Herleitung

Für zwei Wellenlängen gilt:

$$\begin{array}{l}{E}_{\text{λ}}\left(\mathit{t}\right)=\left({\epsilon }_{\text{λ}\text{A}}-{\epsilon }_{\text{λ}\text{C}}\right){\mathit{c}}_{\mathit{A}}\left(\mathit{t}\right)+{\epsilon }_{\text{λ}\text{C}}{\mathit{c}}_{\mathit{A0}}\\ E_{{\text{λ}}^{\text{'}}}\left(\mathit{t}\right)=\left({\epsilon }_{{\text{λ}}^{\text{'}}\text{A}}-{\epsilon }_{{\text{λ}}^{\text{'}}\text{C}}\right){\mathit{c}}_{\mathit{A}}\left(\mathit{t}\right)+{\epsilon }_{{\text{λ}}^{\text{'}}\text{C}}{\mathit{c}}_{\mathit{A0}}\end{array}$$

Die Gleichungen werden in der angegebenen Weise erweitert.

$$\begin{array}{rcll}{E}_{\text{λ}}\left(\mathit{t}\right)& =& \left({\epsilon }_{\text{λ}\text{A}}-{\epsilon }_{\text{λ}\text{C}}\right){\mathit{c}}_{\mathit{A}}\left(\mathit{t}\right)+{\epsilon }_{\text{λ}\text{C}}{\mathit{c}}_{\mathit{A0}}& |\cdot \left({\epsilon }_{{\text{λ}}^{\text{'}}\text{A}}-{\epsilon }_{{\text{λ}}^{\text{'}}\text{C}}\right)\\ E_{{\text{λ}}^{\text{'}}}\left(\mathit{t}\right)& =& \left({\epsilon }_{{\text{λ}}^{\text{'}}\text{A}}-{\epsilon }_{{\text{λ}}^{\text{'}}\text{C}}\right){\mathit{c}}_{\mathit{A}}\left(\mathit{t}\right)+{\epsilon }_{{\text{λ}}^{\text{'}}\text{C}}{\mathit{c}}_{\mathit{A0}}& |\cdot \left({\epsilon }_{\text{λ}\text{A}}-{\epsilon }_{\text{λ}\text{C}}\right)\end{array}$$

Elimination von ${\mathit{c}}_{\mathit{A}}\left(\mathit{t}\right)$ durch Subtraktion der Gleichungen ergibt:

$$\left({\epsilon }_{{\text{λ}}^{\text{'}}\text{A}}-{\epsilon }_{{\text{λ}}^{\text{'}}\text{C}}\right)\cdot E_{\text{λ}}\left(\mathit{t}\right)-\left({\epsilon }_{\text{λ}\text{A}}-{\epsilon }_{\text{λ}\text{C}}\right)\cdot E_{{\text{λ}}^{\text{'}}}\left(\mathit{t}\right)=\left({\epsilon }_{{\text{λ}}^{\text{'}}\text{A}}-{\epsilon }_{{\text{λ}}^{\text{'}}\text{C}}\right)\cdot {\epsilon }_{\text{λ}\text{C}}{\mathit{c}}_{\mathit{A0}}-\left({\epsilon }_{\text{λ}\text{A}}-{\epsilon }_{\text{λ}\text{C}}\right)\cdot {\epsilon }_{\text{λ}\text{C}}{\mathit{c}}_{\mathit{A0}}$$

Durch Umformulieren ergibt sich:

$$\begin{array}{rcl}{E}_{\text{λ}}\left(\mathit{t}\right)-\frac{{\epsilon }_{\text{λ}\text{A}}-{\epsilon }_{\text{λ}\text{C}}}{{\epsilon }_{{\text{λ}}^{\text{'}}\text{A}}-{\epsilon }_{{\text{λ}}^{\text{'}}\text{C}}}{E}_{{\text{λ}}^{\text{'}}}\left(\mathit{t}\right)& =& {\mathit{c}}_{\mathit{A0}}\frac{{\epsilon }_{\text{λ}\text{C}}\cdot {\epsilon }_{{\text{λ}}^{\text{'}}\text{A}}-{\epsilon }_{\text{λ}\text{C}}\cdot {\epsilon }_{{\text{λ}}^{\text{'}}\text{C}}-{\epsilon }_{\text{λ}\text{A}}\cdot {\epsilon }_{{\text{λ}}^{\text{'}}\text{C}}+{\epsilon }_{\text{λ}\text{C}}\cdot {\epsilon }_{{\text{λ}}^{\text{'}}\text{C}}}{{\epsilon }_{{\text{λ}}^{\text{'}}\text{A}}-{\epsilon }_{{\text{λ}}^{\text{'}}\text{C}}}\\ E_{\text{λ}}\left(\mathit{t}\right)& =& \frac{{\epsilon }_{\text{λ}\text{A}}-{\epsilon }_{\text{λ}\text{C}}}{{\epsilon }_{{\text{λ}}^{\text{'}}\text{A}}-{\epsilon }_{{\text{λ}}^{\text{'}}\text{C}}}{E}_{{\text{λ}}^{\text{'}}}\left(\mathit{t}\right)+{\mathit{c}}_{\mathit{A0}}\frac{{\epsilon }_{\text{λ}\text{C}}{\epsilon }_{{\text{λ}}^{\text{'}}\text{A}}-{\epsilon }_{\text{λ}\text{A}}{\epsilon }_{{\text{λ}}^{\text{'}}\text{C}}}{{\epsilon }_{{\text{λ}}^{\text{'}}\text{A}}-{\epsilon }_{{\text{λ}}^{\text{'}}\text{C}}}\end{array}$$