One has to manufacture a quartz plate cur parallel to its optical axis and not exceeding `0.50mm` is thickness. Find the maximum thickness of the plate allowing plane-polarized light with wavelength `lambda = 589 nm`
(a) to experience only rotation of polarization plane,
(b) to acquire circular polarization after passing through that plane.

Explanation

For quartz
`{:(n_(e),1.553),(n_(0),1.544):}}` for `lambda = 589 nm`.
In a quartz plate cut parallel to its optic axis, plane polarized light incident normally from the left divides itself into `O` and `E` waves which move in the same direction with different speeds and as a result acquire a phase difference. This phase difference is
`delta = (2pi)/(lambda) (n_(e) - n_(0))d`
where `d =` thickness of the plate. In general this make the emergent light elliptically polarized.
(a) For emergent light to experience only rotation of polarization plane
`delta = (2k + 1)pi, k = 0, 1,2, 3.........`
For this `d = (2k + 1) (lambda)/(2(n_(e) - n_(0))`
`= (2k + 1) (.589)/(2 xx 009)mu m = (2k + 1)(.589)/(18) mm`
The maximum value of `(2k + 1)` for which this is less than `0.05` is obtained from
`(0.50xx 18)/(0.589) = 15.28`
Then we must take `k = 7` and `d = 15 xx (589)/(18) = 0.4908 mm`
(b) For circular polarization `delta = (pi)/(2)`
modulo `2pi i.e. delta = (4k + 1) (pi)/(2)`
so `d = (4k + 1) (lambda)/(4(n_(e) - n_(0))) = (4k + 1) (0.589)/(36)`
Now `(0.50 xx 36)/(0.589) = 30.56`
The nearest integer less than this which is of the from `4k + 1` is `29` for `k = 7`. For this
`d = 0.4749mm`