Polarization

What polarization is

Light is an electromagnetic wave whose electric field $\vec{E}$ and magnetic field $\vec{B}$ are perpendicular to each other and to the direction of propagation. Light is emitted by atoms: if one atom emits a wave along the $x$-axis, its electric field may be along the $y$-axis, but another atom emitting along the $x$-axis may have its electric field along ANY direction in the $y$-$z$ plane, with its magnetic field perpendicular to it. Light from the Sun, from stars, or from a bulb therefore has electric fields in all possible directions in the plane perpendicular to propagation.

• Unpolarized light: the electric field takes all possible directions in the plane perpendicular to the direction of propagation. It is drawn as double headed arrows in many directions, for a beam coming towards us.
• Plane polarized light: the electric field is constrained to one particular direction. It is drawn as double headed arrows along one direction only.
• Definition the board marks: the phenomenon of restriction of the vibrations of light waves to a particular plane perpendicular to the direction of wave motion is called polarization of light.

Polarizer, polaroid, planes

• Certain materials transmit only those light waves whose electric field lies along one particular direction; these are polarizers. The direction along which the electric field of the emergent wave is oriented is called the polarizing axis.
• A polaroid is a kind of synthetic plastic sheet used as a polarizer. Uses: (i) to produce and to analyse polarized light; (ii) in sunglasses, to reduce the intensity of the partially or fully polarized light reflected into the eyes from shiny surfaces (glare).
• The plane containing the electric field vectors of plane polarized light is the plane of vibration. The plane perpendicular to the plane of vibration (and to the electric field vector) is the plane of polarization.
• If the polarized light is passed through a second polarizer with its axis perpendicular to that of the first, NO light emerges. If the angle is less than $90^\circ$, only the component of the electric field along the second axis passes.

Only transverse waves can be polarized

In a transverse wave the oscillations can be along any direction in the plane perpendicular to the direction of propagation, so they can be restricted to one direction. In a longitudinal wave, for example sound, the particles oscillate only along the direction of propagation, so there is nothing to restrict. This is why polarization proves that light is a transverse wave.

Derivation of Malus' law

Consider an unpolarized wave of angular frequency $\omega$ and wave vector $k=\dfrac{2\pi}{\lambda}$ travelling along the $x$-direction, with electric field of magnitude

$$E=E_{0}\sin(kx-\omega t)$$

The intensity is proportional to $|E_{0}|^{2}$, and the direction of $\vec{E}$ may be anywhere in the $y$-$z$ plane.

Through the first polarizer

Take one particular wave whose electric field makes an angle $\phi$ with the axis of the first polarizer. The component $E_{0}\cos\phi$ passes; the perpendicular component $E_{0}\sin\phi$ is obstructed. The intensity of this wave after the polarizer is proportional to $|E_{0}\cos\phi|^{2}$.

For unpolarized incident light $\phi$ takes all values from $0$ to $180^\circ$, so we must average $|E_{0}\cos\phi|^{2}$ over that range. Since the average of $\cos^{2}\phi$ is $\dfrac{1}{2}$,

$$I_{1}\propto \dfrac{1}{2}E_{0}^{2}\qquad\Rightarrow\qquad I_{1}=\dfrac{I_{0}}{2}$$

The intensity of an unpolarized wave is reduced to half after passing through a polarizer, whatever the orientation of the polarizer.

Through the second polarizer

The emerging wave is linearly polarized. Let its electric field be along the $y$-direction:

$$\vec{E_{1}}=\hat{j}\,E_{10}\sin(kx-\omega t),\qquad I_{1}\propto |E_{10}|^{2}$$

Let the polarizing axis of the second polarizer make an angle $\theta$ with the $y$-direction. Only the component $E_{10}\cos\theta$ passes, so the transmitted amplitude is

$$E_{20}=E_{10}\cos\theta$$

and the transmitted intensity is

$$I_{2}\propto |E_{20}|^{2}=|E_{10}\cos\theta|^{2}$$

$$\boxed{\ I_{2}=I_{1}\cos^{2}\theta\ }$$

This is Malus' law, after E. L. Malus (1775–1812), who discovered it experimentally. Malus' law gives the intensity of a linearly polarized wave after it passes through a polarizer.

• $\theta$ is the angle between the polarizing AXES of the two polarizers.
• $\theta=0^\circ$: parallel setting, $\cos^{2}\theta=1$, the intensity is unchanged by the second polarizer.
• $\theta=90^\circ$: crossed setting, $\cos^{2}\theta=0$, no light emerges.
• Three polarizers in succession, unpolarized light entering: $I_{3}=\dfrac{I_{0}}{2}\cos^{2}\theta_{1}\cos^{2}\theta_{2}$.

