Prism and Dispersion

A prism is a wedge of refracting material. Two refractions, one at each face, deviate the ray. When the prism is at minimum deviation it gives a clean formula relating the deviation to the prism angle and refractive index.

Concept

Geometry:

• Prism angle $A$: angle between the two refracting faces.
• Deviation $D$: angle between the incident and emergent rays.
• At minimum deviation $D = D_m$, the ray passes symmetrically through the prism. The refracted ray inside is parallel to the base.

At minimum deviation,

$i_1 = i_2 = \frac{A + D_m}{2}, \quad r_1 = r_2 = \frac{A}{2}$

Applying Snell's law at the first face:

$\mu = \frac{\sin\!\left(\dfrac{A + D_m}{2}\right)}{\sin\!\left(\dfrac{A}{2}\right)}$

Derivation

In the prism, two facts hold for any orientation:

1. $r_1 + r_2 = A$ (geometry of the wedge: triangle of refracted ray and the two normals).
2. $D = (i_1 - r_1) + (i_2 - r_2) = i_1 + i_2 - A$.

At minimum deviation, by symmetry $i_1 = i_2$ and $r_1 = r_2$, so $D_m = 2i - A$ and $A = 2r$. Hence

$i = \frac{A + D_m}{2}, \quad r = \frac{A}{2}$

Snell's law at face 1 gives the prism formula.

Dispersion

Refractive index depends on wavelength: $\mu_{\text{violet}} > \mu_{\text{red}}$ in ordinary glass. Different colours are deviated by different amounts, splitting white light into a spectrum.

Angular dispersion (between violet and red):

$\theta = D_v - D_r$

Dispersive power:

$\omega = \frac{\theta}{D} = \frac{\mu_v - \mu_r}{\mu - 1}$

where $\mu$ is the average (yellow) refractive index. $\omega$ is dimensionless and depends only on the material.

Worked Example

A prism of angle $A = 60^\circ$ shows $D_m = 30^\circ$. Find $\mu$.

$\mu = \frac{\sin\!\left(\dfrac{60 + 30}{2}\right)}{\sin\!\left(\dfrac{60}{2}\right)} = \frac{\sin 45^\circ}{\sin 30^\circ} = \frac{0.7071}{0.5} = 1.414$

So $\mu \approx \sqrt 2$.

Common Confusions

• The prism formula is valid only at minimum deviation. For other angles, $D$ depends on $i_1$ in a more complicated way.
• "Dispersion without deviation" can be achieved with a flint+crown combination chosen so that net $D = 0$ but net angular dispersion is nonzero (used in spectroscopes).
• "Deviation without dispersion" is achieved by an "achromatic" combination — net colour dispersion zero but net deviation nonzero.

Key Takeaways

• Prism: $r_1 + r_2 = A$, $D = i_1 + i_2 - A$.
• At minimum deviation: $\mu = \frac{\sin((A+D_m)/2)}{\sin(A/2)}$.
• Dispersion arises from $\mu$ depending on wavelength.
• Dispersive power $\omega = (\mu_v - \mu_r)/(\mu - 1)$.