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Phase variations in microwave cavities for atomic clocks

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Abstract
Phase Variations in Microwave Cavities for Atomic Clocks
Ruoxin Li
2007
We analyze the phase variations of the microwave field in a
how these variations affect the frequency of an atomic clock.
microwave fields in
T E 0n
T E 0n
microwave cavity and
We analytically solve for the
cavities which are used in atomic fountain clocks.
The analytic
solutions show significant new terms that are not present in previous two dimensional treatments.
The new terms show that cavities with small radii, near 2.1 cm for a 9.2 GHz cavity, have smaller
phase shifts than cavities with larger radii.
We also show that the three dimensional phase
variations near the axis of the cavity can be efficiently calculated with a rapidly converging series
of two dimensional finite element calculations. We use finite element methods to study the large
fields and phase shifts associated with the holes in the cavity endcaps. The effects of the phase
variations on atoms traversing a cavity are analyzed using the sensitivity function and we present a
cavity design that has small phase shifts for all atomic trajectories. For two tt/2 pulses, the
proposed cavity has transverse variations of the effective phase that are ±0.1 prad and produce no
systematic frequency offset for a nearly homogeneous and expanding cloud of atoms.
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Phase Variations in Microwave Cavities for Atomic Clocks
A Dissertation
Presented to the Faculty of the Graduate School
of
Yale University
in Candidacy for the Degree of
Doctor of Philosophy
by
Ruoxin Li
Dissertation Director: Professor Kurt E. Gibble
December 2007
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UMI Number: 3293341
Copyright 2007 by
Li, Ruoxin
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©2007 by Ruoxin Li
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iii
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Table of Contents
Abstract............................................................................................................................................... i
Title......................................................................................................................................................i
Table of Contents............................................................................................................................. iv
List of figures....................................................................................................................................vi
List of tables................................................................................................................................... viii
Acknowledgments........................................................................................................................... ix
Chapter 1.............................................................................................................................................1
Introduction.........................................................................................................................................1
1.1 Spatial phase variation and distributed cavity phase shift...................................................... 1
1.1.1 Two-level quantum system............................................................................................. 1
1.1.2 Fountain clocks and the Ramsey fringe....................................................................... 5
1.1.3 Distributed cavity phase (DCP) error............................................................................9
1.2 M otivation................................................................................................................................10
1.2.1 Applications of atomic clocks......................................................................................10
1.2.2 Previous treatments.......................................................................................................12
1.3 Structure of this work and summary of results..................................................................... 14
Chapter 2 ...........................................................................................................................................18
The analytic solutions for a cylindrical cavity.............................................................................. 18
2.1 The wave equation and the field expansion...........................................................................18
2.2 Small feed versus infinitesimal feed...................................................................................... 19
2.3 The side wall losses................................................................................................................ 20
2.4 The endcap losses....................................................................................................................26
2.5 Discussion of analytic results.................................................................................................30
Chapter 3 .......................................................................................................................................... 34
The finite element m ethod............................................................................................................. 34
3.1 M otivation............................................................................................................................... 34
3.2 An overview of Finite Element M ethod................................................................................34
3.3 Two equivalent approaches.....................................................................................................36
3.4 Two popular elements............................................................................................................. 39
3.4.1 Node-based triangular elements..................................................................................40
3.4.2 Edge-based triangular elements..................................................................................42
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3.4.3 Mixed axisymmetric elements...................................................................................44
3.5
Discretization of the double curl equation........................................................................... 45
3.6
The boundary conditions.......................................................................................................47
Chapter 4 .......................................................................................................................................... 51
The finite element model for a cavity with endcap holes.......................................................... 51
4.1
4.2
4.3
Introduction............................................................................................................................51
Two-dimensional finite element calculations......................................................................52
The effects of endcap holes.................................................................................................. 55
Chapter 5.......................................................................................................................................... 65
The improved cavities....................................................................................................................65
5.1
5.2
M otivation.............................................................................................................................. 65
Model of frequency shift due to distributed cavity phase variations.................................65
5.3
Improved cavities.................................................................................................................. 74
Chapter 6 .......................................................................................................................................... 83
Summary.......................................................................................................................................... 83
6.1 Our principal results.............................................................................................................. 83
6.2
Future projects........................................................................................................................83
6.2.1 Imperfections of fountains......................................................................................... 84
6.2.2 Evaluating DCP error..................................................................................................87
6.2.3 Further improvements.................................................................................................90
Appendix.......................................................................................................................................... 91
Power dependence of DCP errors for cavities without endcap holes.........................................91
Bibliography.................................................................................................................................. 101
V
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List of figures
Figure 1: A schematic of a fountain clock.................................................................................... 6
Figure 2: The Bloch vector evolution during the Ramsey separated oscillatory method. ... 7
Figure 3: A Ramsey central fringes for a clock with T=0.5s.......................................................8
Figure 4: A picture of how the Bloch vector evolves in cavity with a spatial phase variation.
..................................................................................................................................................9
Figure 5: Ho in the vertical cross section of a cavity without endcap holes............................19
Figure 6: A typical cavity with a single feed.............................................................................. 20
Figure 7: The boundary condition and the decomposition........................................................21
Figure 8: The Poynting vector in the vertical and transverse cross-sections.......................... 23
Figure 9: The phase of Hz at the midsection with 1, 2, and 4 feed, respectively.................... 25
Figure 10: The solution for endcap losses only..........................................................................26
Figure 11: Phase for different R.................................................................................................. 31
Figure 12: A triangular element for the node-based finite element..........................................41
Figure 13: A triangular element for the edge-based finite element.......................................... 43
Figure 14: The boundary conditions used for a microwave cavity with cut off waveguide. 48
Figure 15: Contours of |H0jZ|........................................................................................................ 56
Figure 16: The comer at the edge of the endcap aperture......................................................... 57
Figure 17: The mesh grid used in the m=0 calculations............................................................58
Figure 18: Contours of Z,og(|£'9,(F)|) near the wall of a centered endcap hole for the TE0n
mode...................................................................................................................................... 59
Figure 19: Contours of the phase of H z(r ) near the wall of a centered endcap hole for the
TEon mode and the cavity of Figure 18............................................................................. 61
Figure 20: The phase of Hz for different trajectories................................................................. 62
Figure 21: The phase of Hz along a trajectory 1008 from the wall...........................................62
Figure 22: The effective surface resistance................................................................................ 63
Figure 23: The sensitivity function as a function o f time..........................................................68
Figure 24: Different effective phases for different modes........................................................ 76
Figure 25: An improved cavity design........................................................................................79
Figure 26: The effective phase of the improved cavity at optimal power...............................80
Figure 27: 8<X>avg as a function of cavity radius R for cavities that are fed at z=0 and z=±d/3..
81
Figure 28: Atom’s survival fraction (solid), atomic densities on the way up (higher two curves)
and down (lower two curves) as a function of x................................................................93
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Figure 29: Power dependence (PD) of the phases with only longitudinal variation for a a= -l
atomic density on the way up without beam misalignment..............................................95
Figure 30: Power dependence (PD) of the phases with transverse variation for a a = -l atomic
ball on the way up for well-aligned beam for m=0 phases, and for a 1mm misalignment
for the m=l phases with 5% feed imbalance..................................................................... 98
vii
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List of tables
Table 1: Some of the relative systematic uncertaintycontributions (multiplied by a factor of
10"16) of seven primary Cesium fountain clocks................................................................12
Table 2: Phase for the side wall losses...................................................................................... 25
Table 3: Phase for the endcap losses..........................................................................................29
Table 4: Phase expansions for a cavity with R=2.42cm.......................................................... 92
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Acknowledgments
I would like to acknowledge the guidance, council, instruction and support of my advisor,
Kurt Gibble; the assistance and comradeship of past and present lab-mates, especially Russ Hart,
Irfon Rees, Xinye Xu, Lingze Duan and Chad Fertig; the members of the Weis and O ’Hara groups
who generously lent both advice and encouragement, in particular Xiao Li and Fang Fang; the
stimulating contributions to the calculation made by Wenhua Yu; the help of members of the staff of
the Graduate Registrar's and Business offices at Yale; and finally, the financial support from NASA
Microgravity program, the Office of Naval Research, Yale university, and The Pennsylvania State
University.
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D edicated to m y wife.
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Chapter 1
Introduction
As an introduction to this dissertation, we explain the relationship between the spatial
phase variation in microwave cavities used in atomic clocks and the frequency shift of the Ramsey
fringe due to this phase variation. We then show why this work needs to be done.
1.1
Spatial phase variation and distributed cavity phase shift
In microwave cavities, the magnetic field has a small spatial phase variation due to the
power loss in the metal walls. When atoms interact with the microwave field in atomic clocks, an
extra phase difference accumulates between the atomic coherence and the microwave field because
of the spatial phase variation. The extra phase difference is reflected by the shift of the Ramsey
fringe. The shift of the Ramsey fringe due to the microwave spatial phase variation is called the
distributed cavity phase (DCP) error. In this section, we briefly discuss the relationship between
the spatial phase variation and the DCP error. To understand this effect, we start with describing
the quantum physics of a two level system because it is a simple and good approximation to the
clock transition. We then discuss the concept of atomic fountain clocks and Ramsey fringes. In
the final subsection we show that the spatial phase variation is the direct cause of the DCP error.
1.1.1 Two-level quantum system
A two-level system is a system with only two possible quantum states. To derive the
equations of motion of a two-level quantum system, we start with the time-dependent Schrodinger
equation:
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Chapter 1: Introduction
( 1 .1)
and
( 1.2)
where h is the Plank constant divided by 2n, H(t) is the total Hamiltonian for an atom interacting
with a microwave field, r is the coordinate of the electron, Hz(t) is the z component of the
microwave magnetic field and pz is the atomic transition magnetic dipole moment.
The
time-independent atomic Hamiltonian is denoted as Ho, its eigenvalues are £k= hcok , and its
eigenfunctions are^. (?) with 7i^(pk (?) = £k(pk (?)
The eigenfunctions (pk(?)form a complete set. The solutionT*(r,/)of Eq. (1.1) can,
therefore, be expanded in terms of <pk (r ):
(1.3)
k
where Ck(t) is the expansion coefficient, | Ck(t)|2 gives the probability for the system being in state
<pk ( r ) .
Substituting (1.3) into (1.1), multiplying on both sides by (f>*(r) and integrating over
spatial coordinates r , we can get:
(1.4)
and co0 -a > j-c o k .
where
Here the orthogonality between <p,(r )
and (pj ( r ) (i^ j) has been employed.
We consider a system with two internal states 11^ and 12) , where 11^ denotes the ground
state and 12^ the excited state. The initial condition for an atom in the ground state is C2(0)=0 and
C i(0)= 1. Eq. (1.4) becomes:
2
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Chapter 1: Introduction
in ^
r
at
= c A < W n {‘) ^
(1.5)
dt
We take the form of the microwave magnetic field as:
H z (t) = H 0 cos (a>Qt+ Q>(t))
(1-6)
where H0 is the field amplitude, and ®(t) is the time-varying phase. The time-varying phase ®(t)
is the spatial phase variation experienced by atoms moving in the microwave field and its derivative
with respect to time is a frequency detuning relative to (o0. The matrix element TV n can be written
as:
7Cn
c o s ( f iy + <!>(/)) = ^ -Q (f)e
+ c.c.
(1.7)
where
Q _
( 18)
%
is the on-resonance Rabi frequency, Pr is the Bohr magneton. We define the microwave field
detuning from the atomic resonance frequency A=d&/dt. In the rotating wave approximation we
neglect the fast oscillating terms e~2a¥ . Eqs. (1.5) can be rewritten as:
^
dt
^
dt
= - i n e ‘A'C2
2
(1.9)
= -iQ e~ iA,Cl
The Bloch vector is defined as
f AL
- AL
a ( t ) = e T C ’C, + e " '2 C,*C2, - i e 2 C2*C, + i e ‘2 C,*C2, |C,|2 - |C 2|2
V
J
( 1. 10)
We calculate the time derivatives of the three components of the Bloch vector by using Eq. (1.9).
The resultant equations take the following form:
3
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Chapter 1: Introduction
<h\ ( 0
+ Aa2 (f) = 0
dt
- Aa](t) +^ p - - n a 3(t) = 0
, . d a A t)
Cla2(t)+
iK -'- = 0
dt
Eqs. (1.11) are the equations o f motion o f a Larmor precession and can be written as:
da
dt
0
-A
0" V
Q a2 = ClRx a
A
0
0
-C l
0,
( 1. 12)
)
The physical picture is that the Bloch vector a = ( a ,, a 2, a 3) processes about a fictitious
field C1R = ( Q ,0 ,—A) and the Larmor frequency is given by the magnitude of ClR
Cl„ = V a 2 + Q 2
(1.13)
The solution of Eq. (1.12) for a constant fictitious field during time t takes the form of a
3x3 matrix as follows [1]:
r “\ ( t Y
a2( r ) = 7Z{C1,A,t ) a2 ( 0 )
(1.14)
va 3 M ;
cosQ^r
Q2
+ 7 y - ( 1 _ C 0 sQ «r )
llR
o fi/
— :A
— s •i n
QA
o srj^
—fi
( l- c o s Q
where
(1.15)
-^ -sin Q „r
C1D
*
QA
Qi
(1-cosQ ^ r)
c o sfi rt
Cl
.
sinQ^r
ClR
sinQ„r
Cl2
l--p ^ -(l-c o s Q ^ r )
cit
4
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Chapter 1: Introduction
1.1.2 Fountain clocks and the Ramsey fringe
In 1950 Ramsey invented a molecular beam resonance method by sending molecules
through two separate oscillatory electromagnetic fields [2],
[3-4] later based on Ramsey’s method.
A fountain clock was considered
The first atomic fountain was successfully built by
Kasevich et al. in 1989 [5] after the laser cooling and trapping technique became available.
first laser-cooled Cesium atomic spectroscopy was conducted in 1991 [6].
schematic o f a typical fountain clock.
The
Figure 1 shows a
In Ramsey’s method, atoms interact with two microwave
fields separated by a free evolution time T.
The probability of atoms being in one of the
hyperfine ground states after the two interactions is a function of the microwave detuning A.
Generally a clock measurement consists of 5 steps: (1) a sample of atoms are prepared in one of
the hyperfine states (|l)o r|2 ^ as defined in section 1.1); (2) the atomic coherence is prepared
between the two states by applying a “n/2" pulse to the sample when atoms pass through the
microwave cavity for the first time; (3) a phase difference between the microwave field and the
atomic coherence accumulates while atoms fly outside the cavity (with no microwave magnetic
field applied) during the free evolution time T; (4) the phase difference is converted into a
population o f atoms being in one of the two quantum states by applying a second “n/2" pulse
when atoms pass through the microwave cavity for the second time; (5) the atomic population in
one of the two quantum states is measured by fluorescence; (6) the above process is repeated
many times while the detuning of the microwave field is scanned and the frequency which
maximizes the fluorescence is found to be the clock transition frequency, and a local oscillator is
locked to work at this frequency.
During the two atomic passages, for a small detuning A <SCf 2 ,
the evolution matrix 7Z (Q , A, r ) can be simplified as that for a fictitious field along the axis 1’,
about which the Bloch vector precesses by an angle fix:
5
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Chapter 1: Introduction
I
I
I
I
I
I
I
A
I
Figure 1: A schematic of a fountain clock. The atoms are initially trapped in a MOT, cooled down to
a couple of micro Kelvin and prepared in either the ground or excited state of the two clock transition
hyperfine levels. Then they are launched vertically upwards. On the way up, atoms pass the cavity
the first time (for time x) and interact with the microwave magnetic field. A coherent superposition
of the two hyperfine states is prepared during this passage. The atoms keep going up and exit the
cavity and then come back down under gravity. During the flight (for time T »x) outside the cavity,
the atomic coherence precesses relative to the rotating frame of the microwave field and the phase
difference between the magnetic field and the atomic coherence accumulates. When the atoms pass
the cavity for the second time (for time x), this cumulative phase difference is converted into the
atomic population on either of the two energy levels.
' 1
0
0
^
/£ (Q ,0 ,r ) = 0
cosQ r
sinQ r
v0
-s in Q r
cosQ ry
(1.16)
Similarly, during the free evolution time T, the field is zero outside the cavity, thus fi=0,the
fictitious field is then equal to the detuning A and is along the axis 3’.
vector precesses about the axis 3 ’ by an angle -AT.
Therefore, the Bloch
The evolution matrix becomes
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Chapter 1: Introduction
Bloch vector a
Fictitious field Q
te /2 x
1’
-Q,
II
1
III
’
tt/2 x
Figure 2: The Bloch vector evolution during the Ramsey separated oscillatory method. I, II and III
correspond to the first passage, free evolution and the second passage, respectively. The initial
positions of the Bloch vector at each stage are represented by the dashed arrows and the final position
by the solid ones. For a n il interaction on the first clock cavity passage (I), the Bloch vector
precesses 90° about the 1’ axis. During the free evolution time, the Bloch vector precesses in the
l ’-2’ plane by an angle 8<|)=TxA (II). Another n il interaction on the second passage (III), the phase
angle accumulated in the free evolution time is converted into the atomic population difference, and
the 3’ component of the Bloch vector is read out by the measurement following the Ramsey
interrogation sequence.
y£(0,A ,r)
v
cos \T
-sin A f
0^
sinAF
cosAF
0
0
0
1
(1.17)
In the context of the evolution matrix, a Ramsey sequence becomes a product of a set of
three evolution matrices, which represent the two microwave interrogations and the free evolution
between them, as expressed in Eq. (1.18).
Figure 2 shows the Bloch vectors for the atom at each
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Chapter 1: Introduction
$ 0.8
»5
3
C/J
o
p
T3
i-i
O
0.6
a*
p
CT*
0 .4
^
0.2
5
-4
-3
- 2 - 1 0
2
1
3
4
5
A / ( 2 n ) (H z)
Figure 3: A Ramsey central fringes for a clock with T=0.5s.
of these stages.
"a, (T + 2r)"
0
1
p
a
o
>
p,
o
P
A
ll
a 2 (T + 2r)
M » ))
va3(T + 2 r )y
a 2 (0)
(1.18)
Va3(0),
The probability of atoms being in the excited state is the projection of the Bloch vector onto the axis
3’. After the second passage, it is
1
P(/) =
f
t
1
\
.(0
(1.19)
(0)y
In the simplest case, we consider that the two microwave pulses have a constant amplitude Q and
Qx=n/2.
Figure 3 shows a Ramsey fringe for a free evolution time T=0.5s and two n/2
interrogation pulses. The fringe is a periodic function of the detuning and its amplitude is the
largest for two n/2 pulses. The line width o f the fringe is 1/T, the inverse of the free evolution
time.
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Chapter 1: Introduction
4L
1
Figure 4: A picture o f how the Bloch vector evolves in cavity with a spatial phase variation. The
phase gradient provides a torque along axis 3’, which introduces an extra phase difference between
the atomic coherence and the microwave field. As a result, the Ramsey fringe is shifted by this
extra phase difference.
