Numerical Methods in Photonics

8.3.7.2  Exterior Problem for the Scattered Field

As for the Helmholtz equation, the scattered field in the exterior domain satisfies Maxwell’s equations in weak form

$\begin{array}{l}\underset{{\text{Ω}}_{\text{ext}}}{{\displaystyle \int }}dxdy(\nabla \times v^{\ast })\cdot \frac{1}{\text{μ}}(\nabla \times E_{\text{s}})-\underset{\partial {\text{Ω}}_{\text{ext}}}{{\displaystyle \int }}ds\cdot \left(v^{\ast }\times \left[\frac{1}{\text{μ}}\nabla \times E_{\text{s}}\right]\right)\hfill \\ -{\text{ω}}^2\underset{{\text{Ω}}_{\text{ext}}}{{\displaystyle \int }}dxdy{v}^{\ast }\epsilon E_{\text{s}}\text{=}\text{0}\hfill \end{array}$

and the same formula holds true for the complex extension of all quantities

$\begin{array}{l}{\displaystyle \underset{{\mathrm{\Omega }}_{\text{ext}}}{\int }d{x}^{c}d{y}^c({\nabla }^c\times v^{\ast }{c}^{})\cdot \frac{1}{\left|{\text{μ}}^c\right|}\left({\nabla }^c\times {\text{E}}_s^c\right)}-{\displaystyle \underset{\partial {\mathrm{\Omega }}_{\text{ext}}^c}{\int }d{s}^c\cdot }\left(V^{*c}\times \left[{\nabla }^c\times E_{\text{s}}^{\text{c}}\right]\right)\\ -{\text{ω}}^{\text{2}}{\displaystyle \underset{{\mathrm{\Omega }}_{\text{ext}}^c}{\int }d{x}^{c}d{y}^c{V}^{*c}\epsilon E_{\text{s}}{c}^{}}=0.\end{array}$

In Maxwell’s case, the entire mapping can be traced back to the material mappings (8.61) and (8.62); hence, we have to replace

$\begin{array}{lll}{({\mathrm{\mu }}^c)}^{-1}\hfill & =\hfill & \frac{1}{\left|J\right|}{J}^T{\mathrm{\mu }}^{-1}J\hfill \\ {\epsilon }^c\hfill & =\hfill & \left|J\right|J^{-1}\epsilon J^{-T}.\hfill \end{array}$

Therefore, the formula for the complex extended field given in original spatial coordinates is

$\begin{array}{l}\underset{{\text{Ω}}_{\text{ext}}}{{\displaystyle \int }}dxdy(\nabla \times v^{\ast }{c}^{})·\frac{1}{\left|J\right|}{J}^T{\text{μ}}^{-1}J({\nabla }^c\times E_s^c)-{\omega }^2\underset{{\text{Ω}}_{\text{ext}}}{{\displaystyle \int }}dx\hfill \end{array}$