Beam dynamics effects on the muon
anomalous precession frequency in the
Fermilab Muon g − 2 Experiment
Candidata:
Maria Domenica GALATI
Matricola 531370Relatore:
Prof. Marco INCAGLI
Relatore Interno:
Prof. Simone DONATI
Abstract 6
1 Introduction 7
1.1 From Schrödinger to Dirac . . . 7
1.2 Magnetic moment of a Dirac fermion . . . 8
1.3 Deviations from g = 2: the anomalous magnetic moment . . . . 8
1.4 Why aµ? . . . 10
2 The E989 Fermilab Muon g −2 Experiment 11 2.1 Measurement principle of aµ . . . 12
2.2 Beam production . . . 14
2.2.1 Pion decay . . . 14
2.2.2 Muon decay . . . 15
2.3 The Storage Ring . . . 18
2.4 Detectors . . . 20
2.4.1 Calorimeter and tracker system . . . 20
2.4.2 Auxiliary detectors . . . 22
3 Beam Dynamics 23 3.1 Weak Focusing Betatron . . . 23
3.1.1 Betatron oscillations . . . 23
3.1.2 Hill’s equation . . . 25
3.2 The Fast Muon Kicker . . . 26
3.3 The Electrostatic Quadrupoles (ESQ) and Beam Collimators . . . 27
3.3.1 Requirements for ESQ and Beam Collimators . . . 28
3.3.2 Aliasing and CBO . . . 29
3.3.3 Lost Muons . . . 31
4 Faraday Magnetometer 33
4.1 Measurement principles . . . 33
4.2 Setup components and optimization . . . 35
4.3 Calibration and data acquisition . . . 37
4.3.1 λ/2in scan . . . 38
4.3.2 Noise measurement with a λ/4 . . . 39
4.4 Apparatus calibration . . . 41
4.5 Magnetometer simulation . . . 42
4.5.1 Polarization Ray Tracing in two dimensions . . . 42
4.5.2 Polarization Ray Tracing in three dimensions . . . 45
4.5.3 Simulation’s strategy and analysis . . . 47
4.6 Comparisons between simulation and experimental data . . . 48
4.6.1 λ/2in scan . . . 48
4.6.2 λ/4 scan . . . 48
4.6.3 Presence of a static magnetic field . . . 49
4.7 Two-crystal periscope . . . 52
4.8 Run 1 results . . . 54
5 Lost Muons studies 57 5.1 Incorporating muon losses into the fitting function . . . 57
5.2 Calorimeter coincidences . . . 59
5.3 Tracker Muon Identification . . . 59
5.4 Lost Muons preselection . . . 60
5.4.1 Energy and time analysis of the selected events . . . 61
5.5 Optimized Lost Muons selection . . . 62
5.6 Monte Carlo simulations analyses . . . 64
6 Measurement of the anomalous precession frequency ωa 71 6.1 Data selection . . . 71
6.2 Event-based methods . . . 71
6.3 Data corrections . . . 73
6.3.1 Superposition of pulses in the same crystal . . . 73
6.3.2 Finite beam length . . . 73
6.3.3 Superposition of pulses in the same calorimeter . . . 74
6.4 Fit procedure . . . 75
6.4.1 5 parameters fit . . . 76
6.4.2 9 parameters fit . . . 76
6.4.3 12 parameters fit . . . 77
7 Corrections to the measured ωa 83
7.1 Eddy currents effect on the ωa fit . . . 83
7.2 Muon losses . . . 84
7.2.1 Phase-momentum correlation determination . . . 84
7.2.2 Lost muon-momentum correlation determination . . . 85
7.3 Pitch correction . . . 85
7.4 Electric field correction . . . 87
7.5 Phase Acceptance . . . 88
7.6 ωa Run 1 results . . . 90
8 Measurement of the muon magnetic anomaly aµ 91 8.1 Magnetic field determination . . . 91
8.2 Computing aµ . . . 93
Conclusions 96 A Polarization Ray Tracing Matrices for the optical elements of the periscope 97 A.1 One-crystal periscope . . . 97
A.2 Two-crystal periscope . . . 100
Abstract
The measurement of the muon magnetic anomaly, aµ = gµ
−2
2 , is one of the most
accu-rate tests of the Standard Model (SM).
Recently, the international theory community published a comprehensive SM predic-tion for the muon anomaly, aSM
µ = 116591810 × 10
−11, with a precision of 0.37 parts
per million (ppm) [1]. The experimental measurement of aµ has become increasingly
precise through a series of innovative improvements employed by three experimental campaigns at CERN [2, 3, 4], and more recently at the Brookhaven National Labo-ratory (BNL) [5]. The BNL result, with its 0.54 ppm precision, differs from the SM by 3.7 standard deviations. Because the BNL result hints at undiscovered physics not included in the SM, the Fermilab E989 Muon g − 2 Experiment was constructed to independently confirm or refute that finding. The first measurement of the muon anomaly will be presented to the scientific community on April 7: the E989 experiment has reached a precision of 465 parts per billion (ppb) and it aims to reduce it to 120 ppb, once the full statistics will be collected.
In the E989 experiment, positive 3.1 GeV/c muons are injected into a 14 m diameter storage ring (SR), where both muon’s spin and momentum vectors precede. The dif-ference between the spin frequency and the cyclotron frequency is called “anomalous precession frequency”, related to aµ through ωa = aµmqB, where B is the dipole
mag-netic field inside the SR. Therefore, aµ can be extracted by accurately measuring ωa
and B.
This thesis describes how ωa is measured. In particular, I will show the analysis I have
carried out on two specific systematic effects due to beam dynamics, that can alter the
ωa value if not properly taken into account.
A kicker system produces a ∼ 250 G magnetic field parallel to the ring dipole field that steers the muon beam onto the designed orbit. The large high voltage field of ∼ 120 kV in the kickers’ plates induces eddy currents that produce a magnetic field that lasts few tens of µs, well into the ωaanalysis region. This spurious field modifies the main dipole
field at the ppm level, thus modifying the measured ωa, if not corrected. I will describe
how this field can be measured by a Faraday Magnetometer which I have contributed to build and characterize in the laboratories of the Istituto Nazionale di Ottica (INO) in Pisa, and that will be installed inside the g − 2 SR during summer 2021.
The second effect that I will discuss is related to muons which disappear not because of their decay into positrons, but because of their interaction with the material near the storage region, and in particular with the beam collimators. Muon losses are due to the beam betatron oscillations and they are present mostly, but not only, in the microseconds after the injection. This effect distorts the spectrum of the muons decay and, if not properly corrected, can result in a systematic modification of the muon precession frequency. A study both on data and on Monte Carlo simulations has been performed to better understand this loss mechanism and to include it into the final ωa
measurement.
Introduction
1.1
From Schrödinger to Dirac
The Schrödinger equation " i~∂ ∂t + ~2 2m∇2− ˆV # ψ = 0, (1.1.1)
being first order in the time derivative and second order in the spatial derivatives, is clearly non-Lorentz-invariant, and therefore cannot provide a description of relativistic particles.
The first attempt to construct a relativistic theory of quantum mechanics was based on the Klein-Gordon equation
1 c2 ∂2 ∂t2 − ∇ 2 ! ψ+m 2c2 ~2 ψ = 0, (1.1.2)
which is second order in both space and time derivatives, and can be expressed in the manifestly Lorentz-invariant form
∂µ∂µ+
m2c2
~2 !
