# Michelson Interferometer

If it is possible for an interferometer itself to be famous, then this would be the one. The Michelson interferometer is used for a wide variety of applications, with a particular usefulness in detecting small distance changes. I, Optics Girl, have actually used a Michelson interferometer to detect changes in interference patterns for a strained vs unstrained material. If this prospect doesn’t get you excited, then I don’t know what will!

INTRODUCTION TO THE SETUP

The basic setup for a Michelson interferometer is shown in the figure below.

In our analysis, we will denote “Movable Mirror” as “M1” and “Fixed Mirror” as “M2”, and assume that the mirrors are totally reflective i.e. the Fresnel coefficient for reflectivity is $R =1$. We will get to the movable vs fixed differentiation in a bit. The beam-splitter plate ideally splits the total incident amplitude into a 50% transmitted and a 50% reflected beam, so that equal intensities of light are propagating in each arm. The compensator plate is also placed in the first arm so that the light in each arm passes through the thickness of a glass plate 3 times (in arm 1, twice through the compensator plate and once through the beam splitter; in arm 2, three times through the mirror). The beam is also reflected twice and transmitted once in each arm, giving the same phase constants for each arm. Thus, assuming that the two mirrors M1 and M2 are equidistant from the beam splitting plate, there should be constructive interference at the observation screen.

BEAM SPLITTERS AND PHASE SHIFTS

We can describe the phase shifts in a Michelson interferometer in the same way that we described the reflected and transmitted waves in a dielectric stack back in Section 3 of this portion of the website: using the reflection and transmission coefficients in a scattering matrix of the following form:

$$S = \begin{bmatrix} r & it\\ it & r \end{bmatrix}$$

This matrix was formed for a reference plane for a multi-layer dielectric reflector, and it contains the 90$^{\circ}$ phase shift of the reflected and transmitted waves with respect to one another. Thus, in the case of the Michelson, we can figure out the

reflected and transmitted intensities in a similar manner. We can do this by considering the above figure. If M2 is allowed to move a distance d from its origin, then the maximum phase difference between the two arms would be given, as usual, by the $k$-vector multiplied by the optical path difference. In this case, since the light would travel a total distance of 2d when the arm is at its furthest point, the path difference is given as:

$$\delta = \frac{2\pi}{\lambda}2d = 2\frac{\omega}{c}d = \omega\tau$$

where $\tau = 2\frac{d}{c}$ is the optical delay between waves coming from the two arms.

Keeping with our notation of the scattering matrix, which occurs at the beam splitter plate, the field reflected back towards the source is given as:

\begin{align} E_R & = r(rE_0)+it(ite^{i\delta}E_0 \\ & = (r^2-t^2e^{i\delta})E_0 \\ & = [r^2-(1-r^2)e^{i\delta}]E_0 \end{align}

Here, we can note the fact that if the two path lengths are equal (d = 0), then $\delta$ equals 0 and the reflected electric field would be given as:

$$E_R = (2r^2-1)E_0$$

We can also see here that if $r^2 = 0.5$, which is the case for a 50% beamsplitter, then the reflected field back towards the source is zero.

INTENSITY PATTERN ON OBSERVATION SCREEN

An important thing to note here is that the each fringe pattern corresponds to a single wavelength, so a broadband source would produce different fringe patterns for each wavelength.

https://iopscience.iop.org/article/10.1088/0143-0807/36/2/025019#:~:text=Schematic%20representation%20of%20a%20Michelson,and%20(b)%20generates%20straight%20or