Friday, 27 March 2015

Use of an Optical

interferometer to detect

sound waves as an

analogue for

gravitational waves.

Report by Thomas Acton Partners: Salas Luffy, Edoardo Paluan, Gareth Ellis

Supervisor: Riccardo Sapienza

2

Abstract

This aim of this experiment is to use sound waves to imitate the chirp and burst patterns produced

by gravity waves. We are using sound as an analogue to gravity waves so we can confirm the setup

is capable to detecting analogous waves and hopefully learn about the difficulties in detecting

gravity waves. By using a tuning fork resonating at 440Hz we tried to imitating a chirping

gravitational wave. We successfully detected it at 439.8Hz ± 0.06 Hz using an ArduinoTM and at

402.9Hz ± 201.4Hz using an oscilloscope. The explanation for the difference in results is due to the

folding frequency as a result of the number of data points each device is able to record. The

ArduinoTM can record as many as it wants, at a lower sample rate, while the oscilloscope is limited

to 2500 points. The ArduinoTM is capable of detecting frequencies the oscilloscope is ‘unaware’ of,

while the oscilloscope is capable of detecting much higher frequencies than the ArduinoTM.

The oscilloscope was unable to detect claps whereas the ArduinoTM detected 5 constituent

frequencies consistent across multiple claps at: 576.8Hz ± 3.96Hz, 613.5Hz ± 2.02Hz, 657.9Hz ±

6.60Hz, 704.05Hz ± 1.20Hz and 805.4Hz ± 7.79Hz. The claps are analogous to a burst of gravity

waves from a supernova. It was concluded the setup is capable of detecting these ‘fake’ gravity

waves but only inside a limited frequency window because of the slow sample rate on the

ArduinoTM.

3

Table of contents

1. Title page

2. Abstract

3. Table of contents

4. Introduction

a. Introduction

b. Setup used

c. Interference

5. Introduction

a. Interference patterns

b. Fringe condition

c. Refractive index change

6. Introduction

a. Detection

b. AdruinoTM

7. Introduction

a. AdruinoTM

b. Oscilloscope

c. Gravity waves

8. Introduction

a. Gravity waves

9. Results

a. Background

10. Results

a. 440Hz tuning fork

11. Results

a. 440Hz tuning fork

12. Results

a. 440Hz tuning fork

13. Results

a. Piezo-electric crystal (900Hz):

14. Results

a. Piezo-electric crystal (2000Hz):

15. Results

a. Clap

16. Results

a. Clap

17. Results

a. Tone generator (2kHz + 2.5kHz + 3kHz + 7kHz):

18. Results

a. Tone generator (2kHz + 2.5kHz + 3kHz + 7kHz):

19. Analysis and conclusion

20. Analysis and conclusion

21. Bibliography

4

Introduction

The aim of this project is to use a Michelson interferometer to detect sound waves. By doing so we

hope to show how a similar set-up could be used to detect gravitational waves, which is currently

being used by the Laser Interferometer Gravitational Wave Observatory (LIGO). Conceptually the

two are similar because the change in the density of the air, caused by the sound waves creates the

same effect as an increase in path length; which would happen from a stretch of space-time if a

gravitational wave was to pass over the experiment.

The Michelson interferometer was most famously used in the Michelson-Morley experiment of

1887, which helped Einstein with his formulation of the special theory of relativity1. The apparatus

works by splitting light from a laser into two separate beams of equal amplitude and directing them

down arms towards mirrors, recombining them and sending the beam in the direction of a detector.

Figure 1 below shows the setup we used, and how it works.

The apparatus should be setup such that there is a slight path difference between Mirror 1 and 2 so

to cause an interference pattern at the detector, shown in figure 2. The optical path difference (OPD)

is calculated geometrically. Take figure 3 as an example we can see that light hitting mirror B has to

travel an additional distance of 𝐴𝐵⃗⃗⃗⃗ ⃗ + 𝐵𝐶⃗⃗⃗⃗ ⃗ further, which is the same as 2 × 𝑑𝐶𝑜𝑠(𝜃), where d is the

difference in arm length. As a result we can define the optical path difference in equation 1, where n

is the refractive index of the medium.

𝑂𝑃𝐷 = 2𝑛𝑑𝐶𝑜𝑠(𝜃) = 𝛿

(1)

Figure 1: A simple diagram of a Michelson interferometer setup.

