Towards A Transportable Atomic Fountain

Towards A Transportable Atomic Fountain

by

Paul David Kunz

B.S., University of Texas at Austin, 2004

A thesis submitted to the

Faculty of the Graduate School of the

University of Colorado in partial fulfillment

of the requirement for the degree of

Doctor of Philosophy

Department of Physics

2013

This thesis entitled:


written by Paul David Kunz

has been approved for the Department of Physics

Jun Ye

Steven Jefferts

Date

The final copy of this thesis has been examined by the signatories, and we find that both the

content and the form meet acceptable presentation standards of scholarly work in the above

mentioned discipline.

iii

Kunz, Paul David (Ph.D., Physics)


Thesis directed by Dr. Steven Jefferts

For many years atomic fountains provided the most accurate measurements ever

made: up to sixteen decimal places of the Cesium ground state hyperfine splitting. This

thesis project sought to bring this unprecedented precision, afforded by atomic fountains,

into a transportable package. Such a devise would be valuable to many applications

involving precise timing and frequency comparisons. For example, the active field of optical

clock research could benefit from a stable and robust transportable frequency reference,

since frequency comparisons at the sixteenth decimal place are presently highly

impractical beyond distances of 100 km. Outside of the scientific community the

applications become more numerous. This is especially true if further refinement

culminated in a commercially available version of a transportable atomic fountain.

The bulk of this research concerned the development of a suitable laser system. A

design based on electronically phase-locking independent slave lasers to a single master

laser would provide a simple, robust, compact, and low-power system for operating an

atomic fountain. We investigated newly available distributed feedback (DFB) and

distributed Bragg reflecting (DBR) diode lasers, and found each to be lacking in one or

another critical factors. The DFB laser diodes proved to have unreliable lifetimes,

unacceptable for an atomic fountain that is meant to be robust and reliable. The DBR lasers

have an internal phase delay that severely limits the achievable phase coherence between

the lasers, and this was found to be detrimental to the requisite sub-Doppler laser cooling

in a fountain. We present measurements characterizing the limitations of the current

system, and propose solutions for further investigation.

Dedication

To my family, with all my love and gratitude.

v

Acknowledgements

First, I would like to thank my advisor, Steve Jefferts, and “co-advisor” Tom Heavner

for offering me the opportunity to work at NIST under their guidance. They have been

tremendously patient and supportive, and their doors were always open as I constantly

peppered them with questions.

Steve is a great experimentalist and problem solver with abundant energy. He is a

walking encyclopedia of electronics knowledge, a talented and experienced machinist, and

generally a wealth of practical knowledge who advised me not just on research but on

everything from fixing my car to the plumbing in my house. Steve is incredibly generous,

for example, immediately offering up his house when my family and I were evacuated for a

month after the Fourmile fire. Finally, I have to say he also really knows how to have a good

time, whether it’s captaining a sailboat around the Caribbean, flying his plane to remote fly

fishing spots, or scuba diving in the Mediterranean. He is an inspiration, professionally and

personally.

Tom spent many hours helping me in the lab, tweaking optics, tracking down noise,

coding LabView, and thoroughly editing my writings. I’m grateful for his organizational

skills, and meticulous style as an experimentalist, both attributes for which I needed a

mentor of his caliber. I also always appreciated his dry, witty, sarcastic sense of humor; I’m

chuckling now, remembering a number of his quips.

vi

Having Vladi Gerginov in our lab as a guest researcher in the Fall of 2011 was

absolutely a highlight of my graduate experience. He was a true pleasure to work with, and

a motivating presence at a critical juncture in the project.

I want to thank Stephan Barlow, who never hesitated to offer a helping hand. With

his vast experience he is a trove of knowledge especially when it comes to chemistry,

vacuum techniques, mechanical design/engineering, machining, or just perspective on life

as a research scientist.

Neil Ashby has been a much appreciated presence in my time at CU/NIST. Not only

has he coordinated my PREP appointment and organized graduate poster sessions, but he

was a huge help with my Comps II paper on relativistic effects in the GPS system. He has

also done numerous favors such as writing last-minute letters of reference when I was in

need, and not to mention serving on this thesis committee. He is an extraordinary teacher,

and genuinely caring person.

I greatly appreciate the help I received from Terry Brown and Carl Sauer of the JILA

electronics team, Francois-Xavier Esnault, Eric Blanshan, Liz Donley, Ricardo Jimenez-

Martinez, and John Kitching of NIST’s Atomic Devices group, Craig Nelson, Archita Hati, and

Dave Howe of NIST’s Time & Frequency Metrology group, and Doug Gallagher who runs

NIST’s machining facilities.

I have made some wonderful friends through the CU graduate physics department.

We’ve grown a lot from our initial bachelor-hood days of first-year gradschool. I hope that

we can continue to keep in touch wherever life takes us: Ron Pepino, John Gaebler, Ron

Propri, Mike Thorpe, and Ricardo Jimenez-Martinez.

vii

I would never have made it to where I am without the unending love and support of

my family. I’m grateful to be close with my aunts, uncles, and cousins and you all have

played a role in helping me achieve this degree. Dad, I don’t have the words to thank you

enough for everything, from raising me (single-handedly from age 10) to cooking and

caring for Cy these last weeks while I finished writing this thesis, and everything in

between. Mark and Lou are an inspiration and tremendous source of support, who also

enabled the finishing of this thesis by providing precious child care; thank you for that and

so much more. Dave and Janie were god-sends during the cross-country move this summer.

Without all of these favors and generosity, I would never have finished. Please know that

my gratitude goes much deeper than simply appreciating your acts of kindness. You all are

remarkable people and I’m blessed to have you as family.

Aubra and Cy, I love you more than you can ever know. Aubra, thank you for your

patience throughout this long process. Thank you for the motivation and the

encouragement you gave me along the way. Thank you for keeping me sane. And thank you

also for editing this thesis. Cy, thank you for your incomparable hugs, which you always

seemed to offer when I needed them most; you’re an angel.

viii

Contents

1 INTRODUCTION ............................................................................................................................................... 1

1.1 THESIS STRUCTURE ........................................................................................................................................................... 1

1.2 BRIEF HISTORY OF CESIUM FREQUENCY STANDARDS AND FOUNTAINS .................................................................. 5

1.3 COMPARING FREQUENCY STANDARDS ........................................................................................................................... 6

1.4 FREQUENCY TRANSFER .................................................................................................................................................. 10

1.5 A TRANSPORTABLE RUBIDIUM FOUNTAIN ................................................................................................................. 12

2 ATOMIC PHYSICS INTRODUCTION ......................................................................................................... 15

2.1 ALKALI ATOMIC STRUCTURE ......................................................................................................................................... 15

2.1.1 Gross Structure ................................................................................................................................................. 16

2.1.2 Fine Structure ................................................................................................................................................... 18

2.1.3 Hyperfine Interaction .................................................................................................................................... 22

2.1.4 Spectroscopic Terminology ......................................................................................................................... 23

2.2 INTERACTION WITH ELECTROMAGNETIC WAVES ...................................................................................................... 26

2.2.1 Rabi Oscillations .............................................................................................................................................. 29

2.2.2 Density Matrix .................................................................................................................................................. 31

2.2.3 Vector Model ..................................................................................................................................................... 32

2.2.4 Ramsey Interrogation ................................................................................................................................... 34

2.3 ADDITION OF ANGULAR MOMENTA AND WIGNER SYMBOLS................................................................................... 35

2.4 LINE STRENGTHS (RATE EQUATIONS – INCOHERENT INTERACTIONS) ................................................................. 37

3 LASER PACKAGE ........................................................................................................................................... 41

3.1 INTRODUCTION ................................................................................................................................................................ 41

3.2 BACKGROUND ................................................................................................................................................................... 43

3.2.1 NIST F1 & F2 ..................................................................................................................................................... 49

ix

3.2.2 PARCS – A Laser System Designed For Space ....................................................................................... 54

3.3 SEMICONDUCTOR DIODE LASERS ................................................................................................................................. 56

3.4 LASER HOUSING AND BEAM SHAPING ......................................................................................................................... 60

3.5 OPTICAL ISOLATION ........................................................................................................................................................ 64

3.6 LASER SYSTEM CONFIGURATION .................................................................................................................................. 66

3.7 DAVLL MODULE............................................................................................................................................................. 73

4 POLARIZATION ENHANCED ABSORPTION SPECTROSCOPY (POLEAS) FOR LASER

STABILIZATION ....................................................................................................................................................................... 76

4.1 INTRODUCTION ................................................................................................................................................................ 76

4.2 THE SPECTROSCOPY SIGNALS ....................................................................................................................................... 78

4.2.1 Doppler-Free Saturated Absorption Spectroscopy ............................................................................ 79

4.2.2 Polarization Spectroscopy ........................................................................................................................... 82

4.3 THEORY ............................................................................................................................................................................. 83

4.3.1 Classical Model ................................................................................................................................................. 84

4.3.2 Quantum Mechanical Model ....................................................................................................................... 86

4.4 EXPERIMENTAL APPARATUS ......................................................................................................................................... 96

4.5 EXPERIMENTAL RESULTS ............................................................................................................................................... 97

4.6 DISCUSSION ................................................................................................................................................................... 100

4.7 SUMMARY ...................................................................................................................................................................... 104

5 OPTICAL PHASE-LOCK LOOPS .............................................................................................................. 105

5.1 INTRODUCTION ............................................................................................................................................................. 105

5.2 LOOP CHARACTERIZATION ......................................................................................................................................... 107

5.3 LASER TRANSFER FUNCTION ..................................................................................................................................... 114

5.4 LOOP FILTER DESIGN .................................................................................................................................................. 120

6 PHYSICS PACKAGE .................................................................................................................................... 132

6.1 INTRODUCTION ............................................................................................................................................................. 132

x

6.2 THE COOLING REGION ................................................................................................................................................. 133

6.2.1 Glass Cooling Region ................................................................................................................................... 134

6.3 DETECTION REGION ..................................................................................................................................................... 135

6.4 STATE SELECTION ........................................................................................................................................................ 136

6.5 THE RAMSEY MICROWAVE CAVITY ........................................................................................................................... 137

6.5.1 Introduction ................................................................................................................................................... 137

6.5.2 Cavity Geometry ............................................................................................................................................ 139

6.5.3 The Cavity Endcaps ..................................................................................................................................... 145

6.5.4 Exciting The TE011 Cavity Mode ........................................................................................................... 148

6.5.5 Construction Of The Ramsey Cavity ...................................................................................................... 149

6.5.6 Fountain Frequency Errors Associated With Microwave Cavities ............................................ 151

6.6 THE ASSEMBLED PHYSICS PACKAGE ......................................................................................................................... 153

7 CONCLUSION ............................................................................................................................................... 156

7.1 OUTLOOK ....................................................................................................................................................................... 157

8 BIBLIOGRAPHY .......................................................................................................................................... 159

xi

Figures

Figure 1.1. Relationship between accuracy and size of present day frequency standards.

This demonstrates the open niche for a transportable atomic fountain. ................ 9

Figure 2.1. Vector representation of angular momentum coupling. This picture is from

http://en.wikipedia.org/wiki/Azimuthal_quantum_number ................................... 21

Figure 2.2. Energy level diagram of rubidium (not to scale). ........................................................... 25

Figure 2.3. Vector model of Ramsey interogation. P is pseudo-field vector, Ω is the Rabi

frequency, ω is the microwave cavity frequency, τ is the interaction time, T is

free-evolution time, Rz is proportional to the population difference between the

two states. ...................................................................................................................................... 35

Figure 3.1. Illustration of a simplified atomic fountain, showing a laser-cooled cloud of

atoms launched up through the Ramsey microwave cavity and subsequently

detected via fluorescence. Taken from

http://nist.gov/pml/div688/grp50/primary-frequency-standards.cfm ............. 43

Figure 3.2. Typical laser frequency detunings for fountain cooling and launch sequence. .. 45

Figure 3.3. Photo and diagram of commercial solid-state Verdi pump laser. This laser is

used to pump the Ti:Sapph laser shown in Figure 4. From

http://repairfaq.ece.drexel.edu/sam .................................................................................. 49

Figure 3.4. Diagram of Coherent MBR-110 Ti:Sapphire Laser(from the User Manual). The

pump light comes from the Verdi laser, shown in Figure 3. This laser system

works well in a laboratory environment, but is not suited for transportable

applications. .................................................................................................................................. 51

xii

Figure 3.5. Double-pass AOM diagram and photo of modules with custom aluminum

baseplates. These modules allow fine rapid control over the frequency and

amplitude of the light, but they consume much table real estate and electrical

power. Diagram is from [7]. .................................................................................................... 52

Figure 3.6. For comparison, the PARCS laser system designed to operate on the

International Space Station uses diode lasers and double-pass AOMs. My design

greatly reduces the number of optical components. ..................................................... 55

Figure 3.7. Laser housing designed for easy laser replacement, to accommodate the

unreliable DFB laser diodes. ................................................................................................... 61

Figure 3.8. DBR Laser housing design, using modified Fiberport for beam collimation,

increased mechanical stability. .............................................................................................. 62

Figure 3.9. A disassembled Thorlabs Fiberport. After modification, it is used for collimating

the free-space laser beam. ....................................................................................................... 63

Figure 3.10. A sketch showing the laser system concept. The master laser and DAVLL

module, on the left, provide a stable optical frequency reference for the three

slave lasers. The slave lasers perform laser-cooling and launching of the atoms

in the fountain. ............................................................................................................................. 65

Figure 3.11. Diagram of offset-locking scheme. This provides fine rapid control over the

frequency of the light and consumes much less space, electrical and optical

power, and optical components compared to double-pass AOM modules. ......... 66

Figure 3.12. Modular laser system configuration with fiberoptical components is compact,

mechanically stable, and eases system testing and optimization. ........................... 69

xiii

Figure 3.13. The up and down beams are overlapped using a polarizing beam splitter (PBS),

sent through an acousto-optical modulator and mechanical shutter, separated,

and coupled into separate fibers. The wave plate controls the separation ratio.

This module was particularly useful for diagnosing the cause of inadequate

laser cooling. ................................................................................................................................. 71

Figure 3.14. Sketch diagramming the dichroic atomic vapor laser lock (DAVLL) method.

The permanent magnets Zeeman shift the atomic sublevels causing the vapor to

absorb two separate frequencies (hence dichroic) depending on the light’s

polarization. The polarization components are separated by the wave plate and

polarizing beam splitter, and the electrical signals are subtracted resulting in a

dispersion-shaped curve for locking a laser. .................................................................... 73

Figure 3.15. DAVLL module with a Rb vapor cell held in black Delrin with bar magnets that

create a ~120 G field parallel to the beam path. A balanced photodetector is in

the bottom of the module with signal output BNC connector visible. .................... 74

Figure 4.1. Basic Doppler-free saturated absorption spectroscopy configuration. Diagram

from MacAdam et al. [3]. .......................................................................................................... 80

Figure 4.2. Evolution of polarization spectroscopy apparatus. a.) 1976 Wieman and Hansch

[32] b.) 2003 Yoshikawa et al. [34] c.) 2006 Tiwari et al. [35] .................................. 83

Figure 4.3(a) 87Rb D2 level diagram (not to scale) showing population optically pumped by

σ+-polarized light to the 2,2 state. The numbers and dashed arrows show the

allowed transitions and their relative strengths. (b) Predicted spectra of the

POLEAS apparatus. The dashed lines are the individual signals (with Doppler-

backgrounds removed and vertically offset for clarity) that would be detected

xiv

by the two arms of the circular analyzer, and the solid line is the full POLEAS

output signal of 87Rb D2 2F F manifold. Frequency units are natural

linewidths (Γ = 2 π 6 MHz). ..................................................................................................... 94

Figure 4.4. The polarization enhanced absorption spectroscopy apparatus. PBS is polarizing

beam splitter; λ/4 is quarter-wave plate; Pol is polarizer; M is mirror; BPD is

balanced photodetector. ........................................................................................................... 96

Figure 4.5. (a) Theoretically predicted spectra. (b) Experimental data. Dashed line is

Doppler-free saturated absorption spectra (D-F SAS), and solid line is

polarization enhanced absorption spectra (POLEAS) taken under similar

conditions. Units are the same for both figures. ............................................................. 99

Figure 5.1. Schematic diagram of optical phase-lock loop (OPLL). .............................................. 107

Figure 5.2. Common loop filter design with proportional and integral gain. ........................... 109

Figure 5.3. Bode Plots for the loop filter shown in Figure 2. The integrator boosts the gain at

low frequencies where there is a larger phase stability margin. This increases

the hold-in range by compensating for slow thermal and acoustic perturbations

which tend to require a large dynamic range. ............................................................... 111

Figure 5.4. Bode plots of the closed-loop transfer function with loop filter shown in figure 2.

This is a second-order, type 2 loop. ................................................................................... 112

Figure 5.5. Magnitude Bode plot of the error transfer function. ................................................... 113

Figure 5.6. Fiberoptic-based Mach-Zehnder interferometer setup for measuring laser

frequency modulation response.......................................................................................... 116

Figure 5.7. Laser frequency modulation response. Notice the 90 degree phase delay at less

than 1 MHz. .................................................................................................................................. 119

xv

Figure 5.8. Power spectrum of locked beatnote between master and slave lasers using

proportional and integral gain loop filter. This corresponds to a phase error

variance of 2 rad2. ..................................................................................................................... 121

Figure 5.9. A passive implementation of a phase lead compensator. .......................................... 122

Figure 5.10. Bode plots showing the effect of phase lead compensation. The increased

phase margin enables higher loop bandwidths and hence improved phase-

locking between the lasers. ................................................................................................... 123

Figure 5.11. The front and back sides of the final OPLL circuitry. This box includes a fast

photodetector, phase/frequency discriminator, and tunable loop filter. For

scale, the tapped holes in the table are a 1 inch grid. .................................................. 124

Figure 5.12. Fast photodetector subsection of my final OPLL circuit board. Light is incident

on the biased photodiode. The dc signal follows the lower path and is used as a

monitor. The ac signal follows the upper path through two stages of RF

amplification before passing to the phase/frequency discriminator section of

the loop (see Figure 13). ......................................................................................................... 125

Figure 5.13. Phase and frequency discriminator subsection of my final OPLL circuit. An RF

reference signal is input to the lower branch, while the upper branch is the beat

signal from the photodetector. The two signals are compared by the AD9901

chip, which outputs a differential error signal. ............................................................. 127

Figure 5.14. Loop filter subsection of my final OPLL circuit. Here the error signal is tuned

with adjustable proportional and integral gain and phase-lead compensation,

ultimately outputting a correction signal that is fed-back to the laser’s current.

.......................................................................................................................................................... 129

xvi

Figure 5.15. Power spectra of locked beat note using final circuit with lead filter. Phase

coherence improved, with residual phase error down to 0.6 rad2. ....................... 130

Figure 6.1. Stainless steel spherical cube used for the cooling region. Picture from

www.kimballphysics.com. ..................................................................................................... 133

Figure 6.2. Glass cooling chamber and rubidium oven. .................................................................... 134

Figure 6.3. Detection chamber with large welded windows and ports for vacuum pumping.

.......................................................................................................................................................... 136

Figure 6.4. State selection cavity integrated into the Conflat flange above the detection

region. ............................................................................................................................................ 137

Figure 6.5. Cylindrical cavity, showing notation. ................................................................................. 139

Figure 6.6. Solutions for cylindrical cavity dimensions yielding rubidium hyperfine

frequency. ..................................................................................................................................... 144

Figure 6.7. Quality factor of TE011 mode as a function of cavity geometry, a is radius and d

is height. ........................................................................................................................................ 144

Figure 6.8. Diagram of cavity vertically bisected, not to scale. ....................................................... 148

Figure 6.9. Two endcaps and sidewall with ........................................................................................... 150

Figure 6.10. Assembled Cavity and sectored diagram. ...................................................................... 150

Figure 6.11. Physics package design with glass cooling chamber. ............................................... 153

Figure 6.12. Assembled physics package with stainless steel spherical cube cooling

chamber. ....................................................................................................................................... 155

Figure 7.1. Time-of-flight fluorescence signal from launched sub-Doppler cooled atoms. 156

xvii

1

1 Introduction

1.1 Thesis Structure

The goal of this thesis project was to build a transportable laser-cooled Rubidium

fountain frequency standard. While many commercially available frequency standards are

portable, they do not possess the accuracy or potential stability inherent in a laboratory

grade laser-cooled atomic fountain. On the other end of the spectrum, optical clocks are

becoming technically mature, but they still typically require quite a large amount of high

quality laboratory real estate for the ultra-stable narrow linewidth laser systems used to

probe the optical clock transitions.

Several decades ago, this situation was similar for the laser-cooled Cesium fountain

microwave clocks. Since then, there has been much technological progress, particularly in

semiconductor laser development, laser stabilization, compact direct-digital synthesis

(DDS) radio-frequency (RF) sources, and embedded controllers. It is now feasible to take a

system such as NIST-F1 (the US primary frequency standard), which occupies a sizeable lab

space with several optical tables and several racks of electronics, and shrink it down into a

high-performance system occupying the space of only a single rack with a small-sized

physics package. In fact the PHARAO clock is already in operation [1]. While this is

transportable, it might be considered more of a mobile laboratory.

2

The layout of this thesis is as follows. First, in Chapter One, the introduction, I

continue the motivation, initial guiding decisions, and context for this project. I will discuss

the niche into which such a device fits as compared to existing atomic frequency standards.

A very brief history of Cesium atomic clocks will explain how laser-cooling

prompted the development a laser-cooled atomic fountains and why their performance so

significantly exceeds that of thermal beam devices. Cesium holds special status in the

world of time and frequency due to the definition of the SI second. I discuss the decision to

use rubidium instead of cesium for this project.

The stability of frequency standards is given in terms of an Allan variation, and I

introduce the concept and definition to be explicit. From this, the commonly quoted

fundamental fountain stability equation is rapidly derived and can be used to predict the

device stability. Comparing the stability and transportability of various frequency

standards reveals an opening for a transportable atomic fountain. With its predicted

stability and size a transportable atomic fountain could satisfy demands that currently

must rely on active hydrogen masers.

Though research on next generation optical frequency standards is developing

rapidly, frequency-transfer technology is lagging, and comparing state-of-the-art optical

standards therefore presents significant challenges. The accelerating demand for improved

frequency-transfer offers another avenue of possible applications for a transportable

fountain.

With the motivation established, I then outline the original design goals of the

project. The strategy to make a transportable device focused on two major subsystems: the

3

laser-system package and the physics package. By building a laser and optics package

based around phase-locking semiconductor laser diodes using RF beat-note techniques, we

aimed to eliminate much of the optical table real estate used by double-pass acousto-

optical modulator (AOM) systems as well as the power-hungry RF systems needed to drive

the AOMs. The availability of modern DDSs and embedded controllers make it possible to

generate all the RF frequencies and sweeps (frequency and amplitude) required to cool,

launch, and post-cool the atoms using small electronic systems.

The fountain physics package can be made substantially smaller than a laboratory

device such as NIST-F1 by having a much shorter toss height. In a fountain, the atoms

spend most of the Ramsey time at apogee, so relatively long interrogation times are

possible in a short fountain.

The remainder of the thesis describes the work undertaken towards building such a

laser-cooled atomic fountain system. Chapter Two reviews some of the relevant atomic

physics. This will serve as a starting point for an incoming student, presenting key concepts

and notations that can be further pursued in the mountains of literature. It also serves as

background for the modeling of the polarization-enhanced absorption spectroscopy

(POLEAS).

Chapter Three describes the lasers upon which my system was built. I begin by

comparison with previous laser systems, pointing out the advantages of the strategies I

decided on. The different lasers that I used and the evolution of the laser housing design

are addressed. The laser diodes played a key role in this project. While the newly available

internal grating structure lasers at 780 nm initially appeared perfect for my application, the

4

lifetime of these lasers proved unreliable. An alternative laser emerged on the market

promising much longer lifetimes; however, the insufficient phase-coherence between the

locked lasers proved detrimental to sub-Doppler laser cooling. I will show that ultimately,

the transportable fountain will only be realizable with the use of a more stable, phase-

coherent laser system.

Chapter Four presents the novel polarization-enhanced absorption spectroscopy

method that I created and used to frequency lock my master laser. It is similar to other

spectroscopy-based laser stabilization schemes, such as saturated-absorption spectroscopy

and polarization spectroscopy, with the advantage that it provides much improved signal-

to-noise ratio for the relevant cycling-transition peak.

Chapter Five covers the development of the optical phase-locked loops (OPLLs)

used to control the slave lasers. This was a key design concept to avoid using multiple

double-pass AOM modules, which seemed infeasible for the transportable specifications I

was aiming for.

Chapter Six covers the physics package. The design and construction of the Ramsey

microwave cavity, the detection chamber, and the cooling chamber is discussed.

Finally, I provide a concluding chapter to summarize the lessons learned over the

course of my research, and to suggest future avenues to pursue in the quest to construct a

transportable atomic fountain.

5

1.2 Brief History of Cesium Frequency Standards and Fountains

In the broadest strokes, the history of the atomic fountain can be simplified down to

two critical steps, both of which resulted in Nobel Prizes. The first step was Norman

Ramsey’s seminal 1950 paper on his “method of separated oscillating fields” [2]. This

technique resulted in an immediate, factor-of-20 improvement over the previous state-of-

the-art method developed by Ramsey’s graduate advisor, Isidor Rabi. They were

investigating interactions between atoms and newly developed coherent sources of

electromagnetic radiation. Rabi’s magnetic resonance machines sent a beam of atoms (or

molecules) through an RF interaction region that induced atomic transitions. The linewidth

of the magnetic resonances is inversely proportional to the duration time of the interaction.

