thermal noise measurement technique

8/8/2019 Thermal Noise Measurement Technique

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Thermal noise measurement technique

B. J. J. Slagmolen, X. Brokmann, D. A. Shaddock, M. B.

Gray, and D. E. McClellandDepartment of Physics, Faculty of Science,

The Australian National University, A.C.T., 0200, Australia.

E-mail: [email protected]

Abstract. We report the measurement of the transfer function of a flexure suspension

system which is used in the ANU thermal noise experiment. This suspension hinges

over four thin machined flexure membranes ensuring uniaxial motion. We have

succesfully measured the transfer function of the suspension using a novel high dynamic

range measuring technique.

Submitted to: Class. Quantum Grav.

1. Introduction

Long baseline interferometer gravity wave (GW) detectors are predicted to be limited

by three fundamental noise sources[1]. In the low frequency range (up to 10 Hz) seismic

noise will be the dominant noise source. In the frequency range from 10 Hz to 500 Hz

the detector is limited by the thermal noise in the suspension and test masses. Above

this frequency range the detectors are limited by quantum noise. The standard way

of predicting the thermal noise is to measure the Q-factor of the relevant system. By

making assumptions about the type of loss, the fluctuation-dissipation theorem can be

used to estimate the thermal noise spectrum[2]. Usually structual loss of the system is

assumed.

Various groups around the world (VIRGO[3]/LIGO[4]/ACIGA[5]) have projects

for the direct measurement of thermal noise. Such a measurement will include all

displacement noise in the system, and will provide a complete characterisation of the

thermal noise behaviour.

The aim of the thermal noise experiment at the ANU is to create a facility forthermal noise measurements. The first approach is to set up a simple system in which

thermal noise can be measured and to compare the results with theoretical predictions.

To this end we use a simple suspension system with a relatively high thermal noise level.

All other noise sources in the experiment must be suppressed below the thermal noise

level. The methods we developed to suppress laser frequency and intensity noise are

described in [6, 7] along with details of the readout system. Seismic noise is suppressed


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Thermal noise measurement technique 2

0 1 2 3 4-22

-20

-18

-16

-14

-12

-10

Frequency (log) [Hz]Displacem

entNoise(log)(m/Hz)

(a) (b)

180m thick flexure

incident

laser beam

Figure 1. (a) Design of the flexure suspension with incident laser beam coming

from the right on to the flat mirror. The dark part hinges on the four 180 m

thick flexures. (b) Predicted thermal noise spectrum of the flexure suspension with a

resonant frequency of 20 Hz.

by hanging the experiment from an isolator designed by researches at the University ofWestern Australia[8]. Acoustic noise is avoided by housing the experiment in a vacuum

( 106mbar).

In these proceedings we focus on the mirror suspension system of which the thermal

noise is to be measured. We employ a thin flexure similar to those proposed for use

in a future GW detector[9]. The flexure suspension transfer function is measured for

verification of the designed parameters. The transfer function was recorded in air using

a novel high dynamic range measurement technique. We discuss the design of the

suspension and report a measurement of its transfer function to verify it predicted

behaviour.

2. Flexure suspension design

Our experiment is focused around the simple uniaxial flexure suspension, illustrated in

fig. 1(a). Thermal noise induces position fluctuations of the flexure suspension. To

measure these position fluctuations the suspension is incorporated in a Fabry-Perot

test cavity of which the resonance condition is monitored. Attached to the flexure

suspension is the back mirror of the test cavity. The uniaxial nature of the suspension

aids in maintaining mirror alignment thereby eliminating the need for an autoalignment

system.With the avaliable laser power and readout system, a low resonant frequency

(f0 20 Hz) is required to have enough frequency range before the thermal noise

runs into the test cavity shot noise, 3 kHz. The flexure suspension is made out of a

piece of Beryllium Copper which has a width w of 15 mm, and a Young modulus Ey of

131 109 Pa. The length l of the flexures are 1 mm, and have a suspension height h of

10 mm. With a suspended mass of20 g, the flexure membrane thickness, d is related


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Laser

Split photodetector

Sum / difference

split detector signal

Fixed mirror

Flexure to test

/

(a)

0 0.5 1 1.5 2 2.5 3-1

-0.5

0

0.5

1

1.5

2

fringeposition

flexure position [A.U.]

Sumoutput

Subtractedoutput

(b)

Figure 2. (a) The Michelson impulse response experiment layout. The mechanical

flexure resonator is excited by hitting it with a small object. The Michelson output

is detected on a two element split photo detector connected to a digital oscilloscope

to record the ringdown response. (b) The sum (in-phase) and difference (quadrature)

Michelson output signal response.

to the resonant frequency by

d3 = 122

f2

0mlhEyw

(1)

setting f0 20 Hz results in a flexure thickness of 180 m. Note that it is assumed

that all four flexures are identical and parallel so that the entire system behaves like a

single flexure of a resonant frequency of 20 Hz.

3. The impulse response

To verify the actual flexure suspension parameters with the proposed design, we obtained

the transfer function from the impulse response of the flexure suspension.

The measurement was performed using a Michelson interferometer with one of theend mirrors being the flexure suspension, see fig. 2(a). A photo detector was placed

at the unused port of the beamsplitter. The flexure suspension was excited by hitting

it with a small object. To obtain the ringdown signal the intensity response can be

recorded from the output of the photo detector. Recording this response would allow

us to obtain the transfer function. The limiting factor in using the intensity response is

that the measurement can not track when the flexure displacement is more than half a

wavelength. This limits us to a short ringdown signal before it runs into the noise.