ψ = 0. (1.1.3)
However, it can be demonstrated [6] that the Klein-Gordon equation implies that the negative energy solutions, that in classical mechanics can be dismissed but not in quantum mechanics, have unphysical negative probability densities. Therefore it can be concluded that the Klein-Gordon equation does not provide a consistent description of single-particle states for a relativistic system.
The problems with the Klein-Gordon equation led Dirac to search for an alternative formulation of relativistic quantum mechanics. The resulting wave equation, that can be expressed in the covariant form
(iγµ∂
µ− m)ψ = 0, (1.1.4)
1.2
Magnetic moment of a Dirac fermion
A particle satysfying the Dirac equation has intrinsic angular momentum s = 1
2. It can
be shown [6] that the operator ˆµ giving the intrinsic magnetic moment of a particle satysfying the Dirac equation is given by
ˆµ = q
mˆs, (1.2.1)
where q and m are respectively the charge and mass of the particle, and } = c = 1.
Figure 1.2.1: The classical magnetic moment of a current loop with angular momentum L = mvr.
Classically, the magnetic moment associated with a current loop, as shown in Figure 1.2.1, is given by the current I multiplied by the area of the loop πr2, hence
µ= πr2 qv
2πrˆz =
q
2mL, (1.2.2)
where |L| = mvr. Thus, the relationship between the magnetic moment µ and the intrinsic angular momentum S of a Dirac fermion differs from the corresponding ex-pression in classical physics by a factor of two. This is usually expressed in terms of the gyromagnetic ratio g (or Landé factor) defined such that
µ= g q
2ms, (1.2.3)
where the Dirac equation predicts g = 2.
1.3
Deviations from g = 2: the anomalous magnetic
moment
In 1947, motivated by anomalies in the hyperfine structure of hydrogen [7, 8], Schwinger [9] proposed an additional contribution to the electron magnetic moment from a radia-tive correction (Fig. 1.3.1(a)), i.e. the coupling of a particle to virtual fields, obtaining an “anomaly”
ae =
α
2π '0.00116..., (1.3.1)
is the so-called anomalous magnetic moment (of the electron).
This prediction was soon confirmed by Kusch and Foley [10] who measured the Zeeman spectra of the gallium atom in a constant magnetic field and determined the g−factor for the electron to be
ge = 2 × (1.00119 ± 0.00005). (1.3.3)
In 1957, the first muon spin rotation experiment, carried out by Garwin et al. [11], determined that gµ= 2 within 10%, which was confirmed in the same year, with higher
precision, by Cassels at al. [12]. Two years later, in 1959, a more precise experiment by Garwin et al. [13] determined that the muon had a magnetic anomaly that agreed with the Schwinger prediction of α/2π, thereby demonstrating that the muon behaved like a “heavy electron” in a magnetic field.
In 2020 the international theory community published a comprehensive Standard Model prediction for the muon anomaly [1], finding
aSMµ = 116591810(43) × 10−11. (1.3.4)
This result is based on calculation performed in a perturbative expansion in the fine-structure constant α and is broken down into pure Quantum Electrodynamics (QED), electroweak, and hadronic contributions. The largest contribution comes from QED and has been evaluated up to, and including, O(α5) with negligible numerical
uncer-tainty. The electroweak contribution is suppressed by (mµ/MW)2, where MW ' 80
GeV is the mass of the W boson. Hadronic contributions are the most difficult to calculate and are responsible for almost all of the theoretical uncertainty. The leading hadronic contribution appears at O(α2), and is due to hadronic vacuum polarization
(Fig. 1.3.1(c)), and at O(α3), and is due to hadronic light-by-light scattering
contribu-tion (Fig. 1.3.1(d)).
(a) (b) (c) (d)
Figure 1.3.1: Feynman diagrams of representative Standard Model contributions to the muon anomaly. From left to right: 1st-order QED and weak processes, leading-order hadronic (h) vacuum polarization, and hadronic light-by-light.
1.4
Why a
µ?
Although the electron magnetic anomaly has been measured to fractions of a part per billion [14], a precise measurement using muons can be more relevant to constrain new physics. A new particle (boson) contributing to the anomaly in a virtual correction would have an effect which, in general, due to the chirality flip in the boson emission, can be proportional to the square of the mass ratio
αN P ' m l M 2 , (1.4.1)
where ml is the lepton mass and M is the mass of this hypothetical new heavy particle
beyond the Standard Model. Therefore, the relative contribution of heavier virtual particles for muons with respect to electrons scales as (mµ/me)2 ' 43000, giving the
muon anomaly a significant advantage when searching for effects of new physics. More-over, whilst the τ lepton has a mass mτ > mµ > me, thus having a greater sensitivity
than the muon to new physics, an experiment that uses τ leptons is not practical with current technology, both for the τ short lifetime (O(290×10−15s)) which would require
a significant Lorentz boost, and because the τ particles do not have a single dominant decay mode as muons do.
The measurement of aµ has become increasingly precise through a series of
innova-tive improvements employed by three experimental campaigns at CERN, and, more recently, at the Brookhaven National Laboratory (BNL), as summarized in Table 1.4.1.
Experiment Years Polarity aµ×1010 Precision [ppm] Ref.
CERN I 1961 µ+ 11450000(220000) 4300 [2] CERN II 1962-1968 µ+ 11661600(3100) 270 [3] CERN III 1974-1976 µ+ 11659100(110) 10 [4] CERN III 1975-1976 µ− 11659360(120) 10 [4] BNL 1997 µ+ 11659251(150) 13 [15] BNL 1998 µ+ 11659191(59) 5 [16] BNL 1999 µ+ 11659202(15) 1.3 [17] BNL 2000 µ+ 11659204(9) 0.73 [18] BNL 2001 µ− 11659214(9) 0.72 [19] Average 1999-2001 11659208.0(6.3) 0.54 [19]
Table 1.4.1: Summary of aµ results from CERN and BNL, showing the evolution
of experimental precision over time.
The BNL result, with its 0.54 ppm precision, differs from the Standard Model value by 3.7 standard deviations.
The E989 Fermilab Muon g − 2
Experiment
Figure 2.0.1: Image of the storage ring as prepared for Run 1.
The new Muon g − 2 Experiment in operation at the Fermi National Accelerator Lab-oratory (FNAL) aims to measure the muon anomalous magnetic moment with an unprecedented precision of about 140 parts per billion (ppb). The experiment follows the BNL concept, using the same 1.45 T superconducting storage ring (SR), but it benefits from many substantial improvements which will be detailed in the following. The Fermilab Muon Campus delivers 16 highly polarized, 3.1 GeV/c, ∼ 120ns long, positive muon beam bunches every 1.44 s into the SR, where a fast pulses-kicker mag-net deflects them into a 9-cm-diameter storage volume on their first turn. The SR has a 149.2 ns cyclotron period. Four sections of electrostatic quadrupole plates provide weak focusing and vertical containment.
The muon spins precede in the magnetic field at a rate somewhat larger than the cy-clotron frequency, since aµ is not equal to zero.
This Chapter will describe the measurement principles of aµ (Sec. 2.1), as well as the
employed experimental methods, from the production of the muon beam (Sec. 2.2) to the detectors (Sec. 2.4). A separate chapter (Chap. 3) is dedicated to the beam dynamics, including how the quadrupoles, kickers, and collimators are used.
