Techniques in molecular spectroscopy: from broad
bandwidth to high resolutionby
Kevin C. Cossel
B.S., California Institute of Technology, 2007

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2014This thesis entitled:
Techniques in molecular spectroscopy: from broad bandwidth to high resolution
written by Kevin C. Cossel
has been approved for the Department of Physics

Prof. Jun Ye

Prof. Eric Cornell

Date

The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.

Cossel, Kevin C. (Ph.D., Chemical Physics)
Techniques in molecular spectroscopy: from broad bandwidth to high resolution
Thesis directed by Prof. Jun Ye

This thesis presents a range of different experiments all seeking to extended the capabilities
of molecular spectroscopy and enable new applications. The new technique of cavity-enhanced direct frequency comb spectroscopy (CE-DFCS) provides a unique combination of broad bandwidth,
high resolution, and high sensitivity that can be useful for a wide range of applications. Previous
demonstrations of CE-DFCS were confined to the visible or near-infrared and operated over a limited bandwidth: for many applications it is desirable to increase the spectral coverage and to extend
to the mid-infrared where strong, fundamental vibrational modes of molecules occur. There are several key requirements for CE-DFCS: a frequency comb source that provides broad bandwidth andhigh resolution, an optical cavity for high sensitivity, and a detection system capable of multiplex
detection of the comb spectrum transmitted through the cavity. We first discuss comb sources withemphasis on the coherence properties of spectral broadening in nonlinear fiber and the developmentof a high-power frequency comb source in the mid-infrared based on an optical-parametric oscillator(OPO). To take advantage of this new mid-infrared comb source for spectroscopy, we also discuss
the development of a rapid-scan Fourier-transform spectrometer (FTS). We then discuss the first
demonstration of CE-DFCS with spectrally broadened light from a highly nonlinear fiber with theapplication to measurements of impurities in semiconductor manufacturing gases. We also cover
our efforts towards extending CE-DFCS to the mid-infrared using the mid-infrared OPO and FTS
to measure ppb levels of various gases important for breath analysis and atmospheric chemistry
and highlight some future applications of this system.
In addition to the study of neutral molecules, broad-bandwidth and high-resolution spectraof molecular ions are useful for astrochemistry where many of the observed molecules are ionic, forstudying molecules such as CH+
5 with highly non-classical behavior, and for tests of fundamental

iv
physics. We have developed a new technique – frequency comb velocity-modulation spectroscopy– that is the first system to enable rapid, broadband spectroscopy of molecular ions with highresolution. We have demonstrated the ability to record 150 cm−1 of spectra consisting of 45,000
points in 30 minutes and have used this system to record over 1000 cm−1 of spectra of HfF+ in thenear-infrared around 800 nm. After improvements, the system can now cover more than 3250 cm−1
(700-900 nm). We have combined this with standard velocity-modulation spectroscopy to measure
and analyze 19 ro-vibronic bands of HfF+ .
These measurements enabled precision spectroscopy of trapped HfF+ for testing time-reversal
symmetry. For this experiment, we perform Ramsey spectroscopy between spin states in themetastable 3 ∆1 level to look for a permanent electric dipole moment of the electron with what
we believe is the narrowest line observed in a molecular system (Fourier limited with 500 ms ofcoherence time). The long coherence time is a major advantage of using ions, but there are also
some added complexities. We discuss various aspects metastable state preparation, state detection,and spectroscopy in a rotating frame (due to the necessary rotating electric bias field) that wereparticular challenging. In addition, we discuss limits to the coherence time – in particular, ion-ion
collisions – as well as the sensitivity of the current measurements and provide a path towards a
new limit on the electric dipole moment of the electron.

Dedication

To my dad, who was an amazing role model and always encouraged my interest in science.
In some way, he probably always knew that I would achieve this.

vi

Acknowledgements

None of this work would have been possible without a lot of help from a great group of people.
Both Jun and Eric have been great advisors and have tried to teach my how to think like a physicist.
If I only succeed in learning a fraction of this, it will still serve me very well! The members of both
Jun and Eric’s labs are all extremely talented and it has been a pleasure to work with them. I can’t
hope to name all of the people who I taught me various tips or who lent me critical optics but I would
like to thank in particular Mike Martin, Mike Thorpe, Dylan Yost, Arman Cingöz, Tom Alison,
Florian Adler, Matt Swallows, Craig Benko, Piotr Maslowski, Alexsandra Foltynowicz, Bryce Bjork,
AJ Fleisher, Laura Sinclair, Dan Gresh, Huanqian Loh, Matt Grau, and Will Cairncross. I wish
the best of luck to them and have no doubt that they will all be extremely successful. In addition
to all of the graduate students and postdocs in the labs, the JILA staff that I worked with in
the electronics shop (Terry, James, Carl) and instrument shop (Hans, Kim, Tracy, Blaine) are allindispensable for all of the experiments. In addition, Pam, Diane, Krista, Lauren, and all of theother staff do a great job keeping everything running smoothly. Outside of JILA, I am grateful to
Scott Diddams and Nate Newbury at NIST for many interesting discussions, Axel Ruehl, Ingmar
Hartl and the crew at IMRA for help with lasers, Ronald Holzwarth, Tobias Wilken, Thomas Udem
and others at MPQ for hosting me for a month, and Mitchio Okumura and Thinh Bui at Caltech for
great collaborations. My parents have provided amazing support and always gave me opportunities
to learn something new. I would also than Mike Martin, Carly Donahue, Laura Sinclair, and Galen
O’Neil for many fun times outside of the lab. Last, but definitely not least, my wife Eleanor has
provided immeasurable support, encouragement, and joy.

Contents

Chapter
1 Introduction

12 Frequency comb sources9

2.1

Mode-locked lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1

2.2

2.3

2.4

Mode-locking Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Indirect Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1

cw-laser Based Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.2

Typical comb sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Nonlinear fiber optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.1

Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.2

Broadening mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.3

Coherence measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Optical parametric oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Detection techniques
3.1

57

Cavity-comb coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1.1

Tight locking scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.1.2

Swept coupling scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2

Detection sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3

Detection techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

viii

3.4

3.3.1

VIPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.3.2

Fourier-transform spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Velocity-modulation spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.4.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.4.2

Comb-vms

3.4.3

System performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.4.4

Single-frequency vms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.4.5

Extensions of comb-vms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4 Applications of DFCS
4.1

4.2

100

Trace detection in Arsine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.1.1

Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.1.2

Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.1.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Mid-infrared comb spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.2.1

Measurement of individual molecular species . . . . . . . . . . . . . . . . . . 116

4.2.2

Instrument performance limits . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.2.3

Multi-line fitting advantage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.2.4

Determination of absolute concentrations of a gas mixture . . . . . . . . . . . 121

4.3

Future applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5 Velocity-modulation spectroscopy of HfF+
5.1

5.2

133

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.1.1

Diatomic molecular spectra primer . . . . . . . . . . . . . . . . . . . . . . . . 134

5.1.2

Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

5.1.3

Lambda doubling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

ix
5.3

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6 Precision spectroscopy of trapped ions
6.1

6.2

6.3

6.4

eEDM Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.1.1

Measurement Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6.1.2

Current Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.1.3

Overview of JILA eEDM measurement . . . . . . . . . . . . . . . . . . . . . . 180

Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.2.1

Ion Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

6.2.2

Photodissociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

6.2.3

Rotating bias fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Coherent Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
6.3.1

Laser stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

6.3.2

Transfer theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

6.3.3

Transfer parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

6.3.4

Transfer efficiency and timescale . . . . . . . . . . . . . . . . . . . . . . . . . 218

6.3.5

Transfer spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

Ramsey Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
6.4.1

6.5

6.6

171

Sensitivity Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

Coherence Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

6.5.1

Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

6.5.2

Field inhomogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

6.5.3

Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

6.5.4

Collision background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

6.5.5

Coherence time from collisions . . . . . . . . . . . . . . . . . . . . . . . . . . 241

Systematic Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
6.6.1

Non-reversing magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

x
6.6.2

Perpendicular magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

6.6.3

Field inhomogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

6.6.4

Other effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

6.7

eEDM measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

6.8

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

6.9

Useful numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

7 Conclusion

268

Bibliography269

TablesTable
2.1

Mirror coatings for OPO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1

Comparison of cavity-enhanced comb systems.

4.1

Detection limits for a collection of important molecules within the spectral range of. . . . . . . . . . . . . . . . . . . . . 65

our mid-IR FC-FTS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.2

Many trace species in human breath. . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.3

Atmospheric trace species. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.1

Possible states and configurations arising from s2 and sd electron configurations. . . 139

5.2

Fitted constants for observed transitions in 180 HfF+ in cm−1 . . . . . . . . . . . . . . 150

5.3

Matrix elements for the Λ-doubling in 3 ∆1 . . . . . . . . . . . . . . . . . . . . . . . . 158

5.4

Derived constants for observed states in 180 HfF+ . . . . . . . . . . . . . . . . . . . . 164

5.5

Summary of electronic states measured in HfF+ . . . . . . . . . . . . . . . . . . . . . 165

5.5

Summary of states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

5.5

Summary of states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

5.6

Calculate dipole matrix elements between states in HfF+ . . . . . . . . . . . . . . . . 168

5.7

Franck-Condon factors (vibrational overlap integrals) for 3 Π0+ - 3 ∆1 . . . . . . . . . . 169

5.8

Franck-Condon factors for 3 Φ2 - 3 ∆1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

5.9Franck-Condon factors for 3 Π0+ - 1 Σ+ . . . . . . . . . . . . . . . . . . . . . . . . . . 169

5.10 Franck-Condon factors for 3 Π1 - 1 Σ+ . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

xii
5.11 Franck-Condon factors for 1 Π1 - 1 Σ+ . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.12 Franck-Condon factors for 3 Σ0+ - 1 Σ+ . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.1

Trap characterization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

6.2

Estimated statistical sensitivity of the first generation measurement using HfF+ . . . 234

6.3

Systematic error summary.

6.4

Useful numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

6.5¯ δ∆, and Stark shift for different values of Erot , from theory . . . . . . . . . . . . . 267∆,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Figures

Figure1.1

Location of fundamental vibrational bands and spectra of some example molecules. .

5

2.1

Comparison of comb sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2

Sources based on a Yb:fiber laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3

Spectral broadening of a Yb:fiber laser using suspended core fiber. . . . . . . . . . . 24

2.4

Raman shifting and broadening of Er:fiber laser. . . . . . . . . . . . . . . . . . . . . 26

2.5

Pulse propagation in fibers with negative and positive GVD . . . . . . . . . . . . . . 30

2.6

Soliton dynamics in fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.7

Simulation of Raman shifting of an Er:fiber laser in SMF-28 . . . . . . . . . . . . . . 33

2.8

Simulation of supercontinuum generation in highly nonlinear fiber near 1.55 µm . . . 35

2.9

Coherent supercontinuum generation spanning over 1.5 octaves from a Yb:fiber laser38

2.10 Simulation results for the optical spectrum and coherence properties of a Yb:fiber
laser based supercontinuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.11 Dependence of coherence with Raman gain . . . . . . . . . . . . . . . . . . . . . . . 42
2.12 Mid-infrared OPO comb based on periodically-poled lithium niobate (PPLN) . . . . 47
2.13 OPO crystal Holder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.14 PPLN crystal design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.15 OPO performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.16 Stabilization of mid-IR OPO and measurement of idler-comb linewidth . . . . . . . . 55

xiv
3.1

Detection schemes for CE-DFCS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2

VIPA Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.3

Setup of Fourier-transform spectrometer. . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.4

Circuit diagram for mid-IR auto-balancing detector . . . . . . . . . . . . . . . . . . . 81

3.5

Sketch of velocity-modulation spectroscopy (VMS). . . . . . . . . . . . . . . . . . . . 86

3.6

Setup for comb-vms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.7

Setup for cw-vms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.8

Comparison of coherent subtraction and single-direction measurements. . . . . . . . 97

4.1

Sketch of trace detection in arsine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.2

The spectrum of 10 ppm methane (CH4 ), 10 ppm carbon dioxide (CO2 ), 10 ppm
hydrogen sulfide (H2 S), and 2.5 ppm water (H2 O) in nitrogen gas from 1.75-1.95 µm. 110

4.3

Determination of detection limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.4

Spectrum of trace water in arsine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.5

Spectra of several species measured using the mid-IR OPO, FTS, and multi pass cell. 118

4.6

Multispecies detection using mid-IR comb. . . . . . . . . . . . . . . . . . . . . . . . . 123

5.1

All data obtained with cw- and comb-vms.

5.2

Molecular orbital diagram for valence orbitals of HfF+ . . . . . . . . . . . . . . . . . 138

5.3

Angular momentum coupling in diatomic molecules corresponding to different Hund’s. . . . . . . . . . . . . . . . . . . . . . . 135

cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.4

Example transitions with Λ-doubling in one or both states . . . . . . . . . . . . . . . 145

5.5

Loomis-Wood plot showing branches of the 3 Π1 ← 1 Σ+ (3, 1) transition. . . . . . . . 148

5.6

Isotope shifts for each observed transition in HfF+ scaled to approximately grouptransitions by electronic states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5.7

Residuals for three fits to the 3 Π0− ← 3 ∆1 (1, 0) band. . . . . . . . . . . . . . . . . 155

xv
6.1

An eEDM violates both parity (P) and time (T) reversal symmetry. The solid blackarrows show the effect of a P- or T-reversal operation on an electron with both aspin and an electric dipole moment due to a charge distribution asymmetry. . . . . . 173

6.2

Sketch of levels used in the JILA eEDM measurement . . . . . . . . . . . . . . . . . 181

6.3

The eEDM induced Stark splitting (∆E) for J = 1 hyperfine levels of the 3 ∆1state of HfF+ : between F = 3/2 mF = ±3/2 levels (solid line), between F = 3/2
mF = ±1/2 levels (doted line), and between F = 1/2 mF = ±1/2 (dashed line). . . . 183

6.4

Sketch of eEDM experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

6.5

Histogram of Hf+ number measured using counting or integrating . . . . . . . . . . . 193

6.6(1+10 ) resonance-enhanced multiphoton photodissociation. . . . . . . . . . . . . . . . 196

6.7

Rotational state distribution of trapped ions in X 1 Σ+ v 00 = 0 state: 20 ms after
trapping (blue) and 220 ms after trapping (red). The first REMPD photon is tuned
near 35,976 cm−1 . We observed a redistribution towards higher rotational states
after 220 ms of wait time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

6.8

Multiple dissociation pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

6.9

Population lifetimes in trapped HfF+ . (a) X 1 Σ+ v = 0, J = 0 and (b) 3 ∆1 v =0, J = 1. The X 1 Σ+ lifetime exceeds many seconds after the initial fast decay (100ms) due to rotational re-distribution as described in Section IV. The 3 ∆1 lifetime is2.1(2) s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

6.10 Characterization of magnetic field gradient. . . . . . . . . . . . . . . . . . . . . . . . 204
6.11 Adiabatic transfer from 1 Σ+ to 3 ∆1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
6.12 Stabilization cavity for Raman transfer lasers . . . . . . . . . . . . . . . . . . . . . . 209
6.13 Sketch of the laser stabilization setup for the transfer lasers. . . . . . . . . . . . . . . 210
6.14 Transition dipole moment calibration for the 3 Π0+ ← 1 Σ+ (v = 1, J = 1) ← (v =0, J = 0) transition.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

6.15 One photon lineshape for the transfer laser. . . . . . . . . . . . . . . . . . . . . . . . 217
6.16 Fraction of population remaining in 1 Σ+ J = 0 versus transfer laser on duration. . . 220

xvi
6.17 Stark spectroscopy of 3 ∆1 J = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
6.18 Possible two-photon transfer pathways . . . . . . . . . . . . . . . . . . . . . . . . . . 224
6.19 Ramsey spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
6.20 Depletion from the 3 ∆1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
6.21 Coherence time versus fringe frequency. . . . . . . . . . . . . . . . . . . . . . . . . . 238
6.22 Ion-ion collision. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
6.23 Collision trajectory and solid angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
6.24 Solid angle versus scattering angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
6.25 Plot of the estimated coherence time limit from collisions as a function of collisionenergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
6.26 Coherence time versus ion number and Erot . . . . . . . . . . . . . . . . . . . . . . . . 247
6.27 Simulation of Ramsey sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
6.28 Switches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
6.29 All eEDM measurements to date. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

Chapter 1

Introduction

Spectroscopy – loosely defined as the study of the energy levels in atoms and molecules –has existed for over two centuries now, see [1] for an interesting history. Despite the long history,
it is still an active area of research. The invention of the grating spectrometer by Fraunhofer in1814 led to the first high-resolution (at the time) studies of the spectra of molecules. This was later
improved upon by Rowland who developed new ways of making gratings – including curved gratings– in the 1880’s. These inventions led to quantitative studies of the band structure of molecules by
Rowland, Smyth, Herschel and other. Emperical descriptions of the patterns in the spectra weredeveloped by Deslandres in the late 1800’s, but there were no good theoretical descriptions. The
beginning of the 20th century saw the discovery of resonance fluorescence in iodine by Wood [2, 3]
as well as the initial developments of quantum theory. Armed with the added information fromresonance fluorescence, a quantum description of the rotational and electronic structure of spectra
was accomplished by Heurlinger and Lenz, among others. This shows how spectra could provide
information about the structure of molecules, which is still important today. Attempts to fully
understand the electronic structure of molecules resulted in the development of molecular orbital
theory, spearheaded by Mulliken [4]. The field of molecular spectroscopy continued to advance
rapidly with measurements in a variety of molecules with increasing resolution, which led to theobservations of many new effects and continued improvements in the theoretical description ofspectra, see, for example [5].
Further advances have typically been driven by the development of new technologies. For

2example, the introduction of commercial Fourier-transform infrared (FTIR) spectrometers by Digilab (now part of Agilent) in 1969 started the development of bench-top infrared spectroscopy.
Because of its ease of use and extremely broad bandwidth, FTIR is now a ubiquitous technique forchemical analysis of mixtures, environmental science, and laboratory studies of molecular spectra[6, 7] and has even been deployed on a satellite for studying earth’s atmosphere [8]. Recently,
Fourier-transform spectroscopy has even been extended to the vacuum ultraviolet region [9].
Another frontier has been the quest for ever higher resolution. The invention of the laser in1960 revolutionized the field of high-resolution molecular spectroscopy and further improvements
occurred with the development of sub-Doppler saturated absorption spectroscopy and Doppler-free
two-photon spectroscopy [10]. Transitions of some molecules were discovered near the emission
lines of HeNe (e.g., CH4 and I2 ), Nd:YAG (e.g., I2 ) and CO2 (e.g., SF6 and OsO4 ) lasers, which
provide fractional resolutions of better than 10−10 [11]. Continued improvement of tunable lasers
and frequency stabilization methods, including the transfer of frequency stability using opticalfrequency combs [12, 13, 14, 15], as well as molecular cooling techniques also enabled this resolution
to be achieved in other wavelength regions.
As the resolution continues to improve, more and more effects become visible. In addition
to the usual molecular constants for vibration and rotation terms such as spin-spin, spin-vibration,and spin-rotation interactions need to be included to describe high resolution spectra [16]. High
resolution spectroscopy can provide insights into the structure of even very large molecules [17, 18].
It can also be used to observe dynamics such as tunneling and interconversion, which in turn canbe used to learn about reactive radical species [19, 20], proton exchange [21], and hydrogen bonding[22].
At the highest resolutions, molecular spectroscopy can be used directly – or indirectly via
laser stabilization – as a probe of fundamental physics such as observing recoil doublets due tothe interaction with light [11], measuring the speed of light [23], testing for parity violation inmolecular spectra [24, 25], measuring the Boltzmann constant [26], checking temporal variation of
fundamental constants [27], or improving tests of special relativity with new Kennedy-Thorndike3
experiments [28]. Variations of Ramsey spectroscopy can also be used to probe microwave – orlower frequency – transitions for tests of time reversal symmetry [29, 30, 31] or temporal variationof fundamental constants [32, 33, 34].
A final direction has been the desire for increased sensitivity to detect of trace species or weak
spectral lines. This was achieved with the development modulation techniques such as frequencymodulation spectroscopy [35, 36, 37] as well as the use of cavity-enhanced techniques [38, 39, 40].
The combination of these two techniques led to the development of noise-immune cavity-enhanced
spectroscopy (NICE-OHMS) [41, 42], which can provide sensitivity below 1 cm−3 . Ultra-sensitive
cavity-enhanced spectroscopy has also recently been applied to the detection of 14 CO2 with thepotential to perform radiocarbon dating out to 20,000 years [43].
Recent efforts have attempted to combine these previously separate directions into one systemthat can provide broad bandwidth, high resolution, and high sensitivity. This led to the developmentof cavity-enhanced direct frequency comb spectroscopy (CE-DFCS) [44, 45, 46, 47]. The spectrumof a frequency comb consists of a series of regularly space spectral lines underneath a broad spectralenvelope; the frequency of one line is given by νn = nfrep + f0 , where n is an integer (typically
104 − 106 , frep is known as the repetition rate (typically 100 MHz - 10 GHz), and f0 is an offset
frequency that ranges from −frep /2 to frep /2. The spectral envelope can be hundreds or eventhousands of nanometers broad while the width of a single line – or comb mode – can be kilohertz
or even hertz, which shows the incredible combination of spectral bandwidth and resolution possible.
Initially, CE-DFCS was confined to the visible and near-infrared spectral regions where comb
sources were easily obtainable, and the experiments focused on demonstrating various possiblecombinations of resolution, bandwidth, and sensitivity. The first three chapters of this thesis
discuss further developments of CE-DFCS that sought to extend the range and applicability. To dothis, we wanted comb sources that could operate over a broad spectral bandwidth and that operated
in new spectral regions. Many of the molecular transitions in the near-infrared are overtones andcombinations of different vibrational modes. Because of this, the strength of these transitions is
considerably weaker than the strength of a fundamental vibrational transition in the mid-infrared,

4
so the sensitivity can be improved by pushing to the mid-infrared or to the visible/UV where
many molecules have strong electronic transitions. Figure 1.1 shows the regions of fundamental
vibrational bands and shows spectra from some example molecules, which illustrates the motivationfor extending CE-DFCS into the mid-infrared. In particular, we see how a broadband mid-infrared
spectroscopy system would enable spectroscopy of many different species.
In Chapter 2, we discuss different comb sources and then discuss our efforts to increase thespectral range of comb sources used for CE-DFCS. In particular, the ultrashort pulses that are
typical of most frequency comb sources enable very efficient nonlinear optics. By tightly confining
ultrashort pulses in highly nonlinear fiber, it is possible to shift the laser spectrum or to broaden
the spectrum to simultaneously cover a wide spectral range. One challenge with this though is thatthe generated light many not always be fully coherent. We cover some basics of nonlinear effects infiber and then discuss some of our experimental and theoretical studies of the coherence propertiesof the generated light, which enabled coherent broadening covering 1.5 octaves of simultaneous
spectral bandwidth [49]. We also use nonlinear effects to push comb spectroscopy to the midinfrared spectral region with the development of a high-power comb source based on an opticalparametric oscillator (OPO) [50]. We discuss the development and characterization of our system,
which covers the 3-5 µm range. This system enabled a series of measurements, some of which we
discuss later, and a system based on the same design is now being constructed to cover the 6-12
µm region.
In order to take advantage of the broad bandwidth and inherent high resolution of the frequency comb source, we need a readout system that is capable of recording many spectral channels
rapidly, as discussed in Chapter 3. One option for this is a 2D spectrometer based on a virtuallyimaged phased array (VIPA) etalon [51], which provides several thousand simultaneous spectral
channels and microsecond time resolution. We have demonstrated a VIPA spectrometer in the midinfrared to use with the mid-infrared comb source [52]. Unfortunately, the VIPA spectrometer also
has several disadvantages for very broad bandwidth spectroscopy: in particular, the total spectral
range of a single VIPA etalon is limited by the optical coatings and the simultaneous bandwidth5

C–H (bend)

C–C

C–H
C=C

C–X
C–O

O–H

C=C

C=O

Absorbance for 1 ppm-m (10-3)

C–N
4

Propene (CH3CH2CH3)
Acetylene (C2H2)
Acetone (CH3COCH3)
Formaldehyde (HCHO)Benzene (C6H6)
CHCl2F
Isopropanol (CH3CHOHCH3)
Methylamine (CH3NH2)

3

2

1

0
500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Wavenumber (cm )-1Figure 1.1: Location of fundamental vibrational bands and spectra of some example molecules in
the mid-infrared. Spectra from the PNNL Northwest Infrared (NWIR) database [48].

6
recorded is limited by the camera used for detection. To circumvent these problems, we have developed a scanning Fourier-transform spectrometer (FTS) that is capable of recording spectra in
the near- and mid-infrared with below 150 MHz resolution [46].
Chapter 4 covers the results from two different comb spectroscopy systems. One system wasdesigned to test the application of CE-DFCS for the detection of trace water and other contaminantsin arsine (AsH3 ) [53], which is a critical gas for semiconductor manufacturing. In this system, we
demonstrate for the first time direct frequency comb spectroscopy in a region accessed using highly
nonlinear fiber – in this case, 1.75-1.95 µm (5710-5130 cm−1 ). Furthermore, this was the first
demonstration of CE-DFCS with a focus on an industrial application (i.e., trace detection in astrongly absorbing process gas), where the bandwidth is critical for distinguishing impurity signals
from background absorption, along with high resolution for making unambiguous identifications.
Next, we discuss our initial demonstrations of frequency-comb-FTS based on a high power OPO
that operates in the important mid-infrared window from 2100 cm−1 to 3700 cm−1 [50]. The systemprovided for the first time a combination of resolution, spectral range, sensitivity, and acquisition
speed that is sufficient for detecting trace quantities of a wide range of molecules under real-world
conditions [46]. This system also provided the basic tools for later demonstrations of shot-noise
limited CE-DFCS in the near-IR [54] as well as sensitive detection of hydrogen peroxide in thepresence of water in the mid-infrared [55]. Because the FTS only provides time resolution of atbest one second, we also used the VIPA spectrometer with mid-infrared CE-DFCS to demonstrate
rapid detection of reactive chemical intermediates [56]. We conclude this section with an outlook
on potential future applications of comb spectroscopy.
While a nearly uncountable number of neutral molecules have been studied in detail using
spectroscopy, the study of molecular ions is much less developed despite their importance in stratospheric chemistry [57] and astrochemistry [58, 59]. We have developed a new technique for broad
bandwidth spectroscopy of molecular ions by combining CE-DFCS with velocity-modulation spectroscopy [60]. The description of this system is given in Chapter 3, while a detailed analysis of thespectra of HfF+ measured with this system [61] is given in Chapter 5. In addition to it’s importance7
for a test of time-reversal symmetry using trapped ions, HfF+ is challenging for molecular theory
calculations due to the large number of electrons and the inclusion of d and f shell electrons. Ourdata provided critical information for improving and testing the theory of heavy atoms.
The final chapter covers the application of precision spectroscopy using molecular ions tofundamental physics with a measurement of the electric dipole moment of the electron (eEDM)[62, 63, 64, 65], which is a test of time-reversal symmetry. Trapped molecular ions – in this case
HfF+ – can potentially provide very long coherence times for Ramsey measurements, which in turnenables high precision. Partially for this reason, molecular ions have recently attracted considerable
interest for a variety of precision tests of fundamental physics [66, 67, 68, 69, 70]. In this chapter,
we first provide a brief introduction to eEDM measurements and then provide some details about
our ion trap and ion detection systems. We also discuss state detection using resonance-enhanced
multiphoton dissociation and preparation of a desired quantum state using a variation of adiabatic
passage that uses trap motion combined with state-sensitive depletion. For the eEDM measurement,
we need to apply bias electric and magnetic fields, which we do in a rotating frame. We demonstrate
Ramsey spectroscopy between Zeeman sublevels in this rotating frame and discuss limitations to thecoherence time from inhomogeneous fields and ion-ion collisions. Finally, we analyze the sensitivity
and potential sources of systematic errors and demonstrate an initial measurement of the eEDM.
Overall, the techniques developed here should provide the basis for many exciting new applications. Further developments of CE-DFCS to increase the sensitivity and extend the wavelength
coverage further into the mid-infrared could enable studies of many important reactions and reactive
intermediates with applications to atmospheric and combustion chemistry as well as fundamental
chemical kinetics. The combination of bandwidth, resolution, and sensitivity of CE-DFCS also
has exciting prospects for experiments outside of the lab. Our frequency-comb velocity-modulation
system is the first system capable of rapid, sensitivity spectroscopy of molecular ions over a broad
bandwidth. Extensions of this technique could enable survey spectroscopy of many molecular ion
species with applications to astrochemistry, physical chemistry, and fundamental physics. The
combination of survey spectroscopy of ions with the techniques pioneered in Chapter 6 for pre-

8cision measurements in trapped ion clouds could result in a new limit on the eEDM and could
also provide new systems for other tests of fundamental physics such as parity violation, quantum
electrodynamics, and variation of fundamental constants.

Chapter 2

Frequency comb sources

In a broad sense, a “frequency comb” [14, 71, 15] can be defined in the frequency domain
as a collection of narrow, equally-spaced lines with a well-defined phase relationship between thelines. More specifically, each line or mode in the comb should be at a known, and controllable,
frequency. Frequency combs can be generated by a variety of sources; we divide these into three
general categories: mode-locked lasers, indirect sources, and cw-laser based sources. Mode-locked
lasers can produce frequency combs directly at the output of the laser. Indirect sources use nonlinear optical effects to modify or shift the the spectrum of a mode-locked laser. cw-laser basedsources use non-linear or electro-optical materials to generate multiple sidebands from a cw-laser.
We will provide an overview of how each of these sources work and also briefly discuss some common
sources and their advantages and disadvantages. We also provide references to reviews or bookswith more information whenever possible.
The frequency comb spectrum provides several benefits for spectroscopy applications whencompared to more commonly used sources such as incoherent (thermal and LED) light or cw-lasers[44, 46, 72]. First – as we will see – the comb spectrum can be very broad, even matching or
exceeding some incoherent sources. At the same time though, under the broad spectral envelope
there are many narrow lines at well defined frequencies, thus matching the spectral resolution
attainable with cw-lasers. In this respect, a frequency comb can be thought of as an array ofmany thousands of cw-lasers operating simultaneously, each with a precisely defined frequency.
Additionally, the spectrum of a frequency comb can be efficiently matched with the resonances of

10
an optical Fabry-Perot cavity for significantly increased detection sensitivities. Finally, the highspatial coherence of laser sources enables long path-length beam propagation with little loss of
power due to divergence. This can be used for example for multi-kilometer length absorption
measurements in the atmosphere [73].
The comb source used in a particular spectroscopic application is chosen based on the requirements of that application. The first consideration is typically the necessary spectral coverage ofthe comb source. For molecular spectroscopy, the strongest transitions, thus the highest detection
sensitivity, exist in the visible to ultraviolet (electronic transitions) and in the mid-infrared above
3 µm (fundamental vibrational modes). Many interesting atomic transitions occur below 200 nm.
This has pushed the development of comb sources towards these spectral regions. For multi-species
detection capability, typically a broader spectral bandwidth is desired; however, for a given laser
power, a broader bandwidth results in less power at any given spectral line, potentially decreasing
the signal-to-noise. Another consideration is the spacing between comb lines. This not only affects
the power-per-line for a given power and bandwidth, but also presents a tradeoff between instantaneous spectral resolution and the amount of scanning necessary to cover the full spectrum. The
necessary power depends also on the efficiency of the read-out method. Finally, for applications
outside of the spectroscopy lab, the robustness and portability of the system must also be considered. These considerations have led to the development of a wide range of comb sources. Figure
2.1 shows the spectral coverage and power of a variety of different sources currently demonstrated.

2.1

Mode-locked lasers
A frequency comb in the spectral domain transforms to a train of equally spaced pulses inthe time domain. The time between each pulse is given by the inverse of the frequency spacing
between adjacent lines in the frequency domain. This frequency is called the repetition rate (frep ),and is typically set by the optical path-length L of the laser resonator. The inverse of the durationof each pulse sets the frequency domain spectral bandwidth. There is one additional degree of
freedom, which is the pulse-to-pulse phase shift of the oscillating carrier electric field relative to the11Power per mode (μW)

103

HHG [120]
Ti:sapphire + THG
Ti:sapphire + SHG

10-2
10-3

102

10-4
50

101

100

10 GHz Ti:sapph + PCF [129]
1 GHz octave Ti:sapph [128]Ti:sapphire

Yb:fiber + nonlinear fiber [49]
10 W Yb:fiber
Er:fiber + nonlinear fiber [52]Er:fiber
Tm:fiber [111]Cr:ZnSe [88]PPLN OPO [50]
PPLN DFG [96]Degen. OP-GaAs OPO [100]GaSe DFG [95]

200

100
10-1
10-2
200

400

800
1600
Wavelength (log2 scale) [nm]

3200

6400

12800

Figure 2.1: Comparison of various comb sources. Solid lines represent example spectra from eachsource, dotted lines represent the approximate tuning range for tunable sources.

12
pulse envelope (the carrier-envelope phase shift, ∆φCE ). The carrier-envelope phase shift arises
from cavity dispersion which results in a difference between the group velocity vg and the phasevelocity vp at the pulse center frequency:∆φCE = ωL

1
1
−
vg
vp


.

(2.1)

This causes in a shift of the comb modes in the frequency domain such that the frequency of themodes is now given by νm = m ∗ frep + f0 . Here m is an integer mode number and f0 is the offset
frequency (also called the carrier-envelope offset frequency) due to the carrier-envelope phase shift,f0 = ∆φCE /(2π) ∗ frep .
Lasers that emit such a train of short pulses are called mode-locked lasers and were originallyinvestigated solely for ultrashort pulse generation. Over the past few decades, mode-locking has
been demonstrated by a variety of different methods in many different lasers, including solid-statelasers, dye lasers, fiber lasers, and diode lasers. Twenty years after the initial demonstration of amode-locked laser, the ability to completely control and stabilize the frequency domain structureof such a laser was achieved, resulting in the first fully phase-stabilized frequency comb [74]. Such
precise control has only been reliably accomplished in a small subset of all mode-locked lasers,
and we focus our attention on the most common of those gain media: namely Ti3+ :sapphire,
Yb3+ :fiber, Er3+ :fiber, Tm3+ :fiber, and (to some extent) Cr2+ :ZnSe. Several mechanisms exist formode-locking these lasers [75, 76, 77] and can be broadly divided into active or passive methods.

2.1.1

Mode-locking Mechanisms

2.1.1.1

Active mode-locking

Active mode-locking works by forcing multiple laser resonator modes to have a well-definedphase relation between them [77]. This can be accomplished with an intra-cavity electro-optic or
acousto-optic modulator that is driven at frep (i.e., the laser mode spacing). Thus, laser light at
one mode will have sidebands located at adjacent modes. These sidebands will experience gain andwill also obtain sidebands, which results in a cascaded generation of laser modes with fixed phase.

13
While active mode-locking is very robust, it is difficult to produce extremely short pulses using
only active mode-locking because there is no strong pulse shortening mechanism [78]. Because ofthis limitation, most comb sources rely on some form of passive mode-locking possibly in addition
to active mode-locking for self-starting.

2.1.1.2

Saturable AbsorberMode-locking can also be achieved by modifying the temporal response of the laser cavity tofavor pulse formation over cw lasing. One way to accomplish this is to incorporate an intra-cavity
absorbing medium with an absorption coefficient given byα=α0
1 + IIs(2.2)

where α0 is the zero-power absorption coefficient, I is the intra-cavity intensity, and Is is thesaturation intensity (defined as the point where the absorption coefficient is α0 /2). This has
the effect of absorbing the leading edge of a pulse and increasing the effective gain for the peakof the pulse. The carrier lifetime of common saturable absorbers is in the range of picoseconds
to nanoseconds, which would limit the achievable pulse duration to this range; however, whencombined with gain saturation – which lowers the gain for the trailing edge of the pulse – shorter
pulses are possible.
The most common saturable absorbers are semiconductor-based quantum wells (such as GaAs
or InGaAsP) grown on the surface of Bragg reflecting mirrors [79]. These are called either saturable
Bragg reflectors (SBRs) or semiconductor saturable absorber mirrors (SESAMs) and are typicallyused as an end-mirror in the laser cavity. These saturable absorbers often have a saturation fluenceof about 20 µJ/cm2 , which gives a saturation energy of 250 pJ for a beam radius of 20 µm andnecessitates tight focusing for effective mode-locking. In order to overcome some of the limitations(such as carrier lifetime) of semiconductor-based saturable absorbers and extend the applicability
to other systems, saturable absorbers based on materials such as carbon nanotubes and graphene
have been demonstrated [80, 81, 82].

14
2.1.1.3

Kerr effect

An optical field (E) traveling through a material causes a polarization of the material given
by P = χe E, where χe is the material susceptibility [83]. For strong fields, χe is itself dependent
on the field strength and is approximated by χe ≈ χ(1) + χ(2) E + χ(3) E 2 + .... For a material withinversion symmetry (and amorphous materials), χ(2) = 0. Using I = (cn0 0 |E|2 )/2 and n2 = 1 + χe ,we obtain for small χ(3) that n ≈ n0 + n2 I, where n2 is the nonlinear index and is proportionalto χ(3) . This (instantaneous) modification of the index of refraction as a function of intensity is
known as the Kerr effect.
In solid-state lasers such as Ti:sapphire [84, 85] and Cr:ZnSe [86, 87, 88], the Kerr effectresults in a variable nonlinear phase as a function of radial position in the the beam [89, 77]φnl (r, t) =

2π
λ



2

2

n2 dI(t)e−(2r /w0 ) ≈



2π
λ





r2
n2 dI(t) 1 − 2 2
w0

(2.3)

for a thin material of thickness d and a Gaussian beam of waist w0 . This parabolic phase front
results in an effective lens of focal lengthf=w02
4n2 dI0(2.4)

where I0 is the peak pulse intensity. This lens can be used as an effective saturable absorber formode-locking (called Kerr-lens mode-locking or KLM) in several ways. First, with a hard aperture,
the transmission through the aperture will increase with more lensing, so the net gain will be higher
for pulsed operation. Even without a hard aperture, the presence of the Kerr lens modifies thecavity parameters, thus a cavity near the edge of stability can be made more stable for pulsed
operation. KLM lasers may not be self-starting; however, typically only small perturbations arenecessary to initiate mode-locking. In general, KLM lasers provide the shortest pulse durations
achievable directly from the gain medium.
The Kerr effect also occurs in Yb- and Er-doped gain fibers, where it results in a nonlinear
rotation of elliptically polarized light. This polarization rotation can be used in a ring cavity withpolarization selective elements to achieve mode-locking by increasing the transmission through an15intracavity polarizer for pulsed light relative to cw light [90]. Polarization-rotation mode-locking canbe used to create all-fiber lasers with no free-space sections and provides for reliable, self-starting
mode-locked operation [91].

2.2

Indirect Sources
The ultrashort pulsed output of a mode-locked laser results in a high peak intensity per pulse.

This provides large nonlinear effects in many materials, which can be used to extend or shift the
spectrum of a mode-locked laser. Frequency combs can thus be generated in spectral regions thatare difficult or impossible to fully cover with cw-lasers such as the extreme ultraviolet (below 100nm) or the mid-infrared (2-10 µm). In addition, frequency combs can simultaneously cover multiple
octaves of spectral bandwidth using non-linear optics, far exceeding the tuning range of any cwlaser, while still maintaining the high resolution of a cw-laser. This flexibility makes indirect comb
sources well suited for new applications in spectroscopy.
Many crystals do not possess inversion symmetry and therefore exhibit χ(2) nonlinearity. Thisnonlinearity can be used for second-harmonic generation (SHG), sum-frequency generation (SFG),
difference-frequency generation (DFG), and parametric generation [83]. In SHG, two pump photons
are combined to form one photon at twice the frequency; similarly in SFG, two pump photons at
different frequencies are combined to produce one at the sum frequency. DFG again works in thesame way, except that the difference frequency is produced. Parametric generation is the reverseof SFG, where one pump photon is down-converted into two lower energy photons (a signal andidler). Phase-matching between all three frequencies must be satisfied in all of these processes; thiscan be accomplished by using different axes of a birefringent crystal and tuning the input angle andpolarization relative to the crystal axes or by periodically poling a crystal such that the phase of the
produced light is periodically changed to add coherently (similar to Bragg reflection). The phasematching can be tuned by rotating the crystal, or in the case of periodically-poled materials, it ispossible to create a “fan-out” poling-period that varies linearly across one dimension of the crystal
so that the crystal just needs to be translated. In addition, longer periodically-poled crystals can

16
be used without severely limiting the angular acceptance (and thus the phase-matched spectral
bandwidth), resulting in higher conversion efficiencies.
The most common approaches for producing mid-infrared frequency combs beyond 3 µm areto use either DFG or an optical parametric oscillator (OPO, see Section 2.4). DFG combs havebeen demonstrated using the spectrum generated directly from a Ti:sapphire laser [92]; however,
the achievable powers were very low. More power can be obtained by using two synchronized
Ti:sapphire lasers [93], but this is experimentally more complicated. Multi-branch Er:fiber lasers
enable mW-level, tunable DFG with one branch used with nonlinear fiber to provide tunable,
shifted light and the second branch used to provide high-power, unshifted light [94, 95]. Up to
100 mW has recently been acheived using a fan-out periodically-poled crystal and a Yb:fiber laser[96]. In this case, some of the light from the fiber laser was sent through nonlinear fiber, andthe red-shifted Raman soliton (see Section 2.3 for more information on nonlinear fiber optics)
was mixed with the remaining unshifted pump light to generate the difference frequency. DFG
systems are convenient and compact, but the power limitations can hinder some applications. For
higher power, it is necessary to use an OPO, in which the signal and/or idler light produced byparametric generation is resonant with a cavity containing the nonlinear crystal. This greatly
increases the conversion efficiency, but also adds some additional complexity. For comb generation,
the cavity is generally pumped synchronously, that is the OPO cavity FSR should match an integer
multiple of the laser repetition rate, and the cavity length is actively controlled since this sets thef0 of the generated comb [97, 50, 98]. It is also possible to run an OPO with the signal and idler
degenerate (a “divide-by-two” system), which can be used to produce near-octave spanning spectral
bandwidth in the mid-infrared [99, 100]. This broad spectrum presents a drawback though as well,since the power per spectral element is usually low. The upper limit to the attainable spectral
range is set partially by the absorption edge of crystals used (commonly lithium niobate), so toreach longer wavelengths, other crystals must be used. These are usually angled-tuned crystals
such as AgGaSe2 [101]; however, recently, periodic patterning of GaAs has been developed, which
could enable significantly higher powers. In order to use GaAs for mid-infrared generation, the17pump wavelength must be above about 1.6 µm, thus the increased interest in Tm:fiber systems.
Very recently, an octave-spanning mid-infrared spectrum up to 6.1 µm was demonstrated with a
Tm:fiber laser and degenerate OPO using orientation-patterned GaAs (OP-GaAs) [102], although
the average power was only about 30 mW.
Large nonlinear effects can also be obtained by tight confinement in small-core optical fiber[103, 104] (such as photonic crystal fiber [105, 106], microstructure fiber [20, 107], and highly
nonlinear fiber [108]). These fibers not only result in high intensities but also provide a longinteraction length over which to accumulate effects. In addition, the dispersion profile of the fiber
can be tailored by adjusting the mode size and adding dopants, which provides even more control
over the nonlinear effects present. Because of the flexibility, nonlinear fibers have found a wide
range of applications. In addition to visible light production [109], Er:fiber lasers broadened withnonlinear fiber have also been used to seed a Tm:fiber amplifier, which can produce >3 W average
power femtosecond pulses tunable around 2 µm [110, 111]. This source can be used to pump an
OP-GaAs OPO in the 5-12 µm spectral region (where sources for comb spectroscopy are lacking)or can be used to generate tunable THz radiation using OP-GaAs [112]. Broadening in nonlinear
fiber can also be used to coherently link different wavelength regions. For example, a broadened
Yb:fiber comb would perhaps be the ideal source to link a highly stable cw laser at 1.55 µm to
the Strontium optical clock transition at 698 nm if the Yb:fiber laser can provide coherent light at
both spectral regions simultaneously. Finally, in an application of comb spectroscopy to trace gas
analysis in semiconductor processing gases (Section 4.1), an Er:fiber laser can be used to provide
tunable 1.8-1.9 µm light with reasonable powers from a compact system. Currently, fibers withlarge nonlinearities are readily available throughout the visible and near-infrared and have recently
been demonstrated in the mid-infrared [113, 114, 115, 116]. Nonlinear fibers enable flexible comb
generation in many wavelength regions using robust (fiber-based) pump lasers; however, care must
be taken so that the coherence of the pump is preserved in the nonlinear processes, as discussed in
Section 2.3.
With extremely high peak electric fields, like those attainable at a tight focus in an optical

18
cavity, the perturbative expansion to the polarizability breaks down and the material (in this caseusually a jet of noble gas) is ionized. The electron is accelerated away from the atom for someperiod of time, until the electric field reverses and accelerates the electron back toward the atom.
The electron can then recombine with the atom, emitting all of the excess energy from the fieldin high-order (odd) harmonics of the original laser frequency. This process, called high-harmonic
generation (HHG), can be used to produce combs in the vacuum ultra-violet (VUV, 100 to 200nm) and extreme ultra-violet (XUV, below 100 nm) [117, 118, 119] with over 100 µW per harmonicorder[120].

2.2.1

cw-laser Based Sources

Currently, cw-laser based comb sources have not seen many applications to spectroscopy andso we only briefly mention them. It is possible to make a frequency comb simply by applying astrong frequency modulation, which results in sidebands at harmonics of the modulation frequency.
With very strong modulation it is possible to put optical power into high orders, resulting in a
comb in the frequency domain. However, in practice, the achievable spectral bandwidth is small
and thus does not provide much of an advantage over cw-laser spectroscopy. A comb can also be
generated by driving a Raman transition between rotational levels in a molecule using two (pulsed)
laser frequencies. With enough Raman gain, it is possible to cascade this process and produce anoctave-spanning comb [121]. Currently these combs have only been demonstrated with large mode
spacing (the rotational spacing of the molecules used), which potentially limits the usefulness forspectroscopy.
Recently, frequency combs based on parametric frequency conversion of cw-lasers in microresonators have been been demonstrated [122, 123, 124]. When a cw-laser is injected into ahigh-finesse microresonator, the high intracavity intensity results in cascaded four-wave mixing(FWM) and can generate a comb with a frequency spacing set by the free-spectral range of the
microresonator. These sources show some interesting potential for spectroscopy applications due totheir compact size and inherent simplicity; however, they are currently limited to repetition rates

19
above about 20 GHz, which is a potential drawback. In addition, the inherent noise properties of
the generated comb are not yet well understood or controllable [125, 126].