To overcome this we generated a zero crossing error signal for the arm length

difference of the interferometer and recorded this simultaneously with the intensity. Thiserror signal was 90 degrees out of phase with the intensity at photo detector as shown in

fig. 2(b). With this signal we tracked the suspension displacement over more than half

a wavelength. By taking the atan(error/intensity), where error and intensity are the

normalised error signal and interferometer output intensity respectively, the Michelson

phase difference was determined unambiguously. As a result there was no measurement


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limitation on the range of displacement that could be recorded. This allowed large

impulse responses to be obtained.

We obtained the error signal using a variant of the tilt locking technique[10]. By

introducing a small tilt in both interferometer mirrors we generated a small component

of the Gaussian TEM01 beam profile at the output. The Michelson interferometer output

was detected with a two element split photo detector, which had a sum and a difference

output. The detector was aligned so each half of the TEM01 mode was detected on one

element of the photo detector. The sum output provided us with the intensity response

and the difference output provided us with the error signal.

With this approach the recording length of the ringdown of the signals was

increased, which increased the dynamic range of the measurement. The measurement

was now only limited by the sampling rate of the digital oscilloscope and the linear

displacement range of the mechanical flexure resonator.

3.1. Transfer function measurement

After setting up the Michelson inteferometer the flexure suspension was excited and the

added and the subtracted signals from the split photo detector were recorded on a digitaloscilloscope. Both signals were downloaded and stored for analysis. From the ringdown

signal we obtained the resonant frequency and the Q-factor of the mechanical resonator.

In fig. 3(a) the recovered ringdown signal is shown indicating a total displacement of

4.5 m. A 30 kHz anti-aliasing filter was applied for both the sum and difference

signals before recording. Taking the Fast Fourier Transform of the ringdown generates

the transfer function, see fig. 3(b). As can be seen from the graph, the dynamic range

of this method is around 80 dB.

The transfer function had a Lorentzian profile. The measured response was curve

fitted with a theoretical profile from which the resonance frequency and the Q-factorwere extracted. The resonant frequency was found to be 185 Hz with a Q-factor of800.

The experiment was performed in air which sets a theoretical limit on the resonator Q-

factor at approximately 900[1].

The measured resonant frequency of 185 Hz was an order of magnitude higher

than the design resonant frequency. After closer examination of the four flexures it

was apparent that the individual flexures were not exactly parallel to each other. This

resulted in a much stiffer flexure, and hence a higher resonant frequency.

4. Conclusion

We measured the transfer function of the uniaxial suspension system to compare system

properties with design values. The results where recorded using a novel measurement

technique, where both the Michelson response and error signal were obtained. This

allowed the flexure suspension position to be measured over many wavelengths with

interferometric precision. The ringdown response was obtained with 80 dB of dynamic


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0 1 2 3 4-3

-2

-1

0

1

2

3

time [s]

displacement[m]

100

102

104

10-6

10-4

10-2

100

frequency [Hz]

transferfunction[A.U.]

(a) (b)

Figure 3. (a) The experimentally recorded ringdown of the mechanical flexure

suspension showing the flexure displacement of 4.5m. (b) The Fast Fourier

Transform of the ringdown response gives the transfer function. The dynamic range

of the measurement was around 80 db.

range thus allowing accurate determination of the transfer function. This technique is

generally applicable to other mechanical systems.

The measured resonant frequency of the flexure suspension was found to be

185 Hz, which was an order of magnitude above the design value due to non-parallelism

of the individual flexures.

An experiment is under development to measure the flexure suspension ringdown

response under a vacuum to determine whether the gas damping of the air interfered with

the current measurement results. With a good high dynamic range transfer function

measurement the thermal noise spectrum can be calculated direclty from the fluctuation-

dissipation theorem.

5. Acknowledgements

This research was performed under the auspices of the Australian Consortium

for Interferometric Gravitational Astronomy, supported by the Australian Research

Council.

References

[1] P. R. Saulson, Fundamentals of Interferometric gravitational wave detectors, World Scientific,

1994.[2] P. R. Saulson, Thermal noise in mechanical experiments, Phys. Rev. D 42, 2437 (1990).

[3] M. Bernardini it et al, Plane parallel mirrors Fabry-Perot cavity to improve Virgo

superattenuators, Phys. Lett. A, 243, (1998).

[4] A. Gillespie and R. Raab, Thermal Noise In the Test Mass Suspensions of a Laser Interferometer

Gravitational-Wave Detector Prototype,Phys. Lett. A, 178, 357-363 (1993).

[5] M. B. Gray, B. J. J. Slagmolen, K. G. Baigent and D. E. McClelland, The ANU Thermal Noise

Experiment, 3rd Amaldi Conference Proceedings, (1999).


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[6] B. J. J. Slagmolen, D. A. Shaddock, M. B. Gray and D. E. McClelland, Frequency stability of

spatial mode interference (tilt) locking, send of to JQE.

[7] B. J. J. Slagmolen, D. E. Shaddock, M. B. Gray and D. E. McClelland, Laser stabilization for

the interferometric measurement of the thermal noise of suspended test masses ,9th Marcel

Grossmann Meeting, Rome July 2000.

[8] J. Winterflood et al, in Proc. of 2nd TAMA International Worshop on Gravitational Wave Detection,

(Universal Academy Press, Tokyo, 1999).

[9] D. G. Blair, L. Ju, and M. Notcutt, Ultrahigh Q pendulum suspensions for gravitational wave

detectors, Rev. Sci. Intstrum., 54, 7, (1993)[10] D. A. Shaddock, M. B. Gray, and D. E. McClelland, Frequency locking a laser to an optical cavity

using spatial mode interference, Opt. Lett. 24, 21, (1999)

thermal noise measurement technique

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