Fabry-Perot-Interferometer

The Fabry-Pérot interferometer was developed in 1897 by the French physicists Charles Fabry (1867-1945) and Alfred Pérot (1863-1925). It is an optical resonator made up of two partially transparent mirrors. If the distance between the mirrors cannot be changed (for example in the case of a glass with mirrors vapor-deposited on both sides), these structures are referred to as Fabry-Perot etalon. An incoming light beam is only guided (transmitted) through this structure if there is a constructive interference (wave crest on wave crest), i.e. a certain resonance condition is met.

This allows the Fabry-Perot interferometer, inter alia. Use as an optical filter, which filters out a narrow-band spectrum consisting of only a few wavelengths or one wavelength from broadband radiation consisting of many wavelengths. Mirror shifts also make it possible to adjust the spectral properties of the transmitted radiation, since the resonance condition changes with a changed mirror distance L!

1st case: angle of incidence α = 0, i.e. perpendicular incidence of light

A distinction must be made here between two cases:

a.) The wavelength λ remains the same and the length L of the space changes

Constructive interference always occurs when the following condition is met: 2 · L = n · λ. If L increases, then one always obtains constructive interference with integer multiples of the wavelength and the transmission through the Fabry-Perot interferometer is at a maximum.

For example, if the light source consists of 2 different wavelengths, the second intensity pattern is obtained when the length L changes. In this way, for example, the individual modes of a laser can be examined.
For a change from one constructive interference to the next, a change in the gap of exactly λ / 2 is necessary, as stated above. In the visible spectral range, this is about shifts in the range of 300 nm! How can you implement such small shifts?

One possibility is to use a piezo crystal. These can be found, for example, in piezo tone generators (buzzers). If you press a piezo crystal together, a so-called piezo voltage occurs due to the shifting of the centers of gravity of the charges. This can be in the range of high voltage if you only think of the piezo igniter in lighters. Conversely, if a voltage is applied to a piezo crystal, it changes its geometry (so-called electrostriction). This is exactly what we need here. To do this, I simply drilled a hole in the middle of a piezo buzzer.

b.) The wavelength λ changes and the length L of the space remains the same
You need this case if you want to use the Fabry-Perot interferometer as a color filter. In the case of constructive interference, the following condition applies again: 2 · L = constant = n · λ

Whether or not a certain color gets through the interferometer depends on this interference condition. If it is fulfilled, the specific wavelength λ can pass the Fabry-Perot interferometer, otherwise it cannot. So you only get constructive interference or high transmission for certain wavelengths or frequencies. How far are these wavelengths / frequencies apart? The derivation is attached as a picture.

The “allowed” frequencies therefore always differ by c / (2 · L). This means, for example, that the longer the gap between the Fabry-Perot interferometer, the closer the allowed frequencies are.

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Which mirrors do you use for this interferometer? Let us assume that the surface mirrors have a reflectivity of 95% (R = 0.95). This means that only 5% of the incident light even gets into the space between the interferometer. When the “first” beam passes through the second mirror, the intensity drops again to 5%. So there is only an intensity of 0.05 · 0.05 = 0.0025 ∼ 0.25%. The “second” and “third” etc. rays increase the intensity a little, but in total a very low intensity is to be expected. When using a laser with, for example, 100 mW, you only get around 0.25 mW at the output.

One could now suggest using mirrors with a lower degree of reflection, e.g. with R = 0.5. This naturally increases the intensity of the laser beam passing through it. But this also has a major disadvantage. The lower the reflectivity, the wider and flatter the intensity profiles become. The so-called finesse is a measure of the sharpness of the intensity maxima. The higher this is, the sharper the interference pattern or the intensity profile. Now, with decreasing R, the finesse F also decreases, as you can see in the attached figure.

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A note on the resolution of a Fabry-Perot interferometer. In the case of a diffraction grating, the maxima of two almost identical wavelengths (for example the sodium double line with λ = 589 nm and λ = 589.6 nm) are extremely close together. This is due to the fact that there is a path difference of only 1 λ for the maximum of the 1st order, correspondingly of only 2 λ for the maximum of the 2nd order, and so on.

But what about the Fabry-Perot interferometer? Now here the distance L between the two mirrors is much, much larger than the wavelength of the light. For example, L is 3 mm and λ = 589 nm = 0.000589 mm. But what effects does this have on the position of the maxima? Due to the large distance L, even with minimal differences in wavelength, all possible phase shifts between 0 ° (constructive interference), 180 ° (destructive interference) and 360 ° (again constructive interference) can occur. For example, if L is just so large that there is constructive interference for one sodium line with λ = 589 nm, then in principle all possible phases can now occur for the second sodium line. This is due to the fact that, in contrast to the grating, the Fabry-Perot interferometer does not have low orders (n = 0, 1, 2 ...), but extremely high ones. At L = 3 mm and λ = 600 nm, for example, the order is already n = 10,000 corresponds to full wavelengths! Therefore, when the space L is enlarged, the maximum of the second line can in principle lie anywhere between the peaks of the first wavelength and does not have to be as close to the first line as in the case of the diffraction grating.

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2nd case: angle of incidence α > 0, i.e. oblique incidence of light

What do the conditions look like when the incidence of light is at an angle? I have made a figure which explains these conditions.

