= Angular isotropy test with planar irradiance measurements
An \b[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 $E(\mathbf n)=\int L(\boldsymbol\omega)\max(0,\mathbf n\cdot\boldsymbol\omega)\,d\Omega$. 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.<Integrating Sphere Theory and Applications by Labsphere>[§3.1.3, Cosine Receptor]{ref}
An original restricted-model example makes the inference quantitative. Assume a <dipole anisotropy of a radiance field>,
$$
L(\boldsymbol\omega)=L_0(1+\epsilon\mathbf a\cdot\boldsymbol\omega),\qquad
|\epsilon|\leq1,\quad |\mathbf a|=1.
$$
Hemisphere integration gives $E(\mathbf n)=\pi L_0[1+(2\epsilon/3)\mathbf a\cdot\mathbf n]$. For $L_0=1\,\mathrm{W\,m^{-2}\,sr^{-1}}$, $\epsilon=0.30$ and $\mathbf a$ along the vertical axis, upward/downward-facing receivers give $1.2\pi$ and $0.8\pi\,\mathrm{W\,m^{-2}}$. Their difference divided by their sum is $0.20$, implying $\epsilon=0.30$ 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.
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