5

From equation 1 it is clear there are 3 variables which control 𝛿: this can be done by varying the

refractive index, changing the angle of incidence or the distance between mirrors. As a change of

angle would mess up the optical alignment that leaves two options. To cause a change in the

distance of the mirrors a piezo-electric crystal can be placed behind one and used to oscillate it. This

is something we will investigate on-top of using sound waves to change the density of the air along

the arm.

The dark fringe condition can be expressed as:

2𝑛𝑑 = 𝑚𝜆

𝛿 = (2𝑚 + 1)𝜋

Where we set 𝜃 = 0. This is because dark fringes occurs when the recombined waves are odd

integer multiples of 𝜋 out of phase. The order of the central fringe is thus:

2𝑛𝑑

𝜆0= 𝑚

From this we now have an equation which lets us calculate the change in fringes from a change

in 𝛿.

2

𝜆0∆(𝑛𝑑) = ∆𝑚

Sound waves change the refractive index by compressing the air. The waves travel longitudinally

and as the wave passes through the arm air molecules are pushed into each other which raises and

lowers the density of air and affects the refractive index. The change in the path difference can be

calculated by the change in the refractive index and the length the change is over.

Figure 2: An interference pattern

caused by an optical path difference

from an interferometer. The ideal

size of the detector is marked as to

detect the greatest change of

intensity when the path difference

changes.2

Figure 3: A diagram used to help

explain where the OPD equation comes

form.

(2)

(3)

(4)

(5)

6

∆(𝑛𝑑) = (𝑛𝑛𝑒𝑤 − 1)𝐿

To detect the clearest signal we are interested in causing the largest change of intensity on the

detector. To do this we setup the experiment so that the central fringe occupies as much of the

detector as possible, like in figure 2. To achieve this we use a telescopic arrangement of two lenses

which collimate the beam to the correct width before it enters the beam splitter.

Every different 𝛿 will have a different effect on the intensity of light hitting the detector as different

amount of fringes will be present. The larger the 𝛿 the more fringes will be created and spread over

the detector. The smallest 𝛿 we can detect is a change of 𝛿 =𝜆

4, as this corresponds to the change

from the first central bright fringe to dark fringe. The different effect on the intensity created by

different 𝛿 allows the detector to be used to detect signals containing more than one frequency. The

output from a signal containing more than one frequency will vary like the graph on the left in

figure 4, which contains 2 very distinct frequencies. By carrying out a Fourier transform on the

signal we are able to split the signal into its component frequencies (see right graph in figure 4) for

analysis.

To collect measurements from the detector we used two systems: an ArduinoTM Due and an

oscilloscope. Both devices are connected to the detector through electronic circuitry which

translates the current produced by the photodiode in the detector into a voltage and adds gain. The

ArduinoTM Due has an analogue read function which allows it to read the a voltage on one of the

general purpose input/output (GPIO) pins between 0-3.3 V with a resolution of 4096 points (12

bits)3, this translates to a resolution of 0.8 mV/point. We sampled at intervals of 461 𝜇𝑠 which is a

sampling frequency of 2170 Hz.

Theoretically the maximum sample rate which could be achieved is around 1MHz, this would be

done by hacking the board and changing the prescaler on the Master Clock (MCLK) of the

Analogue to Digital Converter (ADC), which runs at 1Mhz to 13. This allows the maximum sample

speed of 1𝜇𝑠 and the ARMTM Cortex-M3 processor on the board, which runs at 84MHz can execute

dozens of necessary instructions during that time.

Figure 4: An example of a signal containing more than one

frequency and its Fourier transform splitting it into the

component frequencies which make it up.

(6)

7

In addition to using the ArduinoTM Due we had an oscilloscope to measure the signal graphically in

real-time. The oscilloscope is capable of a much higher sample rate and voltage resolution but

cannot record more than 2500 data points before its memory fills up and it has to be output.

General relativity states that the curvature of space-time is proportional to the energy content at that

location.

𝑅𝜇𝜈 =1

2g𝜇𝜈𝑅 =

8𝜋𝐺

𝑐4𝑇𝜇𝜈

Gravitational waves are predicted as ripples of space-time, which should occur when large changes

to the mass-energy density happen. There are two types of signals which one might hope to acquire:

Chirps and bursts. Chirped signals would be produced by the spiralling of a star into its compact

binary partner, a neutron star or black hole. The signal would evolve over time before reaching a

maximum and then dropping off. A burst signal would be produced by a supernova event. Current

models predict that detections of these burst signals might possible as far out as the Virgo Cluster of

galaxies4. A spherically symmetric system will not produce any gravitational waves and a

quadrupole moment is needed to produce gravitational waves. Events such as supernovas, neutron

star merges and other ‘rare’ celestial events would have their own signature gravitational wave.

