Measures of the intensity of electromagnetic radiation

1. Nov 26, 2012

Wox

I've always struggled with the commonly used measures of the intensity of electromagnetic radiation and it's catching up to me lately. Suppose $\bar{P}(R,\phi,\theta)$ is the Poynting vector of an electromagnetic field (in spherical coordinates) with norm $I(R,\phi,\theta)=\|\bar{P}(R,\phi,\theta)\|$ in $J.m^{-2}.s^{-1}$ (strictly speaking it should be the ensemble average of the norm). Then the flux $\Phi$ through a surface $S$, for which $\Psi$ is the angle between Poynting vector and surface normal at $(R,\phi,\theta)$, is given by (in $J.s^{-1}$)
\begin{align}\Phi_{S}&=\iint_{S} I(R,\phi,\theta)\cos\Psi\ \text{d}A\\ &=\iint_{S} I(R,\phi,\theta)\cos\Psi\ R^{2}\ \text{d}\Omega\\ &=\iint_{S} I(R,\phi,\theta)\cos\Psi\ R^{2}\sin\theta\ \text{d}\phi\text{d}\theta \end{align}
In radiometry, this is called the radiant flux. From this measure of intensity, several other measures are derived, but I'm a bit puzzled how. Take for example the radiant intensity $I_{e}$: the power per solid angle (in $J.sr^{-1}.s^{-1}$). From the equations above I understand that
$$I_{e}=\frac{\text{d}\Phi_{S}}{\text{d}\Omega}=I(R,\phi,\theta)\cos\Psi\ R^{2}$$
but how is this independent from surface $S$ (which is the point of using other measures than the radiant flux)? Secondly, while you're at it, why not using
$$J=\frac{\text{d}^{2}\Phi_{S}}{\text{d}\phi \text{d}\theta}=I(R,\phi,\theta)\cos\Psi\ R^{2}\sin\theta$$
which I also don't see to be independent of $S$, but at least you don't need to remember the $\sin\theta$ when integrating it. Then there are measures like the radiance in $J.m^{-2}.sr^{-1}.s^{-1}$ which is defined as
$$L_{e}=\frac{\text{d}^{2}\Phi_{S}}{\cos\Psi\text{d}A \text{d}\Omega}$$
I can write that down but I can't say I understand what it means. Can anyone shed some light on these issues?

Last edited: Nov 26, 2012