博士論文
Doctoral Dissertation

Search For
Vacuum Magnetic Birefringencewith a high repetitive pulsed magnet（高速繰り返し磁石を用いた真空複屈折の探索）
令和 2 年 12 月博士（理学） 申請
A Dissertation Submitted for the Degree of Doctor of Philosophy

December 2020東京大学大学院理学系研究科
物理学専攻
Department of Physics, Graduate School of Science,
The University of Tokyo

上岡修星
Shusei Kamioka

Abstract
This theses reports search for the vacuum magnetic birefringence (VMB) witha high repetitive pulsed magnet. VMB is an anisotropy of the refractive index of
the vacuum in the magnetic ﬁeld. VMB is predicted to be induced by the virtualelectron-positron pair in the vacuum. The light new particles which could couple
to 2 photons, including ALPs and MCPs, could also induce VMB. VMB is a goodprove for the precision test of QED and search for the new physics, but has notbeen observed yet.
The strongest limit was obtained by using the rotating permanent magnets. In
that experiment, the unexpected noise which is not correlated with the magnetic
ﬁeld was observed, and it is found that stronger magnetic ﬁeld is required to observe
the VMB. One of the solutions is the application of the pulsed magnet. In theprevious experiment with the pulsed magnet, the eﬀect of the disturbance of themagnet to the optical system was observed, and the slow repetition rate of themagnet resulted in the low statistics.
In this thesis, we proposed the experiment aiming for the ﬁrst observation of
VMB induced by QED with the high repetitive magnet and the high sensitive
optical anisotropy detection system. The experimental setup is designed to isolate
the optical system from disturbance of the magnet. The operation ﬁeld of thepulsed magnet is 8.2 T and repetition rate is 0.05 Hz, which is 25 times faster thanthe previous experiment. The dedicated study is performed for the study of the
intrinsic birefringence noise of the optical system at higher frequency region up
to 1 kHz, which is important for the experiment with pulsed magnet, and where
few studies has been reported. The sensitivity√ of the optical anisotropy detection−19
system is reached to around 1 × 10
[m/ Hz]. The sensitivity is about 5 times
better than the sensitivity around 10 Hz which is used in the previous experimentwith the permanent magnets. With the developed setup, the search for VMB withthe pulsed magnet was performed. With total 26000 pulses, no signiﬁcant signal
was observed. The obtained limit is|kCM | < 1.8 × 10−20 [T−2 ].

(1)

The strongest magnetic ﬁeld and the fastest repetition rate of the pulsed magnet
as VMB search has been achieved. The statistic is about 100 times larger than theprevious experiment with the pulsed magnet. This result shows the establishmentof the VMB search with the repetitive pulsed magnet.

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Contents
1 Introduction
1
1.1 Nonlinear interaction of photons in vacuum . . . . . . . . . . . . . . 1
1.2 Vacuum Magnetic Birefringence (VMB) . . . . . . . . . . . . . . . . 2
1.3 VMB mediated by new physics beyond the standard model . . . . . 4
1.3.1 Axion and ALPs . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.2 Milli-Charged Particles . . . . . . . . . . . . . . . . . . . . . 6
1.4 Previous experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.1 Laboratory experiments . . . . . . . . . . . . . . . . . . . . . 8
1.4.2 Astrophysical observations . . . . . . . . . . . . . . . . . . . . 9
1.5 Aim of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 VMB search with pulsed magnet
2.1 Overview of the experimental method . . . . . . . . . . . . . . . . .
2.2 Polarimetry in this measurement . . . . . . . . . . . . . . . . . . . .
2.3 Signal enhancement with a Fabry-Pérot Cavity . . . . . . . . . . . .
2.4 Frequency response of the Fabry-Pérot cavity . . . . . . . . . . . . .
2.5 Noise Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Pulsed magnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Basic properties and pulse shape . . . . . . . . . . . . . . . .
2.6.2 Repetition rate . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Candidates of background . . . . . . . . . . . . . . . . . . . . . . . .
2.7.1 Cotton-Mouton eﬀect of residual gas . . . . . . . . . . . . . .
2.7.2 Cotton-Mouton eﬀect of mirrors . . . . . . . . . . . . . . . .
2.7.3 Faraday eﬀect of the residual gas and mirrors . . . . . . . . .
2.8 Target speciﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Experimental setup
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Optical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Overview of optical system . . . . . . . . . . . . . . . . . . .
3.2.2 Master laser . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Fabry-Pérot cavity . . . . . . . . . . . . . . . . . . . . . . . .
3.2.4 Frequency feedback system . . . . . . . . . . . . . . . . . . .

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3.2.5 Auto locking system . . . . . . . . . . . . . . . . . . . . . . .
3.2.6 Polarizer and Analyser . . . . . . . . . . . . . . . . . . . . . .
3.2.7 Mirror chambers . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.8 Signal detector . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.9 Input intensity stabilization . . . . . . . . . . . . . . . . . . .
Evaluation of the sensitivity to the optical pass length diﬀerence . .
3.3.1 Measurement of the phase retardation of the mirror . . . . .
3.3.2 Sensitivity measurement . . . . . . . . . . . . . . . . . . . . .
3.3.3 Evaluation of known noise source . . . . . . . . . . . . . . . .
3.3.4 Mirror rotation dependence . . . . . . . . . . . . . . . . . . .
3.3.5 Comparison of known noise source between the measured
value and target speciﬁcation . . . . . . . . . . . . . . . . . .
3.3.6 Discussion of the measured sensitivity . . . . . . . . . . . . .
Pulsed magnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Single race-track magnet . . . . . . . . . . . . . . . . . . . . .
3.4.2 Dipole magnet . . . . . . . . . . . . . . . . . . . . . . . . . .
Magnet operation system . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Overview of the magnet operation system . . . . . . . . . . .
3.5.2 Magnet bench and liquid nitrogen container . . . . . . . . . .
3.5.3 Charging unit and the capacitor bank . . . . . . . . . . . . .
3.5.4 Current transformer . . . . . . . . . . . . . . . . . . . . . . .
3.5.5 Liquid nitrogen serving system . . . . . . . . . . . . . . . . .
3.5.6 Control board . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.7 Evaluation of the eﬀect of the disturbance . . . . . . . . . . .
Other components . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.1 Electromagnetic noise reduction . . . . . . . . . . . . . . . .
3.6.2 Laser displacement sensor . . . . . . . . . . . . . . . . . . . .
3.6.3 Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.4 Data logger . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Run and analysis
4.1 Summary of data acquisition . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Run time . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Run cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Data acquisition cycle . . . . . . . . . . . . . . . . . . . . . .
4.2 Analysis method . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Calculation of the ellipticity . . . . . . . . . . . . . . . . . . .
4.2.2 Calculation of the signal shape . . . . . . . . . . . . . . . . .
4.2.3 Parameter estimation . . . . . . . . . . . . . . . . . . . . . .
4.2.4 Time and frequency windows . . . . . . . . . . . . . . . . . .
4.3 Validation run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Summary of the measurement . . . . . . . . . . . . . . . . . .
4.3.2 Parameter estimation . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Measurement of the pressure dependence . . . . . . . . . . .

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3.3

3.4

3.5

3.6

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4.4

4.5

4.3.4 Systematic uncertainties . . . . . . . . . . . . . . . . . . . . .
4.3.5 Comparison with previous experiment . . . . . . . . . . . . .
4.3.6 Mirror Rotation dependence . . . . . . . . . . . . . . . . . . .
4.3.7 Conclusion of the validation run . . . . . . . . . . . . . . . .
Vacuum run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Summary of the measurement . . . . . . . . . . . . . . . . . .
4.4.2 Cancellation of the noise on It . . . . . . . . . . . . . . . . .
4.4.3 Parameter estimation . . . . . . . . . . . . . . . . . . . . . .
4.4.4 Systematic uncertainties . . . . . . . . . . . . . . . . . . . . .
Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Discussion and prospect
96
5.1 Discussion of the experimental result . . . . . . . . . . . . . . . . . . 96
5.1.1 Signiﬁcance of this work . . . . . . . . . . . . . . . . . . . . . 96
5.1.2 Comparison with previous experiment . . . . . . . . . . . . . 96
5.2 Upgrade for future experiment . . . . . . . . . . . . . . . . . . . . . 97
5.2.1 Upgrade of the magnet . . . . . . . . . . . . . . . . . . . . . 98
5.2.2 Upgrade of the ﬁeld length . . . . . . . . . . . . . . . . . . . 99
5.2.3 Improvement of the repetition rate . . . . . . . . . . . . . . . 100
5.2.4 Gain on kCM from the upgrade of the magnet system . . . . 100
5.2.5 Upgrade of a Fabry-Pérot cavity . . . . . . . . . . . . . . . . 100
5.2.6 Reduction of the systematic uncertainty . . . . . . . . . . . . 102
5.2.7 Improvement of the analysis procedure . . . . . . . . . . . . . 102
6 Conclusion

104

A Optical cavity and PDH method
105
A.1 Fabry-Pérot cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
A.1.1 Transmittance and reﬂectance . . . . . . . . . . . . . . . . . . 105
A.1.2 Finesse, free spectrum range and photon lifetime . . . . . . . 107
A.2 PDH method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
A.2.1 a basic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
A.2.2 Basic properties of PDH method . . . . . . . . . . . . . . . . 110
B Birefringent Fabry-Pérot Cavity
112
B.1 Mirrors as a birefringence material . . . . . . . . . . . . . . . . . . . 112
B.2 resonance frequency splitting . . . . . . . . . . . . . . . . . . . . . . 113
B.3 Frequency response of a Fabry-Pérot cavity against the ellipticity . . 115
C Charging unit
119
C.1 Basic speciﬁcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
C.2 Operation sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
iv

D Discussion of the possible birefringence noise source
122
D.1 Evaluation of the correlation with other possible noise source . . . . 122
D.2 Comparison with thermal noise . . . . . . . . . . . . . . . . . . . . . 125
D.3 Comparison with the lower ﬁnesse conﬁguration . . . . . . . . . . . . 126
D.4 Comparison with previous experiment . . . . . . . . . . . . . . . . . 128
D.5 Summary of the noise excess . . . . . . . . . . . . . . . . . . . . . . 128
E Thermal noise130
E.1 General consideration . . . . . . . . . . . . . . . . . . . . . . . . . . 130
E.2 Eﬀect of thickness ﬂuctuation . . . . . . . . . . . . . . . . . . . . . . 131
E.2.1 Brownian noise . . . . . . . . . . . . . . . . . . . . . . . . . . 131
E.2.2 Thermoelastic noise . . . . . . . . . . . . . . . . . . . . . . . 131
E.3 Eﬀect of birefringence ﬂuctuation . . . . . . . . . . . . . . . . . . . . 133
E.3.1 Thermorefracrive eﬀect . . . . . . . . . . . . . . . . . . . . . 133
E.3.2 Photoelastic eﬀect of longitudinal strain . . . . . . . . . . . . 133
E.3.3 Photoelastic eﬀect of transverse strain . . . . . . . . . . . . . 134
E.3.4 Photo-thermo-photo-elastic eﬀect . . . . . . . . . . . . . . . . 137
Acknowledgements

139v

List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
1.7

Box diagram of QED . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Contribution of ALPs . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Summary of the exclusion region of ALPs by terrestrial experiments
7
Experimental setup of PVLAS experiment with the superconducting
magnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Experimental setup of PVLAS experiment with the permanent magnet 9
Experimental setup of BMV experiment . . . . . . . . . . . . . . . . 10
Limit on kCM obtained by the previous experiments. . . . . . . . . . 10

2.1
2.2
2.3
2.4
2.5
2.6

Schematic of the experimental setup . . . . . . . . . . . . . . . . . .
Basic setup of polarimetry . . . . . . . . . . . . . . . . . . . . . . . .
Transfer function of Fabry-Pérot cavity again VMB signal . . . . . .
Schematic view of the magnet system . . . . . . . . . . . . . . . . .
Example of the square of the magnetic by pulsed magnet . . . . . .
Noise spectrum of the design sensitivity . . . . . . . . . . . . . . . .

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26

Schematic view of the experimental setup . . . . . . . . . . . . . . .
Picture of the experimental room . . . . . . . . . . . . . . . . . . . .
Schematic view of the input optics . . . . . . . . . . . . . . . . . . .
Schematic view of the output optics . . . . . . . . . . . . . . . . . .
Picture of the input optics . . . . . . . . . . . . . . . . . . . . . . . .
Picture of Mephisto . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measurement of the ﬁnesse . . . . . . . . . . . . . . . . . . . . . . .
Picture of the mirror mount . . . . . . . . . . . . . . . . . . . . . . .
Schematic view of the frequency feedback system for a Fabry-Pérot
cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.10 Residual frequency noise estimated from in loop error signal . . . . .
3.11 Schematic view of an auto locking system . . . . . . . . . . . . . . .
3.12 Designed gain of the circuit at diﬀerent locking stage . . . . . . . . .
3.13 Input chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.14 Output chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.15 Picture of signal photo detector . . . . . . . . . . . . . . . . . . . . .
3.16 Evaluation of input intensity stability . . . . . . . . . . . . . . . . .
3.17 Measurement of the phase retardation of the mirrors . . . . . . . . .

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3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9

vi

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3.18 Measurement of the phase retardation of the mirrors under diﬀerencealignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.19 Voltage signal of Ie and Ie . . . . . . . . . . . . . . . . . . . . . . . . 41
3.20 Power spectrum of measured noise . . . . . . . . . . . . . . . . . . . 42
3.21 Power spectrum of the estimated possible contribution of known
noise source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.22 Block diagram of feedback system . . . . . . . . . . . . . . . . . . . 44
3.23 Transfer function from Vext to Ve . . . . . . . . . . . . . . . . . . . . 44
3.24 Power spectrum of measured α ﬂuctuation . . . . . . . . . . . . . . . 46
3.25 Setup of the birefringence measurement without the Fabry-Pérot cavity 47
3.26 Noise spectrum of optical pass diﬀerence without cavity . . . . . . . 48
3.27 Noise spectrum of optical pass diﬀerence at diﬀerence rotationalalignment of the mirror . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.28 Comparison between designed value and measure value of known
noise source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.29 Comparison between measured noise and the estimated expected
noise spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.30 Schematic view of the single racetrack magnet . . . . . . . . . . . . . 53
3.31 Picture of the single racetrack magnet . . . . . . . . . . . . . . . . . 54
3.32 Measured timing proﬁle of the magnetic ﬁeld . . . . . . . . . . . . . 55
3.33 Measurement of the ﬁeld length . . . . . . . . . . . . . . . . . . . . . 56
3.34 The ﬁeld map the longitudinal magnetic ﬁeld . . . . . . . . . . . . . 56
3.35 Schematic view of the dipole magnet . . . . . . . . . . . . . . . . . . 57
3.36 Picture of the dipole pulsed magnet . . . . . . . . . . . . . . . . . . 58
3.37 Measured magnetic ﬁeld shape . . . . . . . . . . . . . . . . . . . . . 59
3.38 Measurement of the ﬁeld length . . . . . . . . . . . . . . . . . . . . . 60
3.39 Schematic view of the magnet system . . . . . . . . . . . . . . . . . 60
3.40 Setup inside the liquid nitrogen container . . . . . . . . . . . . . . . 61
3.41 Picture of the current transformer . . . . . . . . . . . . . . . . . . . 62
3.42 Schematic view of liquid Nitrogen serving system . . . . . . . . . . . 62
3.43 Picture of control board . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.44 Evaluation of the stability of the Fabry-Pérot cavity against theoperation of the pulsed magnet . . . . . . . . . . . . . . . . . . . . . 64
3.45 Position dependence of the intensity noise of Ie . . . . . . . . . . . . 65
3.46 Schematic view of vacuum system . . . . . . . . . . . . . . . . . . . 67
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8

Time chart of the DAQ cycle . . . . . . . . . . . . . . . . . . . . . .
Expected signal shape . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of the power spectrum densities . . . . . . . . . . . . . . .
Average of the error signal . . . . . . . . . . . . . . . . . . . . . . . .
Determination of the upper frequency window . . . . . . . . . . . . .
Determination of the lower frequency window . . . . . . . . . . . . .
Example of the measured 1st data of validation run . . . . . . . . . .
Example of the measured 2nd data of validation run . . . . . . . . .
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4.9 Example of the parameter estimation . . . . . . . . . . . . . . . . . .
N2
4.10 Distribution of kCM. . . . . . . . . . . . . . . . . . . . . . . . . . . .
N2
4.11 Pressure dependence of kCM. . . . . . . . . . . . . . . . . . . . . . .
4.12 The ﬁeld map of the transverse direction along the beam path (simulation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.13 Static ellipticity for each measurement . . . . . . . . . . . . . . . . .
4.14 Cotton-Mouton eﬀect at diﬀerent rotational alignment . . . . . . . .
4.15 Cotton-Mouton eﬀect at diﬀerent rotational alignment normalizedby Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.16 Example of the waveform of 1st data in the vacuum run . . . . . . .
4.17 Example of the noise on It . . . . . . . . . . . . . . . . . . . . . . . .
4.18 Distribution of Itspike . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.19 Example of the parameter estimation . . . . . . . . . . . . . . . . . .
4.20 Distribution of kCM . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.21 Distribution of kCM . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.22 Beam path dependence of the ﬁeld map . . . . . . . . . . . . . . . .

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5.1
5.2
5.3
5.4

Limit on |kCM | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Limit on ALPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Schematic of the upgrade . . . . . . . . . . . . . . . . . . . . . . . . 99
Cross section of the magnet . . . . . . . . . . . . . . . . . . . . . . . 101

A.1
A.2
A.3
A.4

Schematic of a Fabry-Pérot cavity . . . . . . . . . . . . . . . . . . . 105
Transmitted power from a Fabry-Pérot cavity as a function of phase 106
Schematic of the PDH method . . . . . . . . . . . . . . . . . . . . . 109
Example of PDH signal . . . . . . . . . . . . . . . . . . . . . . . . . 109

B.1 Example of the signal attenuation by birefringence of mirrors . . . . 114
C.1 Picture of the charging unit . . . . . . . . . . . . . . . . . . . . . . . 120
C.2 Schematic view of the charging unit . . . . . . . . . . . . . . . . . . 121
D.1 Example of the displacement noise and coherence spectrum . . . . . 123
D.2 Coherent spectrum between displacement noise and birefringence noise123
D.3 Coherence spectrum of the birefringence noise and pointing noise of
the reﬂected light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
D.4 Setup for the pointing ﬂuctuation measurement . . . . . . . . . . . . 125
D.5 Coherence spectrum between birefringence noise and measured pointing noise of the mirror . . . . . . . . . . . . . . . . . . . . . . . . . . 126
D.6 Comparison between measured noise spectrum and thermal noise
calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
D.7 Optical pass length diﬀerence measurement with lower ﬁnesse cavity
and noise budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
D.8 Comparison with previous experiment . . . . . . . . . . . . . . . . . 128

viii

List of Tables
1.1

Summary the experimental parameters of the previous experiment . 11

2.1
2.2
2.3

Coeﬃcient of Cotton-Mouton eﬀect of each gas . . . . . . . . . . . . 24
Parameters of the optical system . . . . . . . . . . . . . . . . . . . . 25
Parameters of the magnet . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1
3.2
3.3
3.4
3.5

Speciﬁcation of the signal photo detector . . . . . . . . . . . . . . . .
Speciﬁcation of the single coil magnet . . . . . . . . . . . . . . . . .
Speciﬁcation of the dipole magnet . . . . . . . . . . . . . . . . . . .
Speciﬁcation of the current transformer . . . . . . . . . . . . . . . .
Speciﬁcation of the absolute pressure gauge . . . . . . . . . . . . . .

36
54
58
61
66

4.1
4.2
4.3
4.4
4.5
4.6
4.7

Summary of the data set . . . . . . . . . . . . . . . . . . . . . . . . .
Summary of the typical parameters during validation run . . . . . .
Systematic uncertainties . . . . . . . . . . . . . . . . . . . . . . . . .
N2
kCM
measured by various experiment . . . . . . . . . . . . . . . . .
Summary of the typical parameters during vacuum run . . . . . . .
Example of the Γ dependence of the various eﬀect . . . . . . . . . .
Systematic uncertainties . . . . . . . . . . . . . . . . . . . . . . . . .

68
77
84
84
89
94
95

5.1
5.2

Summary of the experimental parameter . . . . . . . . . . . . . . . . 98
Summary of the gain on kCM by the upgrade of the magnet system.
It is assumed that the observed birefringence noise of the FabryPérot cavity is the same after the upgrade. . . . . . . . . . . . . . . 101

C.1 Speciﬁcation of the charging unit . . . . . . . . . . . . . . . . . . . . 119
E.1 Material parameters of the mirror . . . . . . . . . . . . . . . . . . . . 132
E.2 Material parameters of the mirror . . . . . . . . . . . . . . . . . . . . 132
E.3 Material parameters of the mirror . . . . . . . . . . . . . . . . . . . . 132

ix

Chapter 1

Introduction
This thesis describes a search for Vacuum Magnetic Birefringence (VMB) using ahigh-ﬁnesse Fabry-Pérot cavity and a high repetitive strong pulsed magnet. VMB
is an anisotropy of the refractive index of the vacuum in a magnetic ﬁeld. This
unique property of the vacuum is predicted by quantum electrodynamics (QED).
In addition to that, new particles which could couple to 2 photons including AxionLike-Particles (ALPs) and Milli-Charged Particles (MCPs) also contribute to VMB.
In this section, a theoretical background of VMB based on the nonlinear electrodynamics is shown ﬁrstly. The contribution from the new physics is also describedin Section. 1.3. Previous experiments for VMB search are shown secondly. Finally,
the aim of this thesis is described.

1.1

Nonlinear interaction of photons in vacuum

In the classical electromagnetism where superposition principle holds, the light can
not interact with each other in vacuum. This is not the case when considering thequantum eﬀect. According to QED, the vacuum is ﬁlled with virtual particle andanti-particle pairs which exists very shot time. The interaction between photons
could occur mediated by the virtual electron-positron loop in the vacuum. This
interaction is expressed by a ”box diagram” in Fig. 1.1. In 1930, an eﬀective
Lagrangian of electrodynamics including this nonlinear interaction was derived [1].
The Lagrangian is called Eular-Heisenberg Lagrangian and expressed as follows,
 p


Re
cosh
es
2
(F
+
iG)
1
ds
2
 p
 G − (es)2 F − 1 ,LEH = −F− 2
exp(−m2e s) (es)28π 0 s3
3
Im cosh es 2 (F + iG)
(1.1)
1
µν
2
2
where me is the mass of electron and F = Fµν F = 2 (E −B ) and G = Fµν F̃ µν =12 (E · B). When the electromagnetic ﬁeld is much smaller than critical ﬁeld, EularZ ∞



1

2

1.2 Introduction

Heisenberg Lagrangian can be further approximated as follows,
1
LEH = (E 2 − B 2 ) + Ae [(E 2 − B 2 )2 + 7(E · B)2 ],
2

(1.2)

where a coeﬃcient Ae is
Ae =2α2 ℏ3= 1.32 × 10−24 [T−2 ].
45µ0 m4e c5(1.3)

The electromagnetic ﬁeld should satisfy the weak ﬁeld condition which is deﬁned
as follows,m2e c3= 4.4 × 109 [V/m],eℏ
m2 c2
B ≪ Bcr = e = 1.3 × 1018 [T].
eℏE ≪ Ecr =

(1.4)
(1.5)

The ﬁrst terms of Eq. (1.2) is identical to the classical electromagnetic Lagrangian which leads to classical Maxwell equation. The last term of Eq. (1.2)
is understood as the correction to the classical electromagnetism. It contains thenonlinear term of electromagnetic ﬁeld. This Lagrangian indicates even when theelectromagnetic ﬁeld are small and there is no medium, there exists nonlinear eﬀectof the electromagnetism in the vacuum.
This Lagrangian predicts a lot of unique properties of the vacuum at macroscale including birefringence of the vacuum which is discussed later, dichroism ofvacuum [2], diﬀraction of the light in the vacuum [3] and four wave-mixing in thevacuum [4]. However, so far, no one of them have been observed.

1.2

Vacuum Magnetic Birefringence (VMB)

The refractive index of the vacuum taking into account the quantum eﬀect canbe derived by using Eular-Heisenberg Lagrangian. The electric ﬂux density andmagnetic ﬁeld can be calculated as follows,
∂LEH
,
∂E
∂LEH
H=−
.
∂B
D=

(1.6)
(1.7)

The permittivity tensor ϵ and permeability µ of the medium is deﬁned as follows,
∂D
,
∂E
∂B
µ=
.
∂H
ϵ=

(1.8)

3

1.2 Introduction

Figure 1.1: Box diagram of QED. The virtual electron-positron pair mediates theinteraction between four external photons.
The refractive index tensor is calculated from these values as
√
ϵµ
n= √
,
ϵ0 µ0

(1.9)

where ϵ0 and µ0 is the permittivity and permeability of classical vacuum. Refractive
index is the dimensionless number which describe the eﬀective speed of the light inthe medium.
In 1960s, the refractive index of the vacuum under a slowly varying external
magnetic ﬁeld was calculated [5]. The refractive index of the vacuum is not only
shifted from 1, but also have an anisotropy. This is called Vacuum Magnetic Birefringence (VMB). The refractive index parallel to the external magnetic ﬁeld (n∥ )and the refractive index perpendicular to the external magnetic ﬁeld (n⊥ ) become
as follows,n∥ = 1 + 7Ae B 2 ,(1.10)n⊥ = 1 + 4Ae B 2 .

(1.11)

The amount of the anisotropy (∆n ) is expressed as∆n = n∥ − n⊥ = 3Ae B 2= 4.0 × 10−24 × (B [T])2≡ kCM × (B [T])2 ,(1.12)

where B is the magnetic ﬂux density of eternal magnetic ﬁeld which is perpendicular
to the light pass.

4

1.3 Introduction

This eﬀect is understood as the magnetization of the vacuum. VMB is originally predicted in 1960s. Since then a lot of experiment including the laboratory
experiments and astronomical observations were performed to observe VMB, but
it is not observed yet. In Section 1.4, the summary of the previous laboratory
experiments is described.

1.3

VMB mediated by new physics beyond the standard model

In addition to the birefringence induced by the virtual electron-positron loop, particles which could couple to 2 photos could also induce VMB as they could intermediate the interaction between light and magnetic ﬁeld. Their contribution couldchange the value of VMB from QED calculation. Therefore, measurement of VMB
is also sensitive to new physics predicted beyond the standard model.

1.3.1

Axion and ALPs

Axion is the pseudo scalar particle which is introduced to solve the strong CP
problem in QCD. The CP violating terms of QCD appears as following. One isoriginated from the quark mass matrix, and the other is originated from the vacuum
structure of QCD. Thus it is natural that the CP of QCD is violated, but so far
it is measured that the CP of QCD is not violated at least with the precisionof 10−10 [6, 7]. This ﬁne-tuning problem is called as a strong CP problem. To
solve this problem, it is proposed to introduce an additional global U(1) symmetry
called UPQ (1) to the standard model [8]. This additional symmetry acts to choose
preferred vacuum where CP of QCD is not violated. The Axion is a pseudo NambuGold Stone boson generated by the spontaneous breaking of the additional U(1)symmetry [9, 10]. The Axion acquires mass mA by the mixing with π 0 .
 10

√
mπ fπ
10 [GeV]zmA =∼ 0.6 meV ×,
fA 1 + z
fA(1.13)

where fA is the symmetry breaking scale of the UPQ (1), mπ is the pion mass, fπ isupion’s decay constant, and z ≡ m
md is the quark mass ratio between u and d [11].
This π 0 -mixing introduce the coupling to 2 photons,
1LAγγ = − gAγγ Fµν F̃µν A,
4(1.14)

where gAγγ is the coupling constant between the axion and photon. It is written
as follows, 10α
10 [GeV]−13
gAγγ =Cγ ∼ 10×,(1.15)
2πfA
fA

5

1.3 Introduction

Figure 1.2: Feynman diagram which expresses the contribution of the ALPs to
VMB. The ALPs mediated the interaction between light and magnetic ﬁeld.
where α is ﬁne structure constant. Cγ is the model dependent constant which is
written as
24+z
E
−
,
(1.16)Cγ =N
31+zwhere E is the electromagnetic anomaly and N is the color anomaly. Dependingon the model, Cγ changes between 0.07 and 7. This interaction mediates a mixingof axions and photons in external electromagnetic ﬁelds. It is referred to as thePrimakoﬀ eﬀect [12].
Axion-Like-Particles (ALPs) are pseudo Nambu-Goldstone bosons. ALPs are
predicted by a lot of theories beyond the standard model [13]. In addition tothe original QCD axion, they generally arise from the spontaneous breaking ofadditional global symmetries. ALPs are predicted in many models including supersymmetric theories, string theories, and conformal standard models [14–16].
Generally, ALPs could couple to 2 photons. The interaction can be expressedas1Laγγ = − gaγγ Fµν F̃µν a(1.17)
4
where a is ALPs whose mass is ma and gaγγ is coupling constant between ALPs
and photon. There is no speciﬁed relation between ALPs’s mass and their coupling
between photons. Therefore, ALPs search are conducted to cover whole ma − gaγγ
plane.
As ALPs couple to 2 photons, they could contribute to VMB via a processexpressed in Fig. 1.2 [17]. From the diagram, the birefringence induced by ALPs
is calculated as follows,


2
gALPs
B2
sin2x
∆nALPs =1−,(1.18)
2x
2m2ALPs
L m2

where x is deﬁned as x = B 4ωALPs . LB is the length of the magnetic ﬁeld, and ω isthe angular frequency of the electric ﬁeld. Fig. 1.3 shows the summary of the previous ALPs search by various terrestrial experiments around 10−4 < ma < 1 eV.

