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Absorption Calculation

Loss Density

We start from Poynting's theorem in the absence of free currents,

\[\nabla \cdot \boldsymbol{S} = -\frac{\partial u}{\partial t} = \boldsymbol{E} \cdot \frac{\partial \boldsymbol{D}}{\partial t} - \boldsymbol{H} \cdot \frac{\partial \boldsymbol{B}}{\partial t}\]

where \(\boldsymbol{S}\) is the power flux density, \(u\) the energy density, \(\boldsymbol{D}=\varepsilon_0 \boldsymbol{\varepsilon_r}\boldsymbol{E}\), and \(\boldsymbol{B}=\mu_0 \boldsymbol{H}\).

For harmonic fields \(\boldsymbol{E}(\boldsymbol{r},t)=\mathrm{Re}\left[\hat{\boldsymbol{E}}(\boldsymbol{r}) e^{-i \omega t}\right]\), the time average of this equation can be written as

\[\langle \nabla \cdot \boldsymbol{S} \rangle = -\frac{\varepsilon_0 \omega}{2} \mathrm{Im}(\hat{\boldsymbol{E}}^* \cdot \boldsymbol{\varepsilon_r} \cdot \hat{\boldsymbol{E}})\]

Splitting the relative dielectric tensor into its Hermitian and anti-Hermitian parts, \(\boldsymbol{\varepsilon_r}=\boldsymbol{\varepsilon}_r^H + i \boldsymbol{\varepsilon}_r^A\), where

\[\boldsymbol{\varepsilon}_{r}^H = \frac{1}{2}(\boldsymbol{\varepsilon}_{r} + \boldsymbol{\varepsilon}_{r}^\dagger),\]
\[\boldsymbol{\varepsilon}_{r}^A = \frac{1}{2i}(\boldsymbol{\varepsilon}_{r} - \boldsymbol{\varepsilon}_{r}^\dagger),\]

the time-averaged power loss density (units W/m³) becomes

\[Q = \left\langle\frac{\partial u}{\partial t}\right\rangle = \frac{\varepsilon_0 \omega}{2} (\hat{\boldsymbol{E}}^* \cdot \boldsymbol{\varepsilon}_r^A \cdot \hat{\boldsymbol{E}})\]

Power Flux Density

The time average of the power flux density (Poynting vector) is given by the energy density times the group velocity,

\[\left\langle\boldsymbol{S}\right\rangle = \left\langle u \right\rangle \boldsymbol{v}_g\]

Using the Hermitian part of the dispersion tensor \(\boldsymbol{\mathsf{D}}^H=\frac{1}{2}(\boldsymbol{\mathsf{D}} + \boldsymbol{\mathsf{D}}^\dagger)\), it can be shown that this quantity can also be written as

\[\left\langle\boldsymbol{S}\right\rangle = -\frac{\varepsilon_0 \omega}{4} \,\mathrm{Re}(\hat{\boldsymbol{E}}^* \cdot \frac{\partial\boldsymbol{\mathsf{D}}^H}{\partial {\boldsymbol{k}} } \cdot \hat{\boldsymbol{E}})\]

Absorption Coefficient

The absorption coefficient (units 1/m) can now be expressed in terms of the power loss density (units W/m³) and the power flux density (units W/m²) as

\[\alpha = \frac{Q}{2|\langle\boldsymbol{S}\rangle|}\]

Writing \(\hat{\boldsymbol{E}} = \hat E \hat{\boldsymbol{e}}\) where \(\hat{\boldsymbol{e}}\) is the unit polarization vector, the magnitude of the E field drops out and we obtain

\[\alpha = \frac{\hat{\boldsymbol{e}}^* \cdot \boldsymbol{\varepsilon}_r^A \cdot \hat{\boldsymbol{e}}}{|\frac{\partial}{\partial {\boldsymbol{k}} }\mathrm{Re}(\hat{\boldsymbol{e}}^* \cdot \boldsymbol{\mathsf{D}}^H \cdot \hat{\boldsymbol{e}})|}\]

Optical Depth

The absorption coefficient \(\alpha\) is related to the (dimensionless) optical depth \(\tau\) as

\[\frac{d\tau}{ds}=\alpha\]

where \(s\) is the arc length, and the linear absorption power density (units W/m) along the ray trajectory is given as

\[\frac{dP}{ds}=P_0\alpha e^{-\tau}\]

where \(P_0\) is the initial power of the ray.

Weakly Relativistic Dielectric Tensor

For the hermitian part of the dispersion tensor appearing in the denominator of the absorption coefficient,

\[\boldsymbol{\mathsf{D}}^H = \boldsymbol{\varepsilon_r}^H - n^2 \boldsymbol{I} + \boldsymbol{n}\boldsymbol{n}\]

raytrax follows Travis1 by using the weakly relativistic dielectric tensor taken from Krivenski and Orefice2.

It is implemented in raytrax.physics.dielectric_tensor.weakly_relativistic_dielectric_tensor.

Fully Relativistic Dielectric Tensor

For the anti-hermitian part of the dielectric tensor appearing in the numerator of the absorption coefficient, the fully relativistic dielectric tensor is required, which is computed using the integral approach3.

  1. Marushchenko, Nikolai B., Yu Turkin, and Henning Maaßberg. "Ray-tracing code TRAVIS for ECR heating, EC current drive and ECE diagnostic." Computer Physics Communications 185.1 (2014): 165-176. doi:10.1016/j.cpc.2013.09.002 ↩

  2. Krivenski and Orefice, J. Plasma Physics (1983), vol. 30, part 1, pp. 125-131, doi:10.1017/S0022377800001045 ↩

  3. Bornatici, M., Cano, R., De Barbieri, O., & Engelmann, F. (1983). Electron cyclotron emission and absorption in fusion plasmas. Nuclear Fusion, 23(9), 1153-1257. doi:10.1088/0029-5515/23/9/005 ↩