Using the tables of the Appendix, calculate the difference of refractive indices of quartz for light wavelength `lambda = 589.5 nm` with right-hand and left-hand circular polarizations.

Explanation

Light polarized along the `x`-direction (i.e. one whose electric vector has only an `x` component) and propagting along the `z-`direction can be decomposed into left and right circualry polarized light in accordance with the formula
`E_(x) = (1)/(2) (E_(x) - iE_(y)) + (1)/(2)(E_(x) - iE_(y))`
On passing through a distance `l` of an active medium these acquire the phases `delta_(R ) = (2pi)/(lambda) n_(R ) l` and `delta_(1) = (2pi)/(lambda)n_(1)l` so we get for the complex amplitude
`E' = (1)/(2)(E_(x) + iE_(y)) e^(i delta_(R)) + (1)/(2) (E_(x) + iE_(y)) e^(i delta_(L))`
`= e^(l(delta_(R) + delta_(l))/(2))[(1)/(2)(E_(x) + iE_(y))e^(i delta//2) + (1)/(2)(E_(x) - iE_(y))e^(-i delta//2)]`
`= e^(l(delta_(R) + delta_(l))/(2)) [E_(x) cos ((delta)/(2)) - E_(y) sin((delta)/(2))], delta = delta_(R ) - delta_(L)`.
Apart from an over all phase `(delta_(R) + delta_(1))//2)`(which is irrelevent) this represents a wave whose plane of polarization has rotated by
`(delta)/(2) = (pi)/(lambda) (Deltan)l, Deltan = |n_(R) - n_(1)|`
By definition this equals `alpha l` so
`Deltan = (alpha lambda)/(pi)`
`= (589.5 xx 10^(-6) mm xx 21.72 deg//mm)/(pi) xx (pi)/(180) (rad)`
`= (5895 xx 21.72)/(180) xx 10^(-3)`
`= 0.71 xx 10^(-4)`