A monsoon shower has just passed over your town in the late afternoon. The Sun comes out low in the west behind you, while the rain still falls in the east. On the dark clouds in front of you stands a rainbow. It is a perfect arc of colours, red on the outside and violet on the inside. Walk a few steps, and the rainbow moves with you.
One glass, many refractive indices
So far each material has had one refractive index. Measured carefully, it has a slightly different index for each colour. For a typical glass, violet light has \(\mu \approx 1.52\), yellow \(1.50\) and red \(1.48\). The index is larger for shorter wavelengths.
In 1836 Augustin Cauchy found a simple rule that fits glass well: \(\mu = a + b/\lambda^2\), where \(a\) and \(b\) are constants of the glass. Violet light, near \(400\,\text{nm}\), has a smaller \(\lambda\) than red, near \(700\,\text{nm}\). So \(b/\lambda^2\) is larger for violet, and so is \(\mu\). Since \(v = c/\mu\), red light travels fastest in glass and violet slowest.
A prism deviates each colour by its own amount. Violet, with the largest \(\mu\), is bent most, towards the base; red is bent least. White light entering the prism leaves as a fan of colours. The splitting of white light into its colours is dispersion, and the band of colours is a spectrum.
Why does a window pane not make colours? In a parallel-sided slab every colour leaves parallel to the incident ray, as the second lesson showed. The colours are shifted sideways by slightly different amounts, but they travel on together and overlap again. The light stays white. A prism's faces are not parallel, so its colours leave in different directions.
In 1666 Isaac Newton made a small hole in the shutter of a dark room and let a beam of sunlight through a glass prism. On the far wall the round spot of sunlight became a long band of colours. He then let one colour through a slit into a second prism. The second prism bent it, but no new colours appeared. Newton concluded that white light is a mixture of colours, each bent by its own amount, and that the prism only separates them. His report to the Royal Society in 1672 started a long quarrel, but the experiment was right.
White light passes through a glass prism and spreads into a band on a screen. Which colour is at the end of the band nearest the base of the prism, and why?
Rank these colours by their speed inside glass, fastest first.
How strongly a glass spreads colours
For a thin prism each colour is deviated by \((\mu - 1)A\), with its own \(\mu\). The angular dispersion is the angle between the violet and the red rays that leave it: \[ \delta_v - \delta_r = (\mu_v - \mu_r)A. \] The deviation of yellow, \(\delta_y = (\mu_y - 1)A\), in the middle of the spectrum, is called the mean deviation.
A thicker prism both deviates and disperses more. To compare glasses fairly, divide one by the other. The ratio is the dispersive power \(\omega\), a property of the glass alone, since \(A\) cancels.
For the glass above, \(\omega = (1.52 - 1.48)/(1.50 - 1)\), which is \(0.04/0.5 = 0.08\). A thin prism of \(5°\) made of it deviates yellow light by \(0.5 \times 5° = 2.5°\). It spreads violet from red by \(0.04 \times 5° = 0.2°\). Check: \(0.2/2.5 = 0.08\), the dispersive power.
To keep the numbers simple, imagine a material with \(\mu_r = \sqrt2\) for red and \(\mu_v = \sqrt3\) for violet. A \(60°\) prism of it is set at minimum deviation for each colour in turn. Find the two minimum deviations and the angular dispersion.
Red: \(\sin i = \sqrt2\sin 30° = 1/\sqrt2\), so \(i = 45°\) and \(\delta_m = 2 \times 45° - 60° = 30°\).
Violet: \(\sin i = \sqrt3\sin 30° = \sqrt3/2\), so \(i = 60°\) and \(\delta_m = 60°\).
The angular dispersion is \(60° - 30° = 30°\). Real glass disperses far less: the \(60°\) prism of Fig 6.1 spreads red from violet by only \(3.5°\).
A flint glass has \(\mu_v = 1.65\), \(\mu_r = 1.59\) and \(\mu_y = 1.60\). What is its dispersive power?
Cancelling one, keeping the other
Put two thin prisms together, one pointing up and one pointing down. Their deviations subtract, and so do their dispersions. If the two glasses were the same, both would cancel together. With two different glasses we can cancel one and keep the other. The usual pair is crown glass, with \(\mu = 1.5\) and \(\omega = 0.08\), and flint glass, with \(\mu = 1.6\) and \(\omega = 0.1\).
A crown prism (\(\mu = 1.5\), \(\omega = 0.08\)) of angle \(6°\) is combined with a flint prism (\(\mu = 1.6\), \(\omega = 0.1\)) turned the opposite way. The mean deviation of the pair is zero. Find the angle of the flint prism and the net angular dispersion.