2.2.2

Typical comb sources

Mode-locked Ti:sapphire lasers were used for the first realization of fully stabilized frequency
combs, and they are still probably the most widely used comb sources. These are usually Kerr-lensmode-locked and are sometimes actively mode-locked as well. The Ti:sapphire gain bandwidth isextremely broad, which enables the generation of ultrashort (10 femtosecond) pulses and correspondingly large spectral bandwidth (covering about 700 nm to 1050 nm) directly from the laser.
It is even possible to generate octave-spanning spectra directly from the laser with intracavity SPM[127, 128]. Ti:sapphire combs can be made with repetition rates ranging from < 100 MHz up to 10
GHz [129], providing large flexibility for different applications. Because the ultrashort pulses resultin high-efficiency nonlinear processes, Ti:sapphire lasers also provide the highest power combs inthe visible to UV range by using SHG [130] or supercontinuum fiber. One drawback is that thefree-space cavity limits the robustness of these lasers. In addition, the pump laser is still expensive
and relatively bulky, further limiting the field applicability of the system.
Yb:fiber lasers [75] produce combs directly in the 1000 to 1100 nm spectral region. Theselasers are typically mode-locked using a saturable absorber and are limited in spectral bandwidth(and thus pulse duration of about 80 fs) by the gain bandwidth of the fiber. One common cavity
configuration consists of a linear cavity with the pump diode coupled to the gain fiber using awavelength-division multiplexer (WDM), one cavity mirror (the output coupler) being a Bragg
grating written into fiber, and a short free-space section containing a waveplate, focusing lens andan SBR as the second cavity mirror (see Figure 2.2). While this design does contain a small freespace section, it is still robust since the pump is entirely fiber-coupled and the cavity is mostly
fiber. Additionally, the pump diode is compact and fairly inexpensive. It is also possible to use
polarization rotation mode-locking; however, in this case free-space intracavity gratings are requiredfor dispersion compensation. Yb:fiber combs have been built with repetition rates up to 1 GHz.

20

Amplified Yb:fiber Laser

918 nmFiber stretcher
WDM

47

Large-mode
area Yb:fiber

FBG

PZT

High-harmonic Generation

Grating
Compressor

Yb:fiber

SBR

51

56

63

71

82

97

119

nm

order

976 nm

Optical Parametric Oscillator

Fiber Supercontinuum Generation
locking electronics

idler (i)
PZT

p+s
p-SC

p-SC
DM

PCF

p+i
2 s DM

h/2
p

PBS

p+s
p+i
2s

pump
(p)

DM

p

fan-out
PPLN

0Intensity (dB)

PD2

PD1

simulatedmeasured
-40

-80

600

800

1000

1200

1400

1600

1800

Wavelength (nm)Figure 2.2: Sources based on a Yb:fiber laser. The red-outlined box shows a linear Yb:fiber oscillator
and stretched pulsed amplifier. This type of system can be used to produce frequency combs
covering many different spectral regions with nonlinear optics. The bottom left box shows a tunable
comb in the mid-infrared using an optical parametric oscillator [50]. By using a small core nonlinear
fiber (an example cross-section is given in [20]) it is possible to cover over 1.5 octaves coherently(center box) [49]. As shown in the box on the right, an XUV comb down to 50 nm can be produced
using cavity-enhanced high-harmonic generation [120]. PZT is a piezo-electric transducer, FBG isa fiber Bragg grating, DBR is distributed Bragg reflector.

21
Yb:fiber based amplifiers are very power scalable due to the high Yb-doping concentration possible
in large mode area fibers. In fact, chirped-pulse amplifiers have been used to produce a comb with
80 W average power at 150 MHz [131]. As shown in Figure 2.2, the high average power capabilities
have enabled comb generation spanning over 1.5 octaves in the near-infrared in highly nonlinear
fiber (Section 2.3) [49], in the XUV down to 50 nm with high-harmonic generation [120], and in the
mid-infrared with parametric generation (Section 2.4) [50]. One drawback of Yb:fiber lasers is thatthey are not very tunable without external spectral broadening due to the narrow gain bandwidth.
Er:fiber lasers [132, 133] mode-locked (typically) using polarization rotation, have become
very popular for a few reasons. First, since they produce combs near 1550 nm they can take advantage of all of the advancements in telecommunications technology. This makes them inexpensive
and fairly easy to build. In addition, they can be made entirely out of fiber, without any free-spacesections, which makes them very robust and portable [134, 91] even being used in a drop tower
experiment with deceleration of 50 g [135]. It is also possible to split the comb output into multiple
branches, and amplify each branch separately, which provides a large amount of flexibility. Withhighly nonlinear fiber, Er:fiber lasers can cover from 1000 nm to over 2100 nm (Section 2.3). When
combined with SHG as well, they can be used to provide tunable combs throughout the visible
region [109]. They have also been used for mid-infrared comb generation using DFG or an optical parametric oscillator. However, currently Er:fiber lasers are limited to typically 500 mW perbranch. Also, for spectroscopy, their spectral coverage without broadening or frequency conversion
is not ideal. Due to the low doping concentration of the Er:fiber and the necessary dispersion
compensation, the repetition rate is typically 100-250 MHz, although a repetition rate of 1 GHz
has been demonstrated [136].
Two new comb sources have recently been developed that push the wavelengths toward the
mid-infrared. Tm:fiber combs [137], which operate between 2 to 2.1 µm, function in many ways
similarly to Yb:fiber lasers and with many of the same advantages and disadvantages of thoselasers. Currently, they can provide about 3 W of power at repetition rates up to about 100 MHz,although the power can be increased with careful thermal management. It is also possible to seed

22
a Tm:fiber amplifier with the shifted output of an Er:fiber comb [111]. The primary advantage ofthese lasers is that they allow for the use of new nonlinear crystals for frequency conversion to the
mid-infrared. Cr:ZnSe lasers are similar in may respects to Ti:sapphire except that they operatearound 2.5 µm. The fractional gain bandwidth is even larger than that of Ti:sapphire, which gives
the potential for ultrashort pulse generation in the mid-infrared and thus broad spectral bandwidth
directly from the laser. Currently though, challenges such as cavity dispersion control [138] have
limited the utility of these lasers.

2.3

Nonlinear fiber optics
The observed spectrum from the nonlinear fiber results from a complex interplay of multiple

χ(3) processes such as self-phase modulation (SPM), cross-phase modulation (XPM), stimulated
Raman scattering (SRS), and four-wave mixing (FWM) as well as the interaction of dispersion
with each of these [139, 140]. Because of this complexity, fiber super-continuum sources are often
modeled numerically using a generalized nonlinear Schrodinger equation [141]. Several comprehensive reviews [104, 142] and books [103, 143] have been written about nonlinear effects in fiber, andwe provide a summary in Section 2.3.2. Briefly, the dominant nonlinear process depends on thefiber group-velocity dispersion (GVD, parameterized by the coefficient β2 ) near the wavelength of
the injected pulse. For normal (i.e., positive) GVD, the primary broadening mechanism initially
is SPM, which results in a nearly symmetrical spectral broadening, with some additional contributions from XPM and FWM especially close to the zero-GVD point. SPM and XPM both arise
from the Kerr effect discussed before; however, in this case the nonlinear phase shift occurs in
the temporal domain. Since a time-varying phase results in an instantaneous frequency shift, thisresults in a frequency chirp and spectral broadening. This can also be understood as a temporalfocusing, i.e., pulse compression. In XPM, electric fields at one frequency cause phase shifts onfields at another frequency. FWM is a general term for a χ(3) process that converts two input photons (at the same or different frequencies) into two photons at new frequencies with the same totalenergy. It dominates near the zero-GVD wavelength and also plays a role as the spectrum broadens23
and phase-matching becomes more likely. In the anomalous (negative) GVD region, the injected
pulse can initially form (quasi-) stable wavepackets called solitons. As these solitons propagate,
they shift toward longer wavelengths (i.e., red shift) due to stimulated Raman scattering where aphoton from the shorter wavelength portion of the pulse spectrum is converted into a longer wavelength (lower energy) photon and an acoustic phonon. In addition, as the solitons propagate, theyemit so-called dispersive waves at short wavelengths (in the normal GVD region) where the phasevelocity matches that of the soliton. These dispersive waves are analogs to the Cherenkov radiation
emited by charged particles. Finally, the solitons and dispersive waves can couple through XPM,
resulting in additional spectral broadening.
Many of these features can be observed in the measured spectra from a broadened Yb:fiber
laser shown in Figure 2.3. The fiber used here is the IMRA suspended core fiber [20], and wemeasure the spectrum as a function of power coupled into the fiber. As the power is increased, we
initially see one Raman soliton on the red side of the spectrum and then this soliton progressively
shifts to the red and multiple solitons appear. On the blue side, we initially see peaks below 700
nm, which are the dispersive waves. Between 700 and 900 nm the behavior is more complicated: at
the highest power, this region is completely filled, but we have found that this region potentially has
significant amplitude noise (see below). This fiber setup was used as part of the stabilization of themid-IR OPO (Section 2.4). We have investigated the coherence properties of a similar broadening
setup that covers 1.5 octaves, as discussed in detail below.
Nonlinear fiber optics can also be used for comb spectroscopy around 1.8-2 µm with an
Er:fiber laser (Section 4.1). Initially, we used a long piece of single-mode polarization-maintaing
fiber (e.g., Fibercore HB-1500, which has a slightly smaller core compared to standard SMF-28)
to Raman shift a significant portion (50% or potentially even more) of the initial power to beyond
1.8 µm, as shown in Figure 2.4(a). The final wavelength of the shifted light could be controlled byvarying the input power, polarization, or initial chirp (using a prism pair). This has the advantage
of providing high power concentrated in a single, controllable spectral region; however, we foundsome issues with the coherence of the shifted light, as discussed more below. Because of this, we

24

5 0 0

6 0 0

7 0 0

8 0 0

9 0 0

1 0 0 01 1 0 01 2 0 01 3 0 0

1 4 0 0

1 5 0 0

1 6 0 0

1 7 0 0

6 0

P o w e r in fib e r

5 0
4 0

1 4 m W
2 5 m W
3 7 m W
6 2 m W
8 7 m W
1 2 4 m W
1 5 0 m W

3 0
2 0
1 0
0

d B

-1 0
-2 0

F ib e r le n g th :~ 3 0 c m

-3 0
-4 0
-5 0

E a c h s p e c tru mo ffs e t 1 0 d B
fo r c la r ity

-6 0
-7 0
-8 0
5 0 0

6 0 0

7 0 0

8 0 0

9 0 0

1 0 0 01 1 0 01 2 0 01 3 0 0

1 4 0 0

1 5 0 0

1 6 0 0

1 7 0 0W a v e le n g th ( n m )Figure 2.3: Spectral broadening of a Yb:fiber laser using IMRA suspended core fiber.

25
used supercontinuum generation in highly nonlinear fiber for this project (Figure 2.4(b)).

2.3.1

Numerical simulations

To better understand the spectral broadening process and supercontinuum coherence properties, it is necessary to perform numerical simulations of the pulse dynamics within the nonlinear
fiber. The propagation in fiber can be accurately described by a generalized nonlinear Schrödinger
equation (GNLSE) [144, 141],hi∂A (z, t) α
+ A − iF −1 β (ω) − ωβ1 − β0 Ã (z, ω) =
2 

∂z
Z


i ∂
0 2 0
0
dt .
iγ 1 +A (z, t) R t A z, t − t
ω0 ∂t

(2.5)

The left side describes all linear and dispersive effects where A(z, t) is the field envelope as a functionof position z and time t, α is the absorption coefficient, and the inverse Fourier transform accountsk∂for dispersion. In this, Ã (z, ω) is the Fourier transform of A (z, t); βk ≡ ∂ωk β (ω) ω , where β (ω)
0is the propagation constant, and ω0 is the carrier frequency. The dispersion term is frequently
approximated by a Taylor series, sohi X ik+1 ∂ k A
iF −1 β (ω) − ωβ1 − β0 Ã (z, ω) ≈βk k .
k!
∂T

(2.6)

k≥2

The terms on the right side of Equation 2.5 take into account the relevant χ(3) effects, namely
Raman scattering and the Kerr effect, through the nonlinear coefficient γ = ω0 n2 (ω0 )/cAef f (ω0 ).
Here n2 is the nonlinear refractive index of the fiber core and Aef f is the effective area of theguided mode. The time derivative accounts for intensity dependence of the group velocity, an effectthat leads to self-steepening and optical shock formation (steep leading pulse edges). This sameformalism can be used to describe the Kerr effect. Shock wave formation occurs on a characteristic
time scale given by τsh ≈ 1/ω0 . The integral term on the right side of Equation 2.5 accounts forthe delayed Raman response of the fiber material whose temporal impulse response is given by
R (t) = (1 − fR ) δ (t − te ) + fR hR (t) [103], where the Raman fraction, fR , sets the ratio of Kerr
to Raman non-linearity, hR (t) is the time-domain Raman response function, and te is an electric

26

(a)

1.0

Normalized Intensity

0.8

0.6

0.4

0.2

0
1500

1550

1600

1650

Normalized Intensity(b) 1.0

1700
1750
Wavelength (nm)

1800

1850

1900

0.8

0.6

0.4

0.2

0

1000

1200

1400

1600

1800

2000

2200

Wavelength (nm)Figure 2.4: Comparison of Raman shifting fiber and highly nonlinear fiber. (a) Raman shifted
spectrum from an Er:fiber laser after propagating through ∼8 m of polarization-maintaining fiber.
(b) Spectrum after ∼7 cm of highly nonlinear fiber (OFS Specialty Photonics).

27
delay. In general, it is assumed that the electronic response is instantaneous so that the delay time
can be neglected. The response function and Raman fraction (fR ≈ 0.18) are determined by fitting
the frequency domain Raman gain spectrum [144, 145]. This fit assumes that the gain spectrum
can be described by a single Lorentzian model (which assumes that the Raman response is only
due to a symmetric Si–O–Si stretch); however, this neglects what is known as the ’Boson peak’in amorphous glasses. A potentially more complete model for hR is given in [146], which gives ahigher fit value of fR ≈ 0.245 for silica fiber. For the simulations here we have opted to continue
to use the simpler and more frequently used model with fR ≈ 0.18 for standard SMF 28. Dopingof the fiber core, for example with GeO2 , will increase fR , which will be important below.
We numerically solved the GNLSE using a symmetric split-step Fourier algorithm. Thisalgorithm assumes that the dispersion and non-linearities act independently of each other, so foreach distance step of size h, dispersion acts for the first length h/2, then the non-linearity is applied,
and then dispersion is applied for the second half step of distance h/2. This can be thought of as
neglecting the commutator between the dispersion and non-linearity operators, which results in anerror of order h2 , thus the algorithm converges by decreasing h. For the dispersion step, we havethe following equation (neglecting all non-linearities):
∂A
= D̂A
∂z

(2.7)

where D̂ is the dispersion operator, which can be obtained by comparing with the GNLSE (Equation
2.5). Equation 2.7 can be easily solved by switching to the frequency domainihA(z + h/2, T ) = F −1 exp −h/2D̂(−iω) F(A)(2.8)

where D̂(−iω) is obtained by replacing each time derivative in the dispersion operator with −iω.
While dispersion is easy to apply, non-linearities are more subtle. There are many possible
methods that can be used, each with trade-offs in terms of accuracy and computational difficulty.
First, there are several ways of computing the non-linearity operator. Agrawal [103] typicallyapproximates |A (z, t − t0 )|2 with a Taylor expansion (to first order) about t0 = 0 which reduces the28
convolution integral to just an integral of the Raman response function. Furthermore, second orderderivatives are dropped to give three terms. This is the simplest form of the non-linearity operator;
however, its accuracy is questionable for ultrashort (< 100 fs) pulses, so it was not used in thisstudy. The method used here is to compute the convolution in the frequency domain as an inverse
Fourier transform of a product of Fourier transforms [147]. Note that if the FFT algorithm is used
(as it typically is) the convolution will need to be scaled by a factor proportional to 1/N , where
N is the number of frequency/time grid points. Similarly, the time derivative can be evaluatedusing another inverse Fourier transform/Fourier transform pair, or by using a numerical gradientalgorithm.
The simplest way to apply the non-linearity operator is by multiplication with exp(hN̂ )A;
however, this is not the most accurate method. Another method is to say that in reality, the nonlinearity should have been integrated over the distance h, and then to approximate this integral as
the average value at its endpoints. Now, the value of A(z + h) (which is needed to find the nonlinearity operator at that point) is not known, but can be solved iteratively by first using the initialvalue of A, calculating the final A after applying all dispersion, then using this as the new valueof A(z + h) and repeating until the final field converges. This is ideally very accurate, although is
potentially computationally intensive. The method used here is to solve the differential equation
given by∂A
= N̂ A∂z(2.9)

using a fourth-order Runge-Kutta algorithm. This algorithm seems to be robust while maintaining
high accuracy.

2.3.2

Broadening mechanisms

Our first task now is to understand the dynamics of pulse propagation in optical fibers.
We first look at an 83 fs FWHM Gaussian pulse with 35.2 kW peak power in standard SMF28. Additionally, we neglect all dispersion except GVD and assume that γ is ∼1 W−1 /km. Figure
2.5(a) and (b) show the pulse evolution with -22 ps2 /m dispersion (i.e., in the anomalous dispersion

29
regime) while (c) and (d) show the propagation with the GVD sign flipped (normal dispersion).
From this simple simulation we can see several important features. First, we can see the dramaticdifference between dynamics with anomalous and normal dispersion. In the normal dispersion
case the dynamics are primarily just the result of GVD induced pulse broadening as well as SPM
spectral broadening (which results in a symmetrically broadened frequency spectrum). However,in the anomalous dispersion case the dynamics are clearly different. In this case, the evolution isdominated by the dynamics of solitons.
If we inject a pulse with a given peak power and pulse duration into fiber with anomalous
dispersion, the pulse will rapidly evolve into a soliton with order given bys
P0 γT02N=|β2 |(2.10)

where P0 is the peak power and T0 is the pulse width (not FWHM – with the given parameters
T0 is 50 fs). In Figure 2.5, the power is sufficient to form a second-order soliton (N = 2). To
confirm that soliton dynamics are important, Figure 2.6(a) and (b) show a similar pulse with peak
power high enough to give a third-order soliton, while Figure 2.6(c) and (d) show what happens if
N << 1 so that soliton dynamics are not relevant.
The most distinctive features of Raman solitons are the strong temporal and frequency shifts
seen in Figures 2.5(a)-(b) and 2.6(a)-(b). Each Raman soliton experiences a different group delay
and thus temporally separates from the center seed pulse. Furthermore, it continues to experiencea frequency shift toward the red from intrapulse Raman scattering (where the blue frequency
components of the pulse seed the red frequency components through the Raman gain of the fiber).
This frequency shift, ∆νR (z), is called the Raman induced frequency shift (RIFS) or the soliton
self-frequency shift (SSFS). Neglecting higher order dispersion and losses,∆νR (z) ≈−4TR (γP0 )2 z,
15π|β2 |(2.11)

where TR is a Raman delay parameter, which is about 3 fs for fused silica [103]. Another noticeable effect is an oscillating spectral and temporal profile. This is further confirmation of soliton
dynamics, as these oscillations occur in soliton propagation without non-linear effects.

30

Figure 2.5: N = 2 Gaussian pulses propagating in fiber with only GVD and non-linear effects
included. (a) and (b) Temporal and spectral profile with -22 ps2 /m GVD. (c) and (d) Temporaland spectral profile with +22 ps2 /m GVD.

31

Figure 2.6: Higher and lower power pulses propagating in SMF-28. (a) and (b) Temporal andspectral profiles of an N = 3 soliton; (c) and (d) Temporal and spectral profiles of a pulse with
N << 1.

32
Within a short distance into the fiber, a higher-order soliton (including one with non-integerN ) is believed to break-up into N̄ fundamental solitons in a process termed soliton fission [142].
This fission process can be explained by stimulated Raman scattering dumping enough power fromthe high-order soliton into a frequency shifted component to form a stable fundamental Raman
soliton. The temporal width and power of the kth soliton (k = 1, 2, ..., N̄ ) areT0
and
2N + 1 − 2k
(2N + 1 − 2k)2
P0 .
Pk =
N2
Tk =

(2.12)

Each soliton then experience RIFS according to Equation 2.11 with P0 replaced by Pk . We can see
this in the N = 3 case, Figure 2.6(a) and (b), where two red-shifted solitons are seen at the longer
propagation distances.
While instructive, these calculations did not necessarily use reasonable parameters. For
example, third-order and higher dispersion terms were ignored and the peak power was chosen as
to satisfy the given soliton number condition without regard to how much peak power was required.
We now consider what happens if we use experimentally relevant values. For these simulations weassumed values reasonable for a standard Er:fiber oscillator followed by a single-mode Er:fiber
amplifier. In particular, we assumed 400 mW average power at 250 MHz repetition rate and 70 fs
FWHM sech2 pulses. The center wavelength was assumed to be 1550 nm.
We first simulated propagation of this pulse through SMF-28. The result is shown in Figure
2.7. The given pulse parameters correspond to N ≈ 1 so we would expect to see some soliton
dynamics involved. Indeed this is the case, as we see the clear formation of a single Raman soliton,
which experiences RIFS along the fiber. After propagating through 8 m of fiber, the wavelength has
shifted to around 1850 nm, which is similar to the shift observed in Figure 2.4. As the soliton shifts
to longer wavelengths, |β2 | increases, which results in an increasing temporal pulse width (according
to Equation 2.10). This results in a lower Raman gain and causes the rate of wavelength shifting
to decrease at longer wavelengths.
For comparison, we also simulate the propagation of this same pulse in dispersion-shifted

33

1950

18

1900

16

1850
1800

12

1750
λ nm

(t−β1z) ps

14

10

1650

8

1600

6

1550

4

1500

2
0
0

1700

1450
2

4
Distance [m]

6

8

0

2

4
Distance [m]68Figure 2.7: Simulation of Raman shifting of an Er:fiber laser in SMF-28.

34
fiber (i.e., highly nonlinear fiber, HNLF), which has a zero-dispersion wavelength typically above
1500 nm and a smaller core size than SMF 28. This results in γ ≈ 21 W−1 /km. We useddispersion parameters from [148]. The results from this simulation are shown in Figure 2.8. Using
our knowledge of propagation in SMF-28, we can attempt to deconstruct the dynamics of thesupercontinuum formation in HNLF.
Since we are still pumping the HNLF slightly in the anomalous dispersion region, we still
expect to see soliton dynamics. If fact, given the small GVD and higher non-linearity of the fiber,
we are injecting an N = 48.7 soliton into the fiber; however, the distance at which soliton fission
occurs is given roughly by LD /N where LD = T02 /|β2 | is the dispersion length. This estimate
implies that soliton fission would occur in about 11 cm; therefore, we do not expect to see a large
number of ejected solitons. We can however still attribute the long wavelength edge of the spectrum
to a Raman soliton undergoing RIFS. Instead of seeing an ejected soliton after the fission distance,
which would happen almost immediately in SMF-28, we are instead seeing the high-order soliton
just starting to break-up. However, since the fission distance is so long compared to the length
over which the non-linearities take to become apparent, the process is distorted because of the
non-linearities. In particular, the long wavelength region results from the Raman shifting of thehigh-order soliton. As this soliton is rapidly shifted due to the RIFS, it is still undergoing a fission
process, resulting in further breakup into two pulses with different pulse energies. The higher
energy soliton experiences more RIFS, whereas the low energy component would not experience as
many nonlinearities. In addition, the initial spectrum broadens rapidly due to SPM, which would
serve as a further driving effect for the soliton breakup.
We also see a significant amount of structure on the short wavelength edge. Some of this iscertainly due to SPM (note the symmetrical initial broadening); however, SPM would most likely
not lead to such a strong edge to the spectrum. An additional source of this is coupling from theRaman soliton components into a dispersive wave. This coupling occurs through a phase resonance|u(z,t)|2/P0

35

1.5
1
0.5
0
−0.15

−0.1

−0.05

0
0.05
(t−β1z) ps

0.1

0.15

|U(z,ν)|21
0.5

10*log (|U(z,ν)|2)

0
1000

1500

λ nm

2000

2500

20
0
−20
1200

1400

1600

1800
λ nm

2000

2200

2400

Figure 2.8: Simulated broadening of an Er:fiber laser through 3 cm of HNLF. Dark blue is theinitial pulse, light blue is after the fiber. The top panel is the temporal evolution and the bottom
two panels are the spectral evolution (normalized, linear scale and log scale, respectively).

36
where the dispersive wave is formed at frequency ωDW given by [142]β (ωs ) − ωs /vsg + (1 − fR )γPs = β (ωDW ) ωDW /vsg

(2.13)

where Ps is the peak power of the soliton coupling to the dispersive wave and vsg is the group velocityof the soliton. Since these are changing as the soliton propagates, the generated dispersive wave
could evolve along with its generating soliton. The temporal profile in the first panel of Figure 2.8
corroborates this explanation. In this, we see two Raman solitons slightly delayed from the main
pulse and a broad pedestal around the solitons, which is the dispersive wave.
In addition to understanding the broadening process, we care about the coherence of thegenerated light. As we will see in the next section, we can use a combination of experiments
and simulations to check the coherence of the nonlinear processes occurring in the fiber. This
was the first systematic experimental and numerical study of supercontinuum coherence over wide
bandwidth.

2.3.3

Coherence measurements

We have tested the coherence properties of the broadened spectra from a Yb:fiber laser byperforming free-running beat experiments with several single-frequency lasers. The frequency comb
was based on a Fabry-Perot-type Yb-similariton oscillator mode-locked with a sub-ps lifetime saturable absorber mirror, which are known to exhibit excellent passive stability (details about theoscillator are published elsewhere [149]). While it has been demonstrated that fiber oscillators can
produce high quality frequency combs [150], pulse durations in the 100 fs range make it challenging
to realize super continua with a high degree of coherence over a large spectral region [151], as thesensitivity of the broadening process to noise can lead to large spectral phase fluctuations [152].
Due to this consideration, care was taken to reach near transform-limited nJ-level pulses suitable
for supercontinuum generation. The 152 MHz pulse train was amplified in a Yb-doped double-clad
phosphosilicate fiber with 12 µm core diameter pumped at 976 nm. To avoid self-phase modulation and hence to minimize amplitude-to-phase noise conversion in the fiber amplifier, a linear

37
chirped-pulse amplification scheme was chosen, and it has been demonstrated that this amplification scheme is compatible with low-noise phase control [153, 154]. With fiber-based compensation
for higher-order dispersion contributions, necessary to reach the Fourier-limit, the pulses could berecompressed to < 80 fs with a grating compressor as characterized by frequency resolved optical gating. After coupling the pulse train with a pulse energy of 1.2 nJ into a highly nonlinear
suspended core fiber (SCF), the optical spectrum spanned from 570 nm up to 1.66 µm at the -30
dB-level as shown in Figure 2.9 (a).
We used the measured pulse energy coupled into the fiber, input spectrum, and fiber dispersion in our simulation from Section 2.3.1 (dashed red line in Figure 2.9(a)). The input pulses
were modeled as unchirped hyperbolic secant pulses of 80 fs duration (full width at half maximum) with a center wavelength of 1050 nm. The nonlinear fiber used had a nonlinearity (γ) of
about 40 W−1 km−1 and also an increased fR , due to germanium doping, from the standard valueof 0.18 for fused silica. We found that the simulated spectrum matched measured spectrum bestfor fR = 0.28. The injected soliton order, N , was about 19.2. Because many of the input andfiber parameters were well known, the simulation was able to very accurately match the measured
spectrum.
The measured beat notes are shown in Figures 2.9(b-d). Interestingly, we noticed a significant
amount of amplitude noise in certain spectral regions, which was identified with the help of dualbalanced detection to cancel out common mode noise [155]. After balanced detection, we obtained
beat notes at 698 nm and 1020 nm with a signal-to-noise ratio (SNR) of > 30 dB at 100 kHz RBW.
As shown in Figures 2.9(b) and (c), the amplitude noise floor could be suppressed by up to 20 dB
at 698 nm and 8 dB at 1020 nm by using dual-balanced detection compared to the measurementwith a single detector. This observation supports the existence of comb amplitude noise without
the presence of phase noise in these spectral regions. In other spectral regions (e.g. at 1540 nm)
we observed that dual-balanced detection was not necessary to obtain a similar SNR (shown in
Figure 2.9(d)). More than 30 dB SNR was also obtained for beat signals at 771 nm and for theν −2ν heterodyne beat signal representing the carrier-envelope offset frequency, f0 . Self-referencing

38

ν − 2ν
Intensity (dB)0simulated spectrum
measured spectrum

(a)

-40

ZDWL

-80

600

PWL
1000

800

1200

1400

1800

1600

Wavelength (nm)

-20

-40

-20

0

20

Fourier frequency (MHz)0single detector
balanced detectors(c)single detector

(d)

-20

-20
-40

0

Power (dB)single detector
balanced detectors(b)Power (dB)

Power (dB)

0

-20

0

20

Fourier frequency (MHz)

-40

-20

0

20

Fourier frequency (MHz)Figure 2.9: Coherent supercontinuum generation spanning over 1.5 octaves from a Yb:fiber laser.
(a) Simulated (dashed) and measured (solid) super-continuum spectrum at the output of the suspended core fiber. The zero dispersion wavelength (ZDWL) and pump wavelength (PWL) are
shown with arrows. (b) Beat at 698 nm with (dashed) and without (solid) balanced detection. Up
to 20 dB suppression of the amplitude noise floor is observed. (c) Beat at 1020 nm with (dashed)
and without (solid) balanced detection. (d) Beat at 1.5 µm without balanced detection, exhibitinga signal to noise ratio comparable to the balanced cases.

39
the comb was possible with ν = 1150 nm to ν = 1300 nm (and the corresponding second harmonic,2ν) providing the freedom to detect other beat notes in this region.
The optical heterodyne beat signals help demonstrate that the free-running supercontinuum
is coherent beyond the 1 µs time scale, permitting coherent beats with high SNR to be observed
across the entire spectrum. In addition, a beat frequency between the self-referenced frequency
comb stabilized to the Sr clock at 698 nm and a cavity-stabilized 1.54 µm cw laser was recorded
for a long duration. By fitting the down-mixed beat signal with a chirped sinusoid, we deduce a
coherence time of > 200 ms corresponding to > 3 × 107 pulses from our 152 MHz pulse train [49].
In order to model quantum-limited supercontinuum coherence, we performed multiple simulations with a quantum shot-noise seed perturbing the input pulse, A (t), such that A (t) →A (t) + δA (t). In this case, δA (t) is described stochastically by hδA∗ (t) δA (t + τ )i = (hν/2)δ (τ )[156, 157], and we treat it as a Gaussian-distributed random variable in both quadratures. We also
include the effects of spontaneous Raman scattering along the fiber according to the procedure of[156]. The first-order coherence, g (ω), defined asg (ω) = rhÃi (ω) Ã∗j (ω)ii6=j2

2,(2.14)

h Ãi (ω) ih Ãj (ω) i
was calculated numerically in 50-GHz spectral bins by an ensemble average of 100 independently
simulated spectra [151]. Figure 2.10(a) shows g (ω) along with the calculated spectrum, and it canbe seen that there are significant drops in the coherence in isolated spectral regions that still posses
significant optical power. Another interesting note is that the coherence degrades smoothly acrossthe first Raman soliton.
We also considered the different contributions to g (ω) arising from amplitude and phasenoise separately by calculating the variance of the argument and amplitude of Ã (ω). As evident in
Figure 2.10 (b), certain spectral regions feature decreased coherence arising from amplitude noise
but exhibit only negligible phase noise contributions, which agrees with the experimental results.
Previous experimental studies, where supercontinuum coherence was demonstrated by the use ofa modified Young experiment with two sources, could not make such a clear distinction [158].

40

0.5

-40

0
102 (b)-80
10-2phase noise
fractional amplitude noise

1

10-4

10-2
10

10-6

-4

600

800

1000

1200

Wavelength (nm)

1400

1600

1800

10-8

Fract. amplitude noise

Phase noise (rad)0coherence g
simulated spectrum(a)Intensity (dB)

Coherence g

1

Figure 2.10: Simulation results for the optical spectrum and coherence properties after broadening
in the suspended core fiber. (a) Simulated spectrum (dashed, right) and calculated quantum noiselimited pulse-to-pulse coherence (solid, left). There is a significant loss of coherence on the red sideof the soliton as well as in narrow spectral regions throughout the spectrum. (b) Contributions ofamplitude (dashed, right) and phase (solid, left) noise to the total coherence function. In many
spectral regions, the presence of amplitude noise does not presage the presence of phase noise,indicating that degradation of g (ω) is not necessarily indicative of a loss of phase coherence. In
particular, many of the narrow dips are related to amplitude noise, not phase noise.

41
Numerical simulations of up to 1000 pulses show that the total phase standard deviation does notchange, indicating that that the quantum noise-induced phase deviation is bound at some level
and is not a random walk, which would degrade relative linewidth. This is consistent with othertests of comb coherence that reach the same conclusion [159]. Supercontinuum generation can thus
be seen as a white phase noise process without a random walk so there is no net accumulation ofphase noise. This further confirms that the observed linewidth in the out-of-loop comparison was
not compromised by the supercontinuum.
Numerically, we identified several parameters responsible for a significant loss of coherence.
In particular, we sew a dependence on the fractional Raman contribution (fR ) that had not beenstudied previously. We calculated the average coherence of the first soliton (i.e., from 1450-1750 nm)
and saw a decrease in coherence from 0.9 to 0.7 for fR varying from 0.28 to 0.2 with the soliton center
wavelength fixed by increasing the fiber length, as shown in Figure 2.11. The loss of coherence wasmost dramatic on the red side of the soliton and was caused primarily by increasing phase noise
not amplitude noise. We also noticed reduced coherence with increasing pulse energy and withincreasing pulse duration (both increase the soliton order) as has been observed previously [160];
however, we saw a coupling between these parameters and fR that has not been noted previously.
With a higher value of fR , the coherence was significantly less sensitive to pulse duration (slight
loss of coherence in the soliton at 120 fs) and energy than with low values of fR , where we saw
strong loss of coherence in the first soliton and also low coherence across the spectrum with 90 fs
pulses. These results demonstrate that in addition to the complex interplay between soliton fission
and modulation instability (lower soliton order results in rapid fission and higher coherence) there
seems to be a competition between intrapulse stimulated Raman scattering (resulting in coherent
red-shifting of the soliton) and amplification of noise (primarily on the red side) due to modulation
instability as the fundamental soliton propagates. Overall, these simulations indicate that higher
fractional Raman contribution (e.g., by higher Ge-doping) can be used as an additional tool forincreasing the coherence of supercontinua, especially in cases where the pulse duration cannot be
further decreased.

42

Spectrum

Intensity (dB)

0
−20
−40
−60
−80

fr = 0.28; 24 cm
fr = 0.2; 33 cm
600

800

1000
1200
wavelength (nm)

1400

1600

1400

1600

Coherence

1
0.8

g1

0.6
0.4
0.2
0

600

800

1000
1200
wavelength (nm)Figure 2.11: Simulations of the spectrum (top panel) and degree of coherence (bottom panel,defined in the text) for two different Raman fractions, fR = 0.28 (solid blue) and fR = 0.2 (dashedgray). The simulations are for a Yb:fiber laser pumping suspended core fiber as discussed in the
text. The fiber length was chosen so that the soliton shift of the first soliton was the same in both
spectra. The solition shows a significant loss of coherence on the red side of the spectrum; the lossof coherence is worse for a smaller value of fR .

43
The simulated loss of coherence on the red side of the Raman soliton shown above was also
observed in measurements using an Er:fiber laser. We used two branches of the same Er:fiber
laser (one frequency shifted using an AOM) to pump a short piece of HNLF (for supercontinuum
generation) and a long piece of polarization-maintaining fiber for Raman shifting. We then observed a spectrally-filtered beat note between the Raman soliton and the supercontinuum after amonochromator. When the Raman-shifting fiber was too long, there was a very noticeable, broadband increase in noise on the beat note, especially on the red side of the soliton (soliton centeraround 1800 nm). This was true even if the power was decreased so that the total soliton shift
was also reduced (soliton center around 1700 nm); however, this was not observed for a shorter
fiber length, which resulted in a soliton center of 1750 nm. We believe that the coherence degradation occurs as the RIFS slows down (e.g., above 1800 nm in Figure 2.4). This suggests that itshould be possible to use a different Raman-shifting fiber (e.g., with higher Raman gain) or evensplice together a combination of Raman-shifting fibers with progressively smaller core diameters tomore rapidly shift to longer wavelengths. Preliminary results from R. Holzwarth and co-workers
at Menlo Systems indicate that this is indeed the case and that coherent shifting beyond 1900 nm
is possible using polarization-maintaining fiber for 1310 nm (OFS Truephase 1310 Sensor), which
has a smaller core diameter.
The results of this section show that it is possible to use nonlinear fiber optics to generate
coherent light in a variety of wavelength regions using fiber-based pump lasers. However, there canalso be a loss of coherence in the nonlinear processes that must be considered. We have seen thatnumerical models are capable of reproducing many of the experimental results and provide moredetails into the broadening processes as well as potential issues with coherence.

2.4

Optical parametric oscillators
Two nonlinear systems are important for the generation of light in the mid-infrared: difference-

frequency generation and optical parametric oscillators (OPOs). Since we desired a high-power
mid-IR comb, we chose to use an OPO. An OPO uses parametric conversion inside of a resonator

44
to efficiently convert light from one wavelength (the pump, p) to two longer wavelengths (signal s
and idler i). This process satisfies both energy conservation, ωp = ωs + ωi , and phase matching,∆k = 0 for maximum conversion efficiency. In a periodically-poled crystal, the phase matching is
written as∆k = 2π

np ns ni
1
−
−
−
λp
λs
λi Λ


,

(2.15)

where Λ is the poling period of the crystal. The theory of OPOs has been covered in many references[161, 83, 162], so we only reproduce the pertinent results here. If we assume that there is no pumpdepletion, we can obtain equations for the signal and idler fields (in MKS units) as a function of
position, z:sinh(Γz) −i ∆kz∆k∗sinh(Γz) + iκs Ei (0)e 2
Es (z) = Es (0) cosh(Γz) + i
2Γ
Γ




sinh(Γz) −i ∆kz∆k∗sinh(Γz) + iκi Es (0)e 2 .
Ei (z) = Ei (0) cosh(Γz) + i
2ΓΓHere Γ =p(2.16)

κs κi − ∆k 2 /4 is the gain per unit length with κ a coupling constant given byκs(i) =

ωs(i) def f
Ep (0),
ns(i) c(2.17)

where def f is the effective non-linear coefficient of the material (typically given in picometers per
Volt, note that def f is reduced by a factor of about π/4 when using periodically-poled crystals). If
we now enclose a crystal of length L in a cavity with total round-trip reflectivity of Rs(i) , the field
after one round trip is Es(i) (L)eiφs(i) Rs(i) , including an additional phase shift due to propagation.
To obtain the threshold for steady-state oscillation, these fields must equal Es(i) (0). Assuming that∆k = 0 and that the phase shift is an integer multiple of 2π (due to cavity resonance), we can write
this set of coupled equations as








 Es (0)  Es (0)  = M
,∗∗Ei (0)
Ei (0)with cosh(ΓL)Rs
M=
−iκi sinh(ΓL)
RiΓ

(2.18)

iκs sinh(ΓL)
Rs Γcosh(ΓL)Ri

.

(2.19)

45
This means that M must have an eigenvalue of 1, so det |M − I| = 0 and finally(Rs cosh(ΓL) − 1)(Ri cosh(ΓL) − 1) − Rs Ri sinh2 (ΓL) = 0 →(2.20)

1 + Rs Ri(1 − Ri )(1 − Rs )cosh(ΓL) ==1+
Rs + Ri
Rs + Ri

where we used the identity cosh2 (x) − sinh2 (x) = 1. We can now expand cosh(ΓL) for small ΓL toarrive at the threshold condition(ΓL)2 =

2(1 − Ri )(1 − Rs ).
Rs + Ri(2.21)

This also works for a singly-resonant OPO by taking Ri → 0. Notice that a singly-resonant OPO
2will have a threshold that is higher by a factor of about 1−R. Finally, we can substitute for Γ, andiremembering that I = nc0 |E|2 /2, to find the threshold intensity for a singly-resonant OPO,
ideal
Ith
=

np ns ni 0 c3 2(1 − Rs ).
Rs
2ωs ωi d2ef f L2

(2.22)

This equation can be modified to account for both the spatial and temporal overlap of thepump and signal by replacing L with an effective interaction length Lef f and including coupling
factors gs (spatial) and gt (temporal) [163, 164]Ith =np ns ni 0 c3
2(1 − Rs ).
22
Rs2ωs ωi def f Lef f gs gt(2.23)

The coupling factors, assuming ideal Gaussian spatial and temporal beams, are given by
s
τp2
gt =
and
τs2 + τp2
gs =

wp2
.
ws2 + wp2

(2.24a)(2.24b)√
Here τs(p) = Ts(p) / 2 ln 2 is the pulse duration for the signal (pump) pulses with FWHM intensity
duration Ts(p) and ws(p) is the beam radius for the signal (pump). The effective gain length isπLc
Lef f = Lt erf,(2.25)
2Lt
where Lc is the crystal length and 2 −1/2

δvs
1
1
2
 2


√
+
δv
+
i τs2
τs2
τ2
π  δvs
1
1

 p 
Lt =
+ δvi2
+ 2 −
2
2
2
τs
τs
τp
2 1 + 1
τs2

τp2

(2.26)

46
is the temporal overlap length. Here δvs(i) =ngs(i)ngp−cc

is the group delay walk-off for the signal(idler) in (velocity)−1 . The group velocity is given by ncg and ng = n − λ dndλ is the group index.
τs
for short signal pulse durations.
Equation 2.26 simplifies to Lt ≈ δvs

The phases of the pump, signal, and idler fields are related to each other by φp − φs − φi =−π/2. If we take the time derivative of both sides, we find that φ˙p = φ˙s + φ̇i . This expression canalso be interpreted as a conservation of energy. Since φ̇ ∼ f0 , we arrive at a useful relationshipf0p = f0s + f0i .

(2.27)In order to extend frequency comb spectroscopy to the mid-IR, we built a synchronouslypumped OPO based on a fan-out periodically-poled lithium niobate (PPLN) crystal that covers
the 2.8 µm to 4.8 µm spectral region [50], a schematic of which is shown in Figure 2.12(a). The
main part of the OPO consists of a five-mirror linear cavity that is singly resonant for the signal.
We chose this cavity design because we wanted to have a small retro-reflecting mirror to provide
easy cavity-length adjustment (to match the cavity length to the comb frep ) and fast feedback
bandwidth to the cavity length with a high-speed PZT. The mirror M4 was added so that there
was the option of two output couplers (M1 and M4) and to reduce the overall footprint. M2 andM3 both have a 20-cm radius-of-curvature and are spaced by 20.5 cm. A summary of the cavity
mirror coatings in given in Table 2.1.
We can calculate the intracavity mode using ABCD matrices [165, 166]. The beam waist,
ω(z), at a given position, z, is given byλ1− Im(q) ==2πω (z)
B

s
1−



A+D
2

2
(2.28)

where λ is the wavelength of the light. The beam radius of curvature, R(z), is given by
Re(q) =

1
D−A
=
.
R(z)
2B

(2.29)In order to find the q-parameter, the ABCD matrix can be calculated by using the propagation
1
through a medium with index of refraction n with length d, 
0

d
n

1

, and the reflection from a

47

(a)
DDS

locking electronics

PD2

PD1
p+s
p-SC

p-SC
DM

PZT
M5

λ/2
p

PCF

p+i
2 s DMs reson
M3

OSA

PBS

i

p+s
p+i
2s

DM

p

M1
fan-out
PPLN

ant

M4

M2

i
p+s
p+i
2s

Wavelength (μm)

Rel. Intensity (dB)(b)

pamplified
fs-Yb:fiber laser

MID-IROUTPUT

0
-20
-40
-60

Frequency (THz)

Normalized Intensity

(c)
1.0
0.5

0.0

2.5

3.0

3.5

4.0

Wavelength (μm)

4.5

5.0

Figure 2.12: Mid-infrared OPO comb based on periodically-poled lithium niobate (PPLN). (a)Schematic of the optics layout, see text for details. p, s, and i indicate pump, signal and idler
respectively; M1-M5 (yellow) are the OPO cavity mirrors; DM is dichroic mirror; PD1 and PD2
are photodiodes; PCF is a photonic crystal fiber for pump supercontinuum (p-SC) generation; OSA
is an optical spectrum analyzer. (b) A composite spectrum of all outputs (wavelengths shorter than
2 µm were recorded on the OSA, the idler output was measured using a scanning monochromator).
Arrows above each peak indicate the tuning direction, labels give the assignment. (c) Individuallynormalized idler spectra recorded using the scanning monochromator for different crystal positions.

48

 1
spherical mirror with curvature R, 
− R2


0
 (in this definition, R is positive for a concave mirror
1

and negative for a convex mirror). The ABCD matrix for the focus between the two curved mirrors,
assuming that the center of the crystal is at the focus, is given by:








Lc
2n  10 1 2L2−5 − t



−2
0
1
R



L2−3 −Lc0 1 2L1−3 − t 1
2
 1



0
1
0
1
0

L2−3 −Lc2
 1

 A B  1


=
0 1
0
1
C D


Lc
1 L2−3 − Lc + n   1


−2
0
1
R



Lc
2n 

1

.

