Circular polarization

(a)

Show that the expression

$$\vec{E}(x,t) = \hat{y}E_{\scriptscriptstyle 0}\cos(\omega t - kx) - \hat{z}E_{\scriptscriptstyle 0}\sin(\omega t - kx)$$ (1)

is a solution to the wave equation for $\vec{E}(x,t)$.

(b)

In our discussion of simple harmonic motion we introduced the complex representation by writing the solution to the Equation of Motion as $\Re\mathfrak{e}\left[\underline{A} e^{i\omega t}\right]$, with $\underline{A}= A_r + i A_i$. Using a similar technique, show that the solution (1) can be written in the form

$$\vec{E}(x,t) = \Re\mathfrak{e}\left[\left(\hat{y}+i\hat{z}\right) E_{\scriptscriptstyle 0}e^{i(\omega t - kx)}\right]\;,$$ (2)

where $\underline{\hat{\epsilon}}=\hat{y}+i\hat{z}$ is the complex polarization vector. Such a wave is called ``circularly polarized''.

(c)

The (equivalent) expressions (1) and (2) represent the $\vec{E}$-field of an EM wave travelling along the $x$-axis. Derive an equation for the magnetic field of this wave, $\vec{B}(x,t)$ both in real and in complex representation.

(d)

Draw two snapshots of the waveform, at times $t=0$ and $t=T/4$, i.e., for each of these times draw $\vec{E}(x,t)$ and $\vec{B}(x,t)$ (both magnitudes and directions) for at least 5 equally spaced positions in the interval $0 \le x \le \lambda$. Can you now tell why this type of E & M wave is called ``circularly polarized''?