electromagnetic radiation - Michelson interferometer with and without Gaussian beam

In this video from MIT OpenCourseWare for a Michelson interferometer are demonstrated circular fringes.

I’m curious, how the mode of the laser influences the shape of the fringes. Can one conclude from the circular pattern to a Gaussian beam with TEM00 profile?

Answer

The fringes you see on the top photo and their circular pattern are nothing to do with the laser output mode and are the classic pattern one sees when a noncollimated beam from a point source is input into a well aligned Michelson. The quadratic terms in the fringe shapes betoken defocus arising from an imbalance between the two arm lengths and the decenter and linear components betoken lateral offset between the beams and / or tilt between the beams.

Suppose we see a point source at the Michelson's output port. If the field of view is close to uniformly lit, the field copy from one mirror (looking at one EM field component and assuming that the electric and magnetic fields are all aligned) can be written, to second order in the transverse co-ordinates $(x,\,y)$:

$$\frac{1}{R-\frac{\Delta}{2}}\,\exp\left(\frac{2\,\pi\,i}{\lambda}\frac{\left(x-\frac{x_0}{2}\right)^2+ \left(y-\frac{y_0}{2}\right)^2}{2\,\left(R-\frac{\Delta}{2}\right)} + \frac{2\,\pi\,i}{\lambda}\left(R-\frac{\Delta}{2}\right)-i\,\frac{k_x}{2}\,x - i\,\frac{k_y}{2}\,y\right)\tag{1}$$

whilst the second copy can be written:

$$\frac{1}{R+\frac{\Delta}{2}}\,\exp\left(\frac{2\,\pi\,i}{\lambda}\frac{\left(x+\frac{x_0}{2}\right)^2+ \left(y+\frac{y_0}{2}\right)^2}{2\,\left(R+\frac{\Delta}{2}\right)} + \frac{2\,\pi\,i}{\lambda}\left(R+\frac{\Delta}{2}\right)+i\,\frac{k_x}{2}\,x + i\,\frac{k_y}{2}\,y\right)\tag{2}$$

where:

1. $(x_0,\,y_0)$ are the components of transverse offset between the beams;


2. $R$ is the distance from the point source at the system input along the pathlength of the two beams through the instrument;


3. $\Delta$ is the difference between the pathlengths for the two beam copies; and


4. $k_x,\,k_y$ represent relative tilt between the beams owing to residual misalignment in the instrument.

Now work out the interference between the terms in (1) and (2), assuming $\Delta$ is small compared to $R$ (i.e. that the point source is some distance from the Michelson's input). You'll get a quadratic equation for the fringe lines which shows they are generally laterally offset circles. You'll find the quadratic terms vanish if the arms are balanced, leaving linear fringes of tilt and offset.

To see how the laser mode affects the result, you can replace my expressions with tilted and offset versions of the general equations for a propagating Gaussian mode, as defined, for example, in the Wikipedia page for Gaussian Beam.