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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3293341 Copyright 2007 by Li, Ruoxin All rights reserved. INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3293341 Copyright 2008 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ©2007 by Ruoxin Li All rights reserved. iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. 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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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D edicated to m y wife. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. <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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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) 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 81 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 ---- 1-----1—^~i-----1-----1-----1-----1-----1-----1-----1 i---- ; CL 0.1 Q_ 0 h CL CO i i CO v v ^ i i i i i i i i i i i i i .j / \ i i i i i i i m=1 i iv i i i i i i i i i i ; <d> a g CL s / ^ - 0 i 1 i 3 5 7 9 11 "i 13 b 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. Bibliography [1] C. Audoin and J. Vanier, “The Quantum Physics of Atomic Frequency Standards”, Adam Hilger (1989). [2] N. Ramsey, “A molecular beam resonance method with separated oscillating fields”, Phys. Rev. 78, 695 (1950). [3] J. R. 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Bahoura, “A Cesium Fountain Frequency Standard: Preliminary Results”, IEEE Trans. Instrum. Meas. 44, 128-131 (1995). [63] C. Fertig, J. Bouttier, and K. Gibble, “Laser-cooled Rb Clock”, Proc. 2000 IEEE Freq. Contr. Symp., 680-686 (2000). [64] F. Chapelet, S. Bize, P. Wolf, G. Santarelli, P. Rosenbusch, M.E. Tobar, Ph. Laurent, C. Salomon, and A. Clairon, “Investigation of the distributed cavity phase shift in an atomic fountain”, Proceedings of the 20th European Frequency and Time Forum (2006). [65] R. Li and K. Gibble, "Distributed cavity phase and the associated power dependence", Proceedings o f 2005 IEEE Freq. Contr. Symp, 29-31 (2005). 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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