6.9 Polarization

Polarization

Recall that we can add waves so lets take a plane wave traveling in the z direction and break it into components. $${\vec{E}}_x=E_{0x}\cos (kz-\omega t)\widehat{ı}$$ $${\vec{E}}_y=E_{0y}\cos (kz-\omega t+\delta )\widehat{}$$ Where $\delta$ is some arbitrary phase between the two components. The electric field for the wave is $$\vec{E}={\vec{E}}_x+{\vec{E}}_y$$ Now there are a number of different cases that arise.

If $\delta =0,2\pi ,4\pi ,\dots$ Then we can write the field as $$\vec{E}=(E_{0x}\widehat{ı}+E_{0y}\widehat{})\cos (kz-\omega t)$$

The field is linearly polarized, that is the E field lies along a straight line.

Likewise If $\delta =\pi ,3\pi ,5\pi ,\dots$ Then again it is linearly polarized, but is now "flipped" $$\vec{E}=(E_{0x}\widehat{i}-E_{0y}\widehat{j})\cos (kz-\omega t)$$

Circularly polarized light is a particularly interest example. Let $\delta =-\pi /2$ and $E_{0x}=E_{0y}=E_0$

Then $${\vec{E}}_x=E_0\cos (kz-\omega t)\widehat{ı}$$ $${\vec{E}}_y=E_0\cos (kz-\omega t-\pi /2)\widehat{}$$ $${\vec{E}}_y=E_0\sin (kz-\omega t)\widehat{}$$ or $$\vec{E}=E_0\left[\cos (kz-\omega t)\widehat{ı}+\sin (kz-\omega t)\widehat{}\right]$$

The direction of $\vec{E}$ is changing with time. For example consider the case at $z=0$ . Then

$$\vec{E}=E_0\left[\cos (-\omega t)\widehat{ı}+\sin (-\omega t)\widehat{}\right]$$

$$\vec{E}=E_0\left[\cos (\omega t)\widehat{ı}-\sin (\omega t)\widehat{}\right]$$

In this case the electric field undergoes uniform circular rotation. For light coming out of the page, it will have the motion shown in thefigure

Suppose $\delta =\pi /2$ then you get $$\vec{E}=E_0\left[\cos (kz-\omega t)\widehat{ı}-\sin (kz-\omega t)\widehat{}\right]$$ This now has the opposite rotation, it is Right Handed.

The most general case of polarization has $\delta$ arbitrary and $E_{0x}\ne E_{0y}$ and is elliptically polarized.

Natural light

Natural light is emitted with the $\vec{E}$ field in a mixture of random directions. This in unpolarized light. At any particular instant $\vec{E}$ has a particular direction, but that direction changes rapidly and randomly.

Malus' law

A polarizer is a device that takes incident natural light and transmits polarized light. For example a linear polarizer will take incident light andselect only that component of the light that has its $\vec{E}$ field lined up along the transmission axis. Suppose there is a linear polarizer that transmits light along a particular axis. This is followed by asecond linear polarizer that has its transmission axis at a different angle with $\theta$ being the angle between the transmission axes. Since $I\sim E^2$ then at the second polarizer $$I(\theta )=I(0){\cos }^2\theta$$ where $I(0)$ is the Irradiance of the light hitting the second polarizer.

Lets consider unpolarized light hitting a linear polarizer. Then half the Irradiance will get through. If this is followed by a second polarizer at ${90}^0$ then no light will pass through the second polarizer. Now what happens if a third polarizer is placed between them?