Here La−b is the length between mirrors Ma and Mb and Lc is the length of the crystal (7 mm).
This gives an expected waist of 55 µm.
The OPO is pumped with an amplified Yb:fiber comb (frep = 136 MHz, 10 W average power,∼100 fs pulse duration) coupled into the cavity through M2 (designed with T > 97% at the pumpwavelength). The MgO-doped PPLN crystal (HC Photonics) is 1 mm high, 10 mm wide, and 7 mmlong and has a poling period that varies from 25.5 to 32.5 µm across the width. In order to reduce
losses and lower the threshold, the crystal is AR coated for the signal (R < 3% from 1400-1900 nm)
as well as for the pump (R < 1% at 1080 nm). The crystal is heated to 71 ◦ C to reduce potential
photo-refractive damage from the high intensity pump. We hold the crystal in a U-shaped copper
mount with indium foil above and below the crystal. The crystal is held in place with a top platethat bolts to both sides of the U mount, see Figure 2.13. The pump laser is focused with a 150 mm
lens to a beam waist of ∼100 µm. This gives a Rayleigh range of ∼3.7 cm, so the length of thecrystal is short compared to the Rayleigh range.
The poling period of the crystal was chosen so that the idler covers approximately 3-5 µm.
Figure 2.14(a) shows the calculated gain profile (i.e., Γ) as a function of poling period, which
was used to determine the correct poling period. The phase matching was calculated using the
Sellmeier coefficients for the index of refraction from [167]. The gain and bandwidth both increaseas the poling approaches degeneracy (i.e., signal and idler wavelengths both equal to twice the pump

49

Figure 2.13: OPO crystal holder. A thermo-electric cooler (TEC) is used to heat the crystal. The
crystal is clamped in place between two sheets of indium foil. Marked dimension are in mm.

(a)

50

6

Wavelength (μm)

5

4

3

2

24

26

28
30
32
Poling period (μm)

34

Signal/Idler Wavelength (nm)(b)

Temperature (˚C)
Figure 2.14: Calculations for PPLN OPO. (a) Calculated gain profile as a function of poling period(white is higher gain). The gain and bandwidth both increase as the poling approaches degeneracy(i.e., signal and idler wavelengths both equal to twice the pump wavelength). (b) Phase matching
curves provided by HC Photonics for different poling periods as a function of temperature.

51
Table 2.1: Mirror coatings for OPO
MirrorCoating

R for signal

Dispersion

M1,3,5HR signal, GVD optimized

R > 99.9%

0 → −600 fs2 1400-1600 nm−600 → −100 fs2 1600-1800 nm

M21064-nm enhanced silver∼98%

Unknown (assume ∼0?)

M4

Output coupler

Rs ∼ 99.9%

Unknown

Xtal

AR pump/signal/idler

Rs,i < 3%, Rp < 1%

Unknown

wavelength). The temperature dependence of the phase matching is shown in Figure 2.14(b), which
was provided by HC Photonics.
Rough alignment of the OPO cavity was accomplished with a 1.55 µm laser coupled inthrough M1. In addition, a significant amount of doubled pump light is produced in the crystal;this green light can also be used to align the cavity by using it to overlap the retro-reflection frommirrors M1 and M5 and by optimizing the contrast of the observed fringes while scanning thecavity length. Since the green light has the same repetition rate as the pump laser, it can be used
to tune the length of the OPO cavity to match the repetition rate by maximizing the fringes. Dueto the different index-of-refraction for green versus the signal wavelength, the cavity length needs
to be adjusted from the optimum position for the green fringes. Also, care must be taken in thealignment as there are multiple sets of green light fringes caused by various stray reflections. Afterthe alignment is optimized, the pump-laser power can be increased and the cavity length scanned
until idler light is observed. Figure 2.15(a) shows the measured output power and photon conversion
efficiency as a function of crystal position (i.e., output wavelength) with 8.47 W of pump power.
The output power is over 1 W from 3 µm to 4 µm. The normalized output spectrum measured
using a scanning monochromator for different crystal positions (corresponding to different idler
wavelengths) is shown in Figure 2.12(c). We obtain a simultaneous bandwidth (FWHM) of up to
0.3 µm (300 cm−1 ).

52Signal Wavelength (μm)
1.8 1.7 1.6
1.5
1.4

Output Power (W)

1.0
0.8
0.6

0.8

0.4
0.2

0.0
2.5

3.0
3.5
4.0
4.5
Idler Wavelength (μm)

0.0
5.0

Photon Conversion Efficiency

1.6

(b)
1.6

Idler Power (W)

(a)

0.8

0.0

0

2

4
6
Pump Power (W)8Figure 2.15: OPO performance. (a) Measured output power (black triangles, left axis) and photon
conversion efficiency calculated from output power (grey circles, right axis) as function of idler
wavelength. The output power peaks at almost 1.5 W near 3 µm. The grey dashed line shows thetrend expected from the cavity mirror reflectivity (max level adjusted to match the data). (b) Idler
power versus pump power, measured at λidler = 3.01 µm. This gives a threshold of 1.7 W and an
initial slope efficiency of 0.29.

10

53
To determine the threshold, we measured the output power at 3.01 µm as a function of pump
power as shown in Figure 2.15(b). This gives a threshold of 1.7 W and an initial slope efficiencyof 0.29. The slope efficiency drops to 0.16 at about 4 W, which we attribute to the onset of anadditional (unknown) conversion process; after this, however, we see no further signs of nonlinear
losses up to 9.17 W of pump power (yielding an output power of 1.5 W). We estimate the thresholdusing Equation 2.23 to be ∼1.7 W using reasonable parameters. The effective nonlinear coefficientfor PPLN is around 14 pm/V [168]. We assume that the pump pulse duration (FWHM) is 120 fs
and the signal duration is 200 fs. This duration assumed some uncompensated dispersion in thecavity. Since Lt = 1.4 mm (from Equation 2.26) is significantly shorter than the crystal length,
the effective interacting length is entirely limited by Lt . This means that a shorter signal pulse
duration actually results in an increased threshold.
Now that we can generate mid-infrared light from a synchronously pumped OPO, we want toturn this into a frequency comb, which means that we need to be able to control both f0 and frep .
Since the OPO is synchronously pumped, frep is set by the pump laser. There is not an easy way tocontrol f0i directly without an additional reference laser in the mid-IR; however, we can make useof Equation 2.27 and control both f0p and f0s . In order to do this, we need feedback signals for both
f0p and f0s . Luckily, in addition to the signal and idler light, the OPO produces a variety of othercolors due to various non-phase-matched conversion processes. Figure 2.12(b) shows an output
spectrum from M1 measured with an optical spectrum analyzer, plus the idler spectrum measured
with a monochromator. The arrows indicate the relative tuning direction of the corresponding
peak, which was essential for identification. Of particular interest are the peaks originating frompump plus signal (p + s) and pump plus idler (p + i): if we form the beatnote between each of thesesignals and broadened pump light, we obtain f0p and f0s . We do this by picking off about 400 mWof pump power before the OPO and sending this to a nonlinear fiber for spectral broadening (seeFigure 2.3). This pump supercontinuum (p-SC) is overlapped both spatially and temporally in two
separate interferometers with p + i and p + s out of M4 using dichroic mirrors and delay stages. We
were able to obtain a high signal-to-noise beatnote on the p + s arm, but have not been able to get

54similar results for p + i, probably due to the difficulty in broadening with low noise in the spectral
region from 700-900 nm as discussed in Section 2.3. We have not implemented balanced detection,
which would help reduce amplitude noise from the broadening (see Figure 2.9). However, we canstabilize f0p separately with either an f-2f interferometer or an optical lock to a stable cw laser.
Feedback to f0s is accomplished by tuning the OPO cavity length via a PZT on M5 since f0
depends on ∆φCE which in turn depends on the cavity length according to Equation 2.1. To test
our ability to stabilize the idler, we compared the idler to a narrow-linewidth cw laser as shown in
Figure 2.16. Since we do not have such a laser in the mid-IR, we tuned the idler to 3.09 µm anddoubled it using a thin GaSe crystal to 1.545 µm. We could then form a beat note between one
doubled idler mode and a < 10 kHz linewidth RIO diode laser. In this test, frep of the pump
was locked with an optical lock to a stable 1.064 µm non-planar ring oscillator (NPRO) laser via
feedback to the cavity length (Figure 2.16(a)); since we only wanted to check the idler short-term
linewidth, f0p was stable enough and was left free running. Interestingly, with the idler at 3.09
µm we observe a beat note between the spectrally overlapping p + i and 2s, which gives a rf signal
at 2f0i − f0s . We used this signal to stabilize the OPO cavity length (Figure 2.16(b)). After locking
the OPO cavity we see a stable beat note between 2i and the RIO (Figure 2.16(c), black versus
gray) with a linewidth (at -3 dBc) of 80 kHz, which corresponds to an idler linewidth of 40 kHz
in the mid-IR. The pedestal on the beat note is consistent with the servo bumps on the frep lock.
In this test, the idler linewidth was limited by the feedback bandwidth to the OPO cavity length
PZT. By fully stabilizing the pump laser and replacing the OPO cavity PZT with a high bandwidth
PZT/mirror on a bullet mount [169], we have obtained an idler linewidth of 10 kHz (measured using
a high-finesse cavity).
The development of this OPO comb demonstrated the ability to extend frequency comb technology to the mid-infrared, which has enabled high-resolution spectroscopy in the spectral region
containing the OH and CH fundamental vibrational modes of many molecules. When originallydeveloped, this source was, to our knowledge, the first fully-stabilized mid-IR frequency comb andalso the highest power comb in the mid-IR. Several applications are discussed in the next two

1064 nmYFL
p1065 nmPD4p
m f rep+ f 0p −ν NPRO(a)pcavity length

NPRO

oscillator length

55

OPO

2s
p+i
815 nmPD2

ν np +i − ν n2 s = 2 f 0i − f 0s

(b)

ECDL

i
3090 nm
SHG

1545 nmPD3

ν ki −ν ECDL(c)

Figure 2.16: Experimental setup and rf signals for stabilization of the OPO and measurement of
the idler-comb linewidth. NPRO, non-planar ring oscillator; YFL, Yb:fiber pump laser; ECDL,
external-cavity diode laser; PD2-PD4, photodiodes; SHG, second-harmonic generation crystal(GaSe). (a) In-loop signal for stabilization of frep , 1 kHz RBW. (b) In-loop signal for stabilizationof 2f0i − f0s , 3 kHz RBW. (c) Out-of-loop beat note between 2i and ECDL, with OPO cavity length
unlocked (gray; 0.5 s sweep time; 10 kHz RBW) and locked (black; 2 s sweep time, 3 kHz RBW);
the traces are offset vertically for clarity.

56
chapters as well as in [170, 72, 55, 47, 56]. Furthermore, the same basic design can be used witha Tm:fiber laser pump and OP-GaAs to create high-power frequency combs out to 10 µm andbeyond.

Chapter 3

Detection techniques

In addition to a comb source operating in the desired spectral region, every comb spectroscopy
system needs a detection system for readout. For high sensitivity, we often use an optical cavity toincrease the effective path length. We first discuss some details about coupling the frequency comb
to the cavity. Then we cover different detection techniques. Finally, in addition to spectroscopyof neutral species, one may want to study the spectra of ions. This was motivated by the search
for the electric dipole moment of the electron using molecular ions (Chapter 6). For this, we havedeveloped a frequency comb velocity-modulation spectroscopy system, as described in Section 3.4.

3.1

Cavity-comb coupling
Efficient coupling of the comb light into the cavity is possible because the spectrum of Fabry-

Perot cavity resonance modes and that of the comb are similar in structure. However, while the
comb-line spacing is constant, the frequencies of the optical cavity resonances are separated fromeach other by a frequency-dependent free spectral range (FSR), given by
F SR(ω0 ) =c∂φ
2L + ∂ω
|ω0

(3.1)

∂φ
where c is speed of light, L is the length of the cavity, and ∂ω
represents the intra-cavity dispersion,

originating mostly from the cavity mirror coatings and intracavity absorption. This dispersion termmeans that the separation of consecutive cavity resonant modes is approximately constant only ina limited frequency range. Thus, some care must be taken to minimize the cavity dispersion at

58
the spectral region of interest, typically by operating as close as possible to the maximum of themirror reflectivity. In addition, the dispersion is typically reduced for coatings that have a broader
bandwidth.
The spectral range of most current CE-DFCS systems is limited by the bandwidth of the
optical cavities. Several possibilities exist for increasing the bandwidth while maintain high reflectivity. One possibility is the use of total internal reflection in Brewster angle prism retroreflectors[171]. Cavities using fused silica prisms can provide a finesse of over 100,000 throughout the nearinfrared, but such a system has yet to be combined with CE-DFCS. The dispersion of prism cavities
is significantly worse than standard optical coatings – except near the zero-dispersion wavelength ofthe prism material; however, this can be mitigated by dithering the comb (see below). In addition,
the choice of prism material is critical for high finesse. Our first attempts to use prism cavity in thenear-infrared were limited by water absorption in the prism material. This, along with the difficulty of fabricating prisms from different materials, makes prism cavities challenging to implementin other wavelength regions. Recently, a CaF2 prism cavity with 99.77% reflectivity at 250 nm and99.96% reflectivity at 500 nm was demonstrated [172]. For the mid-infrared, BaF2 would be a goodcandidate material [171], as could the infrared material KRS-5, which can be low loss [173] (unfortunately, it is challenging to find a manufacturer for KRS-5 prisms). Another interesting possibility,
especially in the mid-infrared, are semi-conductor optical coatings, which can have extremely highindex contrast and thus very large bandwidth. For example, reflectivities of 99-99.7% have been
demonstrated with three layer pairs of EuTe/PbTe or PbEuTe/BaF2 [174, 175] and even higher
reflectivity should be possible (although the losses are currently unknown). Other semiconductor
coatings – in this case crystalline GaAs/AlGaAs – have recently been shown to have very highfinesse [176] and are promising for low-loss mid-infrared coatings (potentially 10 ppm near 3 µm).
These two semiconductor coating technologies could potentially be combined to produce low-loss,
extremely broad bandwidth mid-infrared coatings.
In order to maximize the comb-cavity coupling, the comb needs to be spatially mode-matchedto the cavity [166, 165]. In addition, both frep and f0 of the comb need to be matched to the cavity

59
F SR and f0 at the wavelength of interest [45, 46, 47]. Once frep and f0 are optimized, these values
must also be maintained over an extended period of time. This can be achieved in two different
ways: by tightly locking the comb and cavity modes in order to ensure a constant transmission
through the cavity, or by modulation of either one of the comb parameters or the cavity FSR sothat laser light is periodically coupled into the laser cavity (the swept coupling scheme). Each ofthese schemes has advantages and limitations as explained below.

3.1.1

Tight locking scheme

In the tight locking scheme the match between the comb modes and cavity resonances is
actively maintained with high-bandwidth feedback. The error signal for stabilization can be generated with the Pound-Drever-Hall (PDH) [177] method, the Hänsch-Couillaud method [178], orwith a dither lock, where the comb modes are dithered around the cavity modes (or vice-versa)
with amplitude smaller or comparable to the full width half maximum (FWHM) cavity linewidth.
One way to achieve a tight lock is to stabilize the two degrees of freedom of the frequency
comb to an external frequency reference and to implement a feedback loop that actively controls thecavity FSR to match it to frep . A difficulty arises from the fact that for different cavity conditions(e.g. when the pressure in the cavity changes) the optimum comb f0 takes different values, so thelocking electronics must be able to follow these changes. Another limitation arises from the factthat the length of linear cavities is often controlled with a large-travel-range PZT. This, combinedwith the large mass of the cavity mirrors, results in a PZT response bandwidth usually limited
to a few hundred Hz, which is not enough to remove fast (acoustic) frequency noise, although inprinciple this can be addressed by designing a cavity that can accommodate a lower mass mirror.
An alternative scheme is to lock the comb to the resonances of the enhancement cavity bystabilizing the two degrees of freedom of the comb with two servo loops – one for frep and anotherfor f0 . The error signals for each of the feedback loops are derived from different parts of thecavity reflection spectrum [179]. This can be accomplished by applying a modified PDH detection
scheme, using separate photodiodes to detect different parts of the reflected laser spectrum after

60
diffraction from a grating. A detection scheme using three photodiodes was implemented in [179]:
here the PDH error signal from the center of the transmission spectrum was used to control frep
and the difference between two error signals from opposite ends of the transmission bandwidth wasused to control f0 . A similar setup using only two error signals based on cavity reflection was usedin later experiments in the near-IR near 1550 nm [54, 72] and mid-IR near 3800 nm [55]. In thenear-IR, an Er:fiber comb spanning 1510 to 1610 nm was tightly locked to an enhancement cavity
with a finesse of 8000 with a transmission bandwidth of 50 nm centered at 1530 nm, limited by
the dispersion of the dielectric mirror coatings, which were designed for peak reflectivity at 1600
nm. The limiting effect of cavity dispersion is seen in the mid-IR system, where the simultaneously
coupled bandwidth is reduced from 150 nm at 3750 nm (near the center of the 3800 finesse cavity)
to around 75 nm at 3900 nm. A single-point lock in the same system provides a bandwidth ofaround 20 nm. By comparing the single-point lock to the two-point lock, it is also possible todetermine the cavity dispersion [55].
The two-point locking scheme enables broadband, tight locking of the comb to the cavity
resonances, maximizing the enhancement factor and providing constant, high-power signal in cavity transmission. It can therefore be efficiently combined with Fourier-transform-based detection
methods. On the other hand, this scheme requires more complicated locking electronics comparedto the swept coupling scheme and also requires higher bandwidth servo actuators in the laser systems. Moreover, residual frequency noise in the laser is translated into amplitude noise (FM-to-AM
conversion) by the cavity resonances. This FM-to-AM noise conversion is often the primary sourceof noise in cavity-enhanced comb spectroscopy, and much effort has gone in to reducing it by frequency modulation [180], balanced detection (see Section 3.3.2), or laser dither as in the swept
coupling scheme.

3.1.2

Swept coupling scheme

In addition to noise from FM-to-AM conversion, the walk-off between the comb lines andthe cavity modes due to dispersion in the mirror coatings typically limits the spectral bandwidth

61that can be achieved for most of CE-DFCS experiments with the tight comb-cavity lock. The
bandwidth limitation is more severe when a higher finesse cavity is used, resulting in a compromise
between the sensitivity enhancement factor and the simultaneously usable spectral bandwidth.
The negative effects caused by cavity dispersion can be reduced by special low-dispersion cavity
designs. Alternatively, the swept coupling scheme allows the entire spectral bandwidth of a comb
to be transmitted through the cavity [181, 44]. In this case, the comb modes are periodically swept
across the cavity modes (or vice versa) with the amplitude of the sweep larger than the FWHM
linewidth of the cavity resonance (in contrast to a dither lock), resulting in periodic transmission
peaks. By stabilizing the time interval between the transmission peaks (with feedback to either the
comb or the cavity), it is possible to eliminate slow drifts between the cavity and comb modes. The
dispersion problem is largely avoided as the corresponding mode pairs come into resonance rapidly
one after another, thus when averaging over a sweep, the full bandwidth of the source spectrum
is transmitted. In addition, the required bandwidth of locking electronics is much lower than in
the tight locking scheme, since low-bandwidth feedback is sufficient to stabilize the positions oftransmission peaks.
The main disadvantage of this method is a reduction of the useful experimental duty cycle
and transmitted power arising from the limited time over which the comb modes are on resonancewith the cavity. A more subtle effect is connected with the enhancement factor of the absorption
signal. The enhancement of the interaction length is related to the cavity finesse, F , which is
proportional to the inverse of the cavity losses. For fast sweeps, the time that the comb lines
spend on resonance becomes only a fraction of the cavity lifetime. This results in an enhancement
factor that is closer to F/π for rapid sweeps instead of 2F/π for the tight locking case [44], which
decreases the absorption sensitivity assuming the same fractional noise levels. The fact that theenhancement factor is dependent on the sweep parameters implies that a calibration of the effective
length enhancement can be necessary for intermediate sweep speeds, although this is not absolutely
necessary for higher sweep speeds as the enhancement factor approaches F/π. While the swept
coupling scheme can be relatively easily combined with Fourier-transform-based detection methods,

62
the performance is worse than with a tight lock [182], due to the fact that the amplitude modulationof the transmission signal leads to a disturbance of the detected interference signals.
The swept cavity-comb coupling scheme has been used in a variety of experiments [44, 46],
including in the arsine spectroscopy experiment discussed in Section 4.1, and its robustness andreliability has been demonstrated by its application for trace gas detection in the north coast ofFrance [183] and in the Antarctic [184]. Amplitude noise reduction down to the shot-noise levelhas also been demonstrated [185]; however, the transmitted power is still very low.

3.2

Detection sensitivity
Before we discuss detection methods in more detail, we should first provide a backgroundfor comparing systems. The sensitivity of a detection system can be determined from the relative
noise, σ, on the baseline of the spectrum. The minimum detectable absorption coefficient can be
defined as
αmin [cm−1 ] =

σ
Lef f

,

(3.2)

where Lef f is the effective interaction path length with the sample. For a tight comb-cavity lock,Lef f is equal to 2F L/π, where L is the physical cavity length (assuming a linear cavity with theentire length used for absorption); in the case of cavity ringdown, Lef f is reduced to F L/π. In
intermediate cases, i.e., for the swept coupling scheme, Lef f takes a value between those two limitingcases [44]. When the system is white noise limited, σ averages down with the square-root of theintegration time, so the minimum detectable absorption coefficient normalized to 1 s acquisition
time is used for comparison between systems:
1s
αmin[cm−1 Hz−1/2 ] =

σ √

Lef f

T,

(3.3)

where T is the acquisition time of the spectrum. Note that this formula is valid only if the system
is white noise limited on the scale of a second or longer. For broadband systems such as DFCS, afigure-of-merit including the number of simultaneously resolved spectral elements, M , can be used

63
1s, DFCS[170, 186]. This absorption sensitivity per spectral element, αmin
, is defined asrσT
1s, DFCS
−1
−1/2
αmin
[cm Hz]=.
Lef f M

(3.4)

This value can be interpreted as the performance in terms of absorption sensitivity that a cw-laser
system would need, including scanning time, to match the performance of the broadband system.
The fundamental limit to the sensitivity in an absorption measurement is shot-noise in thelaser probe [41]. For a laser power, P , the shot-noise level is given by
s
2eB
σ=
,
ηP

(3.5)

where B is the detection bandwidth, η is the photodiode response, and Lef f is the effective path
length. Frequently, B = 1/T for a measurement time of T . This gives a shot-noise limited absorption sensitivity of
1s, shotαmin[cm−1 Hz−1/2 ] =

r

2e 1
.
ηP Lef f(3.6)

For shot-noise limited CE-DFCS [54], we substitute P ≈ P/N , where N is the number of spectral
elements in the entire comb spectrum, so P/N is the approximate power per spectral element.
Normalizing to the number of spectral elements in the acquired spectrum, M , according to Equation
3.4, we haves
1s, shot, DFCS
αmin
=

2eN π
,
ηP M kF L(3.7)

where k is a factor ranging from 1-2 depending on the coupling scheme (see Section 3.1), F is thecavity finesse and L is the single-pass absorption length in the cavity.
There are many sources of noise that prevent the shot-noise limit from being reached. As
discussed above, FM-to-AM noise conversion will result in amplitude noise significantly above theshot noise in many cases. Another common source of noise in many different experiments is optical
etalons. These can occur within, or between, any optical systems and result in fringes in the
spectrum. Depending on the source of the etalon fringes, they may be stable over the course ofmany measurements, in which case they can be removed in post-processing by fitting or filtering,or they may change during a measurement. These etalons are particularly troublesome because

64they can contribute significantly to the noise level. There are several ways of tracking down andremoving etalons in an experimental setup. Of course, the best option is to not introduce etalons in
the first place, which means that optical components should be AR-coated or tilted slightly where
possible. In particular, mirror substrates can be a persistent source of etalons, so the back surface
should be slightly wedged when possible. The size of an etalon can be deduced from the fringe
frequency, and this can be used to guess the origin of the fringes. Another option is to monitor
the spectrum while tapping on or moving various optical elements and to look for changes in theetalon phase or frequency. By combining these techniques, etalon effects can be mitigated.
Table 3.1 summarizes the sensitivity of various cavity-enhanced comb systems that have been
published. The different detection systems are discussed in more detail in the next section.

Table 3.1: Comparison of cavity-enhanced comb systems.
Ref

[181]
[187]
[188]
[44]
[189]
[54]
[72]
[190]
[60]
[53]
[185]
[180]
[191]
+

Detection system

Grating+CCD
Grating+CCD
Grating+CRDS
VIPA
Dual-comb FTS
FTSFTS
Grating+CCD
VIPA+VMS
VIPA
Grating+CCD
NICE-OFCS
FTS + swept-coupling

αmin

1s
αmin

1s
αmin

(cm−1 )

(cm−1 Hz−1/2 )(cm−1 Hz−1/2 )per spec. elem.

8.3 × 10−7 in 40 ms
1 × 10−8
6.3 × 10−7
8 × 10−10 in 30 s
1.2 × 10−6 in 18 µs
1.7 × 10−10 in 400 s
5.4 × 10−9 in 1 s
8 × 10−10 in 10 s
3 × 10−7 in 30 s
4 × 10−9 in 180 s
3.2 × 10−9 in 18 ms
3 × 10−8 in 0.8 s
1.5 × 10−7 in 1 h

1.6 × 10−7
1 × 10−7
2.5 × 10−10
4.4 × 10−9n/a◦
3.4 × 10−9
5.4 × 10−9
2.5 × 10−9
1.5 × 10−6
4 × 10−8
7 × 10−10
3.4 × 10−8
3 × 10−7

5.9 × 10−9
5 × 10−9
1.3 × 10−11
7.4 × 10−11
n/a ◦3.4 × 10−11
5.4 × 10−11
3.3 × 10−10
4 × 10−8
9 × 10−10
1.5 × 10−11
4.3 × 10−10
4.2 × 10−9 †Finesse

420
3000
4500
28000
8000
8000
3800
18000
100
30000
32000
2300
90000

Resolution(GHz)

Simul. Coverage/
Center Wavelength(nm)/(nm)

6
12
25
0.8
0.4
0.38
0.8
110
0.03+
0.9
13.5
2
6

20/8604/420
15/80025/1600
30/1530
30/1530200/3760
4/430
10/80020/1850
4/430
30/1575
25/1500

Comb-mode resolved so resolution limited by frequency calibration.
Could not continuously average for 1 s.
†
This number is given in the reference as 6 × 10−11 , but they divide by the number of spectral elements, not the square-root.

◦

65

66

3.3

Detection techniques
A variety of possible readout systems exist, as illustrated in Figure 3.1. In some cases, theabsorbing spectrum is sparse enough that the entire comb can be detected on a single photodiodeor dispersed with a grating and measured on a photodiode array, as shown in Figure 3.1(a). The
comb modes are then scanned to record the spectrum [192, 193, 120, 194]. This can be combinedwith comb vernier (where frep and the cavity FSR are slightly detuned so that only every n-thcomb mode is transmitted [195]) or cavity filtering using a high-FSR cavity to obtain comb-mode
resolution; however, the number of spectral channels, and thus the bandwidth or resolution, islimited by the size of the photodiode array.
For some transitions, especially vibrational transitions in larger organic molecules in the
mid-IR or many visible to UV electronic transitions in molecules, the absorption features to bemeasured have a width on the order of 0.5 cm−1 or more. In this case, a low resolution dispersive
spectrometer is suitable, as demonstrated in a comb spectroscopy system for IO, BrO, NO2 andH2 CO [183]. This technique is very similar to incoherent broad-band cavity enhanced absorption
spectroscopy (IBB-CEAS) [196, 197]. For broad spectral features, analysis methods similar to thosedeveloped for differential optical absorption spectroscopy can be used [198, 199].
Instead of measuring changes in transmission to detect the presence of absorption lines, it
is also possible to measure changes in cavity ring-down time in a technique called cavity ringdown spectroscopy (CRDS) [39]. In cw-CRDS, the laser is tuned to resonance and then the lightis rapidly turned off and the temporal decay of light leaking out of the cavity is measured. This
technique has the advantage that it is significantly less sensitive to intensity noise because the decay
is measured relative to the initial intensity only. It is possible to combine comb spectroscopy withCRDS [188], as illustrated in Figure 3.1(b). Here, a grating is used to disperse the light spectrally
in one direction while a rotating mirror is used to provide a time axis. The light is then detected
on a 2D camera. While helping to reduce intensity noise, the number of spectral elements is still
limited to the size of one dimension of the camera.

67
For many applications, it is desirable to record spectra simultaneously over a broad bandwidth with high resolution, which means that more spectral elements are required. Two common
approaches for this are the VIPA spectrometer or Fourier-transform spectroscopy (Figure 3.1(c)and (d), respectively), which we describe in Sections 3.3.1 and 3.3.2.

3.3.1
3.3.1.1

VIPA
Basic operation

A VIPA, or virtually-imaged phased array [200, 201, 202], is a rectangular etalon plate(sketched in Figure 3.2(a) and pictured in Figure 3.2(b)) with a high reflectivity (HR) coatingon the entrance surface (reflectivity > 99.9%) and a partial reflective (PR) coating on the backsurface (reflectivity typically about 97%). Additionally, an antireflection-coated (AR coated) entrance stripe is placed at the bottom of the entrance surface to allow light, which is focused in one
dimension, to be coupled into the device. In contrast to an ordinary Fabry-Perot etalon, the VIPA
etalon is tilted at an angle; because of this and the focusing of the incoming light, it transmits all
the incident wavelengths. The basic idea is that the incoming light has a range of k-vectors in one
dimension due to the cylindrical lens: each k-vector propagates in the etalon with different path
lengths, and thus the etalon resonances are shifted for each k-vector. This results in mapping of
frequency to k-vector, i.e., dispersion. The main advantage of a VIPA is the extremely high angulardispersion, up to 30-40 times better than a traditional dispersion grating. The output of the VIPA
is a pattern of overlapping mode orders that repeat at the etalon FSR - typically 50 to 100 GHz,
depending on the thickness of the plate. To spectrally resolve overlapped orders a cross-dispersion
grating with a resolution better than the VIPA FSR is used. After the grating the light is imaged
on a two-dimensional camera [51].
An example of the resulting image is shown in Figure 3.2(c). The vertical stripes are single
mode orders of the VIPA, which are dispersed in the horizontal direction by the lower-resolution
grating. The yellow horizontal lines show the range of one VIPA FSR on the camera, where the68

Figure 3.1: Detection schemes for CE-DFCS. (a) A grating spectrometer equipped with a detector array. (b) Broadband cavity ringdown setup. A grating provides spectral dispersion in thehorizontal direction and a rotating mirror creates the time scale in the vertical direction for a twodimensional detector array. An acousto-optic modulator (AOM) is used to turn off the laser beam
to initiate the ringdown. (c) A VIPA spectrometer. The cavity output is dispersed vertically witha high-resolution VIPA etalon and is cross-dispersed with a diffraction grating. (d) Femtosecond
Fourier-transform spectrometer. In the Michelson interferometer approach, the cavity output isanalyzed by an interferometer with a fast scanning mirror in one or both arms. This effectively
changes the repetition rate of the frequency comb. The pulses of the combs from the two interferometer arms with different frep interfere on the detector. In the dual comb approach, the output of
the enhancement cavity beats with another frequency comb with frep,2 6= frep,1 on a photodetector.

69

(a)

Cylindrical VIPA
lens
HR
PR

Grating(b)

AR Vertical

diffraction

Horizontal
diffraction
Imaging
Lens

Camera(c)

VIPA dispersion

Single comb modes

VIPA FSR(d)

Grating dispersion

Figure 3.2: VIPA Spectrometer. (a) Sketch of the components. AR, HR, and PR are antireflection, high reflection, and partial reflection. (b) Photo of mid-IR VIPA. (c) Image without comb
modes resolved. Dark spots are molecular absorption lines. (d) Image showing single comb-mode
resolution.

70
spectral information is unique. Thus, a single column within this region contains spectral information from one VIPA FSR. By reading the pixels column-by-column a traditional one-dimensional
spectrum can be created. The dark spots in the image represent CO2 absorption signals for individual ro-vibrational transitions around 1800 nm. It is also possible to resolve individual comb
modes, as shown in Figure 3.2(c) for a cavity-filtered 2 GHz comb in the mid-infrared.
VIPA spectrometers offer several advantages as a readout system:(1) The short integration time for a single image, on the order of tens to hundreds of µs, makethem an ideal tool for studies of transient events [56].
(2) Spectral information is acquired simultaneously across all of the channels. This helps the
S/N and also reduces issues of uncorrelated noise across the spectrum.
(3) The spectrometer does not involve any moving parts and is compact, which makes it particularly attractive for field-deployable devices.
(4) In addition, the approach is compatible with the swept coupling scheme, which reduces theoverall complexity of the experimental setup.
On the other hand, there are also several significant disadvantages:(1) Since the VIPA is an etalon the resulting fringe pattern is sensitive to optical alignment
and to mechanical vibrations, which tends to reduce the S/N .
(2) In addition, it is challenging to completely remove FM-AM noise in the tight-locking scheme
unless an additional modulation – as in velocity-modulation spectroscopy (Section 3.4) –can be introduced.
(3) Calibration of the frequency axis needs to be carefully checked since the pixel-to-frequency
mapping is nonlinear. We typically accomplish by calibrating to known spectra, but thisis not always possible. If individual comb modes are resolved, then the absolute frequency
at one point is all that is required.

71(4) VIPA etalons tend to be relatively expensive and are available only in limited spectral
regions.
Number (2) above is particularly troublesome. Unfortunately, it is not currently possible
to perform balanced detection with imaging arrays as can be done for single-element detectors.
2D cameras capable of lock-in detection do currently exist, as discussed in Section 3.4.2. The
lock-in bandwidth is up to 250 kHz, which is not high enough for frequency-modulation (FM)spectroscopy techniques. However, it should be possible to combine frequency-comb spectroscopy
with wavelength-modulation spectroscopy, which has been shown to give similar results to FM
spectroscopy at modulation frequencies above the noise bandwidth [203, 204, 205] and has been
implemented with cavity enhancement [206]. Currently though, lock-in cameras are only available
based on Si detectors (visible to 1 µm), so additional techniques would be needed for other wavelength regions. Active noise cancellation using an AOM is another option for removing amplitude
noise, but adds complexity. Because the FM-to-AM noise on neighboring comb teeth is correlated,
it should be possible to remove noise by comparing noise on nearby pixels. This only works wellthough if camera images are captured fast compared to the noise, otherwise residual uncorrelated
noise still causes problems. In general, the swept-coupling scheme is used instead of the tight lockwith dispersive spectrometers to reduce the effects of FM-to-AM noise conversion.

3.3.1.2

Design considerations

When designing a VIPA spectrometer, there are several parameters that must be optimized
to achieve the desired resolution and spectral coverage for a given application. A detailed discussionof VIPA spectrometer design and alignment is given in [207], we only highlight a few additional
points here.
The resolution of the VIPA spectrometer is limited by the finesse and FSR of the etalon.
The finesse is determined both by the reflectivity of the coatings and by the number of round-trip
passages within the tilted etalon. Resolution of 500 MHz has been demonstrated with VIPA FSRs
of 25 to 100 GHz. It has proven difficult in practice to achieve higher resolution. One option is

72
to decrease the FSR for a given finesse; however, this has several disadvantages. First, the grating
cross-dispersion must be capable of resolving a single FSR of the VIPA etalon, which is challenging
for small FSR as discussed below. In addition, the spectral bandwidth typically decreases for agiven camera size with decreasing FSR. Another possibility is to instead increase the finesse of theetalon by increasing the reflectivity of the PR coating. The challenge with this approach is that the
number of bounces inside the etalon must also be increased, which means that the etalon needs tobe taller while still maintaining a high degree of parallelism (significantly better than λ/2) between
the faces. Since the etalon is thin (on the order of 1-2 mm), this is further complicated by the factthat the etalon will flex a bit when mounted. In an effort to alleviate this, the etalon can havea thick backer plate optically contacted to provide increased rigidity (this can be seen in Figure
3.2(b)); however, distortions of the VIPA lineshape – which appears as a broad pedestal aroundthe main line – have been observed for VIPAs with a design finesse of 1000. These distortions are
believed to be caused by surface flatness/parallelism issues and limit the finesse to about 200.
The achievable resolution is sufficient to resolve single modes of high repetition rate combs,
as demonstrated with Ti:sapphire combs [51], and used in the velocity modulation experiment in
Section 3.4.2 [60, 208]. Another approach to achieve comb mode resolution with lower repetition
rate lasers is to use an intermediate filter cavity. This is shown in Figure 3.2(d) for a filtered combwith 2 GHz mode spacing; here, the filter cavity was set to transmit every 15th comb tooth ofa femtosecond mid-infrared OPO [52]. The resolution was determined to be around 600 MHz byfitting the individual comb modes.
One note is that the VIPA lineshape – in the ideal case – is a Lorentzian function, nota Gaussian. The broader tails of the Lorentzian lineshape mean that there is the potential forsignificant cross-talk between VIPA resolution elements, which is made worse by distortions in thelineshape. This must be considered for example for high-precision lineshape studies.
As mentioned above, another important consideration is the resolution of the cross-dispersion
system, which is set by the grating dispersion and the focal length, fi , of the imaging lens. The

73
separation in the focal plane, ∆x, for one VIPA FSR is given by∆x =fi∆λFSR ,
Λ cos θd(3.8)

where ∆λFSR is the wavelength separation for one FSR, Λ is the grating pitch, and θd = sin−1 (λ/Λ − sin θi )is the angle of diffraction for an angle of incidence, θi . In order for each FSR to be resolved, thisimust be larger than the diffraction-limited spot size, 4λfπd , for an input beam of size d measured

in the grating diffraction plane. An additional consideration is that the maximum resolution of
the grating also depends on the number of lines illuminated, N , and the illuminated width, d, by∆λ = λ/N = λΛ/d. Typically, d is set by the width of the VIPA, which is commonly around 2 mm.
For an 1800 line/mm grating, this gives a resolution of around 200 GHz at 800 nm, which would
not be enough to resolve mode orders of most VIPAs. Because of this, we sometimes enlarge theoutput from the VIPA in the direction opposite to the VIPA dispersion direction using a pair ofcylindrical lenses.
Finally, we would like the imaging system to be as efficient as possible. There are several
places where losses can occur. Of course, the grating needs to be chosen to be as efficient as
possible at the angle used. This can be accomplished with a blazed grating operating near the
Littrow condition. The losses from the VIPA are typically dominated by clipping of the input andfirst reflected beam on the HR/AR interface. The ratio of output to input power is [45]!22Pout /Pin ≈1−e(δh/2)
2
wv(3.9)

where δh = 2t tan(θv ) is the height of the first reflection on the input surface, θv is the incident
angle on the VIPA front surface (obtained using Snell’s law from the incident angle outside of the
VIPA and the index of refraction of the VIPA), t is the thickness of the VIPA, and wv is the beam
size at the focus of the cylindrical lens. In addition, we would like to concentrate as much power as
possible into one order of the VIPA, i.e., we want as much power as possible in the region between
the lines indicating one FSR in Figure 3.2. We can approximate this by calculating the size of one
FSR in the focal plane,∆yF SR ≈F SRλfi2 n2,cȳ(3.10)

74
where ȳ ≈ 1 − 2 mm is the height by which the interference pattern is shifted out of the VIPA, andalso by calculating the size of the image in the focal planewy =√2fi wi,fc(3.11)

where fc is the focal length of the cylindrical lens and wi is the input beam size. Ideally, wy ≈∆yF SR , and ∆yF SR is less than the size of the camera. This suggests that the power in one FSR
can be increased by increasing the focal length of the cylindrical lens.
VIPA spectrometers can currently cover the visible, near-IR, and mid-IR out to about 4.3µm. As with high-finesse mirrors, the bandwidth of a single VIPA system is limited by the coatings
to typically ∼ 20% of the central wavelength.
3.3.1.3

VIPA alternatives

Another option for achieving high angular dispersion is an echelle grating, which is a coarsely
ruled grating that is used in a high diffraction order [209, 210]. The resolving power, R ≡ λ/∆λ, isrelated to the diffraction order (m) and number of grooves illuminated by R = mN , which actually
only depends on the total illuminated width. The diffraction order sets the free-spectral range ofthe grating, Fλ = λ/m. This is a useful feature for spectral multiplexing because multiple FSRs canbe cross-dispersed as in the VIPA spectrometer. A VIPA has a resolving power of around 300,000,
so to match this we need to illuminate a width of around 10 cm in the visible/near-IR; however,this increases to 50-100 cm in the mid-IR, which would be challenging. In addition, the FSR ofa standard echelle grating will be significantly different than that of a VIPA. Currently, echelle
gratings are readily available with 20 lines/mm and with up to about 75% diffraction efficiency,
which gives m = 2 sin θ/(λΛ) ≈ 125 at 800 nm. This in turn corresponds to Fλ ≈ 6 nm or
around 3 THz (100 cm−1 ), independent of wavelength. In order to resolve 1 GHz, one FSR needs
to be recorded using 3000 pixels – a reasonable proposition in the visible/near-IR but currently
cost-prohibitive in the mid-IR for anything more than one FSR.
In the mid-IR, an echelle spectrometer has been built with a resolving power of ∼30,000 [211].

75
This spectrometer used a custom echelle grating with 4 lines/mm and a camera with 256×312 pixels.
This is still about an order of magnitude lower than that achieved in a VIPA spectrometer. To
reach higher resolving power, new gratings etched into Si or Ge prisms (called immersion gratingsor grisms) are being developed, aiming at resolving powers above 100,000 [212, 213, 214]. The useof internal diffraction in a high-index material immediately increases the resolving power by up ton2 compared to a standard grating of equal size [215]. In addition, the FSR is smaller by n for thesame grating pitch. Progress has been made on the manufacture of these gratings via wet etching[216, 217, 218]; however, currently these gratings are still custom and will still require large cameras
and somewhat large beams for high resolution. Eventually though, this technology may be a viable
alternative to VIPAs, even in the mid-IR, and should be able to operate over a wider wavelength
region than a comparable VIPA spectrometer.

3.3.2

Fourier-transform spectrometer

A different method for obtaining broad bandwidth spectra is to use an interferometer as
the wavelength resolving element: in this approach, a spectrum is obtained by first recording theinterference signal as a function of optical delay between the two arms and then taking the Fouriertransform of the interferogram. Fourier-transform spectroscopy (FTS) is a common technique inmolecular spectroscopy typically using an incoherent light source [6, 7], which can provide extremely
broad spectral coverage (from 2 to 50,000 cm−1 ) and can reach resolutions of about 0.001 cm−1 (30MHz, although resolutions of 1-10 GHz are more typical). The main shortcoming of this method is
the long acquisition time – associated with the use of thermal light sources – and the need for long
averaging times to obtain high sensitivity. The lack of spatial and spectral coherence of thermal
sources makes it very challenging to use a long optical path length (required for high resolution)or to efficiently couple light into an enhancement cavity for high sensitivity. Replacing the thermal
source of an existing Fourier-transform spectrometer with an optical frequency comb offers an
instantaneous increase in spectral brightness and thus reduces averaging times significantly. The
coherent frequency comb also allows an efficient combination of cavity-enhanced spectroscopy and

76
FTS.
As with dispersive spectrometers, FTS using a scanning interferometer as a readout systemfor comb spectroscopy has several advantages and disadvantages. The advantages include:(1) FTS is broad-bandwidth and works with minor changes from the near-IR to beyond 10 µm.
FTS can also work in the visible to UV.
(2) The resolving power of FTS is significantly higher than a VIPA spectrometer.
(3) It is possible to employ balanced detection to reduce amplitude noise – for example, fromcavity FM-AM conversion – by over a factor of 100, which enables the use of the tightcomb-cavity lock.
(4) Broad-bandwidth lock-in detection is possible because the signal is measured using only
one detector.
(5) The spectrum is immediately calibrated.
There are also some significant disadvantages:(1) It is challenging to make the system compact while still maintaing high resolution.
(2) The scanning stage makes FTS more sensitive to vibrations, which would make applications
to field deployable systems more challenging.
(3) The acquisition rate and thus temporal resolution is limited by the scanning cart speed anddata acquisition card as is typically on the order of seconds.
The use of two combs with detuned repetition rates (called dual-comb spectroscopy, shown in thelower part of Figure 3.1(d)) helps with most of these disadvantages, but requires two comb systems.
We have developed an FTS system for near- and mid-IR comb spectroscopy. The frequency
comb light transmitted through a sample cell or cavity is sent into a Michelson interferometerequipped with a fast-scanning delay stage on which two retro-reflectors are mounted. To increase

77

DM

IR

Autobalanced
detector

BS

Ref

Figure 3.3: Setup of Fourier-transform spectrometer. The comb light (IR) and light from a reference
laser (Ref, typically 780 nm or 1064 nm) are co-propogated (offset out of the page) with a halfmirror (HM). The two beams are split by different parts of a quadrant beam-splitter (BS) andthen travel through two arms of a Michelson interferometer. The lengths of both arms are scanned
simultaneously using two corner-cube retroreflectors mounted on a single state. The two output
ports of the interferometer are sent to an auto-balancing detector for the IR. The reference laser isdetected separately (not shown).

