Physics in a nutshell

$\renewcommand{\D}[]{\,\text{d}^{#1} {#2}}$ $\DeclareMathOperator{\Tr}{Tr}$
Stop the war — peace for Ukraine!

# Radiant Properties of Extended Objects

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). (a) Basic geometry of the set-up: A small portion $\D{A}$ of the surface area emits some amount of radiation into a solid angle $\D{\Omega}$ along the direction given by $\vec{\theta}(\vartheta , \varphi)$. The included angle between the normal vector of the surface $\vec{n}$ and $\vec{\theta}$ is labelled $\vartheta$.

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.

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

The total radiant intensity $I$ emitted by a surface $S$ in the direction of $\vec{\theta}(\vartheta,\varphi)$: \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): \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}

In the case of isotropic radiance $L=L(x,y)$ this even simplifies to: \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}