Right-angle prisms can be used to reverse the direction of light, as shown. Find

Question

Right-angle prisms can be used to reverse the direction of light, as shown. Find the index of refraction that will allow all the light to be reflected as shown in the figure, and not leak out the back. The benefit of prisms is that no light is lost. The problem with using prisms is that light from only some angles will be reflected. On the other hand, paint used on highway signs often contains small transparent spherical beads that provide nighttime illumination of the sign's letter by retro-reflecting vehicle headlights beams. In other words, no matter where the light comes from, the light reflects off the back of the bead and emerges exactly anti-parallel to the initial direction. Find the index of refraction of the bead. You may assume that the angle theta of incidence on the surface of the bead is small.

Explanation / Answer

Hi,

(a) In the case of right angle prisms, what matter the most is the angle of incidence of the ray of light. The angle of incidence once the ray is inside the prism must be greater than the critical angle, in that event none of the light will pass through the prism and all of it will be redirected, changing its path by 90°.

This phenomenom is explained by the principle of the total internal reflection which requieres that the index of refraction of the medium the light is in to be higher than the index of refraction of the medium the light is trying to go. Therefore, the index of refraction of the prism should be at least higher than 1.0 (the index of refraction of air).

To answer how big it should be, we must find a suitable value for the critical angle. If we assume the angle is 60° (approximately what we can see in the figure) then we have the following:

sin(c) = 1/n :::::::: n = 1/sin(c) :::::::::: n = 1/sin(60°) :::::::::: n = 1.16

So, for a critical angle of 30°, the index of refraction of the prism should be higher than 1.16

Note: greater critical angles will give smaller index of refraction, but I think 60° is a good approximation of the angle that is shown in the figure.

(b) In the case of a spherical bead it is said that it should have an index of refrection of about 1 plus the index of refrection of the medium around the sphere, as it is usually air (which is 1) the optimal index of refraction should be 2. However, due to factors such as the spherical aberration and Fresnel reflection coefficients (which are higher as the index is higher as well, making the sphere less efficient in terms of concentrating light), the index of refraction tends to be lower than 2. In fact, for comercial applications the index of refraction of the spherical beads vary from 1.5 to 1.9

I hopw it helps.

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