Polarization and Malus's Law

Light is a transverse electromagnetic wave — the electric field oscillates perpendicular to the direction of propagation. Polarization describes the orientation of this oscillation.

Types of Polarization

• Unpolarized light: electric field oscillates in all directions perpendicular to propagation (e.g., sunlight)
• Linearly polarized light: electric field oscillates in one fixed direction
• Circular/elliptical polarization: the electric field vector rotates as the wave propagates

Polarizers

A linear polarizer transmits only the component of light parallel to its transmission axis. When unpolarized light passes through a polarizer, the transmitted intensity is halved:

$I = \frac{I_0}{2}$

Malus's Law

When already polarized light of intensity $I_0$ passes through a polarizer whose transmission axis makes an angle $\theta$ with the polarization direction:

$I = I_0 \cos^2\theta$

This is Malus's Law. At $\theta = 0°$, full transmission. At $\theta = 90°$, complete extinction.

Example: $I_0 = 100\,\text{W/m}^2$, $\theta = 45°$:

$I = 100 \cos^2 45° = 100 \times 0.5 = 50\,\text{W/m}^2$

Brewster's Angle

When light hits a surface at a special angle called Brewster's angle $\theta_B$, the reflected light is completely polarized parallel to the surface. The condition is:

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

$\theta_B = \arctan\!\left(\frac{n_2}{n_1}\right)$

At Brewster's angle, the reflected and refracted rays are perpendicular. Polarized sunglasses exploit this to block glare from horizontal surfaces (road, water).

Example: Air ($n_1 = 1$) to glass ($n_2 = 1.5$):

$\theta_B = \arctan(1.5) \approx 56.3°$

Your Task

Implement Malus's Law and Brewster's angle using math.cos, math.radians, and math.atan.