Polarization of Light

How to use it

The lab has two modes. Polarizers sends a beam through up to three polarizing filters to a detector. The 3D view draws the electric field of the light as arrows. Before the first filter the field points in a random, ever-changing direction: unpolarized light. After a filter it swings only along that filter’s transmission axis, the gold line, and the arrow length shows the field amplitude. Tick a polarizer to put it into the beam, turn it with its slider and read the intensity after every step.

Reflection shows a ray meeting the boundary between two transparent materials. Set both refractive indices and the angle of incidence and see how much light is reflected for each polarization.

Controls

• Polarizers: a checkbox and an angle slider for P1, P2 and P3 (0° to 180°, measured from the vertical). The source is unpolarized or polarized at its own angle. Experiments set up classic cases, and Spin turns the polarizer chosen for the graph.
• 3D view: drag to turn the scene, or focus it and use the arrow keys. Space plays or pauses, Home resets the view. The small circles show the field as seen from the detector, looking back at the source.
• Reflection: drag in the diagram or use the slider. On the focused diagram the arrow keys change the angle by 0.5° and B jumps to Brewster’s angle. Choose unpolarized, s or p light and add a filter to the reflected ray.
• Display: colour of the light (380–750 nm) and animation speed. Settings are remembered in this browser.

How it works

Light is an electromagnetic wave whose electric field oscillates at right angles to its direction of travel. If the field always swings along one line, the light is linearly polarized. An ideal polarizer passes only the part of the field along its axis. If the field meets the axis at an angle θ, the amplitude that gets through is E₀ cos θ, and intensity goes with the square of the amplitude. That is Malus’s law:

In unpolarized light every direction is equally likely, and the average of cos²θ is ½, so the first polarizer passes half the intensity at any angle. Crossed polarizers pass nothing, because cos 90° = 0. Put a third one between them at 45° and light returns: ½ · cos²45° · cos²45° = 1/8 = 12.5 %. The middle filter adds no light; it turns the field, so the last filter no longer meets it at 90°.

Reflection depends on polarization too. A field perpendicular to the plane of incidence is s-polarized, a field in that plane is p-polarized. With Snell’s law n₁ sin θ_i = n₂ sin θ_t, the Fresnel equations give the reflected fractions:

Nothing is absorbed, so 1 − R is transmitted, and unpolarized light reflects (R_s + R_p)/2. R_p drops to zero at Brewster’s angle, where the reflected and refracted rays are at 90°:

From air into glass (n = 1.50) that is 56.3°. Going into a less dense material, beyond the critical angle sin θ_c = n₂/n₁ all light is reflected. The Ray Optics lab traces refraction through lenses and prisms, and the Wave Simulator shows waves in motion.

The beam colour is the hue of the chosen wavelength, computed from the CIE 1931 colour-matching functions and converted to sRGB. It does not change any number here; in real glass the index varies slightly with wavelength.

Things to try

• Load the three-polarizer surprise and untick P2: the detector goes dark. Tick it again and light returns.
• In the same setup, choose P2 for the graph and press Spin. The reading peaks at 45° and 135°.
• With a polarized source, set P2 to 0°, 30°, 45°, 60° and 90°: you get 100, 75, 50, 25 and 0 %.
• In Reflection, send p light from air into glass and go to Brewster’s angle: the reflected ray vanishes.
• Try glass to air: Brewster’s angle is 33.7° and total internal reflection starts at 41.8°.

Frequently asked questions

Why do polarized sunglasses cut glare?

Glare from water and wet roads is reflected light, and reflection favours the s part, whose field is horizontal. For water (n ≈ 1.33) Brewster’s angle is about 53°, where the reflection is completely s-polarized. The lenses have a vertical axis and block it. Tilt your head 90° and the glare returns.

Why does a screen go dark through polarized sunglasses?

LCD screens and many OLED screens have a polarizing film on the front, so their light is linearly polarized. As you rotate a polarizer in front of the screen it dims and turns almost black when the axes are crossed, as Malus’s law predicts.

How can adding a filter let more light through?

Each polarizer passes the part of the field along its axis and sends it on in that direction. Two 45° steps each pass half, while one 90° step passes nothing. The extra filter only changes the direction of the field.

Are real polarizers ideal?

No. Common polarizing film passes roughly 40 % of unpolarized light instead of 50 %, and a crossed pair leaks a little. The lab uses ideal polarizers, so the numbers follow the formulas exactly.

Sources

• Ling SJ, Sanny J, Moebs W (2016). Polarization. University Physics Volume 3, OpenStax, section 1.7. Malus’s law, Brewster’s law and why polarized sunglasses cut glare.
• Brewster D (1815). On the laws which regulate the polarisation of light by reflexion from transparent bodies. Philosophical Transactions of the Royal Society of London 105, 125–159. Brewster’s own account of the angle at which reflected light is fully polarized.