Double slit interference with a phase difference between the slits

A very thin sheet of plastic (n=1.6) covers one slit of a double slit apparatus illuminated by 540nm light. The center point of the screen, rather than being a maximum, is dark. What is the minimum thickness of the plastic?

To get a dark spot at the centre of the pattern, you need the light coming out the two spots to have a phase difference of 180^o (+/- n*360^o).

Until the light hits the plastic, the light heading towards the two slits is exactly in phase. However, with in the glass the light travels slower - ie it has a shorter wavelength λ_plastic=λ/n_plastic.

Lets say that the thickness of the plastic, L, is x*λ (x is not necessarily an integer).

If L is the minimum thickness that creates the dark spot, then L must x+1/2 of the shorter “in plastic� wavelength

L=xλ

L=(x+1/2)λ/n

Then,

xλ=(x+1/2)λ/n

x(1-1/n)=1/2n

x(n-1)=1/2

x=1/2(n-1)

L=xλ=λ/2(n-1) = 450nm