Polarization

Property of light as a transverse wave that allows filtering transmitted intensity according to the electric field orientation.

Conceptual context

Visible light is a transverse electromagnetic wave: its electric field oscillates in a plane perpendicular to the propagation direction. When all oscillations share a single plane, the wave is linearly polarized. Polarization is the property that enables us to filter, analyze and manipulate the orientation of those oscillations.

Within the physical-optics block, polarization complements interference and diffraction phenomena because it reveals an additional dimension of the wave that is invisible when only intensity or color is observed. Understanding this phenomenon is essential for interpreting polarized sunglasses, LCD displays, polarization microscopes and optical-communication techniques.

🟢 Essential level

Natural light oscillates in every direction perpendicular to its travel, with no preferred orientation. A polarizer is a filter that transmits only the electric-field component aligned with its transmission axis. After passing through it, light becomes linearly polarized: its electric field oscillates in a single, well-defined plane.

If a second polarizing sheet (called the analyser) is placed after the first, the transmitted intensity depends on the angle between the two axes. With parallel axes, light passes almost entirely; with perpendicular axes, transmission drops to zero. That angular dependence is the heart of Malus's law.

Reflection can also produce polarization. When light strikes a transparent surface at a specific angle, the reflected beam emerges completely polarized. That angle is called the Brewster angle and depends on the refractive indices of the two media.

An everyday example helps fix the idea: polarized sunglasses remove the annoying glare from a wet road or the surface of a lake. That glare is sunlight reflected close to the Brewster angle, so it arrives mostly polarized along a single direction; the sunglasses' axis is manufactured perpendicular to that direction to block the glare without darkening the whole scene too much.

🔵 Formal level

Basic formulas

The quantitative relationship between $I$ and the angle $\theta_p$ between the wave's polarization direction and the polarizer axis is given by Malus's law:

Malus's law

Here $I_0$ is the intensity of the polarized light reaching the analyser and $\theta_p$ is the angle between the wave's polarization direction and the filter's transmission axis. The squared-cosine dependence means that intensity does not fall linearly but stays high at small angles and drops rapidly as the angle approaches 90 degrees. When $\theta_p$ equals zero, $I$ equals $I_0$; when $\theta_p$ equals 90 degrees, $I$ vanishes (crossed polarizers).

[!NOTE] If the source emits unpolarised natural light, the first polarizer transmits exactly half the original intensity. Malus's law then applies taking that half as $I_0$ for the second polarizer.

Real student questions

Why does inserting a third polarizer at 45 degrees between two crossed ones let light through again? Each polarizer projects the electric field onto its axis. The intermediate filter rotates the polarization direction, providing a non-zero component to the third filter. Without the intermediate sheet the projection onto the second axis is zero.

Do polarized sunglasses work the same way as a laboratory polarizer? Yes, the principle is identical: they absorb the electric-field component perpendicular to their transmission axis. The difference lies in optical quality and material uniformity.

Does polarization affect the color of light? Not directly. Polarization acts on the orientation of the electric field, not on the wavelength. However, in birefringent media different wavelengths may undergo different rotations, producing colors through interference in polarized light.

Can a longitudinal wave be polarized? No. Polarization is exclusive to transverse waves because it requires the oscillation to be orientable in different directions perpendicular to propagation. In longitudinal waves only one oscillation direction is possible.

Final synthesis

Polarization reveals that light is not a scalar wave but a vectorial oscillation with a definite orientation. Malus's law quantifies how a polarizer filters intensity according to the squared cosine of the angle, and Brewster's law identifies the angle of fully polarized reflection from the refractive indices. Mastering both relationships enables the interpretation of everyday phenomena and the design of optical devices based on electric-field orientation control.