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6.2.4 Scaling of the Ordinary Differential Equation System

A simple scaling yields a more compact form for the ODE system (6.19). An appropriate choice is

$\displaystyle \widetilde{\mathbf{E}}(\mathbf{x})=\mathbf{E}(\mathbf{x}), \quad\...
...(\mathbf{x}),\quad \tilde{\chi}(\mathbf{x}) = \frac{1}{k_0^2} \chi(\mathbf{x}).$ (6.14)

By introducing the Kronecker symbolb $ \delta_{nm}^{}$ the ODEs write to

$\displaystyle \begin{aligned}\frac{d \widetilde{E}_{x,nm}(z)}{d z} &= \sum_{p,q...{E}_{x,pq}(z) + g_{nm,pq}^{yy}(z)\widetilde{E}_{y,pq}(z)\right],\end{aligned}$    

whereby the coefficients are given by

$\displaystyle \begin{alignedat}{2}r_{nm,pq}^{xx}(z) &= +k_{x,n}k_{y,q}\tilde{\c...
...&\;\,g_{nm,pq}^{yy}(z) &= -k_{x,n}k_{y,m}\delta_{np}\delta_{mq}.\end{alignedat}$    


... symbolb
The Kronecker symbol is defined as $ \delta_{nm}^{}$ = 1 for n = m and 0 otherwise.

Heinrich Kirchauer, Institute for Microelectronics, TU Vienna