Polarization and Brewster's Law

Polarization of Light

Light is a transverse electromagnetic wave. This means that the oscillations of the electric and magnetic fields are perpendicular to the direction of propagation of the wave. When these oscillations occur in all possible directions perpendicular to the direction of propagation, the light is said to be unpolarized.

Polarization is the phenomenon of restricting the oscillations of the electric field vector of a light wave to a particular plane. When the electric field oscillations are confined to a single plane, the light is said to be plane-polarized or linearly polarized.

Types of Polarization

• Unpolarized Light: Light waves where the electric field vector oscillates randomly in all directions perpendicular to the direction of propagation. Sunlight, light from an incandescent bulb, and light from a candle flame are examples of unpolarized light.
• Linearly Polarized Light (Plane-Polarized Light): Light waves where the electric field vector oscillates in a single plane. This plane is called the plane of polarization.
• Partially Polarized Light: A mixture of polarized and unpolarized light.
• Circularly Polarized Light: The electric field vector rotates in a circle in a plane perpendicular to the direction of propagation. The tip of the electric field vector traces out a circle.
• Elliptically Polarized Light: The electric field vector rotates in an ellipse in a plane perpendicular to the direction of propagation. The tip of the electric field vector traces out an ellipse. This is the most general form of polarization.

Methods of Producing Polarized Light

There are several methods to produce polarized light from unpolarized light:

Polarization by Reflection

When unpolarized light is incident on a smooth, non-metallic surface (like glass, water, or plastic), it gets partially polarized upon reflection. The reflected light is strongest polarized when the angle of incidence is such that the reflected ray and the refracted ray are perpendicular to each other. This specific angle of incidence is known as Brewster's angle.

The reflected light will be completely plane-polarized if the angle of incidence is Brewster's angle. The plane of polarization is parallel to the reflecting surface.

Polarization by Scattering

When light waves strike small particles or molecules in the atmosphere (like air molecules), they are scattered in different directions. This scattered light is generally partially polarized. The light scattered by the atmosphere, which makes the sky appear blue, is partially polarized. If you observe the sky through a polarizing filter, you can notice changes in brightness as you rotate the filter, indicating polarization.

Polarization by Dichroism

Dichroism is the property of certain crystals to absorb light waves vibrating in one direction more strongly than light waves vibrating in a perpendicular direction. Crystals like tourmaline and herapathite exhibit this property. When unpolarized light passes through a thin sheet of a dichroic material, the light vibrating in one direction is absorbed, and the light vibrating in the perpendicular direction is transmitted. This transmitted light is plane-polarized.

Polaroid sheets, commonly used in sunglasses and camera filters, utilize this principle. They are made of long-chain polymer molecules aligned in one direction, which absorb light polarized parallel to these molecules.

Polarization by Birefringence (Double Refraction)

Certain crystalline materials, such as calcite and quartz, exhibit birefringence. When unpolarized light enters such a crystal, it splits into two rays: the ordinary ray (O-ray) and the extraordinary ray (E-ray). These two rays are plane-polarized in mutually perpendicular directions.

The O-ray travels with the same speed in all directions within the crystal, and its vibrations are perpendicular to the optic axis of the crystal. The E-ray travels with a speed that depends on its direction of vibration relative to the optic axis, and its vibrations are parallel to the optic axis.

By suitably cutting and orienting these crystals, one can separate these two polarized rays. Devices like Nicol prisms and Rochon prisms are constructed based on this principle to produce highly polarized light.

Malus's Law

Malus's Law describes the intensity of polarized light after passing through a second polarizing filter, called an analyzer. If a beam of plane-polarized light of intensity $I_0$ is incident on an analyzer, and the angle between the plane of polarization of the incident light and the transmission axis of the analyzer is $\theta$, then the intensity of the transmitted light $I$ is given by:

$I = I_0 \cos^2{\theta}$

If the incident light is unpolarized, and it first passes through a polarizer, the intensity of the polarized light is $I_0/2$. When this light then passes through an analyzer at an angle $\theta$ to the polarizer's axis, the final transmitted intensity is:

$I = \frac{I_0}{2} \cos^2{\theta}$

Brewster's Law

Brewster's Law relates the angle of incidence at which light is completely polarized upon reflection to the refractive index of the reflecting medium. It was discovered by Sir David Brewster in 1812.