Attempting to increase the length of the single interaction to further decrease the

resonance widths failed due to inhomogeneities in the C-fields and RF phase. Ramsey’s

solution was to use two separate interaction zones separated by an arbitrarily long

distance (limited mainly by how well the experimenter can thread the particles through the

two cavities). A number of additional auxillary benefits come with Ramsey’s method as

well, including greatly reduced sensitivity to field inhomogeneities that might be present in

the interaction region.

In an attempt to further reduce the linewidths in atomic beam devices, Zacharias

conceived of the atomic fountain. His idea was to build a vertical thermal beam device with

a single interaction zone. Atoms from the oven on the slow end of the Boltzman

distribution should pass through the cavity on the way up, reach apogee, and return under

gravity through the cavity a second time thus achieving very long Ramsey times of

6

approximately one second. Ultimately, he was not able to get the experiment to work,

presumably because collisions from hotter atoms prevented even the cold atoms from

returning.

The second critical step towards atomic fountains was the development of laser

cooling, specifically the optical molasses. Chu and colleagues used sodium cooled to the

milli-Kelvin level to demonstrate a fountain with a 2Hz linewidth. Subsequently, work by

Chu and Clairon helped demonstrate the cesium fountain. As with Ramsey’s revolutionary

technique, laser cooled atomic fountains benefited from many auxiliary advantages aside

from just long interrogation times. One such advantage is the dramatic technological

simplification of using a single resonant cavity, which alleviates the challenge of

maintaining perfect phase-coherence between the two separated cavities used in a beam

clock. Atomic fountains, until recently surpassed by optical clocks, held the record for

measurements with the highest precision, reporting uncertainties of just a few parts in

.

1.3 Comparing Frequency Standards

Frequency stabilities are typically characterized in terms of an Allan deviation,

which is based on frequency differences between subsequent measurements rather than

frequency differences from the mean value, as with the Standard deviation. It is mostly

used for characterizations over long times, i.e. seconds to years (for short-term

characterizations, less than 1 second, phase noise is more appropriate). As expected, the

Allan deviation is the square root of the Allan variance, also known as a two-sample

variance, which is defined as,

7

22

1

2

2 12

1

2

12

2

y n n

n n n

y y

x x x

(0.1)

where is the measurement time, is the average of the nth fractional frequency over

time , is the phase departure (in units of time), and ⟨ ⟩ denotes time average. This

definition assumes that there is no dead-time between measurements; if there is then the

values of the instability will be biased. These biases have been studied in great detail, and

there are a number of modified Allan variances for dealing with particular cases. The

standard Allan variance is basically a running average of two subsequent fractional

frequency measurements. One of the most powerful features of using the Allan deviation is

its relation to power law spectral density types. It is well known that different noise

processes result in spectral densities that scale as different power laws. These various

categories of noise become apparent on the log-log plots of Allan deviation. Noting the

flavor of the noise in this way can be a useful diagnostic.

The Allan deviation can be written using more familiar terms, yielding an equation

that is widely used for atomic fountains. Given a resonance peak with noise, to obtain the

fractional frequency stability one can use the slope of the sides of the peak to translate the

signal uncertainty, , into the frequency uncertainty, ,

0

0

0

1

1 1

1 1

Δ

y

I

I

max

slope

IK

(0.2)

8

The accounts for the resonance line shape, if it were a triangle then . In the case of

an atomic fountain the derivative midway up the Ramsey resonance gives .

Rearranging equation (0.2) gives,

0

1 Δ

1 1 1

I

y

maxK I

SK QN

(0.3)

where we now see the familiar terms of signal-to-noise ratio ⁄ , and the of the

resonance. Fundamentally an atomic fountain will be limited by the quantum projection

noise, or shot noise of the atoms, which scales as the square-root of the atom number, √ ,

and averages down like √

⁄ . For a fountain one must also note the measurement time, ,

is not continuous, rather there is dead-time in every cycle when the atoms are not under

Ramsey interrogation. Therefore the √

⁄ term becomes √ ⁄ , where is the total cycle

time, . This gives the widely used equation for predicting the stability of a

given atomic fountain,

0

1 Δ 1y

T

N

(0.4)

Comparing existing frequency standards on a plot of device size versus frequency

stability (see Figure 1.1), there exists an open niche between an active Hydrogen maser

(which is commercially available) and an atomic fountain (found only in the form of

laboratory experiment).

9

Active hydrogen masers offer excellent short to medium term frequency stability

(seconds to weeks), but suffer from large (parts in 10-14) drifts over longer times. More

specifically such masers (which weigh ~250 kg, and consume ~100 W of power) have a

short-term instability of 10-13 at 1s and hit their flicker floor of 10-15 in a day, but over

longer times drift at around a few parts in 10-14 / year. Commercial cesium beam clocks

(which weigh ~25kg, and consume ~50W of power) have a fractional instability

somewhere between 2×10-12 and 5×10-13, and hit their flicker floor of ~1×10-14 at about 10

Figure 1.1. Relationship between accuracy and size of present day frequency standards. This demonstrates the open niche for a transportable atomic fountain.

10

days. Cesium based devices hold a special place among frequency references thanks to the

definition of the second made in 1967,

the second is 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 caesium133 atom.

They are therefore known as primary frequency standards, and can establish a level of

accuracy. Commercial standards achieve accuracies of +/- 5×10-13. If higher levels of

accuracy are desired the next step would be to build a cesium atomic fountain. Numerous

cesium fountains have been developed around the world and the best ones can achieve

stabilities of a few parts in 10-16. These atomic fountains take up ~250 cubic meters of lab

space, and ~100kW of power. The factor of 1000 difference in accuracy between

commercial cesium beam standards and laboratory atomic fountains leaves open a sizeable

niche indeed.

1.4 Frequency Transfer

The catch in building a precision frequency standard is that in order to measure its

stability, a second frequency reference is needed which is at least as stable as the first.

Presently the achievements of laboratory frequency standards themselves are outpacing

the capabilities of long distance frequency comparison. Optical clocks are now the most

precise frequency references, achieving uncertainties below 10-16 [3]. The problem is that

there is presently no way to make a comparison at those levels over distances beyond

hundreds of kilometers (and even that is extraordinarily challenging). The only method for

such a comparison within those distances, is to connect the two oscillators via an optical

11

fiber network. This in itself is an active field of research, and the best performance to date

is a 920 km fiber link with fractional frequency instability of 5×10-15 at 1 s of averaging, and

10-18 in ~1000 s, and reaching 4×10-19 at longer times [4]. While this performance is

extremely impressive its cost and complexity prevent it from being a practical real-world

tool in the near future. Furthermore it is not clear that such a network could be extended to

longer distances. This is unfortunate for the numerous laboratories around the world

working on next generation optical “clocks” , as there is no way for them to make direct

comparisons with each other and verify results.

State-of-the-art frequency transfer over longer distances is carried out via satellite

communication. There are a number of ways that this is done, but the best are known as

common view (CV) with carrier phase (CP), and two-way satellite time and frequency

transfer (TWSTFT). The common view method typically uses Global Positioning Satellites

(GPS), while TWSTFT requires purchasing time on an appropriate communications

satellite. In the GPS common view scenario the two remote, ground-based oscillators both

receive simultaneously sent signals from the GPS satellite. By comparing to this common

reference the two ground oscillators can then compare to each other. The short term

stability of the GPS link can be improved by using the phase of the GPS satellite’s carrier

frequency (which is a much higher frequency than the modulated frequency of the normal

signals). This carrier phase method requires much data processing and can be prone to

errors, but for those willing to undertake it the advantage is possible.

Using the TWSTFT method is simpler, although the equipment is more expensive. In

this scenario the two ground oscillators repeatedly send signals back and forth through a

12

geostationary orbiting satellite. Since the two oscillators are sending and receiving signals

simultaneously the path delay of the communication effectively drops out. These two

methods, GPS carrier phase and TWSTFT, have achieved roughly the same levels of

frequency stability (2×10-16 at 30 days, [5]) and they in fact can be used in a

complementary fashion to verify each other’s performance. In addition to clearly and

concisely covering these issues, a review article by Parker [5] makes the point that the

three pillars necessary for global precise timing are currently in balance, equally

contributing to an ultimate uncertainty at the level of a few parts in 10-16. Rapid progress is

being made on one of the pillars (laboratory oscillators, i.e. optical clocks), but the other

two pillars (flywheel oscillators and frequency transfer) do not have a clear path forward.

1.5 A Transportable Rubidium Fountain

With the field of precise timing and frequency metrology as it is at present, we

proposed to build a prototype transportable rubidium fountain. While such a device would

not be a silver bullet for rectifying the state of flywheel oscillators or frequency transfer, it

could make meaningful contributions. Being transportable, it could be an attractive

alternative as a transfer standard. The level of robustness and simplicity necessary for a

transportable standard would also make it useful as flywheel, able to operate

autonomously for long times.

One simple fact that makes a transportable atomic fountain enticing is that as the

atoms are experiencing Ramsey interrogation they spend the majority of their time near

apogee – that is, coming to a stop and then beginning to fall back down. The interrogation

13

times scale as the square root of the height. This means that if a 1 m tall fountain yields a 1

Hz linewidth, then with only a 25 cm tall fountain the linewidth would be 2 Hz.

Our group’s decision to build a rubidium transportable fountain instead of a cesium

one brings one significant advantage. The frequency shift of the clock transition due to Rb –

Rb collisions is roughly 50 times smaller than that with cesium [6]. This means that the

fountain can operate with more atoms in each toss, thereby increasing the signal-to-noise

(and hence the stability), without suffering from the systematic uncertainty due to spin-

exchange collision shift. On the other hand, as previously mentioned, cesium references

hold a special place thanks to the current definition of the second. A rubidium fountain

could still provide accuracy far superior to a commercial beam standard since the relative

frequencies of cesium and rubidium are well understood.

We considered ‘transportable’ to mean a device that could fit into a truck or van,

such that it could be driven to another location, powered up, and with minimal setup time

begin running. Within a physics package of roughly a cubic meter, basic ballistic equations

show that Ramsey times of ~0.4 s are easily achievable (leaving plenty of room for cooling

chamber and microwave cavities).This translates into linewidths of ~2.5 Hz. Assuming

cooling-laser beam radii of 1 cm, and optical power of 10 mW per beam, we would expect

to load an optical molasses with ~108 atoms in ~200 ms. This time to load the molasses is

“dead time” since Ramsey interference cannot take place while the lasers are on. State

selection, which refers to the process of initializing the cloud of cold atoms into the desired

clock state (a magnetically insensitive Zeeman sublevel within one of the hyperfine ground

states), is usually accomplished through a combination of microwave cavity (separate from

14

the Ramsey cavity) and laser beams. The most straight forward scenario assumes that the

cold atoms begin evenly distributed among the Zeeman (magnetic) sublevels, the state

selection cavity transfers a subset of the atoms into the desired clock-state, and then the

laser blasts away all the other uninitialized atoms. A conservative estimate assumes the

fraction of initialized atoms is equal to inverse of the number of sublevels. For the present

case of rubidium, since there are five sublevels, assuming 1/5 of the original atoms in the

cloud become initialized signal atoms is a safe estimate. With relatively large cavity

apertures (diameters ~2 cm) afforded by the TE011 mode, and sub-Doppler atom

temperatures 10-6 K or ~1.5 cm/s (i.e., a few photon recoils = 0.58 cm/s) essentially all of

the atoms will return to the detection region. Plugging the numbers into equation (0.4), a

transportable rubidium fountain that would take up roughly a couple cubic meter, could

expect a short-term instability of 10-13 to 10-14, averaging down into the 10-16’s. This level of

frequency stability surpasses anything presently available in the market and would set the

state-of-the-art for a transportable frequency standard.

15

2 Atomic Physics Introduction

In this chapter, I present an overview of some of the fundamental atomic physics

relevant to this thesis project. I will start by briefly reviewing the atomic structure of

rubidium. I will describe electromagnetic transitions between atomic states, particularly

magnetic and electric dipole transitions. This will explain Rabi oscillations, which form the

foundation of the Ramsey interference signal on which atomic fountains are based.

Ensembles of many atoms are most easily described using the density matrix formalism,

which facilitates the incorporation of relaxation or dephasing processes. Using the density

matrix formalism, I will describe the Bloch vector or “fictious spin” model, which provides a

simple visual picture for understanding Ramsey interference in atomic fountains. I will

also introduce tools and notations (such as Wigner symbols) that are not usually covered in

standard physics classes, but are common in atomic physics publications, and are used in

this thesis to model the polarization-enhanced absorption spectroscopy signal.

2.1 Alkali Atomic Structure

To develop a picture of the rubidium atom, I will quickly build from the gross

structure of single electron atoms to multi-electron alkali atoms, and then add the fine and

hyperfine corrections. This is the level at which atomic fountains operate; using the

hyperfine splitting of the ground state as the clock transition and cycling excitations of

hyperfine levels for laser cooling and optical fluorescence detection. Alkali atoms have long

received special attention within the atomic physics community. They have a simple

hydrogen-like structure with a single valence electron and filled lower energy shells. This

16

means that most interactions involving alkali atoms concern just the one outermost

valence electron. Cesium was the first element discovered using the then-novel technique

of spectroscopy developed by Bunsen and Kirchoff in 1860 [1]. Its long scientific history is

what led to its status in defining the SI second.

2.1.1 Gross Structure

For hydrogenic atoms, consisting of a nucleus and one electron, the wavefunction,

( )r , in Schrödinger’s time-independent equation can be solved for exactly. Schrödinger’s

equation in this case is

2 2

2

0

( )2 4

Zer E r

m r (0.5)

Where m is the electron mass, 2 is the Laplacian operator, Z is the number of

protons, e is the charge of an electron, 0

is the permittivity of free space, and r is the

distance from the nucleus to the electron. Eqn 1.1 is most naturally solved for in spherical

polar coordinates, where the variables can be separated and the solution written as a

product of separate radial (Laguerre polynomials, ( )nL

R r ) and angular components

(spherical harmonics, ( , )Lm

Y )

, , ,L LnLm nL Lm L

r R r Y nLm (0.6)

The eigenvalue energies, n

E , are inversely proportional to principle quantum

number squared, 2n , and independent of the angular momentum quantum number, L (here

Lm , called the magnetic quantum number, is the projection L onto the quantization axis,

17

usually taken as z , such that Lz

L m ). These energy eigenvalues agree exactly with the

results obtained from Bohr’s model of the atom, but they ignore higher order corrections

due to relativistic effects (we will get to some these shortly, including fine and hyperfine

structure). The degeneracy with regards to L is lifted once we extend the analysis to multi-

electron atoms where the Coulomb potential deviates from pure 1 / r behavior.

The notation used here for quantum numbers and operators is fairly standard;

often, characters that represent a single particle (an electron for instance) are lowercase,

and uppercase is reserved for the total or summation over multiple particles. For example,

a single electron’s angular momentum quantum number is often written as l, and the total

orbital angular momentum for a multi-electron atom as L. The principle quantum number,

n, is always lowercase as it just refers to the outermost electron. Since I only provide an

undetailed account of alkali atoms in this thesis (using hydrogenic atoms as a brief stepping

stone), I will not distinguish between the two cases; multi-electron alkali atoms can be

characterized by their single valence electron. As such, I have chosen to use uppercase

characters for the angular momentum quantum numbers, and use the common lowercase

m designation for the projection of those operators – also called the magnetic quantum

numbers.

The time-independent Schrödinger equation for multi-electron atoms can

only be solved using approximation methods. While the Hartree-Fock method provides an

elegant approach to tackling this problem, the calculations can be lengthy. This method

relies on the “central field approximation”, which assumes that each electron moves in a

spherically symmetric potential This spherically symmetric potential is due to the

18

combination of the nuclear attraction and the partial screening by the mean field of all the

other electrons. Hartree showed how to obtain all the individual electron wavefunctions

using this intuitive approximation, and also created a “self-consistent field” method for

iteratively solving the system of equations. Fortunately, there is a far simpler approach that

works particularly well for alkali atoms, and is based on empirical formulas. Replacing the

Coulomb potential with a proportionally reduced, effective Coulomb potential *

Z Z , the

Hamiltonian now reads

2 * 2

2

02 4

C

Z eH

m r (0.7)

This results in energy levels that depend on an effective principle quantum number,

*

Ln n . The quantities

L are known as quantum defects, and depend on the orbital

quantum number L. At large distances (high n ) this “defect” approaches zero, and the

energy levels become essentially degenerate in L again.

This sums up the gross energy level structure of rubidium, which only depends on

the radial part of the wave function. It differs from hydrogen in that the shielding from the

closed inner subshells lifts the degeneracy of the orbital angular momentum.

2.1.2 Fine Structure

At this point we can no longer ignore the particles’ inherent quantum spin. Taking a

step back to single electron hydrogenic atoms, the electron’s spin can easily be accounted

for by tacking on a spin eigenfunction, or spinor, .

L S L SnLm m nLm m L S

q r nLm m (0.8)

19

This is because the Coulomb interaction is independent of spin, and the Hamiltonian

is separable. The energy levels are unaffected, and just their degeneracy increases by a

factor of two. That ceases to be true if the atom is placed in a magnetic field. Each of these

angular momenta, orbital and spin, have an associated magnetic moment

L L

Bg L (0.9)

S S

Bg S (0.10)

Here g is a Lande g-factor, and the Bohr magneton is / 2B

e m . These magnetic

moments interact with a magnetic field, B , and there is an associated potential energy of

the form E B . Atoms create their own magnetic field due to the orbiting motion of the

charged particles; in the frame of the electron, the nucleus orbits around it, creating a

magnetic field that couples to the electron’s spin magnetic moment. This phenomenon is

known as the spin-orbit coupling, or Fine Structure splitting. There are other effects

incorporated into the Fine Structure, such as relativistic corrections to the electron’s

kinetic energy, but these only shift the energy levels and do not split them (i.e. lift the

degeneracy), and I will not address them here. For most of the periodic table the spin-orbit

coupling is a weak interaction relative to the Coulombic interaction, and this is referred to

as the Russell-Saunders or L-S coupling regime. For heavy atoms with large Z this is not

necessarily the case, and the so-called j-j coupling approach is more appropriate. For our

purposes L-S coupling is sufficiently accurate.

The spin-orbit Hamiltonian is

20

( )so

H r L S (0.11)

Where ( )r is the spin-orbit coupling constant. The energy splittings can be

calculated using perturbation theory. This calculation is made much simpler if the

Hamiltonian matrix is diagonal in the initial zero-order basis states. The eigenstates

described above, for the Coulomb potential, do not satisfy this requirement. The above

states, designated by the quantum numbers , , ,L S

n L m m , are simultaneous eigenfunctions of

the operators 2 2, , , ,

c z zH L S L S since all of these operators commute with each other. In other

words, each of those quantum numbers can be measured simultaneously without

obfuscation from the uncertainty principle. This is no longer the case when including the

spin-orbit Hamiltonian; so

H does not commute with z

L orz

S , so L

m and S

m are no longer

“good” quantum numbers, meaning their values are not stationary. Instead, we use the

electron’s total angular momentum J L S , which allows the creation of new wave

functions that are simultaneous eigenstates of the alternate set of operators, 2 2 2, , , ,

c zH L S J J

. The spin-orbit Hamiltonian does commute with 2 J and

zJ , and the new wave functions

therefore constitute a basis set in which the spin-orbit matrix is diagonal.

If we measured the magnitude of a quantum angular momentum vector, say

orbital angular momentum L , we would find the expectation value or eigenvalue of its

magnitude operator, 2L . Since the eigenvalue for 2

L is 1L L , we would draw a vector of

length 1L L . If we measure the projection of L onto the quantization axis to findz

L , the

maximum value that we would get is L m ax

m L . In order for these measurements to be

21

consistent with each other, this means that the vector must not lie along the z axis, rather it

is at some angle with respect to z . If we stopped there, with the vector at the requisite

angle to z , this would imply that it had some definite projection onto x and y as well, but

the uncertainty relations forbid this. Any single measurement of x

L or y

L could have some

value, but on average it must be zero. In other words, it cannot be a stationary state in x

L or

yL if it is a stationary state in

zL

. Hence, we arrive at the cone or

precession interpretation.

Figure 2.1 is an illustration of L-

S coupling, where we see that

L S J . In this case, J and J

m

have stationary values, so that if

the system started in the state

JLSJm it would remain in that

state and we would repeatedly

obtain the same values upon

measurement of those quantum

numbers. On the other hand, if

we tried to measure L

m or S

m we would not get a stationary value, even if the system was

originally prepared in a L S

LSm m state.

Figure 2.1. Vector representation of angular momentum coupling. This picture is from http://en.wikipedia.org/wiki/Azimuthal_quantum_number

22

To keep things in perspective before moving on to the next level of detail, the

fine structure corrections are a factor of 2 smaller than the Coulomb interactions, where

is the fine structure constant

2

0

1

4 137

e

c . (0.12)

The interactions resulting in the Hyperfine structure are more than another factor of

α smaller than the fine structure.

2.1.3 Hyperfine Interaction

The protons and neutrons that make up a nucleus also have intrinsic spin,

and they combine to produce a total nuclear spin, I . It too has an associated magnetic

moment,

N

I Ig I , (0.13)

Where 2N p

e m is the nuclear magneton, which is inversely proportional to the

mass of the proton (rather than the electron). This means that the nuclear magneton is

roughly two thousand times smaller than the Bohr magneton. In a parallel fashion to the

fine structure spin-orbit coupling, the hyperfine structure relies on the interaction between

the nuclear and electron magnetic moments, and is sometimes referred to as spin-spin

coupling. Within the context of the full hyperfine correction there are a number of effects.

These include isotope shifts, which account for the finite mass and volume of the nucleus,

and that shift (but do not split) the energy levels. Furthermore, the nucleus can have higher

23

multipole moments, and these also interact with the electrons’ electromagnetic field. In

particular, the electric quadrupole moment can be just as important as the magnetic dipole

moment in certain contexts. For this thesis, however, I will focus on the magnetic dipole

interaction. The Hamiltonian for this interaction can be written

H F

H J I , (0.14)

where β is the magnetic hyperfine structure constant. As with the fine spin-orbit

interaction, it is convenient to combine the nuclear and electron angular momentum into a

total angular momentum, F J I . Forming new states of these good quantum numbers

FLSJFm simplifies the diagonalization of the hyperfine Hamiltonian. These new states

can be written as a linear combination of the old states, J I

LSJm Im ; I will discuss this

shortly (see “Addition of Angular Momenta” section).

2.1.4 Spectroscopic Terminology

There is short hand and specific terminology commonly used in the lab and

the literature when discussing atomic structure and interactions. The Coulomb interaction

determines the electron configuration, which lists each occupied shell (n), subshell (l), and

the number of electrons (N) in each subshell in the form Nnl . They are listed in order from

lowest energy (innermost) to highest energy (outermost). For historical reasons the orbital

angular momentum subshells go by the letter code of s,p,d,f,g,h,.... for l=0,1,2,3,4,5,... It is also

common to use a shorthand notation for filled subshells by writing the equivalent noble gas

atom. Following this notation, the configuration for rubidium is

24

2 2 6 2 6 2 10 6

1 2 2 3 3 4 3 4 5 5s s p s p s d p s Kr s (0.15)

The total spin (S) and total orbital (L) angular momentum quantum numbers are conveyed

in the term, written as 2 1SL , where again the same code letters are used for the L quantum

number (but capitalized now, as it’s the total). The term for rubidium is

2S . (0.16)

Writing the spin superscript as 2 1S instead of just S might seem odd, but this

form tells us the multiplicity. The multiplicity is meant to represent the number of fine

structure states within the term, as these would typically be resolvable even with dated

spectroscopy methods. For example, in the early 1800s Fraunhoffer observed the famous

sodium “doublet” ( 2P term) as two distinct lines. Unfortunately this connotation of

resolvability only holds when L S , which is not the case for alkali ground states (L=0 and

S=1/2 ). Despite being a “doublet” the rubidium ground state consists of a single J value.

The fine structure, which depends on the J quantum number in the L-S coupling regime, is

explicitly revealed by the so called level. This is written most often in Russell-Saunders

notation as 2 1S

JL , which is just the term with J represented in the subscript. The ground

state rubidium level is

12

2S . (0.17)

25

Despite the somewhat confusing caveat regarding multiplicities mentioned above,

the terminology is still commonly used, with the rubidium ground state referred to as

“doublet S one-half”. Any further designation (e.g., hyperfine state), is usually just written

with the appropriate quantum number, often in the ket notation (e.g. F

Fm ). This atomic

structure and terminology is summed up, for the case of 87Rb , in Figure 2.2.

Figure 2.2. Energy level diagram of rubidium (not to scale).

26

2.2 Interaction with Electromagnetic Waves

This brief review of atomic interactions with electromagnetic fields is carried out in

the semi-classical regime, where the atoms are treated quantum mechanically, and the

fields classically. I will start with the standard two-level atom described by the time-

dependent Schrödinger equation, which will reveal the signature Rabi flopping

phenomenon. I will then recast that same phenomenon in the fictitious spin or Bloch vector

context using the density matrix formalism.

