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A ray of light travelling in air is incident at 30 deg on one face of a glass slab in which the refractive index varies with the coordinate y. Inside the slab, the light travels along the curve y = 4x² (where x and y are in metres). Find the value of 2*mu, where mu is the refractive index of the slab at y = 1/2 m.

1. 2
2. 3
3. 4
4. sqrt(3)

Correct answer: sqrt(3)

Solution

By the invariance of n*sin(theta) for a ray in a graded-index medium, n*sin(theta) = sin(30) = 0.5 at all points. At y=1/2: x = sqrt(y/4) = 1/(2*sqrt(2))... wait, y=4x² => x=sqrt(y)/2 = 1/(2*sqrt(2)) at y=1/2. The slope of the curve is dy/dx = 8x = 8/(2*sqrt(2)) = 2*sqrt(2). The angle the tangent makes with x-axis: tan(alpha)=2sqrt(2). The angle with vertical (normal to the horizontal layers): phi where tan(phi)=1/(2sqrt(2)) => the angle theta (with vertical) satisfies the geometry. Using n*cos(phi)=const leads to mu*cos(phi)=const approach or Snell's invariance gives mu*sin(theta)=0.5.

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