78
the optical path difference corresponding to the movement of this stage, the lengths of both interferometer arms are scanned simultaneously but in opposite directions (Figure 3.3). Light from astable 780 nm (or 1064 nm) cw-laser co-propagates with the comb light in the interferometer andis used as a reference for frequency calibration. To keep the reference and comb beams parallel but
easily separable, we use a quadrant beamsplitter in which the first and third quadrants are coatedas 50:50 beamsplitters for the comb light and the second and fourth quadrants are coated for thereference light. The two beams are split with the upper two quadrants and are recombined on thelower two quadrants. In addition, we found that the retro-reflectors caused more beam deviation
than expected, so the contrast of the interferograms changed by around 50% while scanning acrossthe full range. This can be improved with better retro-reflectors that deviate the beam less.
The comb interferograms at both output ports of the Michelson interferometer are monitored with an auto-balancing detector [219] using either InGaAs photodiodes (for the near-IR) or
HgCdZnTe photodiodes (MCZT, from Vigo Systems). The auto-balancing circuit subtracts thephoto-currents from the two photodiodes and uses an active feedback to keep the DC currents
equal. This subtraction doubles the useful signal compared to a single interferometer output (dueto the fact that the two output signals of the interferometer are complementary [6]) while subtracting away the common mode amplitude noise, such as FM-to-AM conversion by the enhancement
cavity. Both interferograms (from the comb and reference cw-laser) are digitized by a two-channel
analog-to-digital converter at 1 Msample/s with 22-bit amplitude resolution (National Instruments
NI 5922). The high data rate allows us to exploit the scanning speed of the FTS; to avoid digitization noise in the analog-to-digital conversion, at least 20-bit resolution is required.
The auto-balancing detector is based on the circuit developed by Hobbs [219]. For the nearIR, the basic auto-balancing circuit works well, but significant modifications were required dueto the different behavior of the mid-IR photodetectors. Figure 3.4 shows the schematic for the
mid-IR auto-balancing detector. Briefly, the matched transistor pair Q3-A and -B function asa current splitter, with the ratio of currents set by the difference in base-emitter voltages. This
basically acts to divide the current from the reference photodiode, with some of the current shunted

79
to ground via Q3-A. The reduced current from the reference photodiode is subtracted from thecurrent from the signal photodiode and converted to a voltage using the transimpedance amplifier
U2. Transistor Q2 is used to reduce the capacitance at the input of the transimpedance to increase
the bandwidth. The output signal is compared to reference voltage (near 0 V) and used to servo
the splitting ratio of the reference photodiode current to match the current in the two paths at
the summing junction. The primary changes to the Hobbs design for the mid-IR detectors is thephotodiode reverse-bias control, which was critical to providing the highest common-mode noise
cancellation possible. Because the bias voltage is very low for the mid-IR photodiodes (around
0.4 V), small changes in transistor Q2 or Q3 base-emitter voltage can significantly affect the bias
voltage and thus the detector response. Such a change will happen as the photocurrent, andthus the current through the transistor, changes. To keep the photocurrent linearly related tolaser power, the bias voltage is actively controlled to maintain a constant voltage drop across thediode. The primary challenge is making this feedback very fast – ideally an order of magnitude ormore above the signal frequency of around 150 kHz so that oscillations at the signal frequency arehighly suppressed. Basically, this feedback works by using a differential instrumentation amplifier,here using two op-amps but a fast three op-amp circuit would be better, to measure the voltage
drop across the diode and then using an additional op-amp to close the feedback loop. Many of
the capacitor and resistor values were adjusted to optimize the common-mode rejection ratio of
the instrumentation amplifier and to reduce parasitic oscillations due to non-linear behavior of thephotodiodes. In addition, the photodiodes were chosen to be as well matched as possible. Transistor
pairs Q1A-B and Q4A-B are current mirrors, which allow the photocurrent from each diode to bemonitored using transimpedance amplifiers U1 and U6. In addition to the lower bias voltage andparasitic oscillations, the mid-IR detectors also have a shunt resistance that is significantly lower
than that in visible and near-IR detectors. This low shunt resistance means that voltage noisefrom op-amps can cause significant noise at the output of a transimpedance amplifier if additionalcare is not taken. In addition, this shunt resistance could contribute to the oscillations seen in
the feedback. The shunt resistance increases at lower temperature, so we are currently using a

80
detector with an integrated four-stage TE cooler (the manufacturer does not recommend coolingbelow around 180 K). With these detectors, we measure a constant noise floor (above about 20µW) and a S/N at ∼150 kHz that increases linearly with increasing laser power. The S/N is abouta factor of 3 below the shot-noise limit, which indicates that we are detector noise limited. We
are not sure if this limit arises from dark current/Johnson noise in the photodiode or from thefinite bandwidth of the feedback loop. To improve the performance further, it would be good to
remove the feedback loop, which could potentially be accomplished by measuring the photocurrentof each diode with a transimpedance amplifier, and then using a current-to-voltage converter todrive the basic Hobbs circuit. In addition, cooled InSb detectors (for up to 5.5 µm) seem to haveslightly better performance – both a higher detectivity (D∗ ) and higher shunt resistance – thanthe MCZT detectors. InSb photodetectors do have higher capacitance so will be slower, but are
still fast enough. These changes should reduce the detector noise and enable shot-noise limited
detection in the mid-IR.
The resolution of the FTS is set by the maximum optical path delay (OPD) between the twointerferometer arms, which is limited by the maximum scan range (L) of the cart to R = 1/OP D =1/(4L) = 0.0035 cm−1 (105 MHz). This resolution is high enough to resolve individual comb modes
using minimal cavity filtering. Once individual comb modes are resolved, the resolution is set by
the comb tooth linewidth. Using our mid-IR OPO, we could achieve a resolution of 10 kHz, which
would require an OPD of 30 km in a standard FTS!
When FTS is used, the S/N in the spectrum, (S/N )f , is related to the S/N in the timedomain (interferogram), (S/N )t , as
(S/N )f = (S/N )t

√

N
,
M

(3.12)

where N is the number of data points in the interferogram and M is the number of resolved spectral
elements in the spectrum [220, 54]. To calculate the absorption sensitivity per spectral element, we
use Equation 3.4 with σ = (S/N )−1f :1s, DFCS, FTS
αmin[cm−1 Hz−1/2 ] =

σt √
M
Lef f

r

T
.
N

(3.13)

A

B

C

D

7

C30 +V1
0.1uF

R44
DNI

0.1uF
C31 -V1

0.1%

3
+
6
U11
2
AD825ARR36
2.000k 4-V1+V1 R43
DNI

R40
1k

R46
DNI

-

7

8

-V1

+V1

R31
DNI

R10
DNI

Negative Output
4

R29
1k

R34
DNI

0.1uF
C25 -V1

AD825AR

6

2

+ 3

U8

7

0.1%

R25
2.000k

C24 +V1
0.1uF

R38
2.000k

2

6

-V1

7

R28
0.1uF

+V1 C26
0.1uF

0.1%

R27
2.000k

0

7

-V1

R41
1.62k

0.1uF
C27

4

4

R30
1.62k

0.1uF
C7 -V12.000kR24

0.1%

7AD825AR
3
+
6
U7
2
-

C1 +V1
0.1uF

0.1uF
C29 -V1

2.000k
R35

0.1%

AD825AR
3
+
6
U10
2
-

7

C28 +V1
0.1uF

0.1uF
C33

6
3 + U9AD825AR
R33 4

R32
1k

D3

R39
0.1uF

3 + U12AD825AR

2+V1 C32
0.1uF

R45 4
0

R42
1k

D4

7

PD reverse bias control circuit, x2

positive output

8

R26
2.000k0.1%

R372.000k
0.1%

R50
1.0k

R49
1.0k

R48
1.0k

R47
1.0k

6

R16
221

Q4-B

R23
1.0k

Q3-A

6

R17
221

C22
100pF

Q3-B

R22
1.0k

Q1-A

R5
221

R4
27.4

Q4-A

D5

D7

Q2

C23
100pF

R6
221

2

1

R7
1.0k

Anode

Cathode

J3
Reference PD

SP3
RTN

7
-

C11
5pF

6
AD825AR

U2

R12
10k

5

R8
51.1

0.1uF
-V1 C10

4

3 +

2

Anode

2
+V1 C14
0.1uF

CathodeJ2
Signal PD
1

SP4
Logn Output

D8

D6

Q1-B

5

5
4

3

2

C15
1uF

+

-

7

-V1

1

U3

0.1uF
C13

LF356M

6C12
1uF

C5 +V1
0.1uF

3

2

6
LF356M

0.1uF
C21

4

1

5

R14
10k

U6

-V1

4

+

-

7

C17
100pF

6
LF356M

0.1uF
C4

R20
1M

R1
0

C18
0.1uF

+V1

R19
20k

1

5

R2
10k

U1

-V1

4

+

-

R11
10k

3

2

7

C2
100pF

C3
0.1uF

+V1

4

SP14
RTN

C9
1uF

C16
1uF

R13
100k

R21
51.1

CW

SP13
LOCK-IN INPUT

R18
51.1

R3
51.1

SP6
RTN

-V1

R15
12.1k

3

SP7
RTN

C19
4.7uF

C6
4.7uF

+

+

Vo (nV/rt(Hz))
231
190
144
91
170

-V1

3

2

1

+V1

J1

1

1.0

SHEET 11

OF 1

DRAWN BY: T. Brown
DATE: 05/26/2011

DN: YJ133

2

BUILT FOR: P. Maslowski
VERSION:

UPDATE: 8/1/2011

1.0

HOBBS_PHOTODETECTOR

HgCdZnTe PD with 4 Stack TEC

JILA ELECTRONICS LAB

Vs (volts)
8.11
4.39
2.40
1.05
4.55

DATA: @ ~100 kHz using SR780.
Vr (volts)
9.68
9.64
9.58
9.50
5.02

el
ectronics

4-40
M3

2

DESIGN BY: T. Brown

SP5
Difference Output

Vo

SP2
RTN

SP8
Reference Output

Vr

D2
2.5V

D1
2.5V

4-40
M1

SP1
Signal Output

Vs

R9
12.1k

+V1

3

A

B

C

D

81

Figure 3.4: Circuit diagram for mid-IR auto-balancing detector. See text for details. Designed by
Terry Brown.

82
The shot-noise limit is given by σtshot =q

2eNηP T , because the effective bandwidth for each point in

the time domain is N/T [54]. This results in minimum sensitivity that is a factor of√

M worse in

FTS than in a spatially multiplexed system for the same total integration time and signal-to-noise.
As discussed earlier, the main limitation of cavity enhanced spectroscopy utilizing a laser
continuously locked to a cavity is usually the noise originating from frequency-to-amplitude noise
conversion by the narrow cavity modes. Immunity to this type of noise can be obtained by theuse of modulation techniques, as is done for cw lasers in noise-immune cavity-enhanced optical
heterodyne molecular spectroscopy (NICE-OHMS) [41, 42]. Continuously scanned FTS, as we dohere, is a modulation technique as well, since the interferogram appears at an audio frequency– around 160 kHz in our case – limited by the acquisition rate of the data acquisition board.
Thus, the signal is detected at a frequency where the noise is significantly lower than at DC. The
addition of balanced detection reduced the noise in the near-IR by a factor of 600, which enabled
shot-noise limited detection with an optical power of 30 to 100 µW [54]. The deviation from theshot noise limit at powers below 30 µW is due to the decrease of the performance of the autobalancing detector. At powers above 100 µW the intensity noise starts to dominate over the shotnoise. The best shot-noise-limited single element sensitivity obtained for a 6 s acquisition timeis 1.4 × 10−9 cm−1 at 100 µW, corresponding to 3.4 × 10−11 cm−1 per spectral element. At aresolution of 2.3 GHz the measured absorption noise was slightly above that given by the shotnoise limit, due to the drift of the residual etalon fringes. Here a single element sensitivity of
7.6 × 10−10 cm−1 at 1 s was obtained, which translates to 2 × 10−11 cm−1 per spectral element at
1 s. In the mid-IR, a detector-noise limited sensitivity of 5.4 × 10−9 cm−1 Hz−1/2 was obtained,
corresponding to 6.9 × 10−11 cm−1 Hz−1/2 per spectral element for 6000 resolved elements [55].
Very recently, NICE-OHMS has been combined with frequency comb spectroscopy in a techniquetermed noise-immune cavity-enhanced optical frequency comb spectroscopy or NICE-OFCS [180].
The achieved sensitivity in the near-IR is close to the shot noise limit.
Instead of a scanning interferometer, two frequency combs with slightly different repetition
rates can be used, as shown in the lower part of Figure 3.1(d). This technique, known as multi-

83
heterodyne spectroscopy or dual comb spectroscopy [221, 222, 223, 189, 224, 225], provides very
short acquisition times (tens to hundereds of microseconds) for a single spectrum over a very broad
bandwidth. However, as in scanning FTS, the S/N decreases with increasing spectral bandwidth[220]. In addition, the complexity of two independent comb systems makes mid-IR applications ofdual comb spectroscopy more challenging [226, 227].

3.4

Velocity-modulation spectroscopy
Broad-bandwidth precision spectroscopy of molecular ions is useful in a wide range of fields

including fundamental physics, astrochemistry, and physical chemistry. For the eEDM experimentdiscussed in Chapter 6, very little data, especially high resolution data, existed for HfF+ and ThF+
and the uncertainties in the theoretical calculations of excited states were very large (thousands ofwave-numbers), so we needed a technique capable of broad survey spectroscopy to understand andassign the molecular energy level structure. As discussed in Section 6.8 different molecular ions,such as WN+ , could be useful for future versions of the eEDM experiment. Trapped molecularions may also be useful for measurements of nuclear-spin-dependent parity violation and anapole
moments [70, 228]. Sub-millimeter and far-IR spectroscopy of H3 O+ and deuterated isotopolgues
provide high sensitivity to variations of the proton-to-electron mass ratio [68]. Because of it’s
simplicity, spectroscopy of HD+ has already been used to test QED calculations [66]; similarly,
exact calculations including some QED terms exist for HeH+ [229], so this system can test theaccuracy of multi-electron calculations in molecules. Molecular ions such as SiO+ [230], AlH+ [231],
and MgH+ [232] among others are also suitable for direct laser cooling, which will provide more
opportunities for tests of fundamental physics. For many future experiments, broad-bandwidth
precision spectroscopy of molecular ions will play an important role, either for identifying transitions
for follow-up experiments or for providing direct precision measurements.
In astrochemistry, at least 22 ions (both positive and negative) ranging in complexity from H+
3+
to H2 COH+ and C−
8 and even metal containing species such as FeCO [233] have been identified

84
in interstellar and circumstellar gases [59]1 . Because ion-neutral reactions have very fast rate
constants – typically at or near the Langevin limit – and the ion production rate via cosmic
ray ionization is typically high, reactions involving molecular ions are central to understanding
astrochemistry [58, 234]; nonetheless, some rate constants and branching ratios are still not wellknown [235]. For example, the pathway for formation of a very simple molecular ion, CH+ , remains
elusive [236, 237]. In addition, H2 CO+ should be observed based on the currently known interstellar
species, but so far has not [59]. To aid the searches for new species, laboratory measurements areneeded. The development of new microwave and mm-wave observatories – most notably Herschel,
ALMA, and SOFIA – has increased the need for accurate measurements of rotational transitions in avariety of molecular ions [238]. While direct laboratory rotational and THz spectroscopy is possible,
precise measurements of vibrational and vibronic transitions can also provide indirect measurementsof rotational transitions to the accuracy needed [239]; we demonstrate this as well in Chapter 5,
where the precision on our rotational constants enable predictions of low-lying rotational transitions
to within a few MHz. Another long standing problem in astrochemistry is the identity of the diffuse
interstellar bands (DIBs) [240, 241], which are a collection of unidentified bands spanning thevisible and near-IR. Identification of these bands currently requires more laboratory measurementsof optical transitions to try to match with the observations [236, 235, 242]. The inability toidentify the molecules responsible for the DIBs points to a still very incomplete understandingof the processes that govern interstellar chemistry, and it is believed that many more unidentified
species such as positively or negatively charged polycyclic aromatic hydrocarbons (PAHs, a possible
source of the DIBs) and multiply charged ions are yet to be discovered [235, 243].
On the physical chemistry side, precision spectroscopy of H+
3 above the barrier to linearity(near 9913 cm−1 ) provides rigorous tests for ab initio theory [244, 245, 246]. Currently, theory
can reproduce the spectrum up to the visible with 0.1 cm−1 accuracy with the inclusion of nonadiabatic effects. Further improvements in experimental precision will enable further tests of nonadiabatic corrections as well as relativistic corrections. Adding another hydrogen molecule to form
1

A complete list of molecules is here: http://www.astro.uni-koeln.de/cdms/molecules

85
+
H+
5 adds significant further complexity for experiments and theory. The ground state of H5 canbe considered as a proton-bound dimer of H2 and the excited structures are more like H+
3 +H2 sofully understanding the potential energy surface provides information about clusters and about thereaction dynamics [247]. Many carbocations are interesting for both their presence as intermediates
in reactions such as combustion as well as the challenges they present to theory; for example, thespectrum of CH+
5 , a highly non-classical carbocation, still remains unassigned due to both its
complexity and spectral interference from other contaminant species [248, 249, 250]. Similarly, theethyl cation (C2 H+
5 ) is believed to be an intermediate in solution-phase reactions, but its structureis still not well know [251].

3.4.1

Background

One of the primary challenges with performing spectroscopy of molecular ions is separating
the ion absorption from neutral species. One option is to create an ion beam, or to use an ion trap,
where electric fields can be used to mass-select ion species. Typically though these techniques only
produce small numbers of ions and suffer from small absorption path lengths. A new techniqueof NICE-OHMS in an ion beam has recently been demonstrated and shows considerable promisefor high sensitivity ion spectroscopy [252]. High densities of molecular ions can also be createdusing hollow-cathode lamps and glow discharges; however, in these systems it is hard to separate
the ion signal from the competing absorption from neutral species (which are usually present at
significantly higher concentrations than the ion species). Our initial efforts of survey spectroscopyof HfF+ used emission from a hollow-cathode lamp and were not successful because of neutral
signal. In fact, even after we identified some HfF+ transitions using a different technique, we were
not able to clearly identify the transitions in the hollow-cathode lamp data [208].
The development of velocity-modulation spectroscopy (VMS) by the Saykally group solved
this problem and enabled background-free direct absorption spectroscopy of molecular ions. Velocitymodulation spectroscopy works by modulating the voltage polarity of an AC column discharge (seeFigure 3.5). This discharge produces ions with high density and the reversing polarity modulates

Volts

86

Laser

time
10 kHz, 1kW
Laser

+

ν

signalLock-in+νt

Figure 3.5: Sketch of velocity-modulation spectroscopy (VMS). An alternating voltage is applied
to a pair of electrodes, which forms an ac discharge and results in a modulated drift velocity, shown
here by red and blue arrows for the positive and negative half-cycles of the drive respectively. A
laser propagates on along the discharge and is detected using a photodiode. An absorption line of
the ion (black curve in the upper panel) will be Doppler-shifted (red and blue curves) with respect
to the laser (orange line). This results in a modulated absorption signal. The final signal after
lock-in detection at the modulation frequency is the difference between the two Doppler-shiftedpeaks.

87
the drift velocity of the ions. The velocity of neutral species is not influenced by the voltage applied
to the plasma. This modulated drift velocity causes a modulated Doppler shift when probed along
the axis of the discharge, which is analogous to frequency modulation of the laser. Lock-in detection synchronous with the modulation frequency (i.e., at 1f ) yields an (essentially) background-freeabsorption signal.
Lock-in detection can be understood as recording a spectrum during each half cycle and thentaking the difference between these two signals, as illustrated in Figure 3.5. In the ideal case, theneutral signal in each half cycle is identical and cancels out; on the other hand, the absorption lines
in each half cycle are frequency offset and so still yield signal in the difference. It is apparent fromthis simple analysis that neutral signal may not be completely cancelled if the signals in each halfcycle are not identical. This can be caused by asymmetries in the discharge, for example due topressure/flow gradients or electrode characteristics, so some additional tricks are typically used to
improve the neutral suppression.

3.4.2

Comb-vms

While velocity modulation spectroscopy provides high sensitivity for ions, it typically relies
on cw lasers and thus is slow to cover a wide spectral bandwidth. Fourier-transform VMS has
been demonstrated in emission; however, the sensitivity was low [253, 254]. Instead, we choseto combine VMS with frequency comb spectroscopy. By resolving every comb mode of a femtosecond Ti:sapphire laser (Gigaoptics, 3 GHz repetition rate) simultaneously over a wide spectral
bandwidth, we can achieve high resolution and absolute frequency accuracy with over 1500 channels measured at once. Figure 3.6 provides an overview of the experimental setup, more details
can be found in [60, 208]. Briefly, we couple light from the Ti:sapphire comb into the 120 MHz
free-spectral-range enhancement-cavity that contains an ac discharge cell, with every 25th cavity
resonance matched to a comb mode. The comb light transmitted through the cavity is fiber coupled
and then spectrally dispersed using a VIPA spectrometer to resolve every comb mode. We thenimage the light onto a Heliotis C2 “smart pixel” lock-in camera [255, 256], which uses a combination

88
of CCD and CMOS technology to demodulate and read out each pixel at the discharge modulation
frequency. The camera basically has two storage wells per pixel, one for the in-phase signal and one
for the quadrature. For the first quarter-cycle of the demodulation period, charge is accumulated
in the first well, similarly, charge is stored in the second well for the second quarter-cycle. For thethird and fourth quarter-cycles, charge is subtracted from the first and second well, respectively.
This process can be repeated for a number of demodulation cycles before a full image (called aframe) is recorded. Up to 320 frames can be stored on the camera in one set.
A co-propogating cw Ti:sapphire laser serves as a frequency reference for stabilizing the comb
and provides a marker on the two-dimensional spectral image. The cw laser is locked to its internalreference cavity achieving a linewidth of ∼100 kHz, and feedback to the combs repetition rate locks
a single comb tooth to the cw laser. The signal for the feedback originates from a beat note between
the comb and cw laser; after frequency division, the beat note frequency is converted to an error
signal using a frequency-to-voltage converter. This results in a looser lock than would be possible
with a phase detector; however, the lock was significantly more stable to perturbations. Typically,
the beat note is stabilized to about 1 MHz.
The repetition rate is controlled by acting on both the comb-laser cavity length (slow feedback
via a piezoelectric transducer) and the pump power (fast feedback via an acousto-optic modulator)[257, 258], while the offset frequency of the comb is left free-running. We initially tried to lock frep
to an external rf source and stabilize f0 using the pump power; however, we could only tune f0 by
about 20 MHz before the comb lost modelock. This is because, in our operating conditions, pump
power changes affect frep more than f0 , probably because non-linear refractive index effects in the
crystal can significantly change the cavity length since the crystal is a large fraction of the cavity
length in the high repetition rate laser. Since we can only operate the comb in a small range ofpump powers around 5.5 W, we do not have the ability to tune the pump power to a region where
the feedback to f0 is optimal [257]. Instead, we opted for the alternative locking scheme above.
This scheme has the disadvantage that f0 noise is written onto frep , which will make comb teeth
farther from the lock point less stable. During a scan, we observe fluctuations of frep of less than

89
200 Hz, which will cause at most 300 kHz of noise for a comb tooth 1500 modes (the range of asingle acquisition) away from the lock point.
We produce HfF+ by heating 0.5 g of HfF4 powder to about 550 ◦ C in a 1 m long home-built
discharge cell. Helium buffer gas is flowed through the discharge such that the total pressure is about
3 torr. By recording the emission of the discharge using a low-resolution grating spectrometer, wewere able to reliably monitor molecule production. With each loading of the oven we were able torun for about 3 hours. A 2.5 m long bow-tie optical enhancement cavity consisting of two flat, 98%
mirrors (input and output couplers) and two 100 cm radius-of-curvature, low-dispersion, 99.9% highreflectors surrounds the discharge cell. The reflectivity of the input and output couplers was chosen
to match the losses from the Brewster-angled windows and thus provide efficient input coupling.
We use liquid-crystal variable retarders and a polarizing beamsplittler to rapidly (50 ms) switch
the direction of propagation through the discharge tube while maintaining the comb-cavity lock.
This allows us to subtract out slowly varying noise due to drifts in camera pixel offsets and also to
improve the rejection of neutral background absorption. The cavity length is actively stabilized to
an integer multiple (25) of the frequency-comb laser cavity length, which ensures that each comb
component is coupled to a cavity mode. The error signal was based on a dither lock (later a PDHlock) using the spectrally filtered reflection from the cavity. Our feedback loop provides about 20
kHz of bandwidth by using a high-speed, low-range PZT [169] (oscillation frequency around 50
kHz) and a second long-range PZT. We found it critical to feed forward a 10-kHz compensation
voltage onto the lock of the enhancement cavity to compensate for the 10-kHz pickup from thedischarge, an effect seen in other cavity-based velocity modulation experiments [259] as well. By
using a non-polarizing beam splitter to combine the reflection from the two cavity directions andmaking sure that the combined reflections are well overlapped, we were able to switch directions
without losing lock and without changing the locked spectrum.
About 150 cm−1 of spectrum spread over 1500 channels is recoded simultaneously. For eachmeasurement, we average and subtract images for each direction of propagation and also record
the power per comb tooth by applying a calibrated amplitude modulation to the laser. In order

90

Laser beams

10 kHz
500 mA

SpectrometerHe
550 ˚C
HfF4
10 kHzFiber coupler
FI Faraday Isolator
Liquid crystal
variable retarder
Grating
Electronics

Cyl Lens:
x2.5 hor. tele.
fc = 100 mm
VIPA

fi = 250 mm

FI

Lock-in Camera

Computer
Scan out

MBR 110cw Ti:Sapph

95/5
B/S

FI

3 GHz
Ti:Sapph
Comb

AOM

18W Verdi

Wavemeter

Beatnote

Figure 3.6: Setup for comb-vms. We use a Ti:sapphire comb with a repetition rate of 3 GHz,
which is transferred via fiber to the optical cavity. The comb is coupled into one direction of a ringbowtie cavity, which contains an ac-discharge cell with Brewster-angled end windows. The cavity
finesse is about 100 and the length is matched to be an integer multiple (25) of the comb cavity.
We use liquid crystal variable retarders and a polarizing beamsplitter to alternate the direction of
propagation through the cavity. The ac-discharge is driven at 10 kHz with about 500 mA. We flow
helium gas through the tube at a pressure of about 3 Torr. HfF+ ions are created by heating about
0.5 g of HfF4 powder to 550 ◦ C inside the discharge tube. The cavity transmitted light, whichcontains the modulated absorption signal, is analyzed using a two-dimensional imaging system anda lock-in camera. A cw Ti:sapphire laser is used to stabilize the comb and serves as a frequency
reference in the spectra. A fiber-coupled grating spectrometer is used to monitor the discharge
fluorescence.

91
to fully sample the spectrum, we interleave 30 measurements with the cw laser stepped over 3
GHz. This results in a spectrum that covers 150 cm−1 sampled every 100 MHz with an absolute
frequency accuracy of 30 MHz (set by a rubidium atomic transition referenced wavemeter). When
the wavemeter was not calibrated for a particular measurement, we estimate the uncertainty to
be 100 MHz. For strong bands with fully resolved isotope structure, absolute accuracy of thedetermination of the band origin is limited predominantly not by statistical errors but by theabsolute knowledge of the cw Ti:sapphire laser frequency we use as a reference. For our purposes
we were satisfied with the 30-100 MHz (0.001 - 0.003 cm−1 ) absolute accuracy of the wavemeter.
This could be readily improved if needed [239]. For analysis purposes we then interpolate thespectrum onto a fixed 0.001 cm−1 grid. This allows us to easily average or combine a collection ofdifferent scans.

3.4.3

System performance

To track down possible sources of noise, we monitored the intensity noise on a single pixel andacross the camera – i.e., by subtracting the first frame from all other frames – as a function of time.
One major source of noise was drifting baseline offsets caused by discharge noise coupling to either
the laser lock or the cavity. This results in frequency noise, which is converted into amplitude noiseat the discharge frequency. The offsets were reduced with a feed-forward by adding a phase-shifted
and amplitude-controlled signal from the discharge drive to the cavity lock. Etalon fringes werealso a significant source of noise. In particular, we found that one polarizing beam-splitter createdfringes and, most significantly, the fiber to the imaging system caused high-contrast etalon fringes
until it was replaced with a polarization-maintaining, angle-cleaved fiber. Other sources of laser
frequency noise can result in amplitude noise due to FM-to-AM noise conversion by the cavity if thenoise bandwidth is high enough to contribute noise at 10 kHz. Several such sources were discovered:in particular, air currents and water flow. Air currents in the comb pump laser beam path cancause significant frequency noise on the comb light. We removed these by carefully enclosing theentire beam path and sealing any small gaps in the enclosure. In addition, we found several issues

92
related to the cooling water for the comb. The first was vibrations caused by the chiller that weretransmitted along the cooling lines. This was improved by including a short section of soft rubber(Tygon) tubing in the cooling lines, which acts as a vibration damper. In addition, turbulent flow
presumably near the crystal caused laser frequency noise. We reduced this noise by including alarge water reservoir between the chiller and laser to filter pressure fluctuations and by reducing
the cooling water flow.
In addition to technical noise, we also care about the readout noise of the camera and thetotal efficiency of the system. First, we can calibrate the camera to obtain a relationship between
counts and intensity by measuring the light from a modulated LED, which is assumed to be shotnoise limited. By comparing the noise as a function of the average number of counts on a pixel, we
can determine readout noise as well as the relationship between the number of photons and counts.
This calibration gives around 3300 photons/count. When converting to power, it is necessary toinclude the fact that the in-phase or quadrature signal only comes from half of the integration time
and the fact that the first two demodulation cycles (periods) are used for a pixel offset subtraction
scheme so the number of cycles of data is reduced by two. The readout noise is about 1 count per
cycle (or about 0.15 counts in 50 cycles, which is a more typical operating condition) or equivalently
about 15 fW. While the camera noise on a single pixel averages down as expected as a functionof time, we found that variations across the camera do not average down at all. We attribute
this variation to small pixel-to-pixel offsets, which are removed by subtracting images for the two
directions.
For all of the data shown here, we measured the following efficiencies: 30% fiber coupling tocavity, 70% from the fiber output to the cavity, 15% transmitted through cavity, 35% coupling toimaging system, 15% through imaging system, 60% in one VIPA FSR, and ∼30% camera efficiency.
This gives an expected power per comb mode of about 20 nW. The fiber coupling efficiencies andcavity transmission (limited by cavity losses) have been improved since. Some of the losses in theimaging system efficiency are due to VIPA coupling efficiency as well as grating losses. We foundthat the camera efficiency was very dependent on the incident angle due to an angular dependence

93of the micro-lens array on the camera; however, we were still never able to increase the efficiency
beyond 30%.
Without the discharge running, we measured a fractional technical noise floor of about 1×10−5
at 1 s for a single direction. Rapidly taking the difference between two directions reduced this toaround 3 × 10−6 . The noise was increased with the discharge running to about 2 × 10−4 for one
direction. The combination of switching directions, intensity normalization, and rolling mediansubtraction (subtracting from every point in the spectrum the median of the surrounding 20 points)
reduced the noise to 1 × 10−5 at 1 s. The noise averages down as the square-root of time for up to
100 s before significant deviations are observed.
Recording one full spectrum, consisting of 30 interleaved measurements, takes about 30 minutes and results in a single-pass fractional absorption sensitivity of 3×10−7 . Since one scan contains
45000 channels, this equates to a sensitivity of 4 × 10−8 Hz−1/2 per spectral element), which is thesensitivity that a single-frequency laser system would need, in addition to being able to scan 150
cm−1 continuously, to match the performance of the comb-vms system. We recorded spectra over a
total of spectral bandwidth of 1000 cm−1 with both the oven on and off to check for contamination.

3.4.4

Single-frequency vms

In addition, we also scanned some bands using single-frequency velocity-modulation spectroscopy as sketched in Figure 3.7. For these measurements we removed the enhancement cavity
and counter-propogated beams from the cw Ti:sapphire laser through the discharge tube. We thensubtract these two beams using an auto-balancing photodetector, which is discussed in Section
3.3.2. Due to noise from the discharge, we reached a sensitivity of about 5 × 10−8 Hz−1/2 with 1
mW on the detector; this noise seemed to be related to optical pickup from the discharge emission,
which was reduced using irises, and to non-common-mode amplitude noise from acoustical pickup.
In particular, we found that we could minimize the acoustical pickup on the cavity mirrors by tuning the discharge frequency slightly to avoid resonances. The use of more rigid mirror mounts also
decreased the noise. In order to find bands without scanning the full spectral range, we scanned

94
several small 1-cm−1 sections spaced by 10 cm−1 and then move to a new region if no lines were observed. Using this technique we were able to find 15 more bands ranging from 9950 to 14600 cm−1 .
We can continuously scan about 0.5 cm−1 in 4 minutes, which is at least 30 times slower than the
comb-vms system.

3.4.4.1

Coherent Subtraction

We have also investigated a novel technique for differential detection that relies on coherent
interference between the two counter-propogating laser beams instead of subtraction of photocurrents as discussed previously. This is accomplished by coherently splitting and recombining thetwo beams using one non-polarizing beamsplitter to form a Sagnac-type interferometer containing
the discharge cell and cavity if desired (blue inset to Figure 3.8). The signal is then detected
at the destructive interference, or “dark port”, of the interferometer. In the regime where laser
intensity noise dominates, the signal-to-noise using coherent subtraction increases as the splitting
ratio approaches 50:50 between the two beams until detector readout noise or shot noise dominates.
Physically, the fractional signal is increasing while the fractional noise level remains constant and dc
power are both decreasing. We have tested coherent subtraction versus a single direction of propagation using a photodiode and single-channel lock-in detector with our cw laser and demonstrate afactor of 10 gain in the signal to noise (blue data in Figure 3.8). We expect an improvement in the
2

2

T −R
S/N of about R2 +T
2 −2RT , where T and R are the transmission and reflection of the beamsplitter(assuming no losses). Thus, we would expect roughly a factor of 25 improvement with the 52/48
beamsplitter that was used. The lower than expected enhancement could come from losses in the
PBS or alignment differences. We tried to implement a tunable cancellation ratio between the two
directions using with Faraday rotator to provide a small polarization rotation, but beam distortion
made this challenging.
Coherent subtraction, unlike subtraction of photocurrents, is sensitive to phase as well as
amplitude of the light field; thus, the resulting lineshape is dependent on both absorption (approximately first-derivative shaped) and dispersion (approximately second-derivative) and can vary if

95

Laser beams

10 kHz
500 mA

Fiber coupler
FI Faraday Isolator

HeElectronics

550 ˚C
HfF4

10 kHzSpectrometer

Lock-in0˚ 90˚

FI

Wavemeter

Computer
MBR 110cw Ti:Sapph

Scan out

18W Verdi

Figure 3.7: Setup for cw-vms. We use a cw Ti:sapphire laser that is tunable from about 9500 14800 cm−1 (1050 - 675 nm). After fiber coupling between tables, the laser is split with a 50/50 nonpolarizing beamsplitter. Each beam goes through an additional 50/50 non-polarizing beamsplitter(to separate the return light) and is then sent in opposite directions through the discharge cell. Halfof the returning light in each direction is reflected by the beamsplitters and is differenced using an
auto-balancing photodetector [219]. This difference signal is then sent to lock-in detectors to record
signal in-phase and 90◦ out-of-phase with the HfF+ absorption. We placed irises in the beam paths
to reduce the amount of discharge fluorescence observed on the auto-balanced detector; this helped
both to reduce noise and primarily to reduce baseline drifts in the spectra.

96
there are differential phase shifts between the two directions. We performed a simple simulation,
shown as dashed lines in Fig. 3.8, that reproduces the lineshape modification. These simulations
were done by applying a cosine modulation in time to the center frequency for both the absorption (assumed Gaussian lineshape) and dispersion terms (obtained via Kramers-Kronig) and thenselecting the appropriate frequency term from the Fourier cosine transformation. The electric field
for the two directions (denoted cw and ccw) is given by:Ecw (t) = T exp −α(χicw (t) + iχrcw (t)) exp(iπ) exp(iφ)(3.14)


Eccw (t) = T exp −α(χiccw (t) + iχrccw (t)) ,where α is the absorption coefficient and χi (t) and χr (t) are the imaginary (Gaussian) and real partsof the polarizability. The time dependance is given by a modulation of the center frequency of theGaussian absorption lineshape at the discharge modulation frequency, ωmod . The modulation is outof-phase for the two directions. We then write the detector signal as I(t) = |Ecw (t) + Eccw (t)|2 andRextract the lock-in signal by S = I(t) cos (ωmod t)dt. Repeating this at a variety of simulated laser
frequencies generates the lineshape. The amplitude and linewidth of the single-direction simulation
was scaled to the measured value and these values were used for the coherent-subtraction simulation.
The constant phase offset (most likely due to the beamsplitter) added to the coherent subtraction
simulation was chosen to fit the measured lineshape (about π/20 radians). In addition to the phasesensitivity, which could be useful for some applications, coherent subtraction has several otheradvantages. First, the dark port can be used to reduce the optical power seen by the detector thus
avoiding saturation while still maintaining a shot-noise limited signal-to-noise ratio. Second, thesubtraction is performed prior to the detector and is wavelength independent. This means that
coherent subtraction can be used with comb-vms or other dispersive detection systems for increased
signal-to-noise. We did not implement coherent subtraction with the comb-vms system here because
we were close to camera-noise limited, thus the gains would not have been very significant. It wasalso not used for the cw scans because the lineshape would change slightly day-to-day, possibly due
to alignment variations.

97

x10

SingleDirection (x10)
Coherent
Subtraction

Figure 3.8: Comparison of coherent subtraction and single-direction measurements. The signal
to noise for coherent subtraction (using a 52/48 beam splitter) surpasses that of single-direction
measurements by a factor of 10 when the noise is primarily technical light intensity noise. Bothmeasurements of a single N+
2 line were made with the cw laser. The dashed lines are a calculationof the lineshape (see text). The modification in the lineshape for the coherent subtraction is due toa stray differential phase shift of approximately π/20 between the two counter-propagating beams,
which results in a sensitivity to both absorption and dispersion.

98
3.4.5

Extensions of comb-vms

Frequency comb modulation spectroscopy can be applied to both ionic and neutral moleculesby either demodulating at the discharge modulation frequency (ion velocity modulation) or at
twice the discharge modulation frequency (neutral-molecule production yield modulation). This
technique could also be readily extended to the detection of radicals by concentration modulation[6]. Some ions can also be produced directly in a supersonic-expansion discharge source [260], andcan also be detected with concentration modulation. It might also be possible to modulate thedischarge direction and perform velocity modulation in the expansion. A high-sensitivity ion-beam
VMS system has recently been developed [252], which enables mass-selective spectroscopy of ionswith high precision due to kinematic narrowing in the fast ion beam. Such a system should workwith any ion production method. In addition to studying the spectra of stable ions, gas-phaseion reactions can be studied using a selected-ion flow tube [235]. Combing comb-vms with thesetechniques would enable broad bandwidth, high-resolution studies of cold ions and radicals.
For many applications, comb-vms will need to work in a variety of spectral regions. Comb
sources also enable efficient non-linear optical generation, which allows for spectral broadening andaccess to spectral regions ranging from the UV to the mid-IR, as discussed in Chapter 2. Currently,
the lock-in camera is only available with Si detectors and thus only works in the visible to near-IR
out to 1 µm. It might be possible to use similar technology with other detector materials, such as
InGaAs, but this would require significant development work. Some alternatives do exist however.
One option is to use only one axis of the camera for spectral information and then use a rotating
mirror to provide time information on the second axis of the camera, as was done for comb cavityringdown spectroscopy [188] (see Figure 3.1(b)). If timed correctly, every pixel could then record
one-quarter of a cycle of the demodulation frequency, which would then be similar to the lock-in
detection method. Alternatively, cameras with integration times of under 20 µs are available evenin the mid-IR. In this case, the camera could be triggered to only record for one-quarter of a cycle.
This would then be repeated for each quarter-cycle, again providing similar information to the99lock-in system. However, both of these techniques will reduce the amount of on-camera averaging
and will also require more time for the same amount of data. Another option for the mid-IR
is to up-convert the cavity-transmitted light to the near-IR using sum-frequency generation. An
upconversion-based VIPA spectrometer for the mid-IR has been demonstrated using a cw Nd:YAG
laser [261], but the power was very low. Instead of a dispersive spectrometer, it should be possible
to perform comb-vms with FTS. Lock-in detection has been demonstrated with a scanning FTS toenable NICE-OFCS [180], so this should also work with VMS.

Chapter 4

Applications of DFCS

In this chapter, we discuss several implementations of direct frequency comb spectroscopythat highlight new capabilities and explore new spectral regions. First, we look at the application of CE-DFCS to the detection of trace contaminants in arsine gas [53], which is used in the
manufacture of many semiconductor devices. This experiment demonstrated for the first time
direct frequency comb spectroscopy in a region accessed by supercontinuum generation – in thiscase, 1.75-1.95 µm (5710-5130 cm−1 ). This spectral region is mostly unexplored via laser-based
spectroscopy owing to the lack of widely tunable lasers operating here. In fact, DFB diode lasers
have only recently become available in this spectral region [262], but still with limited tuning range.
Two possible broadly tunable sources are optical parametric oscillators [263] or difference frequency
generation [264]; however, these cw systems must be carefully scanned to cover the full spectral region. Our CE-DFCS approach provides high sensitivities across more than 2000 detection channels
distributed over a broad, simultaneous bandwidth in a robust and compact system. Furthermore,this experiment also demonstrated CE-DFCS for the first time with focus on an industrial application (i.e., trace detection in a strongly absorbing process gas), where the bandwidth is critical for
distinguishing impurity signals from background absorption, along with high resolution for making
unambiguous identifications.
In the second section, we present our first implementation of direct frequency comb spectroscopy in the mid-infrared [46] using the OPO comb (Section 2.4) and Fourier-transform spectrometer (Section 3.3.2). As discussed in Chapter 1, the mid-infrared is an important spectral

101region because of the strong fundamental vibrational bands, which provide both increased sensitivity and access to more molecules than in the near-infrared. This first demonstration achieved
part-per-billion detection limits in 30 seconds of integration time for several important molecules
for breath analysis and atmospheric chemistry – including methane, ethane, isoprene, and nitrous
oxide – across the 2.8-4.8 µm spectral region. We also show how DFCS enables precise concentration measurements even in gas mixtures that exhibit continuous absorption bands and how itallows detection of molecules at levels below the noise floor via simultaneous analysis of multiple
spectral features. In addition, the use of a multipass cell – despite the reduced sensitivity – provides
access to the full spectral range covered by the source and demonstrates a general system capableof working from the visible to mid-infrared beyond 10 µm. The source and detection system also
provided the basis for later applications of mid-infrared CE-DFCS [55, 56].
Finally, we discuss some future prospects of comb spectroscopy and illustrate where thebroad bandwidth, high resolution, and high sensitivity of comb spectroscopy could provide anadvantage in certain applications. Specifically, we look at continued industrial applications of trace
gas detection, prospects for breath analysis, and both laboratory and field applications to the studyof atmospheric chemistry.

4.1

Trace detection in Arsine
Arsine (AsH3 ) and phosphine (PH3 ) are important process gases used in the production of

III-V semiconductors via metal organic chemical vapor deposition (MOCVD) [265, 266, 267, 268].
These compounds are used in devices ranging from high-brightness light emitting diodes and highpower laser diodes to solar cells. Trace levels of contaminants present in the process gases canresult in unintentional doping and lattice defects. The inclusion of these dopants gives rise to
additional energy levels in the bandgap of the host semiconductor, leading to undesired changes of
its electrical and optical properties. In particular, oxygen incorporation in III-V semiconductors
has been shown to form a deep recombination level, resulting in a decrease of photoluminescence
efficiency and carrier lifetimes as well as reduced device reliability. A primary source of oxygen

102
impurities is the presence of traces of water vapor in the precursor gases, which has been shown tonegatively impact the semiconductor at the level of 10-100 parts per billion (ppb) [268]. Because of
its ubiquity and low vapor pressure, water is extremely difficult for the manufacturer to completely
remove. Furthermore, water impurities can be introduced from contamination in the transfer lines
at the point of use; therefore, on-line monitoring of water concentrations during semiconductor
growth is desirable. In addition to water vapor, many other impurities including carbon dioxide,hydrocarbons (methane and ethane), hydrogen sulfide, silane and germane must be controlled in
the process gas [266].
Current systems, employing different techniques, capable of detecting water at the sub-100ppb level all have significant drawbacks [266]. For example, Fourier transform infrared (FTIR)
spectrometers require long acquisition times owing to the low spectral brightness of their thermal
light source. In addition, the required sensitivity can only be achieved by using a highly optimized
research system with a long pathlength gas cell, a high sensitivity detector, and an ultrahighpurity nitrogen purge system. Another potential technique is negative ion-atmospheric pressure
ionization mass spectrometry (APIMS) [269], but these expensive systems are large and complex,
not suitable for on-line monitoring and are mainly limited to inert background gases. Current
research has focused on laser-based spectroscopy systems such as tunable diode laser spectroscopy(TDLAS) or cavity ring-down spectroscopy (CRDS) [265, 267, 270], which provide highly sensitive
measurements with rapid acquisition times; however, these systems typically target one or two
absorption lines of the desired impurity species. Thus, multiple contaminants cannot be easily
identified and quantified, while unexpected impurities can significantly degrade the reliability ofthe measurement. In a recent study [268], it was necessary to use gas chromatography with mass
spectrometry, atomic emission spectroscopy, and pulsed discharge ionization detection in addition to
CRDS, to check for all of the critical impurities in arsine. Many of these impurities could potentiallybe detected with one system using CE-DFCS. Previous cw-CRDS experiments measuring water inphosphine at 940 nm [270] and 1400 nm [267] were limited by competing background gas absorption
and by the line strength of the water transition. The 1.75-1.95 µm (5710-5130 cm−1 ) region contains

103bands of several important impurities, including a water band at 1.85 µm (5400 cm−1 ) that isstronger than the 1400 nm band by a factor of 2-3, and should be a somewhat transparent region of
the arsine absorption spectrum [48]. Therefore, it is a compelling range to explore with CE-DFCS.

4.1.1

Experimental Setup

A sketch of the experimental system is shown in Figure 4.1. We used a home-built modelocked Erbium-doped fiber ring laser [207] that provides 130 mW average power at a repetition
rate of approximately 250 MHz. The laser output was then amplified with a single mode Er:fiber
amplifier to produce 81-fs pulses with 400 mW average power. These parameters are readily
achievable with commercially available fiber lasers as well. After the amplifier and polarization
control optics, all of the light was coupled into 10 cm of standard single mode fiber spliced toa 6 cm long piece of highly nonlinear silica fiber (OFS Specialty Photonics) to provide spectral
broadening. The spectrum (covering 1.2-2.1 µm or 8300-4700 cm−1 ) from this fiber is shownin Figure 4.1(b) and Figure 2.4 as optimized for generation of a frequency comb near 1.85 µm;
however, it is easy to change the spectrum by varying the laser polarization and input pulse chirp.
The average power after a 40-nm bandwidth filter centered at 1.86 µm was 17 mW. Because thelaser and nonlinear spectrum generation were both fiber-based, this entire system was compact androbust – requiring almost no adjustment from day to day. We initially tried Raman shifting fiber
to obtain more power, but the coherence of the shifted light was degraded as discussed in Section
2.3.
The optical resonator used for enhancing the absorption detection was a linear Fabry-Perotcavity with one 2-m radius concave mirror and one flat mirror and a peak finesse of 30,000 (mirror
reflectivity ∼0.9999). In order to obtain accurate absorption values, we characterized the mirror reflectivity via wavelength resolved ringdown measurements, see Figure 4.1(c). For this werecorded ringdown traces of an empty (under vacuum) cavity in 10 nm wavelength increments
using a monochromator and a fast extended-InGaAs photodiode. Several measurements for eachwavelength were averaged, and the resulting data were fit with an 8th-order polynomial.

