Physics in a nutshell

$\renewcommand{\D}[2][]{\,\text{d}^{#1} {#2}}$ $\DeclareMathOperator{\Tr}{Tr}$

The quantities radiant energy and radiant power/flux characterise a system as a whole but they are not capable of characterising the radiation itself with respect to a particular location/direction.

Therefore it is useful to introduce a few more quantities that characterise a radiation field locally regarding spatial and directional properties.

The term flux density $E$ refers to the spatial density of radiant power. It is defined as the infinitesimal amount of radiant power $\D{\phi (\vec{r}, \vec{\omega})}$ passing through an infinitesimal surface element $\D{A}$ that is aligned normal to a direction $\vec{\theta}$ and located at a position $\vec{r}$ of interest \begin{align} E (\vec{r},\vec{\theta}) := \frac{ \D{\phi(\vec{r}, \vec{\theta})} }{ \D{A} } \end{align} and has units of power per area [W m-2].

The label $E$ (french: éclairage, meaning lighting/illumination) is primarily intended for radiant flux density incident on a target surface. In this case, the term irradiance is used as well.[1]

Accordingly, the radiant flux density emitted by a source is called radiant exitance or formerly also emittance and denoted by a different letter, $M$ (If you know where this letter originates from, please tell me!).

There is however no common letter for the radiant flux density in general.[2]

### Solid Angle

The solid angle subtended by an object with respect to some location $\vec{r}$ can be thought of as a measure of how much of the visual field of a person located at $\vec{r}$ is covered by that object ("optical size").

For instance, during a solar eclipse one can convince oneself that the moon covers roughly the same portion of one's visual field as the sun does in spite of being way smaller.

Different units are conceivable for such a quantity depending on what value is chosen for 100% covering. A very common unit is the steradian [sr] for which 100% covering corresponds to the surface area of a unit sphere (radius $r=1$): \begin{align} A = 4\pi r^2 = 4\pi \end{align}

This definition is useful since an object's solid angle in units of [sr] corresponds to the projection area $A^\perp$ of this object (the area normal to radial lines at the distance $d$) divided by the square of the distance $d$: \begin{align} \Omega = \frac{A^\perp}{d^2} \end{align} An equivalent definition would be the size of the object's projection area on the unit sphere. [3] [4] [5]

The term radiant intensity $I$ refers to the angular density of radiant power. It is defined as the infinitesimal amount of radiant power $\D\phi(\vec{r}, \vec{\omega})$ passing through an infinitesimal solid angle $\D \Omega$ around a direction $\vec{\theta}$ from a position $\vec{r}$ \begin{align} I (\vec{r},\vec{\theta}) := \frac{ \D{\phi(\vec{r}, \vec{\theta})} }{ \D{\Omega} } \end{align} and has units of power per solid angle [W sr-1].

It is specific for a certain point $\vec{r}$ and a certain direction $\vec{\theta}$ but does not depend on the distance of an observer from that point. The solid angle $\Omega$ is necessary in order to apply the concept of flux since flux is defined with respect to a finite area rather than a single point/line. However, by dividing through that solid angle and in the limit of $\Omega$ becoming infinitesimally small this quantity becomes specific for that particular direction $\vec{\theta}$.

The radiant intensity is most often used as a characterisation of entire sources of radiation which are regarded as point sources (since they maybe cannot be resolved anyway). In that case it is possible to assign a finite amount of emitted radiation to a single point.[6]

### Motivation

In reality, sources of radiation are extended objects rather than point sources and one may want to characterise each point of the surface individually rather than the entire source.

However, the concept of radiant intensity is not useful for that matter. It is defined only for single points. But what is the contribution of a single point on the surface to the total radiant intensity of that object? If it were finite, the total intensity would sum up to infinity due to the infinite number of points on that surface. In the end, one finds that the radiant intensity of each point on that surface is effectively vanishing which is not helpful either.