1.1.3 Distributed cavity phase (DCP) error
Generally, there are traveling waves in microwave cavities which carry electromagnetic
power from the cavity feed(s) and distribute it to the metal walls to compensate the wall losses and
sustain the microwave field oscillation. The phase variation with the traveling waves can be
viewed as a spatially varying phase O(r) to the microwave fields. ®(r) is seen by atoms as a time
varying phase <b(t). We consider an on-resonance microwave field with phase d>(t). Eq. (1.12)
becomes
0
da
dO
dt
dt
0
d® 0
-------
A
dt
(„ \
a\
o
n
-Q
0
a2 = ( M 0 + £ M e )a
/
Q
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( 1.20)
Chapter 1: Introduction
where M 0
0
0
(T
0
0
n
,0
-a
"0
-1
,m £= 1
0
,0
0
0"
0 , and s =
c/(D
correspond to the fictitious field
dt
0,
ClR=(Q, 0, -dO/dt) with a spatial phase variation. Figure 4 shows how the Bloch vector precesses
about ClR and an extra phase shift is induced. This shift o f the Ramsey fringe due to <J>(t) is called
a distributed cavity phase (DCP) error.
1.2
Motivation
1.2.1 Applications of atomic clocks
Current time scales are based on microwave transitions between atomic hyperfine energy
levels. Since 1967, the second has been defined as “the duration of 9 192 631 770 periods of the
radiation corresponding to the transition between the two hyperfine levels of the ground state of the
Cesium 133 atom”. There are a variety of techniques utilizing atomic or molecular transitions to
generate stable frequency standards, including thermal beam clocks, Hydrogen masers, fountain
clocks and the emerging optical clocks. The interested reader is directed to [1,7-9] for knowledge
of the physics of atomic frequency standards.
Atomic clocks have a number of applications, which are essential to the following four
scientific research areas. First off, an accurate clock can contribute to the International Atomic
Time (TAI) and its improvement can help realize a better SI unit of time. The scale unit of TAI is
kept by carefully weighting the data from participating clocks maintained by national laboratories.
The best primary Cs atomic frequency standard gets the highest weight. Therefore, improvements
of the participating clocks can bring a better estimate to the time scale. Secondly, synchronization
and comparison among national and international clocks provide rich options for fundamental tests,
such as the tests of Local Position Invariance (LPI), Local Lorentz Invariance (LLI), Gravitational
Red Shift, and the stability of the fundamental constants. In the following, we will give a brief
10
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Chapter 1: Introduction
description of these fundamental tests. Thirdly, atomic clocks can be used to investigate clock
related physics, such as cold collision properties of Cs and Rb atoms, Stark shift and black body
radiation shifts, and Feshbach resonances of Cs atoms.
These will significantly contribute to the
advancement of clocks, as well as to fundamental physics.
Fourthly, atomic clocks have been
widely used in navigation systems. For example, GPS helps scientists achieve better clock
comparison and synchronization, and also helps drivers find the right route.
We introduce the concepts of using atomic clock in the fundamental tests as follows:
(1)
Test of LP1: LP1 is part of the general Einstein Equivalence Principle (EEP) which in
turn is a foundation of Einstein's theory of general relativity. The EEP predicts a dependence of
clock rates on the local gravitational potential and LPI predicts that the gravitational shift is
independent of the atomic species involved as a reference in the clock. Researchers can make use
of the time dependence of the gravitational potential due to Earth's annual elliptical orbital motion.
They can then compare the frequency variations in time between clocks operating with different
atomic species. If LPI is valid, different clocks’ ticking rates should not be changed by their
positions in the gravitational potential. If LPI is violated, the change of the gravitational potential
is reflected by the fractional frequency shift between two nearby clocks, which provides a
measurable effect.
(2)
Test of LLI: LLI states that there is not a preferred rest frame in the universe. If such
a preferred reference frame does exist, a clock operating with different orientations will have a
frequency change. Evidence of LLI violation can be shown by measuring the frequency shift of an
atomic clock between noon and evening. So far, no such shift is found. But with a more stable
and more accurate clock, the limit of the validity of LLI can be set lower.
(3)
Test of gravitational red shift: light loses energy when moving away from massive
objects, such as the earth, the sun or a black hole. This effect is equivalent to gravitational time
dilation, which predicts that a clock in space will be running faster than one on earth. Therefore,
the comparison between two such clocks can give us a direct measurement of this effect.
li
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Chapter 1: Introduction
Cause of clock
uncertainty
SYRTE-FOl SYRTE- SYRTE-FOM NfST-Fl PTB-CSF1 lEN-CsFl NPL-CsFl
[14]
[10,12]
[16]
[13,18]
[15]
[10,11] F02[ll]
Cold collisions
2.4
2.0
5.8
1
7
12
8
Black-body radiation
2.5
2.6
2.5
2.6
0.7
4
Distributed cavity
phase
<3
<3
<2
<0.3
2
5
<0.3
3
Electronics,
microwave leakage
Total systematic
uncertainty (xl0"16)
3.3
4.3
2.4
1.4
2
<2
3
7.2
6.5
7.7
3.3
9
16
10
Table 1: Some of the relative systematic uncertainty contributions (multiplied by a factor of 1016) of
seven primary Cesium fountain clocks [25], All values are based on the latest publications.
(4)
Test of stability of fundamental constants: variations of fundamental constants with
time and space are allowed in a wide range of the cosmological theories developed to unify
gravitation with quantum mechanics.
Experiments testing the stability of fundamental constants
can thus be seen as tests of the Equivalence Principle and as constraints to theoretical work aimed
at a unified theory.
A variation of fundamental constants may be detected by high precision
comparison o f atomic transition frequencies between different energy levels or between different
atomic species.
The above tests can be done with accurate clocks.
The more precise a clock
measurement can be, the more sensitive an experiment is to the possible effect.
Generally
speaking, better understanding of the causes of systematic errors in atomic clocks can guide the
clock researchers towards further improvements to their experiments. However, current methods
of analyzing one of the systematic errors, the distributed cavity phase shift, have problems.
New
techniques have to be developed.
1.2.2 Previous treatments
In the current generation of laser-cooled atomic fountain clocks, accuracies have advanced
to near and beyond 10"15 [17- 23], A number of systematic errors are important in these and
future clocks. In Table 1, some of the systematic errors of seven primary Cs fountain clocks are
12
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Chapter 1: Introduction
listed. One of the potential errors arises because the phase of the microwave field that excites the
atoms is not constant throughout the microwave cavity [24-25]. This error, distributed cavity
phase shift, was evaluated by assuming that the power dependence of this effect linearly grows with
the power [15, 18]. They are not correct evaluations because the power dependence is not a linear
function of power, but fairly complex, so the numbers in Table 1 for the DCP error are not be
trustworthy.
The most common microwave cavity in fountain clocks is a cylindrical
T E on
cavity
because it has small losses (high Q) and useful field geometry. The calculations of the spatial
phase variation in the microwave cavities are either not accurate for either one of the two reasons:
the calculation was done with a coarse mesh grid using 3D finite element method (FEM) [26], or it
was missing important phase variations because a 2D FEM was implemented by ignoring the
longitudinal variation of the field [27-28],
DeMarchi and collaborators have studied several aspects of this problem [27 - 29]. In [28]
they showed that the losses in the conducting walls of the microwave cavity imply that, in addition
to the large standing wave, there is a small traveling wave component. The superposition of these
two fields can be viewed as a standing wave with a spatially dependent phase. Because the phase
of the field is different for different atomic trajectories through the cavity, a clock may have a
frequency error due to this distributed cavity phase shift. DeMarchi and collaborators studied the
transverse phase variations due to the sidewall losses using two dimensional (2D) finite element
calculations that assumed no variation of the fields along the axis of the cavity [28, 29]. This 2D
FEM calculation reached two conclusions which do not hold for a 3D calculation. One o f the
conclusions is that the phase variation can be made arbitrarily small by adding more and more feeds
at the mid-section of the cavity side walls; the other is that a cavity with a larger radius is better than
one with a smaller radius because the phase variation is reduced in the large radius cavity, therefore
it introduces smaller DCP error than the small radius cavity. Those were made because significant
phase variations were missing in their calculations. Here, we use analytic and finite element
13
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Chapter 1: Introduction
techniques and find new terms of spatial phase variation due to endcap losses. With the new terms,
we have gotten different conclusions. We have found that the phase variation cannot be made
arbitrarily small, and that a small radius (slightly larger than the cutoff radius R=2cm) cavity
actually produces smaller DCP error than cavities with large radii. We also study a number of
aspects of the phase and present a cavity design with small, even vanishing, effects due to the phase
variations.
1.3
Structure o f this work and summary of results
Our approach is to first analytically solve for the three dimensional phase variations due to
the losses in a cylindrical cavity. These analytic calculations result in simple expressions for the
phase in cylindrical cavities and can guide us to better cavity designs. We show that an azimuthal
series of 2D finite element calculations can efficiently produce the three-dimensional phase
distributions for cavities with arbitrary shapes. Because the atoms pass through the center of the
cavity, only two to four two-dimensional calculations are required.
Compared with full
three-dimensional finite element calculations [30], two-dimensional calculations require much less
computing resources.
Computing time is reduced from hours to seconds on current desktop
computers. Finally, we consider the effects of the spatial phase variations on the atoms and then
present an improved cavity design. The contents in each chapter are as follows:
Chapter 1 sets up the framework of atomic fountain clocks, and describes how the DCP
error occurs and why we want to study it.
Chapter 2 develops the analytic solutions of the spatial phase variation for a cavity without
endcap holes. We first break up the total field into a superposition of two standing waves - the big
standing waves E 0 and H () corresponds to the solution for a cavity with perfectly conducting
walls, and the small standing waves f
and g that are excited by the wall losses and the power
coming in the cavity through the feed(s). We decompose the boundary condition on the side walls
14
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Chapter 1: Introduction
into its Fourier’s series corresponding to the azimuthal ((f)) dependence and the longitudinal (z)
dependence, and then solve for the fields which have the same <f>and z dependences. Specifically
for azimuthally symmetric fields, we solve for the total fields in a cavity with perfectly conducting
side wall and lossy endcaps.
f
and g become the difference of the total fields and the standing
waves E 0 and H 0. The full 3D solution is the summation of all the above terms. We show that
only a few terms are needed to get the summation to converge at the center of the cavity where the
atoms pass. We compare the phase variations of different <)) and z dependences and point out that
feeding the cavity at different places on the side wall will change the ratio between those modes, so
their contributions to the DCP error change proportionally.
In this chapter, we show that
DeMarchi’s 2D solution is exactly the summation of all p=l modes in our analytic model, which
corresponds to the solution of a cavity with a feed as high as the cavity and the field in the feed
aperture has a cos(rtz/d) dependence, where d is the length of the cavity. With the analytic solution,
we reach conclusions different from DeMarchi’s: we show that an endcap loss excites an
azimuthally symmetric mode, which cannot be eliminated by adding more feeds at the midsection;
further study shows that this mode has a big DCP error for cavities with much greater radii (for
instance, R=3cm for cavity in NIST-F1) than the cutoff radius (R=2cm).
Chapter 3 is a brief introduction to the finite element method (FEM) that we use to
numerically calculate fields and phases in cavities with endcap holes.
overview of FEM.
We begin with an
We then introduce the basic concepts of FEM, which include two equivalent
approaches to numerically implement the partial differential equations and two popular element
choices.
We explain the shortcomings of the node based element and the advantages o f the edge
based element in computational electromagnetics, and present a new variable transformation to
use the two types of elements together to overcome problems of using the node based elements
alone.
We also point out that with geometry discontinuities, we have to solve for all three
15
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Chapter 1: Introduction
components of the magnetic field by solving the double curl equation V x V x g - & 2g = 0, instead
of the Helmholtz equation V2g - k 2g = 0 as we do in Chapter 2.
In Chapter 4, we more specifically describe the results of numerical calculations for a
cavity of one current primary Cs fountain clock. The presence of endcap holes makes an analytic
solution impossible, so that a numerical calculation must be performed. In this chapter, 3D and 2D
FEM calculations are compared to show that our Fourier expansion method can much more
efficiently and accurately get a full solution than the 3D FEM. We decompose the boundary
conditions and fields in the azimuthal dimension and then solve the Maxwell’s equations. We first
solve for the primary fields E 0 and H o . We find that the primary field Ho,z reverses its sign and
is large near edges of endcap holes. We then generate the boundary conditions for f
from H u.
A magneto static approximation is chosen near a perfectly conducting edge and an analytical
solution for fields in that region is given. This solution shows that Ho diverges as p"l/3, where p
is the distance from the edge. We use a novel FEM to calculate the field near a metallic edge. The
solution exhibits p 'l/3 dependence a few skin depth away from the edge, but does not diverge.
The magnetic field at the metallic edge is as large as at the center o f the cavity. It is also shown
that the phase near edges of endcap holes is large (~130mrad for the NIST-F1 cavity geometry) but
falls off rapidly while getting away from the edge.
Chapter 5 describes an improved cavity design. We begin with the introduction to the
sensitivity function, which we use to analyze the effect of the spatial phase variation on the atomic
transition probability. We then present an improved cavity. The analytic solution guides us to
choose a small radius (2.182cm) rather than a large radius cavity. We use 8 feeds to excite the
fields inside the cavity. These many feeds can make the phase of the higher modes (m=l, 2, ...)
negligible, so that we only focus on the m=0 phase variation. We then introduce a two-stage
cut-off waveguide section with a small aperture, which eliminates atoms that experience large
16
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Chapter 1: Introduction
phase near the edges of the large endcap holes. As a result, all the atoms see a phase variation less
than lmrad anywhere on their trajectories. Finally, with the help of the sensitivity function, we
find that the DCP error can be zero at the optimal power if the feeds are placed at the height of
± d / 3, where d is the length of the cavity.
Chapter 6 is the summary chapter. In this chapter, we summarize our principal results.
Then we discuss the future studies needed to extend this study to the fountain clock evaluation.
17
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Chapter 2
The analytic solutions for a cylindrical cavity
2.1.
The wave equation and the field expansion
In this chapter we present an analytic calculation treating a cylindrical cavity with no holes
in the endcaps. We begin by solving for the fields of a TE0i i cylindrical cavity that has lossy (copper)
sidewalls and perfectly conducting endcaps. We then treat the endcap losses for a cavity that has
perfectly conducting sidewalls. The superposition of these two solutions is the full solution for a
cylindrical cavity with metallic surfaces.
The total electromagnetic field in the cavity satisfies the wave equations:
(V 2 + ^ ) / / ( F ) = 0
( 2 - 1)
C2
(v 2 + £ L ) i ( ?) = o
c
where a e~,M time dependence is assumed for all fields. We perturbatively expand the fields as
in [28], In a standing wave basis, the total field can be written as a superposition of a large
standing waves E0( r ) and H 0 ( r ) , which satisfy the wave Eqs. (2.1) with perfectly conducting
walls, and small standing waves / ( r ) and g ( r ) , that are also solutions of Eqs. (1) while
accounting for the losses.
tf(r) = tf0(r) + (l + i)g(r)
E ( r ) = iE0 ( r ) - ( l - i ) f ( r )
While all the fields are generally complex, Eqs.
(2.2) are written so that
E 0 (r), H 0 ( r ) , f ( r ) , and g ( r ) may still be real. In cylindrical coordinates r = ( r ,^ ,z ) , we take
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Chapter 2: The analytical solutions fo r a cylindrical cavity
Figure 5: Hq in the vertical cross section of a cavity without endcap holes. S is the Poynting vector
which denotes the power fed into the cavity from an external source through the small waveguide.
the primary TE0n field to be:
(2.3)
where, for the TEmnp mode, k p =
7P = y c° 2/ c'2 ~
= 1 ,2 ,3 ,..., R is the radius o f the
cavity, J m(x ) is the Bessel Function of the first kind, and the z component of H 0 ( r , z ) is
field H 0 is shown in the vertical cross section, and we define the coordinate system such that the
midsection is at z=0.
2.2.
Small feed versus infinitesimal feed
Atypical feed in use is a small waveguide which connects the TEon cavity and some power
sources. For example, in Figure 6, a typical feed is a rectangular waveguide placed at the middle
of the side wall. The power comes into the feed through the opening at the far end, a typical mode
(generally H ^O for TM mode excitation in the main cavity should be avoided) is excited which
carries the energy through the waveguide, and then propagate into the cavity through the
connection aperture. Usually the aperture is small compared with the dimension o f the cavity to
19
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Chapter 2: The analytical solutions fo r a cylindrical cavity
Figure 6: A typical cavity with a single feed. The feed is generally a small waveguide connecting
to the cavity with an aperture in the middle of the side wall. The power is fed into the feed through
the other opening (connecting another cavity with the same resonant frequency) or by an antenna in
this waveguide.
achieve a weak coupling and a high Q factor. There are exceptions that a larger aperture is used
(for instance, the feed for the cavity in PTB-CsFl).
cavity is insensitive to the size of the feed.
We find that the field and power flow in the
For simplicity, in the analytic solution, we use
infinitesimal feeds.
2.3.
The side wall losses
To demonstrate an analytic calculation of the phase of H z(r ) , we consider a cylindrical
cavity with side wall losses and perfectly conducting endcaps which is fed by one or more
infinitesimally small power feeds at the cavity midsection. To solve for f (r) and g ( r ) , we start
with the boundary conditions for a locally uniform wave incident on a conductor. In Figure 7, a
20
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Chapter 2: The analytical solutions fo r a cylindrical cavity
(a) 300
■<b
m=0, 1, 2
-
p=1, 3, 5
0.6
Figure 7: The boundary condition and the decomposition, (a) The boundary condition for the parallel
component of the electric field as a function of (j). The boundary condition is expanded into the Fourier
series cos(m(|)) for m=0 (dash-dot), 1 (dashed), and 2 (dotted). One should notice that for side wall
losses only, m=0 must be zero, (b) The boundary condition as a function of z. It’s decomposed in the
cos(p;tz/d) series forp=l (dash-dot), 3 (dashed) and 5 (dotted).
typical boundary condition and its decomposition is shown for a feed with finite size. The left
pane shows the Fourier expansion in <)) and the right pane in z. The electric fields on the surface
of the conductor is E f r ) = (l —i ) Rsh x H 0 ( r ) , where R s = f p {)oC 2 a is the surface resistance,
the skin depth is 5 = -Jlfju^coa , the conductivity of copper is o = 5.8><107/Qm, h is normal to
the metallic surface, and terms second order and higher in 5 are neglected [28]. We define the
surface resistance. From this, we get the boundary condition for f ^ i f ) and the power loss at all
positions on the side walls, excluding the feed:
5 = ^ R e ( £ x ^ * ) = - I / , ( F ) / / 0;( 7 ? ,z ) r = ^
The value of
s| / / 0;Z(i?,z)|2 r
(2.4)
( r ) at the feed(s) is such that the proper power is supplied to the cavity.
We solve for f i r ) and g ( r ) by decomposing the above boundary condition for f ( r )
in a
Fourier series of cos(/w ^) and c o s ^ z ) . The series is an even function of z and even in
(j) about the feed position(s) due to the symmetry of the primary field and the cavity feeds (here we
assume that the feeds are at the midsection as all the current clock cavities in use).
21
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Chapter 2: The analytical solutions fo r a cylindrical cavity
This transformation gives a helpful physical picture of power flow and phase gradients in
cavities. For example, the azimuthally symmetric m=0 mode can be physically visualized as a
circular feed that surrounds the cavity. Because the power fed into the cavity through the side
walls is equal to all of the losses, the net m=0 and p=l power is zero and so this mode is not present
in / (r ) and g ( r ) . Indeed, this mode is the primary TE0ii field for perfectly conducting walls and
it cannot lead to any power flow.
