= Dipole anisotropy of a radiance field
{title2=$L(\boldsymbol\omega)=L_0(1+\epsilon\mathbf a\cdot\boldsymbol\omega)$}
A \b[dipole anisotropy of a radiance field] is a directional first-order departure from a constant radiance:
$$
L(\boldsymbol\omega)=L_0(1+\mathbf b\cdot\boldsymbol\omega),\qquad |\mathbf b|\leq1.
$$
Here $\boldsymbol\omega$ is a unit viewing direction, and the bound ensures nonnegative radiance. Writing $\mathbf b=\epsilon\mathbf a$ 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 $3.0\,\mathrm{W\,m^{-2}}$. The pair means give $\pi L_0=3.0\,\mathrm{W\,m^{-2}}$. Since
$$
\frac{E(\mathbf n)-E(-\mathbf n)}{E(\mathbf n)+E(-\mathbf n)}=\frac23\mathbf b\cdot\mathbf n,
$$
the 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 $\int\boldsymbol\omega L\,d\Omega=(4\pi/3)L_0\mathbf b$. 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 $12\,\mathrm{W\,m^{-2}}$, exactly the value for an isotropic field with the same $L_0$. One all-direction scalar cannot determine its dipole.<International Lighting Vocabulary by CIE>[Entry17-21-054, note2, small-sphere normalization]{ref} Receiver angular response must also be characterized rather than assumed ideal.
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