Discussion
For the electric field of an electromagnetic wave given below, which of the following statements is correct?
\[ \begin{aligned} \vec{E} = &\hat{x}\, E_{0} cos(\omega t) \\ &+ \hat{y}\, 2E_{0} cos\left(\omega t + \frac{\pi}{2}\right) \end{aligned} \]
Answer: C
The field is elliptically polarised with a ratio of major to minor axis of 2.
Write the components: \(E_{x} = E_{0}\cos\omega t\) and \(E_{y} = 2E_{0}\cos(\omega t + \pi/2) = -2E_{0}\sin\omega t\). Then
\[ \begin{aligned} &\frac{E_{x}^{2}}{E_{0}^{2}} \\ &+ \frac{E_{y}^{2}}{4E_{0}^{2}} = 1 \end{aligned} \]
The tip of \(\vec{E}\) traces an ellipse with semi-axes \(2E_{0}\) along y and \(E_{0}\) along x, so the axis ratio is 2. Linear polarisation needs the components in phase or opposite in phase. Circular polarisation needs a quarter-period shift and equal amplitudes, which these are not. A wave with a fixed phase difference between its components is polarised, not unpolarised.
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