Next we consider the p=l modes o ff ( r ) and g ( r ) . Physically, these modes feed power
into the cavity at <|>=0 over the entire height of the cavity with a z dependence
of f ^ m p(r) = f j mp{r) cos (m ^ )co s(& ,z). Thus, if the feed is essentially §(<)>), then all m>0 p=l
modes are excited equally and these construct a narrow feed at <|)=0 and an equal amount of losses at
all other t|). With this picture, the m=p=l mode represents power that is fed into the cavity over the
entire height of the cavity and, in (j), it represents power fed into the cavity for -7i/2<(|)<tc/2 and
power leaving the cavity due to wall losses for tc/2«})<3tc/2. Higher m modes further redistribute
this power as a function of <j>. For p>l, these modes redistribute the power on the sidewalls to
construct a narrow feed, for example, at z=0. Considering the m =l, p=3 mode, it feeds power into
the cavity at -7t/2«j)<7r/2 and z=0 and feeds power to the walls for z~±d/3.
With this picture it is straightforward to solve the wave equation for f ( r ) with the
boundary conditions in (2.4):
00
cc
/(? )= Z I
/„ (? >
m-0 p-\+Smo
r
fr,m,p ( r ) =
5k, f f l l - ( - l ) ' ’ J q{Y\R )
------- ----------- T7—
2? p r
2
t
j
\ ■ ( A\
V p r ) sin W
h
cos t V ) r
J m ( VpR)
(2.5)
\ J° [ r 'R\ Jm\ r f ) c o s { m 0 ) c o s ( k pz ) 0
2 2 ( l + ^ 0j J m ( y pR)
The solution for / ( r ) is transverse because for a cylindrical cavity, there is no coupling
22
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Chapter 2: The analytical solutions fo r a cylindrical cavity
m=1
'[ iiiiiiiliU M
?V a raaV ay aV aaa a a ^
W
i i
l E f z z E E E z E~ z Wz EE
t
m =3
z J //-7 E E E E E / / /
§ / f/z z E EE -Ez£zz£
-if
Figure 8: The Poynting vector in the vertical and transverse cross-sections. A small rectangle is used to
denote the feed.
23
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Chapter 2: The analytical solutions fo r a cylindrical cavity
between TE and TM modes.
Where J m (x ) = dJm ( x ^ / d x is dimensionless, Sm . is a
Kronecker delta function, and only odd p modes are excited if the cavity is fed symmetrically about
z=0. Atomic clocks are principally sensitive to the phase of H z(r) which is
<D = - t a n -1 [ l m ( / / z( F ) ) /R e ( //z( r ) ) ] = —g z( r ) / H 0 Z(F ), to lowest order in the skin-depth 8,
Using the solution for / ( F ) , we useg(F ) = V x f ( r ) / / u 0co to get the
whereH 0 z(F) is real.
magnetic field g ( r ) :
Skxk
g
( / ) = ------ —
=
l - ( - l ) P J0 (y,R)
,,
J
\ -------
\
/
\
\ Y r ) c o s ( m( f ) s \ n ( k z ) r
Sk . k n m 1—( —1)" J a( y, R)
t
\
,
s
i
-— J m( r pr ) s m ( m 0 ) s m ( k pz ) t
Zy p r
1
J m ( r PR )
( 2 .6)
——J
g&Z,m,p\
z (r)>= ------'n.^P ryt/ 1 ^, 0 ~—r\ —T
r>\ m\( y Pr)cos(m<t>)cos(k
)
\ Y!
\ Pz )} z
1
+
J m { y pR)
In Figure 8, the power flow (the Poynting vector) is shown for each m =l, 2, 3 and 4 mode
corresponding to a single feed cavity. One can easily see the clear physical meaning of the Fourier
expansion in ((). For example, through the m=l mode the power gets into the cavity on the right
side (where the single feed is), and comes out on the opposite side.
The other modes have
relatively smaller power flow in the center region because the large amount of power is dissipated
by the wall near the region where it gets into the cavity. One can expect that the m=l phase
dominates the phase variation for a single feed cavity, and it’s true and shown below.
Although y
is purely imaginary for p>l, this g, m p{r ) is real because the modified
Bessel function/m(x ) = i '"Jm( i x ) . The sum over m and p for each component is the same as in
Eq. (2.5).
To lowest order in 8, the phase is:
2 (1 + ^ )
cos(*,z)
24
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K r>
Chapter 2: The analytical solutions fo r a cylindrical cavity
m\p
1
3
0
0
0.004+0.021^
1
56.2r + 11.5r3
0.02r
2
34V
omf
-20.9I3 - 6.4r5
0.0 lr3
4
0.0 lr4
1
o
l/t
3
l‘
i—
On
0
+ 9.4r4
Table 2: Phase for the side wall losses. Contributions to the phase near the center of the cavity in micro
radians from Eqs. (2.7) and (2.8) at z=0 where r is in cm for a cavity resonant at 9.2 GHz with a radius of
3 cm. The p=3 terms are given at z=0. At r= 0.5 cm, summing over all terms, the phase is 0=35.8
prad at <j)=0 and -17.7 prad at <j)=7r.
0.1 prad
steps /
10 prad
steps /
Figure 9: The phase of Hz at the midsection with 1, 2, and 4 feed, respectively. The contour spacing
significantly decreases while the number of feeds increases. This shows that the phase gradient is
dramatically reduced by adding more feeds. Again, the small rectangles are used to denote where the
power is fed into the cavity.
This analytic sum forO (F) is particularly useful because the series converges very quickly
in the region o f interest for atomic clocks. In the usual configuration for a
T E 0n
clock cavity, all
atoms traverse the cavity within a few millimeters of the z-axis since there are holes centered on the
endcaps that have a radius of order ra = 0.5 cm, much smaller than the cavity radius R . For small
r, J m( j pr} ~ y pmf m
which is very small for small r/R and large m. Therefore, only a few
azimuthal modes contribute significantly to the phase near the center o f the cavity. The lowest
orders in the Taylor expansion of Eq. (2.7) dominate.
J m( r Pr )
J o{hr)
r pmr n
2
1+
mm\
rP
-
Yx
m+1
25
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(2 .8)
Chapter 2: The analytical solutions fo r a cylindrical cavity
For p>l, y p is imaginary so J m ( y p^ ) ^ Ce^'’^ grows very quickly.
Further, as we
show below, the effects are smaller because of their cosine dependence with z; for p=3, at z~0 the
phase shift has the opposite sign as for z~±d/3 so the average phase on a cavity traversal due to this
term is smaller than its peak phase shift. In Table 2 we show simple polynomial expressions for
each term of the cavity phase from Eqs. (2.7) and (2.8) for a cavity with a radius of 3 cm. For 2
feeds, the phase is given by the sum over m=2,4,6, . . . .
In Figure 9, the contours of phase in the
center region show the same reduction of the phase when more feeds are used. From Table 2 and
Figure 9 it is clear that cavities with 2 or more symmetric feeds have much smaller phase gradients.
Two or more feeds eliminate the large nearly linear phase gradient at the center of the cavity that
results from transmitting power from the feed on one side of the cavity to the walls on the other side
[28, 29], With 2 feeds, the largest contribution is a quadruple phase gradient due to power flowing
in near <j)=0 and n and flowing out at <()= ±n/2.
Thus, the dominant phase behavior is
O (F) = (34prad) r 2 c o s ( 2 0 ) , where r is in cm.
For 4 feeds, only m=4, 8, 12,... terms
contribute and here, only the m=4 p=l term is significant so <t>(r) <x r 4 cos (4 0 ) [29]. However,
as we show in the next section, the endcap losses lead to terms that are azimuthally symmetric (e.g.
<t>(r) cc r 2) and larger than this term. The azimuthally symmetric terms cannot be reduced by
using a large number of feeds at the cavity midsection.
2.4.
The endcap losses
We now treat a cavity with perfectly conducting sidewalls and lossy endcaps. The losses
on the endcaps at z=±d/2 in a cylindrical TE0n cavity are given by an analogue to Eq. (2.4):
S = - R e [ E x H ' ) - f f ( r ) H „ ; ( r , ± d / 2)z
(2.9)
1
= +
4
This gives the boundary condition for f , ( r , d / 2 ) .
On the endcap, f A r , d / 2 ) is therefore
26
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Chapter 2: The analytical solutions fo r a cylindrical cavity
0,Z
1 1t
f
I. 1 J4
tillilK
Hitt!
m .
.
.
i
i
5. 1
I !HK
0,z
Figure 10: The solution for endcap losses only. The dark color in endcaps represents that they are lossy.
The gray color means that in this case the side walls are considered perfect conductor. The magnetic
field Hz (solid) penetrates into the endcaps due to the finite skin-depth, so that it has less curvature in the
longitudinal direction than
z (dashed). Therefore Hz in the radial direction must have more
curvature than Ho,z to be able to oscillate at the right frequency (9.2GHz). The Poynting vector is
shown as arrows in the cross section of the cavity.
proportional to H 0 r( r , d / 2 ) ; a pure excitation of the TE0| waveguide mode.
Thus power is
transmitted to the endcaps by propagation along the cylindrical waveguide section.
It is simplest to first consider feeding such a cavity with a feed that is the entire height of
the cavity (p=l from the preceding chapter) and azimuthally symmetric (m=0). This “cavity”
therefore has no sidewalls but only lossy endcaps as in Figure 10. Nonetheless, Maxwell’s
equations hold and the fields we seek obey the boundary conditions of this “cavity”. As illustrated
27
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Chapter 2: The analytical solutions fo r a cylindrical cavity
in Figure 10, to solve for / ( r ) and g ( r ) , we look first generate the total solution E( r ) and H ( r ) .
We take the known solution for E ( r ) and H ( r ) and then subtract from it the known solution for
E 0 (r) a n d / f 0(F)to yield f ( r ) a n d g ( r ) .
The skin depth 5 of the endcap material leads to a field in the cavity that has less curvature
in the longitudinal direction and more curvature in the radial direction than the solution for
perfectly conducting endcaps. As shown in Figure 10, to first order in 5, the correspondent m=0
p=l mode of the electric field E is:
E{ r , z ) = i
— J ] [ ( / , + dy}r~^co?,^(kx - d k ) z ^ ( j )
(2.10)
2 Y\
2
2
where dk = k iS / d , (y, + dy^j - co/
/
2
2
—{k] - d k ^ , and d y - k ^ dk / y ^ . Eq. (2.10) is clearly
a solution to the wave equation and one can verify that it satisfies the boundary conditions on the
endcaps. We get f ( r ) by expanding Eq. (2.10) to first order in dk and dy and then subtracting
iE 0 ( r ) from it to get i f ( r ) , which is orthogonal to Eit(r) . Then, g ( r ) = V x f ( r ) / p nO) gives:
k ^$
g z,oAr ^ ) = y r ~ l [ r ^ J Q( y T ) ^ ( k lz ) - k ]rJl ( y ir ) c o s ( k {z ) ] + 0 0H 0f r , z )
where
<D0 = ( d / 2 d ) ^ 4 ^ k ]2 / y ^ ) -
is
the
constant
phase
shift
that
( 2 . 11)
insures
thatg 01( r ,z ) and H 0 ( r , z ) are orthogonal (<D0=64.9 prad for R=3cm and d=2.18 cm). The full
solutions are given below. We again get the phase distribution in this open cavity after dividing Eq.
(2.11) b y H 0 z( r , z ) . The remaining non-zero m=0 p=l field components following Eq. (2.11) for
endcap losses are:
U 0,1(r >z ) =
& k^
S
^ - y [ W o i r s ) c o s( K z ) + h z J \ { h r ) s in ( K z ) \ - ^ E o A r ’z )
(2 . 12)
r '
& k^
3S
gr, oj(r ’z ) = i r : - T [ k CJ o ( r T ) s i n ( k iz ) - y ]z f ( y f ) c o s ( k lz ) \ - - - H
2 d yt
2 d
28
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( r, z)
Chapter 2: The analytical solutions fo r a cylindrical cavity
m\p
1
3
0
0.01 +0.061-2
1
32.9r2 + 6.7r4 -6 5 .8 z 2
197.5r + 40.3r3
0.06r + 0.14r3
2
120.8r2 + 32.9r4
O.Olr1
3
-7 3 .5r3 - 22.5r5
0.05r3
4
-5.6r4 - 1.8r6
0.03r4
Table 3: Phase for the endcap losses. Contributions to the phase near the center of the cavity in micro
radians where r and z are in cm for a cavity resonant at 9.192 GHz with a radius of 3 cm. The m=0 and
p= 1 term is from Eqs. (2.11) and the p=3 terms are given at z=0. At z=0 and r= 0.5 cm, summing over all
terms, the phase is 134.6 prad at (|)=0 and -53.4 prad at <j>=7i.
We now have to correct the power feed of this open cavity since no cavities are fed with a
pure m=0 p=l mode. We now envision the cavity with endcap losses and sidewalls that are perfect
conductors. Eq. (2.11) describes the m=0 p=l power fed from the sidewalls to the endcaps in this
cavity. To satisfy the boundary conditions on the sidewalls, we first add p=l m>0 modes to build up
the Fourier series of a narrow power feed 8((()) [two (four) feeds imply a sum over m=0,2,4, ...
(m=0,4,8,...) modes]. The series are the same as in Eqs. (2.5), (2.6) and (2.7), except that they are
multiplied by a factor o f 2k {2R / y ^ d to account for the difference between the endcap losses in Eq.
(2.11) and the side wall losses in the previous subsection. As in the previous subsection, we now
add in Eqs. (2.5), (2.6) and (2.7), the p>l modes, to the p=l mode to reproduce a small feed 8(z) at
z=0, effectively representing the feeds of physical cavities.
The phase near the center of the cavity can again be expressed as a sum of simple
polynomials as in Table 3. Again, the p>l modes have small effects for large cavity radii. Since
the endcap and sidewall losses are of the same order, multiple feeds (e.g. at z=0) generally reduce
the phase variations [28, 29]. However, the m=0, p=l mode that feeds the power to the endcaps
produces an important phase shift. This term is present for any number of feeds on the side walls.
The —r J {( y, r ) cos (£,z) term in Eq. (2.11) produces a phase shift (j){r) = ( S / l d ^ k ^ r 2 which is
29
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Chapter 2: The analytical solutions fo r a cylindrical cavity
considerable (see Table 3). At first glance, the z J 0
) sin ( &,z) term seems to be less important
as <fi(r) = - ( S / d ^ k ^ tan ( k tz ) has no radial dependence. However, the effect of the endcap
holes and the non-linear response of the atoms give it a nearly equal significance. We discuss both
of these further in the following sections.
2.5.
Discussion of analytic results
Armed with the analytic results of the previous three sections, we can begin to optimize the
geometry of a cylindrical cavity to minimize the phase variations.
The superposition of the
solutions for side wall and endcap losses gives the solution for a real cylindrical cavity. This gives
the bulk of the phase shifts for a cavity with holes in the endcaps, which we treat in the following
chapters using finite element methods.
Clearly multiple feeds are preferable since the m=l p=l terms in Tables 2 and 3 are large.
With 2 or 4 feeds, the large m=l terms vanish to the extent that the feeds are the same and the
conductivity and surface finish are homogeneous. In addition, phase shifts are further minimized by
making sure that the modes that produce large phase shifts are detuned from the TE0i i resonance.
In Eq. (2.7), the distributed cavity phase is proportional to 1j J m {ypR) where J m ( y pR^ = 0 is
the resonance condition. As a particular m, p mode is tuned through the TE0n resonance by
changing the cavity geometry, the phase variations due to this mode become large, reverse, and then
become smaller as J m { f pR) goes to zero, reverses, and then grows.
Previous two dimensional finite element treatments without endcap losses suggested
using cavities with large radii and many feeds [28, 29],
Indeed, all the leading terms in Table 2
decrease with increasing R, and, for m symmetrically placed feeds, <f>(r) cc r m co s(m ^ ) [29].
However, the endcap losses increase with R and, more importantly, produce the azimuthally
symmetric phase deviations that cannot be eliminated with feeds placed at the cavity midsection.
30
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Chapter 2: The analytical solutions fo r a cylindrical cavity
40
30
212
20
TD
CD
10
3
CD
cn
CD
8
012
6
4
2
0
414
2
2.4
2.2
2.6
2.8
3
R (cm)
Figure 11: Phase for different R.
The magnitude of the phase of the microwave field at ra=0.5 cm
due to different TEmp terms ®mpsr7ras from Tables 2 & 3 as a function of the cavity radius R where the
resonant frequency is 9.192 GHz. These are the leading terms for cavities with 2 or 4 feeds. Here,
cos (4 ^ ) . The cavity height ranges
the leading order for the m=4 p=l term is <X>(r) = <t>414
\C j
from d=l5.4 cm to d=2.2 cm for this range of R. For 2 feeds, the phase variations due to the endcap
losses are smaller than the m=2 p=l term. For 4 feeds, the phase variations due to endcap losses
(d>o12 r2/ ra2) dominate for cavity radii greater than 2.1 cm.
In Figure 11, we show the phase shift at a distance of 0.5 cm from the cavity axis for three p=l
terms as a function of the cavity radius R.
For two feeds separated by
sum of side wall and endcap losses yield a leading term |^ (r)| = <t>212
change dramatically for cavity geometries near R=d/2.
at the midsection, the
c o s (2 ^ ) that does not
For four feeds, the m=0 endcap loss term
is an order a magnitude larger than the m=4 term for R=3cm which was analyzed in [29].
These
dependences suggest that cavities with radii close to R-2.1 cm have more favorable phase
31
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Chapter 2: The analytical solutions fo r a cylindrical cavity
distributions for atomic clocks than cavities with large radii.
Below, we study such cavities and
include the effects of the endcap holes using finite element methods.
To compare to previous results, we note that the two-dimensional finite-element treatment
of [28] is precisely the p=l case with no endcap losses. Without the endcap losses, a constant z
dependence is exactly analogous to a solution where all terms have a cos(kiz) dependence. This
corresponds to a single feed or multiple feeds that are the entire height of the cavity and narrow in <j>.
Power flow from the feeds to the walls clearly causes the phase gradients. Khursheed, Vecchi, and
DeMarchi elegantly showed that, when the 2D fields have no dependence on z, the transverse phase
gradients are directly proportional to the transverse Poynting vector [28], However, in three
dimensions, the different kp dependences imply that the various p terms do not add to the transverse
Poynting vector and phase gradient in the same way. Therefore, to the extent that p>l modes are
important, the relative orientation of the transverse Poynting vector and the transverse phase
gradient of H z (?) can be arbitrary and vary smoothly from 0 to n. This three dimensional behavior
shows that phase gradients and power flow are not so directly connected and therefore suggests the
possibility that power could be delivered to the cavity walls with much smaller or even no phase
gradients o f H z{r) .
One clear, albeit difficult, way to achieve infmitesimally small phase
gradients is to feed a cavity with a very large number of feeds that are distributed throughout the
cavity such that each feed supplies only the power that is absorbed near that feed.
This is
essentially exciting only very high m and p modes in addition to the primary m=0 p=l mode.
Below we show a simpler method to avoid the effects of phase gradients on the atoms by choosing
a cavity geometry with small phase shifts, judiciously exciting p>l modes, and considering the
effects of the phase variations on the atoms.
In the dissertation, we focus on the Cesium fountain clock cavities, whose resonant
frequency is 9.1926GHz. However, our result can be used by the Rubidium clock cavities, too.
32
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Chapter 2: The analytical solutions fo r a cylindrical cavity
We just need to scale the dimensions of the cavity by the ratio of the frequency of Cesium clock
transition and that of Rubidium clocks, coCs/coRb, and everything we derived above is perfectly
applicable to a Rubidium clock. In the following chapters, this generalization is always taken into
account.
