watt of radiant power at 555 nm is by definition equal to 683 lumens. Given the CIE 1931 luminous efficiency function V(λ), we can calculate the spectral radiant flux Φ(λ) for plants in watts per nanometer for each lumen as:

Φ(λ)/lumen = [Wrel(λ)] / [683 * Σ(400-700) [V(λ) Wrel(λ) Δλ]]


where Wrel(λ) is the relative spectral power distribution and V(λ) is the luminous efficiency function at wavelength λ.


With this, the photosynthetic photon flux (PPF) per nanometer in micromoles per second per nanometer can be calculated:


PPF /nm = (10-3) * [λ Φ(λ)] / (Nahc),


where:


Na = Avogadro's constant, 6.022 x1023
h = Planck's constant (6.626 x 10-34 joule-seconds)
c = speed of light, 2.998 x 108 m/s
λ = wavelength in meters.


Summing over the range of 400-700 nm yields the photosynthetic photon flux (PPF) per lumen for the given light source:


PPF » 8.359 * 10-3 * Σ(400-700) [λ Φ(λ) Δλ]


Given an illuminance value (lux or footcandles), we can similarly calculate the photosynthetic photon flux density (PPFD) in micromoles per second per square meter (μmol/s-m2) for the given light source.