2.1
Measurement principle of a
µThe equation of motion of a muon orbiting with velocity v in a horizontal plane that is perfectly perpendicular to a perfectly uniform magnetic field B (so that B · v = 0) is
mγdv dt = qv × B ⇒ dv dt = − q mγB × v. (2.1.1)
Therefore, the muon velocity precedes around the direction of the magnetic field with angular frequency
ωc =
qB
mγ, (2.1.2)
known as cyclotron frequency.
In the muon rest frame, the magnetic interaction term in the Hamiltonian is
HB = −µ · B = −g
q
2ms · B, (2.1.3)
and from the Heisenberg equation it follows
ds dt = −g
q
2mB × s. (2.1.4)
Therefore, in the rest frame, the muon spin s precedes with angular frequency
ωs = g
qB
2m. (2.1.5)
In the laboratory frame, an additional term [20], the Thomas precession, needs to be considered, leading to a spin precession frequency
ωs= g
qB
2m + (1 − γ)
qB
γm. (2.1.6)
Thus the rate at which the muon spin rotates with respect to the direction of the momentum is ωa= ωs− ωc = g −2 2 q mB. (2.1.7)
For a Dirac particle (see Eq. 1.2.2) g = 2, thus no relative precession is observed (as illustrated in Fig. 2.1.1(a)), while, due to the muon magnetic anomaly aµ being
~ p
~s
(a) g= 2: the muon spin and
momen-tum vectors remain parallel.
~ p
~s
(b) g > 2: the dashed blue arrow is
the spin vector after one turn.
Figure 2.1.1: Illustration of the motion of the muon spin and momentum vectors for a muon orbiting in a magnetic field when (a) g = 2, so the spin does not rotate relatively to the muon momentum, and when (b) g > 2 [21].
In the E989 experiment, the precession is measured in a beam of muons produced by the Fermilab Accelerator Complex and injected in a storage ring in which a very uniform magnetic field is present. In order to confine the muon beam in the storage region, and to provide vertical focusing, the experiment uses electrostatic quadrupoles (see Sec. 3.3), as magnetic quadrupoles would not allow for a stable magnetic field throughout the ring. Due to the presence of the electric field, and to the fact that the muon beam undergoes radial and vertical betatron oscillations (described in Sec. 3.1.1) around the ideal circular orbit, Eq. 2.1.7 has to be rewritten as follows [20]:
~ ωa = q m " aµB − a~ µ γ γ+ 1 ! (~β · ~B)~β− aµ− 1 γ2−1 !~ β × ~E c # . (2.1.8)
A possible Electric Dipole Moment would add a term which we are not considering here.
The effect of the electric field (third term in Eq. 2.1.8) can be canceled out by choosing a specific value of the Lorentz boost: γm =
q 1 + 1
aµ ' 29.3. This value corresponds
to the so-called “magic momentum” pm'3.1 GeV/c. This technique was used for the
first time in the CERN experiment, in Ref. [4], directed by the Italian physicist Emilio Picasso.
In the ideal case of perfect cancellation of the term related to the electric field, and neglecting the betatron oscillations, Eq. 2.1.7 holds and we can write
aµ=
ωa
B m
q . (2.1.9)
Therefore, to measure aµ, a precise measurement of ωa and B is required. Moreover,
corrections induced by beam dynamics have to be included: these will be discussed in Chapter 7. Chapter 6 will describe how ωa is measured, while in Chapter 8 a brief
2.2
Beam production
The FNAL Muon Campus beamline is constructed to deliver pulses of highly polarized (positive) muons to the E989 Storage Ring. These muons originate from pions produced by focusing a proton beam on a Inconel 600 target. A short description of the decay chain follows.
2.2.1
Pion decay
The charged pions (π±) are the lightest (charged) mesons (m
π± ∼140 MeV/c2). They
decay through the weak interaction to final states with light fermions: electrons (me∼
0.5 MeV/c2), and muons (m
µ ∼106 MeV/c2). They can not decay to taus (mτ ∼1777
MeV/c2), because m
τ > mπ±. The three main decay modes of the π+ are listed in
Table 2.2.1 [22].
Mode Fraction Γi/Γ
π+→ µ+νµ 99.98770 ± 0.00004 %
π+→ µ+νµγ (2.00 ± 0.25) × 10−4
π+→ e+νe (1.230 ± 0.004) × 10−4
Table 2.2.1: π+ main decay modes.
While the positron decay is preferred due to phase space considerations, parity violation of the weak force enhances the decay into a muon, with a ratio given by [6]
Γ(π+→ e+ν
e)
Γ(π+→ µ+νµ) = 1.230(4) × 10 −4
. (2.2.1)
Figure 2.2.1 shows the π+ → l+ν
l decay in the rest frame, where the (anti)lepton and
the neutrino are emitted back-to-back.
~ pl+
π
+ ~ pνl ~sl+ ~sνl hl+ = −1 2 hνl = − 1 2Figure 2.2.1: Helicity configuration in π+→ l+νl decay, where l = e or µ.
Due to the V-A structure of the weak current, the muon is preferably emitted in a left-handed helicity state. In fact, the neutrino can be considered massless and thus it is produced in a left-handed (LH) helicity state. Since the pion is a spin-0 parti-cle, the lepton and the neutrino spins have opposite directions. This configuration is sketched in Figure 2.2.1. By selecting muons with the highest possible momentum in the laboratory frame, their momentum is aligned with the pion boost, and thus all spin
2.2.2
Muon decay
After the pions decay, the produced muons follow the decay
µ+→ e+νe¯νµ (2.2.2)
with nearly 100% probability with a lifetime that, due to Lorentz boost, is τ ' 64.4 µs in the Muon g − 2 Storage Ring. As in the case of the pion decay, the muon decay proceeds through the weak force and it is parity-violating. As a consequence, a LH neutrino νe and a RH antineutrino ¯νµ are produced.
In the rest frame of the muon, the highest energy decay positrons come from decays in which the neutrinos are emitted in the same direction, therefore with their spins in opposite directions, as represented in Figure 2.2.2. With the neutrinos’ spins canceling out, conservation of angular momentum forces the positron to carry the spin of the parent muon, thus, since in a weak decay positrons prefer to be RH, they are preferably emitted in the direction opposite to the muon’s spin direction. Therefore, in the rest frame of the muon, the spin direction of the muon can be monitored by observing the instantaneous direction at which the high energy decay positrons are emitted.
µ
+ ~sµ+ ~ pe+ ~se+ ~ pνe ~sνe ~ pν¯µ ~ sν¯µ hν¯µ = + 1 2 hνe = −12 he+ = −1 2Figure 2.2.2: Helicity configuration in µ+→ e+νeν¯µdecay when the two neutrinos
are emitted in the same direction.
It is possible to show [23] that the differential decay width in the laboratory frame is given by:
dNe+
dy (y, α) =
N(y)[1 + A(y) cos(α)]
3 , (2.2.3)
where:
N(y) = (y − 1)(4y2−5y − 5); A(y) = −8y
2 + y + 1
4y2−5y − 5; (2.2.4)
yis the ratio between the positron energy in the lab frame and its maximum achievable
energy (approximately the energy of the muon, i.e. Emax ' 3.1 GeV); α is the angle
between the muon’s spin and momentum in the lab frame. We will refer to A as the
Figure 2.2.3: The functions N(y) (divided by 5 so that it fits the range [0,1] on the y−axis) and A(y) from Eq. 2.2.4. The product N(y)A(y)2, enlarged by a factor
2 for better visibility, is also showed.