33
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Chapter 3
The finite element method
3.1
Motivation
We have shown in Chapter 2 that we can analytically solve for the fields in cavities without
endcap holes. However, a cavity used in fountain clocks must have an entrance and exit for atoms
to enter and leave the cavity. The edges of the entrance and exit holes scatter the electromagnetic
fields, so the fields are not pure TE or TM modes any more, as in a cavity without holes. Therefore,
all p modes for the same m cannot be separated. This does not allow for an analytical solution, and
a numerical calculation must be implemented. We use Finite Element Method (FEM) to solve for
fields in cavities with endcap holes. In this chapter, we will introduce the FEM by beginning with
an overview. Then we discuss the basic concepts of how one derives the FEM solutions. We
describe a novel way to use mixed node- and edge- based elements to avoid spurious modes
resulting from the use of just the node-based elements.
3.2
An overview of Finite Element Method
In scientific studies, a common belief is that the best way to attack a problem is to reduce
its size, or say, to break it into smaller partitions. This is the idea behind FEM. In a typical
application of FEM, a computational domain is broken into small elements. The continuous
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Chapter 3: The finite element method
system is mapped onto elements and is converted into a discrete system. It is then easier to find a
solution to the simplified system. However, one needs to keep in mind that the solution is just an
approximation.
FEM has evolved for nearly 6 decades. The basic idea of FEM was first mentioned in
1940s by Richard Courant in [31], where he discussed piecewise approximations. John Argyris
then studied FEM for aircraft structural analysis [32],
developed for solving structural mechanical problems.
Their pioneering work was initially
Since the development of a rigorous
mathematical foundation by Strang and Fix [33], FEM has been generalized in a wide range of
scientific disciplines, e.g. electromagnetics, heat transfer, fluid dynamics and chemical engineering.
FEM belongs to a class of numerical modeling which involves partial differential equations (PDE).
Generally speaking, FEM can be applied to any PDE related systems. Its applications are still
expanding.
The major advantage of FEM is its ability of treating complex geometries, especially
through the use of triangular (for two-dimensional problems (2D)) and tetrahedral (for
three-dimensional (3D)) elements.
Moreover, the application of FEM leads to sparse matrix
systems, which is superior to a fully populated system because the former requires much less
memory and computer time than the latter.
The geometrical adaptability and low computer
requirements of FEM have made it one of the most popular numerical methods in all branches of
engineering. Its application to boundary value problems [34,35] renders it particularly useful for
our analysis of microwave cavities in atomic clocks and we focus on 2D problems throughout this
dissertation.
The subdivision o f the domain into small elements is referred to as meshing of the geometry
and is an important part of the FEM solution procedure. For 2D problems, these elements are
typically triangles. A general rule of the size of elements is that the smaller the elements are, the
35
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Chapter 3: The finite element method
more accurate the solution will be. In electromagnetics modeling, by keeping the elements small
enough (typically less than 1/10 of a wavelength per side), the field inside the element can be safely
approximated by some linear or if necessary, higher order expansion. While mapping the fields
onto the mesh, one gets unknown coefficients which may represent the field values at the nodes (for
linear node-based elements) or the average field values over the edges (for linear edge-based
elements).
These coefficients form algebraic equations based on the governing PDEs and
boundary conditions of the original system. Thus the solution of complex PDEs is transformed
into a matrix algebra problem, which can be programmed into standardized computer codes.
FEM also has a weakness, the lack of efficiency of treating open boundary problems. The
most common method to circumvent this is the truncation of outer boundaries.
However,
truncations bring errors into the solution. When a truncation is sufficiently far away from the
domain the error is small, because the field excitation becomes small on the truncated boundaiy so
that its error is suppressed. On the other hand, when the truncation is too far, too much computer
resources need to be assigned to the extra space, which may downgrade the solution if the computer
limitation is reached. Therefore it is always natural to ask “how far should the truncation be from
the domain of interest?” A quick answer is that within the limits of the computational power
available, the truncation should be as far as possible. Fortunately, in the cavities used in atomic
fountain clocks, the openings are at the far ends of the below cutoff waveguides. In the FEM
model, we can use tmncations many cutoff wavelengths away from the cavity, so that the fields
have decayed sufficiently when they reach the truncated boundary. Consequently, any resultant
errors from the truncation would have negligible effect on the solution.
3.3
Two equivalent approaches
The Rayleigh-Ritz and Galerkin’s methods are two standard approaches for solving PDEs
arising in practical engineering and physical problems. While the Rayleigh-Ritz method is useful
36
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Chapter 3: The finite element method
in FEM theory, Galerkin’s method is simpler and is used more in practical situations.
Both
methods project a continuous space onto a finite separable Hilbert space. As mentioned in the
previous section, a PDE related problem is then rephrased to seek a discrete solution set whose
unknowns are the coefficients of the expansion. The Rayleigh-Ritz method seeks a stationary
point of a variational functional.
In general, for operators L, which are self-ajoint and
positive-definite, the stationary point of the following functional F ( u ) =
is a
solution to the equation Lu-fi=0, where («,&) = Ji^a-bdQ. defines the inner product of functions a
and b on the domain Q. The discretization starts with the trial function, u , expanded in terms o f N
basis functions associated with the meshing of the computational domain
u = H u.iwJ = {u )T {w )
H
where w, are the basis functions and Uj are the unknown expansion coefficients. The functional
becomes
F {u ) = ^ { u }T \^\ci{ w } L { w }T dQ. { u } - { u } T ^ { w } f d Q .
(3.2)
This functional is extremized by requiring all partial derivatives with respect to the coefficients,
{«}, to vanish
=K
JQ] M +^ Mr[ Ll w}Lw/ Q]
A single equation is obtained by differentiation with respect to each
-
For i= l,2,
=0 (3-3)
, N we obtain
N equations which can be written as a matrix system
[M ] {«} = {£}
The matrices [M] and {B} are given by
37
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(3.4)
Chapter 3: The finite element method
M
ij =
J[ w
.L w jd C l
(3.5)
Bi = ia wi f d n
Galerkin’s method is one of the methods of weighted residuals [36], which begins with the
residual 1Z=C u - f for a testing function u and seeks a solution forw =u by satisfying the condition
71=0 within the domain. In general, it is possible to find a solution which satisfies the residual
condition only in the weighted sense over the domain Q. A set of weighting functions t, are
introduced for each trial function u: and the residual 1Z is rewritten as
(3.6)
In general, any testing function t, can be used. One popular choice is called Galerkin’s
method. When applying the Galerkin’s method, the testing function is identical to the expansion
function used forw , e.g., t,=vv, and the weighted residual equation is given by
(3.7)
This is identical to Eq. (3.3) derived from the Rayleigh-Ritz method.
Therefore, Galerkin’s
method leads to the same linear system Eq. (3.4) as the Rayleigh-Ritz method.
In practical
situations, one can use either of the two methods without caring about their theoretical origins.
The matrix [M] in Eq. (3.4) is a square of size NxN, very sparse and typically symmetric
unless nonlinear material exists in the computational domain.
Its sparsity is a result of the
orthogonality of the separable Hilbert space, which allows interactions only between adjacent
elements because they share an edge. The nonzero entries provide the relationship among field of
adjacent elements within the computational domain, which is given by the governing PDEs o f a
physical system.
{B} is a column matrix of size N. The entries of {B} store the information of
boundary conditions and sources of excitations. The steps involved in the generation and solution
of an FEM system can be summarized as follows:
(1) Define the problem’s computational domain
38
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Chapter 3: The finite element method
(2) Choose type o f discrete elements and shape functions
(3) Generate mesh
(4) Enforce the partial differential equation over elements to generate element matrices
(5) Apply boundary conditions and assemble element matrices to form global sparse
matrices
(6) Choose solver and solve the matrix system
(7) Post process field data to extract solutions of interest
In this dissertation, we use the commercialized FEM package Comsol Multiphysics.
This powerful tool can do steps (3,5) automatically, one only needs to incorporate the physics into
the model, for example, draw the geometry (step (1)), select the type of elements from the list
(step(2)), find and include the appropriate boundary conditions (step(4)) and then solve the
problem
by
selecting
afterwards(step(6,7)).
preprogrammed
solvers,
and
finally
deal
with
the
solution
In the following sections, we will focus on the choice of elements,
transferring from PDE to matrix equations and boundary conditions.
3.4
Two popular elements
As addressed in section 3.3, a continuous function is mapped onto a discrete space when
FEM is used.
The mapping starts with breaking up the computational domain into elements of
simple shapes.
A set of suitable interpolation polynomials (commonly referred to as shape or
basis functions) are used to approximate the unknown functions within each element.
Once the
shape functions are chosen, it is possible to program the computer to solve for fields in
complicated geometries by solely specifying the shape functions.
interest is the vacuum inside a microwave cavity.
In our case, the domain of
We more specifically define the
39
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Chapter 3: The finite element method
computational domain as the cross section o f microwave cavities in the (r, z) plane using
cylindrical coordinates.
We use two-dimensional triangular elements throughout the dissertation
because they are simple and can form any complex geometry in any shape.
There are many kinds of elements, for example, the nodal elements, the edge elements,
the bubble elements, the discontinuous elements, the density elements, the divergence elements,
and so on.
choices.
In Electromagnetics the node-based and edge-based elements are two popular
We discuss their shape functions and usage in our model.
3.4.1 Node-based triangular elements
Indicated by the name, the node-based elements are triangles in which the solution is
determined by the values at the three vertices or nodes on the edges (for high order shape functions)
of the specific triangles. Shape functions of node-based elements are derived by using Lagrange
interpolation polynomials. For the linear interpolation, the unknowns are the field values at the
vertices of the triangle. For the higher order interpolation, unknowns will include the values at the
middle nodes of each edge for quadratic elements or more nodes for even higher order elements.
For simplicity without losing generality, we only discuss the linear shape functions.
In their final expression, the shape functions will be expressed in terms of the so-called
area coordinates L*, which is the local coordinates of a point in a specific element e. i= l, 2, 3
denotes the three vertices of e and the order of labeling the vertices is fixed for all triangles, which
means either clockwise or counterclockwise, but not both. Let’s consider a point P within a
triangle located at (r, z), as shown in Figure 12 where {rfz*' } is the global coordinates of the ith
triangle node. The area of the smaller triangle formed by points P, 2, and 3 is given by
40
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Chapter 3: The finite element method
P (r,z)
Figure 12: A triangular element for the node-based finite element.
Each node (solid circle) has a
value of the unknown field component(s). P is a point inside the triangle at (p, z), which together
with the three vertices divides the triangle into three smaller triangles. The field value at P can be
expressed as the summation of the values at the vertices weighted by the fraction of the areas of the
corresponding small triangle to the total area of the big triangle.
1
r
A, = - 1
e
e
r2
Z2
e
e
r3
Z3
1
z
(3.8)
The area coordinates If, are then given by
^ _ A, _ AreaP23
1
A
(3.9)
A rea\23
Similarly, If2 and If, are
e _ A2 _ A reaP3\
2
A
^ _ A3
3
Area\23
A reaP\2
A
(3.10a)
(3.10b)
A rea\23
The coordinates of P can then be expressed in terms of L* as
(3.11)
1=1
/=1
The suitable basis functions are just f , it is unity at node i and zero for all remaining nodes within
41
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Chapter 3: The finite element method
the element. The function ue(r,z) can be expressed as
u e ( r , z ) = ' ^ u eiLei { r , z )
(3.12)
1=1
where u* is the value of u at the ith node of element e. Discussion of higher order basis functions
is out of the scope of this dissertation. Interested readers can find additional details in [37-38],
3.4.2 Edge-based triangular elements
Edge basis functions were described by Whitney [39] over 50 years ago. Nedelec [40]
theoretically developed the foundation of using finite elements in the curl space with degrees of
freedom associated with the edges, faces, and elements of a finite element mesh. Since then, twoand three-dimensional shapes, and higher order elements had been constructed [41 - 44].
In electromagnetics, we encounter serious problems when only node-based elements are
employed to represent vector electric or magnetic fields. Nodal basis functions impose continuity
in all three spatial components, which contradicts the discontinuity of field components
perpendicular to interfaces of different materials or to surfaces of perfectly conducting comers.
Special boundary conditions have to be applied in order to satisfy the divergence equation and
spurious modes are observed when modeling with only nodal elements [45].
The special
boundary conditions are difficult to derive and to implement, and sometimes are case dependent.
However, edge-based finite elements, whose degrees of freedom are associated with the edges,
have been shown to be free of the above shortcomings. Edge basis functions require continuity
only along the components tangential to element edges. The feature mimics the behavior of field
components along discontinuous material boundaries and near perfectly conducting comers.
Hence it automatically satisfies the divergence equation whose satisfaction is a key ingredient for
eliminating the spurious modes.
We consider again the triangular element depicted in Figure 13 or the edge-based element
42
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Chapter 3: The finite element method
(r,e, Z!e)
P (r,z)
Figure 13: A triangular element for the edge-based finite element. Each arrow represents a single
vector shape function on each edge.
which is similar to that for the node-based ones. The only difference is that the unknowns are the
field components along the triangle edges. Following Whitney’s [39] procedure, the basis functions
are defined as
W\ = ly ( L y L ) - L ^ L ] ),
i , j = 1,2,3
(3.13)
e
where Wk denotes the basis function for the kth edge formed by nodes i and j of the eth element,
L j is given in Eqs. (3.9) and (3.10) and ly is the length of the edge. The vector field H
'e
inside
the element can, therefore, be expanded as
H e = Y ^ H ke Wl
(3.14)
k =1
where
H kedenotesthe tangential magnetic field along the kth edge.
It can beeasily shown that
the edge-basedfunctions definedabove have the following properties with theelement
V-Wl=0
—
e
(3.15)
Thus V •H = 0 is automatically satisfied. Quite conveniently, all the spurious modes with
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Chapter 3: The finite element method
node-based elements are eliminated by using the edge elements.
3.4.3 Mixed axisymmetric elements
Many microwave devices exhibit a rotational symmetry akin to the interrogation cavity we
used in atomic fountain clocks. Exploiting this rotational symmetry can dramatically save on
computer memory and time. The magnetic field can be expanded in its Fourier series as we do in
Chapter 2:
H (r,$,z) +ig(r,#,z)
= H o A r ’z ) z + H o A r ’z ) r
(3.16)
+ i [ g 0 A r ’z ) z + g o A r ’z ) r
+
{ [g m,r ( r , z ) r + g mz ( r , z ) z c o s( m f ) +
( r , z ) s i n ( m </>)^j
where H is the primary standing wave which corresponds to the field in a perfect conductor cavity,
g is the secondary standing wave induced by the wall losses. It has been proposed that the edge
and the nodal elements are, respectively, employed for the meridian and azimuthal field
components [46-47] to eliminate the spurious modes caused by the poor satisfaction of the
divergence equation if only nodal elements are used. For m > 1, however, as will be shown below,
the curl operator is not correctly modeled. The curl of the three field components can be explicitly
written as
(3.17)
This leads to a 1/r singularity when discretizing Maxwell’s double-curl equation. Analytically, we
know that the field behaves as rm near the z axis. If the field is forced to behave as rm, the
44
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Chapter 3: The finite element method
singularity in the differential equation ought be eliminated. M. F. Wong et. al. [47] introduced a
variable transformation for the case m = l:
gc>m,r ={\ r G m ,r - G
m,<p J
S
(3.18)
£
c>m,z
= rG m ,z
Here edge elements are used for Gp and Gz, and nodal elements for G^. This transformation forces
gm>r and gm,z be proportional to r, which explicitly eliminates the singularity at r=0 and we do not
need to apply any boundary condition along the z axis. We further propose the transformation for
m>l as follows:
g „ , = ('■’ <?„,
G „ ,t )
g .s = '" ~ 'G .s
£— —Vm/~i
(j
0
/ 7 7 ,2
(3.19)
m ,z
While there is another proposal for the transformation as in [46], ours gives a better solution near
the z axis because the field is forced to exactly obey the power law in the radial direction by our
transformation, while the method in [46] only forces the field to be zero at r=0. In the method in
[46], higher order power law behavior in the field is governed by the double-curl equation and then
solved by using FEM so that it is still an approximation.
3.5
Discretization of the double curl equation
In the previous section, we have already chosen the right shape functions for different field
components. We then need to “map” Maxwell’s equations, etc., the double-curl equation
V x ( V x g J - £ 2g m = 0
(3.20)
and the boundary condition which has the form
f\\m = - j ^ V x S m = Z mBC
(3-21)
on to the mesh elements and then solve for the generated matrix equation. Following Galerkin’s
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Chapter 3: The finite element method
approach, we need to solve the equivalent integral equation
j L { v x ( V x 5 j - * !i „ ] r f r = o
V
(3.22)
where V is the domain in which we solve for the microwave fields.
Substituting
g ' m- ( V x V x g m) = ( V x g m) - ( v x g ' m) - V - ( g ' mx V x g m) into
Eq.
(3.22), we get
J ( v * g ' m) { V x g m) - k2g ' m - g m d v + j g ' m - [ n x ( V x 8 n,)_ d t7 = 0
(3.23)
Here dV denotes the volume integral and do the surface one, n is the unit vector perpendicular to
the surface, pointing from the vacuum to the metal, and we have used Gauss’s
theorem j^ V • V^jdV =
• V~^dcr = 0
and the vector equation a - ( i x c ^ = - b - ( a x c ^ .
Substituting the boundary condition into it, we get the weak formula of the double curl equation
d V = § r g \ - ' \ n x ( s c o J mBC)^d<J
(3.24)
Due to the <|) dependence of g , the formula can be further simplified by explicitly performing the
integration over <)>and taking out the common factors on the two side of the equation
J x [ ( v x g ' . ) - ( V x g . ) . k g m-gn rdrdz =
rdrdz (3.25)
r g ' m•
By substituting Eq. (3.19) into Eq. (3.26), the integral becomes
i{ [( mGLs + dGl , J dz) ( mGm,. + dGm, J dz)
dG„
G ml ,z ,
'd r
7 i - dG,my a z - m G " ' ' - r
8
4 ™
^ : , * dGV dX mG^ 8G"Ydr
dr
a
- * 2[ ( * W ~G'm4 )(rG m, - G m4 y G ' m4 Gm 4 + r 2 G'm^GmJ
=
t
{ { rG L,r ~ Gl j ) '' [ n
x
( £0}f m,BC)
] r +
( GL,t ) •
[ « x
}f^drdz
)_
+ ( rGl,z ) ■[« x ( e t o f m,Bc ) \ ) rmdY
46
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Chapter 3: The finite element method
(3.26)
We then define the degrees of freedom of Gmr and Gmz on the edges of each triangle (using
the edge elements), and assign Gm,« to the vertices of the triangles (using the nodal element).
Because the basis functions that are used to approximate the desired solutions are themselves
divergence free, these definitions eliminate the spurious modes whose divergence is nonzero when
only using nodal elements. To finalize the discretization, we need to express the field vector
(^m r’Gm0 >Gmz) in the form of the superposition of the shape function weighted by the
unknowns along the edges (for Gmr, and Gm,z) or at the vertices (for Gm,<^), as in Eqs. (3.12) and
(3.14), then explicitly implement the integral by utilizing the orthogonality of the shape functions
between different elements.
To do this, we use the commercialized FEM software Comsol
Multiphysics. We write Eq. (3.26) into the software and the integral over all the elements is done
automatically.
3.6
The boundary conditions
We use four different boundary conditions for the FEM model.