6

1.3 Introduction

Around these mass region, the other popular method of ALPs search is LightShinning through the Wall (LSW) method [18]. LSW method measures the numberof photons which pass through the photon-shielding wall. The photons producedby the external light source are injected to the photon shielding wall and they areconverted to a real ALPs by the magnetic ﬁeld. ALPs can pass though the wall as
they could interact with matters very weakly. After ALPs pass through the wall,they are re-converted to photon by magnetic ﬁeld and then detected by the photon
detector after the wall. The major diﬀerence between LSW and VMB as ALPs
search is that LSW has the sensitivity to real ALPs where the birefringence could
be induced by real and virtual ALPs. This diﬀerence results in the diﬀerent massdependence of the ALPs sensitivity. Because of this diﬀerence, VMB search has
the better sensitivity to ALPs at around 10−2 < ma < 1−1 eV.
Another terrestrial experiment searching for the ALPs around these mass regions
is search for the four-wave mixing in the vacuum [19]. In this experiment, thephoton-photon scattering of the pulse laser mediated by the ALPs is searched.
The resonance of the cross section of the photon-photon scattering via s-channel
exchange of ALPs is the signal. The signal is also enhanced by injecting anotherstimulated laser to the collision point of the laser. This experiment has the sensitivity in the narrow region around 0.1 eV which is determined by the energy ofthe lasers. The advantage of ALPS search with the VMB experiment against thismethod is that VMB has sensitivity against ALPs in the larger mass range.
The stringest limits at around 10−2 < ma < 1−1 eV so far is mainly obtainedby PVLAS experiment which is one of the VMB search experiments [18]. The
red line in Fig. 1.3 is the obtained limit when VMB sensitivity reaches the QED
expectation value which is the benchmark sensitivity of VMB experiment.

1.3.2

Milli-Charged Particles

Milli-Charged particles are non-quantized small charged particle. A lot of models
beyond the standard model have additional U(1) symmetry. The charge of thisnew symmetry is referred to as paracharge. In general, it is assumed that the SMparticles don’t have paracharge, thus the particles under additional U(1) symmetry
don’t interact directly with ordinary SM particles. From this reason, it is called”hidden sector” where such particles assumed to be resided, whereas our world iscalled ”visible sector” [24]. The only interaction between hidden sector and visible
sector is realised by a mixing of U(1) gauge boson of each sector with mixing angle
χ. Particles which has paracharge g in hidden sector can be seen as a particle
which have an electric charge β = χg from visible sector [25]. These particles in
the hidden sector are called Milli-Charged Particles (MCPs).
MCPs are searched in the parameter space spanned by the mass and charge[26]. They are constrained by a lot of experiment including direct searches [27–29],
indirect observations [30], the CMB observation [31]. As MCPs could also interact
with photons, they contribute to VMB [32]. The contribution depends on whether
they are fermion or boson. In each case, VMB induced by MCPs are calculated as

7ga [GeV-1]

1.4 Introduction

10−3
10−4
10−5
10−6
VMB QED background
PVLAS
Inada et. al (LSW)
NOMAD (LWS)
ALPS (LSW)OSCAR (LSW)
Nobuhiro et. al (FWM)

−7

10

10−8
10−4

10−3

10−2

10−1

1
ma [eV]

Figure 1.3: Summary of the excluded region of ALPs by terrestrial experiments.
The red line is the QED background of VMB method. Black: exclusion region
by PVLAS experiment [18]. Exclusion regions by various LSW method (NOMAD
experiment [20], ALPS experiment [21], OSCAR experiment [22], Inada et. al [23]),and the exclusion regions by four wave mixing (Nobuhiro et. al [19]) are also shown.
follows,

∆nB =


3

2


 − 2 Aβ B
2135 π 1/2 21/3 Γ 3




 28
Γ 1

2
χ− 3 Aβ B 2
4for χ ≪ 1

(1.19)for χ ≫ 1

(1.20)for χ ≪ 1

(1.21)for χ ≫ 1.

(1.22)

6

∆nF =


2


 3Aβ B2 2135 π 1/2 21/3 Γ 3


−


14
Γ 1

χ− 3 Aβ B 246χ and Aβ are expressed asχ=3ℏωβeBℏ

3 ,
F
2 mB,
c4
β

Aβ =

2β 4 α2 ℏ34 ,F
45µ0 mB,c5
βwhere B denotes boson and F denotes fermion.

(1.23)

8

1.4 Introduction

1.4

Previous experiments

1.4.1

Laboratory experiments

So far, a lot of experiments to observe VMB have been performed. They are
categorized by the type of magnets to induce VMB. As discussed in Section 2, thesensitivity of VMB search depends on square of the transverse magnetic ﬁeld andﬁeld length. Thus, requirements for the magnet are long ﬁeld length and strongtransverse magnetic ﬁeld. It is also required to modulate the magnetic ﬁeld toseparate VMB signal from static birefringence of materials. In this section, theprevious experiments are summarized. The obtained limit of kCM in the previous
experiment is shown in Fig. 1.7 , and the experimental parameters are summarizedin Table. 1.1.
VMB searches with a superconducting magnet (∼ 2007)
VMB search with superconducting magnet had performed by PVLAS experiment [33, 34]. Figure 1.4 shows a schematic view of the experimental
setup [35]. They use 5.5 T and 1.0 m superconducting magnet. To modulate the magnetic ﬁeld, the cryostat for the superconducting magnet was
rotated at 0.3 Hz. This experiment ended up with the observation of systematic signals appeared at the signal frequency and limited measurement
time by availability of liquid helium for the magnet. So far, the origin of thesystematic signal has not been understood [35].
VMB searches with a permanent magnet (2009 ∼ 2016)
VMB search with permanent magnet has also been performed by PVLAS
experiment [36]. Figure 1.5 shows the schematic view of the experimental
setup [35]. They use 2.5 T and 1.6 m permanent dipole magnet. Magnets arerotated mechanically around 5 Hz and signal appears at the second harmonics
of the rotation frequency. The advantage of their method is they do not need
any cooling apparatus to operate the magnet. High duty rate and less noisefrom the magnetic system can be expected.
From 2014 to 2016, they performed VMB search with total live time of 3
months. The obtained limit is about 10 times above the QED calculation [35].
This experiment has achieved the best sensitivity for VMB. The sensitivity
is limited by the unexpected wide band noise excess. They concluded thebirefringence noise is induced by the optical system. [37].
VMB searches with a pulsed magnet (2007 ∼)A VMB search with pulsed magnet has been performed by BMV experiment[38]. Figure 1.6 shows the schematic view of the experimental [38]. The ﬁled
length is 0.13 m and the peak magnetic ﬁeld is 6 T. The repetition rate of
their magnet is 6 pulse/hour. The advantage of this method is that stronger
magnetic ﬁeld compared to other method can be used and the signal would
appear at higher frequency where less birefringence noise can be expected.

9

1.4 Introduction

Figure 1.4: Experimental setup of PVLAS experiment with the superconducting
magnet [35]. The ﬁled length is 1.0 m and the peak magnetic ﬁeld is 5.5 T. The
magnet is rotated around 0.3 Hz.

FigureExperimental
setup
of PVLAS-LNL
PVLAS experimentpermanent
magnet [35].
Figure1.5:
15: Top:schematic drawingof thesetup. Bottom:witha photographof the apparatus
above
the
oor
level.
The
lower
optical
bench
is
below
the
beam
supportingthecryostat
and
sits
about
m
The ﬁled length is 1.6 m and the peak magnetic ﬁeld is 2.5 T. The magnet is3rotatedbelow ground level on the concrete ‘raft’.
around 10 Hz.
46

The disadvantage is that the pulsed magnet generates large acoustic andmechanical noise. They observed the birefringence noise due to the acoustic
Figure 19: Schematic view and photograph of the optical bench layout of the PVLAS-FE apparatus.
noise
inducedplate;by PthepulsedAmagnetic
ﬁeld.= quarter-wave plate; TR = transmission;
HWP= Half-wave= Polariser;= Analyser; QWP
EXT = extinction.

general scheme of theobservations
optical setup and a photograph of the apparatus are shown in
1.4.2 AAstrophysicalFigure 19. A Nd:YAG laser (Innolight Mephisto, 2 W power) emitted at = 1064 nm.
beam rst
passed througha quarter-wave
plate(QWP)soreducing
ellipticity
The Thestrongest
magneticﬁeld inthe universe
known
far is thetheinitial
magnetor
whoseof the laser beam. A rst half-wave
plate (HWP) placed before a two stage Faraday isolator
8magnetic ﬁeld is typically 10 T. As magnetors also radiate X-ray or γ-ray, theallowed the adjustment of the power being injected into the Fabry-Perot cavity. The beamthen passed through a lens to match the laser waist with the cavity waist for optimal mode
matching. Two steering mirrors followed by a second HWP brought the beam to the entranceof the vacuum system with the desired alignment and polarisation direction. Between thesecond steering mirror and this second HWP a glass window allowed the sampling of there⇥ected power from the cavity for phase locking the laser to the cavity via the Pound-eported absolute values of the vangence and their uncertainties re-10of the laser frequency. It is then injected into a monomodeoptical ⇥ber before entering a high ⇥nesse Fabry-Pérot cavity of length Lc = 2.27 m, consisting of the mirrors M1
and M2 . This corresponds to a cavity free spectral rangeof ⇤FSR = c/2Lc = 65.996 MHz. The laser passes throughan electro-optic modulator (EOM) creating sidebands at
10 MHz. We analyze the beam re ected by the cavity on
the photodiode Phr . This signal is used to lock the laser frequency to the cavity resonance frequency using
the
1.4 Introduction
Pound-Drever-Hall method [12], via the acousto-optic modulator and the piezoelectric and Peltier elements of thelaser.

version of the PVLAS appararmanent magnets and a Fabryhed a noise oor correspondingat 3 , but ”only when no spu[8]. All over our paper, we giveonding to a con⇥dence level ofates an evidence for a non-zeroments are summarized in Fig. 1.
vacuum CM measurements are
es and that one has to focus not
optical sensitivity and maximi- Figure 2. Experimental setup. EOM, electro-optic moduured, but also on minimizing all lator ; AOM, acousto-optic modulator ; PDH, Pound-Drever; Ph, photodiode; P,
polarizer; A, analyzer.
SeeﬁledtextlengthforFigure1.6: Hall
Experimental
setup ofBMV
experiment[38]. Theis 0.13 me⇤ects by decoupling theappamoredetails. ﬁeld is 6 T. The repetition rate is 6 pulse/hour.
and
the
peak
magnetic
nd by performing an appropriate
BMV (pulsed magnet)|kcm [T-2]| (95% C.L.)

−17
To10
measurethe ellipticity inducedPVLAS
by(permanent
the Cotton-Mouton
magnet)nt a measurement of kCM obtai- e⇤ect one−18needs a transverse magnetic⇥eld as high as posPVLAS-LNL (supercondicting magnet)
ion setup of the BMV (Biréfrin- sible. 10
This is ful⇥lled using pulsed ⇥elds
delivered by one
QED expected value−named19e) experiment at the National magnet,
X-coil,especiallydesignedin our labora10
boratory of Toulouse, France - tory. The principle of this magnet and its properties are
10−20in details in Refs. [9, 13]. It can provide a maxity of this experiment is the use describedThis method allows to provide mum ⇥eld
−21 of more than 14 T over an equivalent lengths in terrestrial laboratories wi- LB of 10
0.137 m [10]. Data have been taken with a maxiitself [4]. Our apparatus is also mum magnetic⇥eld of 6.5 T reached within 1.70 ms while
10−22
ared Fabry-Pérot cavity among the total−durationof a pulse is less than 10 ms as shown in
23[10]. We calibrated our expe- Fig. 3.10
Moreover, we can remotely switch the high-voltage[10], and recently published a connections
in order to set it parallel or anti10−24 to reverse B20102014 repetition
2016
nt of the Cotton-Mouton e⇤ect parallel to the x 2008
direction. The2012
maximumrate[Year]
with the theoretical prediction is 6 pulses per hour.
acquisition and analysis proceWe infer the cavity ⇥nesse from the measurement of theunt the symmetry properties of photon lifetime ⌅ [10]. Its value is regularly checked during
t to the orientation of Figure
the ma-1.7: data
andobtained
we get by⌅ =the1.07
ms. Thecorresponding
Limittaking
on kCM
previous
experiments.
95% conﬁdenceof the cavity birefringence.
is : by the previous experiments
levelTheof kCM⇥nesse
obtainedisshownchronologically.
The⇧c⌅e vacuum magnetic linear
bireF
=
,
(7)
detail of each experiment are explained in Chapter 1.4.
Lcn this paper was obtained withs and a maximum ⇥eld of 6.5 T. We get F = 445 000 with a relative variation that does notmeasurementof thepolarizationof the radiationfrom corresponds
the magnetors
t noise oor ever reached.
It is exceed
6%
at the 3 con⇥dence
level. Thisto can also ben of our innovative experimentalacavitylinewidth
⇤⌃
=
c/2F
L
of
148
Hz.
This
is
one
of
used to search for VMB. In 2016, it is reported
that from the measurementof thec
sharpest
infraredcavityin the world [10].
polarizationthe
froma magnetor
called
RX-J1856.5-3754,
the evidence for VMB has
observed [39]. They measured the polarization of the radiation whose wavelength is555 nm and calculate its polarization degree averaged by the phase. The measured
polarization degree is 16.43% ± 5.26%. They compered the result with the theoretical models. It is concluded that the observed polarization degree favours themodel including VMB. However, other group pointed out that depending on the11

1.5 Introduction

Table 1.1: Summary the experimental parameters of the previous experimentType of the magnetGroup
2 [T2 ]
Beﬀ
Field length [m]Signal frequency
Statistics

Superconducting
PVLAS
30, 5.3
1.0
0.6 Hz63500 sPermanent
PVLAS
6.5
1.6
∼ 10 Hz60 days

Pulse
BMV
20
0.137
∼ 100 Hz0.3 s (100 pulses)

model of the radiation from the magnetor, the observed result can be reproduced
even when the eﬀect of VMB is not included [40].

1.5

Aim of this work

The science target of this experiment is the ﬁrst observation of VMB with highrepetitive strong pulsed magnet. As the demonstration of the principal of theexperimental method, the aims of this work are summarized as following.
Establishment of the VMB search with high repetitive pulsed magnet
From the result of the previous experiment and the magnetic ﬁeld dependence
of dependence of VMB, it is clear that stronger magnetic ﬁeld is necessary
to observe to the VMB induced by QED. One of the solutions is the pulsed
magnet. The world record of the peak magnetic ﬁeld of the pulsed magnet ismore than 80 T [41].
The previous experiment reported the eﬀect of the disturbance to the opticalcavity due to the pulsed magnet. The development of the optical system
dedicated to the vibration reduction and suppression of the disturbance tothe optical system, and establishment of the stable operation of the pulsed
magnet with the optical system is a purpose of this work.
Another merit of this experiment is the application of the high repetitive
pulsed magnet to the VMB search. The disadvantage of the pulsed magnet
is its slow repetition rate due to the joule heating. To overcomes this disadvantage, a compact and short pulse magnet is selected to reduce the joule
heating. The target repetition rate is 0.05 Hz which is 30 times faster thanthe previous experiment. As the sensitivity scales with square root of thenumbers of generated pulses, this repetition rate gives 5 times better sensitivity than the previous experiment even when the magnetic ﬁeld and ﬁeld
length are the same.
Investigation of the birefringence noise at higher frequency
PVLAS experiment observes unexpected birefringence noise [37] which is notcorrelated with the magnetic ﬁeld. The origin of the noise is not understood.

12

1.5 Introduction
So far, the study of the birefringence noise is mainly performed low frequency
typically up to 25 Hz, which is important for the experiment with permanent
magnet. Above the frequency, only limited study has been reported [42]. In
the experiment with pulsed magnet, the noise level around several hundreds
hertz becomes important for the sensitivity. In this experiment, the dedicatedstudy of the evaluation of the birefringence noise up to ∼ 1 kHz is performed.
The frequency response of the possible noise source in the homodyne detection
scheme are derived theoretically taking the birefringence of the mirrors, andits contribution is evaluated experimentally. To obtain the new knowledge ofthe birefringence noise at higher frequency is another aim of this work.

Chapter 2

VMB search with pulsed
magnetThis section describes the experimental method of VMB search with pulsed magnets. First, the basic idea of polarimetry to measure the birefringence is introduced.
After that, a Fabry-Pérot cavity is introduced as optical pass multiplier and thebasic ideas of a pulsed magnet is introduced. Finally, the possible noise source
and the target sensitivity for the observation of VMB eﬀect predicted by QED is
discussed.

2.1

Overview of the experimental method

Figure 2.1 shows the schematic view of the VMB search with a pulsed magnet. A
pulsed magnet is used to induce VMB by a strong magnetic ﬁeld. The anisotropyof the refractive index is measured as a polarization change (ellipticity) of the lighttraveling through the magnetic ﬁeld region. A polarizer and an analyzer are placedwith cross-nicol conﬁguration to measure the polarization change. The light whichpass through the analyzer is called as the extraordinary light (Ie ) and which do notpass is called the ordinary light (It ). Ellipticity is deﬁned as the ratio between Ie
and It . The polarization angle of the polarizer is ﬁxed to 45◦ against the direction of
the magnetic ﬁeld. A Fabry-Pérot cavity which made with 2 high reﬂective mirrors
is placed outside the magnetic ﬁeld region. It is used as an optical pass length
multiplier to enhance the interaction region. The enhancement is proportional to
the factor called as ﬁnesse F . It is also used to generate a static ellipticity Γ. The
ellipticity is written as follows,
Ie /It = Γ2 + 2Γ

2F LB B 2 (t)
kCM ,
λ

(2.1)

where kCM expresses the coeﬃcient of VMB which is deﬁned in Eq. 1.12. In thefollowing section, the detailed discussion of the experimental method is described.

13

14

2.2 VMB search with pulsed manget

2.2

Polarimetry in this measurement

The basic idea of birefringence measurement in this experiment is to measure thepolarization change of the laser with homodyne detection. The basic setup of thismethod is shown in Figure 2.2. One pair of a polarizer and an analyzer are placedwith Cross-Nicol conﬁguration. Between the polarizer and the analyzer, the target
birefringence material where birefringence has time dependent and a wave plate
whose properties is already known is placed. The angle between polarization axisof the polarizer and the fast axis of the birefringence material is ﬁxed to 45◦ andthe angle between the polarization axis of the polarizer and the fast axis of the
wave plate is ﬁxed to θ.
To calculate the change of the polarization, Jones calculus is used. A Jones
vector expresses the phase and amplitude of the electric ﬁeld in 2-dimensional
space. The each component of the Jones vector expresses the electric ﬁeld of two
orthogonal polarization states. The Jones matrices are the operators which acts on
the Jones vectors. The Jones matrices express the operation of the optical elements
against the electric ﬁeld. X-axis is deﬁned parallel to the polarization axis of thepolarizer and input laser travels along z-axis.
The Jones vector of the input laser after the polarizer Ein is
 
1
.
(2.2)
Ein =
0
Pulsed magnet
Length: L
magnetic field B(t)n⊥IeAnalyzer
Polarizern||Photo detector

It

I0
Fabry-Pérot cavity
Length: Lcav
Finesse: F
Static ellipticity: ΓFigure 2.1: Overview of the experimental setup to measure VMB. A strong pulsed
magnet is used to induce large ∆n. A Fabry-Pérot cavity is used to enhance theeﬀective interaction length and to induce the static birefringence. From the output
power of the analyser, the ellipticity is measured.

15

2.2 VMB search with pulsed magnet

Waveplatephase retardation: α45°θAnalyzer

input laser

Ie

It

Polarizer
target material
birefringence: ΔnLength: L

Figure 2.2: Basic setup of the polarimetry for this experiment. The target material
whose birefringence is ∆n, and length is L is placed between polarizers. A waveplateis used to induce the static birefringence. The power ratio of the output of theanalyser is used to measure the ellipticity.
By assuming the phase retardation of the waveplate is α, the Jones matrix of the
wave plate Wwp is −iα/2 2ecos (θ) + eiα/2 sin2 (θ)−isin(α/2)sin(2θ)
.
(2.3)
Wwp =
−isin(α/2)sin(2θ)
e−iα/2 sin2 (θ) + eiα/2 cos2 (θ)
Assuming the length of the target material is L and its birefringence is ∆n(t),the phase retardation Ψ(t) can be written as
Ψ(t) = π∆n(t)L,
λ(2.4)

where λ is a wavelength of input laser. Then the Jones Matrix of the birefringencematerial X is1−iΨ(t)X=.
(2.5)−iΨ(t)
1
The analyser can separate the light whose polarization is perpendicular to thepolarization angle of analyser, called extraordinary light Ee , and whose polarization
angle is parallel to the polarization angle, called ordinary light Et .
The Jones matrix from the input port of the Analyser to output port for the
extraordinary light is


0 0
Ae =
.
(2.6)
0 1
The Jones matrix from the input port of the Analyser to output port for theordinary light is


1 0
At =
.
(2.7)
0 0

16

2.3 VMB search with pulsed manget

By using these matrices, the amplitude of ordinary and extraordinary light can becalculated. Hereafter, it is also assumed that 1 ≫ α ≫ α2 sin(2θ) ≫ Ψ. The Jones
vector of the ordinary light is
Et = At Wwp XEin

1 − i α2 cos(2θ)
∼
.
0

(2.8)

The amplitude of the extraordinary light is
Ee = Ae Wwp XEin0=iΨ(e−iα/2 cos2 (θ) + eiα/2 sin2 (θ)) + isin(α/2)sin(2θ)


0
∼
.
iΨ + i α2 sin(2θ)(2.9)

The ellipticity is deﬁned as the ratio of the power of extraordinary light Ie andordinary light ItIeα(t) = [ sin(2θ) + Ψ(t)]2It2∼ Γ2 + 2ΓΨ(t)L= Γ2 + 2Γπ ∆n(t).
λ(2.10)In the second line
2 of the above expression, the static ellipticity is deﬁned asα2
Γ = 2 sin(2θ) . This expression shows that the linear term of the birefringence appears as a time-dependent term of the measured ellipticity. In the case of
VMB measurement, from the measured ellipticity during applying magnetic ﬁeld,kCM can be calculated as
kCM =

2.3

Ie
λ
(t).
2
2πΓLB (t) It(2.11)

Signal enhancement with a Fabry-Pérot Cavity

In this section, it is shown that a Fabry-Pérot cavity can enhance the polarization
change. The basic properties and the deﬁnition of term about a Fabry-Pérot cavity
is explained in appendix A.
The diﬀerence from the previous section is that 2 high reﬂective mirrors are placed
before and after the target material.
Again, by using Jones matrix, ellipticity can be calculated. The amplitudetransmittance and reﬂectivity of 2 mirrors are t and r respectively, and the length17

2.3 VMB search with pulsed magnet

between 2 mirrors is Lcav . The Jones matrix to express the trip from one mirror
to the other is expressed as
L

R = rei2πν c I(2.12)≡ reiϕ I,

where ϕ is an acquired phase for each trip between 2 mirrors and ν is a frequencyof the input laser. The Jones matrix which represent the input to the ﬁrst mirror
to the second mirror is expressed as
Tin = teiϕ I.

(2.13)The Jones matrix which represents the output from the second mirror is expressedasTout = tI.

(2.14)The Jones matrix of the light transmitted from the second mirror after n-th roundtrip between 2 mirrors is Tout (RX)2n XTin . The output amplitude from the second
mirror is a sum of all n-th round tripped light. Therefore, the amplitude of theordinary light is
Et = At Wwp Tin Tout∞
X(RX)2n XEinn

= At Wwp t e (I − (RX)2 )−1 XEin
2 iϕt2 eiϕ∼ At Wwp
1 − r2 ei2ϕ=−ir2 2Ψ
1−r2 ei2ϕ1−ir2 2Ψ
1−r2 ei2ϕ

t2 eiϕ(1 − α2 sin(2θ))
1−r2 ei2ϕ

0

1

!

!
Ein

(2.15).

When a Fabry-Pérot cavity is on the resonance, ϕ should fulﬁll the condition of
ϕ = mπ (m ∈ Z). Thus the amplitude of the ordinary light during the resonance is
!
α
t2
(1
−
sin(2θ))
2
2
1−r
Et =
.
(2.16)
0
The amplitude of the extraordinary light when cavity is on the resonance is∞
X(RX)2n XEin
Ee = Ae Wwp Tin Tout
n

=

t2
1−r2



0
2

Ψ
i 2r
+ i2sin(2θ)
1−r2



!

(2.17).

18

2.4 VMB search with pulsed manget

The acquired ellipticity isIeα
2r2 Ψ(t) 2(t) = [ sin(2θ) +] .
It
2
1 − r2(2.18)As discussed in Appendix A, Finesse is deﬁned as
F ∼

πr2
.
1 − r2(2.19)

By substituting this expression and Γ to Eq. (2.18), the ellipticity with a FabryPérot cavity is derived as follows,Ie2F(t) = [Γ +Ψ(t)]2Itπ2F∼ Γ2 + 2Γ
Ψ(t)
π
2F L
∆n(t).
= Γ2 + 2Γπλ(2.20)

Compared this result with the ellipticity without a Fabry-Pérot cavity, the ellipticity becomes 2F
π times larger. This can be understood as the incident light travels
eﬀectively Fπ roundtrips between 2 mirrors before it comes out from the second
mirror. The eﬀective length of the target material becomes 2F
π times longer.
With the manufactured mirror, the ﬁnesse more than 300,000 is established,thus the expected enhancement of the ellipticity signal is more than 200,000.
It should be noted that the real mirrors have a small birefringence [43] and it is used
to generate the static birefringence Γ. Due to the birefringence of the mirrors, thediscussion about polarimetry with a Fabry-Pérot cavity is needed to be modiﬁed.
The detailed discussion is shown in Appendix B.

2.4

Frequency response of the Fabry-Pérot cavity

In this section, the response of VMB signal against the time-dependent magnetic
ﬁeld is discussed.
The output of the extraordinary electric ﬁeld of Fabry-Pérot cavity at the given
time t is the sum of the extraordinary electric ﬁeld generated before t.

Eeout (t) = itm


X
2L
2L
2 iω 2Ljc(rm e) × 2Ψ(t −j)Et (t −j) .
cc(2.21)jwhere Et (t) is electric ﬁeld of ordinary light inside the cavity at t, and 2ψ(t −
2L
2L
t
j 2L
c )E (t − c j) is the amplitude of the extraordinary light generated at t − j c .
rm and tm is the reﬂectivity and transmittance of the mirror.
2L
When cavity is on a resonance, eiω c j = 1 and Et (t) = Et (t − 2L
c j). By
performing Fourier transformation to Ψ(t), the above equation is re-written as

19

Gain

2.5 VMB search with pulsed magnet

105

104
1

102
103
frequency [Hz]10

Figure 2.3: Example of the transfer function of Fabry-Pérot cavity against VMB
signal when ﬁnesse is 300,000 and cavity length is 1.5 m.

Z
2 j
Eeout (t) = iEt tm Σ(rm) ×Z

= iEt tm ×
∼ iEt
Z
∝

dΩ2Ψ(Ω)e−i(t−j c )Ω

dΩ2Ψ(Ω)e−itΩ ×tm×
2
1 − rmZ2L

12 e−i c Ω
1 − rm1dΩ2Ψ(Ω)e−itΩ ×2
rm
1 + 1−r2 i 2L
c Ω
2L

(2.22)m1dΩΨ(Ω)e−itΩ .
Ω
1 + i πc
2LF

The last line can be interpreted that Fabry-Pérot cavity acts as a 1st order low-passﬁlter to the VMB signal and its cutoﬀ frequency is fc = 4Fc L . As an example, thetransfer function of Fabry-Pérot cavity to the generated birefringence inside thecavity is shown in Fig. 2.3 when F = 300,000 and L = 1.5 m.
This calculation should be modiﬁed to take into account the mirror birefringenceof the mirrors. The detailed discussion is shown in Appendix B.

2.5

Noise Budget

In this section, possible noise sources are discussed quantitatively. Hereafter, we
discuss the sensitivity about optical pass diﬀerence between 2 polarization which

20

2.5 VMB search with pulsed manget

is the unit of the length. The optical pass length diﬀerence is deﬁned as∆Lψ ≡ L × ∆n =λ Ie
2F 2ΓIt(2.23)As the relative optical pass diﬀerence between 2 polarization is concerned, ﬂuctuations of the absolute length of the optical cavity does not become the noise source.
This is an advantage of this method, as cavity length ﬂuctuation by thermal noise
would not appears as noises in this experiment unlike other optical experiment.
The main noise sources come from the shot noise, the frequency noise, theintensity noise and the detector noise.
Shot noise
The ﬂuctuation of the current on a photodiode for the signal detector obeys
Poisson distribution depending on the DC current on the photo diode. Thisis called shot noise. When laser intensity to the photodiode is I, electric
current on the photodiode is i = qI where q [A/W] is the eﬃciency of thephotodiode. In this situation, the ﬂuctuation of the current ∆i isp√∆i = 2eqI [A/ Hz],(2.24)
where e is elementary charge.
By inserting this expression to Lψ , the optical pass length diﬀerence noisedue to the shot noise can be calculated as,
∆Lψ =

λ
∆Ie
×
2F
2ΓIt
∆ie

λ
q
×
2F
2ΓIt
p
2eΓ2 It /q
λ
=
×
2F
2ΓIt
r
√
e
λ
=
×
[m/ Hz].
2F
2qIt=(2.25)

Intensity noise
The ﬂuctuation of intensity inside the Fabry-Pérot cavity could also becomea noise source. When we denote the relative intensity noise of the ordinary
light as RINIt , the contribution of the intensity noise to the optical passlength diﬀerence is written as follows when the interested frequency region is
below the cut-oﬀ frequency of the Fabry-Pérot cavity,
λ
∆It
Ie
×
×
2F
2ΓIt
It
λ
RINIt × Ie
=
×
2F
2ΓIt
λ
RINIt
=
×Γ×
.
2F
2

∆Lψ =

(2.26)

2.6 VMB search with pulsed magnet

21

Frequency noise√
As discussed in Appendix B.2, the frequency noise ν Hz/[ Hz] appears as
the intensity noise of the extraordinary light when the Fabry-Pérot cavity hasbirefringence. Below the cut-oﬀ frequency of the Fabry-Pérot cavity, it canbe written as
4 Fπ αν∆Ie /Ie =,(2.27)
×F
2
νFWHM
1 − ( π α)where α [rad] is the phase delay during the one round trip inside the cavity
due to the birefringence of the mirrors, and νFWHM is the FWHM of theresonance curve of a Fabry-Pérot cavity.
The contribution of the frequency noise to the birefringence noise can be
written as follows,
λ
Ie
∆Ie
×
×
2F
2ΓIt
Ie
4 Fπ α
λ
Γ
ν
=
× ×
×
F
2
2F
2
νFWHM
1 − ( π α)∆Lψ =(2.28)

Detector noise√
When the signal of the photodetector has the current noise ∆iPD A/[ Hz],
this could also become the noise source. Below the cut-oﬀ frequency, its
contribution can be written as follows,
∆Lψ =

λ
∆iPD
×
2F
2qΓIt

(2.29)The design sensitivity of this experiment is calculated and discussed in Section 2.8.

