$4.(25$ points$)$ In the Young's double slit experiment monochromatic light passes through two slits separated by a distance $\backslash ($ d$=0\mathrm{.}5\backslash$mathrm$\{$~mm$\}\backslash ),$ and the interference pattern is observed on a distant screen $\backslash ($ R$=5\backslash$mathrm$\{$~m$\}\backslash )$ away from the two slits.

$($a$)(5$ points$)$ If the distance between the two consecutive maxima on the screen is $0\mathrm{.}7$ cm $,$ what is the wavelength of light? What is the distance between two consecutive minima?

$($b$)(5$ points$)$ Make a careful plot of the intensity of light on the screen as a function

of distance y from the central maximum, showing at least $5$ maxima $($including the

central one m$=0).$ Label each maximum and minimum and indicate the path

difference and the phase difference for each maximum and minimum in your plot.

$($c$)(10$ points$)$ When one of the slits is covered by a transparent film with a refractive

index of n$=1\mathrm{.}5,$ the zeroth order maximum is observed to be shifted by $2\mathrm{.}2$ fringes.

How thick is the film?

$($d$)(5$ points$)$ The two slits are now illuminated by light containing two wavelengths,

$450$ nm and $600$ nm $.$ What is the least order m at which a maximum of one wave$-$

length will fall exactly on a minimum of the other?