If we are interested in counting all positrons above a certain energy threshold Eth, we
can integrate Eq. 2.2.3 over y going from yth = Eth/Emax to 1, obtaining:
N(yth) = N0(yth)[1 + A0(yth) cos(α)], (2.2.5)
where
N0(x) = (x − 1)(−x2+ x + 3), A0(x) = x(2x + 1)
−x2+ x + 3. (2.2.6) It is important to note that α is not constant over time, but it oscillates from 0 to 2π with frequency ωa, so Eq. 2.2.5 implicitly contains a time dependency. Because of the
exponential decay of muons, the distribution of emitted positrons will actually be
N(yth, t) = N0(yth)e−t/τ[1 + A0(yth) cos(ωat+ φa)], (2.2.7)
where φa is the initial phase of the muons’s spin relative to their momentum, at the
time of injection. Figure 2.2.4 shows N(yth, t) for different values of yth: it is evident
that for higher energy thresholds, the count rate of positrons has a larger oscillation amplitude, but the number of events is lower. It is known from previous studies, in fact, that δωa ωa ∝ q 1 N0A20 . (2.2.8)
Therefore, to obtain the best statistical result, the energy threshold to use is the one that maximizes N0A20: Figure 2.2.5 shows that the optimal threshold is yth ' 0.6,
Figure 2.2.4: Number of positrons produced as a function of time, for three different values of energy threshold. Notice the logarithmic scale on the y−axis.
Figure 2.2.5: The functions N0(x) (divided by 3 so that it fits the range [0,1] on the y−axis) and A0(x) from Eq. 2.2.6. The product N0(y)A0(y)2, enlarged by a
factor 10 for better visibility, is also showed.
The number of positrons detected above a certain threshold (Eq. 2.2.7) represents the number of counts that we expect for muons orbiting exactly on the ideal trajectory, without any side effect coming from beam dynamics or detectors, and it is commonly referred to as 5-parameter formula:
N(t) = N0e−t/τ[1 + Aacos(ωat+ φa)]. (2.2.9)
In Section 6.4 we will see how this formula modifies in presence of systematic effects that affect the measurement of ωa, if not properly taken into account.
2.3
The Storage Ring
The muons produced by the pion beam have a net spin polarization of around 95%. The muons momentum distribution has a RMS of approximately 2% centered around 3.094 GeV/c, and a temporal length of 120 ns. Of these injected muons, only 1 ÷ 2% are stored.
A bunch of muons is referred to as a “fill”: these fills deliver O(104) muons to the
stor-age ring at an averstor-age rate of 11.4 Hz, and only about 500 positrons are detected and counted. Each cycle consists of two groups of eight fills, each every 1.4 s, as illustrated in Fig. 2.3.1.
Figure 2.3.1: Bunch structure: two groups of eight fills are delivered every 1.4 s.
With a 4.5 cm radius storage region and a 7.112 m orbit radius, the E989 ring can at best store muons within approximately 0.5% of the design momentum [24] [25].
The muon beam enters the storage ring through a superconducting inflector magnet which provides a field-free region. The muon beam does not start on the ideal tra-jectory, otherwise, muons will impact on the inflector after one turn. The exit of the inflector is placed 77 mm radially outwards the center of the storage region. Three pulsed kicker plates are used to kick the muon beam into the center of the ring, as it will be discussed in Sec. 3.2.
The magnetic field B0 inside the SR is generated by three superconducting NbTi+Cu
coils around a C-shaped yoke as shown in Figure 2.3.3. They operate with a current O(5 kA) that generates a magnetic field B0 = 1.4513 T. The C-shape faces the interior of the ring, so that decay positrons can travel unobstructed to the detectors placed around the interior of the SR.
Figure 2.3.2: Plan view of the g − 2 SR as seen from above. The beam circulates clockwise.
Figure 2.3.3: A cross section of the storage ring magnet featuring the components used to generate the highly uniform 1.45 T magnetic field in the Run 1 configuration.
2.4
Detectors
2.4.1
Calorimeter and tracker system
After the muons decay in the storage ring, the emitted positrons, having lower energy than the parent muons, curl inward. Twenty-four electromagnetic calorimeter stations (Figure 2.4.1) are placed around the inner radius of the SR to intercept these decay positrons and measure their energies and hit times. They are designed and optimized for the specific need of the ωa measurement, and constantly calibrated over time.
Each calorimeter consists of a 9-column by a 6-row array of PbF2crystals instrumented
with silicon photomultiplier (SiPM) photodetectors that detect the Cherenkov light produced by the positrons energy conversion. Each crystal is a 14 cm × 2.5 cm × 2.5 cm block.
Figure 2.4.1: Rear view of a calorimeter.
The SiPMs are sampled at a frequency of 800 MHz. When a signal in a SiPM exceeds ∼50 MeV, the data-acquisition system stores the 54 waveforms from the calorimeter in a set time window of approximately 40 ns around the event. Decay positron hit times and energy are derived from the reconstruction of these waveforms.
E989 calorimeters are well suited to the need of the Muon g − 2 experiment: PbF2 has
a very high density (7.77 g cm−3), a X
0 = 9.3 mm radiation length and a Molière radius
RM = 22 mm for energy deposition. These quantities ensure that the energy deposit
of a typical decay positron is contained almost entirely within one or two crystals. Incoming positrons deposit energy by producing an electromagnetic shower and emit-ting Cherenkov radiation in few nanoseconds. This fast response is one of the reasons for the choice of a Cherenkov detector. Moreover, the PbF2 crystals have a very low
The refraction index of PbF2 is n = 1.8, therefore positrons will produce Cherenkov
radiation only if they have a minimum energy of
Emin =
q
mec2+ (βγmec)2 '620 keV (β = 1/n) . (2.4.1)
To precisely measure the energy and time of the positrons, all the 24 calorimeters need to be calibrated. Calibration is achieved using a laser system, which monitors the gain fluctuations, ensures performance stability of the detectors throughout long data-taking periods, time-synchronizes all detectors, and can also emulate the signals coming from muon decays. The laser calibration pulses are generated by 6 identical lasers, each one serving 4 calorimeters.
Two tracking stations, each consisting of 8 straw tracking modules, are placed around the storage ring, at approximately 180◦ and 270◦ after the injection point, in positions
unobstructed by quadrupoles or collimators (see Fig. 2.3.2).
The primary physics goal of the tracking detectors is to measure the muon beam profile as a function of time throughout the muon fill. This information is used to determine several parameters associated with the dynamics of the stored muon beam. This is required for the following reasons:
• Momentum spread and betatron motion of the beam lead to ppm level corrections to the muon precession frequency associated with the fraction of muons that differ from the magic momentum and the fraction of time muons are not perpendicular to the storage ring field (see Sections 7.3 and 7.4).
• Betatron motion of the beam causes acceptance changes in the calorimeters that must be included in the fitting functions used to extract the precession frequency. • The muon spatial distribution must be convoluted with the measured magnetic field map in the storage region to determine the effective field seen by the muon beam (see Sec. 8.1).
