Spherical irradiance ($E_{\mathrm{e},\mathrm{o}}$)

Spherical irradiance counts incident radiance from all directions without selecting a receiving-plane normal. Using the projected-area convention specified in the CIE entry's note2,Here $d\Omega$ is an element of solid angle. An ideal small sphere of radius $a$ intercepts radiant flux ${\Phi }_{\mathrm{in}}=\pi a^2{E}_{\mathrm{e},\mathrm{o}}$. The denominator is its diametrical projected area, not its full surface area.^[ref]

For an original isotropic radiance field example, take $L_{\mathrm{e}}=L_0=2\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$ in every direction. Then $E_{\mathrm{e},\mathrm{o}}=4\pi L_0=8\pi \mathrm{W}{\mathrm{m}}^{-2}$. A one-sided plane instead receivesThe factor four follows from the different angular geometries; it is not a unit conversion. Averaging the sphere's intercepted power over its actual $4\pi a^2$ surface also gives $E_{\mathrm{e},\mathrm{o}}/4$, rather than the projected-area-normalized quantity above.

An all-direction receiver must have a characterized angular response. A single flat irradiance detector inherently samples a hemisphere with projected-area weighting, so rotating it or enclosing it in a diffuser does not supply an all-direction value without an appropriate response model and measurement calibration. Report the angular and area conventions with the result; equal units do not make irradiance and spherical irradiance interchangeable.