Refraction and Dispersion in a Prism

1Measuring glass with a triangle

How would you measure the refractive index of a glass so accurately that you could tell one kind of glass from another, or one colour of light from another? Isaac Newton's triangular prism, the one that spreads sunlight into a band of colours, is also one of the most precise instruments in optics. The trick is a special angle at which the light passes through the prism symmetrically: the angle of minimum deviation. This lesson derives the prism formulas, uses the minimum-deviation method to find $nn$, and explains why white light spreads into a spectrum.

2The path of a ray through a prism

A prism has two plane refracting faces that meet at the refracting angle $AA$ (Figure 5.1). A ray strikes the first face at P with angle of incidence $i_{1}i\_1$ and refracts at $r_{1}r\_1$; inside, it meets the second face at Q with angle of incidence $r_{2}r\_2$ and leaves at the angle of emergence $ee$. The ray is bent towards the base at each face, so the total deviation $DD$ is the angle between the incident ray produced and the emergent ray.

Figure 5.1: A ray through a prism of refracting angle A = 60.0° and n = 1.52, drawn to scale for $i_1={40}^{\circ }i\_1\; =\; 40^\backslash circ$.

Two geometric results. (1) The two normals and the two faces form a quadrilateral with two right angles, so the angle between the normals is ${180}^{\circ }-A180^\backslash circ\; -\; A$. In the triangle formed by the normals and the ray PQ, $r_1+r_2+({180}^{\circ }-A)={180}^{\circ }r\_1\; +\; r\_2\; +\; (180^\backslash circ\; -\; A)\; =\; 180^\backslash circ$, so

(2) At P the ray is deviated by $(i_1-r_1)(i\_1\; -\; r\_1)$, and at Q by $(e-r_2)(e\; -\; r\_2)$. The deviations add:

With Snell's law at each face, $\sin i_1=n\sin r_1\backslash sin\; i\_1\; =\; n\backslash sin\; r\_1$ and $\sin e=n\sin r_2\backslash sin\; e\; =\; n\; \backslash sin\; r\_2$, these four equations let you follow any ray.

3Experiment 5.1: minimum deviation and the refractive index

Apparatus: an equilateral glass prism, white paper on a soft board, four optical pins, a protractor.

1. Draw round the prism. Draw a normal to one face and an incident line at $i_1={30}^{\circ }i\_1\; =\; 30^\backslash circ$; stick two pins on it.
2. Looking through the other face, line up two more pins with the images of the first two. Remove the prism, draw the emergent ray and produce the incident ray; measure $DD$.
3. Repeat for $i_1={35}^{\circ },{40}^{\circ },\dots ,{70}^{\circ }i\_1\; =\; 35^\backslash circ,\; 40^\backslash circ,\; \backslash ldots,\; 70^\backslash circ$ and plot $DD$ against $i_{1}i\_1$.

Table 5.1: Illustrative results for a 60.0° prism. The curve in Figure 5.2 is calculated for $n=1.52n\; =\; 1.52$.

Figure 5.2: Deviation against angle of incidence: a U-shaped curve with a single minimum.

Why there is a minimum, and what it means. The curve has one lowest point. Light paths are reversible: if a ray entering at $i_{1}i\_1$ leaves at $ee$, a ray entering at $ee$ leaves at $i_{1}i\_1$ with the same deviation. So every deviation except the smallest occurs at two angles of incidence. At the minimum the two must be the same: $i_1=ei\_1\; =\; e$, and so $r_1=r_2=\frac{A}{2}r\_1\; =\; r\_2\; =\; \backslash frac\{A\}\{2\}$. The ray passes through the prism symmetrically, parallel to the base for an isosceles prism. Then $D_m=2i_1-AD\_m\; =\; 2i\_1\; -\; A$, so $i_1=\frac{A+D_m}{2}i\_1\; =\; \backslash frac\{A\; +\; D\_m\}\{2\}$, and Snell's law gives

Near the minimum, $DD$ hardly changes as $i_{1}i\_1$ changes, so an error of a degree in setting $i_{1}i\_1$ has almost no effect on $D_{m}D\_m$. That is why this is the most accurate way to measure $nn$; with a spectrometer it gives $nn$ to four decimal places.

4Small-angle prisms and the limit of emergence

For a thin prism ($AA$ less than about 10°) and light near normal incidence, all the angles are small, so $\sin \theta \approx \theta \backslash sin\backslash theta\; \backslash approx\; \backslash theta$ in radians. Then $i_1=n{r}_{1}i\_1\; =\; nr\_1$, $e=n{r}_{2}e\; =\; nr\_2$, and

The deviation is the same for every small angle of incidence. Opticians use thin prisms in spectacles to correct a squint, and specify them by the deviation they produce.

When does no light emerge? At the second face the ray goes from glass to air, so it is totally reflected if $r_2>Cr\_2\; \backslash gt\; C$. Since $r_2=A-r_{1}r\_2\; =\; A\; -\; r\_1$ and $r_{1}r\_1$ can never exceed $CC$ (it reaches $CC$ at grazing incidence), no ray can emerge from a prism with $A>2CA\; \backslash gt\; 2C$. For glass with $n=1.50n\; =\; 1.50$, $C={41.8}^{\circ }C\; =\; 41.8^\backslash circ$, so a prism with $A>{83.6}^{\circ }A\; \backslash gt\; 83.6^\backslash circ$ passes no light through its second face. For a 60° prism, emergence needs $r_2\le {41.8}^{\circ }r\_2\; \backslash le\; 41.8^\backslash circ$, so $r_1\ge {18.2}^{\circ }r\_1\; \backslash ge\; 18.2^\backslash circ$ and $i_1\ge {27.9}^{\circ }i\_1\; \backslash ge\; 27.9^\backslash circ$.

5Dispersion

Send a narrow beam of white light through a prism, and it leaves as a fan of colours (Figure 5.3). The refractive index of glass depends on the wavelength: it is largest for violet light and smallest for red. Violet is therefore deviated most and red least. This splitting of white light into its colours is dispersion, and the band of colours is a spectrum.

Figure 5.3: Dispersion: violet is deviated most, red least.

For a thin prism, $D=(n-1)AD\; =\; (n\; -\; 1)A$ for each colour, so

where the subscript $yy$ means yellow light, in the middle of the spectrum. $\omega \backslash omega$ depends only on the glass, not on the prism angle; flint glass has about twice the dispersive power of crown glass, which is what makes the achromatic doublet of lesson 4 possible. A spectrum from a simple prism is impure because the colours overlap; a spectrometer uses a slit and lens (collimator) to make the light parallel before the prism, and a telescope after it, so each colour forms a sharp image of the slit. Spectrometers identify elements in stars and in laboratory samples from the colours they emit.