104

(a)

Feedbackf rep

2D Dispersive
SpectrometerPzT

F ~ 30,000
Er:fiber laser
Valve
Flow controller
Pressure controller

HNLF

InSb Camera

AsH3

N2
H2O

Ref

Impurity

Ref - Sample
Ref
SampleFinal Spectrum Absorption Image(c)log(Intensity)

0

-2

-4
1000

1400
1800
Wavelength (nm)

2200

100
99.98
99.96
99.94
99.92
Residuals (x10-4)(b)Reflectivity (%)

Raw Images

1700

1800

1900

2000

5
0
−5
1700

1800
1900
Wavelength (nm)2000

Figure 4.1: Sketch of trace detection in arsine. (a) Outline of experiment. HNLF: highly nonlinear fiber; PzT: piezoelectric transducer; (b) Spectrum after nonlinear fiber. (c) Measured mirror
reflectivity and residuals after 8th-order polynomial fit.

105
The mirrors were first aligned in mirror mounts and then attached to a stainless-steel gas cellusing Torr seal, after which the mounts were removed. Prior to the final attachment, it is important
to make sure that the input and output laser beam is well centered on the cavity mirrors so thatthe finesse is not reduced due to clipping. Because arsine gas is extremely toxic, the mirrors weresurrounded by a sealed end cap. We also use VCR fittings for the gas line connections because
these are more robust than Swagelok. Finally, the entire system was carefully checked for leaks.
The temperature of the stainless-steel gas tube was actively controlled using heater tape and could
be tuned to adjust the cavity length.
The cavity length (∼60 cm) was adjusted so that the free-spectral range was matched to the
comb repetition rate. In addition, the comb f0 was adjusted via the fiber laser pump power to
optimize the transmission of the comb through the enhancement cavity, see Section 3.1. The usefulspectral bandwidth of the cavity was about 200 nm (600 cm−1 ); however, owing to the varying
free-spectral range frequency due to cavity dispersion, it is not possible to simultaneously match all
comb modes and cavity modes over this bandwidth. To overcome this limitation and to synchronize
the cavity and comb during measurements, we use the swept-coupling scheme (Section 3.1.2). In
this case, the comb modes were dithered (by changing the laser cavity length and therefore therepetition rate) using a triangle waveform with an amplitude of 150 kHz at a frequency of 7.5 kHz
around the cavity modes, and slow feedback to the enhancement cavity length (via a piezo) was used
to keep the time gap between the successive transmission peaks constant. Since our integration time
per individual record was around 150 ms, we integrated over multiple dither cycles. This dither
process effectively allowed us to couple the comb to the cavity over the full spectral bandwidth ofthe mirrors.
The sample gases flowed through the optical buildup cavity for detection. To measure andquantify impurities in arsine, our gas handling system was designed to add trace contaminants toarsine at well-controlled mixing ratios. For this purpose we obtained a calibrated mixture of 10
parts per million (ppm) each of CO2 , CH4 , and H2 S in a nitrogen gas cylinder. In addition, we
added small amounts of water vapor via a diffusion vial (Valco Instruments). Flow controllers were106
used to control the relative arsine and impurity gas flow as well as to control the amount of flow
across the water diffusion vial. Typically, the total pressure in the sample cell was ∼200 torr andwas controlled by a back-pressure controller just after the cell. The gas system was designed tobe very flexible for this experiment and to maintain a high degree of safety due to the toxicity ofarsine gas (e.g., the pumps and arsine cylinder were located in different rooms); however, it was
not optimized for switching speed due to the small diameter of the installed tubing.
The light transmitted from the cavity was analyzed with a VIPA spectrometer (Section 3.3.1).
This spectrometer system provides high resolution (900 MHz or 0.031 cm−1 ) in one dimension, while
maintaining broad bandwidth (∼20 nm or 50 cm−1 , limited by the size of the cameras imaging
sensor) in the orthogonal dimension. Light was coupled into the VIPA etalon (51 GHz free spectral
range) with a horizontal, 6-cm focal length cylindrical lens. The etalon was tilted approximately
2◦ from vertical to provide a high dispersion in the vertical direction. A 1200 lines/mm crossdispersing grating was used to separate the mode orders in the horizontal direction. Finally, thelight was imaged onto the 320x256-pixel InSb focal plane array camera with a 25-cm focal length
lens, resulting in images as shown in Figure 4.1.
Images were collected alternating between sample and reference gases, which provides adifferential measurement of the change in transmitted power with and without absorber. Typically,
we average 20 images (at ∼150 ms integration time per image) with sample gas; then switched toreference gas and averaged another 20 images. We found that 20 images was the maximum numberof images between signal and reference before the noise did not continue to average down. Thisseemed to be caused by drifting etalon fringes and by small changes in the VIPA fringe positions.
This sequence was then repeated to average down to the desired sensitivity.
Because we are measuring high-purity arsine gas relative to pure N2 , there was a significant
index-of-refraction difference between the sample gas and reference gas. We could not fully compensate for this difference by changing the pressure, so frep was adjusted slightly for every switch.
To completely remove gas switching times, we initially tried to use two separate cavities for the
sample and reference; however several technical complications made this challenging to implement.

107
In particular, we found that the alignment to the VIPA would always change slightly, which caused
the fringe patterns to move. This could be removed with a single-mode fiber before the spectrometer. In addition, there was a difference between the finesse of the two cavities, as well as a differencein cavity f0 . Finally, any etalon fringes that differ between the two beam paths will lead to residual
fringes in the difference image.

4.1.2

Data Analysis

The two final sets of images, one with sample gas in the cavity and one with nitrogen referencegas, provided a measurement of ∆I/I0 for each detection channel, i.e., the fractional change in
integrated power due to absorption in the cavity. This quantity is equivalent to measuring thefractional change in cavity ringdown time (∆τ /τ0 ) because the integrated intensity of an exponential
decay is proportional to τ and then the ratio removes the proportionality constant.
The quantity ∆τ /τ0 can be expressed as a function of standard absorption-per-centimeter(α) by using∆τ /τ0 =τ (α) − τ0
τ0

2L
2L
τ (α) =
; τ0 =,2−2αL
c(1 − R e)c(1 − R2 )(4.1)

where L is the cavity length (obtained by counting the laser repetition rate, which is matched tothe cavity FSR), and R is the single-mirror reflectivity [19]. This can be solved for α to give"!#11
1 − R2
α=−
ln
1−
,
(4.2)
2L
R2
1 − ∆I
I0
where ∆I/I0 has been substituted for ∆τ /τ0 . This equation was used to evaluate the per-channel
absorption to give the final spectrum.
We process the images using MATLAB in several steps to generate the final spectrum. First,
we use an image flattening algorithm to remove any background between neighboring VIPA fringes
so that the minimum between fringes is zero. This background arises from both camera backgroundand the VIPA fringes not being infinitely far from each other. After this, we track each fringe andaverage over the width of the fringe so that we now have a 2D image that only contains data from108
the fringes (i.e., the vertical axis is position in the fringe and the horizontal axis is fringe number).
We then take the difference between the sample image and reference image and normalize to thereference image. To account for broad absorption or intensity differences between the images, weflatten the baseline to remove very low frequency structure. In addition we use a high-pass filter toremove pixel-to-pixel noise. The result is shown as the absorption image in Figure 4.1(a). Finally,
we identify the region corresponding to 1 FSR on this image and then concatenate each fringe
together to form a 1D spectrum. Sometimes, an additional background was removed from this 1D
spectrum with a low-order polynomial or median subtraction. We checked to make sure that thisprocessing did not change the intensity of the absorption features.
The wavelength axis was calibrated using known line positions from the HITRAN database[271]. We found that it was necessary to calibrate the wavelength axis using a quadratic fit to atleast five know lines. This calibration was necessary every time the grating was rotated to look at
a new spectral range or if the different regions of the camera imaged were used. Most likely the
non-linear correction arose from the non-linear dispersion of the VIPA and the use of an inaccurate
value of the VIPA FSR. We made sure to use calibration lines that were present in the data or thatwere recorded without changing any part of the optical setup to avoid slight changes in the beam
pointing into the VIPA which would change the calibration.

4.1.3
4.1.3.1

Results
Trace gases in N2

To calibrate our CE-DFCS system, we first recorded the spectrum of the impurity gas cylinderplus 2.5 ppm H2 O in nitrogen (Figure 4.2). The measured absorption spectra are plotted below
the axis as inverted peaks in black. For comparison, known spectral lines from HITRAN [271] (forCO2 , H2 O, and CH4 ) and PNNL [48] (for H2S) are plotted above the axis. This overall spectrum
is a composite of 11 individual spectra, spanning over 700 cm−1 (200 nm) spectral bandwidth witha step size of 0.014 cm−1 . The total acquisition time was approximately 10 hours; however, this109
was severely limited by the switching speed of the gas flow system. By using a gas handling systemdesigned for rapid gas switching (large tube diameter, short lines, high flow rates, etc.) switchingtimes could be significantly reduced, resulting in an estimated acquisition time of ∼15 minutes
per individual spectrum (∼2.5 hours for the full bandwidth) to achieve the current sensitivity. In
addition, a carefully designed system using two separate cavities (one for sample gas and one forreference gas) can reduce common-mode amplitude noise and would require no gas switching, thus
improving sensitivity and further reducing the required acquisition time.
The high resolution of the system can be observed clearly in Figures 4.2(a-c), where three
separate regions of the spectrum are expanded, each with a 30-cm−1 span. Figure 4.2(a) shows areference FTIR spectrum (H2 S, in purple, from PNNL), reference line positions for CO2 (green,from HITRAN), and a spectrum with 1.1 GHz (0.035 cm−1 ) resolution obtained with broadband
CE-DFCS (black line). The current resolution is ideal for the 1-2 GHz wide Doppler and pressure
broadened lines observed here. Since 5-10 comb modes sample each absorption feature, lineshape
distortion is not an issue and single comb mode resolution is not required; however, if desired,resolutions of below 250 kHz (8 × 10−6 cm−1 ) are obtainable with modifications to resolve single
comb lines. In addition, Figure 4.2 illustrates that the relative frequency accuracy of the spectrum
is high; we estimate it to be better than 100 MHz (0.0033 cm−1 ) across the full 700 cm−1 spectral
bandwidth by qualitatively comparing measured line positions with HITRAN. Again, by resolving
individual comb lines, absolute frequencies with sub-kHz accuracies are possible. Indeed, CE-DFCS
can be an effective and accurate tool for future additions and modifications to molecular absorption
databases.
Since we are able to observe many absorption lines for each species, the concentration andminimum detectable absorption may be determined with a modification of the Hubaux-Vos regression method [272], which is similar to methods used by the International Union of Pure and Applied
Chemistry [273] and Semiconductor Equipment and Materials International [274], using a single
measured spectrum at a fixed concentration. All of these methods use a calibration curve of measured absorption versus sample concentration to determine the range of noise at zero concentration.

110

Figure 4.2: The spectrum of 10 ppm methane (CH4 ), 10 ppm carbon dioxide (CO2 ), 10 ppm
hydrogen sulfide (H2 S), and 2.5 ppm water (H2 O) in nitrogen gas. Measured absorption peaks
are shown in black below the axis. For comparison, spectral lines for CH4 , CO2 , and H2 O (linestrength divided by 10) available from the HITRAN [271] database as well as H2 S lines from the
PNNL [48] database are plotted above the central axis. Expanded spectral windows a, b, and cillustrate high-resolution absorption spectra of CO2 and H2 S, H2 O and H2 S, and H2 O and CH4 ,respectively.

111
Instead of measuring a single absorption feature as a function of sample concentration, we determine
the measured absorption as a function of predicted absorption for lines with different strengths.
This result allows us to determine both the noise-equivalent absorption and the concentration of
the sample. In other words, one spectrum allows us to collect noise statistics over a wide range ofsignal strengths. To do this, we first predict the spectrum by approximating the concentration andusing available HITRAN data with modifications due to pressure broadening, Doppler broadening,
spectrometer resolution, and digital filters. We then plot each observed peak height versus thepredicted height of the corresponding peak, as shown in Figure 4.3(a) for CO2 and Figure 4.3(b)for H2 O (note that saturation was observed at higher absorptions, these points were not includedin the analysis). The slope of a linear fit to this comparison data gives a correction factor from theapproximated concentration to the true concentration and the uncertainty in the slope provides theuncertainty in the true concentration. In addition, the minimum detectable absorption is given by
the 3σ standard error of the intercept, which specifies the measured absorption value that can beconsidered a non-zero true absorption with 99.86% confidence. This is shown graphically in Figure
4.3 by the value of the upper confidence interval when it intersects the y-axis. Since the confidence
intervals shown are for the fit, it is not expected that all data should lie within them; the standard
error of the fit is much smaller than the standard deviation of each data point due to the large
number of points. The error on the intercept is roughly the same as the standard deviation foreach data point.
For CO2 in the nitrogen gas, we measure a concentration of 9.7(2) ppm (3σ uncertainty inslope of fit), consistent with the factory-specified impurity level in the cylinder at 10 ppm ± 10%.
In addition, we obtain a minimum detectable absorption (3σ) of 4.5 × 10−9 cm−1 , corresponding
to 325 ppb minimum detectable concentration by using the line strength of the strongest measured
line. This data was recorded with a total integration time (T) of 180 s (time includes reference andsample images, but not gas switching times), which gives a 1-Hz minimum detectable absorptionof 4.2 × 10−8 cm−1 Hz−1/2 . Similarly for H2 O, we measure 2.50(12) ppm, with an absorption sensitivity of 5.5 × 10−9 cm−1 or a minimum detectable concentration of 7 ppb. Since this absorption

112

Figure 4.3: Double logarithmic plots of measured peak absorption versus predicted peak absorption
from the HITRAN database for 10 ppm CO2 in N2 (a), 2.5 ppm H2 O in N2 (b), and 1 ppm H2 O
in arsine (c). Linear fits (dashed) of the data sets were performed up to a measured absorption of
2 × 10−7 , where signal saturation becomes noticeable. The corresponding 3σ (99.87%) confidence
intervals (solid lines) are plotted along with the linear fits. Note that these confidence intervals
are for the fits to the entire data. The 3σ error on the intercept gives the minimum detectable
absorption, the slope of the fit gives a correction factor for the concentration, and the error on theslope gives the uncertainty in the concentration.

113
sensitivity is uniform across our measured spectral range, we project a minimum detectable concentration of 700 ppb for CH4 and 370 ppb for H2 S. We also note the advantage of a dramaticallyincreased dynamic range (larger than the single concentration data in Figure 4.3) obtained fromsimultaneously studying lines with significantly different strengths. Even though strong absorption
peaks (above 2 × 10−7 cm−1 ) provide signals that are saturated at the measured concentration,
at lower concentrations they will be in a linear regime while the weak lines will be too small tomeasure. Additionally, lines that are too weak to measure at the current concentration will be
detectable at higher concentrations. By using the range of water line strengths given in HITRAN
for this wavelength region, we estimate a measurement range for water concentration from 7 ppb
to 100 ppm with this system. The minimum detectable concentrations quoted here are for single
lines, fitting many lines simultaneously would give even lower limits as discussed in Section 4.2.3.

4.1.3.2

Impurities in arsine

After calibrating the system, we filled the high finesse optical cavity with arsine interspersedwith impurities. The same mixture measured in nitrogen was added to a purified arsine gas in aratio of 1:8. Figure 4.4 shows the composite spectrum of trace water recorded in the background ofarsine gas at ∼160 torr. Again, measured absorption features are plotted as inverted peaks below
the axis, with available HITRAN spectral lines on water plotted above the axis. It is apparent thatthe absorption of arsine continues to increase rapidly towards both ends of the spectral window;
past the plotted range, the arsine absorption begins to decrease at ∼1.94 µm, but even at 1.97
µm it was still too strong to collect reliable spectra of trace species. The density of the arsine
absorption obscured the other impurity lines, making it only possible to observe water in thisexperiment. Because of the strongly absorbing background gas, it is clearly a major advantage
to have a spectroscopy system based on CE-DFCS providing broad bandwidth and high spectral
resolution. Such a system allows us to easily identify and quantify water lines that are isolated
from arsine absorption features, as seen by the two zoomed-in bottom panels of Figure 4.4. From
this data (see Figure 4.3(c)) we determine a water concentration of 1.27(8) ppm and an absorption

114
sensitivity of 2.4 × 10−8 cm−1 with an integration time of 600 s, which corresponds to a minimum
detectable concentration for water in arsine of 31 ppb. Arsine is known to have a weak band
centered at 5150 cm−1 , which is probably causing the absorption on the red side of the spectrum.
This first demonstration of CE-DFCS for impurity monitoring of industrial process gases has
shown an absorption detection sensitivity (3σ) of 3.9×10−7 cm−1 Hz−1/2 for water in arsine around1.85 µm (5400 cm−1 ). The 200 nm bandwidth of our system also allowed us to look at absorption
regions of other impurities (CH4 , CO2 , and H2 S), which were obscured by densely spaced arsine
absorption lines. Furthermore, there is significant room for improvement, which would open open
the door for continued industrial applications, as discussed in 4.3. One direction for improvement
could be by extending the spectral coverage to the mid-infrared, which we cover next.

4.2

Mid-infrared comb spectroscopy
In this section we discuss our initial demonstrations of frequency-comb FTS based on a highpower OPO that operates in the important mid-infrared window from 2100 cm−1 to 3700 cm−1 .
Our system is capable of achieving part-per-billion (ppb)-level sensitivity with an integration timeof 30 s or less for a variety of important molecules, such as the greenhouse gases methane, carbon
dioxide, and nitrous oxide [275, 276, 277], as well as isoprene and formaldehyde, which are important contributors to pollution [278, 279]. Furthermore, we demonstrate detection of ethane andmethanol, which are subjects of interest in human breath analysis [280, 281, 282]. These molecules
represent just a small set of trace gasses within the detection range of this new system.
The OPO comb source was discussed in detail in Section 2.4. Idler light from the OPO is
coupled into a Herriott multi-pass cell (New Focus 5611) with 0.3 l volume and a total path lengthof 36.4 m (182 passes), which contains the sample or reference gas. The spatial coherence of thelaser source makes the coupling simple and efficient. The multi-pass cell provides a modest pathlength enhancement and works over a very broad spectral bandwidth; for higher sensitivity, anenhancement cavity could be used instead, as demonstrated in [55]. We record the spectrum after
the multi-pass cell with the scanning Fourier-transform interferometer discussed in Section 3.3.2. To

115

Figure 4.4: Trace water in arsine. The measured spectrum of 1.27 ppm of water vapor in arsine
gas is plotted in black below the axis and the HITRAN reference lines for water are shown in blue
above the axis. The arsine absorption continues to increase both above 5650 cm−1 and below 5350
cm−1 , precluding any additional measurements. The two insets show easily resolvable water lines
even in a background absorbing gas.

116
minimize drifts of the spectrum during the measurement, the OPO comb is phase-stabilized to thepump laser, whose repetition rate is locked to a microwave reference for long-term stability. Usinga phase-locked OPO comb also ensures low intensity noise compared to non-comb supercontinuum
sources.
To extract the absorption spectrum, we measure first the laser spectrum through the sample
gas Is (ν), then purge the cell, fill it back to the original pressure with N2 , and measure the reference
0 (ν)]
spectrum I0 (ν). The spectral absorbance α(ν) is then determined via α(ν) = − ln[Is (ν)/I, whereL

L is the optical path length inside the multi-pass cell. The gas concentrations are extracted fromthe data via a numerical fit of a model spectrum calculated with spectral data (line positions,line strengths, pressure broadening and shift coefficients) from HITRAN [271] or reference spectrafrom the PNNL NWIR database [48] (when HITRAN data was not available) based on a modified
Marquardt algorithm [283]. The parameters for the fitted Voigt profile are a concentration value foreach sample species, a polynomial baseline, and a frequency correction (to account for misalignmentof the beams inside the interferometer, typically on the order of 10−3 cm−1 ). The uncertainty value
for the concentration is extracted from the covariance matrix.

4.2.1

Measurement of individual molecular species

To demonstrate the versatility and wide spectral coverage of our spectrometer we recorded
spectra of a variety of molecular species (each diluted in pure N2 ) in different spectral regions
spanning from 2170 cm−1 to 3520 cm−1 . Figure 4.5 shows measured spectra (black) of nitrous
oxide (9.0 ppm in Figure 4.5(a) and 142 ppm in Figure 4.5(f)), 58 ppm formaldehyde (Figure
4.5(b)), 11 ppm ethane (Figure 4.5(c)), 10 ppm methane (Figure 4.5(d)), and 16 ppm isoprene(Figure 4.5(e)). With the exception of the high-resolution spectrum in Figure 4.5(d), all of these
spectra were recorded with integration times of 30 s or less. Each plot also shows the fitted
spectrum using data from HITRAN or PNNL, as indicated in each graph (plotted in purple andnegative for clarity). These measurements show the instruments ability to rapidly cover a wide
spectral region with high resolution and extract concentrations regardless of the structure of the117
absorption spectrum. For example, isoprene exhibits a continuous absorption band, which makes
quantitative measurements susceptible to errors from baseline drifts. Being able to determine
the concentration of such a sample is extremely valuable since molecules with continuous bands(e.g., isoprene, methanol, ethanol, acetone, etc.) are highly important for many applications. In
additional, removing broad background due to scattering is important for measurements in theatmosphere.

4.2.2

Instrument performance limits

We have made systematic characterizations of the limits of our system in terms of resolution,measurement accuracy, absorption sensitivity, and concentration detection limit. From measurements of the noise floor we obtain an absorption sensitivity of 3.8 × 10−8 cm−1 Hz−1/2 per spectral
element. At a center frequency of 3000 cm−1 , we detect ∼45,000 simultaneous channels at maximum resolution. Using the line strength of the molecular transitions as well as the sensitivity andbandwidth of our instrument, we determine the noise equivalent concentration (NEC) at typicalmeasurement settings of 0.014 cm−1 unapodized resolution, 600 Torr of N2 background pressure,and 30 s total integration time (5 averaged scans for reference and sample). The NEC for a smallcollection of different molecular species within our spectral range is summarized in Table 4.1 (column 3), which shows that ppb-level detection limits are obtained for most molecules in only 30 s.
We further confirmed that the noise averages down for at least 6 min, which reduces the NEC by√
12.
To test the repeatability of our instrument, we take ten independent measurements of amethane mixture certified as 10 ppm (± 5%) in N2 at a pressure of 600 Torr, 0.014 cm−1 resolution,and 30 s integration time and extract the concentration of each dataset from a numerical fit of a
HITRAN reference spectrum. We determine the average CH4 concentration to 10.107(19) ppm,
which is in agreement with the specified concentration of the gas mixture and demonstrates thatthe precision of the entire system (optical and gas handling) is better than 0.2%.

118

-3

-3

-1

1 N2O, 9 ppm, 100 torr, 0.014 cm-1

Absorbance (cm )

x 10

a

0.5
0

x 10
1

-1

-1 HITRAN database
2170

2200

2230

2260

Absorbance (cm-1)

2800

2900

3000

-4

5 C H , 11 ppm, 600 torr, 0.096 cm-1
2 6

c

0

x 10
2

CH , 10 ppm, 100 torr, 0.0056 cm-1

d

4

0
-2

NWIR database

-5
2900

2950

3000

3050

3100

C5H8, 16 ppm, 600 torr, 0.06 cm

HITRAN database

2900

-5

x 10
Absorbance (cm-1)b

NWIR database

2700

-4

x 10

2950

3000

3050

3100

3150

-1

f

-4

-1

e

0
-5

-1

0

-0.5

5

H2CO, 58 ppm, 600 torr, 0.014 cm

x 10
5

N2O, 142 ppm, 600 torr, 0.026 cm

0-5

NWIR database2900

2950 3000 3050 3100
Wavenumber (cm-1)

3150

NWIR database

3300

3350
3400
3450
Wavenumber (cm-1)

3500

Figure 4.5: Collection of absorption spectra throughout the operating range of our mid-IR frequency
comb Fourier transform spectrometer (black) and comparison to HITRAN or PNNL-based fits
(purple, plotted negative for clarity). The trace concentration, nitrogen pressure, and spectral
resolution are designated in the panels; (a) ν3 band of nitrous oxide (N2 O) at a concentration of
9 ppm, a pressure of 100 Torr N2 , and a resolution of 0.014 cm−1 ; (b) formaldehyde (H2 CO), 58
ppm, 600 Torr, 0.014 cm−1 ; (c) ethane (C2 H6 ), 11 ppm, 600 Torr, 0.096 cm−1 ; (d) methane (CH4 ),10 ppm, 100 Torr, 0.0056 cm−1 ; (e) isoprene (C5 H8 ), 16 ppm, 600 Torr, 0.058 cm−1 ; (f) 2ν2 +ν3and ν1 +ν3 band of N2 O, 142 ppm, 600 Torr, 0.026 cm−1 . All spectra are obtained with a single
wavelength setting of the OPO except (a), which is a combination of two measurements at different
central wavelengths.

119

Table 4.1: Detection limits for a collection of important molecules within
the spectral range of our mid-IR FC-FTS. The third column shows thenoise equivalent concentration (NEC) results, which are based on the measured absorption sensitivity of 3.8 × 10−8 cm−1 Hz−1/2 per spectral element under typical experimental conditions of 600 Torr N2 pressure, 0.014
cm−1 unapodized spectral resolution, and 30 s of total integration time (5averages). The fourth column shows the theoretically estimated multi-line
detection limits using the same noise and resolution conditions as thosefor column three. The fifth column displays the experimentally obtained
concentration detection limits from the multi-line fits to measured spectra. The disagreement between the theoretical and experimental values isdue to the fact that some spectra were not measured at the best possible
noise performance of the system. Note that significantly lower detection
limits are achievable with longer integration times (tested to at least 6
min).

Molecule

Theor. multi-line

Expt. multi-line

Band Center

NECdetection limit

detection limit(cm−1 )

(ppb)

(ppb)

(ppb)

CH4

3020

22

2.0

5

C 2 H6

2990

23

1.8

18

C 5 H8

3000

370

4.3

7

H2 CO

2780

54

2.3

40

CH3 OH

2950

350

3.6

40

N2 O

2220

8

0.5

5

CO2

2350

2.2

0.2

0.5

120
4.2.3

Multi-line fitting advantage

An important and already recognized advantage of broadband spectroscopy is that the abilityof detecting multiple absorption peaks over a wide region allows one to achieve a lower detectableconcentration than with single-peak measurements [284, 285, 286]. In general, the multi-line advantage scales with the integrated area under the absorption curve. To understand this, we derive
an equation that allows the estimation of the concentration detection limit obtained with multiline analysis via the known structure of a molecules absorption spectrum. We assume that α(ν)
is the absorption spectrum measured at k discrete frequencies νi with a standard deviation σα ,which is uniform over the spectrum. The model spectrum αM (ν) is calculated for a given sample
concentration C at the experimental pressure and temperature. For simplicity we assume that
there is no baseline in our spectrum and that the model spectrum has exactly the same frequency
axis as the measured spectrum. We would like to find the parameter N (which will be used toestimate the concentration), for which differences between measured and model spectra multipliedby this parameter will be minimized. To do this, we fit the value of parameter N using a linear
least-squares method to minimize the function
F (N ) =kX
i=1[α(νi ) − N αM (νi )]2

(4.3)

so
k

X
dF
2[α(νi ) − N αM (νi )][−αM (νi )] = 0.
=
dN

(4.4)

i=1

Solving this equation gives
PkN=

i=1 α(νi )αM (νi ).
Pk2
i=1 αM (νi )Neglecting error in the model spectra, we can write
"
#
2
k 
k
X
X
∂N
α
(ν
)
σ2
M i
2
σN
=
σα2 =
σα2 = Pk α.
Pk
22∂α(νi )
i=1 αM (νi )i=1 αM (νi )
i=1

(4.5)

(4.6)

i=1

This shows that the multi-line analysis improvement of the detection limit depends on the shapeof the spectrum and scales with its integrated absorption.

121This finding becomes apparent when comparing spectra of methane (Figure 4.5(d)) andisoprene (Figure 4.5(e)), which exhibit clearly different structure. The numbers from Table 4.1
show that the multi-line analysis improves the detection limit by factors of 11 for CH4 and 93 forC5 H8 owing to the much larger integrated area of the continuous isoprene absorption spectrum. To
confirm this finding, we take the uncertainty of the fitted concentrations of the ten measurementsof CH4 , which are extracted by utilizing the entire recorded spectrum, and compare it against theuncertainty when using only one of the strong peaks of the P-branch. The single-peak fits give an
uncertainty of 20-30 ppb, which is consistent with the NEC. In contrast, the multi-line fit yields
an uncertainty of only 7 ppb, which is significantly lower than the NEC and confirms that analysisof the entire absorption spectrum can indeed decrease the detection threshold below the classicallimit given by the single-line S/N ratio. The detection limits obtained from the multi-line fitting
procedure for other measured molecules are summarized in the fifth column of Table 4.1. The fourthcolumn in Table 4.1 shows the theoretical estimates for detection limits under the same conditionsof noise and resolution used for the third column. The scaling of the detection limit indicatedby the theory corresponds well with the experimental numbers: for example, the improvement in
detection limit for C5 H8 is 12 times as high as the one for CH4 due to the much larger integrated
absorption of the continuous spectrum of isoprene. The experimental results shown in the fifth
column of the table differ from the theoretical estimates due to the fact that most spectra were notrecorded during the best possible noise performance of the system, which depended on day-to-dayalignment and was probably limited by etalons or laser noise.

4.2.4

Determination of absolute concentrations of a gas mixture

Most real-world samples measured for atmospheric science, breath analysis, or trace gas monitoring contain more than one species in a particular spectral window. The overlapping absorption
features of different molecules make their unambiguous detection nearly impossible if sufficient
spectral coverage and resolution are not available. The C-H-stretch region around 3000 cm−1 , forinstance, contains absorption features of practically any hydrocarbon with sufficient abundance. To

122demonstrate the capability of our spectrometer, we measure the absorption spectrum of a mixtureof trace amounts of formaldehyde, methanol, and water in nitrogen and determine their absoluteconcentrations. The spectrum was recorded at a resolution of 0.014 cm−1 and an N2 pressure of
600 Torr under a total integration time of 30 s. Here, methanol – as a molecule without a clear
line spectrum – would represent a significant challenge for a cw laser measurement. Figure 4.6(a)shows the measured spectrum of the H2 CO/CH3 OH/H2 O-mixture (black curve) in the spectral
range from 2900-3050 cm−1 and the overall fitted result (gray, plotted negative for clarity). Model
spectra created from HITRAN (for H2 CO and H2 O) and PNNL (for CH3 OH) of the individualcomponents at their fitted concentrations are shown in magenta (H2 CO), green (CH3 OH), andblue (H2 O). The overall fit is remarkably accurate, as demonstrated by the fit residuals plottedin Figure 4.6(b). Although the spectrum contains many overlapping features and some of them(from CH3 OH) do not exhibit sharp lines, the broad bandwidth and high spectral resolution of our
FC-FTS in combination with the multi-line fitting procedure enable precise determination of theconcentrations of all three components. The obtained values are 47.1(1) ppm for H2 CO, 58.30(4)ppm for CH3 OH, and 813(7) ppm for H2 O.
Another important feature of the current system is the large dynamic range in concentration
detections. To demonstrate this, we measured a sample of laboratory air, which contains atmospheric methane; however, the high abundance of water vapor, whose absorption lines strongly
overlap with the CH4 band at 3020 cm−1 , imposes a particular challenge for trace gas detection.
Although the strongest H2 O lines are already close to saturation, the fitting algorithm is able toextract both concentrations, 2.02(1) ppm for CH4 and 1.03(5)% for H2 O.

4.3

Future applications
In the previous two sections, we highlighted some of the capabilities of frequency-comb spec-

troscopy systems for trace gas detection. In particular, we saw in the measurement of contaminantsin arsine gas how CE-DFCS using a VIPA spectrometer provides high-sensitivity measurements(around 10−8 cm−1 Hz−1/2 ) with a simultaneous bandwidth of 50 cm−1 . This one comb source

123

sum of fitstted H2CO

tted CH3OH

tted H2O

Absorbance (cm-1)

x 10-4
4 (a)
3
2
1
0
-1
-2
-3
-4
0.5 (b)
0
-0.5
2900

measurement

residuals

2950
3000
Wavenumber (cm-1)

3050

Figure 4.6: Multispecies simultaneous detection. (a) Measured spectrum of a mixture of formaldehyde, methanol, and water at an N2 pressure of 600 Torr and a resolution of 0.014 cm−1 (black),
as well as the fitted spectra based on HITRAN and PNNL of the three components (H2 CO, magenta; CH3 OH, green; H2 O, blue) and their sum (gray). The fit gives abundances of 47.1(1) ppm
H2 CO, 58.30(4) ppm CH3 OH, and 813(7) ppm H2 O. (b) The residual of the overall fit displays no
remaining structure.

124
can provide spectral coverage across the entire 1-2 µm spectral region. We also saw how the highresolution enables precise measurements of trace species even in the presence of large absorption
from other species. The OPO-based spectroscopy measurements demonstrate the application ofcomb spectroscopy to the mid-IR, which provides access to stronger absorption features for manydifferent species. We showed the ability to achieve low-ppb detection limits for many molecules in
30 s total integration time (including reference and sample spectrum) at an unapodized resolutionof 0.014 cm−1 . The significant decrease in acquisition time compared to standard FTIR is a direct
result of the high spectral brightness and spatial coherence of the frequency comb source. While
the absorption sensitivity of this system is lower than a cavity-enhanced system, the concentration detection limits are still very low due to the strong molecular absorptions and the system
can operate anywhere from around 3-5 µm (2100-3700 cm−1 ). In addition, we saw how the broad
bandwidth provides better detection limits (or correspondingly higher precision on the measured
concentration) as well as the ability to measure mixtures, even when the species in the mixture
has continuous absorption features. We now consider some prospective applications of comb spectroscopy and what might be required for these applications. In particular, we consider potential
applications to gas purity analysis, breath analysis, atmospheric chemistry, and chemical physics.
The arsine gas purity experiment is one example of the application of comb spectroscopy toprocess gas analysis. In addition to arsine gas, there are over 30 other gas used in the productionof semiconductor devices [266]. Depending on what the gas is used for, there are a wide rangeof impurities, including water, CH4 , CO2 , CO, and hydrocarbons, that can all have a negative
effect on the product. Thus, there is interest in systems capable of detecting multiple contaminantssimultaneously in the presence of different process gases. While the first test in arsine was only
able to detect one impurity, we showed how this spectral region could detect multiple impurities
depending on the gas. Several potential improvements could fully demonstrate the capability of
CE-DFCS for process gas measurements. Of course, the gas flow system could be improved toenable rapid switching between sample and reference. In addition, a different readout system thatallows reduction of FM-AM noise, such as FC-FTS, could improve the sensitivity (see Table 3.1)

125
and could enable the use of multiple mirror sets for a wider wavelength coverage. By covering
several regions throughout the 1.1-2.2 µm spectral range, it should be possible to detect multiple
contaminants. For example, CO2 has strong bands from 4800-5000 cm−1 that might be detectablesince we think the arsine absorption that we saw starting at 5350 cm−1 is probably due to a weakarsine band centered at 5150 cm−1 . Other regions in the near-IR might provide other transparency
windows as well, although data on weak arsine bands in the near-IR is limited, so it would be
beneficial to scan the arsine absorption first at lower sensitivity. This system could potentially be
combined with a single cavity using Brewster angle prisms [171] to cover the full range of the source
using a single cavity. Several potential transparency windows also exist in the mid-infrared around3800, 2500, and 1500 cm−1 .
A similar application is monitoring of emissions from industrial applications. For example,
greenhouse gases are regulated by the EPA and typically must be monitored. Fluorinated compounds, such as fluorocarbons, NF3 , and SF6 , are major greenhouse gases and are also significant
emissions from semiconductor manufacturing [287]. Fluorinated compounds also play a significant
role in the removal of ozone at high elevation. A system capable of simultaneously measuringmultiple greenhouse gases and fluorinated compounds would potentially have enormous industrial
applications. All of these compounds absorb in the 800-1500 cm−1 region, which will be within therange of an OPO using OP-GaAs.
The analysis of trace species in breath for disease detection is currently an extremely active
field of research [288, 289, 290, 291]. Besides water, oxygen, nitrogen, and carbon dioxide, breath
contains hundreds of other compounds in varying quantities. Table 4.2 lists the more abundant
ones and typical ranges in human breath. Currently, there are two primary directions for breath
research: detection of a single or a few small species and detection of many volatile organic compounds (VOCs) simultaneously. Several FDA approved breath tests already exist based on thefirst research direction such as measurement of NO for asthma, CO for liver disease (specifically,
neonatal jaundice), and methane for various gastrointestinal issues [292]. There are preliminary indications that other species may be useful for testing for diseases or monitoring treatment, although

126
the evidence is not conclusive: the combination of NO, CO, CO2 , and N2 O has been investigatedfor chronic obstructive pulmonary disease (COPD) [293]; acetone has been shown to have somecorrelation with blood glucose and is possibly an indicator for diabetes [294, 295, 296]; ammonia(and possibly methylamines) show correlation with kidney function [297, 298]. In addition to thesemolecules, breath contains several hundred other VOCs present typically at the 0.1-10 ppb range[299, 300, 301]. Unlike most of the smaller molecules, the physiological origin of many of these is
not known, which makes direct links to certain diseases more challenging. It is believed that at least
some of the VOCs might be caused by oxidative stress – that is, reactions with oxygen-containing
radicals such as OH or O−
2 – which can lead be indicative of various diseases [302, 303, 304]. Because of the unknown origin of most VOCs, studies typically measure many compounds and lookfor correlations with diseases [305] such as tuberculosis [306], cancer [307, 308, 309, 310, 311], and
COPD [312]. Several smaller molecules and VOCs such as CO, H2 O2 , and ethane may also be
indicative of oxidative stress [313, 314, 315].
While many of these studies have determined some correlation with a disease – for example,> 80% sensitivity and specificity have been demonstrated for cancer detection [307] – detecting
molecules and identifying a potential correlation are not the primary challenges in breath analysis.
In fact, these steps are so easy that they can even be accomplished by dogs [317]! Detectionof VOCs is typically performed with gas chromatographymass spectrometry (GC-MS), which has
the disadvantage of not being real-time; however, newer techniques such as proton-transfer reaction
mass spectrometry (PTR-MS) can selectively detect most VOCs with low-ppt sensitivity and < 100
ms time resolution [318, 319, 320]. These systems are currently commercially available and aremoderately portable. The challenges of breath analysis instead are related to the large naturalvariability present in human breath (as can be seen by the ranges quoted in Table 4.2), confounding
variables such as smoking and environmental variations, as well as issues related to breath collection[289, 321, 309, 291]. Looking through many of the references given here, this becomes immediately
apparent as many of the studies show results where the mean value of the measured compound is
statistically different between the test group and a control group, but the measured range overlaps

127

Table 4.2: Typical range for themost common trace species in human breath. From [293, 316, 301,
292]
CompoundConcentration

Methane1-100 ppm

CO1-3 ppm

Acetone0.3-3 ppm

N2 O

300-800 ppb

Ammonia0.2-1 ppm

Methanol∼500 ppb

Isoprene

10-300 ppb

NO

1-100 ppb

2,3-Butanedione

1-100 ppb

Ethane2-20 ppb

Other VOCs

0.1-10 ppb

128
significantly between the two. Because of this, much of the focus is starting to be more towards
full clinical studies and less towards small proof-of-principle studies.
So what can frequency comb spectroscopy contribute to breath analysis? In principle, thesimultaneous detection of a wide range of VOCs simultaneously would be possible, but this requiresvery accurate fitting of overlapping, broadband absorption profiles. This is complicated withouta priori knowledge of all species to include and due to dependences of the absorption profiles
with temperature and pressure and various baseline etalon effects. Given the capabilities of PTRMS, this is probably not the best approach. One potentially beneficial application is to insteaddemonstrate the ability to detect all of the other compound listed in Table 4.2 simultaneously, orat least in one system, with moderate time resolution. Since the required detection limit would
be mostly above 10 ppb, this could be potentially be accomplished with a multipass cell. Such asystem would enable multiple breath tests in one system, which would be beneficial for correlating
different effects and as a diagnostic tool.
Another field that relies significantly on trace gas analysis is atmospheric chemistry. There
are three primary directions for experimental atmospheric chemistry: determination of sources ofpollution and tracking pollution through the atmosphere; studying reactivity in the atmosphere;
studying reactivity in controlled laboratory settings. Some of the different species studied in fieldmeasurements are given in Table 4.3. Many studies focus on understanding several primary atmospheric reaction cycles [199, 322]. The HOx cycle (OH, HO2 , etc.) is responsible for the formationof ozone as well as the oxidation of many VOCs and is initiated by the photolysis or reaction ofozone. This is interconnected with the NOx cycle, which can compete in the oxidation of VOCs
[199, 323]. The products or intermediates of the VOC oxidation reactions can further react, resulting in radical recycling and also in the formation of aerosols. Oxygenated halogens (e.g., IO, ClO,
and BrO) are also important because they may react with VOCs at low elevation in the atmosphereand also cause tropospheric ozone destruction. In addition, because many of the reactions in thesecycles are light-dependent, the dominant cycles and reactions are not only spatially dependent but
are also day-night dependent. In order to understand the global tropospheric chemistry, it is thus

129
important to measure multiple different species from each cycle in different regions and both during
the day and night. Despite considerable advances in understanding atmospheric chemistry, there
still are many surprises [324, 325, 326, 327] and reactions that are not fully understood [328, 329].
In addition to measurements to understand the chemistry of reactive species, various sources of pollution must be measured so that they can be included into models. These include the halogenatedcarbons mentioned earlier related to semiconductor processing, various greenhouse gases (such as
CH4 , CO2 , SF6 , etc.), as well as VOCs, NOx, and SO2 from industrial sources [279, 330, 331] andbiogenic sources.
Currently, most systems for point detection of trace species rely on incoherent broad-band
cavity enhanced absorption spectroscopy (IBB-CEAS), LED-based cavity-enhanced differential optical absorption spectroscopy (LED-CE-DOAS) – both of which are broad bandwidth – or cw-laser
spectroscopy. The advantage of comb spectroscopy for atmospheric measurements can be seen fromthe recent demonstrations of N2 O, IO, BrO, and formaldehyde detection for measurements off thecoast of France and Antarctica [185, 184]. If we compare the sensitivity to IBB-CEAS [334, 335] or
LED-CE-DOAS [336] systems operating in the same spectral region, the comb system provides one
to two orders of magnitude improvement in the sensitivity. In addition to lower detection limits, theimproved sensitivity is also critical for high-speed measurements. This capability is important not
only for measurements in moving vehicles (e.g., airplanes, cars, ships) where the time resolution isnecessary for spatial resolution [330] but also for Eddy covariance flux measurements, which provide
information about surface-atmosphere gas exchange [328, 337]. For atmospheric measurements, itis necessary to deal with background losses due – for example – to aerosols [336], which would cause
problems for cw-laser systems; however, the bandwidth of the comb enables continuous spectralfeatures to be fit as well and so should mitigate this issue.
Laboratory experiments of reactions seek to provide information about the rates of eachreaction [338, 339].1
1

For many kinetics studies, it is also desirable to study multiple species

A compilation of currently accepted reaction rates is available at http://www.iupac-kinetic.ch.cam.ac.uk/

130

Table 4.3: Example trace species in theboundary layer (lowest part of the troposphere). If ranges are given, they correspondto the range from remote to polluted environments. If single values are given they correspond to the average value from measurements. From [332, 199, 333, 331].
CompoundConcentration

CH4
N2 O
NO2
NO3
N2 O5
NH3
Nitric Acid
SO2
H2 S
Isoprene
BrO
IO
CFCl3
CF2 Cl3
CH3 ClSF6Other halocarbon species
CO
HCHO
Glyoxal (CHOCHO)Acetone
Alkanes
Aromatics
Alkenes/Alkynes
Terpenes (e.g. α-pinene)

1.7 ppm
300 ppb
0.01-2000 ppb
1-500 ppb
< 15 ppb
0.02-100 ppb
0.1-50 ppb
0.01-10 ppb
0-800 ppt
0.6-2.5 ppb
0.1-10 ppt
0.1-2 ppt
268 ppt
533 ppt
500 ppt
4 ppt
10-100 ppt
200 ppb
0.1-60 ppb
0.01-1 ppb
0.2-9 ppb
1-300 ppb
0.01-3 ppb
0.01-10 ppb
0.03-2 ppb

131
simultaneously in order to obtain information about branching ratios and rate constants between
multiple reaction paths and to identify unexpected paths [323]. It is also ideal to have high time
resolution to potentially identify intermediates that may contribute to further reactions. Finally,
reactions need to be studied over a range of atmospherically relevant temperatures and pressures[323, 329]. A technique capable of rapid, sensitive detection of many species would thus havewide-ranging applications. We recently demonstrated time-resolved frequency-comb spectroscopy
based on a VIPA spectrometer in the mid-IR combined with a reaction flow cell [329] that enabledmeasurements of the photoproducts of acrylic acid photolysis with a time resolution of 25 µs [56].
In particular, we measured the deuterated-HOCO radical, which is an intermediate in the OH +CO reaction [340]. We also measured several unexpected products that we are still investigating.
With increased sensitivity, perhaps by pushing further into the mid-IR [341], this system could
enable direct measurement of the Criegee intermediate in the ozonolysis of alkenes [342, 343].
Ozonolysis of alkenes is an important reaction in atmospheric chemistry and the stabilization of
Criegee intermediates may help to understand the unexplained OH radical concentrations in theatmosphere [344, 345, 346, 347, 348]. Even without high time resolution, frequency comb systems
combined with atmospheric chambers could provide a new method of sensitively measuring stableproduct branching ratios [349, 350].
A final application of comb spectroscopy is the measurement of stable isotope ratios, which
is useful for breath analysis as well as for atmospheric and geochemistry. For breath analysis, thekinetics of various processes such as metabolism and liver function can be tested by ingesting 13 Clabeled substances and monitoring the appearance of, for example, 13 CO2 [291, 351, 352]. Isotope
ratios in CO2 and CH4 can also provide information about pollution sources and sinks [353, 354],
the history of the atmosphere from ice core samples [355], and the history of other planetaryatmospheres [356]. CE-DFCS could in principle provide a significant improvement in the precision of
isotope ratio measurements compared to other portable techniques (mass spectrometry techniques
are the most precise, but are not portable) because many lines can be measured simultaneously. A
proof-of-principle measurement showed the determination of isotope ratios to 4 parts-per-thousand132
in breath [44]. Improvements could potentially be made by measuring in different bands – forexample, the 2 µm or 4.3 µm bands of CO2 – to measure less abundant isotopes such as 18 O or
17 O. To fully demonstrate the capabilities of CE-DFCS for these measurements, more studies ofthe precision and accuracy – ideally down to the sub-part-per-thousand level [357] – are needed.

