 Angular isotropy test with planar irradiance measurements

An angular isotropy test with planar irradiance measurements rotates a characterized receiver to test whether its readings agree with a specified model of an isotropic radiance field. A one-sided ideal cosine receiver measures $E(\mathbf{n})=\int L(\mathbf{\omega })\max (0,\mathbf{n}\cdot \mathbf{\omega })d\Omega$. Constant radiance gives the same $\pi L_0$ for every normal. Instrument angular response needs characterization; the primary guide describes experimental correction of cosine receptors.^[ref]

An original restricted-model example makes the inference quantitative. Assume a dipole anisotropy of a radiance field,Hemisphere integration gives $E(\mathbf{n})=\pi L_0[1+(2ϵ/3)\mathbf{a}\cdot \mathbf{n}]$. For $L_0=1\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$, $ϵ=0.30$ and $\mathbf{a}$ along the vertical axis, upward/downward-facing receivers give $1.2\pi$ and $0.8\pi \mathrm{W}{\mathrm{m}}^{-2}$. Their difference divided by their sum is $0.20$, implying $ϵ=0.30$ within this chosen model.

Use the same receiver position, background treatment and a stable source monitor while changing orientation. Gain drift or moving into a different spatial field can otherwise imitate an angular effect. A finite set of readings tests the assumed angular family; it does not prove unrestricted isotropy. The six-beam counterexample in isotropic radiance field shows why a few equal axis readings can miss other directional structure.