Theoretically the frequency of the wave is determined by equation (8):

Figure 5: (Left) The ArduinoTM Due with two cables plugged into the

analogue GPIO pins. (Right) The detector plugged into the electronics

box, and the output cable splitting off to the oscilloscope and ArduinoTM.

(7)

Figure 6: Ripples in space-time produced by the gravity waves emitted by the binary system

as one object falls into the other.

8

𝑓 ≅ √𝐺𝑀

𝑅3

Which produces 𝑓𝑁𝑒𝑢𝑡𝑟𝑜𝑛 𝑆𝑡𝑎𝑟 ~ 2𝑘𝐻𝑧 and 𝑓𝐵𝑙𝑎𝑐𝑘 𝐻𝑜𝑙𝑒 ~ 10𝑘𝐻𝑧.

Figure 7 shows how a gravity wave would stretch an object in one direction before returning it to

the original shape and then stretching it again in an orthogonal direction. It is this reason why the

perpendicular arms on the Michelson interferometer make it ideal for detecting gravity waves.

Gravity waves are extremely weakly interacting and barely coupled with matter in comparison to

other waves such as electromagnetism and as a result they should produce a clean, unperturbed

signals unaffected by the large distances and matter they have travelled through towards us.

Unfortunately for the exact same reason they are also extremely difficult. Because the signal is so

weak and noise is the main problem there is the need for very large detectors, because ∆ℎ~ℎ ∙ 𝐿 (∆ℎ

scales linearly with arm length) and long runs to reduce noise as much as possible.

Figure 7: A diagram showing how gravity waves would stretch objects over the period of one

wavelength.

(8)

9

Results

Background:

Figure 8 shows a recording of the background to determine which frequencies are noise in future

test. This sample was recorded over 8.273613 seconds and consist of 18690 data measurements.

A Fourier transform (figure 9) of the background signal was carried out and revealed 6 frequencies

of 9.4Hz, 13.3Hz, 21.5Hz, 30.5Hz, 37.6Hz and 308.7Hz to be present.

Figure 8: The background measurement signal.

Figure 9: A Fourier transform of the background signal.

10

440Hz Tuning Fork

Figure 10 shows two separate sets of data recording the 440Hz tuning fork resonating. The data

contains 17950 data points recorded over 8.273634 seconds for each reading.

Figure 11 shows the Fourier transforms for both sets of data overlaid on-top of each other. The

original frequency of 439.8Hz, overtone of 879.8Hz and undertone of 222.4Hz were detected, along

with 2 other frequencies of 120.1Hz and 723.1Hz.

Figure 10: Two separate sets of data for measuring the signal when a 440Hz tuning fork is

resonating next to one of the arms.

Figure 11: The Fourier transform for both data stets with the frequencies of the signal

labelled.

11

Figure 12 is an enlargement around 440Hz and shows two smaller frequencies which sit at ~ 440 ±

30 Hz and 440 ± 36Hz.

Figure 13 shows the first overtone 879.9Hz, similar to before it has two additional frequencies that

barley register at 880 ± 30Hz and 880 ± 36Hz, if this behaviour was not present around 440Hz tone

I would have missed it and attributed it to noise.

Figure 12: A close up of the Fourier transform at the 440Hz peak, with leaked frequencies

marked.

Figure 13: A close up of the Fourier transform at the 880Hz overtone peak, with leaked

frequencies marked in the same positions as 440Hz

12

Figure 14 shows signal around where the undertone should be at 222.4Hz.

Figure 15 is the signal acquired by the oscilloscope of the 440Hz tuning fork. It uses 2483 data

points each taken 1𝜇s apart. A Fourier transform of this data was only able to detect 2 frequencies:

402.9 and 805.8, no graph of the Fourier transform is shown because no useful data can be taken

from it.

Figure 14: A close up of the Fourier transform at the 220Hz undertone.

Figure 15: The signal recorded on the oscilloscope of the signal when a 440Hz tuning fork is

resonating next to one of the arms.

13

Piezo-electric crystal (900Hz):

Figure 16 shows the signal output from the detector when the piezo-electric crystal was oscillating

the mirror at a frequency of 900Hz. It was taken over 9.061423 seconds using 20039 data points.

Figure 17 shows the Fourier transform of the signal and shows that the actual frequency was

900.6Hz. There was some noise in the signal centred on 860Hz but no sharp peak. Other

frequencies present were at 12.99Hz, 24.03Hz and 36.16Hz which can be attributed to background

noise.

