When light passes through sequential interfaces, there may be a change of phase upon reflection at each interface. Additionally the wavelength changes with each change of the index of refraction. If the thickness of a thin layer is chosen correctly, then it will serve as an anti$-$reflective coating bringing about maximally destructive interference between the first and second reflected waves.

For convenience, the index of refraction for a variety of materials is provided below.

Materials at $20$C for light with a vacuum wavelength of $589$nm$.$Materialn$$Materialn$$Materialnbenzene$1\mathrm{.}501$diamond$2\mathrm{.}419$plexiglas$1\mathrm{.}51$carbon disulfide$1\mathrm{.}628$fluorite $($CaF$2)1\mathrm{.}434$quartz $($crystalline$)1\mathrm{.}544$carbon tetrachloride$1\mathrm{.}461$glass $($crown$)1\mathrm{.}52$quartz $($fused$)1\mathrm{.}458$ethanol$1\mathrm{.}361$glass $($flint$)1\mathrm{.}66$sodium chloride$1\mathrm{.}544$glycerine$1\mathrm{.}473$ice $(0$C$)1\mathrm{.}309$zircon$1\mathrm{.}923$water $($fresh$)1\mathrm{.}333$polystyrene$1\mathrm{.}49$air$1\mathrm{.}00029$ Part $($a$)$

Given that the thin layer of water has thickness dd$,$ let mm be an integer. What is the proper condition for the the layer to be an anti$-$reflective layer?

$$d$=($m$+12)$nwaterd$=($m$+12)$nwater$/$nplexiglasnplexiglas$$d$=($m$+12)$nplexiglasd$=($m$+12)$nplexiglas$/$nwaternwater$$d$=$mnplexiglasd$=$mnplexiglas$/$nwaternwater$2$d$=$m$2$d$=$m$/$nwaternwater$2$d$=($m$+12)$nplexiglas$2$d$=($m$+12)$nplexiglas$/$nwaternwater$2$d$=$mnplexiglas$2$d$=$mnplexiglas$/$nwaternwater$2$d$=$mnwater$2$d$=$mnwater$/$nplexiglasnplexiglas$$d$=($m$+12)$d$=($m$+12)$$/$nwaternwater$2$d$=($m$+12)$nwater$2$d$=($m$+12)$nwater$/$nplexiglasnplexiglas