## HomeExperimentsWave Optics

This article describes the results of my attempts to build a small Michelson Interferometer. The objective of this project is to find out whether it is feasible to use solid state laser modules for wave optic experiments.

• Selection of a solid state laser and integration into an optical setup
• Setup of an test platform which is not too expensive
• Setup of a simple Michelson Morley type of interferometer
• Find a way to evaluate the usability of the test platform

In fact this project does not use scrap parts. Most of the parts are new; the only piece i have acquired as scrap are the basic elements of the optical bench. The Laser Module and the optical parts are new.

Selecting the laser diode has not been done with great care; i simply purchased some powerful 100mW laser diode because it comes with a control unit via eBay. Important was here only to get maximum power at the lowest price including the control unit. Most of the optical parts like mirror's, kinetic mounts have been purchased at Thorlabs.

The idea of the Michelson Interferometer is to split up the beam of a coherent light source (e.g. a laser) into two beams passing a different optical length before they are projected on a screen where the individual waves are summed up. The setup is shown in the figure above. The light emitted by a laser diode is split up into two beams by means of a beam spliter. Each beam (a,b) is reflected b back int the beam splitter which reflect a portion of both beams into the projection optic (c). Since both beams are paralel but have traveled different optical distances they are interfering in path c. The concav lens in the path c is used to spread the interference pattern making them better to observe.

As shown in Figure 1-2, the optical bench has been placed on top of a heay table in order to reduce vibrations. The table has been realized by two 12kg stone tiles carried by three conic legs reducing the contact between floor and table.

## Superposition of two Waves

Let assume that we describe the setup shown in the picture above by means of two plane wave functions, one for each beam) where the observed intensity I at a point x is given by:

 $$I(x) = \mid\psi_1(x)+\psi_2(x)\mid^2$$ (0.0)

The difference in the length of the light paths of both beams is expressed by the $$\Delta x$$ variable. Assuming that the the intensity of both waves are identical we can write;

 $$\psi_2(x) = \psi_1(x + \Delta x)$$ (0.1)

For simplicity reasons we assume that the an apropriate description of the interacting waves are plane waves, which can de be written as:

 $$\psi_i(x) = I_i e^{-i \vec{k} \vec{x} }$$ (0.2)

In the following we will assume that $$I_i$$ to 1. The vector $$\vec{k}$$ respresents the direction in which the wave is propagating

 $$\vec{k} = \frac { 2 \pi }{ \lambda} \vec{n}$$ (0.3)

where $$\vec{n}$$ is the unit vector indicating the propagation direction of the wave.

If the vector n is choosen paralel to the x-axis, the superposition of these two plane waves leads to an observed intensity as given in (1.3):

 $$\begin{equation*} \mid\psi_1(x) + \psi_2(x)\mid^2=I_1^2+I_2^2 + I_1 I_2 (e^{ik \Delta x } + e^{ -ik \Delta x }) \end{equation*}$$ (0.4)

 $$I(x)= I_1^2+I_2^2 + 2 I_1 I_2 cos( \frac{ 2\pi \Delta x}{\lambda} )$$ (0.5)

Since $$cos(x)$$ is a periodic function, the intsity of the super position of both beans will have a minimum and a maximum as specific value depending on the $$\Delta x$$ value.

 $$\begin{equation*} I(x)= \begin{cases} I_1^2+I_2^2 \text{ \qquad\qquad\quad\ \Delta x = n \lambda } \\ \\ I_1+I_2+2I_1 I_2 \text{ \quad \Delta x = \frac{n}{2} \lambda } \end{cases} \end{equation*}$$ (0.6)

This effect is called interference. The picture below shows a computation. ## The results

The recording below shows the correponding observations made using the setup shown above.