Polarized light

The \( \mathbf{E} \) field vector and \( \mathbf{B} \) field vector of an electromagnetic wave form a plane perpendicular to the direction of wave propagation, i.e., \( \hat{\mathbf{ k }} \). Here we only talk about \( \mathbf{E} \) field. For an unpolarized light, its electric field can be decomposed into \( 2 \) directions, here, say \( \hat{ \mathbf{x} } \) and \( \hat{ \mathbf{y} } \). If we assume \( \mathbf{ k } = (0, 0, k_z) \), then $$ \begin{align} \mathbf{ E }_x (z, t) &= E_x \hat{ \mathbf{ x } } = A \sin ( k_z z - \omega t ) \hat{ \mathbf{ x } }, \\ \mathbf{ E }_y (z, t) &= E_y \hat{ \mathbf{ y } } = B \sin ( k_z z - \omega t + \phi ) \hat{ \mathbf{ y } }, \end{align} $$ i.e., there is a phase difference \( \phi \) between \( E_x \) and \( E_y \).

Questions

1. Identify under which condition you will see a linear- or plane-polarized wave?
2. Identify under which condition you will see a circular-polarized wave?
3. Identify under which condition you will see an right hand elliptic-polarized wave? Then for circular-polarized wave? What if \( E_x \) and \( E_y \) vary as \( \cos \) rather than \( \sin \)?
4. What is the amplitude of a total e-field for a plane-polarized wave, and a circular polarized wave, respectively?
5. Look up the physical cause of Brewster angle.