We will be looking at transitions caused by oscillating EM fields, and using the time-

dependent Schrödinger equation

Ht

i . (0.18)

In our case we can split the Hamiltonian into a time-independent part, 0H r , and a

perturbative (small) time-dependent part, int,H r t . The wave function solutions for the

0H r part evolve simply in time as

ni E t

n nu r e , and we already know these

solutions. Writing the state vectors this way, where they evolve in time, constitutes going to

an “interaction representation”. The full solution to equation (0.18) can then be written as

a linear combination of these states with time dependent weighting coefficients, nc t ,

which contain the dynamics due to the perturbation:

n

n n

i E t

n

c t u e . (0.19)

27

As we are only considering two levels, inserting equation (0.19) into (0.18) yields

the following two coupled equations for the interaction amplitude coefficients

0

2 int11 2

i tdi c t c t H e

dt (0.20)

0

2 int12 1

i tdi c t c t H e

dt , (0.21)

where 1

1 u r , and 0 1 2

E E . The probability of finding the system in

state 1 or 2 is given by 2

1c and

2

2c respectively (assuming they have been normalized

as usual).

To go further we must specify the interaction Hamiltonian. The electric and

magnetic components of the radiation wave can be simultaneously characterized by a

single vector potential, usually written as ,A r t . In this way a monochromatic (coherent)

field would be written

0

0

ˆcos

1ˆ

,

2

i k r t i k r t

kA r t A t

eA

r

e

(0.22)

In our case the wavelength of the electromagnetic radiation is much larger than the

size of the atom, so the radiation field can be expanded into multipoles around the atom’s

center-of-mass.

211 ...

2!

ik re ik r ik r (0.23)

28

In this Taylor expansion of the fields, the atom’s interaction strength with each

successive multipole moment decreases by a factor of kr, which for optical frequencies is

roughly 1/1000. The zeroth order approximation, unity, is the electric dipole (E1)

approximation. The first order term includes the magnetic dipole (M1) and electric

quadrupole (E2) terms. Although electric dipole approximation is the most common, there

are cases where its interaction strength vanishes, and this is the case in the fountain

Ramsey microwave cavity interaction. These conditions are summed up by the selection

rules for each multipole interaction, which are based on geometrical considerations of the

polarizations of the fields and the atom’s quantization axis; they are derived from the

Clebsch-Gordon coefficients or Wigner symbols.

It is worth noting that generally magnetic interactions become negligible for

frequencies well below the optical regime. Naively, I picture this in the Bohr model as the

result of a lack of well-defined magnetic moment until the electron has had time to

establish a “current loop” by “integrating” over a few orbits around the nucleus. One can

also perform a brief calculation comparing the electric and magnetic forces on a moving

electron:

1

100E

B

F Eq c

F qvB vv, (0.24)

where the velocity of the electron has been estimated using the Bohr model formula

2

04

ev e m r . From equation (0.23) the interaction Hamiltonian can be

correspondingly expanded

29

2 2

int 0

112

10, ..0 , , .

2r

ME

E

B t d r E r td t rkH E . (0.25)

In this thesis we will only have need for the first two terms of this expansion, the

electric and magnetic dipole interactions.

2.2.1 Rabi Oscillations

Using the electric dipole (E1) approximation, the system of equations (0.20)

and (0.21) can be straightforwardly solved. This calculation is done in many texts and will

not be repeated in detail here. I will mention that the rotating wave approximation (RWA)

is always invoked. This is an extremely common approximation used when the radiation

frequency is close to the atomic resonance frequency. It involves discarding terms

oscillating at the sum of these two frequencies in favor of the lower beat frequency (equal

to the difference). It becomes quantitatively clear by going into an interaction picture. Note

that there are two commonly used rotating frame interaction pictures: one rotates at the

resonance frequency of the atom, and the other rotates at the driving frequency of the

radiation field. Often the latter is used when there is a single radiation field

(monochromatic) frequency driving the system, while the former (the atomic resonance) is

used if there is more than one field frequency.

The solution to this coherently driven (monochromatic) two state system reveals

that the probability amplitudes of the two states oscillate sinusoidally in time. The

frequency of this probability-oscillation is known as the Rabi frequency,0

, and it is often

used to characterize the strength of the interaction (typical values are in the 1-10 MHz

range),

30

0 21 0

0

2 1d E E. (0.26)

In this equation 21

ˆ2 1er is the dipole matrix element, and the

proportionality of the Rabi frequency to the field amplitude is shown explicitly. There is

also a slightly generalized Rabi frequency that takes into account a detuning ( ) of the

driving field frequency ( ) from the atomic resonant frequency, 0

, such that

2 2

0. (0.27)

In terms of these Rabi frequencies, and making the RWA, the Schrödinger equation

becomes

1 10

2 202

c t c td

c t c tdi

t . (0.28)

Finally, the solution for the probability of being in state 2 as a function of time is

2

2 20

2 2sin

2P c t . (0.29)

On resonance this reduces to 2

2 0sin 2P t . As usual, these probabilities must be

normalized, and therefore 2

2 2

11c c . Notice, that when 2t the population goes

completely from one state to the other (called a π-pulse).

This very simplified model, which has assumed a purely two-state atom and

a perfectly monochromatic field and no other de-phasing or relaxation effects (such as

31

spontaneous emission or collisions), is never-the-less quite powerful and Rabi oscillations

can readily be observed in the lab. In order to make a more realistic model that does take

into account relaxation effects I will introduce the density matrix formalism.

2.2.2 Density Matrix

The density matrix enables calculations on ensembles of atoms that interact

with each other and their environment, something that is impractical using state vectors

and probability amplitudes (i.e. full wave functions). This is made possible by averaging

over information that is unimportant. For example, say an atom spontaneously decays, we

rarely need to keep track of what exactly happened to that emitted photon (e.g. what

particular mode it ends up in), we just care that there’s one less excited atom in the

ensemble. Or say an atom collides with a background particle, we do not want to keep track

of the wave function for the background particle.

The notion of coherence and decoherence is also handled naturally in the context of

density operators. Atomic fountains provide a perfect case in point: an ensemble of atoms

is put into a coherent superposition of two states via a dipole interaction, and the phase

evolution of this superposition is randomly perturbed due to collisions and background

radiation. The wave function of such a system is completely intractable, on the other hand,

the density matrix provides a clear and calculable method for modeling this situation.

The general state vector solution to the Schrödinger equation is

n n

n

c

, such that for our two state system we have

1 2

1 2c t c t ; (0.30)

32

this is the same as equation (0.19), except we have not explicitly separated the time

dependence of 0

H and int

H . The corresponding density matrix is

* *

1 2 11

* *

222 1

12

2

1

2

1

21

c cc

c cc c

ct . (0.31)

We see that the density operator has dropped the specific state vectors, and just

kept the probabilities of the two states (the diagonal elements, 11

and 22

) and their

couplings (the off-diagonal elements, which are often called the coherences). There are

many convenient properties of density matrices, but one is that the equations of motion are

succinctly written as

,d

t

iH

d , (0.32)

where the square brackets represent the commutator, H H . This is the

quantum mechanical analogue of Liouville’s equation [2]. It can be verified that this

equation yields the same Rabi oscillating solutions given above when the ensemble is

initiated in a pure state (meaning, for example 11

0 1t , i.e. all in state 1 ).

2.2.3 Vector Model

In this section I use the interaction picture rotating at the frequency of the

applied field, . By using this rotating reference frame the equations and physical

interpretation are greatly simplified. The vector model presented here goes back to

Feynman et al.’s 1957 paper [3] that described a geometrical representation of the two-

state system. We construct three real functions of the complex coefficients 1

c and 2

c , or

33

equivalently the density matrix elements, and these three functions become the

components of a vector R t . The three components of this “fictitious spin” or “pseudo-

spin” or “Bloch” vector are

21 12

21 2

11

1

22

i it

t i

t

ti

U e e

V i e e

W

(0.33)

The W component is the population difference between the two states, and U and V

are the components of the atomic dipole moment that are in and out of phase with the

perturbing field. Taking the time derivative of these vector components, which can be done

using equation(0.32), results in the Bloch equations

0

0

V

V U

U

W

W V

(0.34)

These equations can be compactly written in vector form, and this reveals a nice

visualizable model,

d R t

d tR t (0.35)

Where the Rabi precession or “pseudo-field” vector is 0

,0 , , and the

Bloch or “pseudo-spin” vector is , ,R t U V W . These names come from the resemblance

to magnetic moment precessing in a magnetic field or a spinning top under the torque of

gravity. When the driving field is on resonance points along the U axis and the Bloch

34

vector rotates around this axis in the V-W plane at a frequency 0

(plug in initial

conditions U=V=0, W=-1 to equation(0.34)). This is equivalent to the resonant Rabi flopping

between the two states from the previous section.

2.2.4 Ramsey Interrogation

Atomic fountains rely on Ramsey’s method of separated oscillatory fields.

This process can be broken down into five steps: 1) state preparation or initialization, 2)

first Rabi interaction, 3) free evolution, 4) second Rabi interaction, 5) detection. It is

convenient to visualize the atomic interaction with the vector model described in the last

section, as shown in Figure 2.3. We assume that the detuning is small compared to the Rabi

frequency, 0

, which means that it effectively points along U . For this example we

initialize the atoms into the upper state, 0 0 ,0 ,1R . The microwave cavity produces the

pseudo-field vector 0

,0 , , and the first interaction (atoms are on the way up)

lasts for a time τ. This interaction has the effect of rotating the pseudo-spin vector by angle

0

around the U axis. During the free evolution period, T, there is no driving field (

0

0 ), so the pseudo-field vector is 0 ,0 , . During this time the Bloch vector

precesses about theW axis at a frequency of , in other words it rotates and angle T .

This means that if there is no detuning then the Bloch vector is stationary during this time.

Keep in mind that this is in the interaction picture, where the reference frame is rotating at

frequency . The second microwave cavity interaction (on the way down) is the same as

35

the first. The detection process measures the population difference between the two states,

i.e. projects the Bloch vector onto the W axis.

2.3 Addition of Angular Momenta and Wigner Symbols

In both the fine and hyperfine interactions I mentioned the importance of finding

new, “good”, sets of quantum numbers. This meant combining the interacting angular

momenta, so that instead of looking at each individual piece we could look at the system’s

total angular momentum. Mathematically this entailed a change of basis to allow for

diagonalization of the interaction Hamiltonian. This change of basis is a unitary

Figure 2.3. Vector model of Ramsey interogation. P is pseudo-field vector, Ω is the Rabi frequency, ω is the microwave cavity frequency, τ is the interaction time, T is free-evolution time, Rz is proportional to the population difference between the two states.

36

transformation, and the coefficients of these transformation matrices are the familiar

Clebsch-Gordon coefficients.

The topic of angular momenta addition in quantum mechanics is an important and

elegant one, and whole books are devoted to the subject. I will briefly review some aspects

here due to its critical role in transition selection rules and line strengths, two topics that

are pertinent to this thesis. I will also introduce the Wigner symbol notation, as that is what

I found to be most often used in the literature and I had not been exposed to it previously.

Much of the basis for this section comes from Auzinsh, Budker, and Rochester [4].

Take the fine structure L-S coupling as an example. Both the uncoupled and

coupled bases form complete orthonormal systems, so we can write one in terms of a linear

combination of the other. We want to transform from the uncoupled basis of ofL S

Lm Sm

to the coupled basis of J

Jm . Using the “unit operator trick” for complete orthonormal

systems we can show that

,

,

L S

J L S L S J

L S

m

L S

m

J

Jm Lm Sm Lm Sm Jm

Lm Sm Jm Lm Sm

(0.36)

where the sum is over all the possible ways of combiningL

m andS

m to get the

desiredJ

m . The inverse transformation is formed similarly:

,

.

J

L S J J L

J m

S

J L S J

Lm Sm Jm Jm Lm Sm

Jm Lm Sm Jm

(0.37)

37

Commonly, the Clebsch-Gordon coefficients are taken to be real, and therefore the

forward and inverse coefficients are equal. It has become more common in the literature to

use an alternate form of these transformation coefficients called the Wigner 3j and 6j

symbols. These symbols have high degree of symmetry, allowing for easier manipulation

and permutations. The 3j symbol is used when combining two angular momenta to form

the total, as we have just done. The relation to the Clebsch-Gordon coefficients is the

following:

11 ,

2 1

JL S

S

J

m

L J

L S

L S JLm Sm J m

m m m J, (0.38)

and note the minus sign for J

m . A rule for 3j symbols is that the values in the lower

row must sum to zero (in this case 0L S J

m m m ); otherwise the overall symbol is

zero (meaning that coupling is not allowed). Using the Wigner symbol notation for dipole

couplings is a quick way to reveal the dipole selection rules.

2.4 Line strengths (Rate Equations – Incoherent Interactions)

We have seen that the two-level model of an atom interacting with coherent

(well defined phase) dipole radiation culminates in atomic Rabi flopping. Although this is a

powerful model and Rabi oscillations can readily be observed under the proper laboratory

conditions, its many simplifying assumptions require care and specialized equipment to

produce in the real world. It is much more common that these kinds of coherent effects

decay away extremely rapidly (or are not established in the first place), and are therefore

unobservable. This can be the case, for example, when excited state lifetime is short

38

compared to the Rabi frequency, or any other relaxation mechanism causes the Bloch

vector to decay to its steady state value before making a complete revolution. These

systems steadily (monotonically) approach “thermal equilibrium” or steady state, and as

they do the elements of the density matrix vary more slowly. The off-diagonal terms

become negligible. It can be shown [5] that integrating the off-diagonal terms over time

yields

0

12 22 112

ti

it e i t t , (0.39)

where is the decay rate of 12

, which accounts for spontaneous emission and

collisions. This large class of incoherent states / interactions is described used rate

equations. These are likely more familiar, as they (along with related Einstein coefficients

and the Fermi Golden Rule) are often covered in standard physics classes.

Transition strengths are a very useful concept. They condense much physics

down to a single number that characterizes the likelihood of an atomic transition relative to

others. They are commonly used in the lab and literature, and I will use them later in this

thesis when I describe my master laser spectroscopy lock (the POLEAS method). I will

define them in terms of the Wigner symbols introduced earlier [4], [6].

Electric dipole radiation only couples to the orbital angular momentum, L,

and leaves the radial and spin components unchanged. (This statement is largely, but not

absolutely, true; note that we are working within the LS coupling regime, and with fields at

or below optical frequencies). It is helpful to write the dipole operator in spherical

coordinates,

39

1

1

ˆ q

q

q

d d u , (0.40)

here qu are the components of a spherical unit vector, and the Einstein summation

convention is used. This is convenient because expanding the polarization vector, ˆ , into qu

means that q=-1,0,1 where q=0 corresponds to linearly polarized light and q=+/-1

corresponds to circularly polarized light. At the level of the gross atomic structure the

electric dipole matrix elements can be written as

1

1

1' '

1

' ' ' ' ' ' '

' 1' ' 1 2 ' 1 2 1

'

q

q q

angularradial

q

L m

qradial

a

q

q

ngular

n L m d E nLm u n L d nL L m d Lm

L Lu n L d nL L L

m q m

(0.41)

In general one sums over the q values, but from now on I will just consider the case

of a single pure linear or circular polarization. In our case the radial part just becomes a

scaling factor, since all the states we consider have the same principle quantum number.

The double bar in the radial matrix element is referred to the reduced matrix element. This

is the general pattern for the matrix elements: factor out the geometrical or angular part of

the transition amplitude (which is then just a number that can then be looked up in a

table), leaving a simplified reduced matrix element, sometimes called the dynamic part.

At the fine structure level the dipole matrix elements can be written

' '' ' ' ' '

11

'

J m

qJ m Jm J

J Jd d

m q mJ . (0.42)

40

As I mentioned the electric dipole only interacts with the orbital angular

momentum, so often it is desirable to expand J into L and S.

' '

1 2 1 2 ' 1

1 ''

' ' '

1

' '

''

''

JL S m

J q J

J J

d J J

J JL J Sd

m q mJ L

L S J m LSJm

L L (0.43)

Finally, at the level of hyperfine structure we get,

1 ' ' '

1

' '2 1 2 ' 1

1

' '2 1 2 ' 1

' ' ' ' ' '

1

''

1'

'

'FL S

F

j J I m

q

J

F

J

d

L J SJ J

J L

J F IF F

F J

J Jd

m q m

L S J IF m LSJIFm

L L

(0.44)

Although this looks intimidating at first, it is a great simplification; by simply

referring to a look-up table, we can reduce all of the symbols to a numerical value. These

symbols reveal the selection rules and they are used in the model of the polarization-

enhanced absorption spectroscopy (POLEAS) covered in chapter four of this thesis.

41

3 Laser Package

3.1 Introduction

Successful atomic fountain operation relies critically on the laser system. Laser

cooling served as the key technology that first enabled realization of Zacharius’s fountain

idea; only when atoms are cooled to temperatures of roughly 10 K or less, can enough

atoms return through the microwave cavity to provide sufficient signal strength. A central

question of this thesis project was: is it now possible to build a suitable fountain laser

package based on internal grating diode lasers (DFB or DBR) that are offset-locked to a

master laser. The system design is tantalizingly simple, but ultimately the reliability was

hampered by some key characteristics of the laser diodes.

In the following section, I will begin by outlining what atomic fountains require of a

laser system and briefly review some other fountain laser packages. This will provide

some context for the subsequent description of the strategy for the laser system of this

thesis project.

I then discuss the lasers on which my design was based: semiconductor diode lasers

(SDLs) with internal grating stabilization. These types of SDLs are a relatively new

technology at the 780nm wavelengths used for rubidium. The novelty of the lasers, paired

with their size, output power, and robust internal cavity structure, indicated a promising

potential for use in an atomic fountain system. . Unfortunately, over the course of the

research, it became apparent that there were some serious drawbacks associated with

42

them as well. I will describe the unique advantages and disadvantages of using this internal

grating type of SDLs that I discovered during my investigations.

One of the drawbacks of the lasers that I initially used was their unreliable and often

short lifetime. I redesigned the laser housing to facilitate easier replacement of aged laser

diodes, but I found this to be an inadequate long-term solution. Fortunately we found a

similar, and even newer, laser diode that promised longer lifetimes. I will describe the final

laser housing design that I came up with for this new laser. It proved to be stable over time,

compact, and easily adjustable for optimizing beam profile and collimation.

Lasers’ sensitivity to optical feedback means that significant optical isolation is

necessary. I began with 30 dB isolators, which were adequate for the initial stages of laser

cooling, but I found that ultimately they were a weak point in the system as I worked

towards sub-Doppler atom temperatures.

I will describe the overall design concept and layout of the optical system. The

optical system design evolved towards a modular system that is largely reliant on optical

fiber components. Control over the amplitude of the light is another significant hurdle for

the laser system designer to overcome, and I will describe the avenues I investigated

regarding rapid attenuation and shuttering.

Finally, I will report on my use of the dichroic atomic vapor laser lock (DAVLL)

method to frequency stabilize the master laser. The method has some attractive features,

and I designed a compact module that provided a simple frequency lock. One drawback of

using DAVLL in the present application, especially in the development stages of this project,

43

was the lack of resolution of the hyperfine transitions. Ultimately, I replaced this method

with a novel method, which I address in the subsequent chapter.

3.2 Background

An illustration showing the basic components of an atomic fountain is shown in

Figure 3.1. Illustration of a simplified atomic fountain, showing a laser-cooled cloud of atoms launched up through the Ramsey microwave cavity and subsequently detected via fluorescence. Taken from http://nist.gov/pml/div688/grp50/primary-frequency-standards.cfm

44

Figure 3.1. The standard sequence for operating an atomic fountain, particularly as it

pertains to laser system design, is as follows.

1.) Molasses / MOT:

First, atoms are collected and cooled to the Doppler limit 150 K in

either an optical molasses or magneto-optical trap (MOT). This requires that

each counter-propagating pair of cooling beams be power balanced, have the

correct polarization, and that the frequency be red-detuned from the atomic

cycling transition resonance.

The more light that is available, the more atoms that can be collected

and cooled. The light intensity should be greater than the saturation intensity

( 23.6

satI m W cm ), in which case, the atom number, N, scales as 3.6

d [1], [2]

, where d is the diameter of the laser beams. It is interesting to note that the

scaling factor just stated becomes even more dramatic ( 6

N d ) when the

beams become small (mm scale)[3]. The red-detuning from resonance is

generally 1 to 3 ( 2 6MHz is the natural linewidth).

Repumping light is also necessary to prevent population accumulation

in the lower (dark) ground state. The repump atomic transition frequency is

multiple gigahertz away from the cooling transition, so usually a separate

laser is used. Repumping with a sideband of a strongly modulated cooling

laser has been demonstrated [4], [5], but this places higher demands on the

limited optical power.

2.) Launching:

45

The cold cloud of atoms is launched up through the microwave

cavities by shifting the frequencies of the vertical beams. The horizontal

beams frequencies remain constant while the upward beam shifts higher in

frequency and the downward beam shifts lower. This is called a moving

molasses, as its reference frame is now Doppler shifted, moving at a velocity

of / 2k relative to the lab frame, where ∆ is the frequency difference

between the vertical beams and k is the wavenumber. For a wavelength of

780nm this means the speed scales as 0.78 msv M H z , where the frequency

shift in this case is for each vertical beam relative to the horizontal beams

(i.e. drop the factor of ½).

3.) Sub-Doppler Cooling:

This stage brings the temperature of the atoms down from ~ 150 K

to as low as a few micro-Kelvin. It involves adiabatically decoupling the light

Figure 3.2. Typical laser frequency detunings for fountain cooling and launch sequence.

46

from the atoms, which is done by ramping down the frequency and

amplitude of the laser beams. Though ramping only frequency or amplitude

can be effective, it is optimal to do both in concert. Typical numbers

correspond to ramping the frequency ~10 (to the red) and reducing the

power ~40dB over the course of a couple milliseconds. I discovered that the

phase coherence between the beams is also critical for sub-Doppler cooling,

as I will explain later in the thesis. A plot showing the frequencies of the

vertical lasers for a typical sequence of these first three stages

(molasses/MOT, launch, sub-Doppler cooling) is shown in Figure 3.2.

4.) State Selection and Ramsey Interrogation:

Although there are options for optical state preparation I will not

discuss them here; most often state selection is done with microwaves, and

only uses laser light to blast away uninitiated atoms from the cloud. For

example, a microwave cavity can apply a π-pulse to place some atoms into

the lower clock state, the uninitiated atoms, remaining in the upper hyperfine

level, can then be removed from the cloud by a strong optical pulse. For this

stage the laser light frequency is resonant with the atomic cycling resonance,

and the amplitude must be pulsed on for roughly a millisecond.

5.) Detection:

For this step the light needs to be resonant with the atomic cycling

transition in order to scatter the most photons. One effective detection

scheme is to have four sheets (narrow in the vertical direction and wide in

47

the transverse direction) of light that the atom cloud falls through

sequentially. The first is a standing wave resonant with the cycling transition;

this tells how many atoms were in the upper hyperfine ground state. The

second layer is a traveling wave that is again resonant with the cycling

transition; this blasts away the atoms that have already been detected. The

third layer is a repump beam that optically pumps the population that was in

the lower ground state into the upper ground state. The final layer is a repeat

of the first, a standing wave resonant with the cycling transition; this reveals

how many atoms were in the lower ground state.

There are many paths forward for developing a laser system to accomplish these

tasks. The main alkali atoms of interest (namely Cs and Rb) require near-infrared

wavelengths (approximately 850 nm and 780 nm respectively). This narrows the choice of

laser technology down to two principle options: solid-state Titanium:Sapphire lasers or

semiconductor diode lasers (a third option, frequency-doubled fiber lasers, has also been

proposed [6], but has not found wide-spread use at this time). Both type of lasers are

available commercially in fully operational forms, with Ti:Sapph costing on the order of

$105 and semiconductor on the order of $104. They both provide single mode, single

frequency laser light with linewidths of approximately 100kHz. Ti:Sapph lasers can

provide watts of optical power, and their wavelength can be tuned over 100s of nm.

Semiconductor lasers provide ~100 mW of optical power (which can be amplified to

the watt scale by a semiconductor tapered-amplifier), and can be tuned over a few

nm. Since bare semiconductor diode lasers can be purchased for much less ($102),

many labs choose to build their own external cavity structures to achieve the narrow

48

linewidths of ~100 kHz. The NIST F1 fountain initially used the low cost lab-built

strategy, but now both fountains use a Ti:Sapph laser for cooling and detection, and a

diode laser for repumping. This has greatly reduced the day-to-day laser

maintenance requirements. There are further details to consider when designing a

laser system, and I will discuss this in the subsequent sections.

49

3.2.1 NIST F1 & F2

Clearly some of the laser system design goals for a laboratory Cs fountain, primary

standard, differ from those of a transportable fountain system. For a primary standard,

fountain performance is the key goal, with little regard for the size of the system. But, as the

fountain operation is the same, and the F1 and F2 systems have been greatly refined and

perform as well or better than any fountain in the world, their laser systems serve as a

good foundation for comparison. The laser system for the NIST fountains is based on two

Figure 3.3. Photo and diagram of commercial solid-state Verdi pump laser. This laser is used to pump the Ti:Sapph laser shown in Figure 4. From http://repairfaq.ece.drexel.edu/sam

50

major components: a commercial Ti:Sapph laser, and six double-pass acouso-optical-

modulator (AOM) modules.

A powerful green (532nm) pump laser is required to achieve lasing in the Ti:Sapph

crystal. This green pump laser uses a vanadate crystal (Nd:YVO4 neodynium-doped yttrium

orthovanadate, which lases at 1064 nm) as its gain medium, and is itself pumped by diode

lasers. The diode pump lasers are in the form of a diode bar, consisting of many emitters,

which provide broadband ~808 nm laser light that is collected by an optical fiber array.

The fiber array, with the 808 nm light, is combined into a single fiber and delivered to the

bow-tie ring cavity resonator where the vanadate crystal resides (see Figure 3.3).

