Isotropic radiance field ($L_{\mathrm{e}}(\mathbf{\omega })=L_0$)

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.

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.

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.

The first angular moment of a radiance field is the vectorwith units $\mathrm{W}{\mathrm{m}}^{-2}$ for frequency-integrated radiance. It records a directional imbalance. Here $\mathbf{\omega }$ is the viewing direction toward the arriving light, as in dipole anisotropy of a radiance field; using propagation directions instead reverses the vector. With that convention change, this is the bolometric vector form of the radiative flux, an existing radiation-field moment.

Opposite ideal planar receivers measure this moment without assuming a pure dipole. Since $\max (0,x)-\max (0,-x)=x$, the cosine-response irradiance formula givesThree orthogonal receiver pairs therefore determine the three components. For original differences $(0.60,-0.20,0.30)\mathrm{W}{\mathrm{m}}^{-2}$, the moment has those components and magnitude $0.70\mathrm{W}{\mathrm{m}}^{-2}$. The derivation requires the same spatial field, stable gains and known angular response. The primary optical guide describes cosine-receptor characterization rather than assuming it perfect.^[ref]

A zero first moment does not establish an isotropic radiance field. For example, two equal oppositely directed collimated beams have cancelling vector moments while remaining strongly directional. Conversely, any nonzero first moment rules out isotropy. The scalar all-direction integral $J=\int Ld\Omega$ bounds the moment by $|\mathbf{M}|\le J$ for nonnegative radiance, by the triangle inequality and $|\mathbf{\omega }|=1$.

Within the restricted dipole anisotropy of a radiance field model, $\mathbf{M}=(4\pi /3)L_0\mathbf{b}$ determines its dipole vector. More general angular structure is not reconstructed by these three numbers. Opposed receivers reveal the first moment, not the entire directional distribution.