First, the lossy wall
condition Eq. (3.21) for the power dissipated by the metal; second, the symmetry condition along
the r and z axes; third, the Perfect Matched Layer (PML) condition for the cut-off waveguide
openings while noting equivalent boundary conditions for the openings; and fourth, the field value
at the aperture which connects the small feed waveguide and the cavity. The different boundary
conditions are shown in Figure 14. The lossy wall condition is applied at all the boundaries
comprised of metal walls. The symmetry boundaries are along the r and z axes. The PML is used
at the top of the waveguide aperture. Finally, the field value is set up at the feeding aperture to
account for the power flow into the cavity.
47
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Chapter 3: The finite element method
PML
Lossy walls
symmetry
Figure 14: The boundary conditions used for a microwave cavity with cut off waveguide.
The solid
lines are the interface between the metal and the vacuum. The lossy wall boundary is used there.
The dashed lines stand for the symmetry boundaries. The dotted line is the waveguide opening
where the perfectly matched layer (PML) is used to mimic the absorption. The field at the feed
apperture is specified as the feed boundary condition.
The lossy wall condition has already been included in the integral equations as discussed in
section 3.3.
We focus here on the symmetry, PML conditions and the feed condition.
The
symmetry boundary conditions are used to reduce the size of the total computational domain to
save on computer memory and runtime. Generally, the application of one symmetiy condition can
save one half of the total computer resource. As addressed in section 3.3.3, one does not need to
do anything to set up the axisymmetric condition along the z axis once the specialized mixed nodeand edge- based elements are used. There is an additional symmetry in our model - the symmetry
of the solution about the mid-plane, which is, in the two-dimensional problem, the symmetry about
the r axis.
This symmetry is caused by feeding the power into the cavity through feeds at
midsection on the side walls. The symmetry states that
S m ,r { Z ) =
~ g m ,r ( ~ z )
g m, A z ) = - s m, A ~ z )
g m, A z ) = g m,z{~z )
48
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<3-27)
Chapter 3: The finite element method
Thus g m,,W ( Z = 0 ) = 0 is the boundary condition to be used. The condition for gmz is
automatically satisfied because the divergence equation V •g = 0 has already been enforced by
the choice o f elements. In Figure 14, it is shown that by using the symmetry conditions, the
computational domain is reduced from the full cross section of the cavity to a quarter, which brings
a four fold computer memory and time saving.
A perfectly matched layer (PML) is used for any openings which are unbounded by the
metal.
PML is an absorbing boundary layer for linear wave equations that absorbs almost
perfectly propagating waves of all non-tangential angles of incidence and of all non-zero
frequencies. The concept of a PML was introduced in the context of electromagnetic waves by
Berenger [48], and an axisymmetric FEM formulation of a PML was presented by [49-50], It
has also been suggested that if the frequency is below cutoff of the waveguide, an almost complete
reflection occurs [51]. We calculate the field in a long cutoff waveguide (at least 10 cutoff
wavelengths long for any modes near the resonance) at the top of the cavity by using both the PML
and PEC (perfect electric condition) boundary conditions at the waveguide opening. We found no
significant difference between the two solutions in all domains which is 1mm below the opening.
For simplicity, using PEC on the top of a long cut-off waveguide is acceptable.
The value of the field at the feed aperture is calculated by power balance, which means that
the same amount of power dissipated by the metal wall due to finite conductivity must be fed into
the cavity through the aperture. We have verified that the field far away from the feed (for
instance, at the center of the cavity) is insensitive to the variation of the field across the aperture by
using different field configurations and seeing negligible difference. For m=0, we can adjust the
field at the feed by an argument that the secondary fields f ( g ) must be orthogonal to the primary
fields E ( H ) [28]. We run the FEM calculation by trying different field values at the feed.
49
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Chapter 3: The fin ite element method
Once the solution of the secondary field is obtained, we calculate the inner product of / and E .
We then change the field value at the feed so that the next solution o f/ is progressively more
orthogonal to E . We keep doing this until the portion of E in / is vanishingly small. There
may be a phase difference between the fields in the small feed waveguide and in the cavity.
However, this phase difference is not essential to our model since it simply adds a constant phase to
our solution, and a constant phase does not introduce any frequency shift in atomic fountain clocks.
Following the above discussion, we use a constant f i with a zero phase as the boundary condition at
the feed aperture to approximate the average power fed through the aperture.
50
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Chapter 4
The finite element model for a cavity with endcap holes
4.1
Introduction
Finite element calculations can flexibly handle complicated cavity shapes. In this chapter,
we consider cavities that are a series of cylinders with holes in the endcaps and we study the effects
of the endcaps. While these can be calculated with mode expansions [52], the finite element
method, in addition to its flexibility, is more efficient because its matrices are sparse [53-55].
Three dimensional finite element calculations require far more resources than 2D calculations.
The 3D computing time scales as T3D= T2d3/2 [28]; for current personal computers, this corresponds
to hours vs. seconds for the same geometry in 3D rather than 2D. From the arguments in the
previous chapter, it is clear that, for multiply fed cavities, only two 2D terms in m are significant
and, for a single feed, only four 2D terms in m are needed for an accurate calculation o f the phase
distribution. In this chapter, we first describe our finite element calculations of three dimensional
phase distributions using a series of two dimensional problems. We then perform a number of
finite element calculations to analyze the effects of the holes in the cavity endcaps.
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Chapter 4: The finite element model fo r a cavity with endcap holes
4.2
Two-dimensional finite element calculations
Davies et al. [53-55] and others have shown that the finite element method can be applied
to electromagnetic problems that have a specific azimuthally symmetry such as cos(m<j)). As in
our analytic treatment above, we can decompose the boundary conditions for the losses and feeds in
the cavity into a Fourier series in cos(m<j)). This leaves a series of 2D problems in r and z to solve,
which is far more efficient. Again, we decompose the fields into a primary standing wave and a
small standing wave that accounts for the wall losses as in Eq. (2.2).
We first solve for the primary fields E 0 (r) and H 0( r ) in a cavity that is composed of a
series o f arbitraiy cylinders. For the TE0u mode, since m=0, E 0 (r) has only a <j> component.
Therefore the vector wave equation in cylindrical coordinates is:
'l d
1
8
co
(r — )
7H r H :
r dr < dr V r
8z
c‘
where E 0 ^(r) = 0
EoA r , z ) = 0
on the boundaries since it is parallel to every boundary.
(4.1)
This is
straightforward to implement in a finite element calculation. We then calculate the magnetic field
by f f 0( r ,z ) = V x £ w ( r ,z V / f t f f l .
The calculation for g zm{r,z) with m=0 follows in the same way from Eq. (2.11). The
boundary condition is:
f i ( r ) $ = - R sn x H 0 (r)
(4.2)
on the cavity walls where h is the normal of the metallic surface and H 0 (F) is from the above
finite element calculation. To this boundary condition for the losses, we must also add the value of
/,( F ) at the cavity feed(s), which has <|) dependence. Integrating
52
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Chapter 4: The finite element model fo r a cavity with endcap holes
2K
f *„(?)=
J ffir)cos(m<f>)d<t>
(4.3)
o
gives the total boundary condition, for all m, for a cavity constructed from an arbitrary number of
cylindrical sections. For m=0, the value of
(r) at the feed must be adjusted carefully so that the
net power into the cavity is zero. This eliminates an excessive amount o f the primary field
E{) f i r )
in the solution for f f i r ) since an arbitrary amount o f Eaf i r ) satisfies the same wave
equation with Dirichlet boundary conditions. As in Chapter 2, any residual amount of E 0 ^ ( r) is
removed so that finally
0(r, z ) andE 0 ^( r ) are orthogonal.
We then use the Maxwell equation
to calculate the magnetic fieldg 0 ( r , z ) = V x f i f i r , z)(j) jf JQa>.
The calculation for g m(r) for m>0 is more involved because both g m>0 (r) and
f m>0 (r) have all three components in radial, azimuthal and longitudinal directions with the
presence of the edge of the endcap holes [56]. Therefore the solutions must simultaneously satisfy
the wave equations and V • g m(r ) - 0 . In a different way, we solve the vector wave equation for
g m(r,z)cos(m</>).
0 )2
v x v x [_Sm {r, z ) c o s (w ^ )] - —r [ g m( r , z ) c o s ( w ^ )] = 0
(4.4)
The picture of the azimuthal expansion for our finite element method is slightly different
from that for our analytic solution in Chapter 2. Here, in the 2D finite element calculations, we
solve for all p modes simultaneously. For example, the effective feed for m=0 can be viewed as a
ring (a height Az=2mm) that feeds power into the cavity symmetrically around the circumference at
z=0. Then, the m>0 modes redistribute the feed around the narrow z=0 feed region to construct 1
or more feeds that are narrow in (j). Thus, if the conductivity of the sidewalls is independent o f <(),
53
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Chapter 4: The finite element model fo r a cavity with endcap holes
f<l> m>o( 0 ^ 0 only f°r those values of z where power is fed.
Therefore, the m>0 boundary
conditions only describe the redistribution of power in the midplane. Note that this does not imply
that g z m>o(r) = 0 at all other z; only that the boundary conditions Eqs. (3.3) are zero. Eqs. (4.3)
give a boundary condition f ^
m>0 (?)
= 0 for all z except the feed aperture where the power is
supplied. S in c e /( ? ) = ---- — ^ V xj^gm(r,z )c o s(/w ^ )J , V x ^ ( r ,z ) c o s ( /n ^ ) J = 0 o n
£Q0) m-0
*,z
the cavity sidewalls. On the cavity endcaps, f r m(?) must be 0 to first order in 8 since there is no
^ component of H 0 (r) to drive this current.
Therefore V x |^g m( r ,z ) c o s ( m ^ ) J ^ = 0 is
satisfied on the cavity endcaps.
We
calculate
the
three
dimensional
phase
distribution
in
the
cavity
as
y , —g 2 m(?) I H 0 z(?) as in Chapter 2. Again, since g, m(?) is a solution to the wave equation,
m= 0
g z m(?)
/
r mfor small r and therefore the Fourier series in m converges rapidly.
We use a
triangular lattice with as many as 50,000 triangles. Each calculation for any m takes less than 4
minutes on a 1.5 GHz personal computer with 1 GB of RAM. We have written our own finite
element codes and, more recently, we use commercially available finite element software that has
the flexibility to specify the boundary conditions for / ^ 0(?) or V x g m>0( ? ) '. In the following,
we show solutions for cavities with endcap holes and solutions with extremely dense meshing that
enables us to examine the behavior o f the fields on the scale of the skin depth. The result with m>0
will be published in [57], In the next chapter, we use these methods to optimize cavity geometries.
1 We have used FEMLab from COMSOL, Inc., Burlington, MA for the calculations presented here.
54
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Chapter 4: The finite element model fo r a cavity with endcap holes
4.3
The effects of endcap holes
The cavities for atomic fountain clocks must have holes in the cavity endcaps if the atoms
are to pass through the cavity. From Eq. (2.11), we see that, when there are holes in the endcaps,
the phase of the field does not go to rc/4 as it does near the surface of the endcap. Therefore,
theholes seem to improve the homogeneity of the phase distribution for 2 reasons; 1) g z 0( r ) can
be smaller over the aperture and 2) H 0 z (r) does not go to zero at z=±d/2, as it does on the metal
boundary.
Instead, H 0 z(r) extends beyond the aperture of the endcap hole and into the
below-cutoff waveguide sections that prevent microwave leakage from these cavities. However,
while the phase shifts are no longer so large, we show that they do have a transverse variation that is
large compared to the phase variation near z=0.
In Figure 15 we show the magnitude o f H 0 z(r) in the cavity and, in the inset,
H 0 z (r) near the wall of the endcap hole. While H 0 z ( r ) does not go to zero for all z=±d/2 as it
does in a cavity with no holes, a large fraction of the atomic trajectories (nominally parallel to z)
with r<ra cross the nodes o f H 0 z (r) (dark blue region in Figure 15 inset) that is due to the hole in
the endcap [58], This occurs because the lowest mode excited in the below-cutoff wave guide
section is the TE0i. Therefore H 0 z (r) in the below cutoff section must be reversed near the walls
relative to the center of the waveguide, and hence reversed relative to H 0 z (r ) at the center of the
cavity. Near the nodes H 0 z (r) , the phase of the field goes from 0 to - n with a large phase chirp
where the phase shift is -n/4 at the node of H 0 z (r) . Here we analyze the fields near the walls of
the endcap holes. In the next chapter, we show a design that has much smaller phase shifts in this
region. A large current is induced in the wall of the endcap hole by the magnetic flux through the
hole. Near the wall, the field is very large with a - n phase shift. In fact, for cavity geometries
55
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Chapter 4: The finite element model fo r a cavity with endcap holes
Contours of |H0 z|
Figure 15: Contours of |Ho>z|.
Contours of the amplitude of H 0 .(r) for the TEon mode of a
cylindrical cavity with R=3cm, d=2.176 cm, and ra= 0.5cm.
lossy walls.
The solid boundary lines represent the
The inset is a 0.4 cm square region centered at (ra, <|), -d/2).
inset contains the node of h oz(r) •
The shaded region in the
H(l _(?) has local maxima at the center of the cavity and at r = ra
on the endcaps (see inset).
56
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Chapter 4: The finite element model fo r a cavity with endcap holes
Figure 16: The comer at the edge of the endcap aperture. We solve the electric field near the comer in
the polar coordinates (p, 0), then calculate the magnetic field in the cavity coordinates (r, <(>, z).
near R~d/2, H 0 z ( r ) is slightly greater at the comer than at the center of the cavity. This large
parallel component o f H ( r ) near the surface implies a large loss in the conductor and therefore
large phase shifts nearby. It’s worth noting that, if the cavity is perfectly machined and vertical,
1% of the atoms pass within 12.5 microns of the walls of a 1 cm diameter aperture on the two passes
through the cavity. The natural fountain velocity reversal and misalignments will eliminate the
bulk of the effects. Nonetheless, since these atoms experience 100 times larger phase shifts than
atoms near the center, further study of these effects is clearly motivated.
Within a few hundred microns of the comer, the fields are magneto static and azimuthally
symmetric (m=0).
2/ 2
The wave equations (2.1) become Laplace’s equation as co / c can be
neglected. Because the radius of the holes in the endcaps is much greater than the skin depth, the
solution near the comer is a 2D problem. We transform our three dimensional problem in
cylindrical coordinates (r, (j), z) to a 2D problem with cylindrical coordinates such as
P =
and ta n (0 ) = (z + < / / 2 ) / ( r - r a ) as in Figure 16, where z and r
are measured from the comer on the endcap at z = -d/2. The solution to V2is0^ (/? ,# ) = 0
2/
4/
is £ 0(1>(/5) = a ( p / S y 2 s i n ( j # ) + Z>(/}/e>)/3 s i n ( j # ) + . . . , satisfying the boundary conditions
57
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Chapter 4: The finite element model fo r a cavity with endcap holes
20pm
PML
metal
to
o
T=
3
vacuum
4 triangles/8
Figure 17: The mesh grid used in the m=0 calculations. We first utilize the symmetry to save computer
memory and time, and use perfectly matched layer at the top of the waveguide. We divide the cavity
into 5 overlapping regions with increasingly higher mesh density. In the smallest region, the mesh
density is about 4 elements/skin-depth, and we solve the fields in the vacuum and in the metal altogether,
which gets rid of the divergence of H, f and g.
E o A P ) = 0 on the conducting walls at# = 0 and 3 ^ / .
Here it is convenient to use the skin
depth to make E 0 ^ ^ p ) dimensionless. For cavities with R~d/2, the antisymmetric coefficient b
is about 1% of the symmetric coefficient a. From H 0 (r, z) = V x £ 0<1(r, z)(j) / p Gco, we get:
j r c o s ( j# ) + z s in ( j# ),0 , - r s i n ( j # ) + z c o s(y # )j
-+
_2
d 2/3 ( r 2 + z 2f
'
(4 .5 )
3/V9
{r cos (y #) + z sin (j # ) , 0, - r sin ( j #) + z cos (y #)}
24
gifTTTf
This H 0 (r^ diverges near the comer a s p~K
^ , just as electrostatic fields diverge with the same
power laws near sharp points (e.g. lightning rods). This result is remarkable, because it’s been
argued that the field near the endcap holes is small [18].
The real H ( r ) cannot diverge at the com er nor will a finite element calculation
of H 0 ( r ). On the scale of the skin depth 8, the field must be smooth. We calculate the field near
the cavity aperture using a novel finite element method. As in Figure 17, the solution domain is
58
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Chapter 4: The finite element model fo r a cavity with endcap holes
125
Figure 18: Contours of Log {^Ev (F)|j near the wall of a centered endcap hole for the TE0| i mode. The
field is smooth near the comer and falls exponentially inside the conductor. Each contour step between
dashed and dotted lines represents e ^2. In this region the solution depends very weakly on the cavity
geometry. Here, R=2.55 cm, ra=0.5 cm, and d=2.6 cm.
divided into 5 overlapping regions. The outer boundary condition is generated by the larger region.
In the smallest region, we can have a mesh density as high as 4 elements per skin depth. The finite
element grid includes the conductor with a mesh spacing that is a fraction of the skin depth and we
simultaneously solve the coupled equations for the real and imaginary parts of the total field E ( r ) .
We then iterate the sequence so that the solutions and the derivatives of the solutions are continuous
across the boundary. The wave equations which couple the real and imaginary parts are:
{E ^ ( r ) + f i f i r j ) = 0
J
2
\
(4.6)
V2( £ „ ( ? ) + / „ ( ? ) = o
v7„(F)+^(£M(r>+/M(?))=o
59
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Chapter 4: The finite element model fo r a cavity with endcap holes
Here the first two equations describe the vacuum region and the last two describe the decay of the
fields in the metal.
We show the magnitude of E ( r ) near the comer of the cavity apertures in Figure 18
Many skin depths away from the comer, the fields decay as
distance normal to the interface.
in the metal where x is the
Near a comer, the parallel component of H ( r ) begins to
approach p~x^ until a distance of nearly 1mm from the comer, where H n(r) begins to follow the
dominant TE0i waveguide mode.
At distances less than a skin depth from the comer, the fields
are smooth and the gradient of the field points nearly along the bisector of the two infinite half
planes forming the 2D comer. Neither E ( r ) nor H ( r ) diverges at the comer. In fact, H f r ) is
20% smaller at the comer than one skin depth away from the comer on the metal surface because
the gradients normal to the surface in Figure 18 are also slightly smaller at the comer. Therefore
the local maxima of H f r ) are about one skin depth from the comer.
In Figure 19-Figure 21 we show the phase of H z(r) near the comer of the endcap hole.
Near the comer, the phase shift is large. An atom passing within one skin depth o f the comer sees
the phase of the field go from - n to - n +0.14 and then back to -tc-0.015 within a few skin depths of
the comer (Figure 20). Farther away, at 108 and 1008 from the comer, the phase variation is
smaller (Figure 20-Figure 21).
In Figure 21, one can also see the large phase variations
experienced by all such trajectories that pass through the node of H 0, i f ) just below z=d/2.
The calculations in Figure 19-Figure 21 we use a mesh density as high as 30 triangles per
skin depth. Using this mesh density for the entire cavity is inefficient. To solve the problem, we
divide the cavity into 5 overlapping sub-regions with increasingly higher mesh density. The
60
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Chapter 4: The finite element model fo r a cavity with endcap holes
205
\
205
Figure 19: Contours of the phase of H z( r ) near the wall of a centered endcap hole for the TEon
mode and the cavity of Figure 18.
the phase variations are very rapid.
The contour steps are 10 mrad.