2.6

Pulsed magnet

In this section, the principle of the pulsed magnet [44] and its advantages for the
VMB experiment is discussed.

2.6.1

Basic properties and pulse shape

Figure 2.4 shows the equivalent circuit of the pulsed magnet system. A pulsed magnet is consisted with the resistance and the inductance. The magnet is connectedto the capacitor through a switch. When switch is closed, the electric charge stored
in the capacitor ﬂows to the pulsed magnet. By solving the circuit equation, thecurrent ﬂows in the magnet becomes
r
C −Rt
t
e 2L sin √Icoil = V0.
(2.30)LLC

22

2.7 VMB search with pulsed manget
Pulsed magnet
L

R

V0
SW

Capacitor bankC

Figure 2.4: Schematic view of the magnet system. It is consisted of a pulsed magnet
and a capacitor bank.
At the coil, the magnetic ﬁeld proportional to the Icoil is generated.
Bcoil = geﬀ Icoil ,(2.31)

where geﬀ [T/A] is the ﬁeld eﬃciency. Field eﬃciency depends on the detailed
structure of the coil. Typically, the circuit is designed so that the current ﬂows in
one direction. Thus, the magnetic ﬁeld becomes pulsed shape rather than damped
oscillation. The signal shape is proportional to the square of the pulsed magnetic
ﬁeld when ignoring the cutoﬀ of a Fabry-Pérot cavity. A typical shape and spectrumof the square of the magnetic ﬁeld are shown in Fig. 2.5. The assumed parameters
are C = 6 mF and L = 80 µH. In the previous experiment with rotating magnets,
the VMB signal appears at around 10 Hz. Compared with that, the VMB signal
with the pulsed magnet has higher frequency components. Thus, the sensitivity
is less aﬀected by the optical pass length diﬀerence noise at the lower frequency
components where the observed unexpected noise becomes larger in the previous
experiment. This is ﬁrst advantage of the pulsed magnet.
The maximum magnetic ﬁeld of the pulsed magnet are limited by the destruction of the coil by the Maxwell stress. For a single Cu-wire coil, the maximum
magnetic ﬁeld is roughly calculated to be 20 T. This magnetic ﬁeld cannot be obtained by other DC magnets. Stronger magnetic ﬁeld can be obtained by using
other wire materials. This is the second advantage of the pulsed magnet.

2.6.2

Repetition rate

As the pulsed magnet has the non-zero resistance, the joule heat is generated at
the same time with the magnetic ﬁeld. This heat generation ends up with theincrease of the temperature of the coil and the increase of the resistance. Too hightemperature leads to the destruction of the coil. Magnets are needed to be cooled
down during the pulse intervals. This time scale limits the repetition rate. Typical
repetition rate of the pulsed magnet is 1 mHz.

23

1.2

power spectrum of B2 [a. u.]

B2(t) [a. u.]

2.7 VMB search with pulsed magnet

1
0.8
0.6
0.4
0.2
0
−2

−1

0

1

2

3
4
time [ms]

1
10−1
10−2
10−3
1

10

102
103
frequency [Hz]Figure 2.5: Example of square of the magnetic ﬁeld with pulsed magnet. Assumed
parameter is C = 6 mF and L = 80 µH. Left: Signal shape in time domain. Right:
Signal spectrum in frequency domain.

2.7

Candidates of background

In Section. 2.5, the intrinsic noise of the optical system is discussed. In addition tothat, the candidates of background induced by the magnetic ﬁeld is also exist. As
they occurred at the same time with the magnetic ﬁeld, they could be fake VMB
signals. These background sources give other requirements to the design of theexperimental setup.

2.7.1

Cotton-Mouton eﬀect of residual gas

It is known that under magnetic ﬁeld, gases become birefringence materials. Thisis called Cotton-Mouton eﬀect [45]. The birefringence induced by Cotton-Mouton
eﬀect of gas under magnetic ﬁeld B is expressed as follows,
gas 2
∆ngas
CM = kCM B × P.

(2.32)

gas
where P [Pa] is the partial pressure of the gas and kCM[T−2 Pa−1 ] is the coeﬃcient of Cotton-Mouton eﬀect. As this eﬀect has the same magnetic dependencewith VMB, it is diﬃcult to distinguish Cotton-Mouton eﬀect of gas from VMB.
Therefore, the pressure around the magnetic ﬁeld region is important to suppress
these eﬀects low enough.
gas
In Table 2.1, kCM
of each gas and required partial pressures to suppress its
Cotton-Mouton eﬀect measured by 1064 nm laser is shown [36, 46–48]. CottonMouton eﬀect is used for calibration of the apparatus for every VMB experiment[36, 48–50].

24

2.8 VMB search with pulsed manget

Table 2.1: Coeﬃcient of Cotton-Mouton eﬀect of each gas and partial pressure
where ∆ngas = ∆nvac
name of gas
N2
O2
H2 O
He
Ar

2.7.2

gas
kCM
[T−2 P a−1 ]−2.0 × 10−18 [46]
1.8 × 10−17 [46]
6.7 × 10−20 [47]
2.1 × 10−21 [48]7.0 × 10−20 [36]partial pressure where ∆ngas = ∆nvac
1.6 × 10−6
1.6 × 10−7
6.7 × 10−5
1.9 × 10−3
5.7 × 10−5

Cotton-Mouton eﬀect of mirrors

It is known the mirror itself shows Cotton-Mouton eﬀect and its value is 10−9 rad[51]. To suppress the Cotton-Mouton eﬀect of the mirrors, the magnetic ﬁeld at
the mirror should be small enough. This gives the requirement of the of leakage
ﬁeld at the mirror should satisﬁes the following relation,r
B 2 × LB
B(mirror) <,(2.33)
108
where LB is the length of the magnetic ﬁeld.

2.7.3

Faraday eﬀect of the residual gas and mirrors

It is known that the longitudinal magnetic ﬁeld also interacts with gases or mirrors,
and it results the rotation of the polarization of the incident light. This eﬀect iscalled Faraday rotation. The induced rotation can be expressed in the case of gasesasΘ = kFgas B∥ × P.
(2.34)
where kFgas T−1 Pa−1 is called the Verdet constant. It is known that the mirrorsof a Fabry-Pérot cavity also shows the faraday rotation. As Faraday rotation
is proportional to the longitudinal magnetic ﬁeld, by applying magnetic ﬁeld of
opposite signs, its eﬀect can be canceled. Because the square term of the faraday
rotation has the same magnetic ﬁeld dependence, it is diﬃcult to distinguish its
eﬀect from VMB. Therefore, suﬃciently small longitudinal magnetic ﬁeld and low
pressure is required.

2.8

Target speciﬁcation

From the previous discussions, the target speciﬁcation of this experiment is calculated. The ﬁrst physics target of this experiment is the observation of the VMB
induced by QED, thus the target speciﬁcation is designed to observe the eﬀect of25

2.8 VMB search with pulsed magnet

QED with reasonable data acquisition time. The design speciﬁcation of the opticalsystem is summarized in Table 2.2. The target speciﬁcation of the magnet system
is summarized in Table 2.3. Fig. 2.6 shows the design sensitivity of optical passlength diﬀerence. The power spectrum of the signal shape is superimposed with
arbitrary unit for comparison. The frequency response of each noise source is discussed in Appendix B. From these spectrums, the sensitivity of VMB per one pulse
can be calculated. The target sensitivity is∆kCM = 1.3 × 10−21 [T−2 /pulse].

(2.35)

Assuming repetition rate of 0.05 Hz, the sensitivity reaches QED prediction with20 days’ run.
Table 2.2: Summary of the design parameters of optical system. From these parameters, the noise budget is calculated.
parameter namewavelength λﬁnesse F
FWHM νFWHM
Output power It
Static birefringence Γphase delay of the cavity αFrequency noise ν
Relative Intensity noise RIN
detector noise ∆iPD
eﬃciency of photo diode q

size [unit]
1064 [nm]500,000
100 [Hz]
10 [mW]
0.003
2 [µrad]√
1 [mHz/√Hz]
10−6 [1/ Hz]√
10 [fA/ Hz]
0.5

Table 2.3: Summary of the design parameters of magnet.
parameter namePeak magnetic ﬁeld
ﬁeld length
pulse width
repetition rate

size [unit]
20 [T]1 [m]
2 [msec]
0.05 [Hz]26

2.8 VMB search with pulsed manget

shot noise
detector noise
intensity noise
frequency noise
signal spectrum [a.u.]

Optical passlength difference [m/ Hz]

10−18
10−19
10−20
10−21

10−22
10−23
1

10

102
103
frequency [Hz]Figure 2.6: Power spectrum density of the designed sensitivity. The parameters
are summarized in Table 2.2. The signal spectrum is drawn for comparison.

Chapter 3

Experimental setup
The development of the optical system, the magnet and the establishment of thestable operation of the magnet with the optical system are the key technologies ofthis experiment. This section describes the detail of the developed experimental
setup.
First, the optical system including the Fabry-Pérot cavity and the polarizers aredescribed. The intrinsic noise level of the optical system is evaluated. After thatthe speciﬁcation of the pulsed magnet and the detail of the magnet operation
system are shown. The rest of this chapter explains the other component of theexperimental setup.

3.1

Overview

Figure 3.1 shows a schematic view of the experimental setup. On the 1.2 m× 2.4 m
optical bench, optical system including a Fabry-Pérot cavity and polarizers are
installed. The optical bench is placed inside the clean room which is surroundedby shading sheets for the safety operation of the laser and to eliminate the stray lightfrom outside of the optical bench. The pulsed magnet is installed on the magnet
bench. The magnet is connected to the charging unit outside the laser clean booth
via coaxial cable. Figure 3.2 shows a picture of the experimental room.

3.2

Optical system

3.2.1

Overview of optical system

Figure 3.3 and Fig. 3.4 show the overall setup of optical system. A Non-Planar
Ring Oscillator laser which is called Mephisto is used as a master laser. An AOM
is used for intensity stabilization for input laser. An EOM is used to generate sidebands to lock the frequency of the laser to the resonance frequency of the cavity.
A polarizar is placed just before the input mirror to align the polarization of the
input laser.
27

28

3.2 Experimental setup

Top view of the experimental setup
output optical system

Shading sheet
Magnet bench

Optical bench

Charging unit

Iron shield
Pulsed magnet
Master laser
Input opticalsystemmagnet container

Mirror chamber
Fabry-Pérot cavity

Figure 3.1: Schematic view of the experimental setup. On the 1.2 m× 2.4 m opticalbench, optical system is installed. A Pulsed magnet are installed on the magnet
bench. The magnet is connected to the charging unit via coaxial cable outside thelaser booth.
Total 4 photodiodes are used in this setup. PDin is used to monitor the intensityof the input laser. PDr is used to monitor the intensity of the reﬂected beam by
M1. This information is used to lock the laser with PDH method. PDt is used
to monitor the ordinary beam transmitted through the analyser. PDe is used tomonitor the extraordinary beam after the analyser which signal could appears.
The mirrors of the Fabry-Pérot cavity and the polarizer and the analyzer isplaced inside the vacuum chamber to avoid the polarization change by the air andto decrease the absorption by the air. Figure 3.5 show the picture of the input
optics.

3.2.2

Master laser

The ﬁnesse of a Fabry-Pérot cavity depends on the reﬂectivity of the mirrors. The
highest reﬂectivity is established at 1064 nm among the manufactured mirrors. In
this work, Nd:YAG laser Mephisto is used as a master laser. The wavelength of
the Mephisto is 1064 nm. The maximum output power is 500 mW. Mephsito has
2 port to tune its wavelength. They are used as the actuator of PDH method.
One is the PZT actuator to tune the cavity length of the cavity inside Mephisto.
Its bandwidth is ∼ 50 kHz and its tuning range is ∼ 200 MHz. The other is thetemperature controller of the crystal in the Mephisto. Its tuning bandwidth is∼ 1 Hz and its tuning range is ∼ 1 GHz.

29

3.2 Experimental setup

Clean room

Charging unit

Control panel

Figure 3.2: Picture of the experimental room. A clean room where laser and themagnet are operated, a charging unit of the magnet and control panel of the magnet
and a data acquisition system are placed.
PDin
lense1Laser

lense2Isolator

AOM

PBS

M1

PBS

EOM

lense3 BS

Polarizer

To the magnet

PDr

Figure 3.3: Schematic view of the input optics.

3.2.3

Fabry-Pérot cavity

We use super mirrors manufactured by Advanced Thin Film company. The designed reﬂectivity R is R > 99.999% and the radius curvature is 2 m. The cavity
length is 1.4 m. Figure 3.8 show the picture of the mirror mount. The mirrors arealigned by 3 axes, pitch, yaw and rotation. The ﬁrst 2 degrees of the freedom areused for the spatial alignment of the cavity, and the last one is used to align thefast (or slow) axis of the mirror to the input polarization angle. All axes can becontrolled from outside the vacuum chamber by the pico-motor.
The ﬁnesse is evaluated by a cavity ring down method. The output power fromthe Fabry-Pérot cavity obeys an exponential decay after turning oﬀ the input laser.
The ﬁnesse F is related to the lifetime of the decay τ as F = τLπc . Fig. 3.7 shows

30

3.2 Experimental setup
Lense PDt
Output chamber
M2
From
the magnet

Analyzer

PDe
Lense

Figure 3.4: Schematic view of the output optics.

Figure 3.5: Picture of the input optics.
an example of a cavity ring down measurement result. The measured ﬁnesse is490,000 which is consistent with the expected ﬁnesse calculated from the designed
reﬂectivity of mirror.

3.2.4

Frequency feedback system

As a high ﬁnesse cavity has a severe resonance condition, it is necessary to lock thefrequency of the laser to the resonance frequency of the cavity. Pound-Drever-Hall(PDH) method is used [52] to extract an error signal between them. The theoreticalbackground about PDH method is described in Appendix A.
The schematics of the frequency feedback system is described in Fig. 3.9. An
EOM is driven with 4.5 MHz RF signal to generate sidebands. The reﬂected lightis detected by fast photo detector (PD2) and demodulated by 4.5 MHz RF to
extract error signal. The error signal is sent to the PZT actuator and temperature

31

3.2 Experimental setup

transmitted intensity [a.u.]

Figure 3.6: Picture of Mephisto.

const
6.18
τ [msec] 0.7309

10

1
−0.5

0

0.5

1

1.5
2
time [msec]

Figure 3.7: Measurement of the ﬁnesse.

controller of the Mephisto after the servo circuits and drivers.
Fig. 3.10 shows the residual frequency noise spectrum evaluated from the inloop error signal and gain of PDH method. The rough requirement of frequency
noise is also shown in red line assuming F = 500, 000, I0 = 3 mW, and the phaseretardation per each round trip is 2 µrad. The transfer function to calculate therequirement is shown in Appendix. B. Above 50 Hz, the estimated frequency noise
is below the requirement.

32

3.2 Experimental setup

Pitch

roll

Yaw

Figure 3.8: Picture of the mirror mount. Pitch, yaw and rotation angle can becontrolled via pico-motor.
Mephisto

EOM

To the PZT
actuator

×22
To the temperature
controller

PI

PI

Beam Sampler

4.5MHzServo1

Fabry-Pérot cavity

RF photo detector

LPF

Error signal
Servo2

Figure 3.9: Schematic view of the frequency feedback system for a Fabry-Pérot
cavity.

3.2.5

Auto locking system

Due to the unexpected large disturbance to the cavity or long term-drift of thecavity length, the cavity sometimes loses its resonance. As the PDH signal isnonzero only near the resonance frequency, the laser frequency should be tuned tonear the resonance frequency manually to lock the laser to the cavity again. For theseamless data acquisition, this locking procedure is automatized. The auto-locking
procedure is divided to 3 steps as follows.
Scanning phase33

Hz/ Hz3.2 Experimental setup

Estimated from in loop error signal

1

Rough requirement

10−1
10−2
10−3
103

102

Hz
Figure 3.10: Spectrum of the residual frequency noise estimated from in loop error
signal.

PDt

Laser
PZT actuator

PZT driver

＋

Servo
Error signal
gain switch

triangle wave
generator

start/stop

Comparator

Figure 3.11: A schematic view of an auto locking system.
When the cavity loses its lock, a triangle wave generator is switched on. It canmodulate the input laser frequency more than 1 FSR to ﬁnd the resonancefrequency. During this phase, the temperature feedback is turned oﬀ. The
period of the triangle wave is about 1 s.

34Gain [dB]

3.2 Experimental setup

no resonance
1st stage
2nd stage
final stage

160
140
120
100
80
60
40
20
0
1

10

102

103
104
frequency [Hz]Figure 3.12: Designed gain of the locking circuit at diﬀerent locking state.
Locking phasewhen the frequency of the input laser comes closes to the resonance frequency, some part of the input light transmits the cavity and monitored by
PDt . When the output of the PDt exceeds threshold predeﬁned in the comparator circuit, a TTL signal is send to the triangle wave generator to stop themodulation. Frequency of the laser is ﬁxed to near the resonance frequency.
This helps the initial locking of Fabry-Pérot cavity to start.
Gain boost phase
After the cavity is locked, the gain of the servo circuit is increased step bystep to suppress the frequency noise further. Fig. 3.12 shows the designed
gain of the servo circuit. When the cavity is not locked, the gain of the circuitis black line in Fig. 3.12. After the locking, the gain is increased from the 1ststage to ﬁnal stage in Fig. 3.12. After the gain of the feedback circuit reaches
ﬁnal stage, the feedback to the temperature controller switches on.

3.2.6

Polarizer and Analyser

The gran laser prisms (GLPB2-10-25.9SN-7/30) are used as a polarizer and ananalyzer. The extinction ratio is 3×10−7 , which is small enough for this experiment.

3.2.7

Mirror chambers

Figure 3.13 and 3.14 show the pictures of the input mirror chamber and the output
mirror chamber. To avoid the loss due to the air, the mirrors of a Fabry-Pérot cavity

3.2 Experimental setup

35

MirrorPolarizer

Figure 3.13: Picture inside the input chamber. One high reﬂective mirror and thepolarizer is placed. The rotation angle and the alignment of the mirror and therotation angle of the polarizer can be controlled by the pico-motor from outside of
the chamberis placed inside the chamber. The polarizers are also placed inside the chambers
to avoid the polarization change due to the air. Each chamber has two windows
whose transmittance is 90%.
The chambers are connected only with ﬂexible bellows to the optical bench
and the pulsed magnet and supported by another suspension frame to avoid the
propagation of the mechanical disturbances caused by the pulsed magnet to theoptical bench through the mirror chambers.

3.2.8

Signal detector

The power of the extraordinary light is monitored by PDe . S11499-01 (HAMAMATSU) is used as a photodiode. To detect the small DC power of extraordinary
light, the gain of signal detector is designed to 6 × 107 V/W. To suppress the straylight from other component, an iris and a band-pass ﬁlter are also placed before
the detector. The whole detector is covered with iron shield to suppress the noise
induced by the electromagnetic induction when the pulse ﬁeld is applied. The picture of the signal detector is shown in Fig. 3.15. The speciﬁcation of the signal
detector is summarized in Table 3.1.

36

3.3 Experimental setup

MirrorAnalyzer

Figure 3.14: Picture inside the output chamber. One high reﬂective mirror and theanalyzer is placed. The output light from each 2 ports of the analyzer is led to thephoto-detectors.

Table 3.1: Speciﬁcation of the signal photo detector
parameter
wavelength
eﬃciencyGainbandwidth
sensor area

3.2.9

speciﬁcation
300 ∼ 1100 [nm]0.6 [A/V]
1 × 108 [V/A]
3 [kHz]
5 [mm2 ]Input intensity stabilization

The input power is stabilized with an AOM to suppress the eﬀect of the intensity noise inside the cavity. The requirement for the intensity stability can beroughly estimated assuming that the output intensity from the Fabry-Pérot cavity
is 15 mW, the static birefringence is 1 × 10−5 and its frequency response.
The red line in Fig. 3.16 shows the achieved intensity noise after the stabilizationof the input laser evaluated out of loop. The blue line in Fig. 3.16 is the calculated
requirement. Above 50 Hz, the intensity noise is smaller than the requirement
except for some peaks.

37

3.3 Experimental setup

Relative intensity noise [1/ Hz]

Figure 3.15: Picture of the signal photo detector.

Input (free run)
Input (locked)
Requirement

10−4

10−5

10−6

10−7

102

103
Hz

Figure 3.16: Result of the evaluation of input intensity stabilization. The black lineis the relative intensity noise of input laser during free run. The red line is relative
intensity noise during intensity locking is activated. The blue line is an estimatedrequirement of noise level.

3.3

Evaluation of the sensitivity to the optical passlength diﬀerence

In this section, the sensitivity of the optical pass length diﬀerence without the magnet is evaluated. The study of the birefringence noise has been mainly performed

38

3.3 Experimental setup

at low frequency up to 25 Hz which is important for the VMB search with the permanent magnet. In contrast, the frequency region which is important for the VMB
search with pulsed magnet is around 100 ∼ 1 kHz. Only limited study has beenreported about the study of the birefringence noise at this frequency region [42].
In this chapter, the systematic investigation of the evaluation of the birefringencenoise in wide frequency region is reported. The noise evaluation method in thehomodyne detection system has been established taking into the account the phaseretardation of the mirrors.
This sensitivity determines the ideal sensitivity of this experiment, and it isuseful to compare this sensitivity with the sensitivity during the magnet operation
to discuss the noise induced by the magnet.

3.3.1

Measurement of the phase retardation of the mirror

As discussed in Appendix B, the transmittance of the extraordinary light is attenuated by κ2 (α) ≡ 1+( 2F )12 ×( α )2 due to the birefringence of the mirror where α [rad]π2

is the phase delay of the mirror per round trip. To measure α, α dependence ofthe transfer function of the extraordinary light against the input power modulation
is used. As discussed in Appendix B, the transfer function from the input power
modulation to the extraordinary light can be written as follows,∆Ie = TFP × I0 (f ),(3.1)

where TFP is deﬁned as follows,
TFP (f ) =+Γ2
2(1 + i Fπ αeq )(1 − i Fπ αeq − i ffc )(1 − i ffc )Γ2
2(1 − i Fπ αeq )(1 + i Fπ αeq − i ffc )(1 − i ffc )

(3.2)

,

where F is the ﬁnesse of the Fabry-Pérot cavity and fc is the cutoﬀ frequencyof the Fabry-Pérot cavity.
AOM is used to modulate the input power and its modulation is evaluatedby PD0 . The modulated intensity of the extraordinary light is evaluated by PDe .
The frequency response analyzer is used to calculate the transfer function from themeasured voltage of the PD0 to PDe .
V
TFP= Ve (f )/V0 (f ) ∝ TFP (f ) ≡ p0 × TFP (f ).

(3.3)

V . The measured curve is
Figure 3.17 shows the measured gain spectrum of TFP
V
ﬁtted by TFP (f ). From the ﬁtting result, the phase retardation per each round trip
is obtained as
αeq = 1.5 ± 0.1[µrad].
(3.4)

The evaluation of the sensitivity is performed with this alignment.

39

Measured5pahse[deg]gain[dB]

3.3 Experimental setup

−20

Fit

0

0

−5

−60

−10

−80

−15

−100

−20

−120

−25

Measured

−40

102

10

freq[Hz]

10

Figure 3.17: Gain curve of the transfer function from the input power modulationV .
to the extraordinary power modulation. The measured curve with ﬁtted by TFP
From the ﬁtting, the phase retardation of the Fabry-Pérot cavity is obtained
To conﬁrm that the measured value depends on the relative angle of the mirror
as discussed in Appendix B, the same measurement is performed after rotating theazimuth angle of the input mirror ∼ 90 degree. The result is shown in Fig. 3.18.
In this case the alpha is measured to be
αeq = 5.5 ± 0.1 [µrad].

(3.5)

Small α is preferred to enhance the signal and the suppress the contribution of
frequency noise, thus the sensitivity evaluation during this chapter is performed byre-rotating the mirror angle. These measured α are consistent with the previous
experiment [53].

3.3.2

Sensitivity measurement

Fig. 3.20 shows the measured sensitivity spectrum of the optical pass length difference. The spectrum is the average of 20 sets of the measurement. For eachmeasurement, waveform of Ie and It is acquired. All data are acquired by ADC(PCI-6229, National Instrument). The measurement time is 10 sec and the sampling rate is 10 kHz. The interval of each time series is 2 second. Figure 3.19 showsthe example of the measured voltage signal of the Ie and It . The optical pass length
diﬀerence is calculated as follows,
λ
1
Ie
(f )
2F 2κ2 (α)Γ It
λ
1
Ie
=
(f )
′
2F 2κ(α)Γ It∆L(f ) =

(3.6)

102

40

Measured5pahse[deg]gain[dB]

3.3 Experimental setup

−20

Fit

0

0

Meas

−40

−5

−60
−80

−10

−100

−15

−120

−20

−140
10

102

freq[Hz]

10

Figure 3.18: Gain curve of the transfer function from the input power modulation to
the extraordinary power modulation after rotating the mirror. The measured curveis slightly diﬀerence from the previous measurement. This is because the transfer
function should dependes on the relative angle of the 2 mirrors. For comparison,
the transfer function assuming no phase retardation is also shown in blue line.
where Γ′ is the static birefringence of mirrors after the attenuation, which can becalculated as
r !
Ie
′
Γ ≡
(3.7)
It
DC

and κ(α) is the attenuation factor discussed above. From the power spectrum of
the optical pass length diﬀerence without magnetic ﬁeld, the ideal sensitivity of the
optical system is obtained. The possible noise source of the measured sensitivity is
discussed later.

3.3.3

Evaluation of known noise source

In this section, the expected birefringence noise level is estimated from all the
known noise source. The considered noise source are• intensity noise• frequency noise• shot noise• detection noise• noise independent of the Fabry-Pérot cavity• magnetic ﬁeld noise

1

41

[V]

3.3 Experimental setup

−8.55
−8.6
−8.65
−8.7
−8.75

[V]

−8.8
0

2000

4000

6000

8000

−7.7
−7.71
−7.72
−7.73
−7.74
−7.75
−7.76
−7.77
−7.78
−7.79
0

2000

4000

6000

8000

[ms]

[ms]

Figure 3.19: Time series of Ie (upper) and It (lower). The measurement time is 10
sec and the sampling rate is 10 kHz.
Among them, the frequency noise possibly has 3 diﬀerence origin. They are• frequency noise due to the residual frequency noise of the PDH feedback
system.
• frequency noise due to the detection noise of the PDH method• frequency noise due to ﬂuctuation of the phase retardation of the mirror
Finally, the contribution of total 8 types of the noises is evaluated. Figure 3.21
shows the power spectrum of the noise budget. The detail of the noise sources andtheir estimation methods are discussed below.
Intensity noise
The ﬂuctuation of the internal power of a Fabry-Pérot cavity could become
the noise source. Its contribution is∆L(f ) =λΓ′ Gp It (f ),×2F
2κ (Ie )DC

(3.8)

42

Optical pass length difference [m/ Hz]

3.3 Experimental setup

10−17

Measured Noise

10−18

10−19

1

10

102

103

Hz

Figure 3.20: Power spectrum of the measured birefringence noise in the unit of
optical pass length diﬀerence.
where Gp is a transfer function from It to Ie . The theoretical expression of
Gp is derived in Appendix.B.2, whose behavior is similar to the ﬁrst order
low pass ﬁlter whose cutoﬀ frequency is the cavity pole. Gp can be written
as follows,
Gp = TFP /T0 ,(3.9)
where TFP is a transfer function from I0 to Ie , which is used in Chapter. 3.3.1.
T0 is a transfer function from I0 to It , and can be obtained by the measurement of the frequency response by modulating the input intensity by AOM.
The intensity noise of the internal power is evaluated by the PDt . Its contribution is shown with orange line in Figure 3.21.
Frequency noise
The contribution of the frequency noise to the optical pass length sensitivity
can be calculated from the following equation.
∆L(f )freq =λ1
× G1 × ∆νnoise (f ),×2F2κΓ′ It(3.10)

where ∆νnoise is the spectrum of frequency noise and G1 is the transfer function from frequency noise to the intensity noise of Ie .
Fig. 3.22 shows the block diagram of feedback system. From this diagram,

43

Optical pass length difference [m/ Hz]

3.3 Experimental setup
freq noise
PDH detection noise
freq split noiseInternal power
Expected no cavity limit
Detector noise
shot noise

10−18
10−19
10−20

1

10

102

103

Hz

Figure 3.21: Power spectrum of the possible contribution of the all known noise
source.
the frequency noise can be written as∆νnoise = Verr /GPDH ,(3.11)

where GPDH is the gain of PDH method and Verr is the spectrum of the error
signal. From these equations, ∆L(f )freq can be written as∆L(f )freq =

λ
1
G1
×
×
× Verr (f )2F2κΓ′ It GPDH

(3.12)

From the box diagram, it is also shown the transfer function from the ex1
ternal input port (Vex ) to intensity noise of Ie is also GG. This transfer
PDH
function is measured by using frequency response analyser. Fig. 3.23 shows
the measured transfer function from Vex to the voltage signal of P De . From
this measurement and the spectrum of the error signal, the contribution of
the frequency noise to the optical pass diﬀerence noise can be calculated. The
result is shown by a red line in Fig. 3.21.
Frequency noise due to the PDH detection noise
Since the detection noises of PDH method is also feedbacked to the laser frequency, it also generates the frequency noise. The general origin of detection
noises are known as the residual amplitude modulation (RAM) of EOM, shotnoise of the reﬂected light, and electrical noise of the detector of the reﬂected44

3.3 Experimental setup

Ie
FP
cavityGFP [W/W/Hz]Δν
frequency
fluctuation ν0

External
input
VEX [V]GPDH [V/Hz]

PDH

＋

＋

PZT

Servo

Error signal
Gservo [V/V]

GPZT [Hz/V]

Figure 3.22: Block diagram of feedback system.
light. The transfer function of the detection noise to the intensity noise of theextraordinary light is also Gfreq . The detection noise of the PDH method is
evaluated from the error signal of PDH signal when the input laser frequency
is far from the resonance frequency. The contribution of the PDH detection
noise is shown in green line in Fig. 3.21.

gain[dB]

15
10
5
0
−5
−10
−15
−20
−25
−30

pahse[deg]Frequency noise due to the phase retardation of the mirrors
The frequency noise discussed above is that against the resonance frequencyof the ordinary light. There is another possibility of the frequency ﬂuctuation.
It is the ﬂuctuation of the phase retardation of the mirrors (∆α). Below thecut oﬀ frequency, the contribution of ∆α to the optical pass length diﬀerence

50
0
−50

−100
−150
−200
2

10

3

10
frequency [Hz]

102

103
frequency [Hz]Figure 3.23: Example of the measured transfer function from Vext [V] to Ve [V].
This function is used to estimate the contribution of frequency noise to the opticalpass length noise spectrum.

