A Lloyd's-mirror-type interferometer is built entirely inside a large prism of crown…
PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Polarisation and anisotropic media · Interference and diffraction
Optics and Photonics, Ch Two-beam interference and interferometry
Problem
A Lloyd's-mirror-type interferometer is built entirely inside a large prism of crown glass (n=1.50). A point source S embedded in the glass emits monochromatic, coherent, s-polarised light (electric field horizontal, parallel to the flat top surface of the prism) in all directions, and a distant detector inside the glass, reaching up to the top surface, receives light by two paths: directly through the glass, and after a single internal reflection at the flat glass-air top surface. The critical angle for this interface is θc=arcsin(1/1.50)=41.8∘.
((a)) Consider first a point of the pattern well away from the edge, reached by a ray reflected at an angle of incidence of 25.0∘ from the normal. Using the Fresnel amplitude coefficient for s-polarised light (Chapter 6, under CV5), decide whether this internal reflection adds a phase of 0 or π, and contrast it with the usual (external-reflection) Lloyd's mirror, where the light starts in air and reflects off the denser medium. Then consider the point where the two path lengths are exactly equal, the analogue of the mirror's edge: at what angle of incidence does the reflected ray meet the surface there, what is rs at that angle, and is the edge bright or dark?
((b)) A thin polariser is now placed in the reflected beam's path only, rotating its polarisation from s to p, so that the direct beam arriving at the detector is s-polarised and the reflected beam is p-polarised. Writing each beam as a Jones vector in the fixed (s,p) basis, use the general two-beam formula I=I1+I2+2Re(E1∗⋅E2) to decide whether fringes exist anywhere on the detector now, and explain in one sentence why the reflection coefficients found in part (a) are irrelevant to this conclusion.