Michelson interferometer

The Michelson interferometer [ ˈmaɪkəlsn- ] is an interferometer that was named after the physicist Albert A. Michelson . This measuring instrument became known primarily through the Michelson-Morley experiment , through which the so-called light ether was to be investigated as a medium for the propagation of light. The Michelson interferometer makes use of the phenomenon of interference , which can only be observed with coherent light. Normally, special light sources, usually lasers , are used for interference experiments. In the experiment it can then be split up with a beam splitter and finally brought into interference with itself. The special thing about the Michelson interferometer is that the beam splitter and the partially transparent mirror in which the beams are reunited are the same.

Michelson interferometer with diode laser and beam path shown

The path difference between the superimposed waves must be smaller than the coherence length . A correction plate is therefore installed in the interferometer arm for media with dispersion and light sources with a broad spectrum . The correction plate is made of the same material as the beam splitter and has the same thickness, but is completely translucent. It lies on the dividing side of the beam splitter and is attached in such a way that the path difference between the two partial beams is compensated for.

functionality

Structure of the Michelson interferometer

An interferometer splits a light wave into two parts. These two waves then travel through different lengths, the duration of which is different. There is a phase shift between the two waves. When they meet, interference occurs.

With the Michelson interferometer, the light wave is split up by means of a semi-transparent mirror . The light emitted by the light source is partly let through (marked red) at the semi-transparent mirror ( beam splitter ), but partly reflected by 90 degrees (marked blue). The transmitted and reflected light now hit a (fully reflective) mirror and are reflected back onto the semi-transparent mirror. Again a part is reflected and a part is let through. The two waves (marked in yellow) then overlap behind the semitransparent mirror, causing interference.

If you change the optical path length of one of the two waves, e.g. B. by moving one of the two mirrors, or by changing the refractive index of the medium in one of the two interferometer arms, the phases of the two waves shift against each other. If they are now in phase, their amplitude is added (one speaks of constructive interference), but if they are out of phase, they cancel each other out (destructive interference). Even the smallest changes in the path difference between the two waves can be measured by measuring the intensity of the resulting wave.

Creation of the interference fringes

Formation of the interference fringes

HeNe laser interference pattern 633 nm

Interference pattern mercury vapor lamp

A parallel bundle of rays comes from the light source (plane wave). This is “widened” by a lens arrangement and then runs divergently (diverging) with a new imaginary point of origin G (spherical wave), which lies in the area of the lens arrangement.

This divergent bundle of rays is split into two divergent bundles of rays by the beam splitter. The two bundles of rays are each reflected by a mirror (depending on the structure), brought together again and directed onto a screen. The interference patterns come about because the direct paths “beam splitter mirror 1 beam splitter screen” ( l ₁ ) and “beam splitter mirror 2 beam splitter screen” ( l ₂ ) are of different lengths. The distance from the imaginary point of origin G to the beam splitter is constant ( g ). If two rays from the two bundles of rays hit the same place at the same time (distance d from the center of the interference rings), then they have covered paths w of different lengths . The exact path can be found using the equation

{\ displaystyle w = {\ sqrt {d ^ {2} + (g + l) ^ {2}}}}

to calculate.

With the same distance d from the center of the interference rings, the paths w ₁ and w _{2 are of} different lengths. If d is increased linearly, then w ₁ and w _{2 increase at} different rates . If a plane wave is viewed, a bright spot appears on the screen in the case of constructive interferences; in the case of destructive interferences it remains dark. The interference rings are a result of the Gaussian rays , which are spherical waves from a given length.

Relative distance measurement

The interferometer is therefore suitable for measuring slow changes in the path length difference between the two partial beams, for example the change in position of one of the opaque mirrors, the achievable resolution being on the order of half the wavelength of the light used. With visible laser light, the wavelength is a few hundred nanometers .

To measure, move one of the two opaque mirrors and count the number of interference minima (or maxima) that are passed through during the movement. Each minimum then corresponds to a change in path length by one wavelength, i.e. a change in position of the mirror by half a wavelength. The absolute path lengths or their absolute difference cannot be measured, and neither can the direction of movement. The speed of the measurable change is limited by the achievable counting rate of the minima.

Improvement of the distance measurement

If the direction of movement of the mirror changes, the problem arises that at the extreme points of the sine (the brightest and darkest points of the interference pattern) it is not known whether the movement of the mirror will continue in the same direction or vice versa, since both produce the same signal curve would. Therefore, in this case, a second sensor must be placed at a different point so that both signals are never at extreme points at the same time.