The path difference Δs of the interfering beams is therefore 2 · L · cos (α). As the angle of incidence α increases, the path difference Δs changes. If the Fabry-Perot interferometer is irradiated obliquely with monochromatic light, constructive or destructive interference occurs depending on the angle of incidence α. If there are many different angles of incidence α at the same time, several intensity maxima and minima are obtained in the form of concentric circles.

The above-mentioned with regard to the position of the maxima of spectral lines that are also closely spaced also applies here, of course, to inclined incidence. It is possible that the second sodium D line is always located exactly between two maxima of the first. Or else, they are closer together. It would also be possible that the rings of both spectral lines coincidentally lie exactly on top of one another. But in contrast to the observation with the diffraction grating, where the two lines are necessarily extremely close together, here with the Fabry-Perot interferometer the distance to one another is arbitrary.

As always, I try to implement it experimentally as simply and inexpensively as possible. Fabry-Perot etalons cost a good 1500 euros. I will try it with surface mirror á 5 euros. For this I ordered surface mirrors with 50% reflectance.

Cheap source of supply: https://astro-didaktik.de/astromedia/lenses-mirro...

At the same store you get the biconvex glass lens with a focal length of 39.5 mm. As a monochromatic light source I will use a 5 mW red laser.

All mounting-plates for the mirrors, the lens and laser are made of aluminum and have the dimensions 50 x 50 x 3 mm. The spring washers are placed on the M4 threaded rod in the space between the two mirrors.

You have to glue the two partially transparent mirrors (R = 0.5) to the carrier plates. To do this, I used ordinary thin double-sided tape. But beware, in the center of the two mirrors there mustn't be any tape there. In the center of the mirrors you have to thrill a hole (diameter = 8-10 mm).

The adjustment of the interferometer could actually be done without any major difficulties due to the multiple reflections of the laser beam.

To do this, the laser is focused on the milky plastic plate. If you now look through the interferometer, you can see several light reflections. These are arranged along a line. The task now is to combine these many light reflections into a single light reflection by adjusting the wing nuts. If this is the case, it can be assumed that the interferometer is at least roughly adjusted. For the further test of the interferometer, the optics of the laser are adjusted so that it is no longer focused on the milky plastic plate, but creates a larger light spot on it. If you look through the interferometer, with a bit of luck you should see interference rings.

But watch out: I had a sense of achievement right from the start by being able to see the rings (Muster 1 in the attached picture). As a test, I then applied light pressure with my finger on one side of the interferometer. To my astonishment, the interference pattern, i.e. the rings, has not changed. After some thought, I figured out the cause. The observed interference pattern was not generated by the interference in the space between the two mirrors, but within a glass plate itself, which serves as a support for the mirror coating. Since the thickness of the glass is only about 1 mm, the rings of this interference pattern have a large distance. But we want the interference caused by the 3-4 mm distance between the two mirrors. This ring pattern caused by this (see Muster 2 in the picture) must be significantly narrower. But only this then reacts to pressure with the finger against a plate of the interferometer.

To be able to change the brightness of the 5V-Laser-module I use a LM317. With this voltage regulator I can vary the voltage between 1.25V and 5V.

The second setup is very similar to the first. The difference is that there is now a piezo buzzer under one of the two semi-transparent mirrors. In the middle of it I drilled a hole about 6-8 mm in diameter.
Instead of the finderscope during the first setup, the smartphone is now used here. So that you can see the interference rings clearly, I use a telephoto attachment with 8x magnification. You can get this cheaply on ebay or aliexpress.

The electronics for controlling the laser or piezo is very simple. The voltage for the laser is regulated with an LM317 voltage regulator. The piezo is simply operated with a potentiometer with a variable voltage between 0 and 9V.

The second setup with the piezo now enables the space between the two mirrors to be changed. However, these changes must be in the range of the light wavelength, i.e. around 500 nm. The piezo enables precisely these small changes if it is supplied with voltages in the range 0-9V.
If you now look through the interferometer or the smartphone display, you should see the narrow interference rings again if the interferometer is correctly adjusted. If you then turn the potentiometer and thus change the voltage on the piezo, you should recognize a change in the interference pattern. New rings would have to be added from within, or rings would have to move into the center and then disappear.

Commercial Fabry-Perot interferometers cost around 1500 euros. Both of my homemade interferometers only cost around 50 euros each.

Why did I make such an interferometer at all? Now, firstly, more advanced physics projects pose a challenge for me in that I firstly try to implement them successfully and secondly as cheaply as possible with easily available materials. In this case I have something else to do with the Fabry-Perot interferometer, namely the experimental verification of the Zeeman effect. This deals with the shift in wavelengths of light when the light-emitting gas is in a strong magnetic field. But the effect is extremely small. Even with flux densities around 1 Tesla, the shift in the spectral lines is only around 1/100 nm due to the Zeeman effect! This tiny shift cannot be detected with a normal spectroscope. However, it may be possible with my Fabry-Perot interferometer. I'm curious to see if I can do it. I have already bought an old, used HeNe laser for this project...

If you are interested in other exciting physics projects, here is

my homepage: homepage stoppi

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In this sense, good luck with your physics projects and Eureka