Within a skin depth of the comer,
The phase of H z(r) is -n/4 on the bottom surface of the top
endcap.
61
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Chapter 4: The finite element model fo r a cavity with endcap holes
20Q
(z-d/2)/5
108
-10
Figure 20: The phase of Hz for different trajectories. The phase of Hz(r,z) of Figure 19 for a vertical
trajectory parallel to the z axis and a distance of 1 and 10 8 from the wall of an endcap hole (solid lines).
The phase shifts are large near the comer at z=d/2. The dashed lines represent the phase calculated
using Eq. (2.9) as a boundary condition and a mesh density of four triangles per skin depth.
O (mrad)
(z-d/2)/5
-1000
500
-500
Figure 21: The phase of Hz along a trajectory 1008 from the wall. The phases of H z (r) in Figure 19
for a vertical trajectory parallel to the z axis and a distance of 1008 from the wall of an endcap hole
(solid). The phase shifts are large near the comer at z=d/2. Near 7508 (0.5mm) from the comer, there
is a node of H 0 z (r) and its polarity reverses. The dashed line represents the phase calculated using
Eq. (3.2) and a mesh density of four triangles per skin depth 8. At distances much greater than 8 from
the surfaces, the two solutions converge.
62
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Chapter 4: The finite element model fo r a cavity with endcap holes
0.027
w
0.025
1
0.5
1
z-d/2 (mm)
Figure 22: The effective surface resistance. The Rseff (solid) and the Rs (dashed) defined in Eq. (4.2)
are plotted as a function of the height above the endcap for comparison. The Rseff is slightly larger
than the Rs, especially at the edge, and then the difference becomes smaller when it’s away from the
edge. 1.5mm above the edge, Rseff and Rs are nearly equal.
solutions are iterated so that the solution and its derivative are continuous across the boundaries.
The solution is well known for a plane wave incident on an infinite flat surface and this reproduces
the boundary condition Eq. (4.2) for much of the cavity. Within a few skin depths o f the comer,
Eq. (4.2) is not accurate.
Additionally, in the cutoff waveguide section, H 0 7 (r) does not
resemble an incident plane wave. Rather the wave falls exponentially in z, parallel to the surface.
The effective surface resistance Rseff for this case is 6% to 0.3% larger and this can change
g z {r ) by a larger fraction. To calculate Rseff, we solve the metal-vacuum coupled equation Eq.
(4.6) within small regions (only 40pm long) on the waveguide side walls. Then use the ratio
between
fj/H o,z
at the middle of this region as the effective resistance. In Figure 22, we show the
63
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Chapter 4: The finite element model fo r a cavity with endcap holes
effective resistance and the resistance for a plane wave for comparison.
This difference is
probably due to the incidence angle for the field near the edge. After iterating the solution using
this treatment of the metallic comer, we then solve the metallic problem for small regions in the
cut-off waveguide section. From these solutions, we can determine an effective surface resistance
that varies smoothly along the cutoff waveguide section. We use this effective surface resistance
in turn to solve for f ^ Q(r ) .
In Figure 20-Figure 21 we also compare our calculations of the phase to a calculation that
uses the boundary condition Eq. (4.2). We find that the correct power is dissipated in the comer
region when we choose a mesh density such that the length of a triangle side2 is about 8/4 [59],
Figure 21 shows that an atomic trajectory 1008 from the wall of the endcap hole sees a phase shift
that is quite close to what calculated without properly treating the fields in the conductor. Figure
20 shows that when atoms are 18 from wall of the endcap hole, the fields in the conductor must be
treated properly to accurately reproduce the phase shifts.
2 Here we use first order mesh elements. For most calculations where we take derivatives of the finite
element solutions, we use second order mesh elements.
64
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Chapter 5
The improved cavities
5.1
Motivation
To reduce the perturbation of the frequency of an atomic clock due tophase variations, we
must first understand the effect.
Although the phase variations in the previous chapters for
cavities currently in use can be large, here we will only treat small phase variations because the
improved cavities that we suggest below have phase shifts that are everywhere less than lmrad for
all atomic trajectories. For small phase variations, the sensitivity function [60,61] is very useful
to describe perturbative effects on the frequency of an atomic clock. In this chapter we develop
the sensitivity function and then apply it to improve cavities.
5.2
Model of frequency shift due to distributed cavity phase variations
In this section we present a calculation o f the change in the atomic transition probability
due to the spatial phase variations O(r) of the field of a microwave cavity in atomic fountain clocks.
It is both desirable and possible to have minimal variations of the phase along all atomic
trajectories through the cavity. However, while the cavities in current atomic clocks have small
phase variations throughout most of the cavity, all have large phase variations near the holes in the
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Chapter 5: Improved cavities
cavity endcaps. We perturbatively treat small phase variations which gives a simple and linear
picture of the effects.
Small phase variations in the cavity produce small changes in the transition probabilities.
The effect on the transition probability can be calculated using the sensitivity function, which is the
Green function for frequency perturbations. The sensitivity function S(t) is defined as “the
relative variations of the transition probability 8 P due to an infinitesimal phase step 8 <(>arising at
time t”[30].
The change in the transition probability is expressed as the collective phase gradient
weighted by the sensitivity function. For an atom following a trajectory r(t):
1 r
d<$>\r(t) 1
SP = - [ s ( t )
(5.1)
In fountain clocks, the atoms interact with the cavity field for a short time during the
upward and downward cavity passages and we therefore neglect the transverse motion of the atoms
during the cavity traversal. We begin with a magnetic dipole interaction Hamiltonian that leads to
brjH0 : ( r ) , where po is the vacuum permeability, p z is the atomic
a Rabi frequency f2 =
h
hv
transition m agnetic dipole moment, and the coefficient 77 = -----1— is chosen so that the tipping
angle provided by the fictitious field is approximately nil for b=l in the two atomic passages, in
other words ^ ( r ) = J^Q
Thus b is the number of two nil pulses for the microwave
interrogation, for example, b=l means two ni l pulses,
In deriving the explicit form for S(t), we
start with Eq. (1.21) and the following parameters: (1) the microwave field is detuned by half
linewidth, which means AxT=ti/2; (2) since T»x, Q»A throughout the cavity.
We use the
evolution operator 7?.(t,0) developed in Chapter 1, then the Bloch vector at time t is
a { t ) —/ £ ( f ,0 ) d ( 0 ) .
Eq. (1.21) can be rewritten as
66
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Chapter 5: Improved cavities
r
dO
0
0
^
dt
- X ( t , 0 )5 (0 ) =
dO
dt
o
o
n (f)
(5.2)
-n (/)
o
= (M0 +£-Mf )/£(f ,0)a(0)
In the perturbation theory, Eq.(5.2) is solved up to the first order of e:
7Z ( t , 0) = 7^ ( t , 0) +
(5.3)
dt ' e {t)7^ (t, t ') M f a ( t ', 0)
where 72o(t,0) is the evolution operator under no phase gradient perturbation, e(t)=dO(t)/dt. The
solution for the Bloch vector can then be written with the help of Eqs. (1.16), (1.17) and (1.18):
5o ( 0 =
0<,<r
(0 ,- s in [ 0 ( f ) ] , - c o s [ « ( , ) ] )
sin
n
IT
(t-
t
(sin
)
sin [_0(/)],- ■cos
- s in
s in [^ (f )J ,-c o s [ ^ (f)]
sin [ 0 ( f ) ] , - c o s [^ ( r ) ] cos
T
t
<t <T +T
+T < t< 2 r +T
(5.4)
where a 0 is the Bloch vector with no phase gradient perturbation, we also choose AT=7t/2, which is
common in the experiments that the measurement is made at the half height of the Ramsey fringe,
where the slope of the fringe is the largest, so that it’s the most sensitive place to measure the
probability change. With an initial condition of a (0 ) = (0
0
—l ) , the evolution of the atoms
during the first passage through the cavity is just a (0) precessing about the axis 1 by an angle 0(t),
that is a(7) = (0
pulse where
0
- s in [ # ( t) ]
- c o s [ # ( t) ] ) .
The atoms leave the cavity at time t=x with a n i l
{x^) = n j 2.
The transition probability P(t) can be calculated from Eq. (1.19)
<5-5)
67
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Chapter 5: Improved cavities
♦ S(t)
T
T+x T + 2t
7
-1
Figure 23: The sensitivity function as a function of time. It is zero at the beginning of the first passage,
and then falls to minus one for a n/2 pulse at time x. During the free evolution period (x, x+T), S(t) is
flat because there is no microwave interaction. During the second passage, S(t) is symmetric to that in
the first passage.
where P0 is the transition probability without the phase perturbation.
The extra term is the
distributed cavity phase shift.
Comparing Eqs. (5.3) and (5.5), we can find the explicit formula for the sensitivity, in
terms of the tipping angle 0
S ( t ) = ^ ( t , t ' ) M £^ ( t ' , 0)
- s in [ # ( T + 2t ) - 0 ( T + r ) ] s in [ # ( /) ]
= <- s i n [ # ( r + 2 t ) - 0 [ T + r ) ] s i n [ # ( r ) ]
- s in [ # ( r ) ] s in [ # ( r + 2r ) - # ( f ) ]
0 <t<r
r < t <T +t
T + r <t < T + 2 r
(5.6)
In Figure 23 we show the sensitivity function as a function of time t. The sensitivity
function is a symmetric function for the two clock passages and has the largest slope at the
beginning of the first passage and the end of the second one.
The sensitivity function has been used extensively to describe the effects on the atoms due
to a stochastic evolution of the phase of a local oscillator [60], In contrast, the phase variations in
68
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Chapter 5: Improved cavities
a microwave cavity evolve deterministically and, particularly because of the second passage
through the cavity, the phase always returns at some time to some well known offset (e.g. the phase
o f the field at z=0). It is therefore helpful to integrate Eq. (5.1) by parts [58], We assume that,
well outside the cavity, where H z ( r ) is negligibly small, the phase of H z (F) goes smoothly to the
value of the phase at the center of the cavity (which we could define as 0 without loss of generality).
Therefore the surface term S(t)0(t) of the integration by parts is 0, which leads to
sp=~ H ^ r 's>[/"(')>
(5-7)
To facilitate the analysis of the spatial phase variations, we change variables in Eq. (5.7)
from an integral over time to an integral over vertical position in the cavity. Here we neglect
gravity during the cavity passages since the change in velocity is typically a small fraction of the
velocity vz under normal operating conditions of fountains.
Substituting S(t) in Eq. (5.6),
O(F) = —g z( F ) //f 0z( F ) . For an atom that traverses the cavity at q on the upward passage and
returns downward through the cavity at r2 , we get
^ ( ^ ^ ) = ^ ( s in [ ^ ( ^ ) ] ^ ( ^ ) - s in [ 6'( ^ ) ] ^ I)e/ / ( p2))
(5-8)
where rx and r2 are the radial coordinates in the first and the second passages, &&eff ( r ) is
defined as
eff ( r ) = b?j
cos [_0 (r, z ) ] / / 0 z ( r , z ) ® ( r , # , z ) d z
(5.9)
= -b n '—
cos [< 9 (r,z)]g z ( r,(f),z)dz
with q '=-4r| p0pz/hvz~ 1, and z±a, the distances beyond which the fields are negligible.
In Eq. (5.8),
the minus sign in front of the second term is for the return path of the atomic trajectory, since atoms
reverse its moving direction, the same phase gradient has opposite contribution to the effective
69
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Chapter 5: Improved cavities
phase, thus to the frequency shift. The effective phase 8®e// is defined so that it is equal to the
spatial phase if the latter is constant throughout the cavity. When the spatial phase has non-zero
curvature, the effective phase is the weighted phase that atoms really see while passing the cavity.
The definition of 8®ctl{r,(j)) in Eq. (5.9) gives 8Oeff{r)=O(r)xsin[0(r)] if ®(r) does not
depend on z. Eq. (5.8) therefore reproduces the physically obvious result that adding a phase that
is constant throughout the cavity produces no change in the transition probability. Similarly, if an
atom experiences the same tipping angle and phase as a function of position on the two cavity
passages (e.g. a retrace of its path), the phase distribution again produces no change in transition
probability. Finally, at moderately high power, the power dependence of S®efT(r,<|)) is flat if gz(r)
does not oscillate wildly because the oscillatory behavior o f cos[0(r,z)] with z makes the integral in
Eq. (5.9) scale as 1/b. We see this behavior in the appendix for cavities without holes.
Further, we must consider the density distribution of the cold atoms because there is no
population change, or frequency shift, if the density distribution is the same on both cavity passages.
We consider an expanding ball of atoms whose initial 1/e radius is smaller than the final radius. We
now average Eq. (5.8) over the atomic trajectories. From density and velocity distributions when
the atoms are launched, we get:
(5.10)
where N is the number of detected atoms, ri(2)=ro+ v ti>2, t|(2) is the transverse position of upward
(downward) cavity passage, u is the most-probable thermal velocity, v0 is the mean transverse
velocity, Wd(rd) is the detection probability at position r d, and the integration over the velocity is
constrained by ri=| r 2-v(t2—11)(<ra . We do not treat the small effects of the vertical spatial and
vertical velocity distributions. Since Eqs. (5.8-5.9) are linear in gz(r), our Fourier decomposition
70
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Chapter 5: Improved cavities
of gz(r) as gz,m(r) cos(m<|)) leads directly to a Fourier superposition for S P , where the change in
transition probability (or frequency error) due to each azimuthal Fourier component of the DCP is
simply added to give the total S P .
If the densities of atoms on both passages are uniform and uncorrelated, the average of
§P(ri,r2) over all trajectories in Eq. (5.10) is zero. This corresponds to launching a cloud of atoms
that is infinitely large and infinitely hot, an impractical strategy to minimize DCP errors. Another
interesting limit is zero temperature. In this limit, ri= r2 +v0t which strictly gives no DCP error if
v0=0. More pragmatically, the initial cloud sizes in clocks range from a few mm, to clouds that can
be larger than the cavity apertures. We will therefore concentrate on two initial distributions - a
Gaussian distribution narrower than the cavity aperture and a large Gaussian that is characterized
by a quadratic density variation. In the appendix, we show the power dependences of the DCP
error for a cavity without endcap holes for a large Gaussian initial distribution.
One might expect that a simpler way to express Eq. (5.10) is to only consider the density
distributions during the upward and downward cavity passages.
However, the effects of
correlations between rj and r 2 in Eq. (5.10) are lost with this model. Nonetheless, this simplified
model with uncorrelated cavity traversals offers helpful insight.
(5.11)
Here we have taken a uniform detection probability. For large atomic samples with r0£ra, we
consider a density distribution of n 1(ri)=ni+ niai[(ri-r 0ff)2/ra2-'/2] and, similarly for n2(r2), with a
curvature a 2. Inserting these density distributions into Eq. (5.11), we see that the contributions
from the uniform density terms cancel and only the difference in curvatures on the two passages
leads to a DCP error
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Chapter 5: Improved cavities
(5.12)
which can be viewed as a density distribution of ni(ri)= ni(a.i-a2) [(ri -rofr)2
and n2(r2)=l.
With that picture, the effect of the downward passage in the first term o f Eq. (5.8) is simply the
average of sin[0(r2)] and, in the second term, the average of 5<J>cti(r2). This leads to the suggestion
that, if the density distribution is the same on both cavity passages, there is no population change, or
frequency shift. As a final simplification, since the atomic cloud on the downward passage is
much larger than on the upward passage, n2(r2) is more uniform and has a much smaller azimuthal
variation. It is therefore reasonably accurate to further simplify Eq. (5.12) by taking a 2=0, which
we call
8
Pun , for a uniform n2(r2).
From Eq. (5.12) we can now see some general behaviors of DCP errors, provided the
correlation between iq and r 2 is not large. In Eqs. (5.8) and (5.9) or Eq. (5.12), the second term
averages to zero for all m>0 phase variations. Only the first term contributes to S P and the
behavior of sin[^?(r2) j
amplitude at high power.
in this term is straightforward Rabi flopping, albeit with a smaller
Thus, if the atomic density is not centered, it will have a cos((j>)
component which, when multiplied by an m=l effective phase SOefl(ri), will give a DCP error.
For an initially large atomic sample, it is difficult to imagine m>2 density variations and so these
should generally be small. Below we note that non-uniformities in the detection, Wd(rd), will
produce m=2 DCP errors. For azimuthally symmetric phase variations (m=0), both terms will
generally contribute. Lastly, we note that the correlations introduce an interesting effect of the
second term that we discuss in the appendix. Even if n2(r2) is uniform, the second term in Eq. (5.8)
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Chapter 5: Improved cavities
can be non-zero when averaged in Eq. (5.10) because sin[0(rj)] could have the same m=l
azimuthal dependence as
80
eft(r 2), because of the correlation between ri and r2.
Ultimately, it is the frequency shift of a clock that is important. To get from the average
population change to frequency shift 8 v, we divide Eq. (5.10) by the slope of the Ramsey fringes.
The slope for any atom is
(5.13)
where Av is the Ramsey fringe width. Averaged over the ensemble, the slope of the Ramsey
fringes is:
/
dvdr2
(5.14)
dv
v
which gives a frequency shift of S v = S P / d P / d v . Throughout the rest of the paper we prefer to
discuss 8 P (and S P ) because it is never singular. The frequency shift can be singular because
d P / d v has a series of zeros as the power is increased and we show that measurements of S P at
these powers are sensitive to some DCP errors. The distributed cavity phase shift 8 v can then be
expressed as
(5.15)
dPidv
We can now precisely discuss our normalization of the field amplitude q and q'. We define
a Jt/2 pulse as the first maximum of the Ramsey fringe contrast for uniform and uncorrelated
density distributions on both cavity passages. Formally, q and q ' are the solution of:
(5.16)
Since operating clocks have a density distributions that are narrower, especially for the upward
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Chapter 5: Improved cavities
cavity passage, the optimal Ramsey fringe contrast will not occur at b=l, 3, 5,..., but at slightly
lower powers.
To be explicit, a narrow Gaussian distribution on the upward passage will
experience an average tipping angle greater than n/2 at b=l, as defined by Eq. (5.16), because the
atoms are concentrated near the center of the cavity where the tipping angle is largest.
As an example of the DCP shift, let’s consider a typical fountain clock, whose free
evolution time T=0.5s, so the slope of the central peak of Ramsey fringe is -ji/2. From Eq. (5.8), a
2prad phase difference between the up and down atomic passages approximately causes a
8P=lppm population shift, which corresponds to ~ 0 .7 x l0 '16 fractional frequency shift in the clock.
The best Cs clock so far achieves 3.3xl0"16 [13] accuracy. It can be immediately seen that it is
necessary to control the phase difference at the level of micro radians to reach this accuracy.
However, the phase near the comers of the endcap holes is around a hundred milliradians. If one
atom goes through the center of the cavity and comes back down near the wall o f the waveguide, it
may pick up a hundred milliradians phase difference, which causes ~3.5xl0"12 fractional frequency
shift. O f course, only a small number of atoms see this large phase, and as a preliminary study, we
later see that the large phases above and below the comer surprisingly cancel each other out for a
vertical trajectory.
However, allowing atoms which pass through regions with huge phase
gradients is very dangerous. We have found a way to get rid of atoms which pass closely by the
edges. In the next section, we incorporate as one of the key points to find improved cavities.