The secondary physics goal of the tracking detectors involves understanding systematic uncertainties associated with the muon precession frequency measurement derived from calorimeter data. In particular, the tracking system will isolate time windows that have multiple positrons hitting the calorimeter within a short time period and will provide an independent measurement of the momentum of the incident particle. This will allow an independent validation of techniques used to determine systematic uncertainties associated with calorimeter pileup, calorimeter gain, and muon loss (see Sec. 5.3) based solely on calorimeter data.
Figure 2.4.2 shows the locations of the calorimeters and trackers in a segment of the SR. Two decay positron trajectories are indicated in the figure corresponding to high-and low-energy events.
Figure 2.4.2: Scalloped vacuum chamber with positions of calorimeters indicated. A high- (low-) energy decay positron trajectory is shown by the thick (thin) red line, which impinges on the front face of the calorimeter array [24].
2.4.2
Auxiliary detectors
Several other detectors are placed around the ring (Fig. 2.3.2) and their measurements play an important role for the corrections and the systematic studies [26].
• The T0 detector is a scintillating paddle read by two PMTs. It is placed right after the inflector and is used to precisely measure the beam injection time, important for the kickers’ timing. Integrating the pulse waveform is also possible to know the total number of injected particles into the storage ring.
• The IBMS (Inflector Beam Monitoring Station) is a detector made of a series of scintillating hodoscopes used to measure the beam profile. Two of them are placed before the inflector, one before the hole in the yoke, one right before the inflector. These detectors aim is to keep monitoring the beam profile after the final focusing.
• The Fiber Harps are planes of vertical and horizontal scintillating fibers that can be placed in the path of the muon beam in two different locations, at ∼ 180◦
and ∼ 270◦, along the ring. They can destructively measure the beam profile
and therefore provide a direct measurement of the betatron oscillations (see Sec. 3.1.1) to perform the fast rotation analysis. They are used only in dedicated runs and not during normal data-taking.
Beam Dynamics
It is not possible to follow the direction of the spin of a single muon, while it is possible to correlate it with the positron direction emission at the moment of the muon decay. Thus, the measurement of the precession frequency is performed on an ensemble of muons decaying with a time constant which, at the magic momentum pm=3.097 GeV/c,
is τ = 64.4 µs.
This chapter provides a discussion of the behavior of such a muon beam in a weak-focusing betatron. In particular, the role of the kicker and the electrostatic quadrupoles in the beam dynamics, and their possible effect on the determination of the precession frequency, are discussed.
3.1
Weak Focusing Betatron
3.1.1
Betatron oscillations
For a ring with a uniform vertical dipole magnetic field and a uniform quadrupole field that provides vertical focusing covering the full azimuth, the stored particles undergo simple harmonic motion called betatron oscillations, in both the radial and vertical directions.
Let us now consider a simple betatron motion of a relativistic particle (we can assume
β '1), along the radial (ˆx) and vertical (ˆy) directions.
On the x−axis the Lorentz force and the centripetal force act simultaneously and balance at the equilibrium radius R0, so the resulting force on the orbit is
Fx(R0) =
γmv2
R0 − qvB0 = 0, (3.1.1)
where B0 = By(R0). In general, for a given position r along the radial direction we
would have
Fx(r) =
γmv2
Eq. 3.1.2 can be re-written as a function of the displacement from the central radius, x= r − R0: Fx(x) = γmv2 R0 1 − x R0 ! − qvBy(x), (3.1.3)
where we have used the approximation 1
r ' 1 R0 1 − x R0 ! .
Similarly, the vertical magnetic field By(x) can be approximated as
By(x) ' B0+ ∂B0 ∂x x= B0 1 + 1 B0 ∂B0 ∂x R0 R0x ! = B0 1 − nRx 0 ! , (3.1.4) where n= −∂B0 ∂x R0 B0 (3.1.5)
is the field index, a quantity that determines the focusing strength of the magnetic field.
Finally, using Eqs. 3.1.3 and 3.1.4, Eq. 3.1.2 becomes
Fx(x) = γmv2 R0 1 − x R0 ! − qvB0 1 − n x R0 ! = γmv2 R0 1 − x R0 −1 + n x R0 ! = = γmv2 R20 (n − 1)x = γmω 2 c(n − 1)x = γm¨x. (3.1.6)
The horizontal motion equation is therefore ¨x = −ω2
c(1 − n)x, (3.1.7)
and describes a horizontal (= radial) oscillatory motion with betatron frequency
ωx= ωc
√
1 − n ≡ ωcνx, (3.1.8)
where νx is called horizontal tune.
On the y−axis, only the Lorentz force acts on the beam, thus
Fy(y) = qvBx(y), (3.1.9)
where y is the vertical displacement from the equilibrium position (ye= 0). In order to
derive the magnetic field Bx, we can use the z−component of the equation ∇ × B = 0:
∂Bx ∂y − ∂By ∂x = 0 ⇒ ∂Bx ∂y = ∂By ∂x . (3.1.10)
Integrating this last equation we finally obtain the horizontal magnetic field
Bx(y) = −
n
R0By(R0)y, (3.1.12)
so that Eq. 3.1.9 becomes
Fy(y) = − qvBy(R0)ny R0 = − mγv2ny R20 = −mγω 2 cny = γm¨y. (3.1.13)
The vertical motion equation is therefore ¨y = −ω2
cny, (3.1.14)
and describes a vertical oscillatory motion with frequency
ωy = ωc
√
n ≡ ωcνy, (3.1.15)
where νy is called vertical tune.
As can be seen considering Eqs. 3.1.7 and 3.1.14, stable oscillations will happen if 0 < n < 1. This is the condition of a weak focusing magnetic field.
3.1.2
Hill’s equation
We can move from the time to the space domain by using ¨x = v2 xx 00 and ¨y = v2 yy 00 , (3.1.16)
so that Eqs. 3.1.7 and 3.1.14 become
x00+ νx R0 !2 x= 0, y00+ νy R0 !2 y= 0. (3.1.17)
These lead to solutions of the form
x(s) = x0cos νx R0s ! +x00R0 νx sin νx R0s ! , y(s) = y0cos νy R0s ! + y00R0 νy sin νy R0s ! , (3.1.18)
where s is the longitudinal displacement around the ring, and
x0 ≡ x(0), x00 ≡ x0(0); y0 ≡ y(0), y00 ≡ y0(0). (3.1.19) The dominant terms of the solutions are x0
0R0/νx and y00R0/νy, because of the
propor-tionality to the SR radius R0 x0, y0, and therefore they define the maximum beam
The general form of the transverse motion equations in function of s is called the Hill’s
equation:
d2x
ds2 + kx(s)x(s) = 0. (3.1.20)
Since along the E989 there are different components (quadrupoles, kicker, etc.), the focusing strength given by kx(s) changes as a function of azimuth, and the equation of
motion looks like an oscillator whose spring constant changes as a function of azimuth
s. The equation of motion, i.e. the solution of the Hill’s equation, can be written as
x(s) =qεβ(s) cos(φ(s) + δ), (3.1.21)
where β(s) is one of the three Twiss parameters, and it is related to the phase φ(s) by
β(s)dφ(s)
ds = 1 (3.1.22)
The same equations can be obtained for the y coordinate.
3.2
The Fast Muon Kicker
The muons enter the storage ring through an inflector magnet, aligned to the tangent of the ring, whose role is to locally cancel the main dipole magnet, thus allowing the muons to enter the ring with the correct trajectory. The inflector’s interior aperture is displaced 77 mm from the central radius of the storage region. Consequently, the initial trajectories of the injected muons do not follow closed orbits. A series of three 1.27 m-long kicker magnets, located at 1/4 turn downstream from the inflector, steer the muon beam onto the ideal orbit [27].
