Measures of the intensity of electromagnetic radiation

[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?

 

[eljose79]

Thank you for sharing your struggles with the commonly used measures of intensity of electromagnetic radiation. I can understand the confusion and complications that can arise when dealing with these measures. Allow me to shed some light on the issues you have raised.

Firstly, let's focus on the radiant intensity I_e. As you correctly mentioned, this measure is defined as the power per solid angle and is expressed in units of J.sr^{-1}.s^{-1}. This measure is independent of the surface S because it is a measure of the directional flow of energy from a point source. In other words, it gives us information about the amount of energy that is passing through a unit solid angle in a specific direction. This is why it is also known as the directional radiant flux.

To further understand this, let's consider a simple example. Imagine a point source of light, like a bulb, emitting light in all directions. Now, if we want to measure the intensity of the light in a specific direction, say towards the ceiling, we would use the radiant intensity. This measure tells us how much power is passing through a unit solid angle in that specific direction. It is important to note that this measure is independent of the surface because it only considers the direction of the energy flow, not the surface it is passing through.

Moving on to your second question about the measure J, also known as the radiant exitance. This measure is defined as the radiant flux per unit area and is expressed in units of J.m^{-2}.s^{-1}. This measure is not independent of the surface S because it takes into account the area of the surface that the energy is passing through. In other words, it tells us how much energy is passing through a unit area of a surface in a specific direction. Therefore, this measure is dependent on the surface S.

Finally, let's discuss the radiance L_e, which you mentioned to be the most confusing for you. This measure is defined as the radiant flux per unit area per unit solid angle and is expressed in units of J.m^{-2}.sr^{-1}.s^{-1}. This measure is used to describe the amount of energy emitted or reflected by a surface in a specific direction. It takes into account the area of the surface as well as the solid angle of the energy flow. Therefore, it is dependent on both the surface and the solid angle.

I hope this has helped clarify some of your doubts regarding these