7.7.1 Polarization by reflection — Brewster's law

Statement: the tangent of the polarizing angle is equal to the refractive index of the refracting medium at which partial reflection takes place.

$$\tan\theta_{B}=\dfrac{n_{2}}{n_{1}}$$

Explanation and derivation

• When light is incident at an angle on a boundary between two transparent media of refractive indices $n_{1}$ and $n_{2}$, part of it is refracted and the rest is reflected.
• Consider unpolarized light incident from the rarer medium $n_{1}$ on such a boundary, drawn perpendicular to the plane of the paper. Resolve the incident electric field into a component parallel to the plane of the paper (double arrows) and a component perpendicular to it (dots); both are of equal magnitude for unpolarized light. In general the reflected and the refracted beams do not carry these two components equally, so both are PARTIALLY polarized.
• D. Brewster discovered experimentally in 1812 that at one particular angle of incidence $\theta_{B}$ the reflected wave is COMPLETELY plane polarized, with its electric field perpendicular to the plane of the paper, while the refracted wave is only partially polarized. This angle is Brewster's angle (the polarizing angle).
• At this angle the reflected and refracted rays are perpendicular to each other. Hence, for angle of refraction $\theta_{r}$,

$$\theta_{B}+\theta_{r}=90^\circ\qquad\dots(1)$$

• From the law of refraction,

$$n_{1}\sin\theta_{B}=n_{2}\sin\theta_{r}$$

$$n_{1}\sin\theta_{B}=n_{2}\sin(90^\circ-\theta_{B})=n_{2}\cos\theta_{B}\qquad\dots[\text{using }(1)]$$

$$\therefore\ \dfrac{n_{2}}{n_{1}}=\tan\theta_{B}$$

$$\boxed{\ \theta_{B}=\tan^{-1}\!\left(\dfrac{n_{2}}{n_{1}}\right)\ }$$

Showing that the reflected and refracted rays are perpendicular

In the figure let $AB$ be the incident ray, $BC$ the reflected ray and $BD$ the refracted ray.

• From Snell's law, $\dfrac{\sin\theta_{B}}{\sin\theta_{R}}=n\qquad\dots(1)$
• From Brewster's law, $\tan\theta_{B}=n\qquad\dots(2)$
• From (1) and (2),

$$\dfrac{\sin\theta_{B}}{\sin\theta_{R}}=\tan\theta_{B}=\dfrac{\sin\theta_{B}}{\cos\theta_{B}}$$

$$\therefore\ \sin\theta_{R}=\cos\theta_{B}=\cos(90^\circ-\theta_{R})\ \Rightarrow\ 90^\circ-\theta_{R}=\theta_{B}$$

$$\therefore\ \theta_{B}+\theta_{R}=90^\circ\qquad\dots(3)$$

• By the law of reflection the angle of reflection equals $\theta_{B}$. Since the three angles at $B$ on the straight boundary add to $180^\circ$,

$$\theta_{B}+\theta_{R}+\angle DBC=180^\circ\ \Rightarrow\ 90^\circ+\angle DBC=180^\circ\ \Rightarrow\ \angle DBC=90^\circ$$

$$\therefore\ BD\perp BC$$

Hence at the polarizing angle the reflected and refracted rays are mutually perpendicular.

Polarization by reflection is used to cut glare from reflecting surfaces with polaroid sunglasses. It works only for NON-METALLIC surfaces. A polaroid also halves the intensity of completely unpolarized sunlight.

7.7.2 Polarization by scattering

When sunlight strikes air molecules or dust particles in the atmosphere it changes direction; this is scattering. Blue light is preferentially scattered, which is why the sky looks blue; without scattering the sky would look dark and stars would be visible by day.

Scattered sunlight is partially polarized, in much the same way as reflected light. The degree of polarization depends on the angle of scattering — the angle between the direction of the incident light and the direction of the scattered light. If this angle is $90^\circ$, the scattered light is plane polarized. Light reaching us from different parts of the sky is therefore polarized to different degrees, which is why the brightness of the sky changes as you rotate a polaroid while looking at it.

Standard results

• $I_{1}=\dfrac{I_{0}}{2}$ after the first polarizer, for unpolarized incident light.
• Malus' law: $I_{2}=I_{1}\cos^{2}\theta$, $\theta$ between the two polarizing axes.
• Brewster's law: $\tan\theta_{B}=\dfrac{n_{2}}{n_{1}}$; from air, $\tan\theta_{B}=n$.
• $\theta_{B}+\theta_{r}=90^\circ$; reflected ray $\perp$ refracted ray at the polarizing angle.
• Only transverse waves can be polarized; polarization proves light is transverse.