Beam arrives from the accelerator in 16 120 ns-long bunch trains at a peak rate of 100 Hz, with a repetition rate of ' 1.41 s. The storage ring cyclotron period is 149 ns. These features determine the timing constraints of the kicker system: ideally the kicker pulse should be a square wave 120 ns-long that drops to zero after the first revolution, in order not to disturb the trajectory.
A sketch of the effect of the inflector and kicker magnets on the beam trajectory is shown in Fig. 3.2.1. The kicker magnets have to steer the beam by an angle β ' 10.8 mrad to compensate for the displacement of the inflector. This requires roughly a 1.1 kG-m integrated field local perturbation. To produce this field, the kicker plates are charged at a high voltage of ' 137 kV during the 120 ns of the bunch. This induces eddy currents in the surrounding metal, leading to field perturbations in the storage volume. The fixed NMR probes are shielded from this rapid transient field by the vacuum chambers and do not have the required measurement bandwidth. A Faraday magnetometer was built to measure this transient. A detailed discussion is presented in Chapter 4.
Figure 3.2.1: An idealized sketch of the beam geometry before and after the kicker pulse. xcis the 77 mm displacement of the inflector aperture relative to the central
radius of the storage ring, R0= 7112 mm and β ' 10.8 mrad.
3.3
The Electrostatic Quadrupoles (ESQ) and Beam
Collimators
Vertical focusing in the storage ring is achieved using a suite of discrete electrostatic quadrupoles (ESQ) plates, which occupy 43% of the ring circumference. The applied high voltages provide weak focusing with field index n described sufficiently well1 by
n= R0 vB0
∂Ey
∂y . (3.3.1)
The horizontal and vertical tunes, as previously seen in Sec. 3.1.1, are
νx =
√
1 − n and νy =
√
n, (3.3.2)
and define betatron frequencies
ωx = ωc
√
1 − n and ωy = ωc
√
n (3.3.3)
The field index, as explained in Sec. 3.1.1, also determines the angular acceptance of the ring. The maximum horizontal and vertical angles of the muon momentum with respect to the longitudinal direction can be estimated as (see Section 3.1.2):
θxmax = xmax √ 1 − n R0 , θ y max= ymax √ n R0 , (3.3.4)
where xmax = ymax = 45 mm is the radius of the storage aperture.
1The expression in Eq. 3.3.1 is exact for a uniform set of quadrupole fields that occupy the full
3.3.1
Requirements for ESQ and Beam Collimators
The ESQs used in the E989 experiment must:
• Produce vertically focusing electrostatic quadrupole field in the muon storage region.
• Have operating points in resonance-free region. In fact, resonances in the storage ring will occur if Lνx+ Mνy = N, where L, M and N are integers, which must
be avoided in choosing the operating value of the field index as this minimizes the ability of ring imperfections and higher multipoles to drive resonances that would result in particle losses from the ring. These resonances form straight lines on the tune plane in Figure 3.3.1, which shows resonance lines up to fifth order. The operating points lie on the circle ν2
x+ νy2 = 1.
Figure 3.3.1: Tune diagram displaying all resonances up to order 5 and the oper-ating points used in Run1.
• Have stable operation during extended periods of time in a vacuum of 10−6 torr
or better.
• Be optimized for muon storage efficiency.
• Have the minimum possible amount of material at places where the trajectories of incoming muons and decay positrons intercept the parts of ESQ.
Both the E989 ESQ and Beam Collimators must
• Provide effective scraping of the injected beam to remove muons outside the storage region (see Sec. 3.3.3).
• Be made from non-magnetic materials to not deteriorate the quality of the dipole storage magnetic field.
• The thickness and the shape of collimators must be optimized to efficiently remove muons outside the storage region and at the same time, they have little effect (e.g. multiple scattering, showering, etc.) on the decay positrons.
3.3.2
Aliasing and CBO
For a stationary detector, the beam is sampled at the cyclotron frequency ωc, and
therefore only frequencies < 0.5 ωc can be observed, as per the Nyquist sampling
the-orem. For the range of n accessible to the g − 2 experiment, ωx > 0.5 ωc, therefore
the set of measurements sampled by a calorimeter is not sufficient to determine ωx.
In fact, the same set of measurements would be produced by oscillations of the form
x(t) = cos[(kωc−ωx)t], where k is any integer. The smallest of these frequencies, i.e. the
one where k is closest to ωx/ωc, is the observed frequency. This effect is called aliasing,
and it applies to any discretely sampled signal. The different frequencies (kωc− ωx)
are aliases of ωx. The aliasing region for the horizontal and vertical beam motion is
shown in Figure 3.3.2.
In our specific case, for the horizontal motion, k = 1, and an aliased frequency is measured in the radial direction at a frequency ωCBO = ωc − ωx, known as
Coher-ent Betatron Oscillation (CBO) frequency. As a consequence, the tracking detectors
observe the radial position to oscillate at the reduced frequency ωCBO, as can be
un-derstood by examining Figure 3.3.3. The beam oscillation also has components at 2ωx.
In that case, k = 2 and the observed frequency is 2ωCBO.
The vertical oscillation frequency ωy is less than ωc/2 and thus it is observed in
fixed-position detectors as ωy, unchanged. The components of the vertical width oscillation
at 2ωy, which dominate, are greater than ωc/2 and are observed as ωV W = ωc−2ωy.
The V W subscript refers to the common name of this frequency, the vertical waist [28] [29].
Figure 3.3.2: Aliasing regions for ωx,y in function of the field index n.
Figure 3.3.3: The radial CBO oscillation is shown in blue, the cyclotron wave-length is marked by the black vertical lines. One detector location is shown. Since the radial betatron wavelength is larger than the cyclotron wavelength, the detector sees the bunched beam slowly move closer and then further away. The frequency that the beam appears to move in and out is fCBO.
To recap, the frequencies expected in the positron hit time histogram are shown in Table 3.3.1.
Name Symbol Typical value
Cyclotron frequency fc 6.71 MHz
Anomalous precession frequency fa 0.23 MHz
Coherent betatron frequency fCBO 0.37 MHz
Vertical betatron frequency fy 2.20 MHz
Vertical waist frequency fV W 2.29 MHz
Table 3.3.1: Frequencies expected in the positron hit time histogram. Values are taken from E989 Run 1a and correspond to a quadrupole field index n = 0.108.
During Run 1 data-taking, 2 out of the 32 quadrupole resistors were damaged, resulting in a longer time to reach the plateau value (as shown in Figure 3.3.4). As a consequence, the beam shifted vertically, at the level of ' 2 mm, from injection to beam dump, introducing a time dependence into the CBO-related frequencies, which has to be incorporated into the ωa analyses (see Sec. 6.4.5).
0 50 100 150 200 250 300 s] µ Time [ 0 2 4 6 8 10 12 14 16 18 20 V [kV] ∆ Nominal 1-Step Nominal 2-Step Beam Injection Fit Start Time Damaged 1-Step Damaged 2-Step
Figure 3.3.4: Charging profiles for the 30 nominal quadrupole plates driven by either 1-step (black) or 2-step (dotted red) power supplies. The two damaged resistors (solid orange and dotted blue), connected to the same 1- and 2-step power supplies, exhibit markedly different charging profiles during the data fitting period. The vertical dotted black line at time t = 0 represents the arrival time of the muons in the storage ring. The vertical dashed red line at 30 µs indicates the time at which the precession data fits begin. The resistors deteriorated slightly for Run 1d and the precession fit start time was delayed to 50 µs to compensate for the longer time charging profiles.
