Dispersion & Spectra

Newton let a beam of sunlight through a prism and saw it spread into a band of colours. The colours were already in the white light; the prism only separated them, because glass bends each colour by a slightly different amount.

Refractive index depends on wavelength#

So far we have treated a material’s refractive index as a single number. In fact it changes a little with wavelength. A good approximation is

So μ falls as λ rises: in the visible range it is largest for violet and smallest for red. Materials with a larger refractive index, such as flint glass, also tend to have a larger spread of μ across the spectrum than low-index materials like fluorite.

Dispersion by a prism#

Since each colour has its own refractive index, each bends by its own angle. In a prism, violet is deviated most and red least, so a beam of white light fans out into its colours. This separation of light into its component colours by a medium is called dispersion.

A prism does two things to white light: it bends the beam as a whole, and it spreads the colours. Different glasses do these in different proportions, and that ratio is what lens designers care about.

Mean deviation and angular dispersion#

By convention the overall deviation of a white beam is measured by yellow light, whose wavelength is roughly in the middle of the visible range: the mean deviation δᵧ. If violet and red are deviated by δᵥ and δᵣ, the beam’s angular spread δᵥ − δᵣ is the angular dispersion.

For a thin prism (or small angles), each colour is deviated by δ = (μ − 1)A, so

Dispersive power#

Dispersive power from a lens#

A thin lens has 1/f = (μ − 1)K, where K = 1/R₁ − 1/R₂ depends only on its shape. So μ − 1 ∝ 1/f for each colour, and

Two prisms of different glass, placed point to base, can split colours without bending the beam, or bend the beam without splitting the colours. The second trick is the key to colour-free lenses.

Two opposed thin prisms#

Put two thin prisms in contact with their refracting angles A and A′ pointing opposite ways; their dispersive powers are ω and ω′. They deviate light in opposite directions, so for any ray

Using μᵥ − μᵣ = ω(μᵧ − 1), the net angular dispersion and the net mean deviation are

Dispersion without average deviation#

For the yellow ray to go straight on, δᵧ = 0:

Average deviation without dispersion#

For the colours to come out together, δᵥ − δᵣ = 0:

Both tricks need ω ≠ ω′, so the two prisms must be made of different glasses, typically crown and flint.

Spread the light from a source into its colours and you get a barcode that identifies what made it. Astronomers know what distant stars are made of because of this.

Pure and impure spectra#

Ideally each wavelength lands in its own place, with no overlapping: a pure spectrum, with sharp, distinct colours. This needs:

1. a parallel beam falling on the prism, from a narrow slit placed at the focus of an achromatic lens; and
2. the dispersed light to be focused by a second achromatic lens so all rays of one wavelength meet at one point.

With a wide slit, each part of the slit makes its own spectrum, slightly shifted, and they overlap: an impure spectrum with smeared colours.

Emission spectra#

A source excited by heating or an electric discharge emits light; its dispersed light is an emission spectrum, of three kinds.

Continuous spectrum

A filament lamp, a candle or red-hot iron emits all wavelengths over a range. The colours merge smoothly with no gaps or sharp boundaries.

Line spectrum

Atoms can only have certain energies. An excited atom drops to a lower energy by emitting light of particular wavelengths only. A gas of such atoms gives a few sharp bright lines on a dark background, a fingerprint of the element. Sodium vapour, for example, gives two yellow lines at 589.0 and 589.6 nm.

Band spectrum

Molecules have energy levels grouped in bunches: close together within a bunch, widely spaced between bunches. Their emitted wavelengths are grouped too, and each group looks like a continuous band.

Absorption spectra#

Send white light through a material that absorbs certain wavelengths. The dispersed transmitted light shows dark lines or bands where those wavelengths are missing, on an otherwise continuous bright background.

Beyond the visible: ultraviolet and infrared#

Excited objects can emit outside the visible range of about 380–780 nm. We usually say “light” for the visible and “radiation” in general:

• Ultraviolet: about 15–380 nm. Glass absorbs it, so quartz prisms and photographic plates are used to study it.
• Infrared: about 780–40,000 nm. Rock-salt prisms are used; infrared heats what it falls on, so a heat-sensitive thermopile detects it.
• Beyond ultraviolet come X-rays and gamma rays; beyond infrared, radio waves.

The spectrometer is the instrument that turns “violet bends more than red” into numbers: it measures angles to a fraction of a degree and gives refractive indices to four decimal places.

Three parts#

• Collimator: a tube with an achromatic converging lens at one end and an adjustable slit on a sliding inner tube at the other. Set the slit at the lens’s focus and the light leaves as a parallel beam.
• Prism table: a levelled horizontal platform that rotates about a vertical axis, carrying the prism (or grating) and a circular scale in degrees.
• Telescope: an astronomical telescope focused for parallel rays, with cross-wires and a vernier that reads its angle on the circular scale.

First the collimator, telescope and table are levelled. The telescope is focused on a distant object (for parallel rays), and the slit is moved until its image is sharp in the telescope, so the collimator gives parallel light.

Uses#

The angle of a prism

Lit by a sodium lamp, the prism is placed with its refracting edge toward the collimator, so the beam splits and reflects off both faces. The telescope is turned to catch each reflection on its cross-wire. The angle between the two positions is 2A.

The angle of minimum deviation

Find the refracted image of the slit, then nudge the telescope toward the undeviated direction and turn the table to follow it. At one point the image will go no further: that position gives δm. Remove the prism, sight the slit directly, and the difference of the two readings is the angle of minimum deviation.

Refractive index for each wavelength

Using a source with sharp separate lines (a neon tube, say), measure δm for each colour and compute$\ \mu = \sin\tfrac{A + \delta_m}{2}\big/\sin\tfrac A2$. This maps out μ against λ.

A rainbow isn’t an object in a particular place. It’s a set of directions: every raindrop at the right angle from your shadow sends you its colour.

Inside one drop#

Sunlight striking a falling raindrop refracts as it enters, reflects off the back of the drop, and refracts again as it leaves. Dispersion happens at both refractions, so the colours separate.

Light enters the drop at every height, so it leaves in many directions. But for each colour the exit angle has a turning point: near the ray of maximum deviation, a whole range of entry heights sends light out in almost the same direction. That direction is where the drop returns light most strongly.

For red, the maximum deviation is about 137.8°, so red light comes back at 180° − 137.8° = 42.2° to the line from the sun through the observer. For violet the angle is 40.6°, and other colours lie in between.

From drops to a bow#

Stand with the sun behind you. All drops lying on a cone of half-angle 42.2° around the line from the sun through your eye send you red light; those on a 40.6° cone send violet. So you see circular bands of colour centred on the point directly opposite the sun, with red on the outside. From the ground the horizon hides most of the circle and you see an arc; from an aeroplane you may see the whole ring.

The secondary bow#