Polarization Knowledge-Based Study

Study notes

1. Polarization is a characteristic property unique to transverse waves. Light is an electromagnetic wave composed of mutually perpendicular, oscillating electric ($\vec{E}$) and magnetic ($\vec{B}$) fields propagating through space.

   By convention, the direction of polarization of an electromagnetic wave is defined by the orientation of its oscillating electric field vector. In unpolarized light, such as light emitted by the Sun or an incandescent bulb, the electric field oscillates randomly in all possible planes perpendicular to the direction of propagation. When these oscillations are confined to a single plane, the light is described as linearly polarized or plane-polarized.

   A common method to produce polarized light is selective absorption using a polarizing filter (polarizer). A polarizer has a specific transmission axis that allows only the electric field component parallel to that axis to pass through, absorbing the perpendicular component.

   When unpolarized light of initial intensity $I_0$ passes through an ideal polarizer, the transmitted intensity becomes exactly half of the initial intensity:

   $$I_1 = \frac{1}{2}I_0$$

   If this linearly polarized beam subsequently encounters a second polarizing filter (known as an analyzer) whose transmission axis is oriented at an angle $\theta$ relative to the first, the transmitted intensity $I$ is governed by Malus's Law:

   $$I = I_1 \cos^2\theta$$

   • When $\theta = 0^\circ$, the axes are parallel and $I = I_1$ (maximum transmission).
   • When $\theta = 90^\circ$, the filters are crossed and $I = 0$ (complete extinction).

   When unpolarized light strikes a non-metallic surface (such as water or glass), the reflected and refracted rays generally become partially polarized. The degree of polarization depends strongly on the angle of incidence.

   At a specific angle known as Brewster's angle ($\theta_p$), the reflected light is completely linearly polarized with its electric field vector parallel to the reflecting surface and perpendicular to the plane of incidence.

   This condition occurs when the reflected ray and refracted ray are perpendicular to each other ($\theta_p + \theta_r = 90^\circ$). Using Snell's law, Brewster's Law is expressed as:

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

   where $n_1$ is the refractive index of the incident medium and $n_2$ is that of the transmitting medium.

   Understanding polarization enables numerous technological and optical applications:

   • Polarized Sunglasses: Glare from horizontal surfaces like water and asphalt is predominantly horizontally polarized. Polarized lenses have vertical transmission axes, significantly reducing glare.
   • Liquid Crystal Displays (LCDs): LCD screens modulate light intensity pixel-by-pixel by rotating the plane of polarization using liquid crystals placed between crossed polarizers.
   • Optical Stress Analysis: Certain transparent materials become birefringent (doubly refracting) under mechanical stress, revealing stress distribution patterns when viewed between crossed polarizers.
   • 3D Cinema: Orthogonally polarized images projected simultaneously allow separate images to reach each eye through corresponding polarizing glasses, creating depth perception.