Partially polarized light

Figure 16.13: a) Half wave plate and 16.13: b) Analysis of partially polarized light

Many of the polarizers discussed earlier produce partially polarized light. This is a mixture of polarized and unpolarized light. Consider a mixture with unpolarized light of intensity $I_U$ and light that is linearly polarized along the $y$ axis of intensity $I_P$. A polaroid sheet, referred to as an analyzer, is used to analyze this light (right in the Figure 16.13). The intensity of the unpolarized light becomes after passing through the analyzer whereas the intensity of the polarized light is $I_P \, \cos^2 \theta$ where $\theta$ is the angle between the transmission axis of the analyzer and the $y$ axis. The resulting intensity is

$$I=\frac{I_U}{2}+ I_P \cos^2 \theta \,.$$ (16.11)

The intensity of the transmitted light changes as the analyzer is rotated and the maximum and minimum intensity respectively are

$$I_{max}=\frac{I_U}{2}+I_P \hspace{1cm} I_{min}=\frac{I_U}{2}\,.$$ (16.12)

The degree of polarization is defined as

$$P=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}=\frac{I_P}{I_U+I_P}$$ (16.13)

This quantifies the fraction of the light intensity in the linearly polarized component.

Problems

1. Consider polarized light whose $\vec{E}$ is given to be
   
   $$\vec{E}(z,t)=E_0 [A \hat{i} \cos(\omega t -k z) \hat{i} + B \hat{i} \sin(\omega t -k z) \hat{j} ]\,.$$
   
   
   Calculate the degree of polarization $P$ for [a.] $A=1, B=0$ [b.] $A=1, B=1$ [c.] $A=1,B=2$. What are the corresponding polarization states. What happens if these waves are sent through a quarter wave plate that adds and extra $90^{\circ}$ delay to the $y$ component?
2. Incident light is scattered by an electron. What is the degree of polarization for scattering angles [a.] $90^{\circ}$ [b.] $45^{\circ}$?
3. A birefringent crystal of thickness $d$ has its optic axis parallel to the surface of the crystal. What should be the value of $d$ (in $\mu$m ) if the crystal is to be used as a quarter wave plate for light of wavelength ? ($n_e=1.5334$, $n_o=1.5443$).
4. Linearly polarized light with intensity $I$ is normally incident on a polarizer. The plane of polarization of the incident light is at $30^{circ}$ to the transmission axis of the polarizer. What is the intensity of the transmitted light?
5. For the Wollaston prism in Figure 16.11 with , , $n_e=1.486$ and $n_o=1.658$, calculate the angle $\delta$ between the two rays that come out.
6. Calculate the Brewster's angle $\theta_B$ for glass $n=1.5$.