Problem $1(40$ points$)$: A droplet of oil falls on top of a water surface. It spreads out and eventually forms a thin film with a thickness of $250$ nm $.$ The index of refraction of oil is $1\mathrm{.}4,$ and the index of refraction of water is $1\mathrm{.}33.$ A white light beam $($spectrum range: $\backslash (380\backslash$mathrm$\{$~nm$\}-700\backslash$mathrm$\{$~nm$\}\backslash ))$ hits the water surface. Assume the light beam is perpendicular to the thin film. For simplicity, you only need to consider the interference between the first and the second reflected beams.

$($a$)$ What will be the wavelengths of light that have maximum and minimum reflections, respectively? $(10$ points$)$

$($b$)$ What will be the corresponding maximum and minimum reflectance $\backslash ((\backslash$mathrm$\{$R$\})\backslash )$ for those wavelengths? $(10$ points$)$

$($c$)$ More oil is added to the water surface. What is the maximum thickness of the oil film above which thickness no interference will be observed between the first and second reflected beams? Carefully consider the refractive index of the oil when calculating this maximum thickness. $(20$ points$)$