Irradiance

Property

Irradiance of a surface is also, according to the definition of radiant flux, equal to the time-average of the component of the Poynting vector perpendicular to the surface:

${\displaystyle E_{\mathrm{e}}=⟨|\mathbf{S}|⟩\cos \alpha ,}$

where

• ⟨ • ⟩ is the time-average;
• S is the Poynting vector;
• α is the angle between a unit vector normal to the surface and S.

For a propagating sinusoidal linearly polarized electromagnetic plane wave, the Poynting vector always points to the direction of propagation while oscillating in magnitude. The irradiance of a surface is then given by

${\displaystyle E_{\mathrm{e}}=\frac{n}{2{\mu }_{0}c}{E}_{\mathrm{m}}^2\cos \alpha =\frac{n{\epsilon }_{0}c}{2}{E}_{\mathrm{m}}^2\cos \alpha =\frac{n}{2Z_0}{E}_{\mathrm{m}}^2\cos \alpha ,}$

where

• E_m is the amplitude of the wave's electric field;
• n is the refractive index of the medium of propagation;
• c is the speed of light in vacuum;

• μ₀ is the vacuum permeability;

• ε₀ is the vacuum permittivity;
  • $c=\frac{1}{\sqrt{{\epsilon }_0{\mu }_0}}$
  • $Z_0={\mu }_{0}c$ is the impedance of free space.

This formula assumes that the magnetic susceptibility is negligible; i.e. that μ_r ≈ 1 (μ ≈ μ₀) where μ_r is the relative magnetic permeability of the propagation medium. This assumption is typically valid in transparent media in the optical frequency range.

Point source

A point source of light produces spherical wavefronts. The irradiance in this case varies inversely with the square of the distance from the source.

${\displaystyle E=\frac{P}{A}=\frac{P}{4\pi r^2},}$

where

• r is the distance;
• P is the radiant flux;
• A is the surface area of a sphere of radius r.

For quick approximations, this equation indicates that doubling the distance reduces irradiation to one quarter; or similarly, to double irradiation, reduce the distance to 71%.

In astronomy, stars are routinely treated as point sources even though they are much larger than the Earth. This is a good approximation because the distance from even a nearby star to the Earth is much larger than the star's diameter. For instance, the irradiance of Alpha Centauri A (radiant flux: 1.5 L_☉, distance: 4.34 ly) is about 2.7 × 10⁻⁸ W/m² on Earth.

Solar irradiance

The global irradiance on a horizontal surface on Earth consists of the direct irradiance E_e,dir and diffuse irradiance E_e,diff. On a tilted plane, there is another irradiance component, E_e,refl, which is the component that is reflected from the ground. The average ground reflection is about 20% of the global irradiance. Hence, the irradiance E_e on a tilted plane consists of three components:

${\displaystyle E_{\mathrm{e}}=E_{\mathrm{e},\mathrm{d}\mathrm{i}\mathrm{r}}+E_{\mathrm{e},\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}+E_{\mathrm{e},\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{l}}.}$

The integral of solar irradiance over a time period is called "solar exposure" or "insolation".

Average solar irradiance at the top of the Earth's atmosphere is roughly 1361 W/m², but at surface irradiance is approximately 1000 W/m² on a clear day.

SI radiometry units