= Spherical irradiance
{title2=$E_{\rm e,o}$}
\b[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,
$$
E_{\rm e,o}=\int_{4\pi}L_{\rm e}(\boldsymbol\omega)\,d\Omega,
\qquad [E_{\rm e,o}]=\mathrm{W\,m^{-2}}.
$$
Here $d\Omega$ is an element of <solid angle>. An ideal small sphere of radius $a$ intercepts <radiant flux> $\Phi_{\rm in}=\pi a^2E_{\rm e,o}$. The denominator is its diametrical <projected area>, not its full surface area.<International Lighting Vocabulary by CIE>[Entry17-21-054, note2, small-sphere construction; note5, units]{ref}
For an original <isotropic radiance field> example, take $L_{\rm e}=L_0=2\,\mathrm{W\,m^{-2}\,sr^{-1}}$ in every direction. Then $E_{\rm e,o}=4\pi L_0=8\pi\,\mathrm{W\,m^{-2}}$. A one-sided plane instead receives
$$
E_{\rm plane}=\int_0^{2\pi}\int_0^{\pi/2}L_0\cos\theta\sin\theta\,d\theta\,d\phi
=\pi L_0=2\pi\,\mathrm{W\,m^{-2}}.
$$
The 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 $E_{\rm e,o}/4$, 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.
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