Lichtwelle

Ein-dinemsionale Welle

Linear polarisiertes Licht

\[ \begin{aligned} \vec{E}_1(x,t) &= \begin{pmatrix} 0 \\ 0 \\ \hat{s}\sin \left[ \omega\left(t + \dfrac{x}{c}\right) \right]\\ \end{pmatrix}\\[1ex] % \vec{E}_2(x,t) &= \begin{pmatrix} 0 \\ \hat{s}\sin \left[ \omega\left(t + \dfrac{x}{c}\right) \right] \\ 0 \end{pmatrix}\\ \vec{E}_{1+2}(x,t) = \vec{E}_1 + \vec{E}_2 &= \begin{pmatrix} 0 \\ \hat{s}\sin \left[ \omega\left(t + \dfrac{x}{c}\right) \right]\\ \hat{s}\sin \left[ \omega\left(t + \dfrac{x}{c}\right) \right]\\ \end{pmatrix}\\[1ex] % |\vec{E}_{1+2}(x,t)| &= \sqrt{{\hat{s}}^2\sin^2\left[\omega\left(t+\frac{x}{c}\right)\right] + \hat{s}^2\sin^2\left[\omega\left(t+\frac{x}{c}\right)\right]} \\ &= \sqrt{2\hat{s}^2\sin^2\left[\omega\left(t+\frac{x}{c}\right)\right]}\\ &= \sqrt{2}\hat{s}\sin\left[\omega\left(t+\frac{x}{c}\right)\right] \end{aligned} \]

Zirkular polarisiertes Licht

\[ \begin{aligned} \vec{E}_1(x,t) &= \begin{pmatrix} 0\\ 0\\ s\left(x + \dfrac{\lambda}{4}, t\right)\\ \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ \hat{s}\sin \left[ \omega\left(t + \dfrac{x}{c}\right) \right]\\ \end{pmatrix}\\ % \vec{E}_2(x,t) &= \begin{pmatrix} 0\\ s\left(x + \dfrac{\lambda}{4}, t\right)\\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ \hat{s}\sin \left[ \omega\left(t + \frac{x+\frac{\lambda}{4}}{c}\right) \right] \\ 0 \end{pmatrix}\\ \vec{E}_{2y}(x,t) &= \hat{s}\sin \left[ \omega\left(t + \frac{x+\frac{\lambda}{4}}{c}\right) \right] = \hat{s}\sin \left[\omega\left(t + \frac{x}{c} + \frac{1}{4}\frac{\lambda}{c} \right) \right] \\ &= \hat{s}\sin \left[\omega\left(t + \frac{x}{c} + \frac{1}{4}\cdot\frac{2\pi}{\omega} \right) \right] && | \lambda = T\cdot c = \frac{2\pi}{\omega} c\\ &= \hat{s}\sin \left[\omega\left(t + \frac{x}{c} + \frac{1}{2}\cdot\frac{\pi}{\omega} \right) \right] = \hat{s}\sin \left[\omega\left(t + \frac{x}{c}\right) + \frac{\pi}{2} \right]\\ &= \hat{s}\cos \left[\omega\left(t + \frac{x}{c}\right) \right]\\ |\vec{E}_{1+2}(x,t)| &= \sqrt{ \hat{s}^2\sin^2 \left[ \omega\left(t + \dfrac{x}{c}\right) \right] + \hat{s}^2\cos^2 \left[ \omega\left(t + \dfrac{x}{c}\right) \right] } \\ &= \sqrt{\hat{s}^2 \left(\sin^2 \left[ \omega\left(t + \dfrac{x}{c}\right) \right] + \cos^2 \left[ \omega\left(t + \dfrac{x}{c}\right) \right]\right)} && | \sin^2\alpha + \cos^2\alpha = 1 \; \forall \alpha \in \mathbb{R} \\ &= \hat{s} = \text{Konstant} \end{aligned} \]

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