Physics · Optics · Grade 11

Polarisation of light

Why do polarising glasses “switch off” the glare on water?

The situation

One summer an angler went to the Charvak Reservoir and complained that the sunlight glittering on the water dazzled him — the fish under the water could not be seen at all. His friend gave him polarising sunglasses: the glare almost vanished and the bottom came into view. The same phenomenon works in a calculator screen and in 3D cinema glasses. Why does a polarising filter “switch off” the glare, and not an ordinary tinted piece of glass?

Topic

Polarisation of light. Sunglasses and 3D cinema

Key ideas

Malus's law
Brewster angle
Transverse wave

Formulas

I = I_{0} \cos^{2} θ

Malus's law: polarised light passes through an analyser; $θ$ is the angle between the polarisation direction and the axis of the analyser.

I_{1} = \frac{I_{0}}{2}

An ideal polariser passes half of unpolarised light.

I_{3} = \frac{I_{0}}{2} \cos^{2} θ \sin^{2} θ = \frac{I_{0}}{8} \sin^{2} 2 θ

If a third polariser is placed at an angle $θ$ between crossed polarisers; at $θ = 45 °$ the intensity is $I_{0} / 8$ .

r_{s} = \frac{n_{1} \cos α - n_{2} \cos γ}{n_{1} \cos α + n_{2} \cos γ}, r_{p} = \frac{n_{2} \cos α - n_{1} \cos γ}{n_{2} \cos α + n_{1} \cos γ}

Fresnel formulas: the amplitude of the reflected wave for oscillations perpendicular ( $s$ ) and parallel ( $p$ ) to the plane of incidence; $R = r^{2}$ .

tg θ_{B} = \frac{n_{2}}{n_{1}}

Brewster's law: at this angle $r_{p} = 0$ — the reflected ray is completely polarised. For water $θ_{B} \approx 53.1 °$ .

Theory

Light is a transverse wave

In light the electric field strength vector $\vec{E}$ oscillates perpendicular to the direction of propagation. In the light of a lamp or the Sun the direction of $\vec{E}$ changes very quickly and randomly — this is unpolarised light (in the scene the arrows point in different directions). A polariser passes only the oscillation along its own axis, so after it the arrows oscillate in a single plane and the intensity is halved.

Malus's law and the “third polariser” puzzle

The analyser passes the projection of $\vec{E}$ onto its own axis: $E \cos θ$ . The intensity is proportional to the square of the amplitude, so $I = I_{0} \cos^{2} θ$ . Two polarisers with perpendicular axes (crossed) block the light completely. But if you place a third between them at 45°, the light comes back: the filter in the middle “turns” the polarisation direction by 45°, at each stage $\cos^{2} 45 ° = 1 / 2$ passes, and $I_{0} / 8$ reaches the screen.

Why is glare reflected from water polarised?

According to the Fresnel formulas, oscillations parallel to the water surface — horizontal ( $s$ ) — are reflected more strongly than vertical ( $p$ ) ones. At the Brewster angle ( $\tan θ_{B} = n$ ) the $p$ -wave is not reflected at all: the reflected and refracted rays are then perpendicular to each other. The axis of polarising glasses is vertical — it blocks the horizontally polarised glare, but passes half of the unpolarised light coming from under the water. If you tilt your head by 90°, the glare returns. The refracted ray is also partially polarised — it has more vertical ( $p$ ) oscillations (see the result window). A stack of several glass plates set at the Brewster angle reflects part of the $s$ component at every surface, so the transmitted beam becomes more and more completely polarised.

Types of polarisation and polarimetry

In a monochromatic plane wave $\vec{E}$ can be resolved into two perpendicular oscillations: $E_{x} = a \cos (ω t - k z)$ , $E_{y} = b \cos (ω t - k z + δ)$ . In general the tip of $\vec{E}$ traces an ellipse; at $δ = 0$ or $180 °$ it is linear, and at $a = b$ and $δ = \pm 90 °$ it is circularly polarised (the circular system in the 3D cinema mode). Such states are obtained with $λ / 4$ and $λ / 2$ plates and checked with a rotating analyser — in detail in the “Crystal optics” experiment. For partially polarised light the degree of polarisation $P = (I_{m a x} - I_{m i n}) / (I_{m a x} + I_{m i n})$ is measured by rotating the analyser (this quantity appears in the measurements in the “Water glare” mode). This is polarimetry: astronomers use it to study the surfaces of planets and interstellar dust (elongated dust grains aligned by the Galaxy's magnetic field partially polarise starlight). The polarisation of skylight is covered in the “Scattering of light” experiment, and the polarisation of laser light in “Laser physics and holography”.

LCD screen and 3D cinema

An LCD pixel is a twisted liquid crystal between two crossed polarisers: with no voltage, the helix of molecules rotates the polarisation by 90° and the pixel is bright; with a voltage the molecules stand up along the field, the twist disappears and the pixel goes dark. In 3D cinema the images for the two eyes are sent with opposite polarisations. With linear polarisation, tilting the head by $α$ lets a fraction $\sin^{2} α$ of the other image through; with circular polarisation (a $λ / 4$ plate + polariser in each lens) tilting does no harm. Model: the polarisers and $λ / 4$ plates are ideal, and the liquid crystal is simplified to an ideal rotator.

Did you know?

In 1808 the French physicist Étienne Malus, observing sunlight reflected from the windows of the Luxembourg Palace in Paris through a crystal of Iceland spar, discovered the polarisation of reflected light. He also introduced the term “polarisation”.
Bees can see the polarisation of skylight and find their way by it even when the Sun is behind a cloud. This was discovered by the Austrian scientist Karl von Frisch (Nobel Prize 1973).

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

The eye and spectacles — Grade 7
Shadow and eclipses — Grade 7
Measuring the speed of light — Grade 9
Optical instruments — Grade 9
Interference and diffraction — Grade 11
Laser physics and holography — Grade 11
Luminous flux and illuminance — Grade 11
Infrared, UV and X-rays — Grade 11
Scattering of light — Year 1
Crystal optics — Year 1