45

3.3 Experimental setup
can be written as∆L(f )α =

λ
Γ′
×
×
2F
κ



F
π

2
× α × ∆α(f ).

(3.13)

Though it is diﬃcult to measure the ﬂuctuation of the α directly, its uppereﬀect is estimated as follows.
1
To evaluate the stability of α, the property that the transfer function GG
PDH
is proportional to α is used. The stability of α appears at the stability of the
1
gain of GG.
PDH

When the input laser frequency is modulated at f Hz by injecting the voltage
signal to the external input port of the feedback system, this modulationinduces the intensity noise of the extraordinary light and it can be writtenas,∆Ie (t) ∝ I0 × α(t) × sin(2πf t).

(3.14)

By using a lock-in-ampliﬁer, one can extract the voltage signal proportional
to the amplitude of intensity ﬂuctuation at frequency f , which becomes
Vlockin (t) ∝ I0 × α(t).

(3.15)

When it is assumed that ﬂuctuation of α is small compared to the mean valueof α, ∆α(t) can be estimated as follows,∆α(t) = Vlockin (t)/ (Vlockin )DC × αDC ,(3.16)

where (Vlockin )DC and αDC are the mean values of output voltage of lock-inampliﬁer, and α measured in Section 3.3.1.
Figure 3.24 shows the spectrum of estimated ∆α. The laser frequency ismodulated at 550 [Hz] by the internal oscillator of the lock-in-ampliﬁer. The
output voltage of PDe is sent to the lock-in-ampliﬁer. The black line isdetection limit. It can be seen that above 50 Hz, the spectrum is limited by
the detection noise. This spectrum is used as the upper limit of the ﬂuctuationof α.
The possible contribution of the frequency noise against the resonance frequency of the extraordinary light is shown in blue line in Fig. 3.21 below
50 Hz.
Shot noise
From the DC power at the detector, the contribution of the shot noise iscalculated theoretically by using Eq. (2.25). It is shown with pink line in
Fig. 3.21.

46µ rad/ Hz

3.3 Experimental setup

Measureddetection limit

10−3

10−4
1

10

Hz

Figure 3.24: Power spectrum of the estimated α noise in the unit of radian. The
black line is the detection limit when no modulation is applied.
Detector noise
The noise of the photodetector is also become the noise source of the opticalpass diﬀerence. The noise of the detector is evaluated when no light is irradiated to the photodiode. Its contribution to the total noise is shown in grayline in Fig. 3.21.
Intrinsic noise without the Fabry-Pérot cavity
There are some noise sources which could appear even without the FabryPérot cavity. The candidates of the noise source could be• The birefringence induced before the Fabry-Pérot cavity• The instability of the gain of the photo detector.
• The scattering or the turbulence after the analyzer.
To study the contribution of these noise source, the evaluation of the sensitivity without the Fabry-Pérot cavity is performed. The setup is shown in
Fig. 3.25. In this measurement, a quarter wave plate is used to generate thestatic birefringence instead of the Fabry-Pérot cavity. All the other set up is
the same with the measurement with the Fabry-Pérot cavity. When the slow
axis of the quarter wave plate is slightly tilted θ (θ ≪ 1) against the input
polarization axis, the measured signal becomes as follows.
Ie∼ 2θ2 + 2θΨ(t)It(3.17)

47

3.3 Experimental setup

Although it is impossible to distinguish the rotation and the ellipticity, thesensitivity of the optical pass diﬀerence without a Fabry-Pérot cavity can beroughly estimated as
λ Ie∆Lnocav ∼(3.18)
2π It
.
PDt

Laser

Polarizer

QPW

PDe

Analyzer
Input chamber

Output chamber

Figure 3.25: Schematic view of the set up for a birefringence measurement withouta Fabry-Pérot cavity. Instead of the mirrors for a Fabry-Pérot cavity, a quarterwave plate is mounted in the output vacuum chamber. All other set up is the samewith the measurement with a Fabry-Pérot cavity.

The result of the measurement is shown in the black line in Fig. 3.26. The
known noise source is the shot noise and input power noise. The sum of theestimated noise is drawn with gray line in the same ﬁgure. Even without the
Fabry-Pérot cavity, there are some discrepancy between the measured noise
and the estimated noise especially below 100 Hz. When there are the FabryPérot cavity, assuming these noises are generated outside the polarizers, theyπ
could appear 2F
times smaller in the optical pass length sensitivity. With thisassumption, the contribution of the noise without the Fabry-Pérot cavity canbe estimated. The result is drawn with cyan line in Fig. 3.21 below 100 Hz.
Magnetic noise
Mirrors also show the Cotton-Mouton eﬀect. Therefore, the ﬂuctuation of
the residual magnetic ﬁeld at the center of the mirror could become the noise
source. To estimate its contribution, the magnetic ﬁeld at the center of themirror mount is measured. The hole sensor is inserted to the mirror mount.
Measured DC magnetic ﬁeld is 1.6 [mT]. The measured spectrum density
above 1 [Hz]√ is limited by the sensitivity of the hole sensor and it is less than
0.3 [µT/ Hz].
The coupling constant of the Cotton-Mouton eﬀect of the mirror is measured
to be ∼ 1 × 10−9 [rad T−2 ]. From these values, the expected polarization

48

3.3 Experimental setup
noise induced by the ﬂuctuation of residual magnetic ﬁeld is
λ
× B2
π
λ
∼ 1 × 10−9 × × 2 × BDC × B(f )π √< 3 × 10−26 [m/ Hz].

∆L = 1 × 10−9 ×(3.19)

This is almost 7 order smaller than other noise source.

3.3.4

Mirror rotation dependence

Measured noise
Input power noise
shot noise
detector noise
Total noise budget

10−13

10−14

residual [m/Hz]

optical pass difference [m/Hz]

As the static birefringence of the Fabry-Pérot cavity depends on the relative rotation angle between the two mirrors, the static birefringence can be tuned byrotating the mirrors. To identify the origin of the optical pass length diﬀerence
noise, the noise spectra are measured at diﬀerent rotational alignment of the mirror. Since the birefringence noise does not depend on the static birefringence of themirrors, it can be distinguished from other types of noises by these measurementsof the diﬀerent mirror layouts.
Figure 3.27 shows the result of the measurement. The caption shows the static
ellipticity at each alignment of the mirrors. The static ellipticity is chosen so as tobe larger than the minimum value of the static ellipticity and smaller so that thecontribution of known noise sources are not dominant. From Fig. 3.27, it can beseen that the noise ﬂoor of each measurement agrees with each other. It indicatesthat the observed noise is birefringence noise or at least its behavior can not be
distinguished from the birefringence.

10−13

10−14

10−15
10

102

103
[Hz]

10−15
10

102

103
[Hz]

Figure 3.26: Left: noise spectrum of the optical pass diﬀerence without a FabryPérot cavity. Black line: measured noise. Gray line: Estimated total noise budget
from all known noise source. Right: The residual between measured noise andestimated noise budget.

49

3.3 Experimental setup

3.3.5

Comparison of known noise source between the measured
value and target speciﬁcationm/ Hz

The noise budget of the target speciﬁcation is discussed in Chapter 2.8. The comparison with the target speciﬁcation and the measured noise of each known noise
source is shown in Fig. 3.28 around 100 Hz to 1 kHz. The brown line is the sumof the known noise sources evaluated in the previous chapter. The blue line is thedesign sensitivity discussed in Chapter 2.8.
Above the 500 Hz, the diﬀerence between target speciﬁcation and measured
noise comes from the diﬀerence of the noise which is proportional to the square rootof the output power of the Fabry-Pérot cavity. This is due to the less transmittanceof the Fabry-Pérot cavity against the input power (typically ∼ 10%). This is
assumed to be due to the loss of the mirror. When the loss of the mirrors is 4 ppm,
the transmittance can be explained. The diﬀerence can be compensated by using
low loss mirror or injecting 4 times larger input power into the Fabry-Pérot cavity.
Below 500 Hz, the diﬀerence comes from the intensity noise of the Fabry-Pérot
cavity. The intensity noise should be reduced at most 10 times smaller to reachthe designed sensitivity above 100 Hz. Currently, the origin of the intensity noise
is not understood, however, it can be reduced by the stabilization of the output
intensity by using AOM. Another source of the origin of the diﬀerence comes fromthe frequency noise. Although the frequency noise is almost limited by the detectionIe-5= 5.4 × 10
It

10−17

Ie
= 1.2 × 10-4
ItIe
= 1.9 × 10-4
It

−18

10

Ie
= 2.7 × 10-4
It

10−19

1

10

102

Hz

Figure 3.27: Noise spectrum of optical pass diﬀerence at diﬀerence rotational alignment of the mirror.

50

Optical pass length difference [m/ Hz]

3.3 Experimental setup

Target sensitivity
10−18

Total estimated

10−19

10−20
102

103
Hz

Figure 3.28: Comparison of the sensitivity between designed value and measurevalue of known noise source. The brown line is the sum of the known noise sources
evaluated in the previous chapter. The blue line is the design sensitivity discussed
in Chapter 2.8.
noise below 100 Hz, at higher frequency, the eﬀect can be reduced by strengthening
the feedback gain of the PDH method.

3.3.6

Discussion of the measured sensitivity

Figure 3.29 shows the measured sensitivity and the expected noise which is thesum of all estimated noise. In this experiment, the frequency region from 100 Hz
to 1 kHz is important for the sensitivity considering the timing proﬁle of the pulsed
magnet. As shown in Fig. 3.29, above 700 Hz, the sensitivity can be explained by
the estimated noise and its limiting noise source is shot noise as expected. Below
700 Hz, the clear discrepancy between the measured and the expected sensitivity
can be observed. Its mirror rotation dependence is measured for the ﬁrst time, and
it is observed that the noise behaves as birefringence. The situation and size below
100 Hz are the similar to the previous experiment [37]. It seems to indicate thatthe unknown noise source not depending on the detail of the experimental setup
exists and its eﬀect exists even at higher frequency. In the previous experiment,
it is observed the birefringence noise depends on the modulation depth of PEM,
which is used to modulate the in polarization for heterodyne detection. It becomes
clear that the unexpected noise exists even without PEM. As the noise disappears

3.3 Experimental setup

51

without the Fabry-Pérot cavity, the Fabry-Pérot cavity itself is noise source. As
the investigation of the observed noise excess, the following studies are performed.
• measurement of the correlation between the displacement noise• calculation and comparison with the thermal noise• measurement with the lower ﬁnesse cavity
However, they do not lead to the explanation of the origin of the noise excess. The
detailed discussion about these studies and the comparison with other experimentis summarized in appendix D.
Though the sensitivity is limited by the unexpected noise source, by suppressing
the contribution from the known√ noise source, the sensitivity around from 200 Hz−19
to 900 Hz is ∼ 1 × 10
[m/ Hz]. The sensitivity is 5 times better than thesensitivity at around 10 Hz, which is used in the experiment with permanent magnet. The estimated sensitivity from obtained noise spectrum is 3 times worse thanthe designed sensitivity. The sensitivity is less aﬀected by the noise excess compared to the previous experiment, but the identiﬁcation of the origin of this noise
is important for future work.

52

Optical pass length difference [m/ Hz]

3.3 Experimental setup

Measured−17

10

Total estimated

10−18

10−19

1

10

102

103

Hz

Figure 3.29: Comparison between measured noise and the estimated noise spectrum. Above 700 Hz, the measured noise agrees with the expected noise, and thelimiting noise source is shot noise. Below 700 Hz, the discrepancy between themeasured noise and expected noise is observed. This noise is found to be behavesas the birefringence.

53

3.4 Experimental setup

3.4

Pulsed magnet

2 types of pulsed magnet are developed for this experiment. In this section, thebasic speciﬁcation of the magnet are described. One is a magnet with a single racetrack coil, the other is a magnet with the dipole racetrack coil. Former magnet isused for VMB measurement with strong magnetic ﬁeld. The latter is developed forvalidation run with gases. This is because the dipole magnet has small longitudinalmagnetic ﬁeld compared with the single racetrack magnet which induces faraday
rotation of gases.

3.4.1

Single race-track magnet

Design and speciﬁcation
Figure 3.30 shows the schematic view of the single racetrack pulsed magnet.
The coil is wound by a 1 mm×3 mm copper wire with 15 turn and its lengthis 200 mm. A beam pipe whose diameter is 6.3 mm diagonally passes throughthe center of the magnet. The laser runs through the center of the beam pipe.
The coil is reinforced by the stainless steel 304 backup to suppress the deformation due to Maxwell stress. The total length including the reinforcementis 330 mm.
This magnet has non-zero longitudinal magnetic ﬁeld as the beam pipe runs
through the coil diagonally. This longitudinal ﬁeld induces faraday rotationof gases during the gas measurement. Table 3.2 summarizes the speciﬁcationof this magnet.
current

magnetic fieldcoilbeam pipe

Figure 3.30: Schematic view of the single racetrack magnet. The magnet is consisted with a racetrack shaped coils whose length is 200 mm. At the center ofthe coils, a beam pipe whose diameter is 6.3 mm is located. A beam pipe whose
diameter is 6.3 mm diagonally passes through the center of the magnet.

54

3.4 Experimental setup

Table 3.2: Speciﬁcation of the single coil magnet
Parameter
Wound wire
Coil shape
Field length
Operation ﬁeld
Aperture size
Destruction ﬁeld
Total length
WeightResistanceInductancevalue
Cu (1 [mm]×3 [mm])single racetrack
165 [mm]
8.3 T
ϕ1/4 inch (=6.35 [mm]), thickness 0.5 [mm]
12 [T]
330 [mm]
16 [kg]
9 [mΩ] @ DC
23 [mΩ] @ 750 Hz
40 [µH] @ 750 Hz

Figure 3.31: Picture of the single racetrack pulsed magnet. The total length after
the reinforcement is 330 mm. A beam pipe whose diameter is 5.3 mm passed
thorough the magnet.
Pulse shape and Field eﬃciency and ﬁeld length
The measured time proﬁle of the magnet is shown in Fig. 3.32. The widthis 1.2 ms. Field eﬃciency is the ratio between the generated magnetic ﬁeld
at the center of the magnet and current ﬂowing the coil. The current ismeasured by a current transformer, and the magnetic ﬁeld is measured by
the calibrated pick-up coil inserted to the center of the beam pipe. The ﬁeld

55

3.4 Experimental setup
eﬃciency g is measured asmagnetic field [T]g ≡ B(t)/I(t) = 0.6 [T/kA].

(3.20)

8
7
6
5
4
3
2
1
0
−2

−1

0

1

2
3
time [msec]

Figure 3.32: Measured timing proﬁle of the magnetic ﬁeld. The charging voltage
is 2 kV. The peak magnetic ﬁeld is 8.2 T. The pulse width is 1.2 ms.
The eﬀective ﬁeld length is deﬁned as follows,
R 2
B (z)dz
Leﬀ =,B 2 (0)(3.21)

where z axis lies along the beam pipe and the origin of the z-axis is the centerof the magnet. The ﬁeld map is measured by inserting calibrated pick up coil
into the beam pipe along the beam path. Figure 3.33 shows the square of themagnetic ﬁeld as a function of the position normalized by that at the centerof the magnet. From this map, the eﬀective ﬁeld length Leﬀ
B is calculated tobeLeﬀ = 165 [mm].
(3.22)
The ﬁeld map of the longitudinal magnetic ﬁeld is calculated by using ANSYS.
Fig. 3.34 shows the calculated ﬁeld map. As the ﬁeld eﬃciency becomes larger
near of the coil, the map has 2 peaks due at the corner of the coil.
Heating and repetition rate
As the wire of the coil has nonzero resistivity, the temperature of the coilincreases when a current ﬂows, and the magnetic ﬁeld is generated. Since toohigh temperature leads to the destruction of the coil, the repetition rate of
the pulse operation should be slow enough to maintain the temperature of
the coil under the thermal equilibrium.
To determine the repetition rate, the time evolution of the resistivity is used.
The joule heating is proportional to the resistivity of the coil. The total

56

B2 [arb]

3.4 Experimental setup

1
0.8
0.6
0.4
0.2
−150 −100 −50

050

100 150
L [mm]

B/I [T/KA]

Figure 3.33: The distribution of the squared magnetic ﬁeld along the beam pipe.
The value are normalized by that at the center (z = 0).

0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0

−150 −100 −50

050

100 150
L [mm]

Figure 3.34: Distribution of the ﬁeld eﬃciency of the longitudinal magnetic ﬁeld
calculated by ANSYS.
energy loss due to the joule heating can be obtained from the charged voltage
V2
V2of the capacitance before and after generating the pulse by C 20 −C 21 . Whenthe temperature of the coil is under the thermal equilibrium, this energy loss
becomes constant. By monitoring the energy loss during the operation of themagnet, the repetition rate is set at least longer than ∼ 18 s/pulse.

57

3.4 Experimental setup

Stray ﬁeld
As discussed in chapter 2, the mirrors also show the cotton-mouton eﬀect.
The stray ﬁeld at the mirror should be taken into account to discuss thesystematic eﬀect. The stray ﬁeld 50 cm away from the magnet is less than
1 mT. The Cotton-Mouton eﬀect of the mirror is calculated to be at most 6
times larger than the signal of VMB eﬀect. However, this eﬀect is negligible
considering the current sensitivity.

3.4.2

Dipole magnet

Design and speciﬁcation
Figure 3.35 shows the schematic view of the pulsed magnet. This magnet
is a dipole magnet which is consisted with two racetrack shaped coils. The
coil is wound by a 1 mm×3 mm copper wire with 10 turn and its length is
170 mm. At the center of the coils, a beam pipe whose diameter is 6.3 mm islocated. The laser is at the center of the beam pipe. The total length after
the reinforcement is 260 mm.
Comparing with the single racetrack magnet, the advantage of the dipole
magnet is small longitudinal magnetic ﬁeld. Therefore, this magnet is mainly
used for the validation run with gases. Table 3.3 summarizes the speciﬁcationof the dipole magnet.

current

magnetic fieldcoilbeam pipe

Figure 3.35: Schematic view of the dipole magnet. The magnet is consisted with 2
racetrack shaped coils whose length is 170 mm. At the center of the coils, a beam
pipe whose diameter is 6.3 mm is located.
Pulse shape, Field eﬃciency and ﬁeld length
The timing proﬁle of this magnet is shown in Fig. 3.37. The pulse widthis 1.2 ms. The ﬁeld eﬃciency is evaluated in the same way with the single

58

3.4 Experimental setup

Table 3.3: Speciﬁcation of the dipole magnet
Parameter
Wound wire
Coil shape
Field length
operation ﬁeld
Aperture size
Destruction ﬁeld
Total length
WeightResistanceInductancevalue
Cu (1 [mm]×3 [mm])dipole racetrack
150 [mm]1T
ϕ1/4 inch (=6.35 [mm]), thickness 0.5 [mm]Not measured∼260 [mm]16 [kg]
8.5 [mΩ] @ DC
28 [mΩ] @ 750 Hz
43 [µH] @ 750 Hz

magnetic fieldcoaxial
rod

beam
pipe

260mm

Figure 3.36: Picture of the dipole pulsed magnet. The total length after the reinforcement is 260 mm. Beam pipe whose diameter is 5.3 mm passed thorough themagnet.
race-track magnet. The measured eﬃciency is
g = B(t)/I(t) = 1.0 [T/kA].

(3.23)The position map of the dipole magnet is also measured in the same way
with the racetrack magnet. Figure 3.38 shows the square of the magnetic
ﬁeld normalized by that at the center of the magnet as a function of theposition . From this map, the eﬀective ﬁeld length Leﬀ
B is calculated to be
Leﬀ = 150 [mm].

(3.24)

59

3.5 Experimental setup

As the beam pipe is parallel to the coil, the longitudinal component of themagnetic ﬁeld is 0 by design.

3.5

Magnet operation system

3.5.1

Overview of the magnet operation system

Figure 3.39 shows the schematic view of the magnet system. The magnet is put inthe liquid nitrogen container box which is ﬁxed to the magnet bench. The magnet
is connected to the mirror chamber. The magnet is surrounded by double ironplates. They act as both magnetic shield and soundproof. The current is supplied
from the capacitor bank. The level of the liquid nitrogen is monitored by the levelindicator.

3.5.2

Magnet bench and liquid nitrogen container

To reduce the joule heating, pulsed magnets are operated at the liquid nitrogentemperature. The pulsed magnet and inner iron shield and beam pipe are containedin the liquid nitrogen container box. Figure 3.40 shows the setup inside the liquid
nitrogen box. The liquid nitrogen container is ﬁxed to the magnet bench. The
magnet bench is made with stainless steel 304 and mechanically isolated from theoptical bench.

3.5.3

Charging unit and the capacitor bankmagnetic field [T]The charging unit for the pulsed magnet is originally developed for ALPs searchexperiment at SPring8, the detail of this system is discussed in Appendix C. The
magnet is connected to the 3 mF capacitor via a coaxial cable. The charging unit

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
−2

−1

0

1

2
3
time [msec]

Figure 3.37: Measured magnetic ﬁeld shape. The charging voltage is 200 V. The
peak magnetic ﬁeld is 0.9 T.

60

B2 [arb]

3.5 Experimental setup

1
0.8
0.6
0.4
0.2
0

−100

−50

0

50

100
L [mm]

Figure 3.38: Measured square of the magnetic ﬁeld as a function of the position
along the beam pipe normalized by the square of the magnetic ﬁeld at the centerof the coil (z = 0).

Current transformer
coaxial cable(from charging unit)

LqN2 Level indicator
Liquid nitrogenfrom vessel

coaxial rod

Liquid nitrogencontainer
Iron plate
mirrorchambermirrorchamberMagnet bench
Optical bench

MagnetFigure 3.39: Schematic of the magnet system. On the magnet bench, the liquidcontainer is placed. A magnet is inside the container. The magnet is surroundedby the double iron plate. A coaxial cable, a liquid nitrogen level indicator, and acurrent transformer are also installed.

is developed to charge the capacitor to speciﬁed voltage automatically. During the61

3.5 Experimental setup

Inner iron shield

Pulse magnet

Beam pipe

Figure 3.40: Setup inside the liquid nitrogen container. The pulsed magnet is ﬁxed
to the stainless steel 304 base plate with the inner iron shield and a beam pipe.
Table 3.4: Speciﬁcation of the current transformer
parameter
Sensitivity
Low frequency cutoﬀ
High frequency cutoﬀ
Peak current
Peak isolation voltage

speciﬁcation
0.2 [mV/A]
1 [Hz]
14 [MHz]30 [kA]
5 [kV]

operation of the magnet, the capacitor is charged to ±4.5 kV maximally dependingon the sign of and the size of the magnetic ﬁeld.

3.5.4

Current transformer

The pulse shape and the current ﬂow is measured by a current transformer (CWTMiniHF400) during the data acquisition, and it is shown in Fig. 3.41. Its speciﬁcation is shown in Table 3.4. From the measured current shape and ﬁeld eﬃciency,
the generated magnetic ﬁeld is calculated for each pulse.

3.5.5

Liquid nitrogen serving system

During the operation of the magnet, Joule heating and natural evaporation consume
the liquid nitrogen. For every 2 ∼ 3 hours, the liquid nitrogen container should be
ﬁlled. The height of the liquid nitrogen is monitored by a level indicator made of
Pt-100. When the level of liquid nitrogen become lower than the operation level,
the solenoid valve attached to a liquid nitrogen vessel is opened, and liquid nitrogen62

3.5 Experimental setup

is supplied from the vessel until the level reaches the stable operation level. During
the solenoid valve is opening, an exhaust fan is also activated. It exhausts cold gas
from the clean room to outside. Each vessel can contain 100 L liquid nitrogen. For
2 ∼ 3 times a day, the vessel should be exchanged to another vessel.

3.5.6

Control board

Figure 3.43 shows the control board of the magnet system. It has following functions.
• initial setting• charged voltage indicator (DMR)• pulser

Figure 3.41: Current transformer. The current ﬂowing inside the loop at the rightside of the picture is converted to voltage signal.
withdraw
port
Liquidsolenoid valve
Nitrogenvesselopen/close
DC 24Vvent port
PLC
J222LThigh/low
Nitrogen gas

magnet container

Pt 100×4

Figure 3.42: Schematic view of liquid Nitrogen serving system

3.5 Experimental setup

63• current shape error check
Before the operation of the magnet, the setting is initialized by hand. The charging
voltage for positive charging and negative charging is pre-setted and operation
mode (single pulse or repetition mode) are selected. A digital meter relay (DMR)
reads out the charged voltage of the capacitor. When the readout voltage of DMR
exceeds the preset charging voltage, the signal is sent to the charging unit to stop
the charging. The pulser is used to send the trigger signal for discharging thecharged voltage to the charging unit via optical ﬁber. To prevent from an accident
caused by a damaged coil operation, the shape of the pulse current is checked pulse
by pulse. A Mask function of an oscilloscope (LeCroy wave runner 6050A) is used
for this check. Once a pulse shape diﬀerence from a pre-recorded waveform is foundby the oscilloscope, a stop TTL signal is sent to the capacitor bank.

Figure 3.43: Picture of control board. The capacitor bank is operated via thiscontrol board.

3.5.7

Evaluation of the eﬀect of the disturbanceStability of the cavity
The main challenge of this work is to establish the stable operation of a FabryPérot cavity with a pulsed magnet. In the previous experiment, the resonanceof a Fabry-Pérot cavity is lost 10 ms after the magnetic ﬁeld generation, andits peak ﬁeld is 6.5 T [54].
To study the stability of the setup, test operation of the pulsed magnet is
performed. Fig 3.44 shows the result of the test operation. The top ﬁgureshows generated magnetic ﬁeld, and its peak ﬁeld is 8.2 T. The middle ﬁgure
is frequency ﬂuctuation estimated from the error signal. The bottom ﬁgure

64

Ie [W]

frequency noise [Hz]

Magnetic field [T]

3.5 Experimental setup

8
7
6
5
4
3
2
1
0
−100

−50

0

50

100

150

200
[ms]

1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−100

−50

0

50

100

150

200
[ms]

−50

0

50

100

150

200
[ms]

×10−9
80.5
80
79.5
79
78.5
78
77.5
−100

Figure 3.44: Evaluation of the stability of the Fabry-Pérot cavity against the operation of the pulsed magnet. Top: magnetic ﬁeld. Middle: Estimated frequency
noise. Bottom: output intensity of Ie .
shows the intensity ﬂuctuation of |e . No signs of the lost of the resonance isobserved in Ie , and the estimated frequency ﬂuctuation is suﬃciently smaller
than the FWHM of the resonance, which is 220 Hz. Compared to the previous
experiment, even though the peak ﬁeld is larger, the highly stable operation ofa Fabry-Pérot cavity is established. This stability enables the fast repetitive
operation of the pulsed magnet.
Adjustment of the position of the beam pipe
Although the resonance of the cavity is robust against the disturbance fromthe magnet, the ﬂuctuation of the Ie is observed at the same time with thepulsed magnet depending on the relative position between the beam pipe andthe laser.
The distance of the magnet from the optical bench can be tuned along the vertical direction by the adjuster attached to the bottom of the magnet bench,thus the relative position between the beam path and beam pipe can be

65

Relative intensity fluctuation

3.5 Experimental setup

1.004
start

1.003
+ 0.2 mm1.002

+ 0.4 mm1.001

+ 0.65 mm

1+ 0.75 mm

−1

0

1

2

3

4

5
6
msec

Figure 3.45: Alignment dependence of the intensity noise of Ie . The distanceof the beam pipe from the optical bench is changed along the vertical direction.
At each position, the intensity noise of Ie induced by the magnet is measured.
Every waveform is the average of 50 pulses. The pulse ﬁeld is generated at t = 0,
and its peak is 8.3 T. The vertical position of the magnet is changed from initial
position (black line) to 750 µm higher (blue line). The diﬀerent size of the intensity
ﬂuctuation can be seen, and it becomes smallest at the speciﬁc position (red line).
changed. Figure 3.45 shows the relative intensity noise of Ie during the operation of the magnet measured at the diﬀerence distance of the magnet fromthe optical bench. Every waveform is the average of 50 pulses. The pulse
ﬁeld is generated at t = 0, and its peak is 8.3 T. The vertical position of themagnet is changed from initial position (black line) to 750 µm higher (blueline). The diﬀerent size of the intensity ﬂuctuation can be seen depending on
the position of the magnet, and the intensity noise disappears at the speciﬁc
position of the magnet bench (red line).
As the intensity noise depends on the relative position between beam pathand beam pipe, it is not VMB signal, and not also the eﬀect of the cavity
instability. Currently, it is assumed that these intensity ﬂuctuation is due tothe scatter of Ie by the inner surface of the beam pipe. The position of themagnet bench is adjusted by scanning the level of the magnet bench at theintervals of 200 µm before the measurement.