When unpolarized light is incident on a transparent medium (like water or glass) at a certain angle of incidence, the reflected light is found to be completely plane-polarized. The refracted light is then partially polarized.

The angle of incidence for which this phenomenon occurs is called the Brewster's angle, denoted by $i_B$ or $\theta_B$. At this angle, the reflected ray and the refracted ray are perpendicular to each other.

Derivation of Brewster's Law

Consider unpolarized light incident on the surface separating two media with refractive indices $n_1$ and $n_2$ ($n_1 < n_2$). Let the angle of incidence be $i$. The reflected ray and the refracted ray are perpendicular at Brewster's angle $i_B$. So, the angle of refraction $r$ is $90^\circ - i_B$.

According to Snell's Law:

$n_1 \sin{i} = n_2 \sin{r}$

At Brewster's angle ($i = i_B$, $r = 90^\circ - i_B$):

$n_1 \sin{i_B} = n_2 \sin{(90^\circ - i_B)}$

Since $\sin{(90^\circ - i_B)} = \cos{i_B}$, we have:

$n_1 \sin{i_B} = n_2 \cos{i_B}$

Rearranging the terms:

$\frac{\sin{i_B}}{\cos{i_B}} = \frac{n_2}{n_1}$

This gives:

$\tan{i_B} = \frac{n_2}{n_1}$

This is Brewster's Law.

Special Case: Light traveling from air to a denser medium

When light travels from air (where $n_1 \approx 1$) to a denser medium with refractive index $n$ (where $n_2 = n$), Brewster's Law simplifies to:

$\tan{i_B} = n$

Here, $i_B$ is the Brewster's angle for the medium.

Properties at Brewster's Angle

• The reflected ray is completely plane-polarized. The plane of polarization is parallel to the reflecting surface.
• The refracted ray is partially polarized.
• The angle between the reflected ray and the refracted ray is $90^\circ$.

Applications of Brewster's Law and Polarization

• Polaroid Sunglasses: These sunglasses are designed to reduce glare from horizontal surfaces like water, roads, and snow. Glare is often caused by the reflection of unpolarized sunlight from these surfaces, and the reflected light is horizontally polarized. Polaroid lenses are oriented to block this horizontally polarized light.
• Photography: Polarizing filters are used in cameras to reduce reflections and enhance colors. By rotating the filter, photographers can control the amount of polarized light entering the lens.
• LCD Screens: Liquid Crystal Displays (LCDs) use polarizing filters and liquid crystals to control the passage of light, forming images on the screen.
• 3D Movies: Some 3D movie technologies use polarizing filters to present different images to each eye, creating the illusion of depth. For example, in RealD 3D, the projector uses a circularly polarizing filter, and the glasses are also circularly polarizing, with opposite handedness for each eye.
• Scientific Instruments: Polarized light is used in various scientific instruments, such as microscopes (polarizing microscopes) and refractometers, for analyzing the properties of materials.
• Optical Communication: Polarization can be used to encode information in optical fibers.

Example: Calculating Brewster's Angle

Let's calculate the Brewster's angle for water. The refractive index of water is approximately $n = 1.33$.

Using Brewster's Law:

$\tan{i_B} = n$

$\tan{i_B} = 1.33$

To find $i_B$, we take the arctangent:

$i_B = \arctan(1.33)$

Using a calculator, $i_B \approx 53.03^\circ$.

This means that if unpolarized light strikes the surface of water at an angle of approximately $53.03^\circ$, the reflected light will be completely plane-polarized.

Polarization by Malus's Law - An Example

Suppose unpolarized light of intensity $I_0$ passes through a polarizer. The intensity of the polarized light is $I_0/2$. This light then passes through an analyzer. If the angle between the transmission axes of the polarizer and the analyzer is $30^\circ$, what is the final transmitted intensity?

The intensity after the polarizer is $I_{pol} = I_0/2$.

Using Malus's Law, the final intensity $I$ is:

$I = I_{pol} \cos^2{\theta}$

$I = \frac{I_0}{2} \cos^2{30^\circ}$

We know that $\cos{30^\circ} = \frac{\sqrt{3}}{2}$. So, $\cos^2{30^\circ} = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}$.

$I = \frac{I_0}{2} \times \frac{3}{4}$

$I = \frac{3}{8} I_0$

The final transmitted intensity is $\frac{3}{8}$ of the initial unpolarized light intensity.

Summary of Key Concepts