5.3
Improved cavities
To design a better cavity, it is helpful to examine Eq. (4.4) in the context of our analytic
model for a cylindrical cavity. For any p=l mode with a z dependence of cos(A:|z) , if we take
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Chapter 5: Improved cavities
k
\
gz mp( f ) = ~ ® mp(r,<f>)^J0 ( r]r )c o s (k p z) and neglect the variations of the Rabi frequency
with r, then the phase is independent of z throughout the cavity and the effective phase
is dd)etf {r,(j)) = %p® mp (r,</)) with
=1.
For p=3, the effective phase is smaller than the
phase at z=0 and it has the opposite sign, because the phase shift is larger and negative at z=±d/3.
Eq. (4.4) yields <fj3 = - 0 .2 1 7 . For higher p,
= 0 .1 5 2 and E,p decreases as 1/p. Since <^,^0
for p>l, it is possible to design a cavity that appropriately excites m=0 p>l modes that cancel the
m=0 p=l phase shifts3. Here, we first examine the physics of effective phase shift variations using
our analytic treatment of a cylindrical cavity and then we numerically analyze cavities with holes in
the endcaps. We conclude with a cavity design that has small phase variations that are carefully
adjusted so that they have no effect on a cloud of atoms in an atomic clock. It’s also worthy of
noticing that this design retains the same advantages when the scale is changed as 1/co, e.g. to
construct a cavity resonant at 6.834GHz for 87Rb clocks.
Our analytic models show that using a large number of feeds distributed in <j) eliminates the
phase shifts due to m>0 modes. We therefore consider 4 or more feeds distributed around the
cavity circumference. For more than 4 feeds, the m>0 phase variations are uninterestingly small
and therefore we focus on the m=0 phase shifts. In the previous paragraph, we neglected the m=0
p=l k xz J 0 ( y p ) sin ( £,z) term of g z0] (r, z ) . While this term produces no transverse variation
of the phase, it produces a large phase variation in z. Because the tipping angle 6 (r, <f>,z) is not
constant for all r, the phase variation in z leads to a significant transverse variation of
3 Although modes for even p are not excited, it is prudent to avoid any resonances. For even modes
£,= -0.546 and £= 0.203.
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Chapter 5: Improved cavities
: \TE.
TE
"O
0 -5
2.15
2.2
R (cm)
-0.5
TE,
Figure 24: Different effective phases for different modes. The difference of the effective phase shifts
for an atom traversing a cavity at r= ra=0 .6 cm and r= 0 for various modes as a function of cavity radius
R.
The cavity is resonant at 9.192 GHz and has no endcap holes. For R=2.11 cm, if only the m=0
p=l mode is excited, the phase shifts for all trajectories are small. The nominally unexcited m=l
modes produce large phase shifts, 50 times larger than the depicted curves.
(r,<f)A. In Figure 24, we show the m=0 p=l effective phase difference between r=0 and
r=0.6 cm. A comparison of the m=0 p=l effective phase variation in Figure 24 with the phase
variation in Figure 11 shows that the two terms in Eq. (2.10) are comparable.
4 Because the entire effect o f the k{z J 0
sin ( ^ z ) term depends upon power variations, there is a
significant power dependence. Here , we calculate <5T>t^ (r, (f} where we have adjusted the power to
have a maximum o f the average coherence (maximum magnitude o f a2 in equation (4.1) for atoms uniformly
illuminating the cavity aperture.
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Chapter 5: Improved cavities
In Figure 24 we show the effective phase variations of several important modes for cavity
radii between 2 and 2.2 cm. There are a number of routes to eliminating the effects of the m=0
phase variations. From Figure 24, it seems that a cavity radius of 2.11 cm has very small effective
phase variations due to the m=0 p=l mode. However, we calculate the effects of the endcap holes
below, and their effect is to increase the contribution of the k xz J 0 (/,r)sin (& ,z ) term to the
effective phase. As a result, for cavities with endcap holes, the m=0 p=l effective phase shift
difference o f Figure 24 is negative for all R. Therefore, to use p>l modes to cancel the m=0 p=l
effective phase shift, cavity radii less than
2 .2
cm are required so that the p>l modes have
comparable phase shifts to the m = 0 p=l mode.
To cancel the negative m=0 p=l effective phase shift difference, we need a positive phase
shift difference [Oef((0.6 cm) > Ocfl{0)]. Because the m=0 p=3 effective phase shift difference is
negative, one can feed the cavity at z=±d/3 with four or more azimuthally distributed feeds. This
excites the p=3 modes with twice the amplitude and the opposite sign as do feeds at z=0. Another
possibility is to excite the p=5 modes to various degrees. The exact cavity radius and the feed
positions affect the power dependences and future work will examine these considerations.
In Figure 24 we also show the effective phase shift differences for two m=l modes. The
m=l modes are particularly dangerous because they couple directly to an error in launch direction
or tilt of the fountain. As a function of cavity radius, the m=l p=l phase shift is smallest for R=
2.089cm and is 80% larger for R~3cm. The m=2 p=l mode is also considerable (see Figure 11) but
its weak coupling to fountain tilt and density inhomogeneity causes less concern. We also show
the m=l p=3 mode because the TE M3 has a nearby resonance. A small perturbation to R can easily
avoid the TEn 3 resonance so the m=l p=3 effects are much smaller than the m=l p=l effects.
Around R=2.1 cm there are a number of other TE and TM resonances that should also be avoided.
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Chapter 5: Improved cavities
The m=l p=l term is very large because it represents power transmitted from a feed on one side of
the cavity to the wall on the opposite side. We cannot fully benefit from any m=0 cancellation
unless the cavity losses and feeds are symmetric to better than 1%. One of the clear problem when
constructing cavities is the evaluation of the phase shifts of unintended modes. Clairon et al. [62]
have used two independent feeds that are externally balanced to exaggerate or nominally cancel the
m=l phase shifts.
Our picture of phase shifts arising from individual modes shows another
straightforward path to eliminating the phase variations of any mode. Using a number of feeds at
z=±d/3 (but fed with a single external source), one can probe the m=l p=l and m=l p=3 resonances.
Small adjustments could be made to the cavity’s symmetry to insure that these are not excited. This
can be done electrically and, once the cavity is installed, the symmetry can be monitored using the
AC Zeeman shift o f the atoms [63], When the Ramsey cavity is mistuned, atoms see a different
field intensity depending upon the radio frequency (RF). This causes a frequency shift of the
Ramsey fringe known as the AC Zeeman shift in the clock frequency (also called the cavity pulling
effect). We can deliberately introduce a strong RF sideband into the cavity through the feed. For
the sideband detuning A larger than the Rabi width, the ac Zeeman shift is 8v=Q2/A, where Q is the
sideband’s Rabi frequency at resonance. We scan the sideband detuning and measure the Ramsey
fringe frequency shift, then we know how much of asymmetric modes are excited and the ac
Zeeman shift can be used to cancel the shift due to the m^O terms.
Figure 25 shows an improved cavity.
R=2.128 cm.
The cavity is fed at z=±d/3 and has a radius of
The dominant behavior of the effective phase shifts follows those in Figure 24.
It is therefore critical to carefully choose the cavity radius and feed positions.
the cavity near the endcap holes is equally important.
The geometry of
We choose a large 2.1 cm diameter cutoff
wave guide section followed by a 1.2 cm waveguide section which sets the aperture o f the cavity.
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Chapter 5: Improved cavities
1 .2 cm
.5 cm
4.48 cm
25 prad
steps
2.1 cm
4.256 cm
Figure 25: An improved cavity design. The cavity has a radius of R=2.128 cm and power is supplied
at z=±l.487cm. The endcap holes are constructed with two sections of below cut-off waveguide of
diameter 2.1 cm and 1.2 cm so that the atoms experience no nodes of H z(r) as in Figure 15.
phase contours are steps of 25prad.
aperture.
The
The effective phase shift varies by ±0.1prad over the 1.2 cm
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Chapter 5: Improved cavities
■o
0.1
(0
3
0.05
0
■©■-0.05
i
1------1------1------1 \ i
1----- 1----- 1----- 1 / i------1
0.2 \ 0 . 4
/
0.6
(cm)
Figure 26: The effective phase of the improved cavity at optimal power. The average of the effective
phase is zero.
The comer o f the 2.1 cm section is relatively far from all atomic trajectories and the diameter is
such that no atoms see a node o fH 0z(r) .
Therefore no atoms experience the large phase shifts
near the comer as shown in Figure 19-Figure 21.
The 2.1 cm diameter waveguide section is
sufficiently long to make H 0 z (r) decay in this cut off waveguide to small enough value so that
the large phase shifts near the comer of the 1.2cm section do not significantly affect the atoms.
In Figure 26, we show the m=0 effective phase variations for the improved cavity.
We arrive at our cavity geometry in Figure 25 after more consideration of the effect of the
cavity on the sample o f atoms in a clock. In an atomic fountain, atoms cooled to 1.5pk are
sufficiently hot that the mean radius of a laser-cooled sample increases by a factor of 2 during an
interrogation time of 0.5 s. If the m=l modes are not excited, a fountain tilt or a launch angle error
does not produce an error.
We therefore expect the dominant error due to phase gradients to be
density inhomogeneities during the first passage through the cavity.
We expect the density
variation will be small, and therefore we take the lowest order variation to be quadratic,
n{r) = — ^
nr
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Chapter 5: Improved cavities
* z= + d/3
»feeds
R (cm)
TD
CO
H
co
e
CO
z=0 feeds
Figure 27: 8®avg as a function of cavity radius R for cavities that are fed at z=0 and z=±d/3. These
cavities have below-cutoff waveguide sections as in Figure 25, and these ensure that the phase shifts
are everywhere small. For cavities fed at the midsection, z=0, 8®avg is negative for all R. With
power feeds z=±d/3, the m=0 p=3 mode offsets the m=0 p=l phase shifts. The m=0 p=3 mode has a
larger effects for small R, and opposite effects for z=0 and z=±d/3 feeds. For R=2.128 cm, the cavity
in Figure 25, 5®avg=0.
and negative during the first cavity traversal and negligible during the second.
difference of
We take the
( r ) on the two cavity passages which leaves an m=0 average effective phase
difference of &£>avg =
dX>efj ( r )
( 2r 2
. ra4
V
1A
rdr .
r' a2
For the cavity design in Figure 25,
dd>avg = 0 and the variation of & $ ) ( r ) is less than ±0.1 prad5. In Figure 27, we show the
5 0 avg we get from finite element calculations for several cavity radii where the cavities are fed at
z=0 and z=±d/3. As mentioned above, while the cavity with no endcap holes in Figure 24 has a
large positive m=0 p=l effective phase shift differences or large R, the effects o f the endcap holes
and 2.1 cm diameter below-cutoff waveguide sections prevent 8<Davg from increasing for large R.
5 This design retains 8Oavg=0 when the scale is changed as 1/co, e.g. to construct a cavity resonant at
6.834GHz for 87Rb clocks.
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Chapter 5: Improved cavities
Feeding a cavity at z=±d/3, versus z=0, excites the p=3 mode with an opposite polarity and twice
the amplitude - the effect of the m=0 p=3 mode is clear in Figure 27 for cavity radii near 2.1 to 2.2
cm.
82
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Chapter 6
Summary
6.1
Our principal results
We have derived an analytic solution for the phase variations that occur in the microwave
cavities used in atomic clocks. The analytic solutions have significant new terms that were not
present in previous two dimensional treatments. These terms show that cavities with radii near
R=2.1cm have smaller phase shifts. We have shown that a series of 2D finite element solutions is
efficient and can accurately capture the three dimensional phase variations in microwave cavities.
We have applied the finite element method to cavities with endcap holes. Our analytic solutions
suggest an improved cavity in which the peak phase shifts that the atoms experience would be
1,000 times smaller than in the cavities currently in use. The effective phase shift variations are
less than ±0.1 prad and, with reasonable assumptions about density anomalies, the average phase
shift is vanishingly small.
6.2
Future projects
We have successfully worked out solutions of the fields and phase variations in microwave
cavities in current atomic fountain clocks. However, if researchers wish to extrapolate our results
into their error evaluation, they should take into account the following practical considerations,
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Chapter 6: Summary
which fall into three main categories: the effects of fountain imperfections on clocks, the range of
methods for experimentally measuring DCP errors, and the possibility of further improving the
cavity design. In this section, we discuss these issues based on our current understanding and
some preliminary results we have obtained while writing this dissertation.
6.2.1 Imperfections of fountains
In practical situations, fountain clocks are not perfectly set up. Some of the imperfections
are negligible, others should be carefully treated. In the following we briefly address six of the
most important considerations researchers should pay attention to.
(1)
Feed imbalance: As we know from the earlier chapters, adding more symmetric
feeds at the mid sections of the cavity side walls can eliminate the m=l ,2,... modes so that only the
azimuthally symmetric mode produces DCP errors.
However, the feeds may not be ideally
balanced. The opposite feeds they may not feed the same amount of power into the cavity; they
may have different phases, or they may not be aligned along a straight line. The imbalance
between different feeds excites m^O modes. For example, in the SYRTE-F02 clock, the cavity is
fed by two opposite feeds on the side wall. Researchers independently feed power through the two
feeds and they can control the power flows through the two feeds as well as being only off by less
than 1% in the amplitude, so that the m=l modes are at least reduced by a factor o f 200 compared
with a single feed cavity. This reduction makes the m=l phase gradient the level o f the m=0 phase
and its contribution to the total error is comparable to that of the m=0 modes. Further reduction
may be possible. SYRTE-F02’s independent double feeds make the evaluation of m=l error very
easy because, by shutting off one of the feeds, a full m=l excitation is achieved and dominates the
other
1 modes. The comparison between the measurements by individually shutting off either
84
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Chapter 6: Summary
of the two feeds also gives information pertaining to the fountain alignment. An experiment has
been done [64] and their data showed good agreement with our preliminary calculations.
(2)
Atomic beam misalignment: If an ideally symmetric ball of atoms is launched
vertically along the cavity axis, the density distribution is always azimuthally symmetric. The
m^O spatial phase variation averages to zero. Thus only the m=0 modes contribute to the DCP
error. However, the atomic ball may not be perfectly centered. Once the ball’s center of mass is
offset, the m= 1 ,2 ,... phase will be picked up by the density. To see this explicitly, let’s consider a
Gaussian density distribution with an offset of xo and 1/e width a
(x-xoy + y 2
n (x ,y ) = e
(6.1)
We make the transformation from Cartesian to Polar coordinates by using x=rcos((j)) and y=rsin(<))),
and then expand n(r,<j>) into its Fourier series in <)) to obtain
»(/
CO
I r2+xo / ( i r x 2^
—e 2a~ 7r J 0
+
2
2
n
m=1
I
)
\
' 2r x f
2 cos(m^)
I ° J
J
(6 .2)
Eq. (6.2) clearly shows the cos(m<|)) components of the density distribution for an offset x0. For
any nonzero xo, such terms in Eq. (5.10) do not average to zero. So the measurement o f the m=l
DCP error yields information about the atomic offset at the launch point.
(3)
Fountain tilt: This fountain imperfection is similar to the beam misalignment in the
sense that both of them create cos(m<[>) dependence in the atomic density distribution even for an
ideally spherical ball. When the fountain’s axis is not vertical, an offset is only introduced into the
second passage, but it’s negligible in the first passage. In a real fountain, it’s possible that the
misalignment and the tilt are mixed. Once we understand these separately, the combined effects
can also be easily evaluated.
(4)
Non-uniform imaging: We have shown that non-uniform imaging can produce an
effective m=2 atomic density distribution [65]. Since imaging happens after the second clock
85
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Chapter 6: Summary
passage, the atom ball is much larger than its initial size, and the real density distribution can be
presumed to be pretty uniform. For uniform imaging, the m=2 spatial phase has no effect in the
second passage, although for a cavity with two symmetric feeds, the m=2 spatial phase is still
present with full amplitude and is comparable to the m=0 phase. However an m=2 effective
density makes the m=2 DCP error stand out for that passage. Therefore, both atomic passages
contribute to the DCP error for m=2.
(5)
Inhomogeneity of the conductivity: Because of impurities in the metal, it is possible
that the conductivity of the metallic walls is not a constant throughout the whole cavity. The wall
losses at different positions excite different modes. As shown in Chapter 5, that a feed at d/3
above the mid-plane can induce more p=3 mode than a mid-plane feed does. The microwave field
in a cavity made of metal with a non-uniform conductivity can be viewed as a superposition of the
field in a cavity with a uniform conductivity and the set of fields excited by the excess power
dissipation on the wall. One way to calculate the field excited by the inhomogeneous conductivity
is to divide the wall into many small regions and individually solve for the field induced by the
power coming in (or out) of each region but with no power loss for all other regions.
The
superposition of these solutions gives a lull solution for a cavity with non-uniform conductivity
throughout the walls. The conductivity offsets at the small regions determine the coefficients for
corresponding solutions in the superposition. The Q factor can be used as one of the parameters to
determine how great the inhomogeneity o f the conductivity is because Q is directly related to the
conductivities of the wall. One way we can propose is that we calculate the corresponding Qs for
a few modes with different resonant frequencies, and we then fit those values to the measured
loaded Qs for these modes to set the limit for the inhomogeneity.
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Chapter 6: Summary
6.2.2 Evaluating DCP error
As discussed in section 6.2.1, many imperfections can contribute to DCP error. Generally,
their effects are mixed with one another. It may be possible to study them separately. We try to
give suggestions of how one might combine the theoretical analysis as we do in this dissertation
and experimental measurements to evaluate these effects and provide a correct error estimate. We
will address the power dependence which is a common tool to analyze power dependent clock
frequency shifts; we will analyze the complexity of power dependence for DCP errors and its
implication for error evaluation; and we propose a couple of experiments to explore DCP effects on
atomic clocks.
(1)
Analyzing power dependence: Power dependence is often used to evaluate the
systematic errors of atomic clocks. For example, in [18] it was thought that the DCP error grows
linearly with the power. As a result, larger frequency shifts stand out at higher powers and are easy
to measure. The shifts measured at high power can be scaled down to estimate the proportional
shifts at low power, especially at the optimal power (two rc/2 pulses), which provides an indirect
way of measuring DCP errors at lower powers. However, the power dependence of DCP error is
not a linear function of the power, and it is actually pretty complicated. As an example, let’s
consider the analytic phase variation (see appendix for details). The power dependence of all p=l
modes is a function akin to the slope of the Ramsey fringe versus the power, because the spatial
phases for these modes have no longitudinal variation.
The similarity of the Ramsey fringe slope
makes the frequency shift due to these modes nearly independent of power, and thus the DCP error
cannot be evaluated by only measuring the power dependence.
Another example of the
complexity of the power dependence is that of the phase variation induced by the endcap losses,
with just a longitudinal dependence. The power dependence of the longitudinal phase exhibits
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Chapter 6: Summary
peaks near two 2ji, 4tu,
671,...
pulses, and is small near two odd integer n pulses. The DCP power
dependence has two implications: 1) we can measure DCP shift at two even integer n pulses at high
powers for m = 0 phase variation, in addition to measuring it at odd integer n/2 pulses as previous
researchers usually did; 2) if a cavity (e.g. in PTB-CsFl) does not have independently excited
double feeds as in SYRTE-F02, the m=l DCP shift is most easily evaluated at two n/2 pulses,
where the m=l shift is the largest, but at higher powers the m=0 shift dominates. This illustrates
the complexity of the power dependence, and we believe that a more detailed study can provide a
correct understanding o f the DCP shift and lead to a better measurement.