4.4

Conclusions
In this chapter, we highlighted some of the real-world applications of comb spectroscopy forsensitive detection of multiple trace species simultaneously. We showed how the bandwidth enables
detection of trace impurities in a large absorbing background and measurement of species withouthighly resolved absorption features, both of which are challenging for cw-laser techniques. We thendiscussed some future applications of comb spectroscopy including extensions of the semiconductor
gas purity analysis, breath analysis, atmospheric measurements, and isotope ratio measurements.
In particular, we showed where the broad bandwidth and high sensitivity would provide advantages
such as multi-species detection capabilities, rapid measurements, and high precision.
We are also currently developing a cryogenic buffer-gas cooled CE-DFCS system to study
the spectra of molecules at 4 K. This will provide significantly narrower absorption lines andwill concentrate the absorption strength into a much smaller number of lines, which could be
advantageous for trace gas detection and mixture analysis. In addition, such a system has excitingprospects for studying high-resolution spectra of complicated molecules.

Chapter 5
Velocity-modulation spectroscopy of HfF+

In this chapter, we discuss the survey spectroscopy that we performed on HfF+ using both
frequency comb and cw-laser velocity modulation spectroscopy as detailed in Section 3.4. Trapped
HfF+ is used in the measurement of the electron electric dipole moment (eEDM) discussed in
Chapter 6, but for the experiment to be successful we first needed to know more about the electronic
structure of HfF+ . In particular, we wanted to learn several things from broad-bandwidth velocity
modulation spectroscopy for application to the eEDM experiment:(1) precise spectroscopic constants for the ground 1 Σ+ state and metastable 3 ∆1 state, which
will be used for the eEDM measurement;(2) measurement of the parity doubling (Λ-doubling) in 3 ∆1 J = 1;
(3) transitions from both states that may be useful for laser-induced fluorescence and state
manipulation;(4) an intermediate state for a Raman transition from the 1 Σ+ to the 3 ∆1 .
Unfortunately, the early theory and experimental results were no where near accurate enoughfor application to the trapped ion experiments and only were able to provide a rough guide tointerpreting the HfF+ spectra. Here, we provide an introduction to diatomic molecular spectroscopy
and then apply this to detailed analysis of the data, aimed at the goals above.

134

5.1

Results
Figure 5.1(a) shows the spectrum of HfF+ acquired using the comb-vms system in blue

(offset for clarity) as well as all data obtained with single-frequency vms in red. As illustrated
in the zoomed region shown in the lower panel of Figure 5.1(b), the spectrum from about 1210012300 cm−1 is extremely congested due to the presence of many bands, each with five isotopes, andthe high temperature (500 ◦ C) of our oven, which results in observed J 00 values up to about 70. The
dynamic range of frequency-comb velocity-modulation spectroscopy is demonstrated by our ability
to identify the overlapping 1 Π1 ← 1 Σ+ (v 0 = 0, v 00 = 1) and 3 Π1 ← 1 Σ+ (3,1) bands despite thedifference in linestrengths and an offset in band origin of only ∼1 cm−1 . Since 150 cm−1 sections
are acquired simultaneously when using the comb, relative linestrengths within the region are notinfluenced by variability in oven and discharge conditions, which significantly helps to disentangle
bands. The upper panel of Figure 5.1(b) shows predicted line positions from fits to three different
bands (the fitting is discussed below), which illustrates our ability to resolve each isotope (−◦, 180 Hf;−×, 179 Hf; −∗, 178 Hf; −+, 177 Hf; −·, 176 Hf) for multiple bands. Figure 5.1(b) also demonstrates
excellent overlap between the comb and cw spectra.

5.1.1

Diatomic molecular spectra primer

Before we delve deeper into the analysis of the data, we should take a detour and cover somebasics of the spectra of diatomic molecules. More details can be found in numerous books [358, 359,
360, 361]. As we have already seen, molecular spectra can be extremely complicated. Luckily, thespectra of diatomic molecules are frequently not too complicated, but they are significantly morethan twice as complicated as atomic spectra. In addition to the usual electronic states, molecules
can also vibrate and rotate. So we must consider transitions between combinations of electronic,
vibrational, and rotational states. Throughout this chapter, the upper state in a transition isdesignated with a single prime (0 ) and the lower state with a double prime (00 ). First, we look at
the electronic structure of HfF+ , and then briefly cover important aspects of vibration and rotation

13,936 13,937 13,938 13,939 13,940 13,941
-4

-2

0

(e)
2

13,891
13,89013,889

R

-2

-4

-1.0

-1.5

12,168
12,167
0

0.5

1.0

(b)

1.5

-8

-6

-4

-2

0

2

4

6

8

(a)

10,00010,50012,16911,00012,170

0
-0.5

2

4

(d)

13,888QP

13,89013,880

-6

-4

-2

0

2

4

(c)
6

11,500

4

13,91013,900

Wavenumber [cm-1]

12,00012,50013,00013,92013,50013,93013,94014,00013,95014,50013,960135

Fractional Absorption [x10-5]

Figure 5.1: All data obtained with cw- and comb-vms (red and blue, respectively). All y-axes are
fractional single-pass absorption (×10−5 ) and all x-axes are in cm−1 . In (a) we show all of the HfF+
spectra acquired to date. The comb-vms data is vertically offset for clarity. The lower panel of (b)shows a zoomed region of the comb data where some lines were also measured with the cw system.
The upper panel of (b) plots predicted line positions from the fits; different colors correspond todifferent bands (magenta, 1 Π1 ← 1 Σ+ (0,1); green, 3 Π1 ← 1 Σ+ (3, 1); cyan, 3 Π1 ← 1 Σ+ (2, 0)) while
different symbols correspond to Hf isotopes: −◦, 180 Hf; −×, 179 Hf; −∗, 178 Hf; −+, 177 Hf; −·, 176 Hf.
Inset (c) shows the 3 Φ2 ← 3 ∆1 (0, 0) band (on the left) and part of the 3 Σ−
← 1 Σ+ (1, 0) on
0+
the right. The region in (d) illustrates the cancellation of isotope shifts in ∆v = 0 transitions; thesharpness of P- and Q-branch lines is due to a cancellation between the rotational, vibrational, andelectronic contributions to the isotope shift. The splitting in the (higher-J 00 ) R-branch lines that is
not observed in the (lower-J 00 ) P- and Q-branch lines is due to Λ-doubling in the 3 ∆1 state. Inset(e) shows a prominent band-head from the 3 Φ2 ← 3 ∆1 (0, 0) band as well as weaker lines from the3 Σ− ← 1 Σ+ (1, 0) band.
0+

136
that will be needed to analyze the data.
We can treat the electronic states of the molecule in the united atom picture, where the+molecule can be thought of as a single atom (i.e., H+
2 looks like He ). Thus every electron exists

in a molecular orbital (which can be represented as a linear combination of atomic orbitals) with awell defined value of n and l. In addition, the molecular axis (typically set as the ẑ axis) provides
a quantization axis for the projection of l, giving values of ml between ±l. To first order, theenergy only depends on λ = |l|, so we can write state configurations as the values of (n)lλ for eachelectron. The value of λ gives the number of nodal planes that contain the ẑ axis, or equivalently,
the rotational symmetry about the ẑ axis (λ = 0 is cylindrically symmetric, λ = 1 is symmetric
only with a 360◦ rotation, λ = 2 is symmetric with a 180◦ rotation, etc.). In HfF+ , we only careabout the two valence electrons (excitations of core electrons are significantly higher in energy).
Table 5.1 gives the possible configurations arising from the two lowest energy terms (s2 and sd).
Because the radial extent of the d orbitals is small compared to the internuclear distance, we expectthat the molecular orbitals can be treated approximately as atomic orbitals [362]. Using the lλnotation, the dxy and dx2 −y2 atomic orbitals correspond to dδ, dxz and dyz correspond to dπ, anddz 2 corresponds to dσ.
We can get a rough idea of the expected state ordering from (basic) ligand field theory. “Whatfield theory?!,” you say. Don’t worry, a ligand is the generic term for an atom or molecule boundto a metal. This theory is commonly used to explain octahedral and square-planar complexes withtransition metals but can also be applied to diatomics [363, 362]. The general idea is that theorbitals on the metal ion (in this case Hf2+ ) are perturbed by the electric field of the ligand (in thiscase F− ), which is treated as a point charge. If we now consider the d orbitals in the presence of
this electric field, the orbitals that are closest to the ligand are destabilized the most. This meansthat the ordering of the d orbitals would be expected to be dδ < dπ < dσ, see Figure 5.2. All of
the d orbitals are expected to be destabilized relative to the s orbital, which is much more diffuse
and extends beyond the internuclear distance, thus resulting in more coulomb attraction between
the metal cation core and the anion ligand [362]. Thus, we would roughly expect the ordering as

137
given in Table 5.1. This ordering is also seen in the ligand field treatment of LaF [364], which has
a similar valence orbital configuration (except that La+ has an empty f -shell and Hf2+ has a fullf -shell). In addition to the state ordering, the repulsion of the ligand field causes the orbitals topolarize (i.e., to try to point away from the ion), which distorts the shape of the orbitals. Thisdistortion is equivalent to orbital mixing: the shape of the distortion contains information about
which orbitals are mixed and to what extent. Because the dσ orbital is repelled the most, it is themost polarized – mostly with sσ and pσ orbitals in such a way as to try to point mostly away fromthe ligand. The dπ orbitals polarize by mixing mostly with pπ orbitals. The dδ orbitals however
do not polarize and are thus mostly pure atomic d orbitals.
Molecular vibrations are treated as a simple anharmonic oscillator with quantized energy
levels ωv = ωe (v + 1/2) − ωe xe (v + 1/2)2 , where the v subscript indicates the vibrational quantum
number and ωe and ωe xe are the “equilibrium” harmonic vibrational frequency and anharmonicity
correction, respectively. As expected from a pair of masses connected by a spring, the harmonicqfrequency is related to the force constant, k, by ωe = µk , where µ is the reduced mass. For different
isotopes, k is not expected to change much because it is a function of electronic configuration andbond length, but µ does change; thus, ωe is different for different isotopes. Rotation is treated asa semi-rigid rotor with energy levels F (J) = BJ(J + 1) − DJ 2 (J + 1)2 , with angular momentum
quantum number J, rotational constant B, and centrifugal distortion constant D. B is related to~~the moment of inertia, I, by B = 4πcI
= 4πcµ1
R2, where R is the internuclear separation. In

an anharmonic oscillator R changes (typically increases) as v increases, so B depends on v. The
equilibrium rotational constant, Be , is the rotational constant at the minimum of the molecularpotential curve (at an equilibrium separation of re ): Bv is related to Be by Bv ≈ Be − αe (v + 1/2).
Because of the reduced mass, Be also depends on the isotope (re is basically constant). The
centrifugal distortion constant, D, arises because the molecule stretches as it rotates. We canq3
erelate De , ωe , and Be by ωe = 4B
De .
Since the electric dipole selection rule for rotation is ∆J = 0, ±1, we see three branches foreach vibronic transition: P (J 0 = J 00 − 1), Q (J 0 = J 00 ), and R (J 0 = J 00 + 1). Assuming a rigid

138

Hf2+

Hf2+ - - - F-

dσ

dz2

Fz
Hf

2+

dπ

5d

6s

dδ

dxz

dyz

dxy

y
x

dx2-y2sσ

Figure 5.2: Molecular orbital diagram for valence orbitals of HfF+ . The atomic d orbital corresponding to the primary component of each molecular orbital is pictured. Figures are from:http://en.wikibooks.org/wiki/High_School_Chemistry/Shapes_of_Atomic_Orbitals.

139

Table 5.1: Possible states and configurations arising from s2 and sd electron
configurations. Molecular orbitals are denoted in the configuration column
by lλ, where l is the corresponding atomic orbital. Subscripts are used
to distinguish orbitals with the same λ but different l. When writing the
wavefunction, electron spin is denoted by α or β and the value of mλ is given
by a superscript.
State

ConfigurationΛΩ

Wavefunction

X 1 Σ+(sσ1 )200√[|σ1 (α)σ1 (β)i − |σ1 (β)σ1 (α)i] / 2(sσ1 )(dδ)

21σ1 (β)δ 2+ (β)

-2

-1

±2

±2

σ1 (α)δ 2− (α)
 √σ1 (α)δ ±2 (β) + σ1 (β)δ ±2 (α) / 2(sσ1 )(dδ)

22σ1 (β)δ 2+ (α)(sσ1 )(dδ)-2-2σ1 (β)δ 2− (α)

23σ1 (α)δ 2+ (α)-2-3σ1 (β)δ 2+ (β)

±1

±1

√
[|σ1 (α)π ± (β)i + |σ1 (β)π ± (α)i] / 210|σ1 (β)π + (β)i-10|σ1 (α)π − (α)i

3∆

1∆

3Π

(sσ1 )(dπ)√[|σ1 (α)π ± (β)i − |σ1 (β)π ± (α)i] / 2√[|σ1 (α)σ2 (β)i − |σ1 (β)σ2 (α)i] / 21Π

(sσ1 )(dπ)

±1

±1

1 Σ+

(sσ1 )(dσ2 )

0

0

3 Σ+

(sσ1 )(dσ2 )01|σ1 (α)σ2 (α)i0-1|σ1 (β)σ2 (β)i00√[|σ1 (α)σ2 (β)i + |σ1 (β)σ2 (α)i] / 2

140
rotor, the energies of transitions in each band are given by
P (J 00 ) = ν0 − (B 0 + B 00 )J 00 + (B − B 00 )J 002
Q(J 00 ) = ν0 + (B 0 − B 00 )J 00 + (B − B 00 )J 002(5.1)

R(J 00 ) = ν0 + 2B 0 + (3B 0 − B 00 )J 00 + (B − B 00 )J 002 .
In general, B 0 < B 00 because typically the internuclear spacing is larger in the excited state (due,e.g., to a weaker bond) than in the lower state. In this case there is a place in the R branch where
the difference in energy between adjacent J 00 values crosses zero and the R branch progression turns
around. This is called the band head, and it’s position is approximately
J 00 + 1 = −B 0 + B 00
B̄=−.
2(B 0 − B 00 )
δB

(5.2)

The parity (defined as the sign with respect to inversion of spatial coordinates) of rotational wavefunctions alternates with J (similar to the parity of atomic orbitals, which alternate with l). The
parity operator is equivalent to reflection about a plane containing the internuclear axis, σ̂v in a
diatomic [365]. Because of this alternation, we introduce an additional symmetry label of e/f ,where e corresponds to states that transform as σ̂v ψ = +(−1)J ψ, i.e., states with + parity for
J = 0.
One thing to notice is that both Be and ωe depend on the reduced mass. This means thatthey both have an isotope dependence, although the isotope dependence is different for vibrationq(ωe ∼ µ1 ) versus rotation (Be ∼ µ1 ). Because Hf has five observable isotopes, we were able to use
these isotope shifts to gain information about the observed bands. In particular, near the bandhead the rotational energy is low, so the rotational isotope shift is small at this point (it vanishes
at the origin). Thus, we can use shifts in the band head location for different isotopes to roughly


q
µ180
measure the difference in vibrational number: for example, ν̃180 − ν̃178 ≈ 1 − µ178 750(v 0 −v 00 ) ≈0.4 cm−1 for v 0 − v 00 = 1, where 750 cm−1 is the approximate (average) vibrational constant for
the upper and lower states. In this case, the isotopes bands are progressively blue shifted fromthe 180 Hf band (which is the most abundant isotope). In addition, there are small isotope shifts
in the electronic energies that arise from finite nuclear size effects as discussed more below. One

141
interesting point is that the vibrational and electronic isotope shifts are constant across a band
while the rotational isotope shifts are not. Thus, the isotope shifts between rotational lines aredifferent depending on where in the progression you look: for a ∆v = 0 transition we do not fully
resolve the isotope splitting, so this results in a place where the rotational isotope shift cancels theelectronic and (small) vibrational isotope shift, which is observed as the place where the absorption
line is narrowest.
The states in Table 5.1 are denoted by 2Σ+1 ΛΩ , where Σ is the total spin angular momentum
along the internuclear axis, Λ is the total orbital angular momentum along the internuclear axis, andΩ = Σ + Λ. In addition to the spin and orbital angular momentum, molecules also have rotational
angular moment. There are several primary ways that the angular momentum can couple together,called Hund’s cases, as shown in Figure 5.3. This state notation is only technically valid for statesin Hund’s case (a); a state in case (c) is typically denoted by the value of Ω.
The classification into different cases is determined by the spin-orbit energy (the diagonal elements of the spin-orbit operator) compared to the rotational and electronic energies.
P
The spin-orbit operator is fundamentally a one-electron operator given by ĤSO = i âi~li · ~si =P 1i âi lz,i sz,i + 2 (l+,i s−,i + l−,i s+,i ) , where the sum is over open-shell electrons, âi is the spin-orbit
operator (different for each orbital), and the z axis is the internuclear axis. We can use the configurations in Table 5.1 to evaluate the diagonal elements of ĤSO , for example, h3 ∆3 |ĤSO |3 ∆3 i =− h3 ∆1 |ĤSO |3 ∆1 i = A3 ∆ , where A3 ∆ ≡ hdδ|â|dδi. A3 ∆ is approximately given by ζ5d – the spin2(E3 D3 − E3 D2 ) =orbit coefficient of the atomic 5d orbital in Hf2+ . We can obtain ζ5d = 2l+1

1537 cm−1 using the Hf2+ energy levels from [366].
States with small spin-orbit energies are typically case (b), thus the 1 Σ+ state would fall into
this category. However, there is no distinction between case (a) and case (b) for 1 Σ+ , so we arefree to treat it as case (a). (Note that case (b) states can always be projected to a case (a) basis,see for example [367].) The large spin-orbit interaction technically makes case (c) most appropriate
for the states in HfF+ . In general though, we can write the case (c) eigenfunctions as case (a)eigenfunction perturbed by other case (a) functions (see for example [368]) and then denote the

142

(a)

Spin-orbit energy > rotational energy

Good QN: S, Λ, Σ, Ω, J, (F, mF)

F
J

mF

R

S

I

z’

L
Λ

n

Σ

L = ∑ liS = ∑ si
R = rotation
J=L+S+R
F=J+I
Λ = L•nΣ = S•nΩ=Λ+Σ
mF = F•z’Ω(b)Small spin-orbit energy

N=L+R

S

J=N+S

Good QN: S, Λ, Ne, J, (F, mF)

N

J

R
I

Λ
L

(c)

Spin-orbit energy ~ Electronic energy >> rotational energy

Good QN: Ω, Je, J, (F, mF)
I

JR

Je = L + S
J = Je + R

Ω
L
Je

S

Figure 5.3: Angular momentum coupling in diatomic molecules corresponding to Hund’s cases (a),(b), and (c). The coupling of the nuclear hyperfine, I, with J to form F is only shown for case (a),although similar coupling is possible in the other cases.

143
state by its leading case (a) term. The amount of perturbation is small for the low-lying electronic
states, but for the higher states it can be significant, leading to allowed transitions that are denotedin the case (a) basis as singlet-triplet transitions.
States with |Λ| =6 0 (in Hund’s cases (a) and (b)) are in principle doubly-degenerate due to thetwo projections of Λ onto the internuclear axis (corresponding to two precession directions about
the internuclear axis). This degeneracy is lifted due to perturbations and is called Λ-doubling. If
we include the coupling with S, then states with |Ω| 6= 0 are degenerate in Hund’s cases (a) and(c). The symmetric and antisymmetric combinations of degenerate |+Ωi and |−Ωi are states ofgood parity and have opposite total parity. This degeneracy is removed by a coupling between
rotational angular momentum and electronic angular momentum, which directly couples stateswith ∆Ω = ±1. Thus, the coupling between ±Ω is introduced at 2Ω order in perturbation theory.
While technically this is Ω-type doubling, it is an extension of Λ-doubling when S = 0 and so istypically still called Λ-doubling.
The origin of this Λ-doubling is sometimes referred to as the electron motion changing the
moment of inertia; however, Bob Field considers this “nonsense” as it fails to explain the sign or
magnitude of the Λ-doubling in general. Instead, the origin of Λ-doubling is perturbations due to Σ
states, which shift one parity relative to the other. We can understand this more by considering the~ 2 , where B(R) is the position-depending rotational constantrotational Hamiltonian Ĥrot = B(R)R~ is the rotational angular momentum. We can write R~ = J~ − L~ −S~ and, defining z as theand R~ 2 = R2 + R2 = 1 (R+ R− + R− R+ ), where R± = J± − L± − S± . We make useinternuclear axis, Rxy2of the commutator relationships[J± , J∓ ] = ±i~Jz ,[J± , V∓ ] = ±i~Vz ,[V± , V∓ ] = ±i~Vz ,[L± , S∓ ] = 0

(5.3)

144
with V = L, S to write [361]~2 =Ĥrot = B(R)R(5.4a)(J 2 − Jz2 ) + (S 2 − Sz2 ) + (L2 − L2z )(5.4b)+ (J+ L− + J− L+ )(5.4c)− (J+ S− + J− S+ )(5.4d)+ (L+ S− + L− S+ ).

(5.4e)

Line (b) contains all of the diagonal contributions while lines (b), (c), and (d) are off-diagonalterms called L-uncoupling, S-uncoupling, and rotational-electronic, respectively. These off-diagonal~ = J~ − Ω~ (we have time averaged J~e to give Ω:~ thenterms can be simply understood if we write R~ 2 = J~2 − 2J~ · Ω~ +Ω~ 2 . The dot product term leads to the off-diagonal terms above.
R
For a simple example, we consider the simplest source of Λ-doubling in case (a) 1 Π1 . In thiscase, we need to connect |Λ = 1, Ω = 1i with |Λ = −1, Ω = −1i. This is accomplished with one
application of L-uncoupling to a Σ state and then a second L-uncoupling to |Λ = −1, Ω = −1i.
The magnitude of this perturbation is given by second-order perturbation theory asq∼

Be2 J(J + 1) hΛ = −1, Ω = −1|L− |Σi hΣ|L− |Λ = 1, Ω = 1i.
EΠ − EΣ

(5.5)

The remaining matrix elements are calculated constants. One simple assumption for calculatingthese matrix elements is called pure precession [369] in which we treat the operator as a singleelectron operator and consider the configurations such as given in Table 5.1 as pure configurations:pfor example hπ|l+ |σi =l(l + 1). From this, we see that the Λ-doubling in |Ω| states scales
roughly as J 2Ω . If we consider states with higher multiplicity, in addition to the q term arisingfrom two applications of L-uncoupling we should consider various combinations of L-uncoupling
2 and Hand off-diagnonal terms of HSO , which lead to o and p terms corresponding to HSO
SO BL+ ,respectively. More detailed derivations of Λ-doubling in various states can be found in [369] (1,2 Π),[370] (3,4,5 Π), [371] (all ∆ states), as well as in [360]. We discuss the Λ-doubling in HfF+ in moredetail in Section 5.1.3.

145

(a)
3

Π1

+ (e)
2 (f )- (e)J=1 + (f )

(b)
3

Π0+

2
1

+ (e)

(c)
3

Φ2

+ (e)
- (f )
J=2 + (e)
(f )

3

Δ1

+ (e)
2 - (f )- (e)J=1 + (f )

- (e)

3

J=0 + (e)2 + (e)
1

+

Σ

1

- (e)

J=0 + (e)

3

Δ1

+ (e)
2 - (f )- (e)J=1 + (f )Figure 5.4: Example transitions with Λ-doubling in one or both states. P-, Q-, and R-branchtransitions are shown by green, blue, and red arrows, respectively. Total parity as well as e/f isgiven for each rotational level. (a) 3 Π1 ← 1 Σ+ : Λ-doubling in the excited state only results in adifference between an effective B for P/R and Q. (b) 3 Π0+ ← 3 ∆1 : Λ-doubling in the ground state
only. This transition enables the sign of the Λ-doubling in the ground state to be determined. (c)3 Φ ← 3 ∆ : Λ-doubling in both states results in two of each branch.
2
1

146
The consequences of Λ-doubling on rotational bands are shown in Figure 5.4(a-c), which show
the possible transitions with Λ-doubling present in the excited state, ground state, or both states.
P-, Q-, and R-branch transitions are shown by green, blue, and red arrows, respectively. Rememberthat electric-dipole selection rules only allow + ↔ − transitions, where + and − indicate the overall
parity. The corresponding e/f labels are also shown. Notice that the ordering of + and − changes
in each J level, but e/f labels do not. When Λ-doubling is present in only one state, the result
is a different effective B value for the P/R branches and the Q branch, as can be seen by the factthat the R/P-branch transitions in Figure 5.4(a) always go to the upper doublet state while the
Q-branch transitions go to the lower doublet. If Λ-doubling is present in both states, there areinstead two sets of P,Q, and R branches, as shown in Figure 5.4(c).

5.1.2

Fitting

The first step in trying to fit any transition is to look for patterns. Typically, there are
two patterns that can be identified: a band head and chirped sequences. The band head is pretty
straightforward (a particularly obvious one is shown in Figure 5.1(c) and (e)) and gives a rough idea
about where the band origin is located. Also, due to isotope splittings, the shape of the band head
gives an idea of the difference in vibrational quantum number between the upper and lower states(each isotope band will have a band head in a different spot). Chirped sequences are progressionsof lines whose separation (∆) changes by the same amount each time, i.e., δ∆ is a constant. From
Equation 5.1, we see that for any branch, δ∆ ≈ 2(B 0 − B 00 ). Once a chirped sequence is identified,
we can fit it to a quadratic to determine δB and also to determine whether the lines chirp backto the band head (indicating a P/R branch) or to the band origin (indicating a Q branch). In
many cases, the Q branch is the easiest to identify first. The distance between the band origin andband head can be used with Equation 5.2 and δB to obtain guesses for B 0 and B 00 . From here,
an iterative process can be used to fit the bands by simulating the line positions using guessed B
values, assigning as many lines as possible, and then re-fitting to obtain better constants.
Another tool for identifying patterns in complex spectra is to use a Loomis-Wood type plot.

147
The basic idea of this technique is to plot the difference between unidentified line positions and the
expected positions of one branch. These expected positions often come from one identified branchusing the procedure above, from extrapolation from a partial chirped sequence, or from a guess. For
example, we were able to identify and fit the 3 Π1 ← 1 Σ+ (3, 1) band from a collection of unassignedlines in an extremely congested region of the spectrum. To do this, we first tabulated all of theunassigned lines (along with approximate relative intensities) and then plotted the difference withguessed positions of the Q-branch versus m (which corresponds to J). In this case, B 00 was alreadyknown from a previously fit band and the other parameters were estimated from other fits. We
then varied the origin and ∆B and looked for patterns emerging; in particular, a Q branch should
appear as a horizontal line and other branches should appear as sloping lines. The result of this isshown in Figure 5.5 after the parameters have been adjusted to place one horizontal line at zero.
We immediately see from the Loomis-Wood plot the presence of a second horizontal line located∼ 0.7 cm−1 away, which can be attributed to the isotope splitting between 178 Hf and 180 Hf. In
addition, sloped lines corresponding to the P and R branches can also be seen.
Since we were fitting a variety of different transitions with various combinations of Hund’scases, we chose to fit each band with the general expression given in Eq. 5.6
ν(v 0 , v 00 , J 0 , J 00 , s0 , s00 ) = ν0 + Fv0 (J 0 , s0 ) − Fv00 (J 00 , s00 )(5.6)

where the rotational energies, Fv are
Fv (J, s) = (Bv − skD,v 2kv)J(J + 1) − (Dv − s)J (J + 1)2 .
2
2

(5.7)

We include rotation (B), centrifugal distortion (D), Λ-doubling proportional to J 2 (k) and J 4 (kD )for each vibrational level, v; s is an e/f -symmetry term where s = +1 for e-symmetry and -1 forf -symmetry. k and kD are generic constants for the Λ-doubling since we have fit states with several
different values of Λ and Ω. In principle, the J dependences of k and kD are more complicatedand depend on the spin and Ω, see for example [369, 370, 371, 360, 361]. In Σ0 states, only one
symmetry term is chosen and k and kD are both set to 0. ν0 includes both the electronic and

148

2

difference (cm−1)

1.5
1
0.5

~ 0.7 cm-1

0
-0.5
-1
-1.5
-2

10

20

30

40

J

50

60

70

Figure 5.5: Loomis-Wood plot showing branches of the 3 Π1 ← 1 Σ+ (3, 1) transition. Darker points
indicate larger relative intensity. Potential branches are indicated by the boxed regions. The isotope
splitting between the 178 Hf and 180 Hf Q-branches is shown by the vertical arrow.

149
vibrational energies:
ν0 = Te0 − Te00 + Evib ,(5.8)

Evib ≈ (ωe0 (v 0 + 1/2) − ωe x0e (v 0 + 1/2)2 ) − (ωe00 (v 00 + 1/2) − ωe x00e (v 00 + 1/2)2 ).

(5.9)

Here Te is defined as the energy difference between the minimum of the potential curve from theminimum of the X1 Σ+ curve, and ωe and ωe xe are the usual vibrational constant and anharmoniccorrection, respectively.
Due to the congested nature of the spectra, fits were performed to a manually identified listof isolated lines (frequency only) for each band and for each isotope in bands with well resolved
isotope structure. Figure 5.7(b) shows the residuals from a representative fit; root-mean-squarevalues for the residuals were typically < 0.005 cm−1 with no apparent structure. For most ∆v = 0transitions the fits were performed to estimated isotope averaged line centers and then the rotational
constants were shifted by the reduced mass to the 180 Hf values. The 3 Φ2 ← 3 ∆1 bands exhibitedline doubling at high-J due to Λ-doubling (discussed in detail below), for these bands the center ofthe doublet was used for the fits. Similarly, the 1 Π1 ← 3 ∆2 (0, 0) band exhibited doubling of everytransition (i.e., 6 branches) due to large Λ-doubling in the 1 Π1 ; for this band all six branches were
fit simultaneously by assuming the splitting to each transition was k 0 × J 0 (J 0 + 1). A summary of
the fitted constants for each band is given in Table 5.2.

Table 5.2: Fitted constants for observed transitions in 180 HfF+ in cm−1 . Quoted errors are at 95% confidence level, statistical only.
For bands recorded with the wavemeter uncalibrated we include an additional 0.006 cm−1 uncertainty in ν0 . Internal consistencies inthe data suggest that for the ∆v = 0 bands, which typically suffer from partially resolved isotope structure, systematic errors in thefit quantities may exceed the quoted statistical uncertainties in some cases by a factor of two or three. Values without uncertainties
were fixed in the fits. All constants are given for the vibrational levels involved in the transition and are not equilibrium values. The
0 are generic terms proportional to J(J + 1) in the ground and excited states and J 2 (J + 1)2 in theΛ-doubling terms, k 00 , k 0 , and kD
excited state respectively (see text for more details).
Assignment
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19

3Π

3
0− ← ∆1 (1, 0)
1+
1 ← Σ (0, 1)
3Π
3
0+ ← ∆1 (1, 0)
3Π
1 +0+ ← Σ (0, 0)
1 Π ← 3 ∆ (0, 0)
1
2
3Π
1 +
0+ ← Σ (1, 0)
1 Π ← 1 Σ+ (1, 2)
1
3 Π ← 1 Σ+ (3, 1)◦1
1 Π ← 1 Σ+ (0, 1)
1
3 Π ← 1 Σ+ (2, 0)
1
1 Π ← 1 Σ+ (0, 0)
1
3 Σ− ← 1 Σ+ (0, 0)
0+3Π? ∆v = 0? ∆v = 0
3 Φ ← 3 ∆ (0, 0)2
1
3 Σ− ← 1 Σ+ (1, 0)+0
3 Φ ← 3 ∆ (2, 1)
1
2
3 Φ ← 3 ∆ (1, 0)
2
1? ∆v = 0

ν0

B 00

B0

D00
[10−7 ]

D0
[10−7 ]k 00
[10−4 ]k0
[10−4 ]

0kD
[10−9 ]

δE

9948.624(6)10109.877(6)10137.723(7)10401.723(13)+10852.757(13)+11114.653(7)12136.012(3)12216.901(10)12217.369(2)12304.400(3)13002.189(12)+13254.302(7)13729.918(17)+13822.270(14)+13933.340(12)+13953.799(6)14534.839(7)14613.782(7)14636.162(13)+0.29891(2)0.30333(6)0.29887(5)0.30500(10)0.29915(9)0.30482(4)0.30180(5)0.30335*0.30335(2)0.30481(5)0.30474(20)0.30478(10)0.29860(60)0.29980(10)0.29900(10)0.30483(5)0.29745(8)0.29888(5)0.29620(30)0.28272(2)0.28382(6)0.28277(5)0.28437(11)0.28102(9)0.28281(4)0.27973(5)0.27958(20)0.28115(3)0.28096(5)0.28104(20)0.28967(9)0.28680(60)0.28860(10)0.27720(10)0.28808(4)0.27488(8)0.27590(5)0.27600(30)1.85(5)1.9(5)
1.70(20)2.3(1.2)
1.7(4)
1.71(14)1.80(14)1.8(3)
1.88(8)1.78(12)1(2)1.7(5)3(3)1.9(2)
2.2(3)
1.78(12)1.7(4)
1.71(15)
1.7(1.5)

1.81(4)
1.9(5)
1.66(20)2.2(1.2)
1.7(4)
1.72(13)
1.74(14)1.8(3)
1.81(8)1.78(12)1(2)1.9(5)3(3)2.2(2)
2.1(3)
1.98(13)
1.6(4)
1.59(15)
1.8(1.5)-0.123(6)–-0.14(2)
–
0
–
–
–
–
–
–
–
0
0
†
–
†
†
0

–
-3.77(5)––3.59(3)‡–2.68(4)-3.8(2)
3.69(2)-3.82(1)3.55(4)–-0.2(3)0-0.01(3)
–
0
0
0

–
0
–
–
–
–
7.4(1.3)09.7(7)
0
0
–
–
–
–
–
–
–
–

?
0.383
-0.369-0.0350-0.4030.412-0.7240.372-0.783-0.04-0.0840.005-0.035-0.05
0.457-0.422-0.434-0.02◦

Values assigned using a manual fit to multiple isotopes due to challenges of determining the line centers of the weaker lines in the dense spectrum.
Value fixed to the fitted value from the 1 Π1 ← 1 Σ+ (0, 1) transition.
+
Individual isotopes were not fully resolved, thus the fit was done to averaged line positions. We report isotope corrected values with the error due to theuncertainty in the isotope shifts.
†3
Φ2 ← 3 ∆1 transitions exhibited line-doubling at high J”, but the doublets were not resolved well enough to accurately determine k”.
‡
Each transition was a doublet with a splitting given by k0 × J 0 (J 0 + 1) due to Λ-doubling in 1 Π1 .
*

150

151
Assignments of the observed bands were not particularly straightforward, partially because
the presence or absence of low-J lines usually could not be determined due to the high temperatureof the ions and complicating isotope structure. In fact, only for the 1 Π1 ← 1 Σ+ (0, 0) band near
13000 cm−1 (transition 11) were we able to directly establish that Ω0 = 1 and Ω00 = 0 since weobserved an R(0) line but no P(1) line. One valuable feature of the isotope structure was theability to measure isotope shifts for each transition. We define the isotope shift between 180 HfF+
and 178 HfF+ for a given spectral line asδE = E

180

−E

178

µ180
= (1 −)Erot + (1 −
µ178

r

µ180
)Evib + δTe0 − δTe00 .
µ178(5.10)

Here µ180 and µ178 are the reduced masses, Erot is the rotational energy in 180 HfF+ (i.e., offset
from the band origin), and Evib is the vibrational energy in 180 HfF+ . The last column of Table 5.2
gives the isotope shift between 180 HfF+ and 178 HfF+ measured at the band origin. The isotope
shift provides two useful pieces of information for the assignment. First, since the isotope shift isdominated by the vibrational shift, the sign and magnitude of the shift give the value of ∆v for thetransition. We can also use the isotope shift to group transitions that only differ in vibration: this is−1qdone in Figure 5.6 by plotting ν0 + 1 − µµ180
δE −10000 cm−1 versus ν0 −10000 cm−1 for each178
transition. Transitions that lie close to the same horizontal line originate from the same electronic
states.
We can use Hund’s rules to determine a rough ordering of states from the configurations listed
3
1
3 +
1
in Table 5.1: 1 Σ+ , 3 ∆1 , 3 ∆2 , 3 ∆3 , 1 ∆2 , 3 Π0 , 3 Π1 , 3 Π2 , 1 Π1 , 3 Σ+
0 , Σ1 , and Σ. The Σ and Σ

are expected to be very high in energy and are not used in the assignments. These are only statesarising from s2 and sd configurations; in addition, a d2 configuration is expected to contribute 3 Σ− ,3 Φ, and possibly 3 Π configurations in the higher energy regions. Several bands with no Q-branch

were observed (numbers 4, 6, 12, and 16), which are expected to be 3 Π0+ ← 1 Σ+ and 3 Σ−
← 1 Σ+ ,0+with the 3 Π0+ transitions significantly lower in energy. Transitions with ∆Ω = 0 will give very
weak Q-branches that are only visible at low J, which could look like a missing Q-branch.
From Figure 5.6, we can identify transitions 7, 9, and 11 as all originating from the same152

ν0 + δ E*1865 − 10 000 (cm−1)

5000

16

4000

13

3000

7

11

9

19
18

15
14

17

12

2000
1000

5

2
4

0

8

10

6

1
3

−1000
−1000

0

1000

2000

3000

ν − 10 000 (cm−1)
0

4000

5000

Figure 5.6: Isotope shifts (δE) of each observed transition as numbered in Table 5.2. The isotope
shift on the y-axis is scaled to roughly remove the vibrational isotope shifts; the x-axis is thetransition energy - 10000 cm−1 . Transitions between different vibrational levels but the same
upper and lower electronic states should lie on horizontal lines.

153
electronic states. Since we know that Ω0 = 1 and Ω00 = 0, we can assign the ground state to 1 Σ+ andthe excited state to one of 3 Π1 , 1 Π1 or 3 Σ1 . Furthermore, the splitting between 9 and 11 matches
the splitting between v = 0 and 1 from [372], so transition 11 gives us the B0 for 1 Σ+ . This B
value also confirms the rough assignment of the no Q-branch transitions in the previous paragraphas being 3 Π0+ ← 1 Σ+ (transitions 4 and 6) and 3 Σ−
← 1 Σ+ (transitions 12 and 16), all with the0+lower state of 1 Σ+ J = 0. We can use the vibrational isotope shifts to assign all of these fourtransitions now, in particular, transition 6 is 3 Π0+ ← 1 Σ+ (1, 0). The splitting between transition 4
and transition 3, which has the same value of B 0 , matches the splitting between 1 Σ+ and 3 ∆1 [372].
This gives us an assignment for transition 3 and a value for B0 of 3 ∆1 .
We now assume that states with different Ω but the same values of S and Λ have similar B
values: this is a reasonable assumption because a different value of Ω indicates a different orientationof spin and Λ, which would not be expected to affect the bond strength (and thus length). Since
B 0 for transition 11 is not close to the B 0 for transition 12 (3 Σ0 ), we can assume that transition 11(and thus 7 and 9) is to either 3 Π1 or 1 Π1 . By looking at Figure 5.6 and B 00 values, we see thattransitions 2, 8, and 10 are another series originating from 1 Σ+ and going to the same electronic
state. Given the transition energy, the excited state could be 3 Π1 , 1 Π1 , or 3 Π2 , but since the valueof |k 0 | is similar to transition 11, we can rule out 3 Π2 (the doubling for Ω = 2 should be muchsmaller). Transition 2 and transition 9 both start from the 1 Σ+ v 00 = 1 and have similar vibrational
isotope shifts (so they both go to the same v 0 ) but one must be to 3 Π1 and one to 1 Π1 : we can use
the expected state ordering to assign transition 2 (and 8 and 10) to 3 Π1 and transition 9 (and 7
and 11) to 1 Π1 .
Transitions 15, 17, and 18 all exhibit line-doubling at very high J and all have similar isotope
shifts in Figure 5.6; also, B 00 for 15 and 18 is consistent with 3 ∆1 v 00 = 0 from transition 3. The
high-J doublets indicate that both lower and upper states have doubling, and the fact that the
doubling is only visible at high-J indicates that Ω0 > 1. This implies that the upper state is either
3Π

2 or

3Φ .
2

Since the value of B 0 is different than B for 3 Π1 , we assign the upper state to 3 Φ2(additionally, we expect the 3 Π2 to be a bit closer in energy to the 3 Π1 ). Combining all of these154
results gives the final assignments in Table 5.2. Transitions 13, 14, and 19 remain unassigned.
From the fitted constants for the 16 assigned bands, we can determine constants for theX1 Σ+ , 3 ∆1 , 3 Π0− , 3 Π0+ , 1 Π1 , 3 Π1 , 3 Σ−
, and 3 Φ2 states as given in Table 5.4. T0 and ∆G1/2
0+
are directly obtained from the measurements: T0 is defined as the energy of the v = 0 level ofan excited state relative to the X1 Σ+ v = 0 level, and ∆G1/2 is the energy difference between
the v = 0 and v = 1 levels. All other (equilibrium) constants are extracted from the data. The
rotational constant for a given vibrational level is given to first order in v as Bv = Be − αe (v + 1/2),
where Be is the equilibrium rotational constant. Since we have measured at least two vibrational
levels for each state, we can determine αe ≈ Bv − Bv+1 , and then use this to obtain Be for eachstate. Te , ωe , and ωe xe are determined using Equations 5.8 and 5.9. For the 3 Π1 state, we wereable to calculate ωe and ωe xe directly; for all other states we assumed a Morse potential so that
ωe xe = αe2 ωe2 /36Be3 + αe ωe /3Be + Be (i.e., the Pekeris relationship).
In addition, we have observed isotope shifts of the state origins, called δTe in Table 5.4, that
we attribute to Hf electronic isotope shifts (due to the finite nuclear charge radius) [373, 374, 375].
For bands with well resolved isotope structure (i.e., ∆v 6= 0), δTe0 − δTe00 was determined from
Equation 5.10 either by fitting both ν0180 and ν0178 or by measuring δE near the origin and thensubtracting the vibrational contribution. For ∆v = 0 bands, we found the frequency where thelines were the sharpest, indicating that δE ≈ 0 (as illustrated by the Q- and P-branch lines in
Figure 5.1(d)); this location relative to the band origin is used for Erot in Eq. 5.10, which then gave
δTe0 − δTe00 . Values of |δTe0 − δTe00 | were . 0.1 cm−1 for all transitions. To obtain the δTe values given
in Table 5.4, we set δTe00 (X 1 Σ+ ) = 0 and calculated relative shifts of the other states. The simplistic
model of Eq. 5.10 neglects vibrational-band-specific perturbations or other more complicated leveldependent effects; however, results were consistent across different transitions, which supports theassignment of the effect to electronic states, not individual bands. Except for the X1 Σ+ state theisotope shifts are all fairly similar with an average value of about -0.06 cm−1 (-1.7 GHz), which
agrees well with the value for the Hf+ 5d2 6s2 → 5d2 6s6p of -1.8 GHz [375].

155

Π - Δ (1, 0): no Λ-doubling
Residuals (cm )

0.04

(a)

0.01

0.02
0
R
Q
P

-0.02
-0.04

0

20

40

J’’

60

Π - Δ (1, 0): with kD’’

Π - Δ (1, 0): with k’’
0.015

0.015

(b)

0.005

0.005

0

0

-0.005-0.005

-0.01

-0.01

-0.015

0

(c)

0.01

20

40

J’’

60

-0.015

0

20

40

J’’

60

Figure 5.7: Residuals for three fits to the 3 Π0− ← 3 ∆1 (1, 0) band. The residuals are only plotted for
assigned lines in each branch: R-branch (+), Q-branch(square), and P-branch (*). (a) Fit with noΛ-doubling terms included. Notice the large structure to the residuals. (b) Only k 00 included, i.e.,
only Λ-doubling proportional to J 2 in the lower state. No structure in the residuals; the scatter is
00 included, i.e., only Λ-doubling
primarily due to error in identification of the line center. (c) Only kD
4proportional to J in the lower state. There is some slight structure to the residuals, indicatingthat the primary contribution to the Λ-doubling goes as J 2 .

