33-7 Polarization by Reflection

33-7 Polarization by Reflection#

Prompts

  • How does reflection from a dielectric surface (glass, water) affect the polarization of light? What is the difference between partially and fully polarized reflected light?

  • Define Brewster’s angle \(\theta_B\). At this angle, what happens to the parallel vs perpendicular components of the electric field in the reflected ray?

  • Derive the formula \(\theta_B = \arctan(n_2/n_1)\). Why are the reflected and refracted rays perpendicular at Brewster’s angle?

  • For light in air incident on glass (\(n \approx 1.5\)), what is Brewster’s angle?

  • How do polarizing sunglasses reduce glare? Why is the polarizing axis mounted vertically?

Lecture Notes#

Overview#

  • Reflection polarizes light: Unpolarized light reflected from glass, water, or other dielectrics becomes partially polarized (unequal parallel vs perpendicular components).

  • At a special angle—Brewster’s angle \(\theta_B\)—the reflected light is fully polarized perpendicular to the plane of incidence; the parallel components refract into the medium.

  • Brewster’s law: \(\theta_B = \arctan(n_2/n_1)\); at \(\theta_B\), reflected and refracted rays are perpendicular.

  • Polarizing sunglasses: Block glare by absorbing horizontally polarized light reflected from horizontal surfaces.

Polarization by reflection#

When unpolarized light strikes a dielectric interface, resolve the electric field into:

  • Perpendicular components: perpendicular to the plane of incidence (dots in a side view).

  • Parallel components: parallel to the plane of incidence (arrows in the plane).

In general, the reflected light has both components but with unequal magnitudes → partially polarized. At one particular angle of incidence—Brewster’s angle \(\theta_B\)—the reflected light has only perpendicular components → fully polarized perpendicular to the plane of incidence. The parallel components (along with some perpendicular) refract into the second medium.

Brewster’s angle#

Experimentally, at Brewster’s angle the reflected and refracted rays are perpendicular:

(246)#\[ \theta_B + \theta_r = 90° \]

Combining with Snell’s law \(n_1 \sin\theta_B = n_2 \sin\theta_r\) and using \(\sin\theta_r = \sin(90° - \theta_B) = \cos\theta_B\):

\[ n_1 \sin\theta_B = n_2 \cos\theta_B \]
(247)#\[ \theta_B = \arctan\left(\frac{n_2}{n_1}\right) \]

Here \(n_1\) is the index of the incident medium and \(n_2\) is the index of the reflecting medium.

Important

Subscript convention: \(n_1\) = medium of incident/reflected ray; \(n_2\) = medium of refracted ray. Do not swap them in the formula.

Special case—light from air: If the incident ray is in air (\(n_1 \approx 1\)), then \(\theta_B = \arctan(n)\), where \(n\) is the index of the reflecting medium.

Example: glass and water#

Surface

\(n\)

\(\theta_B\) (from air)

Glass

\(\approx 1.5\)

\(\arctan(1.5) \approx 56°\)

Water

\(\approx 1.33\)

\(\arctan(1.33) \approx 53°\)

Polarizing sunglasses#

Sunlight reflected from horizontal surfaces (water, roads, glass) is partially or fully polarized horizontally (in the plane of the surface). This glare is intense and can obscure vision.

Polarizing sunglasses use lenses with a vertical polarizing axis. They absorb the horizontally polarized component of the reflected light, reducing glare while still transmitting much of the unpolarized light from other directions (sky, objects).

Summary#

  • Reflection polarizes: Reflected light from dielectrics is partially polarized; at Brewster’s angle it is fully polarized (perpendicular to plane of incidence).

  • Brewster’s angle: \(\theta_B = \arctan(n_2/n_1)\); reflected and refracted rays are perpendicular.

  • Polarizing sunglasses: Vertical axis blocks horizontally polarized glare from horizontal surfaces.