An infinitely long cylinder of radius `R` is made of an unusual 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

Since, the material is of refractive index -1, `theta_(r),` is negative and `theta_(r)'` positive
Now `" " | theta_(t)|=|theta_(r)|=|theta'_(r)|`
The total deviation of the outcoming ray from the incoming ray is `4 theta_(t)`. Rays shall not reach the recieving plate is
`" " (pi)/(2) le 4 theta_(t) le (3pi)/(2) " "` [ angles measured clockwise from the y-axis]
On solving `" " (pi)/(8) le theta_(t) le (3pi)/(8)`

Now, `" " sin theta_(t) =(x)/(R)`
`" " (pi)/(8)le sin ^(-1). (x)/(R)le (3pi)/(8)`
or `" " (pi)/(8) le (x)/(R) le (3pi)/(8)`
Thus, for light emitted from the source shall not reach the receiving plate. if `(Rpi)/(8)lexle(R3pi)/(8)`