Q11Engineering Physics

The Michelson Interferometer works by splitting a beam of monochromatic light into two identical beams, which travel down two separate orthogonal arms, reflect off perfectly flat mirrors, and recombine to form an interference pattern of concentric circular fringes or straight parallel lines (depending on the mirror tilt).

Introduction of the Glass Plate

Suppose a highly polished, optically flat glass plate of thickness $t$ and refractive index $\mu$ is inserted perpendicularly into the path of one of the beams (say, the beam traveling toward Mirror $M_1$). Light travels slower in glass than in air. Therefore, passing through this plate increases the "optical path" that the light must travel.

The physical distance traveled through the plate is $t$. The equivalent optical path in a vacuum for this distance is $\mu t$. Meanwhile, the other beam travels a distance $t$ through air (refractive index $\approx 1$), making its optical path simply $t$.

The additional optical path introduced by the glass plate for a single pass is: $\Delta x_{\text{single}}=\mu t-1\cdot t=(\mu -1)t$

Total Path Difference and Fringe Shift

However, in the Michelson Interferometer, the light must pass through this glass plate twice (once on the way to the mirror, and once on the return trip). Therefore, the total extra optical path difference ($\Delta$) introduced between the two interfering beams is:

In an interference pattern, a change in path difference of exactly one full wavelength ($\lambda$) causes the entire pattern to shift by exactly one fringe. If the introduction of the plate causes the central fringe to be displaced by $n$ total fringes, the total optical path difference must be equal to $n\lambda$.

If the experiment considers the final recombined phase shift equating to a single displacement equivalent (e.g., standard formula for finding thickness based on observed shift in a one-way path equivalent), the formula is often written as:

This formula allows the incredibly precise measurement of the thickness of ultra-thin transparent films, or alternatively, the measurement of the refractive index of a gas or liquid placed in a cell within one of the interferometer arms.