One can work around this issue by defining a new quantity $L(\vec{r},\vec{\theta})$ called radiance that is a measure of the radiant intensity that originates from a small unit area $\D{A^\perp}$ (aligned normal to the direction $\vec{\theta}$ of interest) rather than from a single point \begin{align} L(\vec{r},\vec{\theta}) = \frac{\Dsq{\Phi (\vec{r},\vec{\theta})} }{\D{A^\perp} \D{\Omega} } \end{align} and has units of power per area per solid angle [W m-2 sr-1]. [7] [8] [9]

Thereby it becomes possible to quantise how much a particular point on the surface on an extended object contributes to the total objects radiant intensity.

The infinitesimal area element $\D{A^\perp}$ is chosen to be normal to the direction $\vec{\theta}$ as it is the only direction in accordance with the geometry given by $\vec{r}$ and $\vec{\theta}$. If one however wishes to use an arbitrary area element $\D{A}$ for the calculation of $L$, one has to consider that the effective projection area $\D{A^\perp} = \D{(A \cos \vartheta)}$ is dependent on the included angle $\vartheta$.

## Energy Density

### Definition

In general the energy density $u(\vec{r})$ is a local quantity that characterises the spatial density of energy $Q$ at a particular point $\vec{r}$. When considering an infinitesimal volume element $\D{V}$ (enclosing the point $\vec{r}$), it is defined as the ratio of the energy $\D{Q(\vec{r},V)}$ stored within the volume $\D{V}$ and the volume itself: \begin{align} u(\vec{r}) = \frac{\D{Q(\vec{r},V)}}{\D{V}} \end{align}

In the special case of radiation this energy $\D{Q}$ is the sum of the individual energies of all photons located within the volume $\D{V}$. The photons move at the speed of light $c$ and may propagate in all possible directions $\vec{\theta}$.

Due to the linearity of Maxwell's equations, their motions superpose independently. Therefore the total radiant energy density $u(\vec{r})$ can be treated as a compostion of individual contributions from all directions $\vec{\theta}$. As before, the energy flux from a single direction $\vec{\theta}$ is in fact 0, so one has to consider a differential solid angle $\D{\Omega}$ around this direction $\vec{\theta}$.

For the volume $\D{V}$ one could consider a small cylinder of base $\D{A}^\perp$ and length $\D{l}$ aligned along the direction of propagation $\vec{\theta}$. The length $\D{l} = c \D{t}$ can also be expressed in terms of the time $\D{t}$ that it takes for the photons to travel through the cylinder. Then the infinitesimal contribution $\D{u}(\vec{r},\vec{\theta},\D{\Omega})$ to the total energy density $u(\vec{r})$ in this point coming from the solid angle $\D{\Omega}$ is \begin{align} \D{u}(\vec{r},\vec{\theta}) &= \frac{\D[3]{Q}(\vec{r},\vec{\theta})}{\D[2]{V}} = \frac{\D[2]{P}(\vec{r},\vec{\theta}) \D{t} }{ \D{A}^\perp \D{l} } \\ &= \frac{L(\vec{r},\vec{\theta}) \cancel{\D{A}^\perp} \cancel{\D{t}} \D{\Omega}(\vec{\theta}) }{\cancel{\D{A}^\perp} c \cancel{\D{t}} } \\ &= \frac{L(\vec{r},\vec{\theta})}{c} \D{\Omega}(\vec{\theta}) \end{align} Here $L$ is the radiance that is in general a function of location $\vec{r}$ and direction $\vec{\theta}$.

[10]

In the case of isotropic radiance $L \neq L(\vec{\theta})$ the integration yields:[11] \begin{align} u(\vec{r}) &= \int \D{u}(\vec{r},\vec{\theta}) \\ &= \frac{L(\vec{r})}{c} \cdot \int \D{\Omega}(\vec{\theta}) \\ &= \frac{L(\vec{r})}{c} \cdot \int_0^{2\pi} \D{\varphi} \int_0^\pi \sin{\vartheta} \D{\vartheta} \\ &= \frac{4\pi}{c} \cdot L(\vec{r}) \end{align}