(2)
Atomic densities: In current clocks, the initial ball size varies. For example, in
PTB-CsFl the Full Width Half Maximum (FWHM) of the ball at the launch point is 0.8mm while it
is 1.5mm for SYRTE-F02. In Eqs. (5.10-5.12) we see that the atomic densities play an important
role in averaging the DCP error for atomic ensembles. Researchers may guess that the smaller the
initial ball is, the smaller the DCP error will be, because most of the atoms go through the center of
the cavity and avoid the large phase gradient near the edge. In fact, this is not the case because the
total DCP shift is the difference between the shifts that atoms accumulate during each of the two
passages. Although the shift during the first passage may be small, atoms always see larger phase
gradients on the way down when they pass the regions near the edges of the endcap holes due to
thermal expansion, so the difference between the up and down passages is big. On the contrary, if
the initial density is nearly uniform and the ball is large, then on the way down, the density is still
fairly uniform even under the expansion. Therefore, on average, atoms see the same spatial phase
gradients in the two passages, and the DCP shifts in the two passages of a large ball are canceled by
each other, so that the total DCP error is small. By accounting for the dependence on atomic
density, we may be able to find optimal launch densities for different atomic temperatures, which
minimize the DCP error while maintaining enough atoms for detection.
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Chapter 6: Summary
(3)
Measure DCP errors: One may propose ways to evaluate the DCP errors by
combining the numerical simulation and experiment. We now know that the longitudinal phase
gradient of the azimuthally symmetric phase variation dominates the DCP frequency shift at high
powers, especially at two 2n, 4n, and 6n (even integer 71) pulses. This makes the measurement for
the m>0 modes difficult at high powers, because the effects of all the modes add to one another.
However, at optimal power (two rc/2 pulses), the m=0 DCP error is extremely small, while the other
modes make relatively large contributions to the DCP error for that power. Hence the best “place”
for measuring the frequency shift for each m^O mode is at optimal power.
To make the
measurement of m^O DCP shift easier, researchers in SYRTE-F02 utilize their independent
double feed technique to exaggerate the m=l shift. For cavities without independent double feeds,
there are other ways to deliberately amplify the m=l shift. As we saw in section 6.2.1, atomic
beam misalignment can generate m^O density variations even for an azimuthally symmetric initial
atomic density. We can introduce relatively larger misalignment so that the m=0 density variation
is smaller than the m=l density portion. This way, we make the m=l DCP shift larger while
making the m=0 contribution smaller. Further amplification can be achieved by using a small
atomic ball at launch. As in Eq. (6.2), a smaller ball on the way up (a is small) can have a larger
ratio of the m=l density portion to that of m=0 for the same initial ball offset.
To evaluate the effect of conductivity inhomogeneity, we can calculate the power
dependences of DCP errors corresponding to the fields excited by wall losses for each small region
(as discussed in section 6.2.1). We manipulate the power dependence functions to find useful
information. For instance, we can fit the power dependence functions to the slope o f the Ramsey
fringe versus the power. The fitting coefficients are proportional to the conductivities for each
region. If there is a good fit, we actually find a distribution of the conductivity which induces a
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Chapter 6: Summary
power-independent DCP error, thus a measurement of power dependence alone is not enough to
evaluate DCP errors. Furthermore, by comparing the power dependence functions corresponding
to wall losses at different small regions, we can see which region produces most of the DCP error at
specific powers. This may be helpful for further improving cavity designs because it provides
information o f where the feeds ought (or not) to be placed to induce (or eliminate) certain
excitations.
6.2.3 Further improvements
We have seen that the improved cavity dramatically reduces the spatial phase variations
along all atomic trajectories. Further simulation shows that we can obtain more improvements
which make the DCP shift nearly zero at two 2n pulses by slightly changing the feeds position
around their initial positions in that design (z=±d/3). By doing this, more or less of the m=0 p=3
mode is excited while the m=0 p=l mode excitation remains the same. By adjusting the m=0 p=3
excitation, the DCP error from the m=0 modes can be made smaller at two 2n pulses, while it is also
small at two 3n/2 and 5n/2 pulses.
This reduction at high power gives more room to the
measurement for m>0 DCP error. For example, while m=0 DCP error is small at b= l, 3, and 5,
researchers can easily measure m=l DCP error at those powers, instead of being limited at b=l by
their current cavities. We suggest that if we manipulate the radius and length of the first cut-off
waveguide section we may further reduce the effect of m=0 modes.
90
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Appendix
Power dependence of DCP errors for cavities without endcap holes
Power dependence (PD) is often used to evaluate systematic errors o f atomic clocks. It
has been assumed that power dependence is also applicable for DCP errors. We calculate the
power dependence of the DCP shifts and show that power dependence may not be adequate for
evaluating DCP errors.
It is instructive to use the analytic solution for a cylindrical cavity with no endcap holes as
a starting point to understand the power dependence of DCP errors. We will first consider the
power dependence of the spatial phase variations and ignore the radial variation of the tipping angle
0. We then show the additional effects from the tipping angle’s radial variations. In previous
DCP evaluations, it was assumed that measuring power dependence was sufficient to estimate the
DCP error. However, our analytic model shows that the DCP error is independent of power for
phases which have no longitudinal variation. This feature implies that the power dependence
alone is not enough to evaluate DCP errors. Our model leads to a simple expression for the change
in transition probability. Throughout this appendix, we discuss the transition probability instead
of the frequency shift because there is an important diagnostic sensitivity for integer n pulses for
which the frequency shifts are singular (because the slope of Ramsey fringe is zero).
In this section, we rewrite the phase variation to lowest order at the center o f cavities
following the analytic solutions in Chapter 2 as
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Appendix: Power dependence o f DCP errors in cavities without endcap holes
m\p
1
3
0
3.6(r/ra)2 + 0.3(r/ra)4 - 59.4(2z/d)2
1.4 + 0.8(r/ra)2 - 14.0(2z/d)2
1
78.3(r/ra) + 6.1(r/ra)3
1.0(r/ra) + 0.4(r/ra)3 -10.0(r/ra) (2z/d)2
2
29.7(r/ra)2 + 3.1(r/ra)4
0.4(r/ra)2+0.1 (r/ra)4- 4.1 (r/ra)2 (2z/d)2
3
-11.2(r/ra)3 - 1.3(r/ra)5
0.1(r/ra)3+ 0.04(r/ra)5- 1.3(r/ra)3 (2z/d)2
4
-0.5(r/ra)4 - 0.1(r/ra)6
0.04(r/ra)4+ 0.01 (r/ra)6- 0.3(r/ra)4 (2z/d)2
Table 4: Phase expansions for a cavity with R=2.42cm. ra is the radius of the endcap holes, and d is the
length of the cavity. Without losing generality, the phase offset is set to zero at the center of the cavity
for the m=0 p=l mode.
/
®o t = 2 > . , ( ? ) + Z >
f
$0
r
COS (k,z)
\ raJ
COS ( m<j))
V
+ £,£ —tan(&,z)
d
+ ^012
\ raJ
® 0p(r) =
/
+
COS
mpm
m~\
odd p
\
V
\ 2
cos
^0^2
V
(v)
COS (k.z)
a)
p =1
(A.l)
p> 1
where the coefficients are given by
<Dmpm
m
r,'d
!^ ' M
S K 2ra2
® 0p2
2d
P=
1
(A.2)
~
^ f ( r i - r r 2)
p
>i
An example of R=2.42 cm and ra=5mm is shown in Table 4. We will focus on this cavity geometry
throughout this appendix.
It is noteworthy that for cavity radii much greater than the TE0i
waveguide cutoff radius R=2 cm, the large p modes are highly suppressed at the center o f the cavity
for a midsection feed.
However, in Chapter 4, we showed that endcap holes introduce
perturbations near the endcaps that do excite high p modes that produce significant phase shifts for
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Appendix: Power dependence o f DCP errors in cavities without endcap holes
1
^ 0 . 5
0
-5
0
x (mm)
5
Figure 28: Atom’s survival fraction (solid), atomic densities on the way up (higher two curves) and
down (lower two curves) as a function of x. The initial density on the way up for a perfectly aligned
beam (dash-dot) is a parabolic function of x with a= -l for a Gaussian beam with 1cm 1/e width. The
initial density for a beam misaligned by 1mm (dashed) has less curvature at x>0 and more curvature at
x<0 than the well-aligned beam. The free evolution time T=0.5s. Due to thermal expansion, the
density in the second passage becomes more uniform. However, the velocity correlation between the
two atomic passages introduces more curvature to the effective densities for the aligned (dash-dot-dot)
and misaligned (dotted) beams.
the atoms.
There are some common features to the phase variations. For m=0 modes, the lowest
order terms show either only longitudinal or only radial dependence.
For example, in the first
row o f Table 4, -59.4(2z/d)2 for p=l and -14.0(2z/d)2 for p=3 terms have only longitudinal
variation; and 3.6(r/ra)2 for p=l and 0.8(r/ra)2 for p=3 terms have only radial variation. Clearly, the
longitudinal variation dominates the m=0 phases.
However, for all m>0 modes, the phase
variations show a transverse dependence of r"1cos(m<|)) and a longitudinal dependence of cos(p k|
z)/ cos(ki z).
While the p=l modes have a radial phase variation that is independent o f z, the
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Appendix: Power dependence o f DCP errors in cavities without endcap holes
higher p modes have both radial and longitudinal variations. It directly follows that there is no
power variation of 8P or o f the DCP errors for p=l modes. These important features will be
discussed below.
We consider a fountain clock with a large initial atomic ball (1/e radius is 1cm) and the
density distribution n, (r) =
N
, where N is the total number of atoms, ra the
n ra
radius of the endcap holes and cii the curvature factor. We consider a i= -l, the free evolution time
of T=0.5s and the atomic temperature of lpK. In Figure 28 we show the atomic densities of a
large ball for the up and down passages as a function of x and the survival fraction. An atom has
55% chance of being detected without hitting the wall on the way down if it passes right through the
center of the cavity on the way up, and 32% chance if it is initially near the wall of the waveguide in
the first passage. In other words, the correlation between the two passages effectively increases
the density in the downward passage, such that atoms are more concentrated to the center than for
uncorrelated atomic balls.
Therefore, more atoms see the central phase gradient than the
uncorrelated atoms on the way down.
In order to see the most basic features of power dependence, we separately analyze the
power dependences for different phase variations. We start with the longitudinal phase variations
because they produce the dominant effect. Since all phase terms with only longitudinal variations
are from the m=0 modes as shown in Table 4, we assume the atomic ball is well-centered and is
launched vertically. Then this atomic density has no m>0 components and consequently no
frequency shift from the phases of m>0 modes. We show the power dependence of the phases
which has no radial variations and is proportional to zxtan(kiz) (m=0 p=l) or cos(k3z)/cos(kiz)
(m=0 p=3), where k|=ji/d and k3=3n/d. We here consider three different density variations for the
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Appendix: Power dependence o f DCP errors in cavities without endcap holes
cos[0(z)]gz(z) (mnr1)
E
mm)
A,
Q.
Q.
m=Q
P=1
-15
-5
— i—
*
&0.1
m=0
P=3
0
0
2
4
6
8
10
12
14
b
Figure 29: Power dependence (PD) of the phases with only longitudinal variation for a a=-l atomic
density on the way up without beam misalignment. PD for longitudinal m=0 p=l phase. The PD for
correlated two atomic passages (solid), uncorrelated (dashed) and uncorrelated with uniform density on
the down passage (dotted) are all large at two 2k (b=4), 4 k (b=8) and 6n (b=12) pulses. The inset shows
the integrand in Eq. (5.9), cos[0(z)]gz(z) at r=3.5mm for b=2 (solid), 4 (dashed), 8 (dotted) and gz(z)
(dash-dot). The integrand is symmetric about the mid plane (z=0) for b=4, 8 and 12, therefore SOeff is
large, but it is antisymmetric for b=2, 6 and 10, so that 8 0 eff is small at those powers, (b) PD for
longitudinal m=0 p=3 phase. Similar features show up in this longitudinal phase.
downward passage. In all three cases, the initial density distributions are the same, which are
parabolic functions of r with a i= - l. On the way down, the atoms are assumed to form either a ball
correlated with the upward passage, or a ball with a uniform density but uncorrelated with the
upward passage, or a ball with a parabolic density distribution with a.2=-0.76 which is determined
by ai and the thermal expansion at the atomic temperature (lpK), and still uncorrelated with the
first passage. In Figure 29 (a), we see that the power dependences in all three cases behave
similarly: they are large at b=4, 8 and 12, corresponding to two 2k , 4 k and 6 k pulses; and they are
small at b=2, 6 and 10, corresponding to two 7t, 3n and 5it pulses. In the inset in Figure 29 (a), we
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Appendix: Power dependence o f DCP errors in cavities without endcap holes
show cos[0(z)]gz(z) at b=2, 4 and
8
and gz(z) itself as a function of z. gz(z) is a symmetric
function about the mid-plane (z=0 ), while cos[0 (z)] changes its symmetry when the power changes.
It is seen that for b=4 and
8,
cos[0(z)]gz(z) is symmetric because cos[0(z)] retains the same
symmetry. For b=2 (also for b = 6 and 10), cos[0(z)] becomes antisymmetric about the mid-plane,
so that cos[0(z)]gz(z) is antisymmetric. If the integrand in Eq. (5.9) is an antisymmetric function,
the integral 8®efr becomes small, while it is large when the integrand is a symmetric function.
Physically, for two 27t, 4n and 6n pulses, atoms see the same spatial phase gradients in the lower
and upper halves of the cavity during each passage and the effects add up. But for two n , 3 n and
5tc pulses, the effects from the lower and upper halves of the cavity tend to cancel each other. The
large population shift at two 2n, 4n and
671
pulses is a general feature for all the longitudinally
varying phases. Figure 29 (b) shows a similar power dependence.
In Figure 29, we see that the PD in the correlated case is close to but smaller than that in the
case with uniform atomic density on the downward passage, and quite distinct from that with a
non-uniform density on the way down. It indicates that for large atomic ball and m=0 phases, the
assumption of a uniform density on the way down without correlation between two passages can
well approximate the PD with correlated two passages.
To show another important feature of the cavity phase, we now consider the phases with
only radial dependence, but with no variation in the longitudinal dimension. We first neglect the
radial variation of H0,z(r) by taking its value at r=0 for all the atomic trajectories,
H 0. ( r , z ) = v_k] cos(Ar,z)/2 . In other words, the radial variation of the tipping angle 0 is also
neglected and# ( r , z ) = b n / 4 [ l + sin ( k {z )] . Therefore, the radial dependence of the phase
(I) in Eq. (5.9) can be taken out of the integral and Eq. (5.9) can be further simplified as
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Appendix: Power dependence o f DCP errors in cavities without endcap holes
7r r d/2
f ' co s[# (r = 0 ,z ) ] //0z (r = 0 , z ) $ ( r ,cp,z)dz
—d
2 *J
d-d/2
v»
■ *7r/2
7r
v
b-A>
— COSi ((m4>)
7720 ) JI
COS b —(l + sinw) cos ( p u ) d u
4
"V ""
r
k /2
4
= ^
/
!2
(A.3)
The population shift can be calculated by averaging over the atomic cloud as in Eq. (5.10),.
Explicitly calculating the integral in r, we get
SP = (a. - a 2)—— sin( b—1
n ^ f }
2
x £ Jb—cos b —[l + sin(M)]Jcos(/?M)Jw
(A'4)
It is now easy to see that for p=l (and m^O), 8<f>ef( is proportional to sin(0i). The integral in Eq.
(A.4) can be explicitly carried out. The population shift SP is
SP = (a, - a 2)
sin2
(A.5)
While we neglect the radial variation of the tipping angle, the slope of the Ramsey fringe is simply
7t/2xsin2(b7i/2). Surprisingly, the population shift has the same power dependence as the slope of
Ramsey fringe.
From Eq. (5.13), therefore, there is no frequency shift in this case and the
frequency shift at all b ’s is a constant ( a i - a 2)Ompin/l 2rt. Thus the DCP errors are independent of
the power.
If the radial variation of the tipping angle is included, the power dependences change only
slightly. In Figure 30, we show the population shift for m=0 and 1 ((a) and (b), respectively)
phases with only radial variations, and with both radial and longitudinal variations ((c) and (d)).
For analyzing the m=0 power dependence, we still assume that a centered atomic ball flies
vertically up and down in the two passages. For m=l phases, we consider an atomic ball
off-centered by 1mm. In other words, its center of mass is at x=lmm, where x is the Cartesian
coordinate.
We assume the feed imbalance is 5% o f the total power.
We still study the
97
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Appendix: Power dependence o f DCP errors in cavities without endcap holes
E
CL
Q.
Q_
CO
0 ■o "O
0
-
m=0
p=1
0.1
T
T
T
T
T
T
T
CM
T
E
CL
CL
■D
CO
E
-
0.1
m=1
0.2
0.2
CM
p=1
o -
-
S '
CL
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Figure 30: Power dependence (PD) of the phases with transverse variation for a a=-l atomic ball on the
way up for well-aligned beam for m=0 phases, and for a 1mm misalignment for the m=l phases with 5%
feed imbalance. The PD for correlated two atomic passages (solid), uncorrelated (dashed) and
uncorrelated with uniform density on the downward passage (dotted) are shown in all 4 panes. In (a)
and (b), the slope of the Ramsey fringe (dotted with big circles denoting the peaks) is included for
comparison.
three different density distributions for the downward passage. In Figure 30(a), we see that the
population shifts for all three densities behave similarly. The population shifts slowly decrease as
the power goes up; the peaks are near the odd integer b ’s, which corresponds to two odd integer n/2
pulses. The slope of the Ramsey fringe has the similar behavior, too, as shown by the dotted curve
98
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Appendix: Power dependence o f DCP errors in cavities without endcap holes
with solid circles which label the peaks of the slope. The similar shapes of population shifts and
the slope of the Ramsey fringe indicate that the frequency shift just weakly depend on power. In
Figure 30(b), one obtains the same power-independent frequency shift as exemplified by the
similar shapes of population shifts and the slope of Ramsey fringe. In Figure 30(c) and (d), for
p=3 modes, the phases have both variations in the radial and longitudinal dimensions. We see that
the largest effect is at relatively higher power, b=5. This is a general trend that, at high powers,
cos[0(z)] is in phase with the terms which have higher spatial frequency, and therefore the effect of
the DCP is “picked up” by the sensitivity of the atomic response.
From Figure 29 and Figure 30, we conclude that the dominant DCP power dependence is
produced by the azimuthally symmetric and longitudinal phase variations.
The DCP shifts
strongly depend on the microwave power, while the dominant radial phase variations produce DCP
shifts which do not depend upon power.
These two basic features show that power dependence
alone is not adequate for evaluating DCP errors for current fountain clocks. The reasons are as
follow:
(1)
the m=0 DCP shift is always present in the cavity because the m=0 phase variation
cannot be eliminated by feeding power through feed at the mid section of the cavity side walls;
(2)
the m=0 DCP shift dominates the power dependence of the population shift,
especially at high power, so that at high power, m>0 shifts are probably impossible to measure;
(3)
the m>0 population shifts have little power dependences, which give relatively small
power variations of the frequency shifts, therefore power dependence measurement alone cannot
extract useful information for DCP shift at all.
As we proposed in Chapter 5 and 6, if we change the current cavities in use to the improved
cavity, we can dramatically reduce the population shift at b=4 due to m=0 phase variations. As a
99
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Appendix: Power dependence o f DCP errors in cavities without endcap holes
result, we can have more room for measuring m>0 DCP shifts.
We may further manipulate the
m>0 DCP shift so that they are easier to measure. For instance, we may be able to create much
greater m>0 DCP power dependence, so that power dependence can be used to evaluate the DCP
error.
too
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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