Additionally, in the ring resonator there are an etalon, an optical diode, an astigmatic

compensator, and a frequency doubling crystal, all mounted on an Invar plate. The etalon

acts as an optical band-pass filter, and the optical diode assures propagation around the

cavity in only one direction. These two elements are critical for single-frequency lasing.

Finally, the second harmonic generation, yielding the 532 nm green light, is achieved with

the lithium triborate (LBO) crystal. To optimize conversion efficiency in this doubling

crystal, its temperature is stabilized around 150°C. The temperature of the vanadate gain

crystal is maintained at 30°C, and the etalon is also individually temperature controlled

(setpoint between 25°C and 75°C). The excess heat is disposed of through the base plate,

which is water-cooled. This pump laser system outputs up to 10 W of 532 nm light with a

linewidth less than 5 MHz.

51

The Ti:Sapph laser resonator receives the green pump light via a free-space beam.

Like the green pump laser, the Ti:Sapph gain crystal resides in a bow-tie ring cavity

resonator along with an etalon, an optical diode, a birefringent filter, and Brewster plates

(see Figure 3.4). The Brewster plates allow the laser frequency to be tuned by changing the

path length of the cavity without altering the beam alignment. There is also an Invar

reference cavity, to which the laser can be locked, for narrowing the linewidth below 75

kHz. With the 10 W pump laser, described in the previous paragraph, this Ti:Sapph laser

can provide up to 1.5 W of optical power, which is more than sufficient for operating an

atomic fountain. The powerful Ti:Sapph laser is frequency locked to the 852 nm Cs D2

wavelength through saturated absorption spectroscopy, and then divided into the cooling

and detection beams.

Figure 3.4. Diagram of Coherent MBR-110 Ti:Sapphire Laser(from the User Manual). The pump light comes from the Verdi laser, shown in Figure 3. This laser system works well in a laboratory environment, but is not suited for transportable applications.

52

The NIST F2 fountain beams are each sent through a custom built double-pass AOM

module [7], shown in Figure 3.5. AOMs are incredibly useful devices, but they have

drawbacks that hamper their utility especially for a transportable system. They operate

using special crystals (tellurium dioxide TeO2, in this case) whose permittivity is sensitive

to mechanical strain. Applying an acoustic radio-frequency wave across the crystal

effectively creates a bulk grating from which a laser beam can diffract. The diffraction from

Figure 3.5. Double-pass AOM diagram and photo of modules with custom aluminum baseplates. These modules allow fine rapid control over the frequency and amplitude of the light, but they consume much table real estate and electrical power. Diagram is from [7].

53

the photon-phonon interaction imparts the phonon’s momentum to the diffracted light.

Through conservation of energy and momentum, the frequency of the nth-order diffraction

is different from the zeroth-order by n times the applied RF frequency. It is uncommon to

use higher than the first-order diffracted beam as the higher orders are weak and sacrifice

much of the optical power.

One of the challenges in using AOMs is the associated deflection of the beam. The

deflection angle of the diffracted beam changes with the applied RF frequency. In the case

of the NIST F2 fountain, these RF frequencies are manipulated to launch and cool the

atoms, but the subsequent optical fiber-coupling (for delivering the light to the physics

package) means that beam deflections cannot be tolerated. Other drawbacks of using AOMs

include the unavoidable loss of optical power (generally at least 15%), their sensitivity to

misalignment, the physical space required to adequately separate the diffracted beams, and

the amount of electrical power necessary for operation (~15 W per device).

There are two principle reasons for using a double-pass AOM configuration: to avoid

the beam deflection, and to get larger frequency shifts and frequency tuning bandwidth. By

passing twice through the AOM, first forward and then back, the beam is diffracted twice,

thereby receiving twice the frequency shift while undoing the deflection. Unfortunately

with the longer beam path, the sensitivity to misalignment increases, and passing twice

through the AOM increases the loss of optical power. In typical operation it is not unusual

to lose 40% of the light in the double-pass module (that is measured from light entering the

module to light exiting the fiber at the physics package). When optimizing a single module

it is possible to reduce the optical losses to around 25% by adjusting alignment, beam size,

54

and polarization, but this is typically at the expense of the other modules, and furthermore

relaxes over time back to the 40% level. This is despite the thoughtful design of the

modules, the thick custom aluminum base plates, and the temperature-controlled

environment of the laboratory. The design takes advantage of a “cat-eye” configuration (see

Figure 3.5) for increased tuning bandwidth. The cat-eye is accomplished by placing a lens

one focal length from the AOM. The deflected beam, after passing through the lens, is then

parallel to the zero order beam, and hits the retro-reflecting mirror at a right angle. As the

AOM drive frequency is swept, the beam remains at a right angle to the mirror, and is

therefore able to stay at the Bragg angle upon passing back through the AOM.

3.2.2 PARCS – A Laser System Designed For Space

The Primary Atomic Reference Clock in Space (PARCS) mission was a NIST-led

NASA project meant to send a laser-cooled atomic clock to the International Space Station,

where it could be used to study relativity and push the state-of-the-art of frequency

standards. The project was canceled before reaching completion, but designs for a laser

system were developed and serve as a point of comparison for the laser system of this

thesis. The demands on the PARCS laser system are similar, but more extreme than those

for an Earth-bound transportable fountain. The laser system would need to be as compact

and lightweight as possible, have low power consumption, and most importantly be

reliable, since it would inhabit the space station where repairs are prohibitively difficult.

The PARCS clock was a zero-gravity version of an atomic fountain (basically a laser-cooled

beam clock), and so would need to carry out a similar sequence of cooling, launching, post-

cooling, and detection.

55

The laser system design that the PARCS scientists and engineers came up with is

Figure 3.6. For comparison, the PARCS laser system designed to operate on the International Space Station uses diode lasers and double-pass AOMs. My design greatly reduces the number of optical components.

56

shown in Figure 3.6. In brief, they use a system similar to that just described for the NIST

fountains, except in place of the Ti:Sapph laser they use multiple semiconductor diode

lasers. Each diode laser would be frequency referenced to a cesium vapor cell through

saturated absorption spectroscopy. The sweeping and jumping of the lasers’ frequencies

would be carried out with double-pass AOM modules. Unfortunately, prototype testing of

this system did not begin before funding was canceled. The project had collaborated with a

semiconductor laser manufacturer in developing a Distributed-Bragg-Reflecting (DBR)

diode laser that they hoped would be more reliable than an external-cavity diode laser (see

next section on diode lasers).

For the laser system of this thesis, I adopted the newly developed DBR laser

technology, but I wanted to avoid the numerous AOMs. By independently offset phase-

locking slave lasers to a master I could achieve the flexibility for cooling and launching in a

fountain. Without the AOMs I could make the system much more compact, power efficient,

cheaper, and reliable (less susceptible to misalignment).

3.3 Semiconductor Diode Lasers

Semiconductor lasers (SCLs) have become the go-to laser within the wavelengths

that they are available. Their popularity is due to their low cost, size, simplicity, tune-

ability, and low (amplitude) noise characteristics. The spectacular progress of

semiconductor manufacturing techniques has enabled this appealing combination of

attributes. These manufacturing techniques deposit layers of active material between a p-n

junction, and form the gain medium, spatial confinement, and resonant cavity. By running

current through these layers, electrons (holes) are excited from the n-type (p-type)

57

material into the conduction band, where the electron-hole pairs can recombine and emit a

photon. At currents below the lasing threshold the device behaves as a light-emitting diode

(LED); above this threshold a population inversion is created, stimulated emission

dominates, and laser light is emitted.

Most diode lasers are index-guided, meaning that the light’s transverse spatial

confinement in the gain medium is accomplished by a waveguide groove, where the index

of refraction is different than outside the groove (gain-guided is the less popular

alternative). In standard bare laser diodes the resonant cavity is formed by the front and

back cleaved faces of the semiconductor, which can be specially coated for desired

reflectance / transmission coefficients. The cavities are short (<1mm) and hence the

frequency linewidth, even in the Schawlow-Townes limit (few MHz), is too wide for many

atomic physics applications. The actual linewidths (100 MHz – GHz ) of bare SCLs are much

wider than this limit due to various factors such as technical noise and the linewidth

enhancement factor (amplitude-phase coupling of the spontaneous emission noise). These

linewidths must be reduced in order for light to be useful for laser cooling, efficient optical

pumping, or spectroscopy, as the (natural or lifetime-limited) alkali atomic hyperfine

linewidths are roughly 5MHz wide.

Various methods are used to reduce the linewidth of SCLs, and they can generally be

divided into either optical or electrical feedback categories. The most common method is to

use a grating to form an external cavity (usually in either the Littrow or Littman-Metcalf

configuration), where the grating-resolved light is coupled back into the diode, forcing it to

lase on that external cavity mode. This type of external cavity diode laser (ECDL) can be

58

home-built or bought commercially ($10^4). They offer narrow linewidths of ~100kHz, but

often suffer from mode-hops, as the external cavities are challenging to stabilize against

thermal and acoustic perturbations.

An alternative to the ECDL are two related types of diodes called distributed

feedback (DFB) and distributed Bragg reflector (DBR) diodes. These diode lasers have a

grating structure built into the diode itself. The difference between the two is that DFBs

have the grating structure built directly into the gain region, whereas DBR’s place the

grating structure in front or behind the gain region. The resulting linewidths (<1MHz) are

not as narrow as ECDLs, but the microscopic, monolithic structure means that these diodes

are much more robust against mode-hops, and have greater continuous tuning ranges. This

technology was initially developed for diodes operating at telecom wavelengths, but has

recently migrated down to near-infrared (NIR) wavelengths of 780nm – 850nm.

Initially these internal-grating diode lasers appeared to be an enabling technology

that could result in a much more robust, reliable, simple and compact laser system for

operating an atomic fountain. When I began this project there was one company

(Eagleyard) making this style of laser (a DFB) at the 780nm wavelength. As I began testing

the system, I discovered that these lasers were too unreliable in terms of their lifetime to

be used for a transportable atomic fountain. In the meantime, a second company

(Photodigm) had begun selling comparable 780nm DBR laser diodes, which quoted much

longer lifetimes (10^5 hours) than the DFBs. I therefore transitioned to these DBR lasers.

Correspondingly the laser system design, and laser mount in particular, had to be refined

and adapted multiple times, which I describe in the next section.

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The manufacturing process for the DBRs is able to be better controlled than that for

the DFBs, resulting in fewer defects and strains and greater repeatability, hence longer,

more reliable lifetimes. Eagleyard does not provide much information on their

manufacturing process, but Photodigm outlines some of their methods. They tout a

patented single epitaxial growth step [8]. Epitaxial growth means deposition of a layer with

an ordered or crystalline structure, which is a widely used strategy in semiconductor

manufacturing. There are many methods for achieving epitaxial growth, but Photodigm

uses one with a high level of controllability known as the molecular beam method.

Molecular beam epitaxy (MBE) is carried out in ultra-high vacuum, and uses slow

deposition rates resulting in highly pure films. It is likely that Eagleyard uses similar

methods for their DFB lasers, but since the grating is in the active gain region, this requires

a more complex regrowth step. Furthermore, the separated regions of the DBR design

results in less current-induced heating of the grating section, and similarly reduced

propagations of current-induced defects.

Laser diodes are very sensitive electrical devices. Care must be taken to avoid static

discharge to the diode, especially when handling bare diodes (grounding bracelets and

grounded table surfaces are recommended). I constructed a small circuit to protect diode

lasers from power spikes, reverse biases, and similar electrical accidents. I included

inductors to mitigate surges, a parallel diode to provide a current bath for reverse bias, and

a series of diodes (greater impedance than laser diode) to allow an alternate current path

in the forward bias case.

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3.4 Laser Housing and Beam Shaping

A stable apparatus for holding the laser with adequate degrees of freedom for

optimizing spatial mode and beam alignment is important. The housing needs to be

compact, and have sufficient heat sinking capabilities. In this section I will discuss the

evolution of my laser housing designs. Initially I used DFB laser diodes, but I discovered

that they were plagued with unreliable lifetimes. I adapted the laser housing design to

accommodate this limitation. I eventually learned of an alternate comparable DBR laser

diode that promised much longer lifetimes, and I transitioned to this new laser. This

iterative process, and the reliable DBR lasers, allowed me to settle on a laser housing

design that worked quite well.

For the initial design I used a commercial mount made for the TO-9 diode package of

the DFB lasers. It incorporated a “cage system”, which uses narrow steel rods to fix the

transverse degrees of freedom while allowing translations along z (nominally the beam

path direction). I built a small sturdy aluminum box to bolt the laser-mount to, and secure

the electrical connectors for diode current and laser temperature controls. The collimation

lens mounted in a cage-plate that rigidly attached to the laser cage-plate. I found that finer

positioning control was necessary, and modified the plates for plastic tipped set-screws to

position the diode relative to the lens. Obtaining a clean collimated beam with this design

was tedious, and obtaining coupling efficiencies better than 50% into a single-mode fiber

was challenging.

61

The steep beam divergence angles (greater than 20 degrees) and the elliptical beam

shape contribute to the challenges. Because the beam diverges so quickly the options

available for collimation-lenses are limited. I used an anti-reflection (AR) coated aspherical

lens with a large numerical aperture (NA

= 0.68 with 3.1 mm focal length).

When the DFB lasers began dying

at a much faster rate than expected, I

redesigned the laser housing in an

attempt to facilitate easy laser diode

replacement. By shifting the necessary

degrees of freedom to the laser diode

mount itself, the downstream optics could

remain in alignment, see Figure 3.7. When

a laser would fail, I could remove the laser housing, install a new diode, and align the laser

back into place without having to re-align everything else downstream. Placing the laser in

a high-quality mirror mount and three-axis, translation stage (that use flexure joints, rather

than springs), made it easy to fine-tune and optimize, but was ultimately not mechanically

stable. While easing laser diode replacement, this design also increased the acoustic

frequency noise on the laser light, and drifted out of alignment, requiring regular

maintenance.

Figure 3.7. Laser housing designed for easy laser replacement, to accommodate the unreliable DFB laser diodes.

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As I transitioned away from the DFB lasers towards DBR lasers, I developed a laser

diode housing that worked well; see Figure

3.8. With the DBR lasers, which promised

much longer lifetimes, I could forego the

design prescriptions weighted towards

easy laser replacement for something more

mechanically stable and considerably more

compact. I rigidly mounted the laser diode

to a base plate. For the collimation lens,

where I still needed appropriate degrees of

freedom for optimal fiber-coupling

efficiency, I modified a Thorlabs

“Fiberport” (part number PAF-X-5-B). The Fiberport is designed to collimate light coming

out of an optical fiber. To collimate a free-space laser beam I needed to flip the lens and

remove the fiber connector, so that it could be mounted close (~ 1 mm) to the laser diode

and still allow access to the positioning set screws.

Figure 3.8. DBR Laser housing design, using modified Fiberport for beam collimation, increased mechanical stability.

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The disassembled Fiberport is shown in Figure 3.9. The aspheric lens is in a

magnetic holder, which keeps it in contact with the steel plate while allowing freedom to

move in the transverse (x and y) directions. A leaf spring provides the restoring force for

the transverse directions, and small set-screws provide for adjustments. Three additional

screws adjust the angle of the steel plate (and hence the lens), and also control the

longitudinal (z) dimension. The steps to modify the Fiberport into a laser collimator

include: removing the fiber connector, surface grinding the steel plate, and inserting

Belleville washer springs to provide tension against the steel plate. Thinning the steel plate

on the surface grinder is necessary in order to place the lens close enough to the laser. The

modified Fiberport provided sufficient degrees of freedom for optimizing the collimated

beam, it was stable, and also compact. The modifications are simple and quick (provided

Figure 3.9. A disassembled Thorlabs Fiberport. After modification, it is used for collimating the free-space laser beam.

64

access to a surface grinder – I used a strong electromagnet to hold the plate in place during

grinding). With this collimation configuration and an anamorphic prism pair, I was

routinely able to achieve better than 60% coupling efficiency into a single mode optical

fiber.

3.5 Optical Isolation

After a collimating lens, an optical isolator is necessary to reduce optical feedback

into the laser. Optical feedback can cause a variety of deleterious effects for a laser, from

subtly adding noise, to mode-hopping, to chaotic “coherent collapse” [9], where the laser

linewidth expands by orders of magnitude. Even weak optical feedback (~80 dB down

from laser output) can broaden the linewidth by 30% [10]. The effects of optical feedback

depend on the power coupling back into the laser, the distance from the laser to the

reflecting surface, the phase of the feedback, and the other cavities / reflections present. In

practice, for applications such as laser cooling, it is desirable to place a strong optical

isolator close to the laser with as few scattering surfaces between them as possible. The

best solution was a 60 dB optical isolator with 3 mm apertures, which minimized beam

clipping and back-scatter from the face of the isolator. Unfortunately I did not have access

enough of these isolators for each of my five lasers. I initially used 30 dB isolators because

they were small, readily available, and cheaper than others. I noticed some noise and mode

hopping problems from scattering off of an optical fiber tip (despite using angle polished

fiber tips), but these problems could be mitigated with careful alignment. Problems also

arose while optimizing the magneto-optical trap (MOT). Light retro-reflects from the MOT

mirrors back through the fiber and into the laser. The apertures on the 30 dB isolators

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were 1 mm in diameter, and the beam clipped the edges of these apertures, which also

became a concern.

I experimented with ways to improve optical isolator throughput and fiber coupling

efficiency. For example, focusing the beam through the isolator and then using separate

cylindrical lenses to circularize the beam profile and mode match into the single-mode

(polarization-maintaining) fiber were effective approaches but required too much table

space for a transportable system. Ultimately, I found that the optimal approach was also

one of the simplest, and is as follows: 1) collimate with an aspherical lens, 2) pass through

the isolator, then 3) use a pair of mirrors to steer the beam into the fiber. I used an

anamorphic prism pair “in reverse” to improve fiber coupling efficiencies; this reverse

Figure 3.10. A sketch showing the laser system concept. The master laser and DAVLL module, on the left, provide a stable optical frequency reference for the three slave lasers. The slave lasers perform laser-cooling and launching of the atoms in the fountain.

66

configuration reduced the wider aspect of the beam rather than expanding the narrow one.

The prisms were situated in front of the isolator to reduce back-scatter from the beam

clipping the isolator aperture.

3.6 Laser System Configuration

The overall laser system design strategy is based on offset locking internal-grating

diode lasers from a single master diode laser (also internal-grating stabilized). A sketch of

the original concept is shown in Figure 3.10, and the original diagram of the offset locking

scheme is shown in Figure 3.11. A master laser is frequency-stabilized using spectroscopy

Figure 3.11. Diagram of offset-locking scheme. This provides fine rapid control over the frequency of the light and consumes much less space, electrical and optical power, and optical components compared to double-pass AOM modules.

67

on a rubidium vapor cell; in the early diagram we see the DAVLL method employed. This

acronym stands for Dichroic Atomic Vapor Laser Lock, and I experimented with this as an

alternative to the common saturated absorption spectroscopy method. There are some

attractive features of the DAVLL method, but I found it troublesome when I was initially

attempting laser cooling; I will discuss this further below. All the lasers used to operate the

fountain are offset-locked to the master laser by taking a small amount of light from the

master and slave and overlapping the two beams onto a high-bandwidth photodetector.

The beatnote frequency is then compared to a synthesized reference frequency, and finally

the servo signal is fed back onto the slave laser’s current.

In all, I used five lasers: one master, one up beam, one down beam, one horizontal,

and one repump to accommodate the traditional 0,0,1 fountain geometry of my physics

package. This 0,0,1 geometry means that the up and down vertical beams are aligned with

Earth’s gravitational force. The notation is borrowed from solid-state crystal orientations,

and distinguishes from the alternate 1,1,1 fountain geometry that is becoming more

common (in this case the up beams form a tripod and down beams are the mirror image,

such that there is no vertical beam along the atoms’ trajectory). The light for detection can

be taken from the horizontal laser.

I built a mock physics package for evaluating the laser system. This consisted of a

laser-cooling chamber and a detection region below (but no microwave cavity). This

allowed me to establish a MOT and optical-molasses, attempt sub-Doppler cooling, launch

the atoms, and measure their temperature using the time-of-flight method.

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In the early configurations the system relied mostly on free space beams and optical

components, only using optical fibers (single-mode polarization-maintaining or PM) for

final delivery to the physics package. Transitioning to a more modular design, and

incorporating more fiber components such as splitters, combiners, and switches (see

Figure 3.12) made evaluating different components and configurations easier. The modular

system also required less maintenance to keep aligned since the free-space path lengths

were kept short. Eliminating bulky optical components such as mirrors and polarizing-

beam-splitting cubes rendered the system more compact. Using fiber combiners to overlap

the master and slave laser beams for the beat-note signal assured that the wavevectors

were perfectly aligned, yielding optimal beat-note signal strength.

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Foregoing the use of AOMs brings one additional challenge, since in addition to

providing fine control over the frequency of the light, AOMs can also provide fine and rapid

control over the amplitude of the light. Fine control of the light’s amplitude can be achieved

by adjusting the power of the RF signal fed to the AOM. In this case, the frequency and

beam deflection stay constant, and only the optical power diffracted by the AOM changes.

The NIST fountains actively stabilize the optical power that reaches the atoms by using this

mechanism.

Significant attenuations can be achieved by using a spatial beam filter (for example

an optical fiber) in conjunction with turning off the RF power to the AOM. In a single pass

configuration with the first-order beam coupled into an optical fiber this can result in more

than 50 dB of attenuation at the fiber output [11]. Furthermore, double-pass configurations

Figure 3.12. Modular laser system configuration with fiberoptical components is compact, mechanically stable, and eases system testing and optimization.

70

can achieve attenuations greater than 70 dB, and two single-passes in series resulted in

100 dB [11]. These attenuations occur rapidly (≤ 10 ns) compared to mechanical shutters.

The NIST fountains use mechanical shutters in addition to the AOM attenuation since they

were able to detect a small light shift when mechanical shutters are not used.

Initially for this project I used lab-built mechanical shutters based on the designs

developed for the NIST primary standards. These shutters use a microcontroller to control

a stepper motor that actuates the flags, which block the beam. They have proven to be

extremely long lived, with those on NIST F1 logging near ~ 108 cycles and counting. For

both the upward-going and downward-going beams, I used a single shutter that was on

translation stages centered at the intersection of the beams (which crossed at a shallow

angle). The beams passed through long focal length 1:1 telescopes such that the foci were

near the intersection point. The light extinction rates were controlled through a

combination of beam shaping, beam alignment, and shutter alignment. The horizontal

beams were overlapped with the repump light before they launched into the same fiber,

and a second shutter was located in front of this fiber.

Shutters can be quite noisy devices, both from acoustic vibrations and the switching

electronics, and care must be taken that such noise does not corrupt sensitive devices, such

as the lasers. The stability of theshutter-dependent configuration was less than optimal. As

I continued to struggle with insufficient laser cooling, I continued to refine the shuttering

and light attenuation.

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I purchased a fiberoptic switch, made by Agiltron, to evaluate as a means for fine

control of light attenuation. The device is called a NanoSpeed variable optical attenuator

(VOA), and it does not rely on movement of mechanical parts for its operation. It can be

provided with PM fiber. Its rise and fall times are less than 300 ns, and they promise high

reliability. The drawback of this VOA is that it only provides 25 dB of attenuation, less than

what is needed for the fountain. Nevertheless, it performed well when used in combination

with a mechanical shutter for complete extinction. It has the added benefit of efficient use

of optical power, because while one port can be directed to laser cooling the other port can

go to the detection region. In this way the VOA can be used to actively servo the critical

detection light amplitude.

A configuration that proved particularly useful is diagrammed in Figure 3.13. Rather

than crossing the beams, here I used polarizing beam splitters to overlap and then separate

them. The beams enter and leave the module through PM fibers, and the four optical

Figure 3.13. The up and down beams are overlapped using a polarizing beam splitter (PBS), sent through an acousto-optical modulator and mechanical shutter, separated, and coupled into separate fibers. The wave plate controls the separation ratio. This module was particularly useful for diagnosing the cause of inadequate laser cooling.

72

collimators are matched so that the beam profiles also match. With this setup I could easily

optimize the beam overlap by comparing the injection into the same fiber (by orienting the

λ/2 plate to 45 degrees half of the light from each beam comes out both ports of the PBS).

For optimal control over the light attenuation I used an AOM in single-pass configuration

and a mechanical shutter for complete extinction. The decision to use the AOM was for

diagnostic purposes, as I was attempting to discern what was preventing effective sub-

Doppler laser cooling. Ultimately, the light attenuation was found to be less of a problem

than the phase-coherence between the cooling beams. This beam overlap module, with its

waveplates and PBSs, enabled easy tests that led to the root problem. By rotating the

waveplates, the up and down cooling beams could be formed from either the two separate

lasers or a single laser divided in half. Note that I could not launch the atoms in this

configuration, but I could drop them and measure their temperature using the time-of-

flight method. The effect was dramatic; with separate phase-locked lasers the atoms were

near the Doppler cooling limit of ~ 143 µK, but with a single vertical laser they were ~ 10

µK.

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3.7 DAVLL Module

The dichroic atomic vapor laser lock (DAVLL) is a spectroscopy-based method for

frequency stabilizing a laser [12], [13], [14], [15]. I initially used this method to lock my

master laser, since it had a number of attractive features. It can be straightforwardly

implemented in a compact package because it uses a single beam passing in one direction

through the vapor cell (the signal is Doppler-broadened). It does not require frequency

modulation ofthe light to obtain a zero-voltage-crossing error signal, thus the necessary

electronics are simple. Since the signal is Doppler broadened, there is only one zero-

crossing within a broad frequency span (more than 500 MHz), and this could simplify

development of an auto-locking servo.

