Question

An infinitely long cylinder of radius R is made of an unusual exotic material with refractive index –1 (Figure). The cylinder is placed between two planes whose normals are along the y direction. The center of the cylinder O lies along the y-axis. A narrow laser beam is directed along the y direction from the lower plate. The laser source is at a horizontal distance x from the diameter in the y direction. Find the range of x such that light emitted from the lower plane does not reach the upper plane.

Answer 1

Here the material has refractive index – 1, θ_r is negative and `θ_r^'` positive.

Now, `|θ_i| = |θ_r| = |θ_r^'|`

The total deviation of the outcoming ray from the incoming ray is 4θ_t. Rays shall not reach the receiving plate if `pi/4 ≤ 4θ_i ≤ (3pi)/2` [angles measured clockwise from the yiaxis]

On solving, `pi/8 ≤ θ_i ≤ (3pi)/8`

Now, `sin θ_i = x/R`

`pi/8 ≤ sin^-1  x/R ≤ (3pi)/8` or `pi/8 ≤ x/R ≤ (3pi)/8`

Thus for light emitted from the source shall not reach the receiving plane if `(Rpi)/2 ≤ x ≤ (3Rpi)/8`.