d. Now, suppose the hemispherical shell is removed, and we instead observe the interference pattern on the top wall. To the nearest integer, what is the total number of fringes that appear on the top wall? You may assume that d≪L.
Given constants:
## Mirror Mirror on the Wall
Consider a square room with side length L. The bottom wall of the room is a perfect mirror.* A perfect monochromatic point source with wavelength λ is placed a distance d above the center of the mirror, where λ≪d≪L.
<image_1>
*Remember that the phase of light reflected by a mirror changes by 180∘.
Context question:
a. On the right wall, an interference pattern emerges. What is the distance y between the bottom corner and the closest bright fringe above it? Hint: you may assume λ≪y≪L as well.
Context answer:
y=8dλL
Context question:
c. Now suppose we place a transparent hemispherical shell of thickness s and index of refraction n over the source such that all light from the source that directly strikes the right wall passes through the shell, and all light from the source that strikes the mirror first does not pass through the shell.
hemispherical shell
<image_2>
At what y is the fringe closest to the bottom-most corner now? (You may find it convenient to use ⌊x⌋, the largest integer below x.)
Context answer:
y=4dL((n−1)s−λ⌊λ(n−1)s−21⌋−2λ)