Figure 3.14. Sketch diagramming the dichroic atomic vapor laser lock (DAVLL) method. The permanent magnets Zeeman shift the atomic sublevels causing the vapor to absorb two separate frequencies (hence dichroic) depending on the light’s polarization. The polarization components are separated by the wave plate and polarizing beam splitter, and the electrical signals are subtracted resulting in a dispersion-shaped curve for locking a laser.

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An illustration of the basic components is shown in Figure 3.14. A small amount of

linearly polarized light is directed through a Rb vapor cell, which is in a magnetic field that

is collinear with the laser beam. The field Zeeman-shifts the hyperfine magnetic sublevels;

some sublevels shift up in frequency, and others down. This results in two separate

Doppler-broadened peaks each preferentially absorbing either right or left circularly

polarized light. These two orthogonal circular polarizations can be extracted from the

linearly polarized light with a quarter-wave plate and polarizing beam splitter. The signals

from the two detected beams are then subtracted from each other, yielding a dispersion-

like curve that can be fed back to the diode current source for locking.

A photograph showing the compact DAVLL module that I built can be seen in Figure

3.15. The locking signal is optimal for magnetic fields in the range of 75 to 150 G [14]. The

Figure 3.15. DAVLL module with a Rb vapor cell held in black Delrin with bar magnets that create a ~120 G field parallel to the beam path. A balanced photodetector is in the bottom of the module with signal output BNC connector visible.

75

DAVLL systems reported in the literature typically rely on either solenoid electromagnets

or permanent-magnet discs with the vapor cell residing in the center. I compared magnetic

discs (those described by Corwin et al [12]) to rare earth (Neodymium) permanent bar

magnets, and found both provided strong locking signal (approximately equivalent to each

other). The bar magnets simplified the mechanical design and construction, and they

allowed for a more compact module. I integrated a balanced photodetector into the base of

the module.

One of the main drawbacks to the DAVLL method is that the hyperfine transitions

are unresolved due to Doppler broadening. I found this to be a significant impediment as I

began attempting laser-cooling. Typically, achieving a magneto-optical trap (MOT) is the

simplest method to verify laser-cooling since the signal is bright and easily visible with an

infrared camera. Without visible hyperfine transitions in the absorption spectrum serving

as a reference (as in the case of saturated-aborption spectroscopy), I constantly questioned

if the master laser was locked to the correct frequency. I set aside the DAVLL method while

I was trouble-shooting the system and did not come back to it, as I developed a novel

alternate method (POLEAS – see next chapter) that served very well.

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4 Polarization Enhanced Absorption

Spectroscopy (POLEAS) For Laser

Stabilization

4.1 Introduction

Numerous methods exist for frequency-stabilizing a laser, including various cavity-

based approaches, which hold the record for short-term stability [1], and spectroscopy-

based approaches [2], which provide longer-term stability. Here I present a spectroscopy-

based laser stabilization method that I call Polarization Enhanced Absorption Spectroscopy

(POLEAS). This method is related to two other commonly used methods, Doppler-free

saturated absorption spectroscopy (D-F SAS) [3,4] and polarization spectroscopy (PS) [5–

8]. The POLEAS method combines some of the favorable aspects from both D-F SAS and PS,

resulting in a more stable, robust, and compact laser frequency lock. The unique

combination of features offered by POLEAS makes it a superior candidate for many

applications requiring laser frequency stabilization.

Comparing the three Doppler-free, nonlinear spectroscopy methods, POLEAS, D-F

SAS, and PS, there are general commonalities, but also important differences in the details

of what each method offers to laser-locking applications. They each reveal narrow

(<10MHz), non-Doppler-broadened hyperfine spectrum features, due to the velocity-

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selective, counter-propagating beam configuration that they have in common. They also

each use balanced photodetectors (circuits that output the difference between the signals

detected by two separate photodiodes), which results in the cancellation of the Doppler-

broadened background profiles. One important difference between them is that the

POLEAS and D-F SAS output signals consist of resonant absorption or transmission peaks,

whereas the PS signal consists of dispersion-shaped curves at the atomic resonant

frequencies. The PS dispersion signals of some atomic transitions (in particular closed or

cycling transitions) are suitable for directly stabilizing a laser to the atomic resonance. On

the other hand, locking a laser to the top of an absorption peak (as with D-F SAS and

POLEAS) requires an additional stage of modulation to generate a correction signal. This

correction signal is achieved through phase-sensitive demodulation electronics, whereby

the signal is modulated and then demodulated, yielding the derivative of the absorption

peak. While this phase-sensitive detection requires specialized electronic equipment,

which can be costly and cumbersome, it provides the distinct advantage of being relatively

insensitive to changes in the signal amplitude (e.g. due to temperature fluctuations or laser

intensity). The frequency stability of D-F SAS, afforded by its narrow peaks and associated

phase-sensitive detection, largely accounts for its widespread use in laser stabilization

applications.

In this chapter, I first briefly review D-F SAS and PS, as these are common

techniques that form a basis for introducing the POLEAS variation. I then describe the

POLEAS signal, and guide the reader through the theoretical model. I show the simple and

compact apparatus needed to realize the POLEAS method. I report on its reliable

performance when used to stabilize laser frequencies for atomic laser-cooling experiments,

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and point out its relative merits when compared to D-F SAS and PS. In the final section I

address environmental sensitivities of the various spectrometers, and discuss

measurements that indicate their relative long-term stabilities.

4.2 The Spectroscopy Signals

Both PS and POLEAS rely on the effects that polarized atoms have on a probe beam.

An atomic vapor can be polarized using a circularly polarized pump beam. The polarized

vapor has different electric susceptibilities,

and

, for right- and left-handed circularly

polarized probe light. These complex electric susceptibilities can be decomposed into their

real and imaginary parts; the imaginary parts yield absorption coefficients,

and

, and

the real parts correspond to indices of refraction,

n and

n . By using a linearly polarized

probe beam (i.e. equal mixture of right- and left-hand circular polarization) the differences,

and

n n n , result in clear spectroscopy signals. The refraction

difference, n , rotates the plane of optical polarization and is referred to as gyrotropic

birefringence. The PS method detects this polarization rotation of the probe light induced

by the birefringence ofn . On the other hand, the absorption difference, , makes the

probe beam elliptically polarized (by scattering away more of one circular polarization

than the other). The POLEAS scheme detects this ellipticity of the probe light, induced by

the absorption difference, .

The detectors used in the PS and POLEAS schemes, to observe the effects of n and

respectively, closely resemble each other. A balanced polarimeter is used in the case of

PS, to detect the rotation of the plane of polarization. The balanced polarimeter usually

79

consists of a half-wave plate, a polarizing beam splitter (PBS), and a balanced

photodetector. The half-wave plate rotates the polarization-axis of the light to 45 degrees,

relative to the axis of the PBS [6,7]. In this way, the difference in power at the two

detectors is correlated with a polarization rotation from the atomic vapor.

Correspondingly, a circular analyzer (see Figure 2) is used in the case of POLEAS, to detect

the ellipticity of the polarization. This circular analyzer is implemented by replacing the

half-wave plate in the balanced polarimeter with a quarter-wave plate. When the quarter-

wave plate is at 45 degrees, the two circular polarization components are projected onto

the horizontal and vertical axes of the PBS. The PBS then separates these two components

and directs them onto the balanced photodetector. Note that it is possible to achieve either

the balanced polarimeter or the circular analyzer using the same optical components by

adjusting the various rotation orientations, as pointed out by Yashchuk et al. in the context

of magneto-optical spectroscopy [9].

4.2.1 Doppler-Free Saturated Absorption Spectroscopy

Doppler-free saturated absorption spectroscopy (D-F SAS) is the most commonly

used spectroscopic method for frequency-locking a laser. It is a version of pump-probe or

nonlinear spectroscopy, meaning that, due to the modification of the population density of

the atoms by the pump beam (through optical pumping and saturation effects), the

absorbed power has a nonlinear dependence on the input beam power. Actually, saturated

absorption spectroscopy is a misnomer since optical pumping effects dominate over

saturation effects with regards to the observed spectroscopy signals [10,11]. A basic D-F

SAS configuration is given by MacAdam et al. [3], and is shown in Figure 4.1. The simplest

method to lock the frequency of a laser using this method is called “side-locking”; where an

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applied dc offset voltage shifts the signal such that it crosses zero volts, and hence the side

of the peak is used as the frequency correction signal [3]. Though simple, this method is

not particularly stable, since the lock point is sensitive to amplitude fluctuations. More

often a “peak-lock” method is used, which is first-order insensitive to amplitude

fluctuations and therefore more stable. The peak-lock method relies on phase-sensitive

detection of a modulated signal, resulting in a derivative of the spectroscopy signal, which

crosses zero at the peaks’ maximums. This derivative, after some appropriate filtering,

provides the correction signal that stabilizes the laser’s frequency. The stability and sub-

Doppler, hyperfine resolution of the D-F SAS method account for its widespread use and

proven effectiveness for myriad precision atomic physics experiments.

One drawback of the D-F SAS peak lock method is the need for modulation of the

signal. The easiest way to apply modulation to the signal is by modulating the current of the

diode laser, but this means that the modulation is on all of the light, and this is undesirable

for many experiments. There are numerous techniques for circumventing this drawback,

Figure 4.1. Basic Doppler-free saturated absorption spectroscopy configuration. Diagram from MacAdam et al. [3].

81

including alternate modulation schemes and alternate types of spectroscopy (e.g. PS or

DAVLL). One alternative modulation scheme applies an oscillating magnetic field on the

atomic vapor [12–14], which Zeeman shifts the atoms’ magnetic sublevels. By applying the

modulation to the atoms instead of the light, the light remains “clean” for the intended

experiment. Another alternative is to use an external optical modulator, such as an acousto-

optic modulator (AOM), to modulate only the light in the spectroscopy locking system. This

method has been used many times in our labs, in a configuration that sends only the pump

beam through an AOM.

The technique of sending D-F SAS’s pump beam through an AOM works well, but

there are a couple issues to be aware of. First, the frequency of the laser is offset from the

atomic resonance frequency by ½ of the AOM’s RF drive frequency. This is because in the

lab frame the pump and probe beams are now detuned from each other by an amount

equal to the RF drive frequency. Therefore, instead of interacting with the class of atoms

that have zero (longitudinal) velocity, the laser light interacts with a nonzero velocity class,

which is Doppler shifted into resonance with both beams. The second issue to keep in mind

is the spatial deflection of the beam by the AOM. When the dither is applied to the AOM, the

beam is spatial swept at the dither frequency, so care must be taken to overlap the pump

and probe beams as well as possible. If the beams are not centered on each other, the

spatial modulation can distort the correction signal in deceptive ways due to the increased

amplitude modulation that is present in addition to the usual frequency modulation.

82

4.2.2 Polarization Spectroscopy

As the POLEAS method is related to modern versions of polarization spectroscopy

(PS) in many ways, I briefly review the development of PS. Polarization spectroscopy,

originally introduced by Weiman and Hansch [5], has seen a number of refinements over

the years. In its original form, a magnetically shielded vapor cell is placed between two

nearly-perpendicular polarizers, and the probe beam is aligned through this set-up and

onto a detector (see Figure 4.2a). A circularly-polarized pump beam counter-propagates

through the vapor cell (overlapped with the probe beam), and partially polarizes the atoms.

The polarized vapor increases the transmission of the probe light through the crossed

polarizer. The authors pointed out that such PS signals have larger signal-to-background

ratios than SAS. Unfortunately this PS scheme is quite sensitive to amplitude and

temperature fluctuations, which translate into frequency drifts of multiple megahertz.

About a decade ago, two groups independently improved the stability of the PS laser lock

by adding a balanced polarimeter [6,7] (see Figure 4.2b). This method was better at

removing the background and yielded a more symmetric dispersion-like correction signal,

which improved the stability to ~1MHz/h [7]. The frequency drift could be further reduced

by enclosing the PS in a temperature controlled box, but this indicated that the

temperature sensitivity was still too high for many applications. The temperature

sensitivity was likely due to waveplates (rotating the polarization), birefringence of the

vapor cell windows, and other optical components. In 2006 a further modification, termed

bi-polarization spectroscopy (see Figure 4.2c), largely mitigated the temperature sensitivity

and reported frequency drifts of only 0.05MHz over 3 hours [8].

83

4.3 Theory

Figure 4.2. Evolution of polarization spectroscopy apparatus. a.) 1976 Wieman and Hansch [32] b.) 2003 Yoshikawa et al. [34] c.) 2006 Tiwari et al. [35]

a.

84

In this section I give a theoretical description of the POLEAS scheme adapted from

previous models developed for PS and SAS. I start with a simple classical model adapted

from Pearman et al. [6], then proceed to a full quantum mechanical derivation.

4.3.1 Classical Model

The theory for the POLEAS lock can be directly adapted from much of the work

already done on polarization spectroscopy. In this way I will first model the experiment

following the methods laid out in [5,6]. The signals can also be derived from first

principles [15–17] with accuracy that is more than sufficient for the present purposes; this

allows the discrimination of saturation and optical pumping effects [10,11].

I begin with an empirical model based on previous work describing polarization

spectroscopy. The electric field of the probe beam traveling in the z direction can be

written as

0

ˆkzi t

E E e . (0.45)

If the beam is linearly polarized at some angle, , with respect to the horizontal x

axis, then the amplitude vector can be written

0 0

cos 1 1

2 2sin

xi

y

iE e eE E

E i i , (0.46)

where the last form is in terms of circular polarization basis vectors. The birefringence

results in differing absorption coefficients and indices of refraction for the two opposite

polarizations. The windows of the cell can also be birefringent due to stresses from the

85

manufacturing process and pressure gradients. Thus the electric field of the probe-beam

after passing through the vapor cell is

0

1 1exp

2 2 2

i i

i i

R I

e eE E i nL b ib i e

Le

c i i (0.47)

where

1

2 4 2R I

LnL b i b

c

ia

(0.48)

Here, n is the refractive index of the gas, L is the length of the cell, bR and bI are the real and

imaginary parts of the windows’ birefringence, and α is the absorption coefficient of the

gas. In general the values for n, bR, bI, and α differ for the two orthogonal circular

polarizations, as such, when written alone they represent the average over the two

polarizations, but when paired with Δ take the difference of their respective values.

These equations are essentially those given by Pearman et al. [6] for polarization

spectroscopy, but here rather than going directly into the polarimeter, I first send the

probe beam through a quarter-wave plate. This converts the circular components to linear

components, and thereby reveals just the absorptive parts of the signal. This modification

to the model just requires multiplying by the Jones matrix for the quarter wave plate

2 2

2 2

4 4

4 4

cos sin 1 cos sin@

1 cos sin sin cos

1 11 1@ 45

2 1 1 2

i i

i i

i iQ W P

i i

i i e eQ W P

i i e e

(0.49)

86

After passing through the quarter wave plate the electric field is now

4 4

0

4 4

0

2

1 0exp

2 0 12 2

1 01

0 12

R I

R

i i

i nL ba

I

b i ac

Li

L e eE E i nL b ib i

c

E e e e e e e

(0.50)

And signal intensity at the photodetector is then

2 2

0

2

0

1

2

sin h 2I

signal y x

y x

L b

I I I

c E E

I e a

(0.51)

Following the approximations described in [6], including small angle and assuming

Lorentzian lineshape where

0

21 x

(0.52)

with 0

being the max absorption difference at line center, and

0

2x is the

frequency detuning scaled in units of half linewidth, we finally obtain

2 0

0 22 1

IL b

signal I

LbI I e

x (0.53)

4.3.2 Quantum Mechanical Model

A more detailed theoretical model gives insight into what is happening at the atomic

level, and is useful for distinguishing the effects of optical pumping and saturation and

87

predicting the dependence of the signal on beam diameter and intensity. Much of the

quantum mechanical modeling of SAS and PS has been developed and refined by Moon et

al. [10,15,17–25], though many other groups have contributed as well, for example Hughes

et al. [16]. Here I will relay the work done in the literature, summarizing some of the

mathematical details, providing background where helpful, and filtering out parts that are

not directly needed for the present experiment. In the latter publications by Moon et al.,

they provide concise formulas with tabulated results that can be used directly to give the

predicted spectra, but in order to follow the calculation it is necessary to start with their

previous work.

The calculation proceeds in the following way: begin by writing down the optical

Bloch equations, from which we get the rate equations for the various atomic levels. The

rate equations are used to calculate the electric susceptibility. The susceptibility contains

the birefringence (refractive index, n) and the dichroism (absorption coefficient, α ) as

shown in [26]

1

1 R e

m

2

I

n

k

(0.54)

where k is the wavevector. The real part (the refractive index) yields a dispersive curve as

used in polarization spectroscopy, but for the POLEAS method we use the imaginary part

(the absorption). The transmission of the probe beam through the cell then follows the

derivation in the previous section, where for the absorptive part this is known as Beer’s

law

88

L

T e (0.55)

where δ is the frequency detuning.

The equation of motion for the density matrix is concisely written as

,sp

iH (0.56)

where the last term is put in by hand to account for spontaneous emission. From this we

obtain the optical Bloch equations (see for example [27]) as stated in [16],

2

2 2

2 2

2

R

ee ge eeeg

eg gg eg

ge gg ge

gg

R

eg ee

R

ge ee

R

g ege ee

i

i i

i i

i

(0.57)

where and ee gg

are the excited and ground state probabilities, respectively, R

is the

Rabi frequency, and is the decay rate of the excited state. These equations are in the

field-interaction representation as denoted by the tilde over the off-diagonal or coherence

terms, and eg ge

. This field-interaction representation is in a frame that is rotating at the

frequency of the laser field, and the rotating wave approximation is applied (RWA). This

representation is commonly used when an atom is interacting with a nearly-resonant,

monochromatic field. The steady state values can be used for the off-diagonal terms since

the coherences approach those values at times much less than the interaction time (the

89

average time each atom spends in the beam). This allows us to set the time derivative of

the off-diagonal terms to zero, and solve for the diagonal terms (rate equations)

2

2 21 4

ee ggR

ee ee (0.58)

The last term is the rate of spontaneous emission, while the first two terms are

stimulated emission and absorption respectively.

Equation (0.58) now needs to be applied to each of the relevant Zeeman sublevels.

To simplify the notation, and follow that used in the literature, we write the population of a

particular ground state (excited state) Zeeman sublevel as m

FP (

m

FQ ), in place of its density-

matrix element,

, ,F m F m

. If the pump laser beam has a frequency detuning of from the

1F F atomic resonance, and the hyperfine splittings of the excited states is '

e

e

F

F, and

the individual normalized relative line strengths are written as ,

,e e

g g

F m

F mR , then the rate

equations can be written as [17],

1 1 1, ,

1 , ,

1 1 1

1 1

0

,

2 ,

1 1

1,

, ,

1

21 2

0

21 2

2 1 4

2 1 4

ee e e e

e

e e ee

e e e

e

e e

e

eg

e

e

m m qF F mF FF m q F m mm

F F m F m F

F F F F m m

F mF m mm

F F m F

F F

F

F

F

m

F

m

m q m mF FF mm

F F m q F m

m m

P QP R R Q

P R Q

P QQ R

s

kv

Rs

kv

,

, '

e e e

e

g

F m m

F

F F F

Q

(0.59)

90

Here 2

02

R pum p sa ts I I , and 1,0q , which corresponds to

, polarizations

of the pump beam, respectively. The term in the parenthesis is the effective total laser

detuning as seen by an atom that is travelling with velocity, v . Atomic line strengths were

discussed in the atomic physics introduction chapter of this thesis. The normalized relative

hyperfine linestrengths used above are given by,

,

,

2

2 1 2 1 2 1 2 1 2 1

1

1 1

e e

g g e e g e g

e e e e g e

g g g g g e

F m

F mR L J J F F

L J S J F I F

J L F J m q m

F (0.60)

The rate equations of (0.59) can be solved iteratively to first order in the intensity

parameter, 0

s , for all of the atomic level populations. This is a tedious process that

fortunately has been carried out, described, and tabulated elsewhere [17,22]. Once the

populations are known the polarization and hence susceptibility can be calculated.

Recall that the electric susceptibility ( e

) conveys the degree of polarization (P) of a

medium when subjected to an electric field [27,28],

0 e

P E . (0.61)

The polarization is the average over the individual induced atomic dipole moments,

P x N d , (0.62)

91

where the average is taken over a small volume centered at x , and N is average number

density. The atomic polarizability ( ) is another closely related term that has a similar

meaning to e

but at the individual atomic (or molecular) level, such that

0 local

P Nd N E . (0.63)

Here we have to distinguish between the ambient applied electric field and the local

electric field, immediately surrounding the atom, which can be quite different depending

on the atom’s polarizability. In the special case where the local and ambient fields are the

same then e

N . As an aside, the atomic polarizability has seen renewed attention in

recent years as it is related to some of the limiting uncertainties in both microwave

fountains and optical clocks [29–31]. Using the quantum mechanical density matrix

formalism, these quantities translate directly over. Recalling that the expectation value of

an observable, in this formalism, can be written as the trace of a density matrix,

ˆˆO Tr O , the polarization along some axis q is,

ˆˆq q

P N Tr D . (0.64)

The velocity distribution of the atoms must also be accounted for. The atoms’

velocity not only Doppler shifts its resonant frequency, but also affects the interaction time

as the atom traverses the laser beam. Details on the modeling of these effects can be found

in [16,25], but in summary this involves integrating over the velocities weighted by the

Maxwell-Boltzmann distribution,

92

21v

v uf dv e dv

u, (0.65)

where 2B

u k T m is the most probable velocity. In the more recent papers by Moon et

al. [17,19] the Doppler shift (kv) is written in the denominator of the rate equations with

the other frequency detunings, and therefore the velocity effects are not addressed

separately as in the earlier papers.

We can now write down an expression for the susceptibility as given by equation

four from Moon and Noh [19],

2

, 1

m3

2 1

,3

4 2 /

e

e

ee

F m m m q

F F Fv

FmF

F

uR P Q

i

Ndv e

u kv (0.66)

where λ is the resonant wavelength, and m

FP is the total ground state population given by

the sum of the dipole allowed transition steady state solutions,

1, ,

1

,

1

1

1

2

2 2 1

p p p p

p

p

p

p

pp

F m F mm

F F F p p F F p p

F m

F

F

F p

F

F

F p

P P k P k

P kI

(0.67)

The first three terms are the ground state populations as the laser is tuned near the

respective transition, and δp is the detuning from the 1p p

F F transition while Δ is the

frequency spacing between the two designated excited hyperfine levels. The last term

prevents double counting the background ground state population distributed among the

93

sublevels, present even in the absence of a pumping laser beam (for 87Rb this term equals

2/8).

The full expansion of the expression for χ is given by Moon and Noh [19], and they

subsequently describe the calculation of the relevant integrals. The solution of the integrals

results in a function of the form,

1,

1 2 1 1

bL a b

b a i b. (0.68)

The imaginary part of this function corresponds to the absorption and has a

Lorentzian profile, whereas the real part yields a dispersion profile corresponding to the

refractive index. Polarization spectroscopy relies the latter (real), while saturated

absorption and the present POLEAS variation rely on the former (imaginary), given by,

2

2

1 1,

1 4 1 1i

b bL a b

b a b

. (0.69)

For this imaginary part, 1 1 b is the normalized linewidth, and a is the

normalized frequency. Many of the absorption peaks involve the summation of multiple

Lorentzian functions; in their more recent report [10], the authors show a simplifying

approximation for summing multiple Lorentzians into a single Lorentzian. In this way they

were able to succinctly give the solutions for each peak, which they tabulated in the

appendix, [10].

94

I used these developed theoretical models to predict the POLEAS signal for the

Figure 4.3(a) 87Rb D2 level diagram (not to scale) showing population optically pumped by σ+-polarized light to the 2,2 state. The numbers and dashed arrows show the allowed

transitions and their relative strengths. (b) Predicted spectra of the POLEAS apparatus. The dashed lines are the individual signals (with Doppler-backgrounds removed and vertically offset for clarity) that would be detected by the two arms of the circular analyzer, and the solid line is the full POLEAS output signal of 87Rb D2 2F F manifold. Frequency units are natural linewidths (Γ = 2 π 6 MHz).

95

1 2 3 2

5S 2 5P 1 , 2 , an d 3F F transitions of 87Rb atoms. In

Figure 4.3(a), I show the relevant energy level diagram of atoms optically pumped by the

σ+ pump beam. The allowed σ± probe transitions and their relative strengths are

represented as well.

Figure 4.3(b) shows both the two individual signals detected by the arms of the

circular analyzer(the Doppler-broadened background has been omitted and the signals

offset vertically for clarity), and the full POLEAS output resulting from the difference

between the two individual signals.

The resonance features of the individual signals are all positive peaks except for the

2 3 F F resonance, which is a negative peak. The dominant mechanism behind

these features is optical pumping, both between Zeeman sublevels and hyperfine

states [4,14,20]. Hyperfine depopulation pumping is effective in the majority of transitions

and is largely responsible for the increased transmission, resulting in positive peaks. In the

case of the 2 3 F F transition, hyperfine depopulation pumping is ineffective due to

the transition’s closed cycling nature (i.e. there is no direct dipole-allowed decay path to

the dark 1F hyperfine state from 3,3 ). Pumping among the Zeeman sublevels causes

population to build up in the 2,2 stretched ground state sublevel (i.e. the atoms become

polarized). The probe beam absorption is thus enhanced, resulting in a negative peak.