66

3.6 Experimental setup

Table 3.5: Speciﬁcation of the absolute pressure gaugeparameter
Full scale
Lower limit
Accuracy
Resolution
Bandwidth

speciﬁcation
1.333 [kPa]
0.1 [Pa]
0.2%
0.06 [Pa]33 [Hz]

3.6

Other components

3.6.1

Electromagnetic noise reduction

The strong magnetic ﬁeld and current could induce the electromagnetic noise. As
these noise appears at the same time with magnetic ﬁeld, the reduction of theelectromagnetic noise is necessary. For the reduction of the noise, the master laser,
every photo detector is shield by the 1.6 mm iron plate.

3.6.2

Laser displacement sensor

The laser displacement sensor HG-C1100 (Panasonic) is used to measure the vertical distance between the optical bench and magnet bench. The measurement
range is ±35 mm, and measurement center distance is 100 mm. The resolution
is 10 µm. Total four sensor is used. These sensors are used when adjusting thevertical distance between the optical bench and magnet bench.

3.6.3

Vacuum system

Figure 3.46 shows the schematic view of the vacuum system. Each mirror chamberis connected to a turbomolecular pump. A scroll pump is used as a roughing vacuum
pump for the two turbomolecular pumps. The chambers are also connected to thegas cylinder for calibration measurement by gas. All of these components areconnected with bellows hoses to isolate the vacuum chamber from the vibration of
the pumps.
Three pressure gauge are used for this system. One is the absolute capacitancepressure gauge (M-342DC-11) whose range is 1 Pa ∼ 1000 Pa. This gauge is used
for gas measurement to monitor the gas pressure precisely. The speciﬁcation of theabsolute pressure gauge in summarized in Table 3.5. The other 2 are Cold Cathode
Gauge whose range is 10−9 −10−2 hPa. They are connected to each mirror chamber.
They are used to monitor the pressure during the vacuum measurement.

67

3.6 Experimental setup
scroll pump absolute pressure
bellows hose
gage
TMP

TMP
pressuregagegas
cylindermirrorchamberbeam pipe

mirrorchamber

Figure 3.46: Schematic view of vacuum system

3.6.4

Data logger

PCI-6229 and PCI-6225 (National Instrument) are used as data logger throughout
this experiment. The sampling rate is 160 kHz. The full scale is ±10 V and theresolution is 16 bits. The quantization noise VADC is calculated to bep
VADC = 2 × Vfull / 6 × fsampp(3.25)= 2 × 10/ 6 × 160 × 103√∼ 3 × 10−7 [V/ Hz].
This is small enough compared to the intrinsic noise of all photo detectors.

Chapter 4

Run and analysis
In this chapter, the data acquisition and its analysis are described. First, thesummary of the data acquisition is described. Then the basic analysis procedure is
introduced. After that, the measurement and analysis of the Cotton-Mouton eﬀectof the nitrogen gas is described as the analysis validation. Finally, the measurement
and the analysis of VMB is described. The systematic uncertainty is also described.

4.1

Summary of data acquisition

4.1.1

Run time

Table 4.1 summarize the data sets. The validation run was performed with the
dipole magnet at the room temperature on Sep. 30. The charging voltage are 200 V
and -200 V, respectively. Total 800 pulses are generated during this measurement.
The measurement of VMB was performed with single racetrack magnet from Nov.
13 to Dec. 10. The charging voltage are 2 kV and -1 kV respectively. Total
26000 pulses are generated during this measurement. The data sets of vacuum
measurement are divided to 2 groups (B-1, B-2 and C-1, C-2). The sign of thestatic birefringence of the mirrors is reversed for each data set groups. These
information is used to cancel the systematic eﬀect.
Table 4.1: Summary of the data set
run ID
A-1
B-1
C-1
B-2
C-2

data
30/9/20
16/11/20-20/11/20
23/11/23-26/11/20
30/11/20-4/12/20
7/12/20-10/12/20

target medium
N2
vacuumvacuumvacuumvacuum68

magnetdipolesinglesinglesinglesinglecharging voltage
200 V, -200 V
2 kV, -1 kV
2 kV, -1 kV
2 kV, -1 kV
2 kV, -1 kV

pulse
800
6500
8000
6200
5600

4.2 Run and analysis

4.1.2

69

Run cycle

The basic run cycle and the operation procedure is described.
During the measurement, the electric light inside the clean room is turned oﬀ toavoid the stray light. Before stating the repetitive operation of the pulsed magnet,
the charging voltage of the positive and negative magnetic ﬁeld is set by using thecontrol board of the charging unit. After that, the initial charging of the capacitor isstarted, then the capacitor bank is switched to the repetitive operation mode. The
waveform of the magnetic ﬁeld and the output intensity of It is monitored during
the repetitive operation mode. When errors are detected in the waveform of themagnetic ﬁeld or the output intensity of It becomes lower than the pre-deﬁned
threshold voltage, the operation of the charging unit is stopped automatically. The
magnet container is ﬁlled with liquid nitrogen during the VMB measurement andthe liquid nitrogen is served automatically for every 3 hours. The consumption of
the liquid nitrogen is ∼ 200 L for each day.
During the repetitive operation, the charging unit repeats the following operations. The capacitor is charged to positive operation voltage, and when the trigger
signal from the control board is inputted, the charge is released to the magnet.
After that, the charging to the negative charging voltage is started. The charge
is also released by the trigger from the control board, then the charging of thepositive operation voltage is started again. The trigger is also sent to the ADC torecord the waveforms.

4.1.3

Data acquisition cycle

The data acquisition cycle is consisted of the acquisition of the 3 types of data set
which is described below. Figure 4.1 shows the time chart of the charged voltage,
magnetic ﬁeld and data type.
1st data The data contains the waveform of Ve , Vt , Verr and out put voltageof the current transformer. The capacitor is charged to positive operation
voltage and the positive magnetic ﬁeld is generated. The waveform includes
time series -300 ms before and 300 ms after the magnetic ﬁeld.
2nd data The capacitor is charged to negative charging voltage The data contains
the waveform of Ve , Vt , Verr and magnetic ﬁeld. The capacitor is charged tonegative operation voltage and the negative magnetic ﬁeld is generated. The
waveform includes time series -300 ms before and 300 ms after the magneticﬁeld.
3rd data This data set contains the waveform when magnetic ﬁeld is not applied.
This data is used to discuss the expected noise and stability of the cavity.
The data contains the waveform of Ve , Vt , Verr during 600 ms. The interval
between 2nd data and 3rd data is 2 [s].

70

4.2

4.2 Run and analysis

Analysis method

In this section, the analysis procedure is described.

4.2.1

Calculation of the ellipticity

From the voltage output of the 2 photo detectors, the intensity of the ordinary
light (It ) and extraordinary light (Ie ) is calculated as follows,It= Gt Vt(4.1)

Ie = Ge Ve ,(4.2)

where Vt and Ve are the voltage output of the PDt and PDe , and Gt and Ge are
the gain of the each photo detector.
The ellipticity is calculated from the ration of these intensity as follows, 
Ie
Ellipticity ≡
(4.3)
.
It
As discussed in Section 2 and section B, the function form of the ellipticity can be
written as follows,IedB.
= Γ2 + ηB 2 + κB + ξ
It
dt

(4.4)

The 1st term of the above equation comes from the static birefringence inducedby the mirrors. The 2nd term is proportional to the square of the magnetic ﬁeld
Vchargetime
B(t)1st data

1st data
2nd data
3rd data

2nd data

Figure 4.1: Time chart of the DAQ cycle. 3 types of the data are acquired during
the run. ﬁrst data and second data correspond to the data during the positive
magnetic ﬁeld and negative magnetic ﬁeld respectively. Third data corresponds tothe data without magnetic ﬁeld

71

4.2 Run and analysis

and VMB signal could appear in this term. The 3rd and the last terms include
the eﬀect of the Faraday rotation due to the residual gases and the eﬀect of theelectromagnetic noise which would not appear ideally. To get rid of the last 2
terms from the ellipticity, the sign averaged ellipticity H(t) is deﬁned as follows by
averaging the ellipticity of the 1st data and 2nd data weighting by the ratio of thepeak magnetic ﬁeld to cancel the eﬀect which depends on the sign of the magnetic
ﬁeld.
 
  
1
Ie
B+ Ie
H(t) ≡+,(4.5)
2
It 1 B− It 2
where B+ and B− arethe peak magnetic ﬁeld of the 1st data and 2nd data respecIe
tively. It
and IIet
are the measured elliplicity in the 1st data and 2nd data12
respectively.
The eﬀective magnetic ﬁeld used for the ﬁtting is deﬁned as follow,


1
B+
2
≡
Beﬀ
B(t)2+ +× B(t)2− ,
(4.6)
2
B−
where B(t)2+ and B(t)2− are the timing proﬁle of the square of the magnetic ﬁeldof the 1st data and 2nd data, respectively.

4.2.2

Calculation of the signal shape

As discussed in chapter 2, the Fabry-Pérot cavity acts as frequency ﬁlter against
the VMB. It is necessary to calculate the expected signal shape from the appliedmagnetic ﬁeld shape. The cut-oﬀ frequency is calculated from the measured ﬁnesse.
Figure 4.2 shows the expected signal shape. The signal shape without the cavity
ﬁltering is drawn for comparison.

4.2.3

Parameter estimation

As the noise spectrum is not a white noise, it is not suitable to perform a parameter
estimation in time domain. Instead, the parameter estimation is performed in thefrequency domain, assuming the noise spectrum is stationary.
The log likelihood function for parameter estimation is deﬁned as follows,logL(η) = −∆fnX(H̃(fi ) − η × B˜2 (fi ))2
eﬀ

k=1

S(fi ),(4.7)

where ∆f is the sampling frequency, H̃(fi ) is the Fourier component of the mea2 (f ) is the Fourier component ofsured ellipticity H(ti ) whose frequency is fi , B˜eﬀ
isignal shape which is calculated in the previous section, η is the free parameterto be estimated deﬁned above and S(fi ) is the power spectrum density of the ellipticity without the operation of the magnet. This expression can be understood

72

4.2 Run and analysis

as the minimization of the residuals between measured ellipticity and signal shapeweighted by the intrinsic noise level of ellipticity at each frequency bin. the The
power spectrum of all 3rd data during each run are used to obtain S(f ). Figure
4.3 shows the example of S(f ) obtained during the validation run and B 2 (f ).

signal shape [arb]From the discussion of the section 2 and section B, kcm is acquired from the η

Expected signal shape1

Without cavity response

0.8
0.6
0.4
0.2
0
−2 −1

0

1

2

3

4

5 6 7
time [msec]

Ellipticity

Figure 4.2: Waveform of the calculated expected signal shape. The signal shapewithout a cavity response is also drawn for comparisonS(f)B2(f) [a. u.]

10−6
10−7
10−8
10−9
10−10
10−11
1

10

102

103
frequency [Hz]Figure 4.3: Example of PSD of S(f ) and B 2 (f ).

73

4.2 Run and analysis
as follows,
kCM =

λ
η.
4F LB Γκ2

(4.8)

Finesse and κ is measured before the measurement. Γ is the static ellipticity induced by the mirror and deﬁned as the ellipticity just before applying the magnetic
ﬁeld.
The parameter estimation is performed for every cycle during the data acquisition. From the distribution of kCM of all the cycles, the mean values of kCM and
its uncertainties is obtained.

4.2.4

Time and frequency windows

Time window
As shown in Section 3.5.7, the eﬀect of the disturbance appears in the error
signal. The time window used for the parameter estimation should be chosen
not to include the eﬀect of these disturbance to reduce the bias of the parameter estimation due to them. The information of the error signal is used
to decide the time window.
Figure 4.4 shows the waveform of the error signal taking the low-pass behaviorof the cavity response into the account. The waveform is the average of 200
pulses. The magnet ﬁeld is generated at t = 0, and its peak ﬁeld is 8.3 T.
Though it is not obvious when the eﬀect of the disturbance begins to appear,a large peak appears at t = 9.3 ms and it rises around t = 6 ms. The endof the time window is chosen at least not to include this peak, then decided
to 5 ms after the pulse. The beginning of the time windows is chosen 35 ms
before the beginning of the pulse. The total length of the timing window is
40 ms.
Frequency window
As the sampling rate of the ADC is 160 kHz, the Fourier spectrum spreadfrom 0 Hz to 80 kHz. However, the higher frequency region where VMB
signal would not appear is useless for parameter estimation. The frequency
window is chosen to reduce the bias from these frequency regions and reducethe computing time.
The ideal sensitivity as a function of the end frequency of frequency window
can be calculated from the pulse shape and the noise spectrum measured
without the operation of the magnet. The left side of Fig. 4.5 shows theresult of the calculation. X-axis is the highest frequency used for the parameter estimation and Y-axis is the relative sensitivity against that of using all
frequency components. The right side of 4.5 shows the coherence spectrum
between the magnetic ﬁeld and the error signal from 0 to 300 ms after thegenerating magnetic ﬁeld. A peak at around 1.25 kHz is observed. Thisfrequency region has almost no contribution to the sensitivity, thus can be

74

4.2 Run and analysis

Error signal [a. u.]removed without aﬀecting the sensitivity. From these information, the upper
frequency of the frequency window is chosen to 1 kHz. Before the VMB measurement, the large peak appears at 50 Hz at the frequency noise and intensity
noise of Ie . Currently it is assumed due to the malfunction of Mephisto. To
avoid the eﬀect due to this peak, the lower frequency is chosen not to include
this peak. The right side of Fig. 4.6 shows the relative intensity noise of theellipticity signal during VMB measurement. The left side of Fig. 4.6 shows
the relative sensitivity as a function of the lowest frequency using for the parameter estimation. The frequency region around 50 Hz does not contributeto the sensitivity. From this ﬁgure, the lowest frequency for the parameter
estimation is chosen to be 100 Hz.

0.00150.001
0.00050−0.0005−0.001−0.0015
−10

0

10

20

30

40

50
ms

Figure 4.4: The average of the waveform of the error signal. The pulse is generated
at t=0. The 200 pulsed are used for averaging. The waveform is ﬁltered by thecavity response. From this ﬁgure, the end of the time windows is chosen to be 5 ms.

75

Coherence

Relative Sensitivity

4.2 Run and analysis

1

1.2
1
0.8
0.6
0.4
0.2

10−1
103

102

0

103

freq [Hz]

[Hz]

1

1

10

102
freq [Hz]Relative Intenity noise [1/ Hz]

Relative Sensitivity

Figure 4.5: Left: Relative sensitivity as a function of the highest frequency of frequency window. X-axis is the highest frequency used for the parameter estimationand y-axis is the relative sensitivity against that of using all frequency components. Right: coherent spectrum between the magnetic ﬁeld and error signal after
the generating the magnetic ﬁeld.

10−2
10−3
10−4
10−5
10−6
1

10

102

103

Hz

Figure 4.6: Left: Relative sensitivity as a function of the lowest frequency of thefrequency window. X-axis is the lowest frequency used for the parameter estimationand y-axis is the relative sensitivity against that of using all frequency components.
Right: Relative intensity noise of the ellipticity during the VMB measurement. A
large peak at 50 Hz can be seen. Frequency windows is chosen not to include this
peak.

76

4.3 Run and analysis

4.3

Validation run

The analysis validation run is performed by measuring the Cotton-Mouton (CM)eﬀect of gas. The CM eﬀect has the same magnetic ﬁeld dependence with VMB
eﬀect. Therefore, the measurement of the CM eﬀect of the gas has been used as
the validation of the experimental setup in all the previous experiment. In thiswork, the nitrogen gas, which is one of the most common gas as the validation, isused as the target gas. As the CM eﬀect depends on the pressure of the gas, thepressure dependence is measured. The result is compared with the previous result
to validate that the system and analysis works properly.

4.3.1

Summary of the measurement

The dipole magnet is used for analysis validation. The measurement is performedat the room temperature as CM eﬀect depends on the temperature and all theprevious measurement is performed at the room temperature. As the magnetic is
not cooled, the charged voltage is set to be 200 V to avoid the destruction of themagnet due to the temperature increase.
The mirror chamber and the beam pipe is ﬁlled with nitrogen. The pressure
is measured by the absolute pressure gauge whose range is from 10 Pa to 1.3 kPa.
When gas pressure is more than 100 Pa, the output intensity of Fabry-Pérot cavity becomes unstable. This instability is avoided by decreasing the input power.
The measurement is performed at 4 diﬀerent pressure which are 600 Pa, 700 Pa,
850 Pa and 1000 Pa. For each pressure, total 100 cycles of data are acquired. The
measurement time for each pressure is 1 hour.
Finesse and κ are measured before the measurement. No signiﬁcant pressure
dependence of the ﬁnesse is observed.
Figure 4.7 shows the example of 1st data when gas pressure is 1 kPa. At t=0,
the magnetic ﬁeld is applied. You can see the change of Ie after applying magnetic
ﬁeld whereas no change can be seen in It . Figure 4.8 shows the example of 2nd
data when gas pressure is 1 kPa. The change of Ie can be also seen and its sign isthe same with 1st data though the sign of magnetic ﬁeld is the opposite.
The typical parameters used for the analysis is summarized in table 4.2

77

4.3 Run and analysis

Table 4.2: Summary of the typical parameters during validation run

magnetic field [T]parameter
FinesseIt
Ieκ(α)
Peak magnetic ﬁeld
Field length
Pressure
Temperature

1.2
1
0.8
0.6
0.4
0.2
0
−30

value
4.1 × 105
6.1 × 10−4 [W]
55 [nW]
0.86
0.9 [T]
150 [mm]
600 − 1000 [Pa]22 [◦ C]

−20

−10

0

10

20

30
[ms]

−20

−10

0

10

20

30
[ms]

−20

−10

0

10

20

30
[ms]

It [W]

Ie [W]−957.8 ×10
57.6
57.4
57.2
57
56.8
56.6
56.4
56.2
−30

×10−3
0.618
0.616
0.614
0.612
0.61
0.608
0.606
0.604
−30

Figure 4.7: An example of the measured data of validation run. Top: magnetic
ﬁeld. middle: Ie . Bottom: It . The change of Ie after the magnetic ﬁeld can beseen whereas no change can be seen in It . The range of the Y-axis of the middle
and bottom ﬁgure is ±1.5% of their DC power.

78

magnetic field [T]

4.3 Run and analysis

0
−0.2
−0.4
−0.6
−0.8
−1
−1.2
−30

−20

−10

0

10

20

30
[ms]

−20

−10

0

10

20

30
[ms]

×10−3
0.62
0.618
0.616
0.614
0.612
0.61
0.608
0.606
0.604
0.602
−30
−20

−10

0

10

20

30
[ms]

It [W]

Ie [W]−957.8 ×10
57.6
57.4
57.2
57
56.8
56.6
56.4
56.2
−30

Figure 4.8: An example of the measured data of validation run. Top: magnetic
ﬁeld. middle: Ie . Bottom: It .

79

4.3 Run and analysis

4.3.2

Parameter estimation

N2

Ie/It

kCM is estimated from the acquired data as discussed in the previous section.
Figure 4.9 show an examples of parameter estimation. The black line is the sign
averaged ellipticity which is calculated from 1st and 2nd data. The red line is theestimated signal. The estimated signal can explain the measured ellipticity well.
The DC level of the estimated signal is decided from the ellipticity just before
N2
applying the magnetic ﬁeld. From this estimation, kCM
is calculated from Eq. 4.8.
It should be noted that as sign of Γ can not be decided, the obtained value is theN2absolute value of kCM
.

×10−6

Measured

93.4
93.2
93
92.8
92.6
92.4
92.2
92
91.8
91.6

Estimated−15 −10

−5

0

5

10

15 20
time [ms]Figure 4.9: An example of the parameter estimation.Black line: Measured sign
averaged ellipticity. Red line: Estimated signal.
2

N
Figure 4.10 show the distribution of 100 cycles kCMmeasured at 1 kPa. The
distribution is ﬁtted by Gaussian. From the ﬁtting result, the mean value and its
N2
N 2 at 1 kPa iserror of kCM
is obtained. The measured kCMN|kCM| = (2.241 ± 0.007) × 10−15 [T2 ].
2

(4.9)

The systematic uncertainty is not included here.

4.3.3

Measurement of the pressure dependence

As the CM eﬀect depends on the gas pressure, the pressure dependence is alsomeasured. Figure 4.11 shows its result. For each point, the uncertainties of the xaxis is the accuracy of the absolute pressure gauge (0.2%) and drift of the measured
pressure (1 Pa). The uncertainties of the y axis are the statistic uncertainty and80

4.3 Run and analysis

Entry

Distribution of kCM
χ 2 / ndf

35

8.144 / 823.72 ± 3.00

Constant

30
25

Mean

2.241e− 15 ± 7.244e− 18

Sigma

7.033e− 17 ± 5.231e− 18

20
15
10
5
0

2

2.1

2.2

2.3

2.4

2.5

×10−15[T-2]

N2
Figure 4.10: Distribution of kCMmeasured at 1 kPa. The distribution is ﬁtted by
N2
Gaussian. From the ﬁtting result, the mean value and its error of kCM
is obtained.

uncertainty of the ﬁnesse (0.8%). By ﬁtting measured pressure dependence with
N 2 of nitrogen at unit pressure is obtained aslinear function and from its slope, kCM
N2|kCM| = (2.16 ± 0.05) × 10−18 [T−2 Pa−1 ].

(4.10)The uncertainty is the statistic uncertainty. The systematic uncertainties are discussed in next section.

kCM [ 10-15 T-2]

χ2 / ndf

2.8
2.6

0.6432 / 2

p0

0.07711 ± 0.04004

p1

0.002159 ± 5.232e−05

2.4
2.2
2
1.8
1.6
1.4
600

700

800

900 1000 1100
Pressure [Pa]N2
N2
Figure 4.11: Pressure dependence of kCM. The black dot: Measured kCM
at eachpressure. Red line: Result of the ﬁtting.

81

4.3 Run and analysis

4.3.4

Systematic uncertainties

FinesseThe systematic uncertainty of ﬁnesse comes from the uncertainty of the cavity
length. Measured length between 2 mirrors is 1.35 ± 0.04 m. The systematic
uncertainty of ﬁnesse is calculated to be 3%.
Attenuation factor
The attenuation factor is calculated from measurement of α. The uncertaintyof the estimation of α is typically 2%. Thus, the systematic uncertainty of
the attenuation factor is 2%.
Photo detector
The calibration of the photo diodes are performed by the power meter PM160.
From its accuracy, the systematic uncertainty of gain of the photo diodes are
3 %.
Relative angle between polarizer and magnetic ﬁeld
The misalignment of the polarization angle of the polarizer from 45◦ changes
the ellipticity as follows,sin(2(π− ∆θ)) ∼ cos(2∆θ)
4(4.11)The relative angle between input polarizer and magnetic ﬁeld is adjusted byrotating the polarization angle of the polarizer. The angle of the polarizer isadjusted to be 45◦ against the optical bench. This adjustment is performedby using the scale on the polarizer mount whose interval is 2.5◦ . The magnet
is ﬁxed to the magnet bench and its parallelism against the optical bench is±1.5◦ considering the adjustment range of the adjuster of the magnet bench.
The accuracy of this adjustment is assumed to be 4◦ . From this assumption,
the systematic uncertainty is calculated to be 1%.
Current transformer
The speciﬁed accuracy of the current transformer (CWT-MiniHF400) is 2 %.
As the sensitivity is proportional to square of magnetic ﬁeld, the uncertainties
due to the current transformer is calculated to be 4 %.
Field eﬃciency
To estimate the accuracy of the ﬁeld eﬃciency, the ﬁeld eﬃciency is measured
10 times at the center of the magnet. From the variation of measured ﬁeld
eﬃciency, the uncertainties of the ﬁeld eﬃciency is estimated to be 0.4 %. As
the sensitivity is proportional to square of the magnetic ﬁeld, the contribution
to this uncertainty to the uncertainty of kCM is 0.8 %.
Field length
From the deﬁnition of the ﬁeld length, the uncertainty of the ﬁeld length

82

4.3 Run and analysis
comes from the uncertainty of ﬁeld eﬃciency discussed above and the accuracy of the position of the pick-up coil in the beam pipe. The accuracy ofthe position measurement is assumed to be 1 mm. From this discussion, theuncertainty of ﬁeld length is calculated to be 1%.

Position of beam path
As the magnetic ﬁeld proﬁle is not uniform in the transverse direction along
the beam pipe, magnetic ﬁeld depends on the beam path whereas the ﬁeld
eﬃciency is measured at the center of the beam path. Although, Until now,
it is assumed that the beam path coincides with the center of the beam pipe,
they can be diﬀerent each other during the data acquisition.
To estimate the contribution of this uncertainty, the transverse ﬁeld map iscalculated by ANSYS. Figure 4.12 shows the result of the simulation. The
boxes are Cu-wire of the coils whose height is 3.0 [mm] and width is 1.0 [mm].
The center of the ﬁgure corresponding to the center of the beam pipe whose
inner diameter is 5.35 [mm].
The possible position of the beam center is estimated to be as follows. Whenbeam is close to the surface of the beam pipe, the beam acquires additionalloss due to the interruption of beam by the beam pipe. Finesse of a FabryPérot cavity including loss is expressed as
F =π,
1 − R + Ploss(4.12)

where Ploss is power loss between 2 mirrors. From this expression, to obtainF ∼ 500, 000, the loss should be suﬃciently smaller than 6 ppm. The power
loss of the beam by the surface of the beam pipe is estimated to be
2r2(4.13)
Ploss = exp − 2 ,ωwhere r is the distance from the center of beam to the surface of the beam
pipe and ω is beam radius. From these equations, the minimum distancefrom beam center to the surface of the beam pipe is calculated to be 1.6 [mm]whereas the radius of the beam pipe is 2.67 [mm]. Thus, the position of thebeam center is located within ±1 [mm] from the center of the beam pipe.
From the simulation performed above, the position dependence of the magnetic ﬁeld compared to the center of the beam pipe within this region is+1.1 % and −1.0 %. The contribution of this position dependence to theuncertainty of kCM is +2.2 % and −2.0 %.

4.3 Run and analysis

83

Figure 4.12: The ﬁeld map [T/kA] of the transverse direction along the beam path(simulation). The boxes are the cu-wire. In the center region (red) has a good
uniformity.

84

4.3 Run and analysis

Table 4.3: Summary of systematic uncertainties.
parameter
FinesseAttenuation factor
Gain of photo diodes
polarization angle
Current transformer
Field eﬃciency
Field length
Alignment of beam pipe
Total

4.3.5

value
3%
2%
3%
2%
4%
1%
1%
+2.2%, -2.0%
7.6 %Comparison with previous experimentN2
kCM
is measured by other groups with the same wavelength and at the room temperature. Table 4.4 shows the summary of the previous measurement. The obtained
result in this measurement is consistent with the previous result within their uncertainty.

N2
Table 4.4: Comparison of kCMmeasured by various experiments.

Ref.
[55]
[56]
[46]
[49]
This result

4.3.6

N2
|kCM
| × 1018
2.50 ± 0.11
2.17 ± 0.21
2.02 ± 0.16 ± 0.08
2.00 ± 0.08 ± 0.06
2.16 ± 0.05 ± 0.16

Mirror Rotation dependence

To check that the observed signal is a cross term between birefringence of the mirror
and Cotton-Mouton eﬀect, the rotational alignment dependence of Cotton-Mouton
eﬀect is also measured.
Cotton-Mouton eﬀect is measured at diﬀerence rotation alignment of the mirrors including the local minimum of the static ellipticity. For each point, the pulsed
magnet is operated 10 cycles. Mirrors are rotated clockwise before each measurement to tune the static polarization change of a Fabry-Pérot cavity. Figure 4.13
shows the measured static ellipticity before each measurement.

85

4.3 Run and analysis

Etatic ellipticity Ie/It

Figure 4.14 shows the average of the measured ellipticity at each rotationalalignment. Each waveform is the average of 10 cycles data and mean value issubtracted for comparison. The Number in the caption corresponding to the run
number in Fig. 4.13. The size of measured ellpticity due to Cotton-Mouton eﬀect
depends on the rotational alignment of the mirrors. Before and after the local
minimum of the static ellipticity (cyan line), the sign of the polarization change isreversed. This can be understood as the sign of Γ changes by rotating the mirrors.
Figure 4.15 show the waveform of |I|IteΓ|| except for the that of the local minimum of
the static ellipticity. This value should not depend on the rotational alignment of
mirrors. As expected, all waveforms in ﬁgure 4.15 agree each other.
From these measurements, it is conﬁrmed that the measured ellipticity depends
on the rotational alignment of the mirrors as expected.

10−4
10−5
10−6

2

4

6

8
10
run number

Figure 4.13: Static ellipticity for each measurement. For each measurement, data
are acquired for 10 cycles. Mirrors are rotated clockwise to tune static ellipticity
before each measurement.

86

4.3 Run and analysis

1

Ellipticity

×10−6
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−5
0

2
3
4
5
6
7
8
9
10

5

10

15

20

25

30
ms

|Ie|
| It Γ |Figure 4.14: Cotton-Mouton eﬀect at diﬀerent rotational alignment. The numberin the caption corresponding to the run number in ﬁgure 4.13. The waveforms areaverage of 10 cycles and its DC value is subtracted for comparison.

×10−3

0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
−0.01
−5

1
2
3
4
6
7
8
9
10

0

5

10

15

20

25

30
ms

Figure 4.15: Cotton-Mouton eﬀect at diﬀerent rotational alignment normalized by
Γ. All waveforms agree each other as expected.

87

4.4 Run and analysis

4.3.7

Conclusion of the validation run

N2
From the measurement of the CM eﬀect of nitrogen gas, it is shown that kCMcanbe measured consistently with the previous result within its uncertainty. From thisresult, it can be concluded that the system and analysis procedure can detect thesignal of the CM eﬀect which has the same magnetic ﬁeld dependence properly,
thus this system has the potential to detect VMB.
It is also shown that the sign of the static polarization change of the mirrors canbe reversed by rotating mirror and the signal is proportional to the sign of the static
polarization change of the mirrors. This property is used in the VMB measurement
to cancel the systematic eﬀect which is independent of the static polarization of
the mirrors.