156
5.1.3

Lambda doubling

Due to our ability to see high-J lines, we were able to measure precise values for the Λdoubling, parametrized by k and kD in Equation 5.7, in several different electronic levels. In
Hund’s case (a), k can be related to the more standard Λ-doubling parameters defined in [370]for Π states (o, p, and q) and to õ∆ as defined in [371] for 3 ∆ states; k was not measurable inΦ states. The values of k and kD were useful in assigning many transitions and also providing
some insight into inter-state perturbations. For the eEDM measurement, the most important Λdoubling parameter is the splitting of the J = 1 levels in the 3 ∆1 , as this goes into determining
several different potential systematic effects. As discussed in Section 5.1.1, Λ-doubling occurs due tocoupling with electronic states that have non-degenerate (single parity) rotational levels – typicallyΣ states – via off-diagonal spin-orbit and L-uncoupling interactions. The selection rules for theseinteractions have ∆Λ = ±1, ∆Ω = 0 and ∆Λ = ±1, ∆Ω = ±1, respectively. We can now estimate
the Λ-doubling terms by using these two interactions to link Ω → −Ω.
In a case (a) 3 ∆1 state, the Λ-doubling occurs from two applications of the spin-orbit interaction (which has no J dependence) and two applications of the L-uncoupling interaction (∼ J).
So, we expect the Λ-doubling splitting to be given by ±õ∆ × J(J + 1), where the upper and lower
signs refer to the two e/f-symmetry levels and õ is given in [371]. We can make a rough estimateof the value of õ∆ by a simple scaling from that measured in TiO or WC [376]:õ∆ ≈ζ 2 (Hf2+ )B 2 (HfF+ )× õ∆ (WC/TiO).
ζ 2 (WC/TiO)B 2 (WC/TiO)(5.11)

This scaling relationship arises from the application of two spin-orbit operators, which scale as theatomic spin-orbit, ζ, and two L-uncoupling operators, which scale as the rotational constant B[371, 376]; however, it neglects changes in state order and spacing. Using this and ζ5d from earlier
we predict õ∆ ≈ 50 kHz.
A more detailed estimate can be made by explicitly writing out the possible perturbations,as in [377]. We only include states in the sd manifold given in Table 5.1 because the spin-orbit andL-uncoupling effects are, to a good approximation, just single-electron operators. The Λ-doubling

157
parameter is given by [371]
X √
õ∆ =
4 30
i,j,k

Q



S S

(matrix elements)(E0 − Ei )(E0 − Ej )(E0 − Ek ) 


1

1


s
2  (2S − 2)!
S0(2S + 3)!

,(5.12)

where i, j, k are the intermediate states, S = 1 is the spin multiplicity of 3 ∆1 , S 0 is the other spinQ
multiplicity of the perturbing states, and (matrix elements) is the product of matrix elementsof either the spin-orbit or L-uncoupling operators. Table 5.3 shows the intermediate state andoperator combinations from the states in Table 5.1 that contribute to the Λ-doubling as well as thecalculated value of õ∆ from each combination.
Take, for example, the first line in Table 5.3. We need to evaluate first the spin-orbit matrix
element between 3 ∆1 and 3 Π1 , i.e., h3 ∆1 |ĤSO |3 Π1 i. Remembering the definition of ĤSO as a sum
over single-electron operators from Section 5.1.1 and ignoring the s electron, which does not change,
we have that
1
â
â
1
h3 ∆1 |ĤSO |3 Π1 i = √ hδ 2+ (β)| l+ · s− |π + (α)i = √ hδ 2+ | l+ |π + i hβ|s− |αi
2
2
2
2
 
 
1
1 p
1
1 √
=√
l(l + 1) − mπ (mπ + 1) hδ|â+ |δi = √ ζ5d
6 − 2.
2
2 2
2Similarly, h3 Π1 |ĤSO |3 Σ1 i =(5.13)

√

√
3
3
+
+
2 hπ |â|π i = 2 ζ5d . For the L-uncoupling operator, we have√h3 Σ1 |L̂+ |3 Π0 i = hσ2 (α)|Bl+ |π − (α)i = B 6,

(5.14)and h3 Π0 |L̂+ |3 ∆−1 i = 2B. Finally, we use these matrix elements as well as the energy differencesfrom Table 5.5 in Equation 5.12 to evaluate õ∆ . In this case, we do not involve any singlet states
so S = S 0 = 1. Summing all of the contributions gives õ∆ ≈ 17.6 kHz.
The Λ-doubling in the 3 ∆1 can be observed at high-J in the 3 Φ2 ← 3 ∆1 (0, 0) band bycomparing neighboring P-, Q-, and R-branch lines, as shown in Figure 5.1(d). Since the R-branchlines have the highest J, there is noticeable doubling in these lines compared to the P- or Qbranch lines, which indicates that this doubling is not due to isotope splitting. In this band,the lines are not sufficiently well resolved to permit an accurate determination of the Λ-doubling
constant, but by fitting the 3 Π0− ← 3 ∆1 (1, 0) transition, we were able to precisely determine

158

Table 5.3: Matrix elements for the Λ-doubling in 3 ∆1 . Four matrix elements, twoapplications of spin-orbit (HSO ) and two applications of L-uncoupling (HL+ ) are required to link 3 ∆1 and 3 ∆−1 . Each row gives one pathway, with the final value of õ∆
calculated according to Equation 5.12.
Λ=2

Ô

Λ=1

Ô

Λ=0

Ô

Λ = −1ÔΛ = −2õ∆ (kHz)

3∆
1

HSO

3Π

1

HSO

3Σ

1

HL+

3Π

0

HL+

3∆
−1

-3.9

3∆
1

HSO

3Π

1

HL+

3Σ

0

HSO

3Π

0

HL+

3∆

−1

-4

3∆
1

HSO

3Π

1

HL+

3Σ

0

HL+

3Π
−1

HSO

3∆
−1

-3.7

3∆
1

HSO

1Π

1

HSO

3Σ

1

HL+

3Π

0

HL+

3∆

−1

6.5

3∆
1

HL+

3Π

0

HSO

3Σ

0

HSO

3Π

0

HL+

3∆

−1

-4.2

3∆

1

HL+

3Π

0

HL+

3Σ

−1

HSO

3Π

−1

HSO

3∆

−1

-4

3∆

1

HL+

3Π

0

HL+

3Σ

−1

HSO

1Π

−1

HSO

3∆

−1

6.5

3∆

1

HL+

3Π

0

HSO

3Σ

0

HL+

3Π

−1

HSO

3∆

−1

-3.7

3∆

1

HSO

1Π

1

HL+

1 Σ+

HSO

3Π

0

HL+

3∆

−1

6.9

3∆

1

HSO

1Π

1

HL+

1 Σ+

HL+

1Π

−1

HSO

3∆

−1

5.4

3∆

1

HL+

3Π

0

HSO

1 Σ+

HSO

3Π

0

HL+

3∆

−1

8.9

3∆

1

HL+

3Π

0

HSO

1 Σ+

HL+

1Π

−1

HSO

3∆

−1

6.9

159
the sign and size of the Λ-doubling. As shown in Figure 5.7, the best fit was obtained with k 00
00 , implying that the Λ-doubling goes mainly as J(J + 1) as expected (adding k 00 toinstead of kDD

the fit with k 00 resulted in no improvement). This fit gives a value of k 00 = −1.23(6) × 10−5 cm−1 ,so õ∆ = −k 00 /2 = 6.2(3) × 10−6 cm−1 = 185(9) kHz (corresponding to e above f ). This issignificantly larger than the estimated values above, presumably due to unaccounted for interactions
or mixings in the excited state that invalidate the pure configuration approach. The precision onõ∆ is only about an order of magnitude worse than that obtained for TiO using pure rotational
spectroscopy [377], illustrating the ability to obtain high-resolution spectroscopic information frombroad-bandwidth ro-vibronic spectra.
Λ-doubling in the 3 Π0 can arise from interactions with both 1 Σ+ and 3 Σ+ states, which resultin a splitting of the Ω = 0− and 0+ levels that is independent of J. We measured this splitting to
be 189.099(4) cm−1 with the 0+ (e-symmetry) higher in energy than 0− (f ). To the extent that theX1 Σ+ state is a pure (sσ)2 configuration, we can assume that this splitting is from a higher-lying
3 Σ+ state with an (sσdσ) configuration. The spin-orbit interaction between the 3 Σ+ (f ) and the0−3Π

0− can be estimated by( √12 hσ1 (β)π(β)| 12 âl+ s− |σ1 (β)σ2 (α)i)2∆E

We have used hπ(β)|l+ s− |σ2 (α)i ≈

√

3 2
ζ
≈ 4 5d .
∆E(5.15)

6 in the pure-precession model [370, 369]. Using the measured

splitting and the atomic spin-orbit coefficient, we estimate ∆E ≈ 11000 cm−1 , which gives anapproximate location of the 3 Σ+state to be 21200 cm−1 . This is close to the predicted position of
0−21694 cm−1 from the new calculations (see Section 5.2). In a Hund’s case (a) basis, this splitting
corresponds to the more familiar 2(o + p + q) [370].
The cause of the Λ-doubling in the 3 Π1 state is primarily interaction with the 3 Σ+
state
0−
through the 3 Π0 . This type of interaction is characterized by the parameter p in [370]. Withe-symmetry above f in 3 Π0 and 3 Π0 located below 3 Π1 , we would expect e above f in 3 Π1 , whichcorresponds to a negative value of k as observed. The magnitude of k is expected to be about

160
4Bp/AΠ , where AΠ is the spin-orbit parameter for the 3 Π manifold andp ≈ Ch3 Π1 |BL+ |3 Σ+
ih3 Σ+|âL− S+ |3 Π0 i/(EΠ − EΣ )0−0−(5.16)≈ CBζ5d hπ + |l+ |σ2 i2 /(EΠ − EΣ ).

(5.17)

Here, C is a numerical factor dependent on the spin [370]. Again using the pure-precession model,
we estimate k ≈ 8 × 10−4 cm−1 , in reasonable agreement with experiment.
We attribute most of the Λ-doubling in the 1 Π1 to interaction with the nearly-degenerate
nominal 3 Σ−state. It is reasonable that the 1 Π and 3 Σ− are not pure (sσdπ) and (dδ)2 configura0+
tions, which would lead to increased spin-orbit interaction between 1 Π1 and 3 Σ−
. Estimating the0+magnitude of the Λ-doubling in this case is difficult due to uncertainty in the purity of the configurations. Qualitatively though, we would expect the effect of a 3 Σ− state above the 1 Π1 state would
be to push e-symmetry below f as observed. In addition we observed Λ-doubling that was strongly
vibrational-level dependent (3.69 × 10−4 cm−1 for the v = 0 compared with 2.68 × 10−4 cm−1for the v = 1), which is indicative of nearly-degenerate interacting states. We also measured aΛ-doubling term proportional to J 4 (denoted kD ). This term can be explained by substituting∆E → ∆E + (BΠ − BΣ )J(J + 1) in the Λ-doubling denominator; the first two terms in the Taylor
expansion give the J(J + 1) and the J 2 (J + 1)2 components. From this we can estimate thatkD ≈ 3k(BΠ − BΣ )/(EΠ − EΣ ), which gives kD ≈ 2 × 10−8 cm−1 .

5.2

TheoryIn earlier theoretical study of HfF+ [378], the correlation calculations of the spectroscopic

constants were performed in two different ways: in the first series 10 electrons from 5d, 6s shells of
Hf and 2s, 2p of F were correlated while in the second series 5s, 5p outer core electrons of Hf and
1s of F were also correlated. The rest of the 1s − 4d inner-core electrons of Hf were excluded fromthe explicit treatment using the generalized relativistic effective core potential (GRECP) method1[379]. While it was shown that the inclusion of the additional electrons in the 20-electrons case160-electron core GRECP for Hf is available at www.qchem.pnpi.spb.ru

161
contributed significantly to the excitation energies, such 20-electron calculations were performed
only for one internuclear Hf–F distance in a small space of many-electron basis functions due tocomputer limitations. This calculation was then applied for other distances as a “core correction”.
In the present work a new 20-electron relativistic correlation calculation was carried out in order toconsider electronic states that were not investigated in [378] as well as to overcome the bottlenecksof the previous studies of HfF+ . Eventually 23 electronic states (with excitation energies up to
22000 cm−1 ) were considered in the present study instead of ten states in [378].
Some modifications were made in the computational procedure to achieve better accuracy.
We used a direct multi-reference configuration-interaction approach accounting for spin-orbit effects(SODCI) [380, 381] as a method to treat both correlation and relativistic effects simultaneously.
One-component basis functions (orbitals) are required in this method to construct many-electron
spin-adapted functions (SAF’s). Instead of the orbitals obtained within the complete active space
self-consistent field method (used in [378]), the natural orbitals of a one-electron density matrix
averaged over the density matrices of the states of interest were used. The latter density matrices
were calculated within the scalar-relativistic coupled-clusters method with single and double cluster
amplitudes (CCSD) using the cfour code [382]. From numerical investigation [383] it is known
that the use of natural orbitals can provide faster convergence in terms of the number of SAF’s
required to account for a given part of correlation energy.
In the SODCI calculation the same atomic basis sets for Hf and F were used as in ourprevious study [378], i.e., a generally-contracted basis set for Hf consisting of 6 s-, 5 p-, 5 d-, 3 fand 1 g- type contracted functions (denoted as {6,5,5,3,1}) and an ANO-I {4,3,2,1} basis set for
F [384]. However, a correction on the extension of the basis set was applied additionally in thepresent work. To evaluate the correction, two scalar-relativistic calculations were performed using
the coupled clusters method with single, double and perturbative triple cluster amplitudes: (i) in
the same basis set that was used at the SODCI stage, and (ii) with an extended basis set for Hf(produced by uncontracting d- and adding g-, h-, i- type basis functions) and an ANO-L {7,7,4,3}[384] basis for F.

162
To compute potential curves the described calculations were performed at 14 points in therange of 3.0 – 4.2 a0 . As shown in Table 5.5, the new calculations are in remarkable agreementwith all of the experimental results. These improvements should enable more accurate calculationsin other species with heavy atoms, where relativistic effects are extremely important.

5.3

Summary
By using a combination of comb-vms and cw-laser vms, we have performed high-resolution

survey spectroscopy of HfF+ . We measured 19 transitions (16 identified) and obtained constantsfor 9 different electronic states. Because of the ability to see high-J lines, many of the constants are
fit to high precision, including Λ-doubling terms. This information enabled further development
and cross-checks of the theory, which have implications beyond the eEDM measurement: accurate
calculations of lanthanides and actinides are important for understanding the bonding and reactivityof species containing these atoms, see for example [385, 386, 387, 388].
In particular, we accomplished the four goals identified in the introduction: we measured
precise constants for the 1 Σ+ and 3 ∆1 states, among others, including the Λ-doubling in 3 ∆1 . The
splitting in the J = 1 is important for the eEDM measurements because it is the opposite parity
levels in this state that are mixed in an electric field to polarize the molecule. The necessary electricfield for full polarization – i.e., when the Stark effect goes from second order to first order in theelectric field – can be found by equating the Stark energy to the energy difference between the twoparity states: Epol ≈ ωef /2πdmf [389], where ωef is the Λ-doubling splitting (in angular frequency)
and dmf is the molecular-frame electric dipole moment of the molecule (4.3 Debye for HfF+ [62]).
We can use our measurement of õ∆ to estimate that ωef = 2π × 4õ∆ ≈ 2π × 740 kHz, which is atleast an order magnitude larger than previously predicted [62]. Nonetheless, this means that Epolis still under 1 V/cm, which is important for the eEDM experiment as larger fields tend to leadto more issues with systematic errors. We also identified transitions for laser-induced fluorescenceor state manipulation (e.g., 1 Π1 from 1 Σ+ and 3 Φ2 from 3 ∆1 ) as well as an intermediate state
for Raman transfer from 1 Σ+ to 3 ∆1 (the 3 Π0+ ). In the next chapter, we use this transition to163
selectively transfer population to the 3 ∆1 J = 1, see Section 6.3. In addition, Table 5.6 lists thecalculated transition dipole moments between the excited states and the three lowest states, which
are useful for estimating transition strengths.
We are currently using an improved version of the comb-vms system to characterize ThF+ ,which has several advantages over HfF+ for the eEDM experiment. However for this species, the abinitio calculations are even more challenging, leading to larger theoretical uncertainties. The first
spectroscopy of ThF+ using PFI-ZEKE plus some LIF (unassigned) has recently been published[390], but, as was the case with HfF+ , the higher excited states remain uncharacterized. By using
highly nonlinear fiber to broaden the comb spectrum (Section 2.3), we have been able to cover
the full 700-900 nm range with comb-vms, which corresponds to 3250 cm−1 of continuous spectra!
This ability to rapidly cover thousands of cm−1 with high sensitivity and high resolution will be a
powerful new tool for the study of ions and radicals for many applications.

Table 5.4: Derived constants for observed states in 180 HfF+ in cm−1 . Values for T0 are given to the ν = 0 levels, with
the X1 Σ+ ν = 0 level set to 0. Values for Te are to the minimum of the potential curves again with X1 Σ+ set to 0.
∆G1/2 is only given for states where we had a direct measurement. All equilibrium values assume a Morse potential unlessotherwise noted. Quoted uncertainties are 95% and are statistical, i.e., they do not account for any deviation from theMorse potential. δTe = Te180 − Te178 is an electronic isotope shift relative to 1 Σ+ for each state as discussed in the text.

1 Σ+
3∆

1

3∆

2

1Π

1

3Π

1

3Π

0−

3Π

0+
3 Σ−
0+
3Φ
2
+
*

T0

∆G1/2

Te

Be

ωe

ωe xe

αe [10−3 ]δTe

0976.930(10)2149.432(18)13002.189(12)10894.697(14)–10401.723(13)13254.302(7)14910.270(16)784.820(12)
–
–
–
–
–
712.930(15)699.497(9)680.442(14)0991.83(74)–13046.04(21)10933.77(6)+10248.34(36)*10437.44(36)13296.93(28)14963.15(33)0.30558(3)0.29963(5)–0.28186(5)0.28454(8)0.28517(17)*0.28518(13)0.29046(14)0.27778(14)790.76(11)760.9(1.5)–
702.9(4)712.382(21)+–719.5(7)705.5(5)684.5(7)2.97(5)2.78(21 )–2.70(15)
2.51(6)+–
3.3(3)
3.0(3)
2.05(31)1.50(2)1.45(8)–1.42(6)
1.43(4)–
1.63(11)
1.59(10)
1.16(14)0-0.049(12)-0.055(12)-0.071(6)-0.054(7)-0.033(16)-0.047(7)-0.106(7)-0.044(16)Calculated without assuming a Morse potential.
Assuming ωe , ωe xe , and αe are the same as for the 3 Π0+ .

164

165
Table 5.5: Summary of states. Comparison of derived molecular constants from this workand from the experiments of Barker et al. [372] with the old theoretical calculations ofPetrov et al. [378] and the improved theory discussed here. The theoretical values of Be
from [378] were computed from the equilibrium bond length. Experimental uncertainties
are converted to 95% (2σ).
State (configuration)1 Σ+(sσ1 )23∆1(sσ1 )(dδ)

3∆2(sσ1 )(dδ)

3∆3(sσ1 )(dδ)

1∆2(sσ1 )(dδ)

3Π0−(sσ1 )(dπ)

3Π0+(sσ1 )(dπ)

3Π1(sσ1 )(dπ)

3Π2(sσ1 )(dπ)+∆G1/2 .
T0 .
*
B0 .
†

Constant

This Work

[372]

[378]

New theory

Te
Be
ωe

0
0.30558(3)790.76(11)00.304(10)791.2(1.0)

00.3082
751

0
0.309
792

Te
Be
ωe

991.83(74)0.29963(5)760.9(1.5)

993(2)
0.301(10)761.3(2.0)

1599
0.2994
718

1229
0.301
754

Te
Be
ωe

2149.432(16)†0.29915(9)*–2151.7(20)†0.300(10)762.3(2.0)

2807
0.2997
719

2394
0.302
766

Te
Be
ωe

–
–
–

3951(2)0.308(10)761.5(2.0)+4324
0.3004
721

3995
0.301
757

Te
Be
ωe

–
–
–

–
–
–

11519
0.2981
696

10610
0.298
747

Te
Be
ωe

10248.34(36)0.28517(17)
–

–
–
–

11910
0.2848
689

10400
0.286
716

Te
Be
ωe

10437.44(36)0.28518(13)719.5(7)

–
–
–

12196
0.2854
699

10658
0.285
724

Te
Be
ωe

10933.77(6)0.28454(8)712.382(17)

–
–
–

12686
0.2835
687

11058
0.285
712

Te
Be
ωe

–
–
–

–
–
–

14438
0.2848
703

13452
0.287
745

166
Table 5.5: Summary of states. (continued)
State (configuration)

Constant

This Work

[372]

[378]

New theory

Te
Be
ωe

13046.3(3)0.28186(5)702.9(4)

–
–
–

14784
0.2805
679

13493
0.283
699

3 Σ−
0+
(dδ)2

Te
Be
ωe

13296.93(28)0.29046(14)705.5(5)

–
–
–

–
–
–

13773
0.292
716

3 Σ−
1

Te
Be
ωe

–
–
–

–
–
–

–
–
–

14757
0.292
711

3Φ

Te
Be
ωe

14963.15(33)0.27778(14)684.5(7)

–
–
–

–
–
–

15284
0.278
671

Te
Be
ωe

–
–
–

–
–
–

–
–
–

17457
0.277
658

Te
Be
ωe

–
–
–

–
–
–

–
–
–

18312
0.289
641

Te
Be
ωe

–
–
–

–
–
–

–
–
–

19167
0.276
691

Te
Be
ωe

–
–
–

–
–
–

–
–
–

19332
0.279
698

Te
Be
ωe

–
–
–

–
–
–

–
–
–

20074
0.280
748

Te
Be
ωe

–
–
–

–
–
–

–
–
–

20330
0.288
610

Te
Be
ωe

–
–
–

–
–
–

–
–
–

20338
0.277
665

1Π

1

(sσ1 )(dπ)

(dδ)2

2

(dδ)(dπ)3Φ3(dδ)(dπ)

1Γ

4

(dδ)2

3Π

0−

(dδ)(dπ)3Π1(dδ)(dπ)3Π0+(dδ)(dπ)1 Σ+(sσ1 )(pπ)

3Π2(sσ1 )(pπ)

167

Table 5.5: Summary of states. (continued)
State

Constant

This Work

[372]

[378]

New theory

Te
Be
ωe

–
–
–

–
–
–

–
–
–

20769
0.283
740

3 Σ+
1

Te
Be
ωe

–
–
–

–
–
–

–
–
–

21415
0.276
665

3 Σ+
0−

Te
Be
ωe

21200◦
–
–

–
–
–

–
–
–

21694
0.277
658

3Φ

◦

4

Estimated from Λ-doubling in the 3 Π0 (see text).

168

Table 5.6: Calculated molecularframe dipole moments for 1 Σ+ and3 ∆ states as well as transition dipole1matrix elements between states in
HfF+ in e a0 . (?) indicates nonzero, but not calculated; (–) indicates < 0.01.

State

1 Σ+

1 Σ+

-1.2
0.02
–
–
–
0
0.15
0.25
–
0.55
0.23
0.03
–
–
–
–
0.06
0.12
0.70
–
–
0.01
–

3∆
1
3∆
2
3∆
3
1∆
2
3Π
0−
3Π
0+
3Π
1
3Π
2
1Π
1
3Σ
0+
3Σ
1
3Φ
2
3Φ
3
1Γ
4
3Π
0−
3Π
1
3Π
0+
1 Σ+
3Π

2

3Φ

4

3Σ

1

3Σ

0−

Dipole (e a0 )
3∆
1

3∆

-1.4
0.02
–
0.04
0.27
0.27
0.03
0.03
0.03
0.08
-0.02
0.48
–

?
0.07
–
–
0.22
0.01
0.18
–
0.07
0.10
0.46

0.36
0.14
0.39
0.03
?
–
0
0.17

–
0.39
–
–
0.10
–
0.03
–

2

169
Table 5.7: Franck-Condon factors (vibrational overlap integrals) for 3 Π0+- 3 ∆1 . The Franck-Condon factors
are calculated using the rotational
constant for each vibrational level toaccount for anharmonicity; however,
the equilibrium vibrational constantand harmonic oscillator wave functions were used for the calculations.

v”

0

1

v’
2

3

4

0
1
2
3
4

0.68
0.28
0.09
0.03
0.01

0.23
0.28
0.32
0.17
0.07

0.02
0.33
0.09
0.27
0.21

0.00
0.04
0.37
0.02
0.20

0.00
0.00
0.07
0.38
0.00

Table 5.8: Franck-Condon factors for3Φ - 3∆
2
1

v”

0

1

v’
2

3

4

0
1
2
3
4

0.41
0.35
0.20
0.10
0.05

0.38
0.00
0.15
0.21
0.17

0.12
0.29
0.07
0.01
0.10

0.01
0.23
0.12
0.15
0.02

0.00
0.05
0.28
0.02
0.15

Table 5.9: Franck-Condon factors for3Π
1 +0+ - Σ

v”

0

1

v’
2

0
1
2
3
4

0.47
0.35
0.18
0.08
0.04

0.35
0.04
0.21
0.22
0.15

0.07
0.36
0.01
0.07
0.17

3

4

0.00
0.16
0.26
0.08
0.01

0.00
0.01
0.22
0.15
0.14

170

Table 5.10: Franck-Condon factors for3 Π - 1 Σ+
1

v”

0

1

v’
2

3

4

0
1
2
3
4

0.45
0.35
0.19
0.09
0.04

0.36
0.02
0.19
0.22
0.16

0.09
0.34
0.03
0.04
0.14

0.01
0.19
0.20
0.12
0.00

0.00
0.02
0.26
0.08
0.16

Table 5.11: Franck-Condon factors for1 Π - 1 Σ+
1

v”

0

1

v’
2

0
1
2
3
4

0.36
0.34
0.22
0.12
0.06

0.39
0.00
0.11
0.19
0.18

0.14
0.26
0.09
0.00
0.07

3

4

0.02
0.25
0.09
0.16
0.03

0.00
0.05
0.28
0.01
0.14

Table 5.12: Franck-Condon factors for3Σ
1 +0+ - Σ

v”

0

1

v’
2

3

4

0
1
2
3
4

0.67
0.28
0.10
0.03
0.01

0.24
0.25
0.30
0.17
0.08

0.02
0.36
0.07
0.23
0.20

0.00
0.04
0.41
0.01
0.14

0.00
0.00
0.07
0.41
0.00

Chapter 6

Precision spectroscopy of trapped ions

In this chapter, we turn our attention to a different aspect of molecular spectroscopy – testsof fundamental physics with precision spectroscopy of trapped molecular ions. In particular, we
discuss an ongoing experiment to constrain the size of the electric dipole moment of the electron(eEDM).

6.1

eEDM BackgroundAs first pointed out by Purcell and Ramsey [391], a permanent electric dipole moment of anelementary particle would be a violation of parity (P) symmetry — that is, an inversion of spatial
coordinates (~r → −~r). To show this consider an electron, which has a spin about an axis, much likea spinning top. If we apply a parity transformation, nothing changes — the electron still spins inthe same direction. However, if the electron charge distribution is not spherical but is, for example,
ovoid as illustrated in Figure 6.1, then this transformation (denoted by the P-transformation arrowin the figure) changes the charge distribution relative to the spin direction. Such an asymmetriccharge distribution can be represented as the sum of two displaced charge distributions of opposite
charge, which is an electric dipole moment (EDM, with units of electron charge, e, times separationin cm). So we see that under a P-transformation, the dipole moment (red arrow) and the spin
vector (blue arrow) reverse direction relative to each other, violating P-reversal symmetry. This
assumes that the dipole moment is aligned along the spin vector; however, if this were not the case,then more quantum numbers would be needed to describe the electron, and the Pauli exclusion

172
principle would not be valid.
In addition to P-symmetry violation, the existence of an electron EDM (eEDM) would bea direct violation of time (T)-reversal symmetry (i.e., t → −t) [29], as shown with the lower
transformation in Figure 6.1. Under T reversal, the direction of the electron spin reverses, but
the shape of the charge distribution does not change; so the dipole moment does not change. Thisagain results in the spin vector and dipole moment being antiparallel if they were originally parallel,
which is a violation of T-reversal symmetry.
The Standard Model, the current theory explaining fundamental particles and their interactions, has been phenomenally successful and has been tested in many cases to high precision.
However, the Standard Model has some potential deficiencies. One primary issue is that the extent
of P-symmetry violation (which is linked to T-symmetry violation by the CPT theorem) can notaccount for the observed ratio between matter and antimatter in the universe by the theory of electroweak baryogenesis (this is one component of the Sakharov criteria) [392, 393, 394]. In addition,
the existence of dark matter is not accounted for in the Standard Model. Finally, a significant
amount of “fine-tuning” is required to reduce the CP-violating phase θ in quantum chromodynamics to less than 10−10 to fit with current measurements of the neutron EDM [395] (the strong-CP
problem) and to reduce the energy scale of gravity compared to the electroweak interaction by 1034
(the gauge hierarchy problem). Many theories extending the Standard Model, such as the popularsupersymmetry (SUSY) [396], have been developed to address difficulties with the Standard Model,
but no direct experimental evidence supporting these theories exists. Currently, the experimentally measured instances of P- and T-symmetry violation are explained in the Standard Model;
however, the Standard Model does not contain other instances of T-symmetry violation. Thus, if
additional cases of T-symmetry violation were discovered, this would provide direct evidence forphysics beyond the Standard Model [397, 398, 399, 400].
As discussed more below, measurements of the eEDM are performed currently either inparamagnetic atoms or molecules. In this case, the measured EDM can arise from an eEDM
or from an EDM of a proton or neutron appearing as an effective eEDM via electron-nucleon

173

Figure 6.1: An eEDM violates both parity (P) and time (T) reversal symmetry. The solid blackarrows show the effect of a P- or T-reversal operation on an electron with both a spin and anelectric dipole moment due to a charge distribution asymmetry.

174
interaction [399, 401, 402, 403]. This second contribution is the dominant term predicted by theStandard Model and is estimated to yield an effective eEDM of around 10−38 e cm (although someflexibility exists for increasing it to perhaps 10−33 e cm with significant tuning) [402]. The current
experimental limit for the sum of these two contributions is |de | < 8.7 × 10−29 e cm [31], whichmeans that a non-zero measurement will – at least for a while – be direct evidence of physics
beyond the Standard Model. An eEDM arises in new theories because of CP-violating interactions
with new particles. As an example, the size of the expected eEDM in SUSY, which arises from a
one-loop interaction in the Feynman diagram, is given by [404, 396]
α me tan βde ≈ esin φCP ≈4π Λ2 4

2.5TeVΛ2tan β sin φCP × 10−27 e cm,(6.1)

where me is the mass of the electron (in MeV), Λ is the energy scale for new particles, tan β is theratio of Higgs boson masses (SUSY assumes a two-Higgs doublet), and sin φCP is a CP-violatingphase. This is a fairly generic scaling for many theories that introduce an eEDM with one loop.
From here, several assumptions can be made to test different aspects of the theories. First, forsimplicity, it is often assumed that the contribution from electron-nucleon interaction is smaller
than the eEDM signal, so the current limit represents a limit on de . In this case, we can use thecurrent limit with Equation 6.1 to constrain Λ > 8 TeV, assuming tan β and sin φCP are both
∼1. This energy scale is already beyond rough “naturalness” constraints (i.e., not “too-much”
fine tuning), which are around 3 TeV, and are close to the upper limit favored by variations of
SUSY [396]. Alternatively, we can assume that Λ ∼ 1 − 3 TeV (the range allowed from Large
Hadron Collider (LHC) measurements and naturalness), and use the eEDM limit to constrain
sin φCP < 0.01 − 0.1 for tan β = 1. The measured Higgs boson mass1 is also significantly higher
than expected from simple SUSY [406, 396]. This can be partially explained with a larger tan β;
however, this also makes the eEDM constraints on sin φCP smaller.
Taken together, the LHC results for the Higgs mass, Higgs decay to two-photons, and lack of
new particles below 1 TeV combined with the EDM measurements are placing significant constraints
1

See The Review of Particle Physics [405] at http://pdg.lbl.gov/

175
on many SUSY models. The current eEDM limit is already probing potential two-loop contributions[399, 407] and is competitive with the current neutron EDM limit for probing high energy (e.g.,
10-100 TeV) particles in split SUSY [404, 406, 407]. An order of magnitude improvement on theeEDM measurement would start to test string/M-theory predictions, which estimate an eEDM up
to about 5 × 10−30 [408]. In addition to these tests, an improvement in the eEDM measurement (aswell as the Higgs decay to two-photons) can also constrain models of electroweak baryogenesis [409].
Conversely, the EDM limits can also be used to constrain the ratio of the expected Higgs decay
to two-photons relative to the standard model [410]. The eEDM measurement currently limits
this ratio to 1 ± 10−5 , which will be tested with new measurements at the LHC. All of the eEDM
constraints given above have assumed that the electron-nucleon interaction term is negligible. If
this term is included in an analysis of all EDM data, the limit is increased to de < 5.4 × 10−27[411]. Of course, the electron-nucleon interaction term also violates CP-reversal symmetry, so it
also tests for physics beyond the standard model. Interestingly, in the global analysis, a new eEDM
measurement in a different system near the current limit will result in better constraints on bothterms separately than would an improved measurement in the same system. The best option thenis to make an improved measurement in a new system, which is our goal.

6.1.1

Measurement Basics

Conceptually, measuring an eEDM is simple: an eEDM in an electric field will exhibit anenergy difference depending on the orientation of the dipole moment relative to the field. This isanalogous to a magnetic dipole moment — for example, a compass needle — placed in a magnetic
field. The compass needle aligns with the magnetic field because this is the lowest energy configuration; the configuration with the needle aligned opposite to the field is higher in energy by
an amount depending on the strength of the magnetic field and the size of the magnetic dipole.
This energy difference increases with the strength of the applied field. To measure an eEDM, one
needs to measure the energy difference of the two states of the electron in an electric field using
spectroscopy. If the eEDM is assumed to be parallel (or antiparallel) to the electron spin, this is176
equivalent to measuring the frequency difference — given by (2de Eeff )/h where h is the Planck
constant and de is the magnitude of the eEDM — between states with the electron spin parallel
and antiparallel to an applied electric field of magnitude Eeff . Since this frequency difference is
expected to be small, the two spin states are typically split with a magnetic field first, and then
any additional shift due to the electric field can be measured with a Ramsey spectroscopy sequence[412]. Technically most experiments actually use a more general Ramsey-type sequence because
the precession frequency is not measured relative to a local oscillator as it would be in Ramsey
spectroscopy.
We can measure the splitting between spin up and spin down with the following Ramsey-type
procedure:(1) initially prepare a collection of N0 electrons all spin up (|↑i);(2) rotate the spins with a π/2 pulse to √12 (|↑i + |↓i);(3) allow the spins to precess for a time τ , after which time |↑i and |↓i have accumulated aphase difference φ = (∆ω)τ ;(4) rotate the spins again with a π/2 pulse to N↑ |↑i + N↓ |↓i);(5) measure N↑ and N↓ .
If we include a non-ideal contrast due for example to decoherence, we can writeN↑(↓) =

N0
(1 ∓ C cos φ),
2

(6.2)

Nmax − Nmin
,
N0

(6.3)

where C is the fringe contrast:C=with Nmax(min) being the maximum (minimum) population measured by varying τ . Taking thedifferenceN↑ −N↓= −C cos φ provides a measurement of the accumulated phase φ. The fundamental
N0

limit to the precision of such measurement is set by shot noise [413, 395, 400]. The shot-noise-limited

177
uncertainty can be determined by first taking the derivative with respect to φ:δ(N↑ − N↓ )= N0 C sin φ.
δφ√Setting φ ≈ π/2 and using the shot-noise limit δN = N , we have√δ(N↑ − N↓ )
N0
δφ =
≈
,
N0 C
N0 C

(6.4)

(6.5)

We can relate δφ to an uncertainty in the dipole moment, δde since φ = (∆ω)τ and ~∆ω = 2de Eeff :
δde =

~
√
,
2Eeff τ C N0 RT(6.6)

where T is the total time for repeated measurements at a repetition rate R. In addition to statisticalsensitivity, the accuracy of any measurement can also be limited by systematic errors, which are
any signals that mimic the splitting due to the eEDM.
A bare electron will rapidly accelerate out of an electric field, making a measurement of theeEDM potentially challenging. While the first attempts to measure the eEDM did do this [414], amore elegant solution was proposed by Sandars [415, 416]. Due to relativistic effects, an unpaired
electron in a neutral atom with an applied electric field will experience an effective electric field.
D E
Initially, it was expected that this should not be true because E~ = 0 for the electron to be
D
E
D E
~
~
stable. This would seem to imply that de · E = de E~ = 0, so there should be no net effect fromthe interaction (this is known as Schiff’s Theorem). However, relativistic motion leads to lengthD
E D ED E~~~~
contraction, which means both de and E are time dependent. So, de · E 6= d~e E~ and can be
non-zero [417]. Even more remarkably, the electric field that the electron experiences in the atom
can actually be larger than the applied electric field. Because relativistic effects in general increasefor heavier atoms, the magnification also increases with atomic number. Thus, by measuring theenergy difference of two spin states in a heavy atom, it is possible to effectively measure the energy
difference of an electron oriented in two different directions in an electric field.
This concept has formed the basis of multiple generations of experiments, including one of themost precise measurements to date performed by Regan et al. [418]. The researchers used atomic
thallium-205 (Tl) to set a limit on the eEDM of 1.6 × 10−27 e cm. The experiment consisted of four

178
separate thallium-sodium beams, which each had a different combination of propagation direction(vertically up or down) and applied transverse electric and magnetic fields (E and B parallel oranti-parallel). Sodium was used as a comagnetometer to measure magnetic-field-related systematiceffects. For the measurement, atoms emerged from an oven and were collimated to form a beam.
A combination of lasers and a radio-frequency (rf) pulse created a coherent superposition of themF 6= 0 Zeeman sublevels of the ground F = 1 state. After this, the atoms traveled through a
99-cm-long region with a 123 kV/cm transverse electric field and a 0.38 G transverse magnetic
field. After the field region, a second rf pulse transferred some population, depending on the exact
value of the magnetic and electric fields, back to the mF = 0 level, which was then measured withlaser fluorescence. One of the dominant sources of systematic error was due to the motion of theatoms through the electric field combined with stray magnetic fields (termed motional magnetic
field effects). While significantly reduced by the use of four atomic beams with different propagation
directions, these effects still limited the accuracy of the measurement.

6.1.2

Current Limit

The relativistic enhancement factor for Tl is about 600, meaning that the effective electricfield at the electron is 600 times larger than the applied laboratory electric field (the effective
electric field in the Regan et al. experiment was about 80 MV/cm). Large enhancement factors are
possible using polar molecules with one heavy atom [419, 29]. This is due to the low electric fields
necessary to polarize the molecule (a result of the closely spaced levels of opposite parity present inmany molecules). This concept eventually resulted in an improved limit on the size of the eEDM
over the Tl measurement using ytterbium fluoride (YbF) molecules [30] to reach 1.0 × 10−27 e cm.
The experiment in YbF was conceptually very similar to the Tl experiment; however, only
one beam was used instead of the four beams in the Tl experiment and instead the direction of
the applied fields were switched. This simplification is partially due to the significantly reduced
sensitivity of YbF to motional magnetic field effects. In addition, the required electric field wasabout 10 kV/cm instead of 100 kV/cm, while the resulting effective electric field was 14.5 GV/cm,

179
about 200 times larger than that in Tl. The YbF experiment was primarily statistical noise limited
but did also have two systematics above 10−28 e cm [420]. The first of these is attributed touncorrected effects from imperfect E reversal. The second is due to a phase shift that depends onthe detuning of the rf pules. An imperfect E reversal or small bias voltages will result in different
Stark shifts and thus rf detunings for different directions of E, which will result in a correlated phaseshift. The next generation YbF experiment aims to reduce these systematics by better control overE reversal.
A recent experiment using ThO has constrained the eEDM to be less than 8.7 × 10−29 e cm[31]. This experiment was again done in a molecular beam, but the choice of molecule provided
several crucial advantages over YbF. In particular, the use of a 3 ∆1 electronic state requires a verysmall polarizing field (∼1 V/cm) due to the closely-spaced Λ-doublet levels (as discussed in Sections
5.1.1 and 5.3) and a has a small magnetic g-factor since gF ∼ (gL Λ − gS Σ) ∼ α/π [389, 421], where
gF , gL , and gS are the magnetic g-factors for each angular momentum component. The Λ-doublingin the 3 ∆1 state can also be used as an internal comagnetometer, which reduces sensitivity to straymagnetic fields and provides the ability to switch E without reversing any applied fields, an idea
initially developed for PbO [422].
The 3 ∆1 internal comagnetometer is illustrated for HfF+ in the inset of Figure 6.2. In
particular, we measure the splitting in both the upper and lower pairs of mF = ±3/2 levels (in theJ = 1, F = 3/2 manifold). These mF states are split in an applied magnetic and electric field by
3gµB B ± 2de Eeff for the upper and lower pairs, respectively. Here g is the magnetic g-factor for theupper and lower pairs, µB is the Bohr magneton, B is the applied magnetic field, de is the eEDM
(to be measured), and Eeff is the effective electric field (from theory). By taking the difference ofthe splitting in the upper and lower states, we can remove all magnetic field dependence — to theextent that the upper and lower pairs of states have the same g-factor.
Another difference in the ThO experiment compared to the YbF experiment is that state
preparation and readout were all optical transitions. In particular, preparation of the spin-statesuperposition was accomplished by preparing a dark state using a depletion laser instead of by180transferring population to a superposition of spin states using an rf pulse. The ThO measurement
was also statistical-noise limited, although the systematic error contribution was estimated to be 3×
10−29 e cm (the systematic error contributions were limited by the precision with which systematic
shifts were measured). In addition to possible systematic errors from magnetic field gradients [423]and stray magnetic fields, the largest systematic errors were related to differential Stark shifts of thedark state by the preparation laser that were caused by non-reversing electric fields or by different
Rabi rates for the two doublets (due, for example, to interference between E1 and M1 transition
amplitudes).

6.1.3

Overview of JILA eEDM measurement

One option for improving the sensitivity of measurements is to increase the free-evolution timeor coherence time of the measurement. In addition to an improvement in the shot-noise-limited
sensitivity, a longer coherence time also helps to reduce the sensitivity to systematic errors because
the Fourier-limited linewidth is smaller. This increase in coherence time can be accomplished byusing molecular ions, which can be easily trapped for many seconds. Specifically, we focus on HfF+
in the initial experiments. A future version of the experiment could use ThF+ for higher sensitivity,
but preliminary spectroscopy work for that molecule is still in progress.
This choice of molecular ion is motivated by several factors. First, the presence of a heavy
atom provides a large effective electric field, around 23.3-24 GV/cm in HfF+ [424, 421, 425] and38 GV/cm in ThF+ [426]. Note that this is significantly lower than initially estimated in [421]and is lower than the calculated value for ThO. This is because the ThO bond is more ionic than
the ThF+ , which means the s electron experiences a larger field and thus is mixed more with p.
Second, these molecules have long-lived, metastable 3 ∆1 electronic states. The small Λ-doubling of
this state (740 kHz in HfF+ ) results in full polarization with only a 1-V/cm-laboratory electric field,
which reduces sensitivity to systematic errors caused by large laboratory electric fields. As can beseen in Figure 6.3, numerical results for 3 ∆1 J = 1 hyperfine levels confirm that at 1 V/cm thesplitting in the |mF | = 3/2 levels due to eEDM related Stark shift is almost saturated at 2de Eeff .

181

Dissociating states

(c)
(b)

J=1, F = 3/2

3

(a)

Π0+

F3gFmBB+ 2deEeffF-Σ = -1

Σ = +1

Hf2+

Hf2+

EB

Hf2+
Hf

3

Δ1

Σ+

2+

Σ = +1
F-

1

not to scale

3gFmBB- 2deEeff

Σ = -1F-

mF = -3/2 -1/2 1/2 3/2

Figure 6.2: Electronic states used in an eEDM measurement in HfF+ . An eEDM is measured
by measuring the splitting of the J = 1, F = 3/2, mF = ±3/2 sublevels of the low-lying 3 ∆1
excited state, as shown in the inset. The states (a), (b), and (c) are excited states used for: (a)population transfer, (b) state readout via laser-induced fluorescence, and (c) state readout withphotodissociation.

182
This saturation occurs as the molecule becomes fully polarized. These results were obtained as
described in [427].
The basic measurement procedure is as follows. First, we state-selectively produce groundstate HfF+ ions using multi-photon photoionization. This process has been shown to produce
30% of the ions in a single quantum state in the ground 1 Σ+ electronic state [428]. The HfF+
ions are then trapped using a six-electrode linear Paul trap, see Section 6.2.1. To provide a bias
electric field, we apply a sinusoidal potential to each radial trap electrode 60 degrees out-of-phasewith the neighboring electrodes. A magnetic field must also be applied parallel or antiparallelto the rotating electric field (Section 6.2.3). After the electric and magnetic fields are applied,
population is transferred (Section 6.3) to either the upper or lower pairs of states using a twophoton transition through an intermediate 3 Π0+ state (Figure 6.2). This transfer scheme results
in an incoherent population distribution between the mF = ±3/2 levels, so we prepare a pure spinstate by depleting one mF level. The next step in the Ramsey spectroscopy sequence (Section 6.4)
is to prepare a coherent superposition of the mF = ±3/2 levels via a π/2 pulse. After this π/2pulse, the superposition evolves for a free-evolution period during which the two mF levels acquirea differential phase shift proportional to their energy difference and the evolution time. A second
π/2 pulse finishes the Ramsey sequence by projecting back to pure spin states. This maps the phasedifference to a population difference between the two spin states. We then measure the population
in a spin state with depletion followed by resonantly-enhanced multi-photon dissociation (REMPD,
Section 6.2.2).

6.2

Experimental Setup
Figure 6.4 shows the basic setup used for the JILA molecular-ion eEDM experiment. Many

parts of the experiment have been described in detail elsewhere [64], so we only provide a summaryand some added technical details here. We start by producing neutral HfF using laser ablation (532
nm, 10 ns, ∼10 mJ pulses) of a Hf target at the outlet of a pulsed supersonic expansion of 1% SF6 in
Ar (backing pressure of ∼80 psi). Initially, we used a solenoid valve (Parker General Valve), but we

183

1

∆E/(2Eeffde)

0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8
1
1.2
Electric field (V/cm)

1.4

1.6

1.8

2

Figure 6.3: The eEDM induced Stark splitting (∆E) for J = 1 hyperfine levels of the 3 ∆1 state of
HfF+ : between F = 3/2 mF = ±3/2 levels (solid line), between F = 3/2 mF = ±1/2 levels (dotedline), and between F = 1/2 mF = ±1/2 (dashed line).