3.3.3
Lost Muons
Not all muons will remain stored until they decay. This could occur, for example, if a muon with a large betatron amplitude oscillates out of the storage region and hits a collimator, losing energy and being lost. In a storage ring with a perfectly uniform dipole magnetic field and ideal quadrupole focusing, no beam losses occur, but in reality, the higher field multipoles provide a perturbative kick which causes some muons to eventually be lost during the measurement period. These lost muons
distort the shape of the decay time histogram and can bias the ωa determination. To
reduce such losses the beam is scraped to create about 2-mm-wide buffer zone between the beam and collimators. During the first ' 10 µs, the quadrupole plates are charged asymmetrically to shift the beam vertically and horizontally, thus moving the edges on the collimators: this procedure, known as scraping, reduces the population of muons at the boundaries of phase space accepted by the storage ring, and helps to minimize beam loss during the period over which we observe the muon spin precession.
After scraping, the plate voltages are symmetrized to enable long-term muon storage. The optimal thickness of the copper collimators is a compromise between providing efficient scraping of the muons and having a low distortion of the magnetic field and low scattering of the decay positrons.
Lost muons spiralize in the magnetic field, having a lower momentum than the ideal one, and eventually hit the calorimeters. The identification of such category of events and the analysis of the systematic error they introduce in the ωa fit will be presented
in Chapter 5.
3.4
Run 1 datasets
Over the course of the Run 1, the fast kicker and the electrostatic quadrupoles oper-ated at few different set points, in the search for the optimal value, identifying four datasets. Across datasets, the beam parameters, such as the amplitude and frequency of the CBO, varied, requiring ωa to be determined during each operating condition
individually. Table 3.4.1 summarizes the key characteristics of these four datasets.
Run 1 dataset n Kickers voltage [kV]
1a 0.108 137
1b 0.120 137
1c 0.120 130
1d 0.107 125
Faraday Magnetometer
Even though the strength of the kicker field alone is well known, its very fast rise and fall times induce eddy currents in all nearby metal, which in turn create additional magnetic fields whose strengths are difficult to predict. These fields can last on a time scale of tens of microseconds, thus persisting into the fit window of ωa.
Measuring such transient fields is difficult both because of their small amplitude and the presence of strong vibrations in the measurement region. To avoid the creation of addi-tional eddy currents, no metal must be added to the measurement region. Moreover, it is needed a fast response of the measurement apparatus. To overcome these obstacles and measure the kicker transient field, two magnetometers based on the Faraday effect have been built. A fiber magnetometer, built at the Argonne National Laboratory, has been used to measure the effect due to residual eddy currents. An independent device, built on a breadboard and based on the same principle but with different characteris-tics, has been built by INFN. Unfortunately, it was not possible to carry it to Fermilab in the Summer of 2020. It will be hopefully mounted within the Muon g − 2 Storage Ring in the Summer of 2021 and used to perform an independent measurement of the kicker residual field. This is required in view of the final measurement of aµ which
combines the full statistics.
I have contributed to build and characterize the INFN magnetometer, currently in-stalled in the laboratory of the National Institute of Optics (INO) in Pisa.
4.1
Measurement principles
In 1845, Michael Faraday observed that the application of an external magnetic field could influence light propagation through a material medium. In particular, he found that the plane of polarization of linear light incident on a piece of glass rotated when a strong magnetic field was applied in the propagation direction. This was the first experimental evidence that light and electromagnetism are related.
The relation between the angle of rotation of the polarization and the magnetic field in a transparent material is:
where:
• θ is the angle of rotation (in radians)
• B is the magnetic field in the direction of the light propagation (in teslas) • d is the length of medium traversed (in meters) where the light and magnetic
field interact
• V is the Verdet constant for the material. This empirical proportionality con-stant (in units of radians per tesla per meter) varies with both frequency and temperature, and it is tabulated for various materials.
A positive Verdet constant corresponds to anticlockwise rotation when the direc-tion of propagadirec-tion is parallel to the magnetic field, and to clockwise rotadirec-tion when the direction of propagation is anti-parallel. Thus, if a ray of light is passed through a material and reflected back through it, the rotation doubles [30].
Figure 4.1.1: Polarization rotation due to the Faraday effect.
In order to measure a Faraday rotation due to a small magnetic field, a material with a high Verdet constant is required, as well as a precise measurement of the initial and final polarization of the light source. These requirements are met in the experimental setup I contributed to mounting in the INO laboratory, consisting of:
1. a polarized laser;
2. a half-wave plate to set the initial polarization angle (optional);
3. a Verdet crystal made of Terbium Gallium Garnet (TGG), which has a reported Verdet constant of −131 rad T−1m−1: its axis needs to be positioned along the
direction of the magnetic field to be measured (in our case, the vertical direction); 4. a second half-wave plate to rotate the polarization of the light exiting the crystal:
5. a polarizing beam splitter (in our case, a Wollaston prism) that separates the incoming laser beam into two orthogonally-polarized beams (one beam vertically polarized and the second horizontally polarized);
6. two photodiodes to measure the laser power of the two orthogonally polarized laser beams: if we steer a laser beam onto a photodiode, the output voltage will be proportional to the laser power and, for constant beam diameter, to the laser intensity. We decided to use a differential detector to achieve the maximum sensitivity to small magnetic fields.
With a final laser polarization of 45°, the two beams exiting the beam splitter will have the same intensity. If the intensity of the incoming laser beam is I0, then the vertically
and horizontally polarized beams exiting the beam splitter will have intensities Iy =
I0sin2(45◦) = I0/2 and Ix = I0cos2(45◦) = I0/2, respectively (Malus’s Law). In
these conditions, Iy − Ix = 0, i.e. the difference between output voltages of the two
photodiodes, VdiffPD, will be, ideally, equal to zero.
In the presence of an additional vertical magnetic field B in the Verdet crystal region, then Iy 6= Ix and VdiffPD will no longer be zero. In particular, VdiffPD will be a function
of B that can be determined by calibrating the system.
4.2
Setup components and optimization
Figure 4.2.1: Me assembling the magnetometer components on the breadboard.
The following components were used:
• linearly polarized He-Ne laser with a central wavelength of 632.8 nm (Thorlabs HNL020LB);
• dieletric mirrors (Thorlabs BB1-E02 and BB05-E02);
• half-wave plates and quarter-wave plate (Thorlabs WPH05M-633 and WPQ05M-633); • two TGG crystals 3.2 cm long, with = 6 mm;
• Wollaston prism with a 20◦ beam separation (Thorlabs WP10-A);
• a balanced photodetector, composed of two silicon photodiodes (Thorlabs PDB210A/M); • absorptive ND filter (Thorlabs NE06A-A);
• focusing lenses with f = 750 mm and f = 200 mm.
The He-Ne laser used is linearly polarized, and it has been mounted with the polar-ization vector in the vertical direction. A polarizer has also been used to reduce the polarization fluctuations of the laser beam.
Using a balanced photodetector, instead of two separate photodiodes, largely sup-presses any common fluctuation, in particular the intensity fluctuations. The balanced photodiode directly amplifies the difference in current of the two single photodiodes, instead of amplifying the two signals individually and then subtracting them.