Zero mean deviation: the two yellow deviations are equal and opposite.
\((1.5 - 1) \times 6° = (1.6 - 1)A_2\), so \(3° = 0.6A_2\). The dispersion of each prism is \(\omega\delta\).
\(A_2 = 5°\). Each prism deviates yellow by \(3°\). Crown disperses \(0.08 \times 3° = 0.24°\); flint \(0.1 \times 3° = 0.3°\), the opposite way.
Net dispersion \(= 0.3° - 0.24° = 0.06°\), with no mean deviation. Yellow light goes straight through, and the colours fan out around it.
Now the reverse. Choose the angles so that the dispersions cancel: \(\omega_1\delta_1 = \omega_2\delta_2\). With a flint prism of \(4°\), \(\delta_2 = 0.6 \times 4° = 2.4°\), and its dispersion is \(0.1 \times 2.4° = 0.24°\). The crown prism must disperse \(0.24°\) too: \(0.08 \times 0.5A_1 = 0.24°\), so \(A_1 = 6°\). The net deviation is \(3° - 2.4° = 0.6°\), and all colours leave together. Such a pair is an achromatic combination.
A flint prism (\(\mu = 1.6\), \(\omega = 0.1\)) of angle \(2°\) is to be paired with a crown prism (\(\mu = 1.5\), \(\omega = 0.08\)) so that the pair shows no dispersion. What must the angle of the crown prism be, in degrees?
Light in a raindrop
A raindrop is a small sphere of water. A ray of sunlight enters it, bends towards the normal and crosses the drop. It is reflected once from the back surface and leaves through the front, bending again. It comes back towards the side the Sun shines from, turned through a large angle. How far it is turned depends on where it struck the drop.
A ray through the centre, \(b = 0\), comes straight back. As \(b\) grows, the returning angle grows: \(28.3°\) at \(b = 0.5R\) for red. Near \(b = 0.86R\) it reaches a largest value, \(42.4°\) for red. For larger \(b\) it falls again. The rays near the largest value all return at nearly the same angle. So light piles up there, and the drop sends back its brightest light at about \(42°\).
This is the prism's minimum deviation again. The drop turns the ray by \(180° + 2i - 4r\), and that turn has a minimum, just as the prism's deviation had. Near a minimum the deviation hardly changes, so many rays leave together, and the light there is bright.
Each colour has its own largest angle, because each has its own \(\mu\). For red, \(\mu = 1.331\) and the angle is \(42.4°\); for violet, \(\mu = 1.343\) and it is \(40.6°\). So each drop sends red a little farther from the direction opposite the Sun than violet.
The bow in the sky
Stand with the Sun behind you and look towards the rain. The point straight opposite the Sun, the shadow of your head, is the antisolar point. Every drop that lies \(42°\) from that point sends red light into your eye. Those drops lie on a circle of \(42°\) radius around it. Drops on a circle of \(40.6°\) send violet. So the bow is a set of coloured circles, red outside and violet inside.
In the evening, the Sun is low in the west. Where should you look for a rainbow, and which colour is on the outside of the bow?
A rainbow is always a circle of about \(42°\) around the antisolar point. When the Sun is low, in the early morning or late afternoon, the antisolar point is just below the horizon. Then nearly half the circle stands above the ground. At noon, with the Sun high, the antisolar point is deep below the horizon, and the bow sinks out of sight. That is why a monsoon rainbow is a morning or evening sight. From an aeroplane, with the drops all round and below, you can sometimes see the whole circle. A fainter second bow, with the colours reversed, sometimes appears outside the first; its light is reflected twice inside each drop.
Now return to the monsoon evening. The Sun behind you is low, so the circle of \(42°\) around the antisolar point stands high over the eastern clouds. Each drop sends you one colour, set by its angle from that point: red from the outer circle, violet from the inner. When you walk, the antisolar point moves with your head, and so does the bow. Everyone sees their own rainbow, made by their own set of drops.
- Thinking that red is deviated most because it has the longest wavelength.A longer wavelength means a smaller \(\mu\) and a smaller deviation. Violet is deviated most.
- Using \(\mu_y\) in the denominator of the dispersive power without subtracting 1.\(\omega = (\mu_v - \mu_r)/(\mu_y - 1)\). The mean deviation of a thin prism is \((\mu_y - 1)A\).
- Expecting one pair of prisms to cancel both deviation and dispersion.With two glasses of different \(\omega\), one condition can be met at a time: equal deviations, or equal dispersions.
- Looking for a rainbow towards the Sun.The drops return light towards the Sun's side, so the bow is seen with the Sun behind you, around the antisolar point.