The distance measurement by Michelson interferometer is characterized by high resolution and linearity (depending on the wavelength of the laser).

Today's gravitational wave detectors represent the most complex variant of the Michelson interferometer for measuring path length with movably mounted mirrors.

Heterodyne Michelson interferometer

Many of today's Michelson interferometers are designed as heterodyne interferometers. A slightly different frequency is used in the two arms of the interferometer. The recombined beams result in a beat in the detector. At the same time, part of the light at both frequencies is superimposed in a reference detector, i.e. not reflected on the mirrors. The actual measurement is then a measurement of the phase difference between the beat on the detector and that on the reference detector. Since phase angle measurements are possible with significantly better accuracy than the interpolation of the interference signal of a homodyne interferometer, resolutions of 10 pm have already been achieved with heterodyne Michelson interferometers . In addition, the above-mentioned problem with the reversal of direction at extreme points is eliminated, since the phase position of the beat increases over a longer distance with a suitable design and can be clearly determined via the phase difference between the signal arm and the reference detector.

Lasers based on the Zeeman effect or an acousto-optical modulator are usually used to generate the two wavelengths .

Use as a spectrometer

If a broadband IR source is used and the beam is allowed to pass through a measuring cuvette with a substance to be measured in front of the detector , its spectrum can be obtained. In order to pass through the frequency band to be measured, one has to change the position of one mirror x over time, for example with a piezo element , in order to generate different path differences and thus resonance and extinction cases at different wavelengths. The Fourier transformation of the interferogram from the spatial I ( x ) or time domain I ( t ) into the frequency domain provides the spectrum of the substance. ${\ displaystyle I (\ nu)}$

Determination of the refractive index of a gas

To determine the refractive index of a gas, a cuvette filled with the corresponding gas is placed in the partial beam, the path length of which was previously varied (the mirrors now remain fixed). The gas pressure and thus the number of gas molecules through which the light penetrates can be varied with a pump connected to this cuvette. The linear relationship between pressure and refractive index is described as

{\ displaystyle n (p) = n (0) + {\ frac {\ Delta n} {\ Delta p}} \, p}

and takes advantage of that due to the increase in the refractive index

{\ displaystyle {\ frac {\ Delta n} {\ Delta p}} = {\ frac {\ Delta N} {\ Delta p}} \, {\ frac {\ lambda} {2s}}}

can be expressed, this leads to ( n = 1 at p = 0):

{\ displaystyle n (p) = 1 + {\ frac {\ Delta N} {\ Delta p}} \, {\ frac {\ lambda} {2s}} \, p}

Here, N denotes the number of intensity maxima in the interference pattern, p the gas pressure, the wavelength of the laser light used and s the geometric path length of the cuvette. ${\ displaystyle \ lambda}$

Measurement of the wavelengths

The two bundles of rays are still coherent if their optical path difference is smaller than the coherence length of the light source. If the distances between the semi-permeable plate and the mirrors are the same, the rays arriving at the detector have a phase difference of 0. If one of the two mirrors is now shifted by the distance , a path difference arises between the two bundles of rays and the light intensity changes. ${\ displaystyle d}$ ${\ displaystyle \ Delta w = 2d}$

If you now determine the number of interference maxima in a displacement path, the wavelength can be easily calculated, since the following always applies: ${\ displaystyle z}$ ${\ displaystyle \ Delta d}$ ${\ displaystyle \ lambda}$

{\ displaystyle \ Delta d = {\ frac {\ lambda} {2}} z.}

Web links

Commons : Michelson interferometer - collection of images, videos and audio files

Determining a change in length ( Memento from June 22, 2006 in the Internet Archive ) (Real-Video 6.9 MB)
Principle of FTIR spectroscopy
DIY instructions from the Ligo website
Michelson interferometer with single photons

Individual evidence

^ John Lawall, Ernest Kessler: Michelson interferometry with 10 pm accuracy . In: Review of Scientific Instruments . tape 71 , no. 7 , 2000, pp. 2669-2676 , doi : 10.1063 / 1.1150715 .

[1] John Lawall, Ernest Kessler: Michelson interferometry with 10 pm accuracy . In: Review of Scientific Instruments . tape 71 , no. 7 , 2000, pp. 2669-2676 , doi : 10.1063 / 1.1150715 .