A beam of white light is incident on the surface of a diamond at an angle theta_

Question

A beam of white light is incident on the surface of a diamond at an angle theta_a. (Figure 1) Since the index of refraction actually depends on the light's wavelength, the different colors that comprise white light will spread out as they pass through the diamond. For example, the index of refraction in diamond is n_blue = 2.450 for blue light. The indices of refraction for the lower frequency colors are smaller. Thus, blue light and red light are refracted at different angles inside the diamond, as shown in the picture. (Not shown is the orange, yellow, and green light in between the red and the blue.) Note that the angles in the figure are not to scale. A diamond will appear bluish in color if the blue light is internally reflected while the remaining colors are allowed to refract out. Now consider theta_c, the angle at which the blue refracted ray hits the bottom surface of the diamond. If theta_c is larger than the critical angle theta_crit, the light will not be refracted out into the air, but instead it will be totally internally reflected back into the diamond. Find theta_crit. Express your answer in degrees to four significant figures. A diamond is cut such that the angle between its top surface and its bottom surface is alpha. For alpha = 31 degree, find the largest possible value of the incident angle theta_a such that the blue light is totally internally reflected off the bottom surface. Express your answer in degrees to three significant figures.

Explanation / Answer

Part B:

From part A, we know that the value of critical angle is derived as shown below.

crit = 24.09°

= 31°

Let d be the angle in the diamond when the beam strikes the diamond

Subtract the critical angle (Part A) from :

d = – crit

= 31 – 24.09

d = 6.91°

Index of refraction in diamond = nblue = 2.450

Index of refraction in air = nair = 1

Using Snell’s law we have:

nairssin(a) = nbluesin(d )

(1) sin(a) = (2.450) sin(6.91)

sin(a) = (2.450) (0.12)
sin(a) = 0.294

a = sin-1(0.294)
a = 17.09°

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