Radiant Properties of Extended Objects

Radiance

In the previous article the quantity radiance was introduced. This quantity contains all information about the spatial and directional characteristics of the radiation emitted by a surface (provided that it is given for each point on that surface).

According to its definition the radiance can be calculated by considering the flux $\Phi$ emitted by a small area $\D{A}$ around a point $\vec{r}(x,y)$ on that surface into a small solid angle $\D{\Omega}$ around a particular direction $\vec{\theta}(\vartheta,\varphi)$: \begin{align} L(x,y,\vartheta,\varphi) = \frac{\Dsq{ \Phi (x,y,\vartheta,\varphi) } }{\D{A \cos \vartheta} \D{\Omega}} \end{align} $\vartheta$ is the angle between the normal vector $\vec{n}$ of the considered surface element $\D{A}$ and the examined direction $\vec{\theta}$.

As explained in the context of the definition of radiance, the $\cos \vartheta$ appears due to the fact that in general the infinitesimal surface element $\D{A}$ of the object is not aligned normal to the direction given by $\vec{\theta}$. However, the definition of the requires a surface element normal to $\vec{\theta}$ which is why the projection factor $\cos \vartheta$ is necessary.^[1]

If $L(x,y,\vartheta,\varphi)$ is given, other quantities can be derived as well:

Radiant Intensity

The total radiant intensity $I$ emitted by a surface $S$ in the direction of $\vec{\theta}(\vartheta,\varphi)$:^[2][3] \begin{align} I(x,y,\vartheta,\varphi) = \int_S L(x,y,\vartheta,\varphi) \D{A \cos \vartheta } \end{align}

Flux Density

The radiant exitance $M$ (emitted flux density) at a point $\vec{r}$ normal to the surface (into the upper hemisphere):^[4] \begin{align} M(x,y) &= \int_\uparrow L(x,y,\vartheta,\varphi) \cos \vartheta \D{\Omega} \\ &= \int_{\varphi = 0}^{2\pi} \D{\vartheta} \int_{\vartheta = 0}^{\pi / 2} \D{\varphi} L(x,y,\vartheta,\varphi) \cos \vartheta \sin \vartheta \end{align}

Isotropic Radiance

In the case of isotropic radiance $L=L(x,y)$ this even simplifies to:^[5] \begin{align} M(x,y) &= L(x,y) \underbrace{ \int_{\varphi = 0}^{2\pi} \D{\varphi} }_{2\pi} \underbrace{ \int_{\vartheta = 0}^{\pi / 2 } \D{\vartheta} \cos \vartheta \sin \vartheta }_{1/2} \\ &= \pi \cdot L(x,y) \end{align}