4.4

Vacuum run

4.4.1

Summary of the measurement

After the validation run, the measurement of VMB has performed. The single
racetrack magnet is used for this measurement. The mirror chambers and themagnet are evacuated to vacuum. The magnet is cooled down to liquid nitrogentemperature. The typical parameters during this measurement are summarized intable 4.5.
During the measurement, the data set are divided for every ∼ 100 cycles depending of the timing of the serving of the liquid nitrogen, alignment of the mirrors
or intervals of the measurement. For each data set, the power spectrum of the third
data is averaged and used as S(f ) for the data set. During the run, the resonanceof the cavity is lost during the pulse ﬁeld is generated, or even when the resonanceof the cavity is not lost, the large decrease of the internal power of the cavity isobserved during the pulse ﬁeld is generated. These eﬀect is assumed to be due tothe electromagnet noise from the magnet system. These events are not used for the
analysis. 4% of the total acquired data is discarded and not used for the analysis.
During the measurement, the mirrors are rotated to ﬂip the sign of the static
birefringence of the mirrors while keeping the absolute of Γ same. When the signof Γ is reversed, the sign of the VMB signal appears in the opposite direction. As
the absolute sign of Γ is not measured during the run, the sign of Γ during B-1 andB-2 run are deﬁned as positive, and kcm is calculated as follows,kCM (+) =λη.
4F LB |Γ|κ2(4.14)The sign of γ during C-1 and C-2 is deﬁned as negative, and kcm is calculated asfollows,
kCM (−) = −1 ×λη.
4F LB |Γ|κ2(4.15)

88

4.4 Run and analysis

Finally, kCM averaging the reversed sign of Γ is calculated askCM (+) + kCM (−).
(4.16)
2. The aim of this averaging is the cancellation of the systematic eﬀects which do
not depend on the sign of Γ. Figure 4.16 shows the example of 1st data. At t=0,
the magnetic ﬁeld is applied. The applied magnet ﬁeld is strongest among all VMB
experiments. No large destabilization of the locking can be seen after the magnetic
ﬁeld. This is because of the strong feedback system and isolation of acoustic andmechanical disturbances.

It [W]

Ie [W]

Magnetic field [T]

kCM =

8
6
4
2
0
−30

−20

−10

0

10

20

30
[ms]

×10−6
0.165
0.16480.16460.16440.16420.164
0.1638
−30
−20

−10

0

10

20

30
[ms]

−10

0

10

20

30
[ms]

1.654
1.653
1.652
1.651
1.65
1.649

×10−3

−30

−20

Figure 4.16: Example of the waveform of 1st data in the vacuum run. Top: magnetic ﬁeld. Middle: Ie . Bottom: It .

4.4.2

Cancellation of the noise on It

It is found that at the beginning of the pulse, a small and fast intensity noise appears
on It for every shot. Its appearance becomes clear by averaging It . Figure 4.17
shows the example of the waveform of It averaged during 50 cycles. Highpass ﬁlter
whose cutoﬀ frequency is 100 Hz is applied to highlight the fast intensity change.
For each cycle, the It of the ﬁrst and second data is averaged with appropriate
weight to cancel the eﬀect which is proportional to the magnetic ﬁeld. The intensity
decrease (∆Itspike ) is 0.1 % during 100 µs after the pulse. Currently, the origin andthe mechanism of the noise is not understood. At least, as this shape appears on
It , it is not the VMB signal.

89

4.4 Run and analysis

Table 4.5: Summary of the typical parameters during vacuum run
parameter
FinesseIt
Ieκ(α)
Peak magnetic ﬁeld
Field length
Pressurevalue
3.3 × 105
1.6 × 10−3 [W]
100 [nW]
0.98
8.3 [T]
170 [mm]10−4 [Pa]As far as this noise is generated inside the Fabry-Pérot cavity due to the frequency noise, its eﬀect also appears on Ie . The transfer function from It to Ie isa function of the ﬁnesse and α, and it becomes a ﬁrst-order lowpass ﬁlter whose
cut-oﬀ frequency is equals to the cavity pole when α = 0 as discussed in B.2.
Therefore, Its eﬀect can be canceled by dividing Ie with It ﬁltered by the cavity
response. However, when the noise appears only on It , the noise should not be
included when calculating ellipticity. Thus, the correct waveform of the ellipticity
depends on the origin of the noise on It .
To make the result independent on the origin of ∆Itspike , the information of
the sign of the static birefringence of the mirrors are used. Figure 4.18 shows the
distribution of ∆Itspike calculated from the average of every 50 shots during themeasurement. The left side of Fig. 4.18 shows the measured ∆Itspike during B-1
and B-2, whereas the right side of Fig. 4.18 shows that of during C-1 and C-2.
The size and the sign of the distribution of ∆Itspike is independent of the sign ofthe static birefringence of mirrors. From these information, independent on theorigin of the noise, it appears in the same direction and the same size on ellipticity
not depending on the sign of Γ. Therefore, the eﬀect of ∆Itspike is expected to becanceled out in the ﬁnal result after averaging kCM with diﬀerent signs of Γ.

4.4.3

Parameter estimation

The parameter estimation is performed as performed during the validation run.
The parameter estimation is performed in 2 patterns. One is that assuming the
Itspike is generated inside the Fabry-Pérot cavity, and the other is assuming Itspike
appears only on It .
Firstly, the analysis is performed assuming that the noise on It is generated
inside the Fabry-Pérot cavity. In this case, the corresponding noise should appear
on Ie , thus Ie is divided by It ﬁltered by the cavity response.
Figure 4.19 shows an examples of parameter estimation. As the intensity noise
at 50 Hz is dominant in the time series, all the wavefrom including the signal shapeis drawn after applying bandstop ﬁlter whose central frequency is 50 Hz to make90

Relative intensity change

4.4 Run and analysis

0.15

´10-3

0.1
0.05

ΔItspike

0

-0.05
-0.1

-0.15
-2 -1.5 -1 -0.5 0 0.5

1 1.5 2 2.5
ms

Figure 4.17: Example of the noise on It . The waveform is average of the 50 cycle ofIt (100 shots) after the cancellation of the eﬀect proportional to the magnetic ﬁeld.
At t = 0, the magnetic ﬁeld is generated. The sudden change of the output intensity(∼ 0.01% ) is observed at the same time with the beginning of the magnetic ﬁeld.
it easier to compare the waveforms. The black line is the sign averaged ellipticity.
The red line is the estimated signal. Again, the mean value of the estimated signal
is decided from the ellipticity just before applying the magnetic ﬁeld.
The left side of Fig. 4.20 shows the distribution of kCM measured during B-1
and B-2, and the right side of Fig. 4.20 shows the distribution of kCM measured
during C-1 and C-2. Total 12855 cycles are used for the parameter estimation. Its
breakdown is 6007 cycles for positive Γ and 6848 cycles for negative Γ. The distribution is ﬁtted by Gaussian. The result of the ﬁtting is shown in each ﬁgure. The
p-value for each ﬁtting indicates the reasonable values. Thus, it is good assumption that kCM during all the run distributes obeying on the normal distribution.
From the ﬁtting result, the mean value and its error of kCM is obtained for eachhistograms. kCM after the averaging the sign of Γ is calculated from Eq. 4.16, andthe obtained kCM is
kCM = (2.6 ± 5.4) × 10−21 [T−2 ].

(4.17)The result is consistent with 0 within its uncertainty. The statistical uncertainty
for one cycle can be calculated from the result by using the total event as
cycle
∆kCM
= 5.4 × 10−21 ×√12855 ∼ 6 × 10−19 .

(4.18)The ideal sensitivity calculated from the noise spectrum and the magnetic ﬁeld
is 5.7 × 10−19 . Although the obtained sensitivity is slightly larger than the ideal
sensitivity (5 %), the value is almost identical. Thus, it can be concluded the

91

4.4 Run and analysis
spike

spike

Mean

30

Distribution of ∆ It

−0.0001221

Entry

Entry

Distribution of ∆ It

Std Dev 3.625e−05

50−0.000121
Std Dev 1.993e−05
Mean

40

25
20

30

15

20

10
10

5
0

−3

×10
−0.25 −0.2 −0.15 −0.1 −0.05 spike0
∆ It

0
−0.25

−0.2

−0.15

−0.1

×10−3
−0.05 spike0∆ It

Figure 4.18: Distribution of Itspike calculated from the average of every 50 cycles.
Left: Itspike obtained during B-1 and B-2. Right: Itspike obtained during C-1 andC-2. The size and sign of the noise is not depends on the sign of Γ. Therefore, its
eﬀect is expected to be canceled not depending of the origin of this noise.
sensitivity is mainly limited by the intrinsic birefringence noise not depending on
the magnetic ﬁeld as expected.
To show the eﬀect of the noise on It does not aﬀect the sensitivity, the same
analysis is performed assuming that the noise is appears only on It . In this assumption, the mean value of It is used to calculate the ellipticity as the corresponding
noise does not appears on Ie . Figure 4.21 shows the distribution of kCM during
B-1 and B-2 (left) and kCM during C-1 and C-2 (right). Again, each histogram is
ﬁtted by Gaussian, and p-value is reasonable in this analysis again. kCM after the
averaging the sign of Γ is calculated from Eq. 4.16, and the obtained kCM is
kCM = (5.8 ± 5.5) × 10−21 [T−2 ].

(4.19)

Though the result is slightly diﬀerent from the previous analysis (0.6 σ), the results
are consistent with each other not depending on the assumption of the noise origin
of It . The diﬀerence of the central value is included to the systematic uncertainty .

4.4.4

Systematic uncertainties

Table 4.7 shows a summary of the systematic uncertainties during the vacuum run.
The uncertainties which are in common with the validation run is not discussed
here. (See section 4.3.4. )Position of beam path
As the diﬀerence magnet is used than the validation run, the systematic
uncertainty related to the magnet should be discussed. The uncertainty due
to the position of the beam path is evaluated in the same way as discussed
in the validation run. In this case, however, as the beam path is not parallel

92

Ie/It

4.4 Run and analysis
×10−6
80.84

Measured

80.82

Estimated

80.8
80.78
80.76
80.74
80.72
80.7
80.68

−3

−2

−1

0

1

2

3
4
time [ms]Figure 4.19: Example of the result of the parameter estimation. Black line: averaged ellipticity. Red line: Estimated signal
to the coil, not only the ﬁeld eﬃciency but also the ﬁeld map changes at thesame time depending on the position of the beam path.
Figure 4.22 shows the ﬁeld map calculated by ANSYS at the center of thebeam pipe and at the beam path where the B 2 L becomes smallest within±1 [mm] from the center of the beam path. By comparing these 2 ﬁeld maps,
the uncertainty from the beam position is calculated to be 6.4%.
Eﬀect of the noise on It
As discussed above, the analysis procedure depends on the origin of the noise
on It . In this work, diﬀerent analysis is performed assuming that the noise
on It is generated inside the Fabry-Pérot cavity or the noise appears only on
It . The diﬀerence of the center value of each analysis is 0.65 σ against thestatic uncertainty of the parameter estimation. This diﬀerence is used as thesystematic uncertainty of the analysis procedure.
Eﬀect of the asymmetry of Γ
Although the birefringence signal depends only on the sign of Γ, the size ofthe non-birefringence noise depends on the absolute of Γ. The example ofthe Γ dependence of the noise is summarized in Table 4.6. Basically, all of
the noises should depends on |Γ| or |Γ|−1 . Thus, the cancellation of the nonbirefringence noise depends on the asymmetry of Γ between each run and thenoise source of the systematic eﬀect.
To discuss the eﬀect of the asymmetry of Γ, the mean values of Γ are calculated from the distribution of Γ for each run. The mean value of |Γ| of93

4.4 Run and analysis
Distribution of kCM
Entries

800

6007

Prob

0.2394700
820.6 ± 13.0

Constant

600
500

Mean 6.931e−20 ± 7.612e−21400

Sigma 5.898e−19 ± 5.391e−21

300
200
100
0

−2

−1

0

1

2

×10−18

Entry

Entry

Distribution of kCM
900

900
800
700
600

Entries

6848

Prob

0.1589Mean−6.407e −20 ± 7.613e− 21

Sigma

6.295e−19 ± 5.404e− 21

500
400
300
200
100
0

-2

k CM [T ]876.3 ± 13.0

Constant

−2

−1

0

1

2

k CM [T ]Figure 4.20: Distribution of kCM assuming the noise on It is generated inside the
Fabry-Pérot cavity. Left: kCM obtained during B-1 and B-2. Right: kCM obtained
during C-1 and C-2. Each distribution is ﬁtted by Gaussian. From the ﬁtting
result, the mean value and its error of kCM is obtained.
positive Γ and negative Γ is
⟨ | Γ(+) | ⟩= 6.08 × 10−3

(4.20)⟨ | Γ(−) | ⟩= 7.14 × 10−3 ,(4.21)

where the standard error of each mean value is 0.2 % and 0.1% respectively.
Therefore, the asymmetry of about 16% exists between two measurement.
Currently, the dominant noise source of the non birefringence noise during
the measurement is not understood. Thus, the eﬀect of this asymmetry isevaluated as follows.
When it is assumed that the dominant noise source depends on |Γ|, the average of kCM should be calculated as follows instead of Eq. 4.16,
ΓkCM≡⟨ | Γ(−) | ⟩ kCM (+) + ⟨ | Γ(+) | ⟩ kCM (−)⟨ | Γ(+) | ⟩ + ⟨ | Γ(−) | ⟩(4.22)On the other hand, when it is assumed that the dominant noise source depends on |Γ|−1 , the average of kCM should be calculated as follows,
−1

Γ
kCM
≡

Γ(−) −1 kCM (+) + Γ(+) −1 kCM (−).
⟨ | Γ(+) | −1 ⟩ + ⟨ | Γ(−) | −1 ⟩×10−18-2(4.23)The kCM is calculated to be as follows from the measured asymmetry of Γ,ΓkCM= 7.4 ± 10−21 [T−2 ](4.24)

Γ−1
kCM= −3.3 ± 10−21 [T−2 ](4.25)

Depending on the assumption of the origin of the non-birefringence noise,
the central value of kCM varies ∼ 0.9σ compared to the result obtained in94

4.5 Run and analysis
Distribution of kCM
Entries

800

6007

Prob

0.6348700
Constant

600

1000

Entries

Sigma

300

6.05e −19 ± 5.53e −212000.3241884.2 ± 13.1

Constant

Mean 3.124e−20 ± 7.807e−21

400

6848

Prob

800

832.2 ± 13.2

600

500

400

Mean

−1.959e −20 ± 7.846e− 21

Sigma

6.489e−19 ± 5.567e− 21

200

100
0

Entry

Entry

Distribution of kCM
900

−18

−2

−1

0

1

2
k CM [T-2]

×10

0

−2

−1

0

1

2
k CM [T-2]

Figure 4.21: Distribution of kCM assuming the noise on It appears on It only. Left:
kCM obtained during B-1 and B-2. Right: kCM obtained during C-1 and C-2. Each
distribution is ﬁtted by Gaussian. From the ﬁtting result, the mean value and its
error of kCM is obtained.
Table 4.6: Example of the Γ dependence of the noise sources against the birefringence signal
parameter
Intensity noise
Frequency noise
Polarization rotation
Scattering or stray lightElectromagnetic noise on PDt
Electromagnetic noise on PDeΓ dependence|Γ||Γ||Γ−1 ||Γ−1 | or |Γ||Γ||Γ−1 |

Eq. 4.17. This uncertainty of the average of kCM is included to the systematic
uncertainty due to the asymmetry of Γ, and it is assumed that the true kCM
uniformly distribute within this uncertainty.

4.5

Result

The following result is obtained from the analysis described in this chapter. As therelative sign of kCM can not be decided, the limit is obtained for the absolute valueof kCM|kCM | < 1.8 × 10−20 (95% C.L.)(4.26)
The comparison with the previous experiment in discussed in the next section.

×10−1895(B/I)2 [T2/KA 2]

4.5 Run and analysis

worst pathcenter

0.5
0.4
0.3
0.2
0.1
0

−150 −100 −50

050

100 150
L [mm]

Figure 4.22: Beam path dependence of the ﬁeld map. Red: the ﬁeld map at thecenter of the beam pipe. Black: the ﬁeld map when B 2 ×L becomes smallest within±1 [mm] from center.

Table 4.7: Summary of the systematic uncertainties.
parameter
FinesseAttenuation factor
Gain of photo diodes
polarization angle
Current transformer
Field eﬃciency
Field length
Alignment of beam pipe
Analysis procedure
Asymmetry of Γ

value
3%
2%
3%
2%
4%
4.1 %
1%
6.4%
+3.2 × 10−21 [T−2 ]−21+4.9 × 10[T−2 ], −5.8 × 10−21 [T−2 ]Chapter 5

Discussion and prospect
In the ﬁrst section of this Chapter, the limit obtained in the previous chapter iscompared with the other limits. In the next section, upgrades for future experimentare described.

5.1

Discussion of the experimental result

5.1.1

Signiﬁcance of this work

As discussed in the previous chapter, the obtained sensitivity is almost identicalto the expected sensitivity. Thus, it can be concluded that the experimental setupemphasizing on the reduction of the disturbance and analysis procedure works as
expected. It is succeeded to search for the VMB signal at higher frequency region
where intrinsic noise of the optical system is smaller by taking the advantage of thepulsed magnet. At the same time, it has also succeeded the operation of the pulsed
magnet with the optical system during one month. No sign of the destruction canbe observed at the pulsed magnet throughout and after the measurement, this work
demonstrate the feasibility of the long term operation of the pulsed magnet.

5.1.2

Comparison with previous experiment

Figure 5.1 shows the limit of |kCM | obtained for this work and previous experiment,
and Table 5.1 shows a summary of the comparison with the previous experiment.
Compared with the previous experiment, the strongest the magnetic ﬁeld has
applied inside the Fabry-Pérot cavity in this experiment. Compared to the previous
VMB search with pulsed magnet, the obtained statistic is more than 100 times
larger. This is due to the successful establishment of the stable operation of the
Fabry-Pérot cavity with the high repetitive pulsed magnet.
Though the system works as expected, the obtained sensitivity is not reached
the sensitivity obtained by the previous experiment. The diﬀerence between theworld record sensitivity obtained by PVLAS can be explained well by the diﬀerenceof the parameter in Table 5.1. The diﬀerence mainly comes from the size of the96

97

5.2 Discussion and prospect|kcm [T-2]| (95% C.L.)

This work

−17

10

−18

10

10−19

BMV (pulsed magnet)
PVLAS (permanent magnet)
PVLAS-LNL (supercondicting magnet)
QED expected value

10−20
10−21
10−22
10−23
10−24
2006 2008 2010 2012 2014 2016 2018 2020 2022
[Year]

Figure 5.1: Limit on |kCM | obtained by this work and previous experiments. The
values derives from the following references: BMV [38], PVLAS [35].
magnet and statistic. Figure 5.2 shows the limit to ALPs obtained for this workand previous experiments. Though the obtained limit is not better than the limit
obtained by PVLAS, the sensitivity is already better than other type of ALPs
search experiments around 0.01 to 0.1 eV.

5.2

Upgrade for future experiment

Based on the discussion in the previous section, the upgrade plan for the future
experiment is discussed in this section.
The factors which are related to the sensitivity can be classiﬁed as follow,• Peak magnetic ﬁeld
• Field length• Repetition rate
• Reduction of the systematic• Reduction of the birefringence noise

98

5.2 Discussion and prospect

Table 5.1: Summary of the experimental parameterga [GeV-1]

parameter
Type of the Magnet
2 [T2 ]
Beﬀ
Field length [m]statistics √Noise level [m/ Hz]
Obtained sensitivity

PVLAS
DC
6.5
1.6
60 days
6 × 10−19
2.7 × 10−23

BMV
pulse
20
0.137
0.3 s (100 pulses)∼ 1 × 10−19
2.8 × 10−21

this result
pulse
28
0.17
40 s ( 104 pulses)∼ 1 × 10−19
7 × 10−21

10−3
10−4
10−5
10−6
This work
PVLAS
Inada et. al (LSW)
NOMAD (LWS)
ALPS (LSW)OSCAR (LSW)
Nobuhiro et. al (FWM)

10−7
10−8
10−4

10−3

10−2

10−1

1
ma [eV]

Figure 5.2: Limit on ALPs by this work and previous experiments.

5.2.1

Upgrade of the magnet

The destruction ﬁeld of the current magnet measured at the center of the beam
pipe is 12 T. As the tensile strength of the copper is about 300 MPa, the destruction
ﬁeld of the copper wire in the case of the single wire coil is roughly calculated to be
28 T. At the destruction ﬁeld of the current magnet, the maximum magnetic ﬁeld
is 15 T at the inner wire of the coil. Thus, the maximum magnetic ﬁeld is assumed
to be not limited by the material properties of the wire but the reinforcement
structure. As the destruction occurred at the corner of the coil, it is assumed thatthe destruction is induced by the insuﬃcient reinforcement at the corner. The 1st
step of the upgrade of the magnet is to develop suﬃcient reinforce structure.

99

5.2 Discussion and prospect
cavity length 3.5 m
Magnet length 2.6 m (field length 1.7 m)
I0

Optical bench

B = 18T

It = 30mW

Magnet bench

Charging unit
4.5 kV, 3mF ×2

Figure 5.3: Schematic of the setup after the upgrade. Total 8 magnets are operated
by 2 capacitor bank, and its peak magnetic ﬁeld is 18 T.
The improvement of the magnet should include the improvement of the ﬁeld
eﬃciency. This is because increasing the current leads to higher temperature of
the coil, then makes the repetition rate worse. One of the ideas is adopting the
dipole structure. From the simulation by ANSYS, the ﬁeld eﬃciency becomes 1.8
times larger by keeping the inductance of the coil and the ﬁeld length the same.
This design gives 3.4 times better sensitivity per pulse with the same operation
parameter of the capacitor bank.

5.2.2

Upgrade of the ﬁeld length

The improvement of the ﬁeld length is also needed for better sensitivity. The ﬁeld
length is limited by the length of the cavity and the stored energy at the capacitor
bank. Thus, the improvement of these items is needed for the enhancement of theﬁeld length.
The maximum charging energy of the capacitor bank is 30 kJ, where current
operating energy is 6 kJ. Therefore, 3 more magnets can be operated with thecurrent capacitor bank. The required cavity length is at least 2.4 m. The pulse
width can be kept the same when connecting 4 magnets 2 series and 2 parallel.
The stored energy of the capacitor needed to be increased to operate more magnets or to operate with stronger peak magnetic ﬁeld. The current capacitor bankhas two charging unit, and one of the two is not used during this work. Therefore, 4more magnets can be operated by installing another capacitor whose capacitance is
3 mF. when using 8 magnets, required minimum cavity length becomes 3.5 m. The
cutoﬀ frequency of the Fabry-Pérot cavity becomes 2.5 times higher. The input
power to the Fabry-Pérot cavity is needed to be increased 2 times larger to keep
the contribution of the shot noise the same.
Another advantage of using multiple magnets is that the magnetic ﬁeld can begenerated in diﬀerent direction at the same time. When generating the magnetic
ﬁeld in the opposite direction at the same time, the VMB signal is not changed as

100

5.2 Discussion and prospect

it depends on the square of the magnetic ﬁeld, but the background which depends
on the linear of the magnetic ﬁeld can be canceled for one shot. This leads to themore reduction of the systematic noise.

5.2.3

Improvement of the repetition rate

The thermal constant needed to be reduced to improve the repetition rate. Figure
5.4 show the cross section of the single coil racetrack magnet and the thermal resistance of each material is described. The thermal constant is determined from theheat capacity of the coil and thermal resistance of the backup and insulating materials. The heat capacitance can not be reduced as far as using the same structureof the coil. The kapton ﬁlm is necessary for insulation of the coil. Therefore, thereduction of the thermal resistance of the glass epoxy (0.081 k/W) or the stainlessback up (0.07K/W) is required to improve the thermal constant.
The glass epoxy is used for the safety, thus it can be removed totally as the
kapton ﬁlm has enough electric breakdown voltage. It occupies 40% of total thermal resistance. It is not straightforward to estimate the possible reduction of thestainless backup as it also aﬀects the destruction of the coil. The basic idea is
to make the thickness along y-axis thin where Maxwell stress is not strong, andmake holes in the backup. By these upgrades, it is aimed to reduce the thermal
resistance half times smaller. The repetition rate becomes half times smaller due
to this upgrade.
After the upgrade of the capacitor bank, there are still room to generate 20%
larger magnetic ﬁeld for each magnet. This leads to 40% better sensitivity for eachshot. From the numerical calculation, the increase of the ﬂowing current 20% larger
results in the increase of the temperature of the coil 30 K higher than current setup.
After all, the repetition rate could be reduced to 13 s/pulse from 18 s/pulse.

5.2.4

Gain on kCM from the upgrade of the magnet system

Finally, the possible improvement of the sensitivity by the upgrade of the magnet
system is summarized in 5.2. The sensitivity per pulse can be calculated assuming
the birefringence noise level is the same as follows,∆kCM = 9.2 × 10−21 [T−2 /pulse].

(5.1)

This can be interpreted as the sensitivity per second as√∆kCM = 3.4 × 10−20 [T−2 / Hz].

(5.2)

This sensitivity is 2 times better than the obtained sensitivity by PVLAS.

5.2.5

Upgrade of a Fabry-Pérot cavity

From the study of the intrinsic birefringence noise of the Fabry-Pérot cavity, it isfound that the unexpected birefringence noise limits the sensitivity. To upgrade

101

5.2 Discussion and prospect~

0.075mm : 0.039 K/W
mm : 0.081 K/W  _6kV
14mm : 0.070 K/W

SUS
backupステンレス補強DUWUDÍs
(0.07(0.070 K/W)
K/W)
½¡B;T

kaptonfilm (0.039K/W)=NIWÅÑ
(0.039 K/W)

y
>RD<O?C
t0.4mmCI
glass epoxy
sheet t=0.4mm(0.081 K/W)
(0.081K/W)

x

z

100mm

Figure 5.4: Cross section of the magnet. The heat resistance of each material is++SUS7¶$Â4$&w
noted.

·¼Ù 0.41W/(m*K)!
#3XG10Zg%
Table 5.2: Summary of the gain on kCM by the upgrade of the magnet system. It¾·¼# _ª!
is assumed that theÊ#Ú
observed birefringence noise of the Fabry-Pérot cavity is thesame after the upgrade.
&#ß$4!Cu-Ag!77K!300K%´parametercurrent Value
upgradegain on kCM&3ÁCu!7Á
Peak ﬁeld [T]
8.3
18
4.8
Field length [m]Repetition rate [Hz] √Expected sensitivity [T−2 / Hz]

0.17
0.05
1.5 × 10−18

1.4
0.075
6
−20
3.4 × 10

8
1.18
46

the sensitivity of a Fabry-Pérot cavity, the understanding of the mechanism of theobserved noise excess is necessary. It is assumed that the noise is generated insidea Fabry-Pérot cavity. Therefore, the improvement of ﬁnesse has less contribution
to the optical pass length sensitivity. In the same way, the increase of the input
power to a Fabry-Pérot cavity improves the sensitivity above ∼700 Hz where thesignal power is not strong. When the cavity length becomes 3.5 m by the increaseof the magnet, the required output power of the ordinary light is 30 mW to makethe contribution of the shot noise small enough.
The ﬁrst step to understand the unexpected birefringence noise further is to
identify whether the noise origin is the thermal like noise of the mirror or not.
One idea is to study the temperature dependence of the noise. This found to be
diﬃcult by previous experiment [35]. Another idea is to study the beam spot size
dependence. Generally, the eﬀect of thermal noise of mirrors become small whenbeam spot size become larger. The study can be achieved building the shorter
Fabry-Pérot cavity. For example, the beam spot size becomes 0.5 times smallerwhen cavity length is 20 cm with the same mirror. Other idea is to study the102

5.2 Discussion and prospectmaterial dependence of the coating or the substrate of the mirror to see the material
dependence of the noise.

5.2.6

Reduction of the systematic uncertainty

The systematic uncertainty should be also reduced for the future experiment.
The largest systematic uncertainty comes from the uncertainty regarding to theasymmetry of Γ. To reduce this uncertainty, the understanding and the reduction of
the non birefringence noise observed in Fig. 4.20 is necessary. Currently, although
the origin is not understood, it is assumed that the origin is related to the movementof the beam pipe against the beam path as discussed in 3.5.7. For the investigationof the eﬀect of the beam pipe, the dependence of the size of the noise against
the material of the beam pipe or against the radius of the beam pipe should bestudied in the future. The property of the noise can be studied by measuring the|Γ| dependence of the non birefringence noise as the dependence depends on theorigin of the noise as discussed in Chapter 4.4.4.
Another systematic uncertainty comes from the spike noise on It which is discussed in Chapter 4.4.2. The origin of the noise can be identiﬁed by measuring
the It by two independent photodetector. When the noise is generated inside the
Fabry-Pérot cavity, the same noise is expected to be observed for each detector,
whereas no coherence is observed when the noise is generated outside the FabryPérot cavity. The origin of the noise can be also identiﬁed by measuring the noise
on It without the Fabry-Pérot cavity. When noise is generated outside the FabryPérot cavity, the same noise is expected to appear even without the Fabry-Pérot
cavity.

5.2.7

Improvement of the analysis procedure

The analysis procedure can be also improved. Currently, the length of the timing
window is chosen to be 40 ms. The length is chosen so that the distortion of theFourier spectrum by the large peak at 50 Hz becomes small and the spectrum hasenough frequency resolution to remove the eﬀect of the peak. Instead of that, it isalso possible to remove the eﬀect of the peak structures by ﬁltering the waveformwith notch ﬁlters in time domain before the Fourier transformation. This proceduremakes it possible to reduce the length of time window shorter. Although the
expected sensitivity does not depend on the length of time windows as far as thesignal shape is well-included in the time window, it could be beneﬁcial for the
Fourier transformation. Currently, the rectangular window is used as the windowfunction. As the signal is located on the end of the timing window, it is diﬃcult
to use another window function which gradually becomes 0 at the beginning andthe end of the timing window. When the length of the timing window becomes∼10 ms, the signal located at the center of the timing window. At this situation,other window functions can be applied, which would lead to reduce distortion of
the spectrum due to the discontinuity of the waveform.

