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A dipole anisotropy of a radiance field is a directional first-order departure from a constant radiance:Here $\mathbf{\omega }$ is a unit viewing direction, and the bound ensures nonnegative radiance. Writing $\mathbf{b}=ϵ\mathbf{a}$ with unit $\mathbf{a}$ gives the amplitude/direction convention in the title. The angular isotropy test with planar irradiance measurements derives its receiver response.

An original inversion uses opposite receiver orientations. Suppose measured irradiances are $E(+x)=3.4$, $E(-x)=2.6$, and all four $y,z$ readings equal $3.0\mathrm{W}{\mathrm{m}}^{-2}$. The pair means give $\pi L_0=3.0\mathrm{W}{\mathrm{m}}^{-2}$. Sincethe fitted vector is $\mathbf{b}=(0.20,0,0)$. Agreement of the three pair sums is an additional check on this restricted model and a stable measurement chain.

The first angular moment of a radiance field is $\int \mathbf{\omega }Ld\Omega =(4\pi /3)L_0\mathbf{b}$. By contrast, integrating $L$ over all directions gives $4\pi L_0$, independent of $\mathbf{b}$. Thus this example's projected-area-normalized spherical irradiance is $12\mathrm{W}{\mathrm{m}}^{-2}$, exactly the value for an isotropic field with the same $L_0$. One all-direction scalar cannot determine its dipole.^[ref] Receiver angular response must also be characterized rather than assumed ideal.

An isotropic radiance field has the same radiance in every direction at the specified point: $L_{\mathrm{e}}(\mathbf{\omega })=L_0$. Angular isotropy is a local property; it does not by itself assert that $L_0$ is the same at every spatial position. Integrating this constant field gives the orientation-independent one-sided irradiance $\pi L_0$. The spherical irradiance article derives the corresponding all-direction value using its projected-area convention.^[ref]

An original counterexample shows why a few equal detector readings do not establish isotropy. Send six ideal narrow beams toward a point along the positive and negative Cartesian axes. Give every beam a normal-incidence irradiance $q=1\mathrm{W}{\mathrm{m}}^{-2}$. A one-sided planar receiver with unit normal $\mathbf{n}$ then recordsThe three axis-aligned receiver orientations all give $q$. However, a normal along $(1,1,1)/\sqrt{3}$ gives $\sqrt{3}q\approx 1.732\mathrm{W}{\mathrm{m}}^{-2}$. Three mutually perpendicular measurements therefore agree even though the idealized field is strongly directional. Finite narrow beams approximate this construction.

An angular isotropy test with planar irradiance measurements must sample enough orientations to test its stated angular model. A collimated light source and a rotating receiver can check angular response before such a test. Detector-area projection and the instrument's own angular sensitivity must be distinguished from variation in the incident field.