Elliptical polarization

Oscillations of different amplitude combined with a phase difference of $\pi/2$ produces elliptically polarized wave where the ellipse is aligned with the $y-z$ axis as shown in Figure 19.4. The ellipse is not aligned with the $y-z$ axis for an arbitrary phase difference between the $y$ and $z$ components of the electric field. This is the most general state of polarization shown in the last diagram of the Figure 19.4. Linear and circularly polarized waves are specific cases of elliptically polarized waves.

Figure 8.4: Elliptical plarization

Problems

1. Find the plane of polarization of a light which is moving in the positive $x$ direction and having amplitudes of electric field in the $y$ and $z$ directions, $3$ and $\sqrt{3}$ respectively in same units. The oscillating components of the electric field along $y$ and $z$ have the same frequency and wavelength and the $z$ component is leading with a phase $\pi$.

2. Find the state of polarization of a light which is moving in the positive $x$ direction with electric field amplitudes same along the $y$ and $z$ directions. The oscillating components of the electric field along the $y$ and $z$ have the same frequency and wavelength and the $z$ component is lagging with a phase $\pi/3$.
   (Ans: Left elliptically polarized and the major axis is making an angle with the $y$ axis.)

3. Find the state of polarization of a light which is moving in the positive $x$ direction and having amplitudes of electric field in the $y$ and $z$ directions, $1$ and $3^{1/4}$ respectively in same units. The oscillating components of the electric field along $y$ and $z$ have the same frequency and wavelength and the $y$ component is leading with a phase $\pi/4$.
   (Ans: Left elliptically polarized and the major axis is making an angle $\tan^{-1}\sqrt{1+(2/\sqrt{3})}$ with the $y$ axis.)

4. Find out the maximum and minimum values of electric field at point $x$ for the previous problem. (Ans: $E_{max}^2=(3+\sqrt{3})/2$ and $E_{min}^2=(\sqrt{3}-1)/2$.)