= Radiance
{title2=$L$}
{wiki}
<Radiance> resolves both position and direction: it is <radiant flux> per <projected area> and per <solid angle>. For a surface element of <area> $dA$, and a small outgoing cone $d\Omega$ at <angle> $0\leq\theta<\pi/2$ to its outward <unit normal>,
$$
d^2\Phi=L(\mathbf x,\mathbf s)\cos\theta\,dA\,d\Omega,
\qquad [L]=\mathrm{W\,m^{-2}\,sr^{-1}}.
$$
Here $\mathbf x$ locates the patch and $\mathbf s$ is a <unit vector> along the outgoing direction. The factor $dA\cos\theta$ is the patch's apparent <area> viewed along that direction. The same local definition applies to transmitted or received radiation, choosing the positive <projected area> for the crossing direction. https://labs.phys.utk.edu/mbreinig/phys421/modules/m5/Radiometry.html[Marianne Breinig, University of Tennessee, Radiometry: radiance and Lambert's law]{ref}.
For a planar patch of <area> $A$ with spatially uniform, direction-independent <radiance> $L$, integrating over the patch gives
$$
I(\theta)=LA\cos\theta.
$$
Take $L=10\,\mathrm{W\,m^{-2}\,sr^{-1}}$ and $A=0.020\,\mathrm{m^2}$. The <radiant intensities> are $I(0)=0.20\,\mathrm{W\,sr^{-1}}$ and $I(60^\circ)=0.10\,\mathrm{W\,sr^{-1}}$. A sufficiently narrow $0.010\,\mathrm{sr}$ cone therefore carries approximately $2.0\,\mathrm{mW}$ or $1.0\,\mathrm{mW}$, respectively. Both views have the same <radiance> $L$: the oblique view presents half the <projected area>. This is <Lambert's cosine law>, not a loss of <radiance> with viewing <angle>.
\Image[/projected-area-radiance.png]
{title=One uniform patch: halving <projected area> halves <radiant intensity> while preserving <radiance>}
{description=The geometric panels show a cross-section of a rectangular patch with the same depth perpendicular to the page. Parallel projection onto a plane perpendicular to the viewing direction changes its width by $\cos\theta$. The graph gives $I(\theta)=LA\cos\theta$ for the stated patch and marks the exact $0^\circ$ and $60^\circ$ cases.}
<Radiance> is useful because it separates a source's local directional output from the geometry of an observation. Integrating over outgoing directions gives <radiant exitance>; integrating incoming <radiance> at a receiver gives <irradiance>. In the stationary <vacuum> setting of <vacuum conservation of specific intensity>, a resolved uniform source retains its <radiance> along each <light ray>, while the <radiant flux> received from the whole source can decrease as its apparent <solid angle> shrinks.
Back to article page