5.2 Discussion and prospect

103

It could be also possible to suppress the eﬀect of disturbance by analysis procedure. The VMB signal spectrum distributes among the wide frequency region due
to its timing proﬁle, whereas the spectrum of the eﬀects of disturbance would havepeaks when they are generated by the mechanical resonance of the experimental
setup. These peak can be eliminated by applying the band-pass ﬁlter before performing the Fourier transformation. It is also possible to suppress these eﬀects by
not using these frequencies for the parameter estimation.

Chapter 6

Conclusion
The search for vacuum magnetic birefringence (VMB) has performed using the highrepetitive pulsed magnet and the high ﬁnesse Fabry-Pérot cavity in this thesis. The
experimental setup to operate the Fabry-Pérot cavity and the pulsed magnet hasdeveloped. The peak magnetic ﬁeld is 8.3 T which is the strongest magnetic ﬁeld
as VMB search and ﬁeld length is 0.17 m.
Dedicated study of the intrinsic noise of the optical system is performed up to
1 kHz. The noise evaluation method with the homodyne detection scheme is established both theoretically and experimentally.
√ The optical pass length sensitivity
at signal frequency reaches to 1 × 10−19 m/ Hz which is 5 times better than thesensitivity at around 10 Hz, which is used in the VMB search with pulsed magnet.
This result shows the advantage of the pulsed magnet. However, even in the highfrequency, birefringence noise excess is observed, and indicates the existence of thebirefringence noise source not depending on the detection scheme or the detail ofthe experimental setup even at the high frequency region.
The calibration of the system and analysis is performed by measuring the
Cotton-Mouton eﬀect of N2 gas. The obtained result is consistent with the previous
experiments.
The search for VMB has performed from 13. Nov. to 10. Dec. Total 26000
pulses are generated during the operation. No signiﬁcant signal is observed andobtained limit to kCM is|kCM | < 1.8 × 10−20 (95%C.L.)(6.1)

This experiment demonstrates the feasibility of VMB search with the high repetitive
pulsed magnet, especially the stable operation of the Fabry-Pérot cavity with thepulsed magnet and analysis method. With using multiple magnets with stronger
ﬁeld, the improvement of the sensitivity on kCM can be achieved in the future
experiment.

104

Appendix A

Optical cavity and PDH
methodFabry-Pérot cavity is an optical resonator which is composed with 2 high reﬂective
mirrors. Pound-Drever-Hall (PDH) method is a basic method to extract the diﬀerence between a resonance frequency of the cavity and frequency of the input laser.
The extracted information is used as an error signal for the feedback system to lockthe laser frequency to the resonance frequency of the cavity. In this section, ﬁrstly
the basic properties of Fabry-Pérot cavity is explained. After that, a detection
scheme of PDH method is explained.

A.1

Fabry-Pérot cavity

A.1.1

Transmittance and reﬂectance

A Fabry-Pérot cavity is an optical resonator consists of 2 high reﬂective mirrors
facing each other. Fig. A.1 show a schematics of a Fabry-Pérot cavity. A part of
input laser transmitted front mirror. For each reﬂection, a part of transmitted lightis reﬂected by mirrors and the rest is transmitted from mirrors.

t1

Ein

t2
Eout

Er

r1

r2

Figure A.1: A schematics of a Fabry-Pérot cavity
105

106

Transmitted power [arb]A.1 Optical cavity and PDH method

1
0.8
0.6
0.4
0.2
0

−6

−4

−2

0

2

4
6
Phase [rad]Figure A.2: Example of transmitted power from a Fabry-Pérot cavity as a functionof phase ϕ when r2 = 0.99 and t2 = 0.01
The output of a Fabry-Pérot cavity can be written as
X
Eout = Ein t1 t2(r1 r2 e−2ϕ )n
n

(A.1)

t1 t2 eiϕ,= Ein
1 − r1 r2 e−2iϕwhere ϕ = 2π Lc ν is a phase change during traveling along the cavity length. From
above expression, the amplitude transmittance and reﬂectance of a Fabry-Pérot
cavity can be deﬁned as follows.
t1 t2 eiϕ,1 − r1 r2 ei2ϕ
r1 t1 t2 ei2ϕrF P = r1 −.
1 − r1 r2 ei2ϕtF P =

(A.2)
(A.3)

The intensity of transmitted light and reﬂected light from a Fabry-Pérot cavity
can be written as follows,
It = |tF P Ein |2 =Ir = |rF P Ein |2 =t21 t22|Ein |2 ,(A.4)((t21 + t22 )r1 − r2 )2 + 4r1 r2 (r12 r22 )sin2 (ϕ)|Ein |2 .
4r1 r22(1 − r1 r2 )2 ( 1+(1−r
sin
(ϕ))
2
1 r2 )(A.5)

4r1 r22(1 − r1 r2 )2 ( 1+(1−r2 sin (ϕ))
1 r2 )

Hereafter, let us assume r1 = r2 = r and t1 = t2 = t for simplicity. As an exampleof transmitted power from a Fabry-Pérot cavity is shown in Fig. A.2 as a function
of ϕ.

107

A.2 Optical cavity and PDH method

A.1.2

Finesse, free spectrum range and photon lifetimec
From Eq. (A.4),when ν = m × 2L
(m ∈ Z), the transmittance become maximum.
c
The frequency interval between each maximum νF SR = 2L
is called free spectrum
range (FSR).
Finesse is deﬁned as the ratio between FSR and FWHM of each resonancecurve.

F inesse =

νFSR
νFWHM

=

πr
.
1 − r2(A.6)

Finesse is a index of a sharpness of cavity resonance and its depends on the amplitude reﬂectance of each mirror.
A photon life time is a decay constant of an internal power of a Fabry-Pérot
cavity. Let’s assume a steady state Fabry-Pérot cavity and at certain time, theinput laser to a Fabry-Pérot cavity is suddenly turned oﬀ. For each round trip,
the amplitude loss of the light inside a Fabry-Pérot cavity ∆E can be expressed asfollows,∆E = −(1 − r2 )Eintra .
(A.7)
where Eintra is internal amplitude of a Fabry-Pérot cavity. From this expression, adiﬀerential equation about internal amplitude of the light can be derived as follows,
dEintrac= − (1 − r2 )Eintra .
dt
2L

(A.8)

By solving above equation with a initial condition that is Eintra (t = 0)isE0 , thetime evolution of the amplitude inside the Fabry-Pérot cavity become
Eintra (t) = E0 e− 2L (1−r )t .
c

2

(A.9)

Therefore, the time evolution of the output power from a Fabry-Pérot cavity becomes as follows,c2
Iout (t) = Iin (0)e− L (1−r )t .
(A.10)
Now, a photon lifetime τ of a Fabry-Pérot cavity is deﬁned as a decay constant of
the output power of a Fabry-Pérot cavity.
Lc(1 − r2 )
FL
∼
.
cπ

τ=

(A.11)

Thus, by measuring the decay constant of a Fabry-Pérot cavity, ﬁnesse of a FabryPérot cavity can be calculated. This method is called cavity ringdown method.

108

A.2

A.2 Optical cavity and PDH method

PDH method

From the previous section, the resonance condition of a high ﬁnesse Fabry-Pérot
cavity is very severe. Suppose input laser frequency is 300 THz, a cavity length is
1 m and ﬁnesse is 300,000, the FWHM is corresponding to cavity length ﬂuctuationof 3 pm. It is necessary to obtain the diﬀerence between laser frequency andresonance frequency and feedback it to laser frequency or cavity length to keep theresonance condition. PDH method is one of the most basic method to extract thediﬀerence.

A.2.1

a basic setup

A basic setup for PDH method is shown in Fig. A.3. An Electric Optical Modulator(EOM) is used to modulate the frequency of input laser. The amplitude after themodulation can be written as follows,
Ein = E0 exp(iωt + iβsin(Ωt))∼ E0 eiωt [J0 (β) + J1 (β)eiΩt − J−1 e−iΩt ],(A.12)

where Ω is a modulation frequency by EOM, β is called modulation depth and Jm
is Bessel function of m-th order. PDH method uses the reﬂected light from a cavity
to extract the errors signal. As the amplitude reﬂectivity of a Fabry-Pérot cavity isa function of acquired phase for each trip during the cavity length, the reﬂectivity
depends on the frequency of the input laser. In this situation, the amplitude ofreﬂected light can be written as
Er ∼ E0 eiωt [rF P (ω)J0 (β) + rF P (ω + Ω)J1 (β)eiΩt − rF P (ω + Ω)J−1 e−iΩt ]. (A.13)
The power of the reﬂected light has a constant term and oscillation term of afrequency Ω and 2Ω. By detecting the reﬂected with a photo detector and demodulating it with a frequency Ω by a Mixer, the obtained voltage signal can be written
as follows.
VP DH = −χqIin × 2J0 (β)J1 (β)Im[rF∗ P (ω)rF P (ω + Ω) − rF P (ω)rF∗ P (ω − Ω)],
(A.14)
Where q A/W is an eﬃciency of photo diode and χ V/A is the eﬃciency of themixer and gain of photo diode circuit. If ω is near the resonance frequency ω0 , ω
can be written as ω = ω0 + δω. It is also assumed that Ω is large than FWHM. In
this situation, the amplitude reﬂectivity of each frequency can be written as
rF P (ω) ∼ rF P (0) + rF′ P (0)δω,(A.15)

rF P (ω + Ω) ∼ 0.

(A.16)

109

A.2 Optical cavity and PDH method

EOM
Laser

Photo detector
DemodulationRF

PDH signal

Error signal [ard]Error sigal [ard]Figure A.3: Schematic of the PDH method. An EOM is used to modulate theinput laser frequency. The information of the resonance frequency is implemented
on the reﬂected light from the mirror.

1
0.5
0

1
0.5
0

−0.5

−0.5

−1

−1
−0.4 −0.20

0.2 0.4
phase [rad]−0.01 −0.005

00.005 0.01
phase [rad]Figure A.4: Example of PDH signal as a function of a phase acquired during the trip
between 2 mirrors when r2 = 0.999 and t2 = 0.001. Left:PDH signal in wide phaseregion. A sharp peak can be seen around the resonance frequency Right: PDH
signal enlarged around the resonance frequency. PDH signal almost proportional
to the diﬀerence between laser frequency and resonance frequency.
By substituting these expressions, the PDH signal becomes
VP DH ∼ −χqIin × 2J0 (β)J1 (β)Im[r′ (0)]δω= −χqIin ×

8t2 rLJ0 (β)J1 (β)δω(1 − r2 )2 c

(A.17)

Fig. A.4 shows normalized PDH signal. At around the resonance PDH signal
is almost proportional to the diﬀerence between laser frequency and resonancefrequency. Therefore, PDH signal can be used as an error signal for feedback oflaser frequency to sustain the resonance of a Fabry-Pérot cavity.

110

A.2 Optical cavity and PDH method

A.2.2

Basic properties of PDH method

In this section, basic properties of PDH method is shown.
The detection limit
As PDH method uses a power of reﬂected light to extract error signal, its
detection limit is deﬁned by the shot noise of reﬂected light.
By assuming r2 + t2 = 1, the eﬃciency of PDH signal can be written as
VP DH ∼ −χqIin ×

8LF
J0 (β)J1 (β)δω,
πc(A.18)≡ GPDH δν.
ωwhere ν = 2π
and GPDH V/Hz is eﬃciency of PDH method. At resonance,
the DC power of reﬂected light is

IDC ∼ 2|J1 (β)|2 Iin(A.19)The shot noise ishotnoise at the photo diode induced with this PD power ispishotnoise = 2eq2|J1 (β)|2 Iin .
(A.20)
Thus, the minimum frequency diﬀerence which can be extracted by PDH
method isχ × ishotnoiseδνmin =GPDH
r
(A.21)ce=.
8J0 (β)LF qIin
Frequency response
For simplicity, let us assume that r ∼ 1 and ϕ ∼ 0. In this situation, theamplitude reﬂectance of a Fabry-Pérot cavity can be written as
rFP ∼

i2ϕ
.
1 − r2 + i2r2 ϕ(A.22)

When the input laser has small phase noise, the amplitude of the input laser
can be written as
Ein = E0 eiωt+iϕ(t)iωt= E0 e+ iE0 eZiωt′dω ′ ϕ̃(ω ′ )eiω t .

(A.23)

When the drive frequency of the EOM is larger than FWHM of a Fabry-Pérot
cavity, the amplitude of the reﬂected light from the input mirror becomes asfollows,
Z
′
iωt
iωt
dω ′ rFP (ω + ω ′ )ϕ̃(ω ′ )eiω t
Er = J0 rFP (ω)E0 e + iJ0 E0 e
(A.24)+ isin(Ωt)J1 Ein .

111

A.2 Optical cavity and PDH method

When the input laser frequency satisﬁes the resonance condition, the intensityof the reﬂected light can be written asZ′
Ir = I0 |J0 rFP (ω) + iJ0 dω ′ rFP (ω ′ )ϕ̃(ω ′ )eiω t + isin(Ωt)J1 |2 .
(A.25)
By demodulating reﬂected power by sin(Ωt), we can obtain the error signal
asZ′
Verr ∝ 4J0 J1 dω ′ rFP (ω ′ )ϕ̃(ω ′ )eiω t
Z= 4J0 J1
Z
∝

dω ′

′

dω

′

ωL
i 2F
π c

ω′ L
1 + 2Fπ c′ϕ̃(ω ′ )eiω t(A.26)1′
ν(ω ′ )eiω t .
ω′1 + i 2πfc

At the last line in the above equation, the cavity pole frequency fc is deﬁned
as
c
fc =
,
(A.27)
4F L
which deﬁnes a typical time constant of a Fabry-Pérot cavity.
From these expression, one can see that the PDH signal acts as a 1st orderlowpass ﬁlter against the frequency noise and its cutoﬀ frequency is fc .

Appendix B

Birefringent Fabry-Pérot Cavity
This section shows the basic knowledge of the birefringent Fabry-Pérot cavity. As
mirrors have small but non-zero birefringence, a Fabry-Pérot cavity itself behaves
as a birefringence material. The birefringence of the cavity leads a polarization
dependence of the resonance frequency, attenuation of VMB signal and additional
dynamical. The dynamic behavior of the Fabry-Pérot cavity against VMB signal
in the heterodyne detection scheme is originally discussed in [53]. In this section,
dynamical behavior of birefringence cavity in the case of the homodyne detection
scheme with static mirror’s birefringence is derived.

B.1

Mirrors as a birefringence material

To take into account the birefringence of the mirror, it is assumed that each mirror
has phase retardation αi (i = 1,2) and placed with incident angles θi (i=1,2) against
the polarization angle of the input laser. These 2 mirrors can be described as
one equivalent waveplate whose phase retardation is αeq and incident angle is θeq .
αeq [43]. θeq is expressed by αi and θi aspαeq = (α1 − α2 )2 + 4α1 α2 cos2 (θ1 − θ2 ),
(B.1)
α1
+
cos(2(θ
−
θ
))
1
2
1
).
θeq = arccos( q α2
(B.2)
α
α
2
( α12 − 1)2 + 4 α21 cos2 (θ1 − θ2 )To taken into account the birefringence of mirrors, the Jones matrix for the reﬂection by mirror should be modiﬁed. From above discussion, it can be interpreted
for each reﬂection, each mirrors acts as a waveplate whose phase retardation isαα′ ≡ 2eq , and its angle is θeq . Thus, the Jones matrix should be modiﬁed asfollows.
R = reiϕ′′e−iα /2 cos2 (θeq ) + eiα /2 sin2 (θeq )−isin(α′ /2)sin(2θeq ).
′′−isin(α′ /2)sin(2θeq )
e−iα /2 sin2 (θeq ) + eiα /2 cos2 (θeq )
(B.3)
112

113

B.2 Birefringent Fabry-Pérot Cavity

B.2

resonance frequency splitting

A unique feature of mirror birefringence is that the resonance frequency of the cavity depends on the polarization. In VMB measurement, the laser frequency is lockto the resonance frequency of input polarization where the signal light is orthogonal
to the input polarization. Therefore, the signal light is slightly oﬀ resonance and it
leads to the attenuation of the signal. To take into account these eﬀects, the signal
should be calculated with taking into account the mirror birefringence. In basic
calculation is done by PVLAS experiment with the assumption that θeq = 0 [53].
In this experiment θeq is small but non-zero. Therefore, we assume that the θeq is
non-zero but we can ignore higher order term of θeq .
When θeq is satisﬁes 1 ≫ θeq , the Jones Matrix for the reﬂection can be approximated asR ∼ reiϕ′

e−iα /2−isin(α′ /2) × 2θeq′′
−isin(α /2) × 2θeqeiα /2

.

(B.4)

These mirrors are used to generate static birefringence term in ellipticity. Thus,
the amplitude of ordinary light becomes as follows,
Et = At Tout Tin∞
X(RX)2n XEinn

= At t e (I − (RX)2 )−1 XEin
−1

′
′
1 − r2 ei(2ϕ−α )ir2 ei2ϕ (2Ψe−iα + sin(α′ )2θeq )2 iϕ
Ein∼ At t e′′
ir2 ei2ϕ (2Ψeiα + sin(α′ )2θeq )1 − r2 ei(2ϕ+α )
2 iϕ

∼

=

At t2 eiϕ(1 − r2 ei(2ϕ−α′ ) )(1 − r2 ei(2ϕ+α′ ) )
′′
1 − r2 ei(2ϕ−α )−ir2 e−i2ϕ (2Ψe−iα + sin(α′ )2θeq )Ein′′−ir2 e−i2ϕ (2Ψeiα + sin(α′ )2θeq )1 − r2 ei(2ϕ+α )
!
2 iϕ
t e
1−r2 e2ϕ+α′

0

.
(B.5)

The amplitude of extraordinary light becomes as follows,∞
X(RX)2n XEin
Ee = Ae Tout Tinn= At t e (I − (RX)2 )−1 XEin
2 iϕ

∼

0

′

−ir2 e−i2ϕ (2Ψeiα +sin(α′ )2θeq )1−r2 e2ϕ+α′1−r2 e2ϕ−α′t2 eiϕ

(B.6)

!
.

114

B.3 Birefringent Fabry-Pérot Cavity
αFrom Eq. (B.5), the cavity is on resonance when ϕ − 4eq = mπ (m ∈ Z). Whencavity is on the resonance, the measured ellipticity becomes as,

F 2
Ie
′
−iα′
π
+ 2Ψ(t)]2(t) =αeq [sin(α )2θeq e
2F 2
2
It
1 − ( π ) sin ( 2 )αeq≡ k(αeq )2 [Γe−i 2 + ψ]2(B.7)∼ k(αeq )2 (Γ2 + 2Γψ).
In the last line of above expression, the attenuation factor k(αeq )2 is deﬁned as1k(αeq ) = 1+( 2F )2 sin(α. As its depends on the birefringence of the mirrors, the
/2)
eq

π

Transmittance [arb]birefringence of the mirrors induces the attenuation of the ellipticity. This is understood as follows. Due to the birefringence of the mirror, the optical pass lengthof the mirrors depends on the polarization. This leads to the polarization dependent resonance frequency. Now, it is assumed that input polarization is almostaligned to the eigen mode of the resonance frequency. Thus, the ordinary lightand extraordinary light in on the diﬀerence mode of resonance frequency. Thus,
the ellipticity is attenuated when ordinary light is on the resonance. In previous
research, it is measured that αEQ ∼ 10−6 rad [53]. In Fig. B.1, an example of thetransmittance of ordinary light from a Fabry-Pérot cavity and ellipticity signal isshown. The used parameters are F = 300, 000 and αEQ = 4 × 10−6 [rad]. Thesetransmittance can not be 1 at the same time.
Transmittance of ordinary light

1.2

Transmittance of ellipticity

1
0.8
0.6
0.4
0.2
0
−15

−10

−5

0

×10−6
5
10
15
Phase [rad]Figure B.1: Example of the signal attenuation by birefringence of the mirrors.
Red: The transmittance of the ordinary light of a Fabry-Pérot cavity. Blue: transmittance of ellipticity signal from a Fabry-Pérot cavity. The used parameters are
F = 300, 000 and αEQ = 4 × 10−6 rad

115

B.3 Birefringent Fabry-Pérot Cavity

B.3

Frequency response of a Fabry-Pérot cavity against
the ellipticity

In this section ,the frequency response of a Fabry-Pérot cavity towards an ellipticity
which is not discussed in Section 2 is discussed. The frequency response of the
Fabry-Pérot cavity when the birefringence of the mirrors is also considered. The
frequency response of a Fabry-Pérot cavity against the VMB signal with non-zero
mirror birefringence is originally discussed in [53] in the case of the heterodyne
detection scheme. In this section, we derive the frequency response for homodyne
setup. In addition to that, the frequency response against input power modulationand frequency modulation is also derived. Because of the non zero static ellipticity
generated in the mirror, the results are slightly diﬀerent compared to [53].
VMB signal
From Eq. B.4, it is approximated that for one round trip during the mirror,−i(sin(α′ /2) × 2θeq + 2Ψ(t))Eintra appears as the amplitude of extraordinary
light where Eintra is the amplitude of the ordinary light in the cavity. It isalso approximated that after n-th round trip, the relative phase delay which
the extraordinary light acquires against the ordinary light is nα.
The amplitude of the extraordinary light transmitted from cavity at time t issum of the amplitude of the extraordinary light generated before t − n × 2Lcand reﬂected n-th times. When cavity is on the resonance, the transmitted
light can be written as follows,e
2 −αeq j) ×Eout(t) = itm Σ(rme2L
(sin(αeq /2) × 2θeq + 2Ψ(t −j) E tc2 −αeq j= itm Σ(rm
e
) ×


Z
−iω(t− 2Lj)cEt(sin(αeq /2) × 2θeq + 2 dωΨ(ω)eZ
sin(αeq /2) × 2θeq
2Ψ(ω)eiωt= itm E t × (
+
dω
)
2L
−iα
2 e
eq
2 e−i(αeq −2 c ω)
1 − rm1 − rmZ
F sin(αeq /2) × 2θeq
2Ψ(ω)eiωt+dω)= Ein (
π
1 + i Fπ αeq
1 + i Fπ (αeq − 2Lc ω)ZΓψ(ω)eiωt= Ein (+dω).
1 + i Fπ αeq
1 + i Fπ (αeq − 2L
c ω)

(B.8)

From, above equation, the elliptivity can be written as follows
et
Ie /It = |Eout|/|Eout|2= k(α)Γ + 2ΓZ
dωH(ω)Ψ(ω)eiωt ,

(B.9)

116

B.3 Birefringent Fabry-Pérot Cavity
where H(ω) is deﬁnes as
H(ω) =

Fπ(1 − i Fπ αeq )(1 + i Fπ (αeq − 2L
c ω))

F
π

+

.
(1 + i Fπ αeq )(1 − i Fπ (αeq + 2L
c ω))
(B.10)

When αeq is 0, H(ω) reduces to 1st order low-pass ﬁlter as discussed in Chapter 2, and when ω is 0, H(ω) becomes 2F
π k(α) as discussed in Chapter B.2.
intensity noise or modulation
For simplicity, it is assumed that the birefringence inside the mirror is all
static and it is denoted as γ. The extraordinary light from the Fabry-Pérot
cavity can be calculated ase
Eout∼ ite Σ(re2 eiαeq )j × 2γE t eij2ϕ
te
= i2γ
Et .
2
1 − re eiαeq +2ϕ(B.11)

where te and re is the transmittance and reﬂectance against extraordinary
light, and ϕ is acquired phase for each trip. Et is the amplitude of theordinary light inside the cavity and it can be written as follows,Et =tt eiϕEin ,1 − rt2 ei2ϕ(B.12)

where te and re is the transmittance and reﬂectance against ordinary light andEin is the amplitude of the input light. For simplicity, it is also assumed thatre = rt and te = tt . From these equations, the amplitude of the extraordinary
light can be expressed by the Ein and ϕ as follows,E e ∼ i2γ
∼ i2γ

te
tt
Ein
2
i2ϕ
1 − re e 1 − rt2 ei2ϕ
t
t
1−r2
1−r2
E
r2 (αeq +2ϕ)
2r2 ϕ in
1 − i 1−r
1 − i 1−r2
2

(B.13)≡ iΓtFP (ω)hFP (ω)Ein .
In the last line it is deﬁned as Γ ≡ 2F
π and tFP deﬁned in Appendix. A isused. hFP (ω) is deﬁned ashFP (ω) ≡

1
1−i

r2 (α

eq +2ϕ)1−r2.

(B.14)

When the input power is modulated at Ω and its modulation is small, the117

B.3 Birefringent Fabry-Pérot Cavity
amplitude can be expressed as
Ein = E0 + E1 sin(Ωt)


Z
E1
′
′
′
′
= dΩ E0 δ(0) +(δ(Ω − Ω) − δ(Ω − Ω) )eiΩ t
2i
Z


′
≡ dΩ′ E0 δ(0) + Ẽ1 (Ω′ ) )eiΩ t(B.15)

where E1 ≪ 1. When input light is on the resonance and modulation frequency is small enough compared to FSR, the power of the extraordinary
light becomes as follows,
Z

 ′
2
dΩ′ tFP (Ω′ )hFP (Ω′ ) E0 δ(0) + Ẽ1 (Ω′ ) eiΩ t
Ie = Γ
Z2′= Γ tFP (0)hFP (0)E0 + dΩ′ tFP (Ω′ )hFP (Ω′ )Ẽ1 (Ω′ )eiΩ t
Z

′
2 2
2
= k(α)Γ E0 + ΓdΩ′ h∗FP (0)tFP (Ω′ )hFP (Ω′ ) + hFP (0)t∗FP (−Ω′ )h∗FP (−Ω′ ) E0 Ẽ1 (Ω′ )eiΩ tZ′
2 2≡ k(α)Γ E0 + 2 dΩ′ TFP (Ω)E0 Ẽ1 (Ω′ )eiΩ t ,2(B.16)
in the last line, the transfer function from input power modulation to thepower of extraordinary light modulation is deﬁned as
1
TFP (Ω) = Γ2
FF2(1 + i π αeq )(1 − i π αeq − i ffc )(1 − i ffc )(B.17)1.
+ Γ2
2(1 − i Fπ αeq )(1 + i Fπ αeq − i ffc )(1 − i ffc )Ωwhere f = 2π
and fc = 4Fc L .

From above equation, when αeq is 0, it is shown that a Fabry-Pérot acts as
the 2nd order low-pass ﬁlter against the input intensity modulation whose
cutoﬀ frequency is fc . This result is consistent with previous calculation [57].
In the same way, the transfer function from the intensity noise of It to theintensity noise of Ie can be derived as follows,
TFP (Ω) = Γ2
+Γ2

12(1 + i Fπ αeq )(1 − i Fπ αeq − i ffc )12(1 − i Fπ αeq )(1 + i Fπ αeq − i ffc )(B.18).

This expression becomes a ﬁrst order low-pass ﬁlter against whose cutoﬀ
frequency is fc when alphaeq = 0. This result is consistent with previous
calculation [57].

118

B.3 Birefringent Fabry-Pérot Cavity

frequency noise or modulation
In the same way as the intensity modulation, the frequency response against
the laser frequency modulation can be also calculated.
It is assumed that the input laser is injected to the Fabry-Pérot cavity andits frequency is slightly detuned from its resonance frequency and is detuned
phase is ϕ0 . In this situation, the amplitude of the input laser becomes asfollows,
Ein = E0 eiωt (1 + iϕ(t)),(B.19)The output intensity of the extraordinary light can be written as follows,Ze
I = const + df HFP (f )ν̃(f )e−2πf t ,(B.20)
where HFP is the transfer function of the input laser frequency modulation
1 dϕ(t)ν ≡ 2πdt to the intensity of the extraordinary light.
HFP (f ) =
−

Γ2 /f
f
f
F
2F
F
2F
2(1 + i 2Fπ ϕ0 )(1 + i π αeq )(1 − i fc − i π ϕ0 − i π α)(1 − i fc − i π ϕ0 )

Γ2 /f
f
f
F
2F
F
2F
2(1 − i 2Fπ ϕ0 )(1 − i π αeq )(1 − i fc + i π ϕ0 + i π α)(1 − i fc + i π ϕ0 )(B.21).

Appendix C

Charging unit
This Chapter describes the basic properties of the charging unit. which is originallydeveloped for the ALPs search at SPring 8.

C.1

Basic speciﬁcations

The charging unit consists with the capacitor and the control unit. Figure C.1
shows the picture of the overview of the charging unit and that of the capacitor
unit. Table C.1 summaries the speciﬁcation of the charging unit.
Table C.1: Speciﬁcation of the charging unit
parameter
size (control unit)
size (capacitor)
Capacitancemaximum charged voltage
detector noise
number of the control unit

C.2

values [unit]
1 m × 1.5 m × 2 m
2m×2m×1m
3 mF± 4.5 kV
3.5 × 10−21
2

Operation sequence

Figure shows the schematic view of the circuit of the charging unit. The unit has
two type of switches. One is the mechanical relay (Rc ) controlling the charging of
the capacitor, the the other is the thyristors (T1 , T2 ) to control the discharge tothe magnet. Rc controls the start, stop and the sign (positive /negative) of thecharging to the capacitor. The feature of this unit is that it has two anti-parallel
thyristors. They are normally opened, and the direction of the following current
119

120

C.2 Charging unit

Figure C.1: Picture of the charging unit. Right: Overview of the charging unit. It
consists of the control unit and capacitor unit. Left: Picture of the capacitor. The
capacitor unit consists of total 12 capacitors whose capacitance is 0.25 mF.
can be determined by closing one of them selectively. The operation of the charging
unit when the repetitive operation mode is selected is designed as follows,
1. all switched are opened.
2. Rc is closed to charge the positive voltage. Rc is opened when the charged
voltage reaches to the pre-deﬁned value.
3. T1 is closed and the pulse ﬁeld is generated. A part of the current is consumed
as the joule heating, the residual returns to the capacitor. It charges thecapacitor in the negative direction automatically.
4. T1 is opened. Rc is closed to charge the negative voltage.
5. T2 is closed, and the negative pulse ﬁeld is generated. A part of the current
is consumed as the joule heating, the residual returns to the capacitor again.
It charges the capacitor in the positive direction automatically.

