A quartz plate cut parallel to its optical axis is placed between two corssed Nicol prisms so that principle directions form an angle of `45^(@)` with the optical axis of the plate. What is the minimum thickness of that plate transmitting light of wavelength `lambda = 643nm` with maximum intensity while greatly reducing the intensity of transmitting light of wavelength `lambda_(2) = 564 nm`? The difference of refractive indieces indices for extroardianry and ordinary rays is assumed to be equal to `n_(e) - n_(0) = 0.0090` for both wavelengths.

Explanation

Between crossed Nicols, a quartz plate, whose optic axis makes `45^(@)` with the principle directions of the Nicols, must introduce a phase difference of `(2k + 1)pi` so as to transmit the incident light( of that wavelength) with maximum intensity. For in this case the palne of polarization of the light emerging from the polarizer will be rotated by `90^(@)` and will go through the analyser undiminished. Thus we write for light of wavelengths `643 nm`
`delta = (2pi xx 0.009)/(0.643 xx 10^(-6)) xx d(mm) xx 10^(-3)`
`= (18pid)/(0.643) = (2k + 1)pi ......(1)`
To nearly block light of wavelength `564 nm` we require
`(18pid)/(0.564) = (2k')pi..........(2)`
We must have `2k' gt 2k +1`. For the smaller value of `d` we take `2k' = 2k + 2`
Thus `0.643 (2k + 1) = 0.564 xx (2k + 2)`
so `0.079 xx 2k = 0.564 xx 2-0.643`
or `2k = 6.139`
This is not quite an integer bt is close to one. This means that if we take `2k = 6` equations `(1)` can be satisfied axctly while equation `(2)`will hold approximately. Thus
`d = (7 xx 0.643)/(18) = 0.250 mm`