## Poynting theorem for light-matter coupling in an absortive medium

Hello everybody,

I'm focusing on the description of the coupling of a monochromatic short EM wave and on the description of the energy absorption by matter.

The problem is the following. It is well-known that the Poynting theorem in a non-dispersive medium considers the variation of electromagnetic energy $u [J.m^{-3}]$, the transport of power flux using Poynting vector $\overrightarrow{S}=\frac{1}{2}\overrightarrow{E} \times \overrightarrow{H'}$.

Then the balance of the energy contained in the EM field is expressed by :
$\frac{\partial u}{\partial t}+\overrightarrow{\nabla}.Re\left(\overrightarrow{S}\right)=\overright arrow{J}.\overrightarrow{E^{'}}-\omega\varepsilon_{0}.Im(\varepsilon)< \overrightarrow{E} .\overrightarrow{E} >$

The purpose are :
- to calculate the propagated intensity in the medium, which is, according to me calculated by $I= \left\| \overrightarrow{S} \right\| = \sqrt{\overrightarrow{S}.\overrightarrow{S'}}$,
- and to calculate the thermal energy dissipated in the medium, that is expressed through the imaginary part of the dielectric function $\varepsilon \in ℂ$, and thus the power transmitted to the matter by thermal dissipation is given by $Q_{abs}=\omega \varepsilon_{0} Im(\varepsilon) < \overrightarrow{E} . \overrightarrow{E} >$. Of course, $< \overrightarrow{E} . \overrightarrow{E} > = \overrightarrow{E} . \overrightarrow{E'}$, where ' means the complex conjugate.

But the big question for me today is that the Jackson, Stratton and Landau are telling different conclusions for the same problem. I presented the Jackson approach which seems clear.

However, Stratton and Landau explain that for harmonic field, $\frac{\partial u}{\partial t}=0$, where $u \in ℂ$, and thus the thermal coupling with matter is given by $\overrightarrow{\nabla}.\overrightarrow{S}$.

Applied to a concrete case, the $\overrightarrow{\nabla}.\overrightarrow{S}$ term is complex and non-symetric, despite that the system has an axial symetry. The Jackson's approach permit to obtain very symmetric distributions, which are full of physical meaning: dissipation increase in the region which the field losses its energy. However, I think that Poynting vector divergence doesn't play any absorption role in the general case (nore constant, neither harmonic but in the context of a short laser pulse).

I ask the opinion to the forum community because this questions doesn't seem to be fixed. Some experts still disagree. I consider that almost everybody is right. The task is to understand how, e.g. find the correct form convincing everyone.

Have a nice evening,
Thibault

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