Figure 16: The signal recorded when the piezo-electric crystal was oscillating the mirror at a

frequency of 900Hz.

Figure 17: The full spectrum of the Fourier transform of the 900Hz signal, and a blown up

graph of the signal and noise around 900Hz.

14

Piezo-electric crystal (2kHz):

Figure 18 shows two separate sets of data acquired when the piezo-electric crystal oscillates one

mirror at a frequency of 2 kHz. Each reading consists of 19833 data points taken over 9.479423

seconds.

Figure 19 shows the Fourier transforms of each data set laid on-top of one another. There is no

consistency between sets and a lot of noise arround the peaks. The first set of data is a mix of

frequencies between 242.7Hz and 303Hz and contains some high order frequencies arround 487Hz

and 606.8Hz. The other data set detects some frequencies centered arround 183.8Hz with a wide

spread of 40Hz either way.

Figure 18: The signal from two sets of data recorded while the piezo-electric crystal was

oscillating the mirror at a frequency of 2000Hz.

Figure 19: The Fourier transform of both sets of data when the piezo-electric crystal was

oscillating the mirror at a frequency of 2000Hz, overlaid and a few frequencies marked.

15

Claps:

Figure 20 shows the signal produced during 3 separate claps. Each clap consits of 2082 data points

taken over 1.134145 seconds, with a sample rate of 545𝜇s. The following key for the Fourier

transform in figure 21 is: Blue (top left) green (top right) and red (bottom).

Figure 21 (next page) is the Fourier transform of 3 separate claps overlaid on-top of each other.

Peaks which appear across all claps are found at (572.7, 577.1, 580.6)Hz, (615.9, 612.4, 612.4)Hz,

(658.2, 651.2, 664.4)Hz, (704.9, 703.2)Hz and (809.9, 796.4, 809.9)Hz. The lower intensity of the

green data set can be attributed experimentally to a quieter clap.

Figure 20: Three sets of data showing the signal produced by a clap.

16

Figure 21: Three Fourier transforms of the signal of a clap overlaid with peaks consisten

between each clap maked on each graph. The key for the signals in figure 20 are: Blue (top

left) green (top right) and red (bottom).

17

Tone generator (2kHz + 2.5kHz + 3kHz + 7kHz):

Figure 22 is data taken on the oscilloscope to investigate frequencies higher than the ArduinoTM is

capable to detecting. This data consists of 2483 data points taken over 0.009932 seconds with a

sample rate of 4𝜇s. The data is measuring the signal from a tone generator playing a combination of

2kHz, 2.5kHz, 3kHz and 7kHz tones.

Figure 23 (next page) is the Fourier transform for the 4 tone signal. It shows the peak at 2015Hz,

and a smaller peak at 2518Hz. No peak is found around 3kHz and the peak at 7051Hz is only just

distinguishable from the noise. The peak at 7051Hz would have been ignored if we were not

looking for a signal there. The peak at 100.7Hz is indistinguishable from the background noise we

know is present at low frequencies.

Figure 22: The signal recorded when a tone generator plays a sound consisting of 2kHz,

2.5kHz, 3kHz and 7kHz tones.

18

Figure 23: The Fourier transform for a tone generated by playing sound consisting of 2kHz,

2.5kHz, 3kHz and 7kHz frequencies. Each peak has been marked.

19

Analysis and conclusions

If we think back to the introduction, the aim of the experiment was to detect sound waves as an

analogue for gravity waves, which come in a chirped or burst form and to detect the limits of our

setup. The experiment using the 440Hz tuning fork could be considered similar a chirped gravity

wave, and the claps an analogue for a supernova and burst waves.

The 440Hz experiment produced a peak when measured on the ArduinoTM at 439.8Hz ± 0.06 Hz,

where the uncertainty is calculated as half the folding frequency.

𝑓𝑓𝑜𝑙𝑑𝑖𝑛𝑔 =1

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑝𝑜𝑖𝑛𝑡𝑠 × 𝑠𝑎𝑚𝑝𝑙𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙

∆𝑓 =𝑓𝑓𝑜𝑙𝑑𝑖𝑛𝑔

2

We also detected an undertone and overtone at 222.4Hz ± 0.06 Hz and 879.8Hz ± 0.06 Hz. These

would not be present in gravitational waves but are characteristic of sound waves and we expected

to detect them.

Other frequencies present were at 120.1Hz ± 0.06𝐻𝑧 and 723.1Hz ± 0.06𝐻𝑧, and most likely arose

due to imperfections in the tuning fork.

