Q.Obtain the equations for constructive and destructive interference for transmitted and reflected waves in thin films.

Step 1. A film of refractive index $\mu$ and thickness $d$ splits an incident ray at its top surface into a reflected part and a refracted part; the refracted part reflects again at the bottom surface and re-emerges, so both the transmitted and reflected light beyond the film consist of (at least) two coherent contributions that interfere.

Step 2. For light transmitted straight through the film, the second contribution (having reflected once internally, off the bottom then possibly the top) travels an extra optical path of approximately $2\mu d$ (twice the thickness, weighted by the film's refractive index, for near-normal incidence) relative to the directly-transmitted part, with no extra phase shift from either internal reflection (since both are rarer-into-denser-then-denser-into-rarer, not producing the special phase-flip case). So: constructive transmission at $2\mu d=n\lambda$; destructive transmission at $2\mu d=(2n-1)\lambda/2$.

Step 3. For light reflected back off the top of the film, the two interfering rays are the direct top-surface reflection and the ray that has entered the film, reflected off the bottom surface, and re-emerged through the top -- again with extra path $2\mu d$. But this time, the direct top-surface reflection specifically occurs at a rarer-to-denser boundary (air into the film), which picks up an additional phase shift of $\pi$ (equivalent to an extra $\lambda/2$ of path) that the transmitted case never has. …

Track the extra optical path 2(mu)d for both the transmitted and reflected beam pairs, adding the extra pi (lambda/2) phase shift only f …

• Applying the extra lambda/2 phase shift to the transmitted-light case as well, when it only applies to reflected light. …