ThermoOpticalCorrection¶

Type:	section
Appearance:	multiple

The thermo-optical correction describes the temperature ( $T$ ) dependency of the relative permittivity in a linear sense near a specified temperature $T_0:$

$\begin{eqnarray*} \Delta \TField{\varepsilon}^{(\mathrm{rel})} & = & C \left(T-T_0 \right) \end{eqnarray*}$

$C$ is the thermo-optical coefficient. It is a $3 \times 3$ - matrix to cover anisotropic effects. The total relative permittivity is then given by

$\begin{eqnarray*} \TField{\varepsilon}^{(\mathrm{rel})} & = \TField{\varepsilon}^{(\mathrm{rel})}_{T_0} + C \left(T-T_0 \right). \end{eqnarray*}$

For too large temperature differences the linear correction term may loose its validity. To prevent obscure relative permittivity definitions one may restrict the absolute value of the correction term by a threshold value $\delta_{\max}$ ,

$\begin{eqnarray*} \Delta \TField{\varepsilon}^{(\mathrm{rel})}_{ij} & = & \min \left ( \max \left ( \Delta \TField{\varepsilon}^{(\mathrm{rel})}_{ij}, -\delta_{\max} \right ), \delta_{\max} \right ). \end{eqnarray*}$

The JCM - syntax looks like this:

# define the relative permittivity with thermo-optical correction
RelPermittivity {
  # define temperature independent terms
  Constant = ...
  ...
  ThermoOpticalCorrection {
     T0 = ... # set correction free temperature
     C = ... # set thermo-optical coefficients
     CutOff .. # set threshold here

  }
}

Warning

Often, the thermo-optical coefficient is defined as a refractive index - temperature relation, $\Delta n = \alpha (T-T_0).$

In a linear sense both definitions are equivalent:

$\begin{eqnarray*} \TField{\varepsilon}^{(\mathrm{rel})} & = & (n_{T_0}+\Delta n)^2 = (n_{T_0}+\alpha(T-T_0))^2\\ { } & = & n_{T_0}^2 +2n_{T_0}\alpha(T-T_0)+\alpha^2 (T-T_0)^2 \\ { } & \sim & \TField{\varepsilon}^{(\mathrm{rel})}_{T_0}+\underbrace{2 n_{T_0}\alpha}_{C}(T-T_0) \end{eqnarray*}$

The approximation in the last line is due to linearization.