Improving the angular detection sensitivity of a torsion

Improving the angular detection sensitivity of a torsion

Improving the angular detection sensitivity of
a torsion pendulum by an electrostatic spring
Y Z Bai, H Yin, L Liu, D Y Tan, Z B Zhou ([email protected])
Center for Gravitational Experiments, School of Physics, Huazhong Univ. of Science & Technology, Wuhan, China
An electrostatic torsion pendulum aiming at improving the angular detection sensitivity without increasing torque noise floor is presented. Theoretical analysis shows that it
could be used to release requirement of angular measurement, and is useful for gravitational experiments with much higher precision requirement. In this poster, the
principle of the electrostatic pendulum system is introduced, and its sensitivity and noise analysis are presented.

The power spectrum of the thermal fluctuation of the electrostatic pendulum, th,e2

i. Introduction
The torsion pendulum plays a paramount role in the field of precision measurement and
gravitational experiments due to its high-precision sensitivity. Such as investigating the performance
of a gravitational sensor and charge management for LISA.

k
I keff (1 i m ) th,e e,n
keff

keff km ke

2

V
k 0 f a ye (a 3 12a l 2 )
xe
xe e
3
e
6
d
e

th,e

2

4kBT
k m

(keff I 2 ) 2 km 2

Resolution of electrostatic torsion pendulum can be given by

H s,e ( )

1

k

I ikm
2

eff

2

min,e 2

eq
4kBTkm
2

e,n
2

H s,e ( )

Compare the electrostatic torsion pendulum with the typical balance as follows
Ground testing for LISA
(University of Trento)

Charge management
(University of Washington)

ii. A Typical Torsion Balance and Its Potential Sensitivity
A typical torsion balance is very sensitive to probe
force or torque which induced by weak signals, and its
resolution is limited by the thermal noise of the
pendulum and the readout noise, namely, the angular
detection level.

at low frequency: <<0 H ( ) 2 H ( ) 2 s s,e 2 2 min,e min at high frequency: >>0

H ( ) 2 H ( ) 2
s
s,e

2
2

min,e
min

iv. Example
The torsion
pendulum presented by
Washington University to study the charge
effects for gravitational-wave is used as an
example to be discussed, whose main
parameters are listed in Table I, and a
couple electrodes are added as add
additional electrostatic spring, whose
parameters are listed in Table II.

The torsion balance is set in a high vacuum chamber,
where viscous damping can be ignored. The motion
equation of the system in this case can be written as
Figure 1

I km (1 i ) th

where is the structure loss angle, th is a random force with a white spectral density,
and is presented as follows

th 2 4kBTR

R Re[ Z ( )]
Z ( ) iI k i k
m
m

th 2

4kBT
k m
( k m I 2 ) 2 k m 2

The power spectrum of the minimum detectable torque can be obtained as follows

n 2 th 2 eq 2

The schematic of the electrostatic pendulum is shown in
Figure 2, where one pair of electrodes are added in sided
of the test mass, where de is the distance between the test
mass each electrode, le is the vertical distance from the
centre of the electrode to the fiber, Vf is the voltage
applied on the electrodes to adjust the performance the
system, and Se=axe aye is the area of each electrode.

Assuming the angle measurement
noise is 510-8 (1+10-2/f)1/2 rad/Hz1/2.
Resolutions of the torsion pendulum
system with or without an
electrostatic spring are shown in
Figure 3. The electrostatic noise of
electrostatic pendulum induced by
fluctuation of voltage 10 V/Hz1/2 is
within 10-15 Nm/Hz1/2, which can be
neglected. If the requirement of
torque detection is 510-15 Nm/Hz1/2
above 0.1mHz, the requirement of
the angle measurement device is
shown in Figure 4.

The motion equation of the electrostatic torsion
pendulum is given by

Conclusions:

H s ( )

1

k

m

min 2

I 2 ikm

eq 2

2

4kBTkm

2
2

H s ( )
H s ( )

n

iii. Electrostatic Torsion Pendulum

I km (1 i ) th,e e e,n

The electrostatic spring can release the
requirement of the angular detection of a
torsion pendulum, which is very important
for the much higher precise torsion
pendulum experiments, such as to
investigate the effects of a test mass for
LISA and advanced LISA projects.

Figure 2(a)

where e is the electrostatic torque, and e,n is the
random torque induced by applied voltage
fluctuations.

e

0Vf2 a ye
2

1
le axe
2
1
le axe
2

x
x

dx

2
2
d x d x
e
e

e,n 2 0 axe a ye leVf Vf ,n / de2

Figure 3

References:

Figure 2(b) Top view

Figure 4

1.M Hueller, A Cavalleri et al, Torsion pendulum facility for ground testing of gravitational sensors for LISA,
Class. Quantum Grav. 19 (2002)1757-1765.
2.S.E.Pollack, M.D.Turner, S.Schlamminger, et al, Charge Management for gravitational-wave observatories using
UV LEDs, Phys. Rev. D. 81, 021001(R)(2010).
3.Q. L. Wang, H. C. Yeh, Z. B. Zhou et al, Improving the sensitivity of a torsion pendulum by using an optical
spring method, Phy. Rev. A, 80, 043811(2009)

For more information: [email protected]

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