121

C.2 Charging unit

Charging relay (Rc )power supply@ charging unit

Thyristor (T1, T2)

magnetCapacitor

Figure C.2: Schematic view of the charging unit. The unit has 2 switches. One
is the charging relay (Rc ) to control the charging to the capacitor. The other isthyristors (T1 , T2 ) to control the discharge to the magnet.

Appendix D

Discussion of the possible
birefringence noise source
In this section, the possible candidate of the noise source which could explain theobserved noise excess is discussed.

D.1

Evaluation of the correlation with other possible
noise source

There are other possible and reported noise source which induce the optical passlength diﬀerence noise, although the transfer mechanism is not understood. As
there is no theoretical prediction of their transfer function, the coherence between
obtained noise spectrum and possible noise source is studied.
Mechanical vibration of the system
In previous experiment, it is observed the seismic motion of the optical bench
induce the optical pass length diﬀerence noise [58]. To study the possibilityof this coupling, the displacement noise of the system and its coherency withthe birefringence noise is studied. The setup is composed with 5 mechanicallyisolated parts. As an example, the measured displacement spectrum of the
optical bench is shown in left side of Fig. D.1. From these displacement
spectrums, the coherence spectrum with the optical pass length diﬀerence
noise is calculated. The result is shown in the right side of Fig. D.1. Except forthe sharp peaks around 15 Hz and 50 Hz, no strong coherence can be observed.
The coherence between measured noise and quantization noise of ADC whichis totally uncorrelated with the measured noise is drawn for comparison. In
the same way, the coherence spectrum between 4 mechanically isolated partof the setup is measured. The result is shown in Fig. D.2. Again, no strong
coherence is observed between measured noise and displacement noise of thesystem. From these measurements, it can be conclude that the displacement
noise is not the origin of the noise excess.
122

123

D.1 Discussion of the possible birefringence noise source

Input beam pointing
The coherence between the pointing ﬂuctuation of reﬂected light from the
input mirror and birefringence noise is observed in previous experiment [59]with lower ﬁnesse conﬁguration. They assume that the input beam angle
ﬂuctuation couples with birefringence of the coating. To study this coherencein our setup, the beam pointing ﬂuctuation of reﬂected beam is monitoredwith QPD. Figure D.3 shows the measured coherence. Again, no strong
coherence has been observed.

horiz

10−7

vert

−8

10

10

Ratio

10−6

Coherencem/ Hz

Pointing ﬂuctuation on the mirror

1.4

horiz

1.2

vert

1

ADC noise

0.8

10−9

0.6

10−10

0.2

1

0.4
0

10−11
1

10

2

10

Hz

10−1
10−1

110

10
1022

10

103
Hz Hz

Optical bench
Output chamber
Input chamber
Magnet bench

1.4
1.2
1
0.8
0.6

Coherence

Coherence

Figure D.1: Left: displacement spectrum of the optical bench. Right: Coherent
spectrum between the birefringence noise and displacement noise. No signiﬁcantcoherence can be observed except for some sharp peaks. The coherence between
ADC noise and the birefringence noise is also shown as an example of the coherence
spectrum between uncorrelated noise.

Optical bench
Output chamber
Input chamber
Magnet bench

1
0.8
0.6
0.4

0.4

0.2

0.2

0

0
1

10

102

Hz

1

10

102

Hz

Figure D.2: Left: Coherent spectrum between the birefringence noise and displacement noise of horizontal direction at 4 diﬀerence points. Right: coherent spectrum
between vertical motion and birefringence noise. Again, no signiﬁcant coherence
can be observed except for some sharp peaks.

124

D.1 Discussion of the possible birefringence noise source
Another proposed story to generate additional birefringence noise is pointing
ﬂuctuation at the mirror surface couples to non-uniformity of the birefringence of the mirrors’ surface [60]. Assuming that birefringence on the mirror
depends on the point at the surface, the birefringence acquired after the
Fabry-Pérot cavity can be written asγ = γi (X1 , Y1 , X2 , Y2 ),(D.1)

where (Xi , Yi ) is the point of the center of the laser at the surface of each mirror whose coordinate system is ﬁxed to the surface of each mirror. Assumingthat the pointing ﬂuctuation is small enough, the birefringence ﬂuctuation
can be written as
∆γ =

∂γ
∂γ
∂γ
∂γ
∆X1 +
∆Y1 +
∆X2 +
∆Y2 ,
∂X1
∂Y1
∂X2
∂Y2

(D.2)

Coherence

where (∆Xi , ∆Yi ) is the pointing ﬂuctuation at each mirror against the coordinate system ﬁx to each mirror’s surface. It is also assumed that as mirrors
are ﬁx to the optical bench rigidly, the pointing ﬂuctuation (∆Xi , ∆Yi ) is not
so much diﬀerent from the pointing ﬂuctuation (δxi , δyi ) whose coordinate
system is ﬁxed to the optical bench. After all, the ﬂuctuation of birefringencecan be written as follows

Pitch
1

Yaw

0.8
0.6
0.4
0.2
10

102

103
Hz

Figure D.3: The coherence spectrum of the birefringence noise spectrum and pointing noise of the reﬂected light. No strong coherence can be observed except for somesharp peaks.

D.2 Discussion of the possible birefringence noise source

125

∂γ
∂γ
∂γ
∂γ
∆x1 +
∆y1 +
∆x2 +
∆y2
∂X1
∂Y
∂X2
∂Y2

1
∆x1
 ∆y1 ≡A·
 ∆x2 
∆y2

(D.3)

∆γ =

To evaluate the contribution of this eﬀect, the measurement of the pointing
ﬂuctuation and A is performed. The conceptual view of this study is shownin Fig. D.4. The pointing ﬂuctuation at each mirror’s surface transferred
to each QPD by appropriate lense. The coherence spectrum of measured
birefringence noise and pointing noise measured at each QPD is shown in
Fig. D.5. No strong coherence is observed.

Laser

Lense1QPD1

Lense2QPD2Polarizer
M2
M1

Analyzer

PDe

Figure D.4: Schematic view of the set up to study the correlation between the
pointing ﬂuctuation and birefringence noise. QPD1 (QPD2) is used to monitor the
pointing ﬂuctuation at the M1 (M2). Lense1 (Lense2) is used to transfer the spotat the M1 (M2) to the QPD1 (QPD2).
It should be noted that these measurements just show that at least there are nocoherence with birefringence noise and possible noise candidate. The amount of thecontribution of these noise to the birefringence noise is still not clear. For future
works, the measurement of the transfer function of these noise to the birefringencenoise should be performed. In order to perform these measurement, the setup
which can modulate the alignment of the mirrors, input pointing and displacementof the system is needed.

D.2

Comparison with thermal noise

The possibility that the thermal noise could become the dominant noise sourceof VMB search had pointed out by the previous experiment. The measured optical pass length diﬀerence noise is compared with the thermal noise derived in126

D.4 Discussion of the possible birefringence noise source

Appendix E. Among the thermal noise discussed in Appendix E, the largest contribution comes from the photo-elastic eﬀect coupling with the brownian motion.
Figure D.6 show the comparison between measured noise and theoretical predictionof the thermal noise. Although the spectrum has the same frequency dependence,
the observed noise is 10 times larger than the predicted thermal noise level. It
should be noted that the thermal noise derived in Appendix E is assuming theisotropic material. There are still possibility that the unexpected anisotropy of thematerial could induce the birefringence thermal noise.

D.3

Comparison with the lower ﬁnesse conﬁguration

Coherence

Figure D.7 shows the measured optical pass length diﬀerence noise with F ∼ 20, 000
and the estimated noise budget around 40 Hz to 2 [kHz]. The setup for the measurement is the same with the setup in chapter 3 except for the reﬂectance of themirror. From Fig. D.7, it is clear that the noise budget can explain the measured
noise spectrum well. The dominant contribution comes from shot noise and frequency noise. This result supports the validity of the noise estimation around thisfrequency region. This result also supports the story that the unexpected noise isgenerated inside a Fabry-Pérot cavity. If the birefringence noise generated outside
a Fabry-Pérot cavity, the same noise spectrum with Fig. 3.20 should be appearedin this measurement and as the noise ﬂour is proportional to ﬁnesse−1 , it should
dominate the measured noise spectrum.

Input pitch
Input yaw
Output pitch
Output yaw

1.2
1
0.8
0.6
0.4
0.2
0
1

10

102
[Hz]

Figure D.5: Coherence spectrum between birefringence noise and measured pointing noise of the mirror. Again, no strong coherence is observed.

D.4 Discussion of the possible birefringence noise source

127

Optical pass length difference [m/ Hz]

Measured
Thermal noise

10−17
10−18
10−19
10−20

1

10

103

102

Hz

Optical pass length difference [m/ Hz]

Figure D.6: Comparison between measured noise spectrum and thermal noise calculation. Measured noise is almost 10 times larger than the calculated noise assuming
the isotropic material

Measuredfreq noiseInternal power
Detector noise
shot noise
Total

10−15
10−16
10−17
10−18
10−19
102

103

Hz

Figure D.7: Measurement of the optical pass length diﬀerence measurement withlower ﬁnesse cavity and its noise budget.

128

D.4

D.5 Discussion of the possible birefringence noise source

Comparison with previous experiment

The noise excess is also reported in the previous experiment. According to [35],the observed noise spectrum in previous experiment can be explained by sum of 2
diﬀerence function and it can be written as follows,
S(f ) =BA+ √ [m2 /Hz],2
f × (1 + (f /f0 ) )
f

(D.4)

Optical pass length difference [m/ Hz]

where A = 4 × 10−36 , B = 2.1 × 10−37 and f0 = 15. To discuss the possibility
that the noise source of each experiment has common origin, the measured noise
spectrum in this experiment is ﬁtted with Eq. (D.4) including the contribution of
the shot noise at higher frequency by setting A and B as free parameters. The
result is shown in Fig. D.8.The red line is the ﬁtting result by Eq. (D.4) including
the contribution of the shot noise. One can see that the noise spectrum measured
in previous experiment can not explain the noise spectrum in this experiment.

10−17

10−18

10−19
1

10

102

103

Hz

Figure D.8: Comparison with the noise spectrum measured in previous experiment.
The black line is measured noise in this experiment. The red line is the ﬁtting result
by the Eq. D.4 including the contribution of the shot noise.

D.5

Summary of the noise excess

From the discussion above, the characteristic of the observed noise excess is summarized as follows.

D.5 Discussion of the possible birefringence noise source

129• From the mirror rotation dependence, the noise behaves as birefringence.
• The noise is generated by a Fabry-Pérot cavity• Thermal noise due to the photo-elastic eﬀect can not explain the observed
noise level.
• From the comparison with previous result, the noise spectrum depends onthe experimental setup.
• Coherence with vibration noise or pointing noise, which are reported previously is not observed.

Appendix E

Thermal noise
According to the ﬂuctuation dissipation theorem [61], a physical system connectedto the thermal bath receive ﬂuctuation from the thermal bath depending on theenergy dissipation of the system and the temperature of the thermal bath. As boththe substrate and the coating of mirrors have internal mechanical loss and heatloss, physical quantities corresponding to these energy losses ﬂuctuate [62]. Theinternal mechanical loss results in the shape ﬂuctuation and the heat loss resultsin the temperature ﬂuctuation of the mirrors. These thermal ﬂuctuation become
the cavity length ﬂuctuation which is called thermal noise. It is known thermal
noise is the one of the fundamental noise sources of experiments with a Fabry-Pérot
cavity [63] including the gravitational wave detectors [64].
In the previous experiment, the possibility that the thermal noise could be thesource of the observed noise excess is pointed out [37]. According to the study, thepossible candidate is the coupling of the photo elastic eﬀect with strain ﬂuctuationof the mirror. But so far, no theoretical calculation of the thermal noise from theﬂuctuation dissipation theorem has been performed and only qualitative consideration has been made. In this section, the possible eﬀect of the thermal noise to
the birefringence measurement is discussed and the ﬁrst direct calculation of thebirefringence thermal noise due to the photo elastic eﬀect is performed.

E.1

General consideration

In this experiment, what is measured is the optical pass length diﬀerence between
2 orthogonal polarization for each round trip inside the cavity ∆L.
∆L = d × ∆n × sin(2θ),(E.1)

where ∆ n is the birefringence of the material, d is the thickness of the materialand θ is the relative angle between the input polarization and the optical axis of
the material. It is known that the thermal noise results in the ﬂuctuation of the
thickness of the material and the refractive index of the material. The eﬀect ofthese ﬂuctuation is discussed.
130

131

E.3 Thermal noise

E.2

Eﬀect of thickness ﬂuctuation

Due to the thermal noise, the thickness of the substrate and the coating of themirror ﬂuctuate. According to Eq. E.1, the thickness ﬂuctuation only in the birefringence material could cause the optical pass length diﬀerence noise when input
polarization angle against the fast (slow) axis of the mirror is not zero. Although
in the previous study, it is assumed that θ = 0, as the coating of the mirrors
have birefringence and input polarization angle is not zero in this experiment, thethickness ﬂuctuation of the coating is needed to be concerned.

E.2.1

Brownian noise

Brownian noise is the ﬂuctuation of the mirror’s surface due to the internal frictionof the material. The power spectrum of the surface displacement of the coating
measured by the Gaussian laser is calculated as follows [65],2kB T 1 − σ 2
SxB (f ) = √ϕeﬀ ,π 3 f ωY

(E.2)

where Y is the Young modulus, σ is the Poisson’s ratio, ω is the radius of the laserand ϕeﬀ is the eﬀective loss angle of the coating which represent the energy loss inthe coating. One can roughly expect that the ﬂuctuation of the optical pass length
diﬀerence due to this Brownian motion isS∆L;B (f ) ≲ SxB (f ) × ∆n2c [m2 /Hz],(E.3)

where ∆nc ∼ 10−6 is the birefringence of the coating.
SxB (f ) is calculated from the value in Table E.1. From these calculations, theexpected eﬀect of the Brownian motion to the optical length diﬀerence isq√S∆L;B (f ) ≲ 6 × 10−24 /f 1/2 [m/ Hz].
(E.4)
This is negligibly small compared to the shot noise limit sensitivity.

E.2.2

Thermoelastic noise

It is also known that the mirror’s surface ﬂuctuates due to the thermal expansionof the material cause by the temperature ﬂuctuation. This thermal noise is called
thermoelastic noise [66–68]. The power spectrum density of thermoelastic noise of
the coating is calculated as follows,
SxTE (f ) =8(1 + σ)2 α2 d2 kB T 2√[m2 /Hz].
πω 2 κρCf(E.5)

The size of the induced optical pass length diﬀerence can be calculated as follows
like the previous section,q√
S∆L;T E (f ) ≲ 6 × 10−25 /f 1/4 [m/ Hz].
(E.6)
This eﬀect can be also neglected compared to the shot noise.

132

E.3 Thermal noise

Table E.1: Material parameters used to calculate the thermal noise of mirrors [62]Parameter(Symbol)
Thermal expansion coeﬃcient (α)
Thermo refractive coeﬃcient (dn/dt)
Thermal conductivity (κ)
Speciﬁc heat capacity (C)Density (ρ)
Young module (Y)Poisson ratio (σ)
refractivity (n0 )photoelastic coeﬃcient (β)
Loss angle of the coating (ϕc )value
3.6 × 10−6 [1/K]
8 × 10−6 [1/K]
1.38 [W/mK]
745 [J/Kms]
2200 [kg/m3 ]70 [GPa]
0.17
1.45
0.23
3.3 × 10−4

Table E.2: Material paramters used to calculat the thermal noise of mirrors [62]Parameter(Symbol)
Thermal expansion coeﬃcient (α)
Thermo refractive coeﬃcient (dn/dt)
Thermal conductivity (κ)
Speciﬁc heat capacity (C)Density (ρ)
Young module (Y)Poisson ratio (σ)
refractivity (n0 )Loss angle of the coating (ϕc )photoelastic coeﬃcient (β)

value
3.6 × 10−6 [1/K]
14 × 10−6 [1/K]
33 [W/mK]
290 [J/Kms]
7200 [kg/m3 ]140 [GPa]
0.23
2.06
3.3 × 10−4
0.17

Table E.3: Material parameters used to calculate the thermal noise of mirrors
Parameter(Symbol)
Temperature (T )beam waist (ω)
wavelength (µ)

value
300 [K]
0.5 × 10−3 [m]1064 [nm]133

E.3 Thermal noise

E.3

Eﬀect of birefringence ﬂuctuation

E.3.1

Thermorefracrive eﬀect

The refractive index of material depends on the temperature. The thermo-refractive
coeﬃcient is deﬁned as
∂n
β≡
.
(E.7)
∂T
It is known this coeﬃcient couples to the temperature ﬂuctuation of the coating
and generates the eﬀective cavity length ﬂuctuation [69]. It power spectrum is
SxTR (f ) =2kB T 2√β 2 λ2 .
πω 2 πf κρC(E.8)

Though this is not birefringence ﬂuctuation, when the birefringence also depends
on the temperature, the birefringence noise could also appear as discussed in [37].
TRS∆L(f ) ≲

2kB T 2√β 2 λ2 ,πω 2 πf κρC bire

(E.9)

where βbire is deﬁned as∂∆n.
(E.10)∂T
For now, such a properties of the coating has not been reported. As the static
birefringence of the mirror is order of 10−6 , It can be guessed that βeﬀ is at least
TR (f ) issmaller than 10−6 . With these assumptions, the amount of S∆L
βeﬀ ≡

TR
S∆L
(f ) < 1.6 × 10−20 .

(E.11)

This is at least more than 10 times smaller than the measured noise level.

E.3.2

Photoelastic eﬀect of longitudinal strain

The stress applied to the material changes the refractive index of the material. Thiseﬀect is referred as photoelastic eﬀect [70, 71]. In general, this eﬀect is expressedwith the photoelastic tensor pijkl as
ϵ−1
ij = pijkl ∂k ul = pijkl skl ,(E.12)

where ϵ−1
ij is inverse permittivity tensor, ul is the linear displacement and skl
is the stress tensor. Here it is assumed that the photoelastic tensor is an isotropic
tensor which can be written as
pijkl = αδij δkl + βδik δlj + βδil δkj ,(E.13)First, let us discuss the eﬀect of the longitudinal deformation of the mirror’s surface

134

E.3 Thermal noisedue to the thermal noise which is well studied in the context of the gravitational
wave detectors. Let’s assume the light traveling along the z axis and input polarization in along the x axis. In this case the inverse permittivity tensor becomes,

1/n20 + αszz
0
−1
ϵ =
.
(E.14)
0
1/n20 + αszz
The eigen value λi of this tensor is
λ = 1/n20 + αszz ,(E.15)As inverse permittivity tensor has only one eigen value, it is clear that the material
is not birefringent.
From these calculations, it is concluded that the as far as the material isisotropic, the longitudinal strain could not induce any birefringence ﬂuctuation.
So far, no anisotropy of the photoelastic tensor of the coating of the mirror isreported.

E.3.3

Photoelastic eﬀect of transverse strain

In the previous study, it is pointed out that, in general, the eﬀect of photo elastic eﬀect due to the transverse strain ﬂuctuation of the coating should be aboutlcoating 2
times smaller comparing eﬀect due to the longitudinal strain, whereω
lcoating is thickness of the coating which is much smaller than the beam size [72].
The transverse eﬀect is negligible comparing to the longitudinal eﬀect and has notbeen calculated so far. Considering birefringence, however, as the longitudinal effect is canceled out, it is needed to take into account the strain in the transverse
plane of the mirrors.
Again, it is assumed that the photoelastic tensor is an isotropic tensor which
can be written as
pijkl = αδij δkl + βδik δlj + βδil δkj ,(E.16)In this situation, the impermiitivity tensor becomes
1/n20 + (α + 2β)sxx + αsyy
2βsxy
−1
ϵ =
,
2βsxy
1/n20 + (α + 2β)syy + αsxx

(E.17)

where the strain in longitudinal direction is ignored.
The induced birefringence can be calculated from the diﬀerence of the eigan value
λi like previous section and it becomes as follows,
∆nt =

n30
n3 q
(λ1 − λ2 ) = 0 β (sxx − syy )2 + 4s2xy .
22The azimuth angel ϕ of the optical axis of this material satisﬁes




cosϕ sinϕ
λ1 0cosϕ −sinϕ−1
ϵ =.
−sinϕ cosϕ
0 λ2
sinϕ cosϕ(E.18)(E.19)

135

E.3 Thermal noise

When the input polarization is along the x-axis, the measured optical pass length
diﬀerence after traveling ∆z becomesn3∆l = ∆z 0 ×2Z(λ1 − λ2 )I(z)e−2

x2 +y 2
ω2

sin(2ϕ)dxdy ×

2
,
πω 2

(E.20)

From Eq. (E.19), ϕ satisﬁes the relation shown below,
1(λ1 − λ2 ) × sin(2ϕ) = βsxy .
2(E.21)

By using this relation, the opticl pass length diﬀerence can be written as follows,n3∆L = 0 ×2Z

4βsxy I(z)e−2

x2 +y 2
ω2dxdydz ×2.
πω 2(E.22)As this value is ﬁnite in general, the transverse deformation could induce birefringence ﬂuctuation. To estimate the ﬂuctuation of this term, several simpliﬁcations
is assumed. First, as coating is thin, it is assumed that the strain does not dependon z and the mirror occupies the inﬁnite half plane and its surface locates as z = 0.
In addition to that, it is assumed that the material parameter of the coating is samewith that of the substrate except for the loss angle. It is also assumed that theeﬀective traveling length of the laser for each reﬂection is zeﬀ . This is calculated
later.
As ssy is deﬁned as sxy = 12 (∂x uy + ∂y ux ), by substituting this relation andperform integration by parts, ﬁnally, ∆L becomes as follows,n32∆L = 2βzeﬀ 0 ×
×
2
πω 2

Z
(

2
2
4y
4x
−2 x +yω 2 dxdy.
u
+
u
)e
x
y
ω2
ω2

When we deﬁne the weighting vector p⃗ asp⃗ = 2βzeﬀ

n30
2 

×
2
πω 2 

2

2

x +y
4y
× e−2 ω2
ω2
2
2
−2 x +y
4x
ω2
×
e
2
ω

(E.23)

,

(E.24)

0
the optical pass length diﬀerence can be written as
Z∆L = ⃗u(x, y, z = 0) · p⃗ dxdy

(E.25)

According to the FDT calculation and Levin’s theorem [73], the power spectrum
density of ∆L can be calculated byS∆L =2kB T Wdiss,
π2f 2 F 2

(E.26)

136

E.3 Thermal noise

where Wdiss is dissipated power when force pressure F p⃗ sin(2πf t) is applied to thesurface of the mirror.
In the case of the brownian motion, Wdiss can be written as
Wdiss = 2πf Umax ϕ,(E.27)

where Umax is maximum stored energy under the applied pressure. Umax can becalculated by solving the elastic equation when applying force pressure F p⃗ sin(2πf t)
at z = 0. Finally the power spectrum density becomes as follows,2n30
2kB T (1 + σ)(2 − σ)
BPE
∆S∆L = 2βzeﬀ
× 3/2
ϕeﬀ
2
ω3Y
π f
(E.28)
2 
n30
zeﬀ 2 2kB T (1 + σ)(2 − σ)
= 2β
×
× 3/2
ϕeﬀ .
2
ω
ωY
π f
The eﬀective loss angle ϕieﬀ can be calculated as discussed in [65]. Though coating
has larger loss angle, as it is thin, less energy is stored in the coating. Thus theloss angle should be weighted by the ratio of stored energy in each material. The
eﬀective loss angle should be written as follows,
ϕeﬀ = ϕs +

δU
dϕc ,
U

(E.29)

where δU is the stored energy in the coating per unit length and U is the total
stored energy.
In our case, the ϕeﬀ becomes as follows,
ϕeﬀ = ϕs +4(5 − 2σ) d√ϕc .
(2 − σ) π ω(E.30)The evaluation of zef f is performed as follows. In the calculation of the thermorefractive noise, the eﬀective length ﬂuctuation when the refractive index of thematerial of the multi layers ﬂuctuate is calculated as follows [69],λλ n1 n2 π
∆n1
∆n2
∆z =
× δϕ =+ n1
n2.
(E.31)
4π
4π n21 − n22n1n2
where δϕ is the phase delay for each reﬂection and ni is the refractive index of the
odd or even layer. From this calculation, the phase delay acquired by the light
whose polarization is parallel to the fast axis of the material is


∆n2;f
∆n1;f
n1 n2 π+ n1,(E.32)
δϕf = 2n2n1n2
n1 − n22
where ∆ni;f denotes the refractive index ﬂuctuation of the fast axis. And for thelight whose polarization is parallel to the slow axis is


∆n1;s
∆n2;s
n1 n2 πn2
δϕs = 2+ n1,
(E.33)n1n2
n1 − n22

137

E.3 Thermal noise

where ∆ni;s denotes the refractive index ﬂuctuation of the slow axis. As it isassumed that the photoelastic tensor for each material is the same, the angle offast axis for given stress is the same for each material. The acquired ellipticity forthe light whose incident angle is θ is calculated to be∆ψ = (δϕf − δϕs ) × sin(2θ).

(E.34)

By the deﬁnition of the optical pass length diﬀerence (Eq. (2.23) ), the optical passlength diﬀerence due to this ellipticity is
λ(δϕf − δϕs ) × sin(2θ)
2πλn1 n2 π n1n2=×(∆n2;f − ∆n2;s ) + (∆n1;f − ∆n1;s ) × sin(2θ).
2π n21 − n22 n2n1(E.35)∆L =By comparing this equation with Eq. E.1, zeﬀ for even (odd) layer can be determinedasn2 π n1
λ= 4π
× nn21−n2 n2(E.36)

n2 π n2λzeﬀ;2 = 4π
× nn21−n
2 n .
1(E.37)

zeﬀ;1

1

1

2

2

After all, the formula to calculate the optical pass length diﬀerence noise due becomes as follows,2
n31 zeﬀ;1
n32 zeﬀ;2
2kB T (1 + σ)(2 − σ)
BPE
∆S∆L = 2β
+ 2β
× 3/2
ϕeﬀ .
(E.38)
2 ω
2 ω
ωY
π f

E.3.4

Photo-thermo-photo-elastic eﬀect

As mirrors of a Fabry-Pérot cavity has non zero absorption of laser power and thebeam proﬁle is not uniform, the absorbed power induce a temperature gradient.
This temperature gradient results in the thermal stress and birefringence due tothe photo-elastic eﬀect. This phenomenon is called thermal birefringence. The
detailed calculation is studied in [70, 71].
For simplicity, let us ignore the longitudinal strain and the strain has an axial
symmetry against the z-axis. It is also assumed that the photo elastic tensor is an
isotropic tensor which can be written as
pijkl = αδij δkl + βδik δkj + βδil δkj ,(E.39)In this situation, when we use cylindrical coordinate (r, θ, z), the impermittivity
tensor can be written as
2
2
2
2
ϵ−1
xx = (α + 2β)(cos θsrr + sin θsθθ ) + α(sin θsrr + cos θsθθ )(E.40)2222
ϵ−1
yy = α(cos θsrr + sin θsθθ ) + (α + 2β)(sin θsrr + cos θsθθ )ϵ−1xy = 2β(srr − sθθ )sinθcosθ.

(E.41)(E.42)

138

E.3 Thermal noise

As we assume the cylindal symmetry, srθ = 0 and every strain component does notdepend on θ.
The eigen value of this tensor is
λ1= 1/n20 + (α(srr + sθθ ) + 2βsrr )(E.43)

λ2= 1/n20 + (α(srr + sθθ ) − 2βsθθ ).

(E.44)

It is also shown that the azimuth angle of the optical axis is θ.
The optical pass length diﬀerence is
Z 3
2r 2
n0
2
∆L =
2β(srr − sθθ ) sinθ × e−2 ω2 rdrdθdz ×
2
πω 2
= 0.

(E.45)In the previous study, it is also shown the stress induced by the TEM 00 beam hasaxial symmetry. From these calculations, it is shown that as far as the beam proﬁleis axial symmetry, the absorption of the internal power does not induce the opticalpass length diﬀerence noise. It should be noted in the previous study, the thermalbirefringence is not zero. This is because they calculated the square of the electricﬁeld of the extraordinary light whereas we are interested in the linear term.

Acknowledgements
First, I would like to express my gratitude to my supervisor Prof. Shoji Asai (theuniversity of Tokyo) for giving me the opportunity of this work and his continuous
support and encouragement.
I would also like to express my great gratitude to Assosiate Prof. Toshio Namba(ICEPP, the university of Tokyo). He supported this work from the beginning to
the end. His insightful advice always helps me to solve the problems in the bestway.
My special thanks go to Dr. Toshiaki Inada (ICEPP, the university of Tokyo)
and Takayuki Yamazaki (ex-ICRPP) for their excellent contributions in developing
the pulsed magnet system. Without their support, this work could never have beenaccomplished.
My sincere gratitude goes to Prof Kosuke Yoshioka and President Makoto
Kuwata-Gonokami (The university of Tokyo) for their technical support to develop
the laser system. I gratefully thank to Prof. Koichi Kindo, Dr. Akira Matsuo, andDr. Takashi Kawaguchi (ISSP, The University of Tokyo) for their technical advice
on development of the pulsed magnet. I would also thank to Prof. Hiroyuki Nojiri(IMR, Tohoku University) for his advice on the design of the magnet and capacitorbank.
I sincerely appreciate Mr. Xing Fan (ex-the university of Tokyo). He is the pioneer of this experiment. After he left this experiment, he continuously encourages
me and gives a lot of useful advice.
I would like to thank all the staﬀs and students of ICEPP and Dept. of Physics
at the university of Tokyo for their supports.

139

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