Poynting theorem for light-matter coupling in an absortive medium

[Cybertib]

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 :
$\frac{\partial u}{\partial t}+\overrightarrow{\nabla}.Re\left(\overrightarrow{S}\right)=\overrightarrow{J}.\overrightarrow{E^{'}}-\omega\varepsilon_{0}.Im(\varepsilon)< \overrightarrow{E} .\overrightarrow{E} >$

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

 

[eljose79]

Dear Thibault,

Thank you for bringing up this interesting and complex topic. The coupling of a monochromatic short EM wave and the energy absorption by matter is indeed a challenging problem that has been studied and debated by many scientists.

Firstly, I would like to clarify that the Poynting theorem is a fundamental concept in electromagnetism that describes the flow of energy in an electromagnetic field. It is based on the conservation of energy and states that the rate of change of energy in a given volume is equal to the sum of the energy flux into the volume and the work done on the charges inside the volume. This theorem is valid for both dispersive and non-dispersive media.

In non-dispersive media, the Poynting vector, as you mentioned, is given by $\overrightarrow{S}=\frac{1}{2}\overrightarrow{E} \times \overrightarrow{H'}$ and represents the energy flow per unit area. The Poynting vector is a complex quantity, and its real part describes the average power flow while its imaginary part represents the reactive power flow.

Now, in the case of a monochromatic short EM wave, as you correctly pointed out, the Poynting theorem states that $\frac{\partial u}{\partial t}=0$, where $u$ is the energy density in the medium. This means that the energy in the field is not changing over time, and thus the energy absorption by matter is given by the divergence of the Poynting vector, $\overrightarrow{\nabla}.\overrightarrow{S}$. This is what Stratton and Landau refer to in their explanation.

On the other hand, Jackson's approach, which you presented, considers the variation of energy in the field, and thus takes into account the imaginary part of the dielectric function, as well as the current density and the electric field. This approach is more general and can be applied to both dispersive and non-dispersive media.

Regarding the difference between the conclusions of Jackson and Stratton and Landau, it is important to note that they are not necessarily contradictory. They are simply different approaches to solving the same problem. Both approaches have their strengths and limitations, and it is up to the scientist to choose the most appropriate one for their specific case.

In conclusion, I believe that both approaches are valid and can provide valuable insights into the coupling of a monochromatic short EM wave