96

4.4 Experimental Apparatus

The experiments described in this chapter use the following laser configuration: a

780 nm distributed-Bragg-reflector (DBR) diode laser beam is collimated, sent through a

60 dB optical isolator, and coupled into a single-mode polarization-maintaining fiber. The

fiber output collimator yields a Gaussian beam of 21 e diameter 7.5 mm. Immediately after

the output collimator is a polarizing beam splitter (PBS), configured so that the optical

power coming out of the two PBS ports is divided equally (i.e. half S polarized and half P

polarized). Irises and neutral density filters are subsequently used to control the size and

power of each output beam. One beam is sent to a standard Doppler-free saturated

absorption spectroscopy (D-F SAS) configuration [3], and the other beam is sent to the

POLEAS configuration.

The POLEAS apparatus is diagrammed in Figure 4.4. An input beam passes through

a linear polarizer (I use the S port of a PBS as shown in Figure 4.4), and then a zero-order

Figure 4.4. The polarization enhanced absorption spectroscopy apparatus. PBS is polarizing beam splitter; λ/4 is quarter-wave plate; Pol is polarizer; M is mirror; BPD is balanced photodetector.

97

quarter-wave plate to obtain the circularly polarized pump beam. This circularly polarized

pump beam then passes through a pyrex, 7.2 cm long, room-temperature, rubidium vapor

cell. The vapor cell is magnetically shielded by μ-metal such that the magnetic fields are

reduced to below 2 mG in the transverse direction, and below 10 mG in the longitudinal

direction. Following the vapor cell is another polarizer, which not only linearizes the

polarization, but also reduces the power, yielding the desired probe beam. This probe beam

is nearly retro-reflected upon itself by mirrors, but the mirrors are angled just enough that

the probe beam can be spatially separated from the incoming pump beam. The passage of

the probe beam through the quarter-wave plate, after probing the atoms in the vapor cell,

distinguishes POLEAS from PS. This results in the absorption profiles rather than the

dispersion profiles, as described previously. This quarter-wave plate in conjunction with

the subsequent PBS acts as a circular analyzer, measuring the ellipticity of the probe beam.

The two components of the probe beam are directed, by the PBS, onto a balanced

photodetector, which amplifies and subtracts them.

4.5 Experimental Results

The spectra of the 87Rb 2F F hyperfine manifold using the D-F SAS and POLEAS

methods are shown in Figure 4.5. The dramatic increase in signal strength for the

2 3F F cycling resonance is clearly evident. These spectra were obtained with the

same input beam parameters (27 µW, 3 mm diameter), and the same magnetically shielded

vapor cell, so as to make as fair a comparison as possible. The theoretically predicted

spectra are shown in Figure 4.5(a), and the experimental results in Figure 4.5(b). The

agreement between theory and experiment is reasonably good. The slight discrepancies

98

are attributed to residual magnetic fields and the approximately 1 MHz laser linewidth,

neither of which were accounted for in the theory.

99

This enhanced S/N ratio can be leveraged to improve the stability of a laser

frequency lock. I locked the frequency of my laser to the 2 3F F peak using the

POLEAS method. This was done by the common phase-sensitive peak-lock method that is

often employed with D-F SAS signals. Here, the laser- diode’s current is modulated at 10

kHz, and the error signal is filtered using proportional and integral gain. The resulting

Figure 4.5. (a) Theoretically predicted spectra. (b) Experimental data. Dashed line is Doppler-free saturated absorption spectra (D-F SAS), and solid line is polarization enhanced absorption spectra (POLEAS) taken under similar conditions. Units are the same for both figures.

100

correction signal is fed back onto the laser current. The servo loop bandwidth is limited by

the 10 kHz modulation signal, and no attempts were made to increase this bandwidth, as I

am chiefly concerned with long-term stability. In principle, the bandwidth could be

increased up to a significant fraction of the atomic transition linewidth, though applying

such high-frequency modulation to the light would require an external optical modulator

(such as an acousto-optic or electro-optic modulator). I find the POLEAS method of laser

stabilization to be robust and reliable in that the laser remains locked for days, and

provides consistent performance for atomic laser cooling experiments.

While the increased signal strength of the cycling resonance is the most striking

attribute of the POLEAS method, there are other appealing aspects that could prove equally

beneficial. The simplicity of the experimental configuration and the fact that the Doppler

background is automatically removed without the need for a separate beam (as in D-F SAS)

mean that POLEAS can readily be implemented in a compact package. Furthermore, by

enhancing the cycling transition signal and simultaneously reducing the other nearby

spectral features, an auto-locking laser stabilization scheme becomes easier to accomplish.

These are two important factors to consider for remote, autonomous, or space-bound

systems, such as atomic clocks for satellite navigation.

4.6 Discussion

The frequency stability of an oscillator is often characterized by its Allan variance,

[20]. At short times, assuming white noise only, the Allan variance is inversely

proportional to the signal-to-noise ratio (S/N) of the spectroscopic resonance signal,

101

1

S N , [21]. This indicates that the short-term frequency-stability of a laser

locked to the closed 2 3F F transition using the POLEAS method can be significantly

better than that using the D-F SAS method, since the S/N of POLEAS is much greater than

D-F SAS. At longer times, the frequency stability is limited by drifts of the lock point due to

environmental sensitivities of the spectrometer.

Although D-F SAS is routinely used and has demonstrated adequate long-term

frequency stability for many atomic physics experiments, PS appears more susceptible to

environmental perturbations. Temperature sensitivity in PS has been reported to be a main

cause of the observed drift rates of order MHz/hour [7]. Without special attention paid to

mitigating these drifts, PS would not provide consistent, day-to-day, long-term frequency

stability sufficient for many applications. It should be noted that a more complicated, but

related, configuration termed bi-polarization spectroscopy (BPS) has realized reduced drift

rates compared to conventional PS [8].

One important difference between the methods, related to temperature sensitivity,

is the use of wave plates, which are known to be temperature sensitive (of order

4

10 C for zero-order wave plates). Whereas D-F SAS typically uses no wave plates,

PS typically uses one quarter-wave plate and one half-wave plate, and POLEAS uses a single

quarter-wave plate. I measured the dependence of the POLEAS lock-point on the angle of

its quarter-wave plate by locking the laser to the POLEAS signal and using the D-F SAS for a

frequency discriminator. I rotated the POLEAS quarter-wave plate by ±5 degrees around its

nominal 45 degree alignment, and found the frequency-to-angle dependence to be 260 ± 80

kHz/degree, which corresponds to an approximate temperature sensitivity of 104 ± 32

102

Hz/°C (using the typical temperature coefficient for zero-order plates given above). I set up

a PS experiment for comparison, and although its quarter-wave plate demonstrated similar

sensitivities the half-wave plate was found to be roughly one hundred times more sensitive.

The high sensitivity of the PS lock to its half-wave plate is due to the direct impact that the

half-wave plate has on the amplitude balance between the photodetectors of the balanced

polarimeter. This half-wave plate is likely a significant contributor to the frequency drifts

previously reported with PS laser stabilization.

Another difference between the spectrometers related to temperature sensitivities

is that D-F SAS and POLEAS rely on absorption coefficients, while PS relies on

birefringence. Reports in the literature have cited temperature-dependent birefringence of

cell windows as a vulnerable component, due to the mechanical stresses on the windows

from the manufacturing process [7]. The D-F SAS and POLEAS methods have the advantage

of relative insensitivity to changes in the windows’ birefringence.

Magnetic field sensitivities are another aspect that must be considered when using

spectroscopy-based laser stabilization. Generally, magnetic shielding is not used for D-F

SAS, but it is for PS and POLEAS. This is because, with D-F SAS one often locks to a

crossover peak for greatest S/N, and crossover peaks are visible in the presence of fields

typically experienced in the lab. Such visibility is not necessarily true for the 2 3F F

cycling transition. This transition is strongly affected by the mechanisms of optical

pumping and atomic polarization, which rely on the relative orientation of the quantization

and optical polarization axes, and hence are sensitive to external fields. In POLEAS and PS,

103

this cycling resonance provides the greatest S/N, and therefore magnetic shielding is

desirable for consistent, optimal performance.

The effect that magnetic fields have on the spectrometers’ signals depends on their

relative orientation. Transverse magnetic fields tend to reduce the atomic spin polarization,

and correspondingly the signal strengths observed with POLEAS and PS. Longitudinal

magnetic fields result in a resonant Faraday or Macaluso-Corbino effect [22] that can be

observed as a change in the Doppler-broadened background signal, but this has much less

impact on the narrow sub-Doppler features. To investigate this I placed the cell in a

solenoid, and shielded them both with mu-metal. In this way I could vary the longitudinal

field from 0 to 1500 mG, while keeping the transverse magnetic field less than 4 mG. To

measure the POLEAS lock point sensitivity to longitudinal magnetic fields I locked the laser,

as usual, to the 2 3F F cycling peak and used the D-F SAS as a frequency

discriminator. I observed no measurable change in the lock frequency when ramping the

longitudinal magnetic field from 0 to 1500 mG. In fact, with the solenoid providing a

longitudinal field of 750 mG, I could forego the mu-metal shield and yet the amplitude of

cycling peak remained with 88 % of its shielded value (without the shield the transverse

magnetic field varied over the length of the cell, but was everywhere less than 400 mG).

Furthermore, as a very crude test I observed that, by placing a small permanent bar magnet

(aligned longitudinally) near the cell, neither the solenoid, nor the mu-metal shield was

needed. In this case the longitudinal magnetic field varied from 150 to 500 mG over the

length of the cell, never-the-less the amplitude of the cycling peak was 67 % of its shielded

value. This indicates that the POLEAS method can be used without the expensive magnetic

shielding.

104

4.7 Summary

I have demonstrated a variation of pump-probe spectroscopy that is particularly

useful for laser frequency stabilization. The polarization enhanced absorption spectroscopy

(POLEAS) signals provide a significant improvement in S/N over standard D-F SAS for the

important and commonly used 2 3F F cycling transition. This improvement can

directly increase the short-term stability of a laser frequency-lock, given sufficient loop

bandwidth. The long-term stability of the POLEAS method is comparable to D-F SAS, and

significantly better than standard PS. The POLEAS signal is automatically Doppler-free,

without requiring a separate Doppler subtraction beam, and lends itself to straight-forward

compact packaging. Finally, by increasing the amplitude of the desired (cycling) peak while

reducing the amplitude of all other peaks in the manifold, the POLEAS method eases the

implementation of auto-relocking schemes.

105

5 Optical Phase-Lock Loops

5.1 Introduction

The central design strategy for the laser system of the transportable atomic fountain

was to use multiple internal-grating diode lasers whose frequency would be controlled via

optical phase-lock loops (OPLLs). This design brings a tremendous simplification of the

optical system; comparing this strategy to the PARCS design described in Chapter 3, I can

eliminate roughly 70 % of the components, including various fiber collimators, polarizing

beam splitters, optical isolators, atomic vapor cells, photodetectors, half-wave plates,

quarter-wave plates, double-pass acousto-optic modulators (AOMs), and single-pass AOMs.

Eliminating all of these components not only reduces the cost and power consumption, but

eases assembly and increases stability of the optics package.

Phase-locked loops are a ubiquitous piece of technology, and have been studied and

used with RF oscillators since the 1930’s. With the invention of coherent optical sources

the techniques were adapted to optical oscillators (i.e. lasers), creating OPLLs. Despite this

long history, OPLL technology is much less developed than its RF and MW counterparts.

This is due largely to the challenges associated with laser manufacturing, especially as

compared to the tremendous progress with electronic integrated-chip manufacturing that

RF and MW technologies benefit from. Internal-grating semiconductor lasers are, in many

ways, the best candidate for implementing OPLLs since they are compact, rugged, relatively

cheap, and can be frequency-tuned with electrical current. The telecom market has driven

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the development of this technology, which is now migrating to other non-telecom

wavelengths.

In this chapter I will begin by introducing some of the basic tools used to

characterize OPLLs. The most important figure of merit in this case is the residual phase

error variance, which conveys the level of phase coherence between the locked lasers. I

show that the reason for the underlying challenge of phase-locking internal-grating

semiconductor lasers stems from the high loop bandwidths necessitated by their

considerable linewidths (~0.6 MHz in this case).

With the need for large bandwidths established, I then discuss the frequency

modulation properties of my lasers. I explain the mechanisms behind their frequency

tuning, and the source of the phase reversal in their transfer function. I describe the

methods I used to measure the transfer functions of my lasers. I show that this phase

reversal occurs at a relatively low frequency (~0.8 MHz). This internal phase delay of the

lasers themselves is the limiting factor in the bandwidth of the loop, and hence the phase

coherence of the lasers.

I then address the design of the loop filter in the OPLL. I show the results from my

early attempts. The final version significantly improved the phase-coherence between the

lasers by adding a phase lead compensator. I describe my final circuit design, which

integrated a high-speed photodetector, phase/frequency discriminator, and tunable loop

filter into a single compact package. Finally, I show the spectrum of the locked beat-note

indicating the improved phase coherence.

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5.2 Loop Characterization

A diagram of a basic OPLL is shown in Figure 5.1. Beams from the master and slave

lasers are overlapped and directed onto a photodetector. Presuming the frequency

difference between the lasers is within the bandwidth of the detector, the detected

beatnote signal is amplified and then compared with a reference oscillator signal. This

comparison creates the error signal: a voltage related to the phase/frequency difference

between the beatnote and the reference oscillator. The error signal is then appropriately

filtered and fed back to the slave laser. In this way the slave laser is phase-locked to the

master laser (within the bandwidth of the loop) at an offset frequency determined by the

reference oscillator.

Figure 5.1. Schematic diagram of optical phase-lock loop (OPLL).

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The purpose of the loop is to minimize the phase error, e o i

, defined as the

difference between the input phase and the output phase. In the present case of the

heterodyne loop shown in Figure 5.1, this becomes

e s m ref (0.70)

where the subscripts m, ref, and s designate master laser, reference oscillator, and slave

laser respectively. In reality these phase values are not steady state, but vary in time. In the

field of control theory it is most common to use Laplace transforms and work not in the

time domain, but rather in the complex frequency domain, t s i . When the

Laplace variable s is completely imaginary, then the Laplace and Fourier transforms are

equivalent, i.e. 2 fi is . For the basic instructive purposes of this accounting I will

often use f in place of s, and treat the additional factors as implicitly understood.

The open loop (G), closed loop (H), and error (E) transfer functions are

fundamentally related to ratios of the phase in the following way [1]:

1

1

1

o S

e

o

D

i

e

i

f KG f K F f

f f

f G fH f

f G f

fE f

f G f

(0.71)

where KϕD is the gain of the phase detector, Ks is the gain of the slave oscillator, and F(f) is

the transfer function of the loop filter. It is a general feature of voltage-controlled

oscillators (VCOs) and their current-controlled laser diode counterparts [2], that they

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behave as an integrator, hence the 1/f form of its transfer function in “frequency domain”.

The roots of the denominator, 1 G f , (i.e. values of f that give solutions to

1 0G f ) are the poles of the transfer function, and PLLs are often categorized

according to the order of the denominator polynomial. Second-order loops are the most

common, and will be the focus here.

One of the most widely used loop-filter configurations relies on proportional and

integral gain (PI), and this is the original type of loop-filter that I used in my OPLLs, see

Figure 5.2. The transfer function of this loop filter is written

I

P

KF f K

f (0.72)

where KP is the proportional gain factor and KI the integral gain factor with its

characteristic 1/f form. The Bode plots for the magnitude and phase of this loop filter

transfer function are shown in Figure 5.3. If we plug equation (0.72) into equations (0.71),

we get a second-order loop with two integrators (i.e. Type 2), one from the loop filter and

one from the slave oscillator. The magnitude and phase Bode plots for the closed loop

Figure 5.2. Common loop filter design with proportional and integral gain.

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transfer function of this system are shown in Figure 5.4. A plot of the error transfer

function is shown in Figure 5.5. Notice the “high-pass” form of the error plot and the

corresponding “low-pass” form of the loop gain; as the gain decreases the loop is less able

to correct for phase errors.

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Bode plots are useful for visually comparing loop designs and checking loop

stability. If the loop gain is greater than unity at a frequency whose phase shift is 180

degrees, then there is effectively positive feedback and the loop becomes unstable. To

ensure a stable loop there must be an adequate phase margin at the unity gain point. In fact

the slave oscillator acts as an integrator [1]–[3], hence contributing an inherent 90 degree

Figure 5.3. Bode Plots for the loop filter shown in Figure 2. The integrator boosts the gain at low frequencies where there is a larger phase stability margin. This increases the hold-in range by compensating for slow thermal and acoustic perturbations which tend to require a large dynamic range.

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phase shift, therefore in effect the phase shift in the remainder of the loop needs to be kept

below 90 degrees at the unity gain crossover.

As the dominant source of noise is phase noise from the lasers, the most important

performance criterion for an OPLL is the variance of the residual phase error,

Figure 5.4. Bode plots of the closed-loop transfer function with loop filter shown in figure 2. This is a second-order, type 2 loop.

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2 2

2,

e

e

m s fr

t

S f df

S f S f E f df

(0.73)

where S f

is the spectral density of the phase error, and fr designates free-running (i.e.

unlocked) [1], [4], [5]. The spectral densities of the master and slave lasers can be concisely

stated in terms of their respective linewidths, 2

2S f f

[6]. Applying this

relation we get,

2

2

2

1

2

m s E f dff

. (0.74)

Since the error transfer function is inversely related to the loop bandwidth, we see from the

above equation that to achieve a small phase error variance we want narrow linewidths

and large loop bandwidths. This is the most critical guiding principle for the present type of

Figure 5.5. Magnitude Bode plot of the error transfer function.

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OPLLs, and I will restate it in the simplest terms (for a more detailed development see

appendix of Ohtsu [3]).

2

2m s

wB

. (0.75)

From this relation we can immediately see the challenges of electronically phase-locking

semiconductor lasers. The linewidth of my internal grating diode lasers is roughly 1 MHz,

which means for good performance, a loop bandwidth greater than 10 MHz is desirable.

This alone is not prohibitive, but I discovered that in my case the lasers themselves have a

phase delay that severely limits the loop bandwidth.

Using a spectrum analyzer to measure the spectrum of the locked beat note, it is

possible to quickly estimate the phase error variance using the following relation [7], [8]

2

carrier

total

Pe

P

. (0.76)

By integrating the power in the spectrum (over a reasonably large bandwidth) and taking

the log of the ratio of the power in the carrier peak to the total power, a good estimate of

the phase error variance is achieved.

5.3 Laser Transfer Function

We have seen that maximizing the loop bandwidth is a critical step in establishing a

good phase lock. The physical path length of the loop can often contribute a significant

phase delay that degrades the phase coherence [8]–[10]. When using fiber optic

components, it is challenging to attain loop propagation delays less than approximately 10

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ns. For a signal at 10 MHz this path delay results in a significant phase shift of 36 degrees,

therefore minimizing the path length is desirable. In my case the path length was not the

limiting factor; rather, it was the response of the laser.

Semiconductor lasers (especially those with internal gratings) are convenient for

electrical feedback OPLLs because the servo signal can be applied directly to the laser’s

current. Unfortunately there are various mechanisms that contribute to the current driven

frequency tuning, which results in the response not being uniform. At lower frequencies

the dominant mechanism is thermal. As the current increases the laser-gain-cavity heats

up and expands, decreasing the laser’s frequency (i.e. red-shifting the wavelength). At

somewhat higher frequencies the thermal response is less effective, and carrier-induced

effects dominate. These effects act in the opposite fashion, blue-shifting the laser

wavelength as the current is increased. As a result of these competing mechanisms there is

a “dip” in the transfer function, and an associated phase reversal. This accumulation of

phase threatens the stability of the loop, and effectively limits the loop bandwidth. The

effects have been studied [11]–[13], and alternate laser designs, such as multi-section gain

structures, have seen success in mitigating the phase reversal, but they often introduce

undesirable side-effects. Typically this phase reversal occurs somewhere in the range of 0.1

to 10 MHz.

There are a variety of methods for measuring the transfer function of a laser [7],

[14]–[17], and each require specialized equipment, foremost an appropriate frequency

discriminator. I assembled a Mach-Zehnder interferometer for this purpose, following the

methods described by McRae et al. [17]. A diagram of the experimental configuration is

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shown in Figure 5.6. I used PM fiber splitters and combiners to implement the

interferometer. Light from the laser-under-test is coupled into a 50/50 PM fiber splitter,

and one of the fiber outputs is directly mated with one of the inputs of the fiber combiner;

this serves as the short leg of the interferometer. For the long leg, I inserted a 3 m PM patch

fiber, which therefore introduced a delay of roughly 16 ns. After recombination of the

beams, the final fiber output is directed onto a fast photodetector. The sensitivity of the

interferometer, in combination with the strong frequency tuning parameter of the lasers

(~1 GHz/mA), means that the residual laser amplitude modulation (~0.7 mW/mA) is

negligible. I used an Agilent 89410A Vector Analyzer (dc to 10 MHz) to measure the laser’s

frequency response. The analyzer’s source output is split with one part fed directly into the

Figure 5.6. Fiberoptic-based Mach-Zehnder interferometer setup for measuring laser frequency modulation response.

117

channel one input, and the other sent to the laser with the photodetector signal fed to the

input of channel two.

The ability of the analyzer to track the frequency of its source output greatly

simplifies the predicted signal measurement. Rather than an infinite sum of Bessel

functions, the measured signal in this case is proportional to just the 1J x Bessel function,

and can be written [17]

2

124 cos 2 1 sin

m cfi t tE J k T

(0.77)

where the field amplitude is long short

E E E , fm is the modulation frequency, ϕ’ is the

relative phase of the modulation, ωc is the carrier frequency, and T is the is the time delay

difference between the arms of the interferometer. The modulation index, β, which

characterizes the strength of the modulation is

sinm

m

f TaT

f T

(0.78)

where a is the amplitude of the modulation. Further simplification is possible by operating

in the approximately linear regime. This is achieved by keeping the path difference small

(T<<1/fm), and the amplitude of the modulation small. In this case, aT , and the first

order Bessel function is approximately linear. Increasing the amplitude of the modulation,

such that the Bessel function zeros become visible on a spectrum analyzer, can provide

calibration points, since the zero points depend only on T and are independent of factors

such as detector sensitivity. Often it is sufficient to normalize the data to the low frequency

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response and look at the relative change. The interferometer is sensitive to vibrations,

temperature fluctuations, and drifts of the laser frequency, therefore many scans (~50)

were averaged. The measurements could be improved by actively stabilizing the

interferometer, for example with a piezo–controlled fiber stretcher. Alternatively an AOM

could be used to introduce a frequency offset between the two arms, and thereby shift the

measurement up to a quieter region of frequency space, but this requires a higher

bandwidth analyzer and specialized RF electronics.

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Typical data from the frequency modulation response measurements are shown in

Figure 5.7, and we see that the laser accumulates a 90 degree phase delay at modulation

frequencies of roughly 0.8 MHz. This is in agreement with the 0.7 MHz value reported by

another group who use the same lasers [7]. All of my lasers performed similarly. This phase

delay, coupled with the ~0.6 MHz linewidths of the lasers [18], reveals the inherent

challenge in achieving significant phase coherence using OPLLs.

Figure 5.7. Laser frequency modulation response. Notice the 90 degree phase delay at less than 1 MHz.

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5.4 Loop Filter Design

My initial loop filter design was based on a common implementation of proportional

and integral gain. The circuit element is shown in Figure 5.2. This loop filter results in a

type two, second-order loop as discussed earlier. The gain factors are given by

2 2

1 1

1 1

1 1

P

I

RK

R

KR C

(0.79)

where the standard time constants are 1 1 2 2

, R C R C . With careful choice of

components and printed circuit board (PCB) design, I was able to observe the delta

function signature of phase-locking on the power spectrum of the beatnote (see Figure 5.8).

Based on the ratio of the power in this carrier peak to the total power the typical phase

error variance was ~2 rad2.

I was not able to achieve sub-Doppler laser cooling with this level of laser phase-

coherence. The experiments that pointed to the lack of phase-coherence as the culprit are

described in the previous chapter. In short, by reducing the number of slave lasers to two,

one for horizontal cooling beams and one for vertical cooling beams, I did observe sub-

Doppler cooled atoms, but this configuration does not allow launching of the atoms (only

dropping).

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Adhering to the original laser system design goals, I wanted to avoid resorting to

any external optical stabilization cavity to narrow the laser linewidths. Alternatively,

adding a phase lead compensator into the OPLL can increase the phase margin of the loop

and allow for higher bandwidth gain. I was able to test a high-speed loop filter with

adjustable phase lead, and found that the phase coherence could be improved (achieving

phase error variance ~0.75 rad2).

Figure 5.8. Power spectrum of locked beatnote between master and slave lasers using proportional and integral gain loop filter. This corresponds to a phase error variance of 2 rad2.

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A passive lead filter circuit element is shown in Figure 5.9. It is essentially a high-

pass filter, creating a smaller impedance pathway for higher frequencies, while low

frequencies have a greater impedance. The transfer function for a lead filter is

1 2

w ith 1 2

a

lead lead a b

b

i fF K

i f

. (0.80)

The time constants in terms of the circuit elements shown in Figure 5.9 are,

1

1 2

1 2

a

b

R C

R RC

R R

(0.81)

The effect that the lead compensator has on the loop transfer function can be seen in the

Bode plots (see Figure 5.10). On the magnitude plot it boosts the gain at high frequencies,

while the phase plot shows an increase in the phase margin (pushing the 90 degree

crossover to a higher frequency).

For my final loop filter design, I incorporated the fast photodetector,

phase/frequency detector, and adjustable loop filter into a single compact package, shown

Figure 5.9. A passive implementation of a phase lead compensator.

