Reflection and refraction with Huygens’ principle

Refraction of a plane wave

Narrated video. Transcript:

A plane wavefront AB travels in medium 1 and meets the boundary with medium 2 at an angle i. The speed of light is v₁ in medium 1 and v₂ in medium 2.

End A touches the boundary first. End B still has a distance BC to travel. It takes a time τ = BC/v₁ to get there.

Meanwhile, A sends a secondary wavelet into medium 2. In the time τ, this wavelet grows to a radius AE = v₂τ.

Draw a tangent from C to this wavelet, touching it at E. Every point between A and C sends its own, smaller wavelet, and they all touch the same line CE. So CE is the refracted wavefront.

Now look at the two right-angled triangles. In triangle ABC, sin i = BC/AC = v₁τ/AC. In triangle AEC, sin r = AE/AC = v₂τ/AC, where r is the angle of refraction.

Divide one by the other: sin i / sin r = v₁/v₂. The ratio of the sines is a constant, the ratio of the two speeds. This is Snell’s law of refraction.

Put v₁ = c/n₁ and v₂ = c/n₂. Then n₁ sin i = n₂ sin r. If v₂ is less than v₁, then r is less than i: the ray bends towards the normal.

So the wave theory needs light to be slower in the denser medium. Foucault’s experiment later showed exactly this. The corpuscular model had predicted the reverse.

The frequency does not change at the boundary: crests arrive and leave at the same rate. Since v = νλ, the wavelength changes, with λ₁/λ₂ = v₁/v₂. In the slower medium, the wavelength is shorter.

Example: light of wavelength 600 nm in air enters glass with n = 1.5. Its speed becomes 3 × 10⁸/1.5 = 2 × 10⁸ m/s, and its wavelength 600/1.5 = 400 nm. Its frequency stays 5 × 10¹⁴ Hz.

Into a rarer medium, and reflection

Narrated video. Transcript:

Now let light go from a denser medium into a rarer one, say from glass into air. This time v₂ is greater than v₁.

Repeat the construction. The wavelet from A grows faster than B moves, so AE is longer than BC. The refracted wavefront tilts the other way: r is greater than i, and the ray bends away from the normal.

Increase i. The refracted ray swings towards the boundary. At the critical angle i_c, r reaches 90°, so n₁ sin i_c = n₂, or sin i_c = n₂/n₁.

For larger angles, the wavelet from A is so large that no line from C can touch it. There is no refracted wavefront at all. All the light is reflected: total internal reflection.

Example: from glass with n = 1.5 into air, sin i_c = 1/1.5 = 0.67, so the critical angle is about 42°.

Reflection works the same way. A plane wavefront AB meets a mirror at an angle i. End B reaches the mirror at C after a time τ = BC/v.

Meanwhile the wavelet from A grows, in the same medium, to a radius AE = vτ, equal to BC. The tangent from C to this wavelet, CE, is the reflected wavefront.

Triangles EAC and BCA share the side AC, each has a right angle, and AE = BC. So they are congruent, and the angle of reflection r equals the angle of incidence i. This is the law of reflection.

Lenses, mirrors and the Doppler effect

Narrated video. Transcript:

Huygens’ construction also shows how prisms and lenses work. Send a plane wave through a thin prism. Light is slower in glass, and the thick base holds back the lower part of the wavefront most. The wavefront comes out tilted, so the beam bends towards the base.

A thin convex lens is thickest at the centre. The middle of a plane wavefront is delayed the most, so it comes out curved, a spherical wavefront converging to the focus F.

A concave mirror also turns a plane wavefront into a spherical one converging to its focus, this time by reflection.

There is a neat rule here. The time taken from a point on the object to its image is the same along every ray. The ray through the thick middle of a lens travels less distance in air, but more in the slower glass.

Now the Doppler effect. If a source of light moves away from us, each new wavefront starts from a little further away. The waves reaching us are stretched: the wavelength increases and the frequency decreases.

For visible light, a longer wavelength moves the colour towards the red end of the spectrum. So light from a receding source is red-shifted. A source moving towards us gives a blue shift.

When the speed is small compared with c, the fractional change in frequency is Δν/ν = −v_radial/c. Here v_radial is the component of the source’s velocity along the line joining it to us, positive when it moves away.

Example: the sodium line at 589.0 nm is seen in a galaxy’s light at 589.6 nm. Since Δλ/λ = −Δν/ν = v_radial/c, the radial speed is c × 0.6/589.0 = 3 × 10⁸ × 1.02 × 10⁻³ ≈ 3.06 × 10⁵ m/s.

The wavelength has increased, a red shift, so the galaxy is moving away from us at about 306 kilometres per second. Astronomers use such shifts to measure the speeds of stars and galaxies.

Try it: a wavefront crossing a boundary

Interactive simulation. Things to try:

• Press Play (or drag the time slider). The wavefront AB reaches the boundary at A first; wavelets start from each point of AC in turn.
• The green line touching all the wavelets is the refracted wavefront. Check that sin i / sin r = v₁/v₂ = n₂/n₁ in the readouts.
• Swap the media (n₁ = 1.5, n₂ = 1.0) and raise i. Past the critical angle the wavelets have no common tangent.
• Tick “Reflected wavefront”: its wavelets grow at v₁, and the reflected wavefront makes the same angle i with the boundary.

Angle in the glass

A plane wavefront in air (n = 1) meets the flat surface of glass (n = 1.5) at an angle of incidence of 30°. What is the angle of refraction?

1. 19.5°
2. 20°
3. 30°
4. 48.6°

Red shift or blue shift?

The hydrogen line of wavelength 656.3 nm is observed in the light of a star at 656.0 nm. What can you say about the star?

1. It is moving away from us at about 137 km/s
2. It is moving towards us at about 137 km/s
3. It is moving towards us at about 1.37 × 10⁵ km/s
4. It is at rest; the change is too small to matter

Summary

• Snell’s law from Huygens’ construction: sin i / sin r = v₁/v₂ = n₂/n₁
• Wavelength in the two media (ν unchanged): λ₁/λ₂ = v₁/v₂
• Critical angle (denser to rarer): sin i_c = n₂/n₁
• Doppler shift (v ≪ c): Δν/ν = −v_radial/c

• At a boundary, wavelets start from each point the wavefront reaches; their common tangent is the refracted (or reflected) wavefront.
• Light bends towards the normal when it slows down, so the wave theory needs light to be slower in a denser medium.
• The frequency stays the same on refraction; the speed and wavelength change.
• For reflection the wavelets grow at the same speed, which gives i = r. For i > i_c there is no refracted wavefront: total internal reflection.
• A receding source shows a red shift (longer λ), an approaching one a blue shift.