Strange peaks were also located around the principle frequency at 439.8Hz ± 0.06 Hz and the

overtone at and 879.8Hz ± 0.06 Hz, these are either aliases or characteristics of the turning fork as

they both appear at ~ ± 30Hz and ± 36Hz from the central frequency. Further investigation would

be interesting.

When the oscilloscope was used to measure the same system we were surprised to see that it was

significantly outperformed. The Fourier transform shows only two frequencies: 402.9Hz ± 201.4Hz

and 805.8Hz ± 201.4Hz. This massive uncertainty arises from the fact only 2483 data points are

taken despite the increased sample rate.

Out of interest we tested the system using a piezo-electric crystal oscillating at 900Hz and 2000Hz,

at different times, with the later designed to be outside our detection range. We detected a very clear

signal at 900.6Hz ± 0.05Hz, with next to no other noise in the signal. The 2kHz experiment

confirmed that we can perform no useful analysis at frequencies greater than half our sampling

frequency and that the mechanical oscillations produced around 2kHz must have excited a resonant

frequency of out setup which caused a lot of noise across a large window of frequencies.

𝑓𝑚𝑎𝑥 =𝑓𝑠𝑎𝑚𝑝𝑙𝑒

2

𝑓𝑀𝑎𝑥 𝐴𝑟𝑑𝑢𝑖𝑛𝑜 = 1084.60𝐻𝑧

The claps were designed as an experiment to test the ArduinoTM’s ability to detect burst, we knew it

excelled over the oscilloscope when it is able to take long data acquisitions and wanted to see how it

would perform with a quick signal consisting of multiple frequencies. This experiment was

impossible to perform on the oscilloscope because of how small the data acquisition windows are.

During the clapping the Arduino sample rate slowed down to 545𝜇𝑠 giving us a new maximum

frequency we are able to detect of 917.431Hz.

(9)

(10)

(11)

(12)

20

Despite this we were able to detect 5 main frequencies consistent across all claps located at 576.8Hz

± 3.96Hz, 613.5Hz ± 2.02Hz, 657.9Hz ± 6.60Hz, 704.05Hz ± 1.20Hz and 805.4Hz ± 7.79Hz.

Finally the oscilloscope was used again to check how the setup responds to frequencies higher than

the ArduinoTM can detect. The tone generator played sound consisting of 2kHz, 2.5kHz, 3kHz and

7kHz frequencies. Figure 23 shows that only signals detected were at 100.7Hz ± 50.3Hz, 2015Hz ±

50.3Hz and 2518Hz ± 50.3Hz. No peak is found around 3kHz, more experimentation needs to be

done to see if this is a dead zone. The small peak at 7051Hz ± 50.3Hz is almost indistinguishable

from the background noise, and would have been ignored if we were not looking for a signal there. I

suspect that if we had more data points we would have much sharper, accurate peaks at all the

frequencies generated by the tone generator.

In conclusion, the experiment is able to detect chirps (440Hz tuning fork) and burst (claps), as long

as they are within the frequency window allowed by our ArduinoTM’s sample rate. An ideal system

would incorporate the resolution and fast sample rate of the oscilloscope with the long data

acquisition time possible with the ArduinoTM. Before the experiment I underestimated the

importance of a long sample duration and predicted the oscilloscope would be the best tool for the

job, but it is apparent the Arduino is actually far superior and it is only the sample rate which lets it

down. This could be improved as previously suggested by hacking the board, and given more time

we would have attempted to do so. Currently our ArduinoTM setup is incapable of detecting actual

gravity waves from neutron stars or black holes, but using the oscilloscope neutrons stars producing

gravity waves around 2kHz would be possible to detect, if we could eliminate enough noise.

Below 50Hz there was a lot of background noise, originating from the sound of the air-conditioning

in the lab to possible vibrations from the building. To improve the experiment the device could be

placed on an isolated bench and the arms could be covered to provide insulation from other audio

interference. Given how important arm length is I would also like to see the setup changed to use

the longest arms possible given space considerations.

21

Bibliography

1. Albert Michelson; Edward Morley (1887). "On the Relative Motion of the Earth and the

Luminiferous Ether". American Journal of Science 34 (203): 333–345

2. http://upload.wikimedia.org/wikipedia/commons/4/42/Laser_Interference.JPG

3. http://www.djerickson.com/arduino/

4. Phys. Today 52(10), 44 (1999); doi: 10.1063/1.882861