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in Figure 5.11. I consulted with Terry Brown of the JILA electronics shop, who offered many

helpful suggestions. High speed circuit design brings its own set of challenges, and I found

the book by Johnson and Howard [19] particularly useful in this regard. The circuit

schematic is shown in Figure 5.12, Figure 5.13, and Figure 5.14.

Figure 5.10. Bode plots showing the effect of phase lead compensation. The increased phase margin enables higher loop bandwidths and hence improved phase-locking between the lasers.

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The first subsection, Figure 5.12, of the circuit is a high-bandwidth photodetector. I

Figure 5.11. The front and back sides of the final OPLL circuitry. This box includes a fast photodetector, phase/frequency discriminator, and tunable loop filter. For scale, the tapped holes in the table are a 1 inch grid.

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used a photodiode built into a fiber optic connector receptacle. This eliminates any optical

alignment and corresponding beam-steering optics (aside from the obvious initial fiber

injection described in the previous chapter). This rendered the system more stable and

Figure 5.12. Fast photodetector subsection of my final OPLL circuit board. Light is incident on the biased photodiode. The dc signal follows the lower path and is used as a monitor. The ac signal follows the upper path through two stages of RF amplification before passing to the phase/frequency discriminator section of the loop (see Figure 13).

126

compact. The signal from the photodiode is ac coupled to two RF amplifiers in series

(Minicircuits ERA-3SM). It is important to properly bias the RF amplifiers, and there is a

helpful application note on MiniCircuits’ website [20]. It can be challenging to find choke

inductors with a sufficiently high self-resonant frequency; using two in series can be

helpful. The dc component of the signal is available for monitoring the optical power

incident on the photodiode.

127

The next section of the circuit, Figure 5.13, is the phase/frequency detector. The

Figure 5.13. Phase and frequency discriminator subsection of my final OPLL circuit. An RF reference signal is input to the lower branch, while the upper branch is the beat signal from the photodetector. The two signals are compared by the AD9901 chip, which outputs a differential error signal.

128

design of this section is centered on the Analog Devices’ AD9901 digital phase/frequency

discriminator integrated chip. A unique feature of this chip is that it does not exhibit “dead

zones” commonly found with other digital phase discriminator chips, where a region near

the lock point exhibits no gain. I used this chip in the ECL (emitter-coupled logic) mode of

operation. The amplified beatnote signal coming from the photodector and the reference

oscillator signal are each pre-conditioned with ultra-fast comparators (AD96685). This

turns the signals into square waves, which helps the digital phase/frequency discriminator,

as it has trouble discerning edges when the slope is too small. These comparators also

output the proper levels for ECL signals. The output of the phase/frequency detector is a

pulse train whose duty cycle is proportional to the phase difference between the two input

signals. As this pulse train is integrated with a low-pass filter, a resulting error voltage is

created. If there is a frequency difference between the two input signals then this

discriminator chip essentially spends its entire cycle in high or low voltage state, driving

the slave oscillator into lock. I use the differential output configuration from the AD9901 to

maximize signal swing and slew rate, which helps reduce phase jitter.

The final section is the loop filter, Figure 5.14. First the differential signal is

converted to single output. There is then an initial stage of proportional gain that is

adjustable with a trimpot. This provides an easy adjustment for the overall loop gain. The

next stage is a PI filter, as described above. The choice of the capacitor value sets the corner

frequency of this element. I found that a relatively low corner frequency (1 - 10 kHz, i.e.

capacitor 100 – 10 nF) worked best to assure that the loop was sufficiently stable. The low

frequency gain can be adjusted with the trimpot in the PI op-amp feedback, but the

performance of my loop was never very sensitive to this. I always had adequate low

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frequency gain for good lock acquisition and hold-in range. There is also another low

Figure 5.14. Loop filter subsection of my final OPLL circuit. Here the error signal is tuned with adjustable proportional and integral gain and phase-lead compensation, ultimately outputting a correction signal that is fed-back to the laser’s current.

130

frequency gain boost (and lag filter) following the PI filter, and these assured that the laser

could remain in lock for long periods of time even as the laser frequencies drifted (i.e. large

hold-in range). A lead filter follows the PI filter (in parallel with the lag filter), and the high

frequency gain

with the associated phase lead helped the phase coherence of the locked lasers. I

needed the corner of the lead filter set to a high frequency (~1 MHz, i.e. capacitor of ~100

Figure 5.15. Power spectra of locked beat note using final circuit with lead filter. Phase coherence improved, with residual phase error down to 0.6 rad2.

131

pF) in order to assure loop stability.

With a network analyzer I measured the transfer function of the loop, and verified

the functionality of all the adjustments. I compared these transfer functions to the modeled

circuit (LT Spice) frequency response, and found good agreement. Using this circuit I was

able to improve the residual phase noise variance from roughly 2 rad2 down to 0.6 rad2,

with more than 50 % of the optical power in the phase-coherent carrier peak, Figure 5.15.

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6 Physics Package

6.1 Introduction

The atomic fountain physics package is the vacuum system in which the atoms

undergo Ramsey interrogation. It can be viewed as consisting of three subsections: a

cooling region, Ramsey cavity and drift region, and detection region. The most immediately

distinguishable characteristic between a standard atomic fountain and the transportable

fountain of this thesis is the shortened drift region. This is the volume above the Ramsey

cavity, where the atoms freely evolve in their superposition state. The atoms spend the

majority of their free-evolution time near apogee, as they come to a stop and begin to fall

back down. This means that a relatively short drift region can yet result in long

interrogation times, and hence precise frequency measurements.

The Ramsey cavity is the heart of an atomic fountain, and requires special attention

to design and construction. The bulk of this chapter focuses on the Ramsey cavity. The

design of my cavity follows in the tradition of the Cesium microwave cavities constructed by our

group for the primary frequency standards NIST-F1 and NIST-F2 [1]. In designing a microwave

Ramsey cavity for a fountain frequency standard there are always a number of decisions to be made

on how to optimize for the specific fountain in question. For example, the number and location of

the cavity feeds, the size of the cavity apertures, and the geometry of the cavity all affect the cavity’s

Q and distributed phase, which in turn affects the atoms.

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6.2 The Cooling Region

The cooling region is where the atomic cloud is gathered and cooled in an optical

molasses / magneto-optical trap (MOT). I used a stainless steel “spherical-cube” made by

Kimball Physics for this region, shown in Figure 6.1.

There are six 2.75 inch Conflat ports on the six

faces of the “cube”. I used these ports for the

cooling laser beams. Four anti-reflection coated

Conflat windows are bolted directly onto the four

horizontal ports. Where the eight corners of the

cube would be, there are instead eight mini-Conflat

(1.33 in) ports. On three of these ports I bolted

uncoated viewports for camera access, which

allowed monitoring of the molasses / MOT. One of

the lower mini-ports was used to attach the rubidium oven.

The rubidium oven was made of a OFHC copper tube roughly 20 cm long. One end

had a mini-Conflat gasket brazed on, and the other end was sealed. The tube was reamed

out so that its inner diameter just accepted a glass ampoule of rubidium. With the ampoule

in this tube, the oven is bolted on. After the entire vacuum system is rough-pumped, baked,

and cooled down again, then the ampoule is crushed, releasing the rubidium. A thermistor

is affixed to the copper tube for temperature monitoring. The tube is them wrapped with

heat tape, which is powered through a variac transformer. I did not find that it was

necessary to actively control the oven temperature.

Figure 6.1. Stainless steel spherical cube used for the cooling region. Picture from www.kimballphysics.com.

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6.2.1 Glass Cooling Region

I designed an all glass cooling region shown in Figure 6.2. The glass chamber was

fabricated by Allen Scientific Glass, who specialize in such work. The motivation for the

glass cooling region was the compact size, as it measured approximately 15 cm3. Not only does

this fulfill the requirement for compactness, but it also enhances the possibility of using a MOT

which increases atom number. Most of the best primary standard fountains operating around the

world use optical molasses rather than a MOT. One important reason for this is that the magnetic

trapping fields used in a MOT can cause significant systematic shifts that are difficult to accurately

characterize. This small, all-glass cooling chamber means that the MOT’s magnetic trapping coils

can be much smaller and use much less current, thereby reducing their effect on the rest of the

system.

Figure 6.2. Glass cooling chamber and rubidium oven.

135

One challenge that the compact glass chamber presents is access from the rubidium oven.

The oven structure is also shown in Figure 6.2. The oven needs to dispense rubidium vapor into

the cooling chamber, but we want to minimize the background vapor in the detection region (which

resides just above). For this I designed the oven with a 90 degree elbow to aim the vapor down. An

extended graphite tube was placed between the cooling and detection regions to reduce the particle

conductance. The oven was mounted to a small bellows that serves two purposes: first, it acts as a

thermal break between the oven and the rest of the physics package, and second, it allows some

flexibility when positioning the oven. I assembled to setup in air (i.e. not under vacuum) to test its

functionality. I wanted to make sure that I could adequately heat the tip of the oven dispenser, and

also position the oven so that it did not obstruct the beam bath or thermally short to the rest of the

chamber. It functioned well under these tests, but unfortunately I was not able to test it under full

vacuum conditions. This would be an avenue for further study should this project carry on.

6.3 Detection Region

In designing the detection region, I sought a compromise between compactness and

flexibility for evaluating a variety of fountain schemes. The essential goal of the detection

region design is to maximize capture of signal photons (while sampling the cloud evenly),

and minimize noise from scattered light. There are two main culprits that contribute to

scattered light noise: background rubidium vapor, and chamber surfaces. To minimize

background rubidium vapor I placed a port to an 8 L/s ion pump directly on the detection

chamber, and installed plates of graphite in the bottom of the chamber. To address

scattered light, I placed the windows through which the detection laser beams pass, far

away from the atoms’ path. Conversely, the photodetector windows are close to atoms’

path to maximize the solid angle for light collection. I designed the chamber with 3 inch

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welded windows, which allowed much optical access and flexibility for optical system

design. Welded windows also avoid the “wasted space” consumed by bulky Conflat flanges.

A photograph of the chamber is shown in Figure 6.3.

6.4 State Selection

My state selection scheme differs from the standard method employed by most

atomic fountains. Often a fountain has two nominally identical microwave cavities where

the lower one is used for state selection and the upper one for Ramsey interrogation. I

avoid the extra space and measurement dead time consumed by the state selection cavity

by designing a state selection cavity formed directly in the stainless steel vacuum flange of

the detection region Figure 6.4. This choice of state selection cavity means that the atomic

Figure 6.3. Detection chamber with large welded windows and ports for vacuum pumping.

137

pumping transitions are not the

usual mF = 0 but rather mF =

+/-1. Traditionally state

selection is achieved in the

following way. As the atoms

enter the state selection cavity,

their population is evenly

distributed among all mF

magnetic sublevels; the cavity

then induces a -transition that transfers the mF = 0 sublevel population to the adjacent

hyperfine (F) level, leaving the rest of the atoms to be blown away by a resonant laser

pulse. Alternatively, my scheme will optically pump the atoms into a stretched state of the

lower hyperfine level (F = 1, mF = +1); from there the compact state selection cavity

induces a population transfer to the desired measurement state (F = 2, mF = 0). In this way I

also greatly increase the measured signal, because I get essentially all atoms into the

measurement state.

6.5 The Ramsey Microwave Cavity

6.5.1 Introduction

The Ramsey microwave cavity is central in an atomic fountain. While control and

preparation of the atoms has afforded great strides in metrology, precise control over the

microwave fields is concurrently required, and care must be taken to avoid limitations due

to faulty cavity design and construction.

Figure 6.4. State selection cavity integrated into the Conflat flange above the detection region.

138

As the frequency uncertainty in fountain measurements has pushed below the 10-15

level, the design, construction, and evaluation of the Ramsey cavities has been studied in

greater detail. Although cavity design issues have not been a limiting source of uncertainty,

they can be significant and will likely become more so, as next generation microwave

fountains overcome other limitations. The largest source of uncertainty for the NIST F1

(room-temperature) fountain is associated with the blackbody shift, and hence motivated

the cryogenic NIST F2 fountain. Another significant source of uncertainty is due to

collisional shifts, but here again paths forward for reducing this effect are known (for

example by using juggling schemes to achieve low densities yet retain strong S/N).

Assuming the limitations of these (blackbody and collision) effects can be mitigated, the

largest uncertainty will be due to microwave effects such as microwave leakage and

distributed cavity phase.

The microwave cavity ultimately limits the achievable size reduction when building

a compact fountain physics-package. Methods do exist for making very small microwave

resonators [1], for example by filling them with a dielectric. But such types of small

resonators compromise many of the benefits of larger cavities, such as mode purity, Q, and

phase flatness.

In the following sections I will describe the basic formulation used for designing and

constructing the portable fountain’s Ramsey cavity. After finding the basic dimensions of

the resonator special considerations for the endcaps are discussed. I describe the bent

waveguide design for the coupled cavity feeds, and briefly cover the manufacturing

139

process. I will review the main concerns with fountain cavity design, and summarize some

of the findings regarding modern fountain Ramsey cavities.

6.5.2 Cavity Geometry

Atomic fountains use a cylindrical microwave cavity resonating in the TE011 mode.

A primary concern is phase flatness; ideally all atoms would experience exactly the same

microwave phase as they pass through the cavity. In reality the cloud of atoms spreads out

as it travels, so each atom traverses a slightly different path through the cavity on the way

up and down. The TE011 mode offers the most uniform phase across the apertures of the

cavity, therefore the apertures can be made relatively large to allow more atoms through

while maintaining minimal phase differences across the apertures. Relatedly, this mode has

low losses and therefore high quality factor Q.

In a cylindrical waveguide, general solutions to Maxwell’s equations can be written

with transverse components in terms of longitudinal components,

Figure 6.5. Cylindrical cavity, showing notation.

140

2

2

2

2

c

c

c

c

z z

z z

z z

z z

iE

k

iE

k

iH

k

iH

E H

E H

E

E H

k

H

(0.82)

with the cut-off wavenumber 2 22

ck k and assuming forward travelling waves i z

e . By

separating the variables and applying boundary conditions the solutions, in terms of Bessel

functions, can be found. The solutions for fields of TEnml modes using the notation shown in

Figure 6.5 are,

0

0

0

2

0

2

2

0

2

cos sin

cos cos

sin cos

sin sin

nm

z n

nm

n

nm

nm

n

nm

nm

n

nm

nm

n

nm

p lH H J n

a d

H p lH J n

p a d

H p lH J n

p a d

H

z

a z

a n z

ik p lE J n

p a d

H pE J

p a

a n z

ik a

sin sin

0z

l

d

zn

E

(0.83)

Here nmp is the mth root of the differentiated Bessel function n

J x (e.g. 0n nmJ p ) ,

22

nmk p a is called the propagation constant, and .

141

The first step in designing the cavity is to calculate the dimensions that give the

desired resonant frequency. Analogous to the rectangular resonator where the easy-to-

visualize resonant frequencies are

22 2

2 2

mnl

mnl

ck c mf

x y z

n l

, (0.84)

the cylindrical cavity resonances for TE modes are

2 2

2

nm

nml

pc lf

a d

. (0.85)

Note that for the TM modes it is necessary to replace nmp with

nmp (the roots of the

undifferentiated Bessel function); these various roots are tabulated, and can be found, for

example, in Jackson’s Classical Electrodynamics. For the TE modes, the root nmp with the

smallest value is11p , indicating that TE111 is the fundamental mode. In fact, the TE21 and

TM01 modes are also above cutoff at the driving frequency; and TM111 is degenerate with

the desired TE011. The TM modes are perpendicular to the driving field, so they are much

weaker. Furthermore, the TM01 modes are spectrally far (gigahertz) from resonant. The

apertures in the cavity endcaps help suppress undesired modes, especially perturbing the

TM111 resonance away from the driving frequency. Coupling to other modes would

generally be a concern at perturbation locations, such as the apertures, but the boundary

conditions, which demand that no tangential electric fields exist in the aperture walls,

severely inhibit TE modes with 0n .

142

Once the radius and height of the cavity are determined the quality factor, Q, can be

calculated from the field solutions. A plot of the radii verses heights that give the resonant

frequency for rubidium is shown in Figure 6.6. For this portable fountain, we wanted to

keep the height relatively short. The plot of the Q in Figure 6.7shows that, after peaking at

2d a , as a/d ratio gets bigger the Q decreases. Compromising the Q somewhat is

acceptable; in fact it can be desirable to avoid the highest Q values since the higher the Q,

the more stringent the temperature stabilization for the cavity will need to be. Choosing a

height of 3 cm gives a radius of about 3.92 cm, and these dimensions in an evacuated

copper cavity yield a 25000Q for the TE011 mode. To calculate the Q the field solutions

from equation (0.83) can be used to find the energy stored in the cavity and the power

dissipated in the walls. The time-averaged energy stored in the electric and magnetic fields

are equal, so the total energy can be written as,

2

22 2

0 0 0

22

2

elecCavityVolume

d a

W W E dV

E dE d dz

(0.86)

and the power loss in the walls is given by

2

tan

2 2 2

0 0

2 2 2

0 00

2

2 0

2

}

{

s

losswalls

ds

z

d

z

a a dz

z z d

RP H dS

RH H ad

H H d

(0.87)

143

where sR is the surface resistivity of copper at 6.8 GHz, and

tanH is the tangential magnetic

field at the wall surfaces. The unloaded Q is the ratio of these times the frequency, and also

familiarly is the frequency over the full width at half max, Γ,

loss

WQ

P

. (0.88)

There is another interpretation that is physically intuitive: Q , up to a geometrical factor of

order unity, is proportional to the ratio of the cavity volume to the volume of the walls into

which the fields penetrate (the skin depth).

144

Figure 6.6. Solutions for cylindrical cavity dimensions yielding rubidium hyperfine frequency.

Figure 6.7. Quality factor of TE011 mode as a function of cavity geometry, a is radius and d is height.

145

6.5.3 The Cavity Endcaps

The pieces that form the top and bottom of the cavity are called the endcaps, and

they require a few further design considerations. Obviously apertures must be made for the

atoms to pass through. The apertures are a small perturbation to the TE011 mode, and do

not change the original dimensions of the cavity. The apertures do introduce the potential

for microwave leakage from the cavity. The presence of resonant fields outside of the cavity

can lead to frequency errors in various ways depending on the location of the leakage, the

power, the modes present, the trajectory of the atoms [2]. The apertures through the cavity

endcaps serve as below-cutoff waveguides, causing the fields to decay exponentially and

thereby minimizing leakage effects.

The fields in the vicinity of the endcap-apertures have been examined in detail, for

example by Ashby et al. [3], and there are a few findings to point out. Near the edges of the

apertures, effects that resemble diffraction appear in the models; very near the edge the

field can even reverse sign. Importantly, such effects do not lead to frequency shifts in the

fountain (though they can lead to some loss of contrast in the Ramsey fringes). Although

their model did not account for losses in the walls, which can cause small frequency biases

via distributed cavity phase, tests for such shifts at highly elevated microwave powers have

indicated that these shifts are in the range of a couple parts in 10-16. The impacts of

manufacturing imperfections, such as mis-centered apertures, were also studied. Although

a cavity with perfectly centered endcap-apertures (below-cutoff waveguides) can only

excite TE01 modes in the below-cutoff waveguides, in reality some excitation of other

modes, especially the fundamental TE11 mode, is expected. It was shown that the fractional

146

excitation of the TE11 mode due to a mis-centering of the below-cutoff waveguide by a

distance, off

, is approximately given by

11 22

01

8

5

cav off

bcw

aFE

a p

(0.89)

where cava is the cavity radius and

bcwa is the radius of the below-cutoff waveguide. This

means that with a generous aperture radius of 0.9 cm, and assuming a worst case mis-

centering of 0.01 cm, the ratio of the fractional excitation of TE11 to the TE01 mode is

roughly -44 dB. Making the below-cutoff waveguide 3.5 cm long then means that the TE11

mode will be down by about 88 dB, and this is the mode that we would worry most about.

The equations used to calculate the attenuations along the below-cutoff waveguide are

given in the paper by Ashby et al. [3] in units of dB per unit length,

2

,

0

1

10Attenuation 20

n m c

bcw

p

a

Log e

(0.90)

where c

is the cutoff wavelength (for TEnm modes ,

2c bcw n m

a p ), and 0

is the free-

space wavelength. Using this equation we also see that the TE01 mode is attenuated by 122

dB at the end of the below-cutoff waveguide.

Chokes are also incorporated into the cavity design (choke gaps are shown in Figure

6.8). The original motivation was for a mode-filter that would shift the resonant frequency

of the TM111 mode, which is degenerate with the TE011 mode in a perfect cylindrical

cavity. In atomic fountain cavities the apertures for the atom clouds serve as a significant

147

perturbation, affecting the TM111 mode much more than the TE011 mode. The chokes

further separate the modes’ resonances. The differences in how the apertures and chokes

affect the TE011 and TM111 modes can be visualized by noting how the surface currents

differ for the two modes. The TE011 current lines form concentric circles around the z axis,

and are undeviated by the apertures and choke gaps. On the other hand, to imagine the

TM111 mode’s current lines, center the cavity on the origin of a Euclidian rectangular

coordinate system, a complete current loop forms in the y-z (and x-z) plane. Notice that the

apertures and chokes require deviations in these current loops. The choke gaps appear as a

parallel plate capacitor. The chokes in fact serve an ulterior motive as well. In the

manufacturing process, when the endcaps are brazed to the sidewalls, the chokes prevent

the braze compound that bleeds through the joint from perturbing the TE011 cavity mode.

148

6.5.4 Exciting The TE011 Cavity Mode

The decision of how to excite the cavity can have a significant impact on the

fountain’s accuracy. It is important that the feeds coupling to the cavity are balanced, in

order to prevent power flows and phase gradients. One possibility that has been used is to

excite coupling areas (of equal size) on opposite sides of the cavity with independent

(mutually isolated) feeds that each use attenuators for controlling the power and a phase

adjuster to maintain equal phase. Independent measurements must be carried out to

assure equal power and phase of the two feeds.

An alternative is to use coupled feeds, which is the arrangement chosen for the

present design. The cavity has three feeds placed in the mid-plane of the z dimension, and

spaced equally in the φ dimension. To maintain ultra-high vacuum (UHV) inside the cavity,

Figure 6.8. Diagram of cavity vertically bisected, not to scale.

149

ceramic windows are brazed into feed holes. A rectangular waveguide resonator is then

“wrapped” around the outside of the cavity such that the standing-wave anti-nodes are

centered on the feeds. The field passes through the windows, and excites the Ramsey

cavity. Only a single drive cable is needed in this design. With this simplicity some

flexibility is sacrificed. For example in primary fountain standards, sometimes power

imbalances our purposely introduced to evaluate sensitivities to distributed cavity phase

shifts. Since the portable fountain of this thesis is not meant to be a primary standard or

achieve record breaking accuracy, but rather designed for robust stability, these trade-offs

are justified.

6.5.5 Construction Of The Ramsey Cavity

Once the cavity is designed, its construction occurs in stages and requires skilled

machining and brazing to achieve an adequate final product that is highly non-magnetic

and UHV compatible. First the individual pieces are machined from oxygen-free high

thermal conductivity (OFHC) copper, see Figure 6.9. The pieces are then shipped to a

specialty brazing facility where they are precisely positioned and brazed together in a

protective atmosphere and without flux that would later outgas and degrade the fountain

vacuum. The cavity is then shipped back to the laboratory where we perform a first tuning.

The resonant frequency needs to be just below the rubidium resonance so that on the

assembled fountain, under vacuum, it can be heated to rubidium resonance a few Kelvin

above room temperature. The tuning is an iterative process whereby the frequency is

measured, a brief acid etch is performed, the temperature is allowed to re-equilibrate, and

the frequency re-measured. After the cavity is tuned, it is shipped back to the brazing

facility so that the ceramic windows can be brazed into place. The fully constructed Ramsey

150

cavity with the drift tube can be seen in Figure 6.10. It returns to the lab for a final tuning

and is then ready to be assembled into the fountain physics package.

Figure 6.9. Two endcaps and sidewall with

Figure 6.10. Assembled Cavity and sectored diagram.

151

6.5.6 Fountain Frequency Errors Associated With Microwave Cavities

Sources of fountain frequency error related to microwave cavities, such as

distributed cavity phase and microwave leakage have been much studied as atomic

frequency standards have continued to improve over the years. Investigations of these

effects predate fountain standards, going well back to thermal beam standards, and

continue to this day. DeMarchi carried out much of the early cavity work for fountains,

showing the relation between spatial phase variation and power-flow within the cavity,

and suggesting a cavity design for the TE011 mode [4], [5]. At their root, microwave related

frequency biases are caused by atoms interacting with out-of-phase components of the

microwave field. Microwave leakage effects are due to such interactions occurring outside

of the cavity. Distributed cavity phase effects occur inside the cavity. Additionally any

impurity in the applied field, such as frequency spurs in the spectrum, can also cause

errors.

Fountain frequency biases stemming from distributed cavity phase (DCP), are a

result of the atoms experiencing a different average phase on their way down than on their

way up through the cavity. A spatial phase distribution transverse across the cavity can

result in such a phase difference when the atoms fail to exactly retrace their upward path.

Much work has been carried out in modeling the fields and phase shifts in fountain

microwave cavities [4], [5], [6], [7], [8]. Indeed, large phase deviations can be found near

the aperture edges [3]. One important point to keep in mind regarding potentially large

phase deviations, is that for frequency biases what matters is the effect these phase

deviations have on the atomic wave function. For example, near the apertures of the cavity,

152

the field is sufficiently weak that, despite the large phase deviations, the effect on the

atomic wave function is relatively small.

