Brewster's angle and the scattering of light

Glare on a lake

Look at a lake on a bright day, through the lenses of polaroid sunglasses. Turn your head sideways, and the glare changes. The light that the water reflects is partly polarised, with its vibration mostly along the surface. A Polaroid with its pass direction up and down takes it out.

Brewster's angle

When unpolarised light is reflected by a transparent surface such as glass or water, the reflected light is partly polarised. David Brewster found that at one particular angle of incidence the reflected light is completely plane polarised, with its vibration parallel to the surface (at right angles to the plane of incidence). This is Brewster's angle $i_B$.

At this angle, the reflected ray and the refracted ray are at right angles. So $i_B + r = 90^\circ$. Snell's law gives $\sin i_B = n\sin r = n\sin(90^\circ - i_B) = n\cos i_B$. Dividing,

This is Brewster's law for light that goes from air to a material of index $n$. For light that goes from a medium of index $n_1$ to one of index $n_2$, $\tan i_B = n_2/n_1$.

• Air to water ($n = 1.33$): $i_B = 53.1^\circ$.
• Air to glass ($n = 1.50$): $i_B = 56.3^\circ$, and the refracted ray is at $r = 33.7^\circ$.

The refracted light is only partly polarised. At Brewster's angle only a small part of the light is reflected: about $7\ \%$ of the unpolarised light for glass of index $1.50$, and about $4\ \%$ for water. All of the reflected light is plane polarised. The rest goes into the material.

The graph shows why. The red curve is for vibration in the plane of incidence. It falls to exactly zero at Brewster's angle. So all the reflected light has the other vibration. (This curve comes from the Fresnel equations, which you need not remember. They are beyond the syllabus, and the graph is only to show you what happens.)

Set the refractive index to $1.50$ and move the dot to Brewster's angle. What is the reading for the red curve? It is zero. Now change the index to $1.20$ and see how Brewster's angle moves.

A worked example. A glass of refractive index $1.55$ is in air. Find Brewster's angle and the angle of refraction. Then find the angle at which light inside the glass is polarised by reflection at the glass-air surface.

1. Brewster's angle. $\tan i_B = 1.55$, so $i_B = 57.2^\circ$.
2. The angle of refraction. $r = 90^\circ - i_B = 90^\circ - 57.2^\circ = 32.8^\circ$.
3. From the glass to air. $\tan i_B' = \dfrac{n_2}{n_1} = \dfrac{1}{1.55} = 0.645$, so $i_B' = 32.8^\circ$.
4. Check. This is the same as the angle of refraction in step 2. A ray that goes in at Brewster's angle is refracted at $32.8^\circ$. If it came back along the same ray, it would meet the other face at Brewster's angle again.

Why is the sky blue?

Sunlight is white, a mixture of all colours. Why does the clear sky look blue, and the Sun look red at sunset? The air is made of molecules, much smaller than the wavelength of light. Such small particles scatter light in all directions. This is Rayleigh scattering, and the strength of the scattering goes as

Blue light, with a short wavelength, is scattered much more than red light. For light of $450$ nm and $700$ nm, the ratio is $\left(\dfrac{700}{450}\right)^4 = 5.86$. When you look at the sky away from the Sun, the light that reaches your eye is the sunlight that the air has scattered, and it is mostly blue. Violet is scattered even more, but the Sun gives less violet light, the upper atmosphere absorbs some of it and the eye is less sensitive to it, so the sky looks blue.

At sunrise and sunset the sunlight passes through a long stretch of air, much more than at noon. Most of the blue is scattered out of the beam on the way. The light that is left, and the Sun itself, look orange or red.

Clouds look white because the droplets in them are far larger than the wavelength. Large droplets scatter all colours almost equally.

The scattered light is also partly polarised, most of all in the direction at $90^\circ$ to the Sun. A Polaroid filter on a camera, turned the right way, can darken the blue sky in a photograph.

Practise with new numbers

Try it in Python

The program finds Brewster's angle for water and for glass. Then it finds the shares of light that are reflected for the two kinds of vibration at four angles.

Common mistakes

• Using $\sin i_B = n$. Brewster's law uses the tangent: $\tan i_B = n$.
• Saying the refracted light is plane polarised too. At Brewster's angle the reflected light is fully plane polarised, and the refracted light is only partly polarised.
• Mixing up the vibration directions. The reflected light at Brewster's angle vibrates at right angles to the plane of incidence, which is parallel to the surface.
• Thinking that the sky is blue because the air is blue. The air is clear. The sky is blue because short wavelengths are scattered most.

Remember these

• Brewster's angle: $\tan i_B = n$ (air to a material), and $i_B + r = 90^\circ$.
• At $i_B$ the reflected light is completely plane polarised, and the reflected and refracted rays are at $90^\circ$.
• Rayleigh scattering by small particles: strength $\propto 1/\lambda^4$. It explains the blue sky and the red sunset.
• Scattered light is partly polarised, most at $90^\circ$ to the beam.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. At Brewster's angle, what is the angle between the reflected and the refracted ray?

2. Why do polaroid sunglasses cut the glare from a road after rain?

3. Why does a cloud look white while the sky looks blue?

4. For what material does the Brewster angle equal $45^\circ$ for light that goes from air to it?

References

• OpenStax, University Physics Volume 3, Chapter 1 (The Nature of Light): polarization by reflection (Brewster's angle) and scattering.
• NCERT Physics Class 12, Chapter 10 (Wave Optics): Brewster's law and the uses of polarised light.