[Physics] Refractive index of prism

opticsrefraction

The formula for the refractive index of a prism is:
$$\mu = \frac{\sin \left(\frac{A + D_m}{2}\right)}{\sin (A/2)}$$

However, this requires me to find out the angle of minimum deviation ($D_m$). This would require me to take several readings, plot a graph, and then find the minimum point of the graph.

If I have two readings of the angle of emergence and angle of incidence, is there a way I can find the refractive index of the prism?

Best Answer

The following diagram should help:

enter image description here

We know of course the relationship between $\alpha$ and $\beta$ because the angle $A$ is known: $$\alpha + \beta = A$$

Next, we have the relationship between $i$ and $\alpha$ from Snell's law:

$$\frac{\sin{i}}{n}=\sin\alpha$$ and similarly for $\beta$:

$$\frac{\sin{e}}{n}=\sin\beta$$

This gives you three equations with three unknowns: $\alpha,~\beta,~n$.

I would recommend solving this numerically, or graphically. Simple example of a graphical solution:

enter image description here

I calculated the expected exit angle given an input angle of 45° and a refractive index of 1.543 - then calculated what exit angle I would get if I didn't know the refractive index. The error goes to zero when you have the right index...

Python code used to generate this:

# prism simulation
from math import pi, asin, sin
import numpy as np
import matplotlib.pyplot as plt

A = 60*pi/180. # 60 degrees, in radians

# pick an angle of incidence
i = pi/4

# pick a refractive index
n = 1.543

# compute the exit angle
alpha = asin(sin(i)/n)
beta = A - alpha
e = asin(sin(beta)*n)

print 'exit angle is %.1f deg'%(e*180/pi)

# calculate n graphically from just i and e
na = np.arange(1.4,1.6,0.001) # possible values

# what would the intermediate values alpha, beta be?
aa = np.arcsin(np.sin(i)/na)
bb = A - aa
ee = np.arcsin(np.sin(bb)*na)

plt.figure()
plt.plot(na, 180*(ee-e)/pi)
plt.plot([n,n],[-20,20])
plt.plot([1.4,1.6],[0,0])
plt.xlabel('possible value of n')
plt.ylabel('error in exit angle')
plt.title('graphically estimating n')
plt.show()
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