To experimentally determine the level of frequency uncertainty due to DCP it is

common to increase the microwave amplitude much higher than in normal operating

conditions. This provides leverage for increasing the potential effect by more than an order

of magnitude over normal operating microwave amplitudes. The potential distortion to the

Ramsey fringe pattern has been calculated, and shown to be power dependent. It can be

challenging to unambiguously distinguish DCP from other microwave effects (such as

leakage), and the scaling with amplitude is not simple. The DCP distortion to the Ramsey

fringe has been modeled theoretically, for example by Jefferts et al. [7]. They found that

asymmetric impact on the Ramsey fringe is

2 2 1

0

0

2

0

1sin 2sin cos

L RP P b

b b

(0.91)

where and L RP P are the atomic transition probability on the left and right sides of the

central fringe, respectively, 0b is equivalent to the angle imparted to the Bloch vector by

the Rabi microwave interaction (i.e. it’s 4 at optimal superposition conditions), and

and are proportional to the imaginary part of the magnetic field. The effect on the

Ramsey fringe is not simple; the distortion changes signs and amplitudes as a function of

the microwave power. The same theoretical model can be applied to microwave leakage

effect. Leakage above the Ramsey cavity has a different net effect on the fountain signal

than leakage below the Ramsey cavity.

153

6.6 The Assembled Physics Package

The physics package is shown in Figure 6.11 and

Figure 6.12. It measures less than 0.75 m tall, and consumes a footprint less than

0.25 m2. After it is fully assembled I initiate the pump-out process. If care has been taken to

keep all components clean, and the graphite has been pre-baked (~500o F for more than

four hours), this process typically takes approximately five days. First a turbo pump is used

to bring the pressure down to 10-6 Torr. The system is then prepped for baking at ~100o C.

Foil is used to protect the windows from contaminants and scratches. Thermistors are

affixed at various locations around the physics package, particularly near the anti-

Figure 6.11. Physics package design with glass cooling chamber.

154

reflection coated windows to assure that the coatings are not damaged by excess heating.

The system is wrapped in heat tape, and insulated with foil. I bring the temperature up

slowly as I monitor the pressure gauges and thermistors.

155

Figure 6.12. Assembled physics package with stainless steel spherical cube cooling chamber.

156

7 Conclusion

Over the course of this project to develop a transportable atomic fountain a number

of unexpected challenges were discovered. Chief among them was achieving sufficient

phase coherence between the lasers, necessary for effective and robust sub-Doppler laser-

cooling. The relatively large linewidths of the lasers require large loop bandwidths in order

to achieve high levels of phase coherence. Unfortunately the lasers also have an internal

phase delay that limits the loop bandwidth.

With the final version of my OPLLs, I was able to observe launched, sub-Doppler

cooled atoms. The time-of-flight atomic fluorescence signal is shown in Figure 7.1, and

Figure 7.1. Time-of-flight fluorescence signal from launched sub-Doppler cooled atoms.

157

indicates temperatures of 20 µK (well below the Doppler limit of ~150 µK). The time-of-

flight method measures the radial expansion of the atomic cloud due to thermal motion to

extrapolate the temperature according to

2 2 2 2

2 23

f i therm

B

i

R R v t

k TR t

m

(0.92)

where Rf is the final radius, Ri is the initial radius, m is the atomic mass, and kB is Boltzman’s

constant. The final radius is measured by the fluorescence signal as the cloud falls through

a “sheet” (a beam profile that is thin in the vertical dimension) of resonant laser light. The

launched, ultra-cold atoms were observed in a test-bed physics package that did not

include the Ramsey cavity. After transitioning to the final physics package, I was unable to

observe launched, sub-Doppler cooled atoms again. This was due to insufficient phase

coherence between the cooling lasers, as indicated by the ultra-cold atom temperatures

achievable when I used a single cooling laser (can only drop, not launch, the atoms in this

case). I suspect that unstable optical feedback slightly increased the phase noise in the

lasers diminishing the attainable phase coherence.

7.1 Outlook There are options available for improving the inter-laser phase coherence, based on

narrowing the combined laser linewidth and/or increasing the loop bandwidth. Increasing

the loop bandwidth requires adding external optical phase modulators with high

bandwidths. For example, EOSpace makes an electro-optical modulator integrated into a

fiberoptic package, that could be an attractive option.

158

There are numerous ways to narrow the lasers’ linewidths. One method that

appears straight-forward and requires minimal new equipment, would be to use a bi-

polarization spectroscopy lock [1] for the master laser, with a high bandwidth feedback

loop (such as the final one I used for my slave lasers) [2]. Recent promising results on

narrowing laser linewidths with fiberoptic external cavities and optical feedback [3], [4]

offer additional possibilities for improving phase coherence. Finally, the laser design and

fabrication is developing at a rapid pace (in the last year new 780 nm DBR lasers that are

more than twice as powerful (180 mW) have become available), so it is possible that in the

near future supplementary line narrowing or modulation bandwidth with be unnecessary.

Once the laser system phase coherence is improved and made reliable there is much

possibility for experimentation with the physics package that I developed. Testing

alternative state selection schemes to improve signal-to-noise ratio is one avenue. I

designed the detection region with the possibility of foregoing the standard cooling region

altogether. In this scenario an optical molasses is formed in the detection region and loaded

from a small 2D+MOT. This allows the background rubidium pressure to remain low for

detection. Not only does this result in a more compact system, but it reduces the

measurement “dead time” in each cycle.

159

8 Bibliography

[1] P. Laurent, M. Abgrall, C. Jentsch, P. Lemonde, G. Santarelli, A. Clairon, I. Maksimovic, S. Bize, C. Salomon, D. Blonde, J. F. Vega, O. Grosjean, F. Picard, M. Saccoccio, M. Chaubet, N. Ladiette, L. Guillet, I. Zenone, C. Delaroche, and C. Sirmain, “Design of the cold atom PHARAO space clock and initial test results,” Appl. Phys. B, vol. 84, no. 4, pp. 683–690, Sep. 2006.

[2] N. F. Ramsey, “A Molecular Beam Resonance Method with Separated Oscillating Fields,” Phys. Rev., vol. 78, no. 6, pp. 695–699, Jun. 1950.

[3] T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency Ratio of Al+ and Hg+ Single-Ion Optical Clocks; Metrology at the 17th Decimal Place,” Science, vol. 319, no. 5871, pp. 1808–1812, Mar. 2008.

[4] K. Predehl, G. Grosche, S. M. F. Raupach, S. Droste, O. Terra, J. Alnis, T. Legero, T. W. Hänsch, T. Udem, R. Holzwarth, and H. Schnatz, “A 920-Kilometer Optical Fiber Link for Frequency Metrology at the 19th Decimal Place,” Science, vol. 336, no. 6080, pp. 441–444, Apr. 2012.

[5] T. E. Parker, “Invited Review Article: The uncertainty in the realization and dissemination of the SI second from a systems point of view,” Rev. Sci. Instrum., vol. 83, no. 2, pp. 021102 –021102–7, Feb. 2012.

[6] Y. Sortais, S. Bize, C. Nicolas, A. Clairon, C. Salomon, and C. Williams, “Cold Collision Frequency Shifts in a ^{87}Rb Atomic Fountain,” Phys. Rev. Lett., vol. 85, no. 15, pp. 3117–3120, Oct. 2000.

[7] G. Kirchhoff and R. Bunsen, “Chemische Analyse durch Spectralbeobachtungen,” Ann. Phys., vol. 189, no. 7, pp. 337–381, 1861.

[8] B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules. Prentice Hall, 2003. [9] R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, “Geometrical Representation of the

Schrödinger Equation for Solving Maser Problems,” J. Appl. Phys., vol. 28, no. 1, pp. 49–52, Jan. 1957.

[10] M. Auzinsh, D. Budker, and S. Rochester, Optically Polarized Atoms: Understanding light-atom interactions. OUP Oxford, 2010.

[11] P. R. Berman and V. S. Malinovsky, Principles of Laser Spectroscopy and Quantum Optics. Princeton University Press, 2010.

[12] H. J. Metcalf and P. V. D. Straten, Laser Cooling and Trapping. Springer, 1999. [13] K. Lindquist, M. Stephens, and C. Wieman, “Experimental and theoretical study of the

vapor-cell Zeeman optical trap,” Phys. Rev., vol. 46, no. 7, pp. 4082–4090, Oct. 1992. [14] K. E. Gibble, S. Kasapi, and S. Chu, “Improved magneto-optic trapping in a vapor cell,”

Opt. Lett., vol. 17, no. 7, pp. 526–528, Apr. 1992. [15] G. W. Hoth, E. A. Donley, and J. Kitching, “Atom number in magneto-optic traps with

millimeter scale laser beams,” Opt. Lett., vol. 38, no. 5, pp. 661–663, Mar. 2013.

160

[16] C. J. Myatt, N. R. Newbury, and C. E. Wieman, “Simplified atom trap by using direct microwave modulation of a diode laser,” Opt. Lett., vol. 18, no. 8, pp. 649–651, Apr. 1993.

[17] A. B. Deb, A. Rakonjac, and N. Kj�rgaard, “Versatile laser system for experiments with cold atomic gases,” J. Opt. Soc. Am. B, vol. 29, no. 11, pp. 3109–3113, Nov. 2012.

[18] F. Lienhart, S. Boussen, O. Carraz, N. Zahzam, Y. Bidel, and A. Bresson, “Compact and robust laser system for rubidium laser cooling based on the frequency doubling of a fiber bench at 1560 nm,” Appl. Phys. B, vol. 89, no. 2–3, pp. 177–180, Sep. 2007.

[19] E. A. Donley, T. P. Heavner, F. Levi, M. O. Tataw, and S. R. Jefferts, “Double-pass acousto-optic modulator system,” Rev. Sci. Instrum., vol. 76, no. 6, pp. 063112–063112–6, Jun. 2005.

[20] L. Jiang, M. Achtenhagen, N. V. Amarasinghe, P. Young, and G. Evans, “High-power DBR laser diodes grown in a single epitaxial step,” p. 72301F–72301F, Feb. 2009.

[21] R. Tkach and A. Chraplyvy, “Regimes of feedback effects in 1.5- #181;m distributed feedback lasers,” J. Light. Technol., vol. 4, no. 11, pp. 1655 – 1661, Nov. 1986.

[22] K. Petermann, “External optical feedback phenomena in semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron., vol. 1, no. 2, pp. 480 –489, Jun. 1995.

[23] W. M. Klipstein and D. G. Enzer, “Mitigation of the light shift in laser cooled clocks without mechanical shutters,” in Frequency Control Symposium and PDA Exhibition, 2002. IEEE International, 2002, pp. 484–487.

[24] K. L. Corwin, Z.-T. Lu, C. F. Hand, R. J. Epstein, and C. E. Wieman, “Frequency-Stabilized Diode Laser with the Zeeman Shift in an Atomic Vapor,” Appl. Opt., vol. 37, no. 15, pp. 3295–3298, May 1998.

[25] V. V. Yashchuk, D. Budker, and J. R. Davis, “Laser frequency stabilization using linear magneto-optics,” Rev. Sci. Instrum., vol. 71, no. 2, pp. 341–346, Feb. 2000.

[26] A. Millett-Sikking, I. G. Hughes, P. Tierney, and S. L. Cornish, “DAVLL lineshapes in atomic rubidium,” J. Phys. B At. Mol. Opt. Phys., vol. 40, no. 1, p. 187, Jan. 2007.

[27] J. M. Reeves, O. Garcia, and C. A. Sackett, “Temperature stability of a dichroic atomic vapor laser lock,” Appl. Opt., vol. 45, no. 2, pp. 372–376, Jan. 2006.

[28] T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics, vol. 6, no. 10, pp. 687–692, Oct. 2012.

[29] W. Demtröder, Laser Spectroscopy. Springer, 2008. [30] K. B. MacAdam, A. Steinbach, and C. Wieman, “A narrow-band tunable diode laser

system with grating feedback, and a saturated absorption spectrometer for Cs and Rb,” Am. J. Phys., vol. 60, no. 12, pp. 1098–1111, 1992.

[31] O. Schmidt, K.-M. Knaak, R. Wynands, and D. Meschede, “Cesium saturation spectroscopy revisited: How to reverse peaks and observe narrow resonances,” Appl. Phys. B, vol. 59, no. 2, pp. 167–178, Aug. 1994.

[32] C. Wieman and T. W. Hänsch, “Doppler-Free Laser Polarization Spectroscopy,” Phys. Rev. Lett., vol. 36, no. 20, pp. 1170–1173, May 1976.

[33] C. P. Pearman, C. S. Adams, S. G. Cox, P. F. Griffin, D. A. Smith, and I. G. Hughes, “Polarization spectroscopy of a closed atomic transition: applications to laser frequency locking,” J. Phys. B At. Mol. Opt. Phys., vol. 35, no. 24, p. 5141, Dec. 2002.

161

[34] Y. Yoshikawa, T. Umeki, T. Mukae, Y. Torii, and T. Kuga, “Frequency Stabilization of a Laser Diode with Use of Light-Induced Birefringence in an Atomic Vapor,” Appl. Opt., vol. 42, no. 33, pp. 6645–6649, Nov. 2003.

[35] V. B. Tiwari, S. Singh, S. R. Mishra, H. S. Rawat, and S. C. Mehendale, “Laser frequency stabilization using Doppler-free bi-polarization spectroscopy,” Opt. Commun., vol. 263, no. 2, pp. 249–255, Jul. 2006.

[36] H.-R. Noh, G. Moon, and W. Jhe, “Discrimination of the effects of saturation and optical pumping in velocity-dependent pump-probe spectroscopy of rubidium: A simple analytical study,” Phys. Rev., vol. 82, Dec. 2010.

[37] D. A. Smith and I. G. Hughes, “The role of hyperfine pumping in multilevel systems exhibiting saturated absorption,” Am. J. Phys., vol. 72, no. 5, pp. 631–637, May 2004.

[38] T. Ikegami, S. Ohshima, and M. Ohtsu, “Frequency Stabilization of Laser Diodes to the Cs-D2 Line with the Zeeman Modulation Method,” Jpn. J. Appl. Phys., vol. 28, no. Part 2, No. 10, pp. L1839–L1841, 1989.

[39] T. P. Dinneen, C. D. Wallace, and P. L. Gould, “Narrow linewidth, highly stable, tunable diode laser system,” Opt. Commun., vol. 92, no. 4–6, pp. 277–282, Sep. 1992.

[40] S. Pradhan, R. Behera, and A. K. Das, “Magnetic field modulation spectroscopy of rubidium atoms,” Pramana - J Phys, vol. 78, no. 4, pp. 585–594, Apr. 2012.

[41] H. D. Do, M. S. Heo, G. Moon, H. R. Noh, and W. Jhe, “Analytic calculation of the lineshapes in polarization spectroscopy of rubidium,” Opt. Commun., vol. 281, no. 15–16, pp. 4042–4047, Aug. 2008.

[42] M. L. Harris, C. S. Adams, S. L. Cornish, I. C. McLeod, E. Tarleton, and I. G. Hughes, “Polarization spectroscopy in rubidium and cesium,” Phys. Rev., vol. 73, no. 6, p. 062509, Jun. 2006.

[43] G. Moon and H. R. Noh, “Analytic solutions for the saturated absorption spectra,” JOSA B, vol. 25, no. 5, pp. 701–711, 2008.

[44] G. Moon and H. R. Noh, “A comparison of the dependence of saturated absorption signals on pump beam diameter and intensity,” JOSA B, vol. 25, no. 12, pp. 2101–2106, 2008.

[45] G. Moon and H.-R. Noh, “Analytic calculation of linear susceptibility in velocity-dependent pump-probe spectroscopy,” Phys. Rev., vol. 78, Sep. 2008.

[46] G. Moon and H.-R. Noh, “Analytic solutions for the saturated absorption spectra: erratum,” J. Opt. Soc. Am. B, vol. 27, no. 9, pp. 1741–1741, Sep. 2010.

[47] H.-R. Noh and W. Jhe, “Analytic solutions of the susceptibility for Doppler-broadened two-level atoms,” Opt. Commun., vol. 283, no. 14, pp. 2845–2848, Jul. 2010.

[48] G. Moon, M.-S. Heo, S. R. Shin, H.-R. Noh, and W. Jhe, “Calculation of analytic populations for a multilevel atom at low laser intensity,” Phys. Rev., vol. 78, no. 1, p. 015404, Jul. 2008.

[49] S. R. Shin and H.-R. Noh, “Doppler spectroscopy of arbitrarily polarized light in rubidium,” Opt. Commun., vol. 284, no. 5, pp. 1243–1246, Mar. 2011.

[50] G. Moon, H.-S. Noh, and H.-R. Noh, “Effect of Laser Beam Diameter Variation in Saturated Absorption Spectra,” J. Phys. Soc. Jpn., vol. 77, p. 074301, Jul. 2008.

[51] G. Moon and H.-R. Noh, “Theoretical Calculation of the Saturated Absorption Spectrum for a Multilevel Atom,” J. Korean Phys. Soc., vol. 50, no. 4, p. 1037, Apr. 2007.

[52] D. Meschede, Optics, Light and Lasers. John Wiley & Sons, 2007. [53] J. D. Jackson, Classical Electrodynamics. Wiley, 1999.

162

[54] J. Mitroy, M. S. Safronova, and C. W. Clark, “Theory and applications of atomic and ionic polarizabilities,” J. Phys. B At. Mol. Opt. Phys., vol. 43, no. 20, p. 202001, Oct. 2010.

[55] M. S. Safronova, D. Jiang, and U. I. Safronova, “Blackbody radiation shift in the ^{87}Rb frequency standard,” Phys. Rev., vol. 82, no. 2, p. 022510, 2010.

[56] J. A. Sherman, N. D. Lemke, N. Hinkley, M. Pizzocaro, R. W. Fox, A. D. Ludlow, and C. W. Oates, “High-Accuracy Measurement of Atomic Polarizability in an Optical Lattice Clock,” Phys. Rev. Lett., vol. 108, no. 15, p. 153002, Apr. 2012.

[57] L. . Turner, K. . Weber, C. . Hawthorn, and R. . Scholten, “Frequency noise characterisation of narrow linewidth diode lasers,” Opt. Commun., vol. 201, no. 4–6, pp. 391–397, Jan. 2002.

[58] H. Talvitie, M. Merimaa, and E. Ikonen, “Frequency stabilization of a diode laser to Doppler-free spectrum of molecular iodine at 633 nm,” Opt. Commun., vol. 152, no. 1–3, pp. 182–188, Jun. 1998.

[59] D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. Mod. Phys., vol. 74, no. 4, pp. 1153–1201, Nov. 2002.

[60] F. M. Gardner, Phaselock Techniques. Wiley, 2005. [61] A. Yariv, “Dynamic analysis of the semiconductor laser as a current-controlled

oscillator in the optical phased-lock loop: applications,” Opt. Lett., vol. 30, no. 17, pp. 2191–2193, Sep. 2005.

[62] M. Ohtsu, Highly coherent semiconductor lasers. Boston: Artech House, 1992. [63] W. Liang, “Study of optical phase lock loops and the applications in coherent beam

combining and coherence cloning,” 2008. [Online]. Available: http://resolver.caltech.edu/CaltechETD:etd-03272008-143400. [Accessed: 05-Jul-2012].

[64] N. Satyan, “Optoelectronic control of the phase and frequency of semiconductor lasers,” 2011. [Online]. Available: http://resolver.caltech.edu/CaltechTHESIS:04292011-221312708. [Accessed: 06-Jul-2012].

[65] K. Petermann, Laser Diode Modulation and Noise. Springer, 1988. [66] F.-X. Esnault, E. Ivanov, E. Donley, and J. Kitching, “Heterodyne optical phase locking of

megahertz-broad DFB/DBR lasers.” unpublished Optics Express, 2012. [67] M. Zhu and J. L. Hall, “Stabilization of optical phase/frequency of a laser system:

application to a commercial dye laser with an external stabilizer,” J. Opt. Soc. Am. B, vol. 10, no. 5, p. 802, May 1993.

[68] M. Grant, W. Michie, and M. Fletcher, “The performance of optical phase-locked loops in the presence of nonnegligible loop propagation delay,” J. Light. Technol., vol. 5, no. 4, pp. 592–597, 1987.

[69] R. T. Ramos and A. J. Seeds, “Delay, linewidth and bandwidth limitations in optical phase-locked loop design,” Electron. Lett., vol. 26, no. 6, p. 389, 1990.

[70] P. Corrc, O. Girad, and J. De Faria, I.F., “On the thermal contribution to the FM response of DFB lasers: theory and experiment,” IEEE J. Quantum Electron., vol. 30, no. 11, pp. 2485–2490, 1994.

[71] G. S. Pandian and S. Dilwali, “On the thermal FM response of a semiconductor laser diode,” IEEE Photonics Technol. Lett., vol. 4, no. 2, pp. 130–133, 1992.

163

[72] S. Saito, O. Nilsson, and Y. Yamamoto, “Coherent FSK transmitter using a negative feedback stabilised semiconductor laser,” Electron. Lett., vol. 20, no. 17, pp. 703–704, 1984.

[73] H. Tsuchida, T. Tako, and M. Ohtsu, “A Novel Technique for Measuring the Frequency Deviation of Semiconductor Lasers Under Direct Modulation,” Jpn. J. Appl. Phys., vol. 22, pp. L19–L21, 1983.

[74] E. Eichen, P. Melman, and W. H. Nelson, “Intrinsic lineshape and FM response of modulated semiconductor lasers,” Electron. Lett., vol. 21, no. 19, pp. 849 –850, 1985.

[75] W. V. Sorin, K. W. Chang, G. A. Conrad, and P. R. Hernday, “Frequency domain analysis of an optical FM discriminator,” J. Light. Technol., vol. 10, no. 6, pp. 787 –793, Jun. 1992.

[76] T. G. McRae, B. J. Field, and P. J. Manson, “Measurement of the frequency modulation transfer function of a laser using a Mach–Zehnder interferometer,” Appl. Phys. B Lasers Opt., vol. 80, no. 7, pp. 849–856, 2005.

[77] “Infrared | IR |Tunable |Laser diode |wavelengths|TO8 Package| C mount.” [Online]. Available: http://photodigm.com/products-and-laser-wavelengths/. [Accessed: 26-Jul-2013].

[78] H. W. Johnson, High Speed Digital Design: A Handbook of Black Magic. Prentice Hall PTR, 1993.

[79] Minicircuits, “BIASING OF CONSTANT CURRENT MMIC AMPLIFIERS.” http://www.minicircuits.com/app/AN60-010.pdf.

[80] S. R. Jefferts, R. E. Drullinger, and A. DeMarchi, “NIST cesium fountain microwave cavities,” in Frequency Control Symposium, 1998. Proceedings of the 1998 IEEE International, 1998, pp. 6–8.

[81] M. Violetti, C. Affolderbach, F. Merli, J. Zurcher, G. Mileti, and A. Skrivervik, “Miniaturized microwave cavity for rubidium atomic frequency standards,” in Microwave Conference (EuMC), 2012 42nd European, 2012, pp. 1320–1323.

[82] J. Shirley, F. Levi, T. Heavner, D. Calonico, D. Yu, and S. Jefferts, “Microwave leakage-induced frequency shifts in the primary frequency Standards NIST-F1 and IEN-CSF1,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 53, no. 12, pp. 2376–2385, Dec. 2006.

[83] N. Ashby, S. Romisch, and S. R. Jefferts, “Endcaps for TE01 cavities in fountain frequency standards,” in Frequency Control Symposium and PDA Exhibition Jointly with the 17th European Frequency and Time Forum, 2003. Proceedings of the 2003 IEEE International, 2003, pp. 1076–1083.

[84] G. Vecchi and A. De Marchi, “Spatial phase variations in a TE011 microwave cavity for use in a cesium fountain primary frequency standard,” IEEE Trans. Instrum. Meas., vol. 42, no. 2, pp. 434–438, 1993.

[85] A. Khursheed, G. Vecchi, and A. De Marchi, “Spatial variations of field polarization and phase in microwave cavities: application to the cesium fountain cavity,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 43, no. 2, pp. 201–210, 1996.

[86] C. Fertig, R. Li, J. I. Rees, and K. Gibble, “Distributed cavity phase shifts and microwave photon recoils,” in Frequency Control Symposium and PDA Exhibition, 2002. IEEE International, 2002, pp. 469–472.

[87] S. R. Jefferts, J. H. Shirley, N. Ashby, E. A. Burt, and G. J. Dick, “Power dependence of distributed cavity phase-induced frequency biases in atomic fountain frequency

164

standards,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 52, no. 12, pp. 2314–2321, Dec. 2005.

[88] Y. Torii, H. Tashiro, N. Ohtsubo, and T. Aoki, “Laser-phase and frequency stabilization using atomic coherence,” Phys. Rev., vol. 86, no. 3, p. 033805, Sep. 2012.

[89] W.-K. Lee, C. Y. Park, J. Mun, and D.-H. Yu, “Linewidth reduction of a distributed-feedback diode laser using an all-fiber interferometer with short path imbalance,” Rev. Sci. Instrum., vol. 82, no. 7, p. 073105, 2011.

[90] Q. Lin, M. A. Van Camp, H. Zhang, B. Jelenkovi?, and V. Vuleti?, “Long-external-cavity distributed Bragg reflector laser with subkilohertz intrinsic linewidth,” Opt. Lett., vol. 37, no. 11, pp. 1989–1991, Jun. 2012.

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