An infinitely long cylinder of radius `R` is made of an unusal exotic material with refractive index `(-1)`, Fig. 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 emmited from the lower plane does not reach the upper plane.
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Explanation

The set up described in the question is shown in Fig. The full line circle represents cross section of infinitely long cylinder of radius `R`. As the material is of refractive index `(-1)`, therefore, `theta_r` is negative and `theta'_r` is positive.
Also, `|theta_i | = |theta_r|=|theta_r'|`
For the incoming ray along Y -axis from the lower plane, total deviation of the outcoming ray is `4 theta_i`. The rays shall not reach the receiving plate if
`(pi)/(2) le 4 theta_i le (3 pi)/(2)`
(angles are measured clockwise from Y-axis)
or `(pi)/(8) le theta_i le (3 pi)/(8)`
From Fig., `sin theta_i = (x)/( R) = theta_i`, whebn small.
`:. (pi)/(8) le (x)/( R) le (3 pi)/(8)` or `(R pi)/(8) le x le (3 pi R)/(8)`
Hence, we conclude that for values of `x` lying between `(pi R)/(8) and (3 pi R)/(8)`, light emitted from the laser source shall not reach the upper receiving plane.
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