The magnetic field we want to measure in the experimental area is along the vertical direction, while the laser light propagates on the horizontal plane. For this reason, a mechanical structure, named “periscope”, is used to hold the Verdet crystal in the vertical direction. The first version I have tested essentially consists of two mirrors, as shown in Figure 4.2.2: the first mirror reflects the beam along the vertical direction, so that the laser passes through the Verdet crystal, while the second mirror reflects the beam back.
The setup configuration is sketched in Figure 4.2.3. A beam splitter is used to select the vertically polarized component of the laser beam. A first mirror (M1) deflects the
light by ' 45◦ and sends it towards a first half-wave plate indicated by λ/2
in. A
sec-ond mirror sends the light into the periscope which channels the beam into the crystal, parallel to the magnetic field, and reflects it back. The exiting beam is then sent to a second half-wave plate (λ/2out) using a steering mirrors system (M3 and M4). Finally,
the beam is split into two orthogonally-polarized beams whose intensities are measured with the balanced photodetector.
To improve the quality of the beam, I added two lenses to this setup: one, with a very long focal length, before the first half-wave plate, aiming to focus the beam in the periscope region, and the other before the Wollaston prism, to focus the two outcoming beams on the photodiodes.
A Neutral Density filter was placed right after the laser to reduce by a factor of three its power, so to maintain a safer environment in the laboratory during the preliminary measurements.
Figure 4.2.2: Photo of the periscope with the lid off. The measured magnetic field is parallel to the crystal axis.
Figure 4.2.3: Setup schematization (made with Inkscape): upper view. The gray box shows the components mounted on the breadboard. The periscope is ∼ 1 m far from the breadboard.
4.3
Calibration and data acquisition
In order to calibrate the magnetometer in a condition similar to the one present in the storage ring, I built a magnetic coil by wrapping a copper wire around a hollow cylinder of diamagnetic material. With a length l = (6.6 ± 0.1) cm and a number of spires N = 67, the expected magnetic field generated when a current I passes through
the coil is Bcoil = µ0 N I l '12.8 I G A −1 . (4.3.1)
This expected value is compatible with a calibration done with a magnetic probe: a linear best-fit of the magnetic field read by the probe in function of the values of current that passes through the coil returned Bcoil= (12.7 ± 0.1) I G A−1.
Previous analyses (see Sec. 4.8) have shown that the kicker transient field decays ex-ponentially with a time constant ∼ 60µs. By using a square wave generator, it was possible to create a similar condition. We have generated a burst of pulses with a frequency of ' 30 Hz and settable amplitude, as shown in Figure 4.3.1. Figure 4.3.2 shows a zoom on the falling edge of one pulse: we see the classic exponential distortion, due to the presence of a capacitive component, with a time constant ∼ O(µs). We can therefore use an input signal of this type to calibrate our system.
Figure 4.3.1: Burst of pulses used as input signal for the magnetometer.
All the data analyzed in this chapter were acquired through the use of a digital os-cilloscope (Digilent Analog Discovery 2) connected to my pc via usb cable. Its maximum sampling rate is 100 MS/s, with a maximum of 8192 samples per acquisition file.
4.3.1
λ/
2
inscan
During the preliminary measurements, we discovered that the amplitude of the Faraday effect changed as a function of the angle of polarization of the laser. To better see this behavior, we measured VdiffPD as a function of the angle of the first half-wave plate,
λ/2in, which is equivalent to changing the angle of polarization of the laser. The results
are shown in Figure 4.3.3. The points in the plot have been obtained, for each λ/2in
Figure 4.3.2: Zoom on the falling edge of a generated square wave that shows an exponential distortion with fitted τ = (3.339 ± 0.002) µs.
Figure 4.3.3: λ/2in scan with B ' 440 mG.
To better understand this behavior, which is due to different phases acquired by the two components of the laser beam at each reflection, I developed a Python simulation of the apparatus, described in Sec. 4.5.
4.3.2
Noise measurement with a λ/
4
Since the Faraday effect consists in a rotation of the laser polarization, we should not see any effect if the incident light is circularly polarized. Using a circularly polarized light allows us to measure the level of noise, which can then be subtracted from the signal. A simple way to convert a linearly polarized light into a circularly polarized light is to use a wave plate. If linearly polarized light is incident on a quarter-wave plate at 45◦ to the optic axis, then the light is divided into two equal electric
field components, one of which is retarded by a quarter wavelength with respect to the other by the plate: this produces circularly polarized light (see Figure 4.3.4).
Figure 4.3.4: Linear to circular polarization using a quarter-wave plate. (source: HyperPhysics - Light and Vision)
This configuration has been tested by adding a quarter-wave plate, λ/4, after λ/2in
in the setup shown in Figure 4.2.3. With an initial vertical polarization, we measured
VdiffPD for different λ/4 angles, as shown in Figure 4.3.5. As we expected, the Faraday
effect is practically null at a λ/4 angle of 45◦.
Figure 4.3.5: VdiffPD amplitude in function of the λ/4 angle, with a vertical initial polarization of the laser beam and a magnetic field B ' 45 mG.
4.4
Apparatus calibration
To maximize the sensitivity of the magnetometer, we set the λ/2in angle at 22◦,
cor-responding to the maximum in Figure 4.3.3. Moreover, since we aim to measure small magnetic fields, we averaged over many acquisitions (∼ O(10 k)) for each value of the magnetic field. Figures 4.4.1 and 4.4.2 show two examples of acquisition of the mag-netic field (in red) and the signal VdiffPD (in blue). During a series of acquisitions over
periods of time ∼ O(10min), the signal VdiffPD can go out of range because of baseline
oscillations due, for example, to vibrations of the system. This can cause a shift in the average signal, or it can even distort it. The Python script written for this analysis checks that the maximum and minimum of the waveform are within the measurement range of the scope.
Figure 4.4.1: VdiffPD signal with an input magnetic field of ' 900 mG.
Figure 4.4.2: VdiffPD signal with an input magnetic field of ' 9 mG.
Figure 4.4.3 shows the trend of VdiffPD for different values of the magnetic field inside
and a slope of s = (424 ± 1) mV/G. This value depends linearly on the power of the laser: in our case, the power after the ND filter and the beam splitter was measured with a PowerMeter to be ' 1 mW. Therefore, the calibration ratio with this type of magnetometer is
dVdiffPD
dB '424 µV mG
−1mW−1
. (4.4.1)
Figure 4.4.3: VdiffPD signal in function of the input magnetic field B.
4.5
Magnetometer simulation
To test the relevant configurations, I have developed a simulation that consists in repre-senting the optical elements of the magnetometer as matrices and then calculating the evolution of the polarization state when a laser beam goes through all these elements. This is the goal of the Polarization Ray Tracing.
4.5.1
Polarization Ray Tracing in two dimensions
One of the most common descriptions of polarization states is the Jones formalism, formulated by R. Clark Jones in 1941 [31, 32].
The effect of an optical element on a collimated beam of polarized light may always be represented mathematically as a linear transformation of the two components of the electric vector of the electromagnetic wave. Linear optical elements can be represented by (Jones) matrices, and the amplitude and phase of the electric field can be represented by a (Jones) vector. When light crosses an optical element, the resulting polarization of the emerging light is found by taking the product of the Jones matrix of the optical