184
found overall improved performance with a disk-piezo based valve (Physik Instrumente P-286.23).
This valve allowed us to reduce the pulse duration to typically ∼35 µs and simultaneously increased
our final ion yield. When installing this valve, the poppet tension was adjusted first so that thevalve barely sealed to atmosphere when the chamber was pumped with only a roughing pump andwas then tightened about one-half to one turn further. In order to prevent damage to the piezo,
the tension should be kept as low as possible. The piezo is biased at 275 Volts to open the valve.
The gas exits through a 1-mm diameter nozzle. The ablation target is located just downstream ofnozzle and is positioned to be just on the edge of the expanding beam. The timing of the ablation
laser is experimentally adjusted to optimize the final trapped HfF+ yield and is typically ∼110 µs
after the start of the gas pulse.

6.2.1

Ion Trap

The supersonic expansion is collimated using a pair of skimmers: the first skimmer (3 mmdiameter) is located about 11 mm from the source and the second skimmer (2 mm diameter) is
located 9.6 cm farther downstream. This gives an expected transverse translational temperature
based on the collimation (i.e., the maximum velocity a molecule can have to barely pass throughboth skimmers) of 3 K. The first skimmer is also biased at 200 V to remove any ions produced in
the ablation. After this, the neutral HfF is ionized using resonant two-photon vibrational autoionization through a highly-excited Rydberg level [429]. This process produces HfF+ only in the
1 Σ+ v = 0 state, with about 35% of the population in the J = 0 level.

The ions are produced at

the center of ion trap and are immediately stopped with a pulsed electric field. The trapping fields
are turned on immediately after this.
The trap is a linear Paul trap with a standard rf (50 kHz) radial quadrupole confinement andstatic dc axial confinement [430, 431, 64]. As shown in Figure 6.4, the trap consists of six radial
electrodes. In order to produce the quadrupole radial confinement, we drive some of the electrodes
synchronously (e.g., the blue and purple electrodes in Figure 6.4), The motion of a single trappedAblationlaser (532 nm)
Ar (99%) +SF6 (1%)
Pulsedvalve Hf

V = -Vrfcos(ωrf t) +Vrotsin(ωrot t - 60˚)

Hf
+V

Y

V = -Vrfcos(ωrf t) +Ionization
Vrotsin(ωrot t + 0˚)+ Dissociation+ Depletion lasersZ

Top view

X

185

Fast MCP

HfF+
Blab

F
V = 2Vrfcos(ωrf t) +Vrotsin(ωrot t -120˚)Side view

B field over one rotation cycle:
Brot

Imaging
MCP

Bstatic+t

Figure 6.4: Sketch of eEDM experiment. A pulsed beam of neutral HfF is created with a supersonic
expansion and ablation of Hf. After a pair of skimmers, we create HfF+ using photoionization in
the center of a six-electrode linear Paul trap (top shown). The radial electrodes provide botha quadrupole rf confinement (ωrf = 2π × 50 kHz) and uniform rotating electric field (ωrf ≈2π × 253 kHz). A static quadrupole magnetic field gradient leads to a rotating magnetic field dueto the circular micromotion from the rotating electric field. Ions can be detected either with a fast
micro-channel plate (MCP) aligned along the trap x axis or with an imaging MCP.

186
ion along the ith direction (i = x, y, z) is given approximately by [431]hiqixj (t) ≈ [a0j + a1j cos(ωi t + φi )] 1 + cos(ωrf t) ,
2

(6.7)

where a0j arises from a static dc electric field,
a0j =4eE~dc · x̂j2m(ai + 12 qi2 )ωrf

and a1j is the secular motion amplitude with frequency ωi = 21 ωrf(6.8)
qaj + 12 qj2 . The parameters ai

and qi are sometimes called the Mathieu parameters and are related to the confinement by
4eκVdc1ax = ay = − az = −2 ≈ 0.004,
2
mZ02 ωrf
2eVrf
1
qx = −qy = az = −2 ≈ 0.23, and2
mR2 ωrf(6.9)

qz = 0where κ is a geometrical factor and the other parameters are given in Figure 6.4. The kinetic energy
averaged over a secular period, assuming ωrf >> ωi and q >> a, is
1
1
4
Ej = m ẋ2j = ma21j ωi2 +24m

eqj E~dc · x̂jωrf (2aj + qj2 )!2.

(6.10)

Assuming a thermal distribution, Ez ≈ 12 kB Tz and Ex,y ≈ kB Tx,y because of the micromotionenergy, which is equal to the secular motion energy [431].
We detect ions by turning off the trap and kicking the ions to one of two micro-channel
plates (MCP). One MCP is connected to a fast trans-impedance amplifier to provide high temporal
resolution; the other MCP provides a two-dimensional image of the ions. The arrangement of these
two MCPs is shown in Figure 6.4. The fast MCP is aligned along the x-axis of the trap, while theimaging MCP is 60◦ off of the x-axis. When used in combination, these two MCPs can provide
three-dimensional information about the trap, but first we need to obtain a relationship between
the position xdj of an ion on the detector along the j th coordinate axis and the position xtj andvelocity ẋtj of the ion in the trap. In particular, we assume that
xdj = m11,j xtj + m12,j τd ẋtj

(6.11)

187
where τd is the time of flight to the detector and m11,j and m12,j are two coefficients that must be
measured. Ideally, both of these coefficients would be 1, but by measuring them, we can accountfor first-order non-uniformity of the kick or stray fields due to patch charges.
In order to measure the two coefficients, we kick the cloud along one axis by pulsing thevoltage on the electrodes for a duration of 2 µs (with an approximately triangular waveform). To
determine the impulse applied, we measured the kick waveform applied to the electrodes and thenintegrate the applied voltage: for example, the integrated kick applied to one fin for a programmed
1 V, 2 µs pulse (amplified 25× in the trap driver electronics) gives 50.5 V µs. The electric field at
the center of the cloud is obtained by COMSOL simulation for a 1 V potential applied to one fin:√
57.21(1) mV/cm
Ex = 2 3 ×× Vx ≡ kx Vx ;1V
57.21(1) mV/cm
Ey = 2 ×× Vy ≡ ky Vy ;1V
8.75(1) mV/cm
Ez = 2 ×× Vz ≡ kz Vz .
1V

(6.12)√
The factors of 2 3 in x and 2 in y and z arise from the electrode geometry. Assuming that thecloud is stationary before receiving a kick along direction j and that the kick is short enough sothat it does not affect the position of the ions, the in-trap position and velocity after the kick are
xtj (t) =Bjsin(ωj t)
ωj(6.13)

ẋtj (t) = Bj cos(ωj t)
where
Bj ≡

e
m

Z ∞
−∞

Ej (t)dt =

e
kj
m

Z ∞
Vj (t)dt.

(6.14)−∞

By scanning the time between the kick and readout, we measure an slosh of the ion cloud on thedetector with amplitude Adj and phase ϕj :xdj (t) = Adj sin(ωj t + ϕj ) = Adj cos(ϕj ) sin(ωj t) + Adj sin(ϕj ) cos(ωj t).

(6.15)

188
Comparing Equations 6.13 and 6.15, we find that
Adj ωj cos(ϕj )m11,j =
Bj

(6.16)

Adj sin(ϕj ).
Bj τd(6.17)

m12,j =

These measurements can be used to determine the temperature of the ions in the trap. First,
we need to find the width of the cloud in the trap from the measured width on the detector:
σjd.
σt,j = q
m211,j + (m12,j ωj τd )2(6.18)

Because the imaging detector is not aligned with a primary trap axes, there is some projection of
motion along x onto the imaging detector. Taking into account the 30◦ angle between the imaging
detector y 0 axis and the trap x axis, the observed width σimg,y0 is a combination of the motion
projected along the trap x axis, σ̃img,x , and motion along the trap y axis, σ̃img,yσimg,y0 =1q 2
2
3σ̃img,x + σ̃img,y.
2(6.19)

The width due to x motion isσ̃img,x = σt,x

q

2
g11,x
+ (g12,x ωx τimg )2 ,(6.20)

where g11 and g12 are the coefficients for mapping x motion on the imaging detector. Finally thewidth observed on the detector due only to y motion isr
σ̃img,y =

h
i
2
2
2 .
4σimg,y− 3 g11,x+ (g12,x ωx τimg )2 σt,x(6.21)We have measured the transfer functions with several different kicks to obtain the transfer functions given in Table 6.1 for two different configurations, called relaxed and normal. For these two
configurations, the trap is turned on in the given configuration, and then the transfer function
coefficients are also measured in that configuration.
Assuming harmonic confinement and neglecting Coulomb repulsion between the ions (i.e.,
space-charge effects) which distorts the trap profile [432, 433, 434], the trap width along the j-th

189

Table 6.1: Secular trap frequencies, transfer function coefficients, trap width, and trap temperatures for two different trap configurations. In addition, the secular frequencies, widths, andtemperatures are given for an adiabatic ramp starting from the relaxed trap.
ConfigAxis

ωj /2π (kHz)m11 or g11

m12 or g12

σt,j (mm)Temp (K)

Relaxedx

3.5

0.093(8) µs/mm

0.08(1) µs/mm

1.1

14

ximg

n/a

0.56(9)0.65(5)

n/a

n/a

y

4.8

2.7(4)

1.04(16)

1.7

64

z

1.6

0.75(3)0.72(6)

3.1

23

x

4.4

0.079(4) µs/mm

0.08(1) µs/mm

1.2

27

ximg

n/a

0.3(1)

0.60(4)

n/a

n/a

y

5.6

3.0(1)

0.69(8)

1.8

93

z

3.7

1.09(7)0.42(6)

1.9

48

Relaxed +

x

2.8

–

–

1.2

11

ramp

y

4.3

–

–

1.5

44

z

0.7

–

–

4.2

9

Normal190
axis is related to the temperature in that direction by
kB Tj
,
mωj2

(6.22)

2mωj2 σt,j.
kB(6.23)

2σt,j=Rearranging give the temperature
Tj =We can use the measured width on the detector and the transfer function coefficients to determine
σt,j and use this to determine Tj (Table 6.1). Because of the complication of both x and y appearing on the imaging detector, the y measurements are the least reliable, which is possibly why thetemperature seems high. From this, we see that the temperature is lower with a looser initial trap.
The x and z temperature are reasonably close to – although higher than – the expected Doppler
temperature from the molecular beam collimation. We have also confirmed that the temperaturescales approximately with secular frequency during an adiabatic ramp, as expected [435]: it actually seemed that the temperature decreased slightly faster than secular frequency, which could
be an indication of space-charge effects. By initializing the trap in the relaxed configuration andthen adiabatically ramping to trap frequencies of (2.8, 4.3, 0.7) kHz in (x, y, z), we were able tofurther decrease the temperature to (11, 40, 9) K. Currently, the trap becomes unstable when we
ramp to lower secular frequencies; this is perhaps due to ground loops that introduce noise on thetrap electrodes. We have reduced these significantly with optical isolation of digital signals andcareful separation of analog power supplies when possible, but there are still residual effects. It isalso possible that we need to lower the trap rf frequency in order to further decrease the secularfrequencies. We have also tried to reduce the temperature by reducing the size of the ionization
laser beams with no effect (if anything, the temperature is lower with larger ionization laser beams).
Of course, to count the highest number of ions possible for the eEDM measurement, we want
the absolute detection efficiency of the MCP to be as high as possible. It is often said that ion
detection occurs with unit efficiency, but this is not completely correct as the absolute detection
efficiency depends on several different factors. As its name implies, an MCP is an array of small
glass channels that are fused together to form a plate. The input face of this plate is negatively

191
biased (typically up to −1.1 kV across one plate) so that positive ions are accelerated toward thesurface. When the ion hits inside a channel, several electrons are ejected. These electrons areaccelerated towards the output and collide with the wall, which causes a cascade of electrons at
the output. These electrodes are detected, either as a current with a fast transimpedance amplifier(for the fast MCP) or a phosphor screen and camera (for the imaging MCP). Typically, multiple
plates are stacked together (two for the fast MCP, three for the imaging MCP), which increases
the total bias applied and also increases the electron gain. A grounded mesh is located in front of
the MCP front surface to shield the trap from the high voltage.
The first limit to the MCP detection efficiency is the transmission of this mesh. Previously, weused a coarse mesh with about 60% transmission, but have now switched to a fine mesh with ∼90%
transmission. Despite having a large transmission, the pitch of the mesh is small, so we noticed noissues due to worse shielding of the bias voltage. Once an ion is through the mesh, the efficiency of
producing secondary electrons depends on the energy (and thus bias, since this accelerates the ions)and mass-to-charge ratio of the impacting ion. As the impact energy is increased, the efficiency
has been observed to increase and eventually saturate at < 100% [436]. This plateau value is close
to the open-area-ratio (i.e., the ratio of the total area of the tubes to the area of the plate) of
the MCP, typically around 63% for the MCPs that we use. This limit arises because ions that do
not enter a tube and just strike the surface emit secondary electrons that are not collected. The
efficiency can be increased beyond this by applying an additional return bias fields using anothermesh to drive electrons back towards the surface [437]; however, the efficiency improvement seems
to only be marginal under most conditions (e.g., 80% instead of 60%) [438]. Coating the MCP witha material to increase the secondary electron yield could push the efficiency up even higher [438].
We measured the counted ion number (for very low total ion number, any peak over a thresholdvoltage is counted as a single ion) as a function of bias voltage and found that it saturated atabout 2.05-2.1 kV, which suggests that we are reaching the limit of the mesh plus open-area-ratioof the MCP. For measuring larger ion numbers, we cannot count peaks because several ions could
contribute to a single peak. The large noise in peak heights even for a single ion makes it impossible

192
to define thresholds for more than one ion, so instead, we integrate to find the total area under the
peak. We have determined a calibration based on the average integrated area of single ion peaks
and have compared integrated and counted signal at low ion number as a function of bias voltage
to be sure that the two methods agree.
Interestingly, we have noticed that using the integrated signal to measure the ion number
consistently results in slightly higher noise than using counted ion number. On way of seeing thisis to look at the statistics for the number of Hf+ measured with counting versus integrating andcompare this to a Poisson distribution. Histograms of the number of Hf+ measured using counting
and integrating are shown in Figure 6.5. The solid lines are the expected Poisson distribution
given the measured mean Hf+ number. From this, we see that the integrated ion number differs
significantly more from a Poisson distribution than the counted ion number. We have also confirmed
this by comparing the standard error on Hf+ measurements to the calculated shot noise. Typically,
the noise when using counts is around 20% above shot noise, while the noise using integrated Hf+number is around 50% above shot noise.

6.2.2

Photodissociation

Ideally, we would like to be able to measure the population in any desired Zeeman sublevelwithin the 1 Σ+ or 3 ∆1 state, but as was pointed out years ago “you can’t always get what you want”[439]. In order to get what we need, we can use a combination of techniques. Historically, a common
molecular state read-out method is laser-induced fluorescence (LIF) [440], where molecules in thetarget state are excited by a bound-to-bound transition and the subsequent spontaneously-emitted
photons are measured. However, the lack of closed transitions in HfF+ and the low detection
efficiency of photomultiplier tubes at the known LIF transition wavelengths combined to makeLIF count rates very low. The low efficiency of photon detection can be circumvented by directly
detecting ions instead. In neutral molecules, a common detection technique is resonance-enhanced
multiphoton ionization (REMPI) [441, 442], which has been used for efficient measurement ofmolecular quantum state populations. In molecular ions, the difficulty of efficiently and controllably

193

450
countsintegrated

400

Number of occurances

350
300
250
200
150
100
50
0

0

5

10

15

Hf

20

25

30

35

40

+

Figure 6.5: Histogram of Hf+ number measured using counting or integrating. The solid lines are
the expected Poisson distribution with the same mean value.

194
removing the second electron (because of the typically very high second ionization energy) makes
REMPI a challenging technique to apply.
An alternative to REMPI is to instead state-selectively dissociate the molecular ion anddetect the dissociation products, leading to the development of resonance-enhanced multiphoton
photodissociation (REMPD) [443, 444]. Soon after the first demonstrations REMPD was combinedwith ion trapping to perform spectroscopy on trapped molecular ions [445]. Most applications reliedon mass-selectivity to obtain zero-background spectra of excited molecular states, but have not used
REMPD to sensitively measure lower state populations. Recent developments have focused on highsenstivity detection of small collections of trapped ions. In Roth et al. [446] and Højbjerre et al. [67],rotational-state-resolved photodissociation was demonstrated with molecular ions in a Coulomb
crystal. There, high signal-to-noise images of co-trapped atomic ions showed the photodissociatedmolecular ions as a loss of dark ions. While providing high quantum efficiency, this technique only
works with Coulomb crystals. Rellergert et al. [447] instead used one-photon photodissociation
followed by mass-resolved time-of-flight ion detection. This method was sensitive to the distributionof molecular vibrational states but did not provide rotational state selectivity. We demonstrate a
combination of these techniques that provides high quantum efficiency, rotational state selectivity
for multiple vibronic states, and rapid detection without the need for Coulomb crystals. A similar
scheme has recently been demonstrated for rotational-state specific detection of AlH+ as well [448].
We use (1+10 ) REMPD to break apart HfF+ into Hf+ and F and then count the Hf+ products
and remaining HfF+ ions. The first photon in the (1+10 ) scheme drives a bound-bound transitionin HfF+ , making the photodissociation process rotational-state sensitive. Subsequently, the second
photon excites HfF+ from its intermediate state to a repulsive molecular potential, as shown in
Fig. 6.6(a). The two ion species are mass-resolved in time-of-flight to the detector. When no
photodissociation pulse is applied, we see one large ion peak on the MCP centered 30 µs after thereadout kick, which corresponds to the time-of-flight of HfF+ (Fig. 6.6(b)). If the photodissociation
pulses are applied on resonance, a second, smaller number of Hf+ ions appears at an earlier time.
Prior to applying the dissociation lasers, the axial and radial confinement are ramped up to form a195tighter trap (∼ 12 kHz in all three dimensions). This helps to increase the dissociation efficiency,
reduce the background counts, and also improve the mass resolution of the time-of-flight detection.
We also wait for a “focusing” time after dissociation before kicking the ions toward the MCP [65].
To calibrate the photodissociation detection efficiency, we probe the X 1 Σ+ , v 00 = 0, J 00 = 0ions using the R(0) line of the 35,976 cm−1 band shown in Figure 6.7. Note the absence ofa Q-branch, which indicates that the intermediate state is also Ω = 0. We observe that thephotodissociation yield saturates with a laser fluence of 160(30) µJ/cm2 for the first photon andapproximately 50 mJ/cm2 for the second photon. The saturation fluence on resonance, Fsat ishc
related to the peak transition cross section, σ0 : Fsat = λσ
, where0

σ0 =4π 3 g0 µ221 g2.
30 hλ g1(6.24)

Here gg12 is the ratio of the upper and lower state degeneracy, which we assume to be 1; µ21 is theusual transition dipole moment; g0 is the peak of the normalized line shape function g(ω). For
the 266 nm transition to the dissociating curve, we assume that g(ω) can be obtained by mapping
the harmonic oscillator wave function for the intermediate state onto the dissociating potential, as
shown by the dashed gray lines in Figure 6.7(a). The v = 0 wave function is a Gaussian with widthq~∆r = 2πcω, where ωe is the vibrational constant (in cm−1 ). With ωe ≈ 650 cm−1 (estimatedeµfrom the higher states in Table 5.5), ∆r ≈ 0.055 Å. The slope of the dissociating resonance has been
estimated to be k = 80000 cm−1 /Å [63], which gives a width of the photodissociation resonanceof ∆ν̃ = k∆r = 4400 cm−1 . The peak of an (area) normalized Gaussian with spectral width ∆ν̃,1
g0 = √2π2πc∆ν̃. If we assume an electronic dipole moment of µ21 ≈ 0.5 ea0 , σ0 ≈ 7.8 × 10−19 cm2 .

This gives an estimated saturation fluence of ∼ 300 mJ/cm2 , which is in reasonable agreement withthe experimental result given the large number of assumptions.
From the ratio of total HfF+ counts on the MCP to the Hf+ counts that are detected as
products of state-sensitive photodissociation, we derive an efficiency of 32(2)%. This includes a
dissociation efficiency of 41(2)%, which we measure from the fractional loss of HfF+ signal whenthe (1+10 ) pulse is applied, and a 78(2)% transport efficiency of the dissociated ions to the detector.

196

(a)

(b)

3.5

Δν

Hf++F
266 nm
Δr

Ω=0
3

Σ+

1

Δ1

HfF+

2.5
2
1.5
1

Hf+

0.5

HfF+

270-285 nm

Ion signal (V)

3

0
15

20

25
30
Arrival time (μs)

3540Figure 6.6: (1+10 ) resonance-enhanced multiphoton photodissociation (REMPD). (a) Photodissociation pulses consist of two photons of different color. The first photon excites molecules fromtheir initial state (one of X 1 Σ+ v 00 ∈ {0, 1} or 3 ∆1 v 00 = 0 to an intermediate state, and the second
266 nm photon dissociates the molecules into Hf+ and F. The spectral width (∆ν) of the resonanceto the repulsive potential, indicated by the vertical, green Gaussian is related to the radial size(∆r) of the vibrational wavefunction of the intermediate state (gray Gaussian). (b) Mass-resolved
time-of-flight ion detection signal, recorded without photodissociation (red) and with photodissociation (blue). The gray dashed line indicates the distinction between Hf+ (to the left) and HfF+
(to the right).

197

60
50

Hf+ Ion Count

R(0)

20 ms
220 ms

R(1)
R(2)

40
30

P(2)
P(3)

20

P(1)

P(4)
10
0

P(5)

−12

−10

−8

−6

−4

−2

0

2

4

First photon frequency − 35976 cm-1
Figure 6.7: Rotational state distribution of trapped ions in X 1 Σ+ v 00 = 0 state: 20 ms after trapping(blue) and 220 ms after trapping (red). The first REMPD photon is tuned near 35,976 cm−1 . We
observed a redistribution towards higher rotational states after 220 ms of wait time.

198
The detector itself counts 57% (90% throughput from the grounded copper mesh in front of thedetector, 63% from the open area of the detector, and 100% quantum efficiency for a given open
channel) of the incident ions. Therefore the overall efficiency, including detector loss, is 18(2)%.
Compared to LIF detection [428], this is an improvement by a factor of more than 200.
Population in the 3 ∆1 v = 0, J = 1 can be detected by dissociating through the sameintermediate state using the 35,006 cm−1 band. The yield on this transition saturates at a fluenceof 70(10) µJ/cm2 . We measure a photodissociation efficiency of around 13% (not including detectorloss) from 3 ∆1 . Currently, we do not understand why the photodissociation efficiency is lower forthe 3 ∆1 than for the 1 Σ+ . The dissociation efficiency is lower for the P-branch than for the Rbranch, which could indicate that state degeneracies play a role (the ratio of excited state to lower
state degeneracy is smaller for the P-branch). To check this (and other effects) we have trieddissociation through other intermediate states but observed no difference in maximum number ofdissociated ions.
The dissociation can be increased by applying multiple dissociation pulses (at a 10 Hz rate)and holding the dissociated Hf+ in the trap between pulses. This is shown for the both 1 Σ+ and3∆

1 in Figure 6.8(a) and (b), respectively. Unfortunately, we find a rapid loss of about 50% of the

dissociated Hf+ within 100 ms (most of the loss was within several ms), which reduces the total
efficiency from multiple dissociation pulses. We experimented with different trapping configurations
to reduce this loss, thinking that perhaps the trap was too loose during the hold time (we expect the
Hf+ to have about 2000 cm−1 of energy, and the trap depth during the hold time was about 2000
cm−1 ), but increasing the trap depth during the hold time did not help. If we include this loss, wefind that we dissociate about 350 HfF+ from 1 Σ+ and about 80 from 3 ∆1 out of a total ion numberof ∼ 3000 HfF+ . This corresponds to about 85% of the expected number in X 1 Σ+ v = 0, J = 0 and
40% of the 3 ∆1 v = 0, J = 1 (including the transport efficiency of 78% from the previous paragraph);
the reason that the total dissociation is lower for 3 ∆1 is still unknown. We are currently replacing
the pulsed YAG laser that pumps a dye laser for the first photodissociation step as well as provides
the 266 nm light to have a firing rate of 30 Hz, so we will be able to apply multiple dissociation

199
pulses faster.
Over the first 200 ms of trapping, we observe a redistribution towards higher rotational statescompared to the initial rotational state distribution from the autoionization (see Figure 6.7). We
believe this state redistribution is due in part to Langevin collisions of HfF+ ions with the neutral
argon atoms that are injected as the supersonic beam’s carrier gas. It could also be caused bycollisions with some initially hot ions. Figure 6.9(a) shows rapid initial loss from the J = 0 state(due to a relatively high argon pressure) followed by a much smaller loss rate after the argon
has been pumped out of the vacuum chamber. The amount of fast loss was reduced by adding an
additional turbo pump on the trap chamber and by decreasing the open duration of the pulsed valve(which was possible by switching to the piezo valve instead of the solenoid valve). We observed
the fraction of ions initially lost from the J = 0 decrease from about 70% to about 40% whenthe valve duration was lowered from 100 µs to 40 µs. In addition, the timescale (i.e., 1/e time)of the loss increased with decreasing pulse duration, which is consistent with a collisional processsince the slope is related to the gas density. However, no further improvement was observed witheven shorter pulse durations, and below about 35 µs, the initial fraction of ions in J = 0 decreased
rapidly (presumably due to worse cooling in the supersonic expansion). Currently, we believe thatthe pump-out speed in the trap center is limited by the mirrors that surround the trap electrodes(which were initially used for fluorescence detection and also serve as the ground plane for the iontrap). Removing these and further increasing the pumping speed may preserve a larger fraction of
the ions that occupy the desired rotational state for subsequent experiments; however, we have notyet been able to confirm that this initial heating is currently due to collisions.

6.2.3

Rotating bias fields

For the eEDM measurement, we need to be able to apply parallel (and anti-parallel) electricand magnetic bias fields to our trapped ions. At first glance, this seems impossible, because astatic electric field will apply a force to the ions and accelerate them out of the trap, which turns
out to be non-ideal for obtaining a long coherence time. We can instead be more creative and

200

(a)

Hf+

150
100
50
0

1

2

3

4

5

1

2
3
4
Number of dissociation pulses

5(b) 50

Hf+

40
30
20
10
0

Figure 6.8: Dissociated Hf+ from the (a) X 1 Σ+ v = 0, J = 0 and (b) 3 ∆1 v = 0, J = 1 measured
as a function of the number of dissociation pulses. For this measurement, the dissociation shutter
was triggered 260 ms after the start of the trap for a variable duration (the dissociation lasers arepulsed at a 10 Hz rate), and the trap duration was fixed to 710.1 ms. The total ion number was
about 3000 HfF+ .

201

(a)

Measured Hf+

100

(b)

10
1

= 7.1(8) s

0.1

Measured Hf+

100
10= 2.1(2) s

1

0.1

0

2

4

6

8

Trap time (s)

10

12

14

16

Figure 6.9: Population lifetimes in trapped HfF+ . (a) X 1 Σ+ v = 0, J = 0 and (b) 3 ∆1 v = 0, J = 1.
The X 1 Σ+ lifetime exceeds many seconds after the initial fast decay (100 ms) due to rotational
re-distribution as described in Section IV. The 3 ∆1 lifetime is 2.1(2) s.

202
rapidly rotate the bias electric field. This will cause an additional, rotating micromotion, but if theelectric field rotation frequency is significantly higher than the trap rf then the trap motion androtating micromotion do not influence each other. In addition, the rotation frequency is chosen to
be incommensurate with the trap frequency to prevent coherent coupling. One concern, though,is that inevitable non-uniformities in the rotating bias field give rise to a ponderomotive potential[449]. This is the same effect that leads to a net trapping pseudopotential from the rf electric fields
in the Paul trap. The ponderomotive potential from a field oscillating with angular frequency ω,eφpond = 4mω
E 2 . This results in a force, Fpond = −e∇φpond , which can cause several problems.

First, the ponderomotive potential adds with the trap pseudopotential and can shift the trap centerand add additional confinement or de-confinement. In addition, the ponderomotive force can cause
an impulse kick and trap slosh as the field is turned on. To try to avoid these issues, we designed
the trap to be as symmetric as possible. We also monitor the voltage applied to each trap electrodeand adjust the phase and amplitude of each drive to be as close as possible to the ideal given in
Figure 6.4.
~ =The radius of the rotating micromotion at any position in the trap is given by rrot (R)~eErot (R)~ is the position of the micromotion circle from the center of the trap, Erot (R)~ is, where R
2mωrot

the value of the electric field, and ωrot is the rotation rate. For Erot of 20 V/cm and ωrot = 2π ×253.7912 kHz, rrot ≈ 0.4 mm, corresponding to a tangential velocity of vrot = eErot /(mωrot ) ≈ 620
m/s.
One option for creating a magnetic field that is parallel to the electric field is to flow current
through the trap electrodes when driving them to produce the electric field, but it is difficult to
produce a very uniform field with this technique. Instead, we opted to use the rotating micromotion
to our advantage and use it to create a rotating bias field from a static field gradient. To do this, we~ = B 0 (xx̂, y ŷ, −2z ẑ).
use a pair of anti-Helmholtz coils to produce a radial quadrupole gradient, B
Over one rotating micromotion cycle, we can write the magnetic field experienced by the ion as~~ rot (t) +a sum of a static and a time dependent component, as shown in Figure 6.4: B(t)= B
~ static (R),
~ where B~ rot (t) = ∇r B
~ · ~rrot (t) and ~rrot (t) is the ion position from the center of theB203
rotating micromotion circle.
We can understand this by time averaging the instanteous magnetic field projected onto the~ · E/|~ E|.
~ The electric field along x̂ can be written as a combination of thequantization axis, i.e., B
applied Erot and the trapping field:
VrfκVdc
E~ = −Erot (cos(ωrot t)x̂, sin(ωrot t)ŷ, 0ẑ) − 2 cos(ωrf t)(xx̂, −y ŷ, 0ẑ) − 2 (−xx̂, −y ŷ, 2z ẑ). (6.25)
RZ
The position, (x, y, z), is given by Equation 6.7 with the addition of a rotating micromotion termED
~ · E~ /|E|~ using Equation 6.7 for (x, y, z) and assuming
xrot = rrot cos ωrot t. We can now evaluate B~ ≈ Erot and Edc = 0. We also assume that the trap is symmetric in x, y (i.e., ωx = ωy ≡ ωrthat |E|and a1x = a1y ≡ a1r , which is not exactly correct but should be a good approximation). After time
averaging over each motional frequency separately and using Equation 6.9, we find2 ω22ma21r ωr2 mrrot
ma21z ωz2 ma21r ωz2 4ma21r ωz2 ωr2 
zBrot = B 0 rrot +
+
+
+
+
2
eErot
2eErot2eErot2eEroteErot ωrf2 ω22(Ex + Ey + Ez ) mrrot(Ex + Ey ) ωz2 4(Ex + Ey ) ωz2 z≈ B 0 rrot +
+
+
+
.
2
eErot
2eErot2eErot ωr2
eErot
ωrf

(6.26)We chose to use an applied quadrupole magnetic field because the uniformity should be very
good. Our anti-Helmholz coils have a radius of 13.6 cm and are separated by 23.6 cm. We cancalculate the field expected from this configuration using the results from [450] and estimate thequadratic correction to the gradient to be around 0.1%/(cm)2 in x assuming perfect coil alignment.
In addition, we have roughly measured the field from the coils before they were installed on thevacuum system, as shown in Figure 6.10. We see no evidence of inhomogeneity of the magnetic
field gradient that is outside of the measurement error. This shows that our assumption about theuniformity of the applied gradient is valid.

6.3

Coherent Transfer
Stimulated Raman adiabatic passage (STIRAP) is a powerful technique for state-selectivelytransferring population between electronic states in atoms and molecules with high efficiency [451,452]. By adiabatically eliminating the intermediate state, spontaneous emission can be completely

residuals (G)magnetic field (G)residuals (G) magnetic field (G)residuals (G)magnetic field (G)

204

0.2
0

-0.2
-0.4

-0.6
0
0.02
0.01
0
-0.01
-0.02
0

1

2

3

4

5

6

7

8

1

2

3
4
5
x position (mm)

6

7

8

10

12

14

6
8
10
y position (mm)

12

14

0.4
0
-0.4
-0.8

0
0.02
0.01
0
-0.01
-0.02
0
1
0.5
0
-0.5
-1
-1.5

2

4

2

4

6

8

0

2

4

8

10

12

0.02
0.01
0
-0.01
-0.02
0

2

4
6
8
z position (mm)

10

12

6

Figure 6.10: Characterization of magnetic field gradient.

205
removed. While STIRAP can be performed with pulsed lasers, for high spectral resolution (andthus sub-level selectivity) cw-lasers are required. However, in an inhomogenously broadened system– for example, due to Doppler shifts – it is not typically possible to transfer all the population using
a standard STIRAP scheme. In the case of trapped ions (in particular, not Coulomb crystallized),oscillating Doppler shifts are present due to harmonic motion in the trap, which makes the use of
STIRAP challenging. With our typical trapping conditions, we have a temperature of about 10 K
in the axial direction (we want to use the axial direction because of the added micromotion androtating micromotion in the radial plane), which yields a Doppler linewidth of about 30 MHz fora single-photon transition around 1 µm. For a two-photon transition to the 3 ∆1 , we expect the
Doppler linewidth to be reduced by a factor of 10, since the difference in energy between 1 Σ+ and3∆−1
1 is about 970 cm .

For STIRAP to work over the full 3 MHz of bandwidth, we would want

to use pulsed lasers with a duration of below 400 ns. However, the splitting between neighboring
Stark manifolds is only around 10 MHz, so we also need pulse durations longer than 100 ns to makesure that we can still resolve the Stark manifolds. This is a challenging pulse duration range toobtain with high power, making STIRAP difficult to apply. Another method, called Stark-induced
adiabatic Raman passage (SARP), uses the Stark-shift from a high-power pulsed laser to sweep
the two-photon detuning through resonance [453]. Using this method, high transfer efficiency has
demonstrated with the need for an off-resonant intermediate state [454], but it will be challenging
to achieve the desired resolution.
Instead we developed a different scheme for efficient Raman adiabatic population transfer inan ion trap shown schematically in Figure 6.11. The initial and target states are coupled using
two cw lasers detuned by an amount ∆ from an intermediate resonance. The oscillating Doppler
shifts result in a two-photon detuning δ, which varies as a function of time for each ion. We use
this to adiabatically sweep δ for each ion across resonance, which results in some transfer acrossthe avoided crossing (Figure 6.11(c)). In one-half of a period of secular trap oscillation along theaxis of the transfer lasers, every ion will cross resonance once and only once, yielding in principle
100% population transfer. However, if the transfer efficiency for one sweep across resonance is not206high enough, this process can be repeated many times, thus equilibrating the population among
the initial and target levels.
The two transfer lasers propagate along the z direction so that there is no radial micromotion(or rotating micromotion in the presence of Erot ). Because the transition is a Λ-type, the twolasers co-propagate so the two-photon transition is Doppler free. To maximize the possible transfer
efficiency, we ramp the trap in the axial direction to give trap frequencies of around (12.5, 10, 1) kHzin (x, y, z). This ramp allows us to use smaller transfer laser beams to increase the intensity andprovides a low temperature and secular frequency along the laser direction, which helps to increase
the adiabaticity.

6.3.1

Laser stabilization

For any Raman transfer scheme to be robust, both of the lasers need to be frequency stabilized
such that their relative coherence time is longer than the length of the transfer. Also, for ease ofuse, they should stay on two-photon resonance for a long period of time. One way of achieving
these goals is to lock both of the lasers to different teeth of a stabilized frequency comb; however,this adds the additional complexity of another laser system and the locks required to stabilize thecomb. Instead, it is possible to use the modes of a stable Fabry-Perot resonator as a “comb” andlock each laser to a different mode. This has the advantage of reducing the complexity, but it
has the disadvantage that the absolute frequency of any given cavity mode is not known a priori.
We can circumvent this problem by using a cavity with a mode spacing (i.e., FSR) large enoughso that we can guarantee using a wavemeter that we always lock each laser to the same cavity
mode. Another disadvantage is that the cavity modes cannot be actively stabilized to an arbitrary
frequency with high feedback bandwidth and the mode frequencies are sensitive to environmental
perturbations such as pressure and temperature. For these reasons, it is necessary to build a cavity
that is passively stable and to lock each laser with an adjustable frequency detuning from a cavity
mode.
The stable cavity consists of two curved mirrors (radius-of-curvature of 50 cm, 0.16% and

207

(a)

(b)

3

Π0+

z

x

3

Δ1

1

Σ+
Adiabatic eigenvalues

(c)
1
0.5
0
−0.5
−1
0

0.5
ω t/(2π)1zFigure 6.11: Adiabatic transfer from 1 Σ+ to 3 ∆1 . (a) Energy level structure of the three states
involved. (b) Laser direction and polarization. (c) Adiabatic eigenvalues of the two-level Hamiltonian (Equation 6.28) over one period of axial trap oscillation, t = 2π/ωz . The inset shows an
avoided crossing in the vicinity of ωz t = (n + 1/2)π.

y

208
0.12% transmission at 900 and 986 nm) mounted on an ultra-low expansion (ULE) glass spacer.
The spacer is about 7.5 cm diameter and 11.5 cm long and has a hole bored out of the center forthe laser as well as a perpendicular hole for venting. The mirrors are attached with a thin layerof Crystalbond 509: to do this, a mixture of Crystalbond and acetone (20% by weight) is injected
between the mirror and spacer (the mirror can be rotated to get a thin, uniform layer with nobubbles), and the acetone is evaporated under a heat lamp (low heat) for at least 24 hours. After
this, the cavity is placed in a temperature-controlled vacuum chamber as shown in Figure 6.12(a).
The cavity rests in the center of a 6” Conflat Tee on Viton hemispheres (half of a Viton ballfrom McMaster-Carr). Viton is used because the extremely low mechanical Q-factor provides good
vibrational damping. Two ends of the tee are closed with standard AR-coated windows (Thorlabs,
B coating) and the upper part of the tee is connected to a valve (for initial pumping with a turbo)
and an ion pump to reach a final pressure of ∼ 2 × 10−7 Torr. The entire chamber is supported byclamps isolated from the optical table with sorbothane rubber for additional vibrational isolation.
Several temperature controllers are used to reduce temperature gradients across the chamber.
The center portion of the vacuum chamber is wrapped with a Minco flexible heater (resistance of526 Ω), which is secured with thermally conductive tape. A precision thermistor (Measurement
specialties 44008RC, 30 kΩ) is epoxied next to the flange of the tee (away from heater) and is usedas the feedback thermister for one control loop. An AD590 temperature sensor is mounted oppositeof the thermistor as a temperature monitor. In addition, two small Minco heaters (resistance of
44 Ω each) are mounted in series on each end flange on opposite sides of the window. A thermistor
and AD590 are mounted above and below the window, respectively, and are used in two more
feedback loops (one for each side). The entire chamber is heated to about 28◦ C, with the sides
slightly warmer than the center. All feedback is done with JILA temperature controllers. Finally,
the entire chamber (except for the windows) is wrapped with Thinsulate insulation. The finishedchamber is shown in Figure 6.12(b).
Since the cavity has an FSR of ∼ 1.3 GHz, an ideal setup would require an offset tuning of
650 MHz so that the laser frequency could be continuously tunable; however, this is technically very

209

(a)

Ion
Pump
Thermistor
AD590
Minco Heater
End view

Minco heater around outside

Al clamp

Viton

ULE

Sorbothane(b)

Figure 6.12: Stabilization cavity for Raman transfer lasers. (a) Sketch of vacuum chamber. See
text for details. (b) Picture of completed cavity wrapped in thermal insulation (white).

210

Figure 6.13: Sketch of the laser stabilization setup for the transfer lasers. AOM: acousto-optic
modulator; DM: dichroic mirror; Pol: Glan-Thompson polarizer; EOM: electro-optic modulator.
The feedback loop is shown in gray. φ: phase shifter.

211
difficult because the low efficiency and alignment sensitivity of high-frequency AOMs makes themhard to double-pass. Since we want to be able to change the two-photon detuning (for example, tomeasure the hyperfine splitting, etc.), we need to be able to tune one laser continuously by morethan 50 MHz without having to adjust the alignment, which is not possible with a single-pass AOM
setup because of the changing deflection angle with changing frequency. The final setup was chosen
as a compromise between flexibility, tuning range, technical difficulty, and available supplies.
A sketch of the setup used for offset locking the lasers to a cavity mode is shown in Figure 6.13.
A small amount of light from each laser is sent to the locking setup via fiber to prevent alignment
drifts. For the T laser, an 800 MHz AOM located before the fiber coupling is used to provide coarse
frequency tuning (this requires re-alignment for large frequency steps). After the fibers, the lightfrom S and T is sent to a 300 MHz and 200 MHz double-passed AOM, respectively. The AOM
frequencies were chosen by first determining the location of the transitions using a wavemeter andthen taking the distance to the closest cavity mode. The double-pass setup [455] consists of a PBS,
followed by a lens to focus light into the AOM; the focal length (f ) of this lens is chosen such thatthe diffraction-limited beam waist, w = 4λf
πD for an input beam of diameter D, is comparable to,but a bit smaller than, the active area of the AOM – typically < 300 µm for a 200 MHz AOM.
After the AOM, the light is collimated with another lens located one focal length away from AOM(f1 = 75 mm). For good double-pass efficiency, it is important to adjust the position of this lensto nicely collimate the beam. After the lens, the light goes through a quarter-wave plate (fast axis
45◦ from the polarization axis) and then retro-reflects off of a mirror located a distance f1 from thesecond lens. The alignment of this mirror is adjusted until light is observed at the output port of
the PBS, although some care needs to be taken to make sure that the beam profile is still nice asa poor exit beam profile can indicate that the light is not diffracting a second time in the AOM.
After the PBS, the light from T and S both pass through half-wave plates before beingcombined using a dichroic mirror; they then go through a linear (Glan-Thompson) polarizer andfinally an EOM (at 12.5 MHz) to create the sidebands used for the Pound-Drever-Hall (PDH) lock.
The polarizer sets the polarization to be parallel to the electric field in the EOM crystal to reduce

212
residual amplitude modulation which can cause noise and offsets in the lock [456]. After the EOM,the light passes through a PBS and quarter-wave plate prior to the cavity. The reflected light fromthe cavity exits the PBS, is split with another dichroic, and is detected on two PDA8A amplified
photodetectors. The cavity transmission is imaged onto a camera to optimize the cavity alignment
and lock to the correct mode.
All of the AOMs are driven with a Minicircuits ZHL-1-2W amplifier (max output power 33dBm). The source for each double-pass AOM is one channel of an Analog Devices AD9959 DDS(clocked externally at 20 MHz with an internal ×20 for a final 400 MHz internal clock). The outputof the DDS is amplified and then doubled since the maximum output frequency of the DDS is 200
MHz given the clock frequency and in practice needs to be a bit lower than that because of aliased
frequency components at 400 MHz minus the output frequency. The PDH error signal is detectedby mixing the rf signal from the photodetector with the monitor output from the EOM drive (after
a phase shifter which is adjusted to optimize the error signal magnitude and symmetry). We cancharacterize the locked laser linewidth by measuring the noise spectrum of the locked error signal
compared to amplitude of the PDH error signal [457]. We estimate a linewidth of around 10 kHz.

6.3.2

Transfer theory

If the hyperfine and Stark structure of 3 Π0+ and 3 ∆1 can be neglected, the transfer problem
takes the form of a three-level Λ system. The finite lifetime of the 3 Π0+ state is modeled by a loss
rate γ. For the three-level system, we use the effective single-particle Hamiltonian0Ω12 (t)/2
0






H(t) = ~ Ω12 (t)/2 ∆(t) − iγ Ω23 (t)/2 ,




0
Ω23 (t)/2
δ(t)(6.27)

23
where ∆(t) = ∆0 + ∆1 cos(ωz t + φ), δ(t) = δ1 cos(ωz t + φ) and δ1 = ω12ω−ω∆1 . Due to the finite
12

size of the transfer laser beams and the radial ion motion, the one-photon Rabi frequencies Ω12 andΩ23 are also functions of time.

213
Using the Hamiltonian (6.27), we have numerically simulated the time-evolution of a single
particle in a harmonic trap, initially in the state |ψ(t = 0)i = |1i = |1 Σ+ i. We use a time step ∆tthat is small compared to an axial trapping period, and successively apply the evolution operator
exp (iH(t)∆t/~) to evolve the state forward in time.
If the one-photon detuning ∆0 is large relative to δ and Ω12,23 , we can perform adiabatic
elimination by requiring that the population of state |2i does not change in time [64]. In thisapproximation, we obtain the Hamiltonian

H=−



Ω21

Ω1 Ω2
~ 

.

4∆0
2
Ω1 Ω2 Ω2 − 4∆0 δ(t)(6.28)

The adiabatic eigenvalues of the Hamiltonian (6.28) evolve periodically with the trap frequency ωz ,as shown in Fig. 6.11(c).

6.3.3Transfer parameters

An intuitive estimation of the transfer efficiency can be obtained by considering the avoided
crossing in Figure 6.11(c). In the limit of large initial and final two-photon detuning, the transfer
probability for a single sweep (total time T ) can be estimated from the Landau-Zener formula [458]"2 #PLZ (T ) = 1 − exp −2π

Ω13
2

δ̇

,

(6.29)

where Ω13 = Ω12 Ω23 /(2∆0 ) is the two-photon Rabi rate on resonance and δ̇ is the two-photon
detuning sweep rate. The factor of 1/2 in Equation 6.29 arises because the Rabi rate is twice theoff-diagonal coupling in the Hamiltonian. For trapped ions at a given temperature, the sweep rate
can be approximated by assuming a linear sweep from negative to positive Doppler shift in one-halfof a trap oscillation period. This givesδ̇ ≈ (2∆ω)/T = 4

vRMS(ω12 − ω23 )νz ,c(6.30)

where ∆ω is the maximum Doppler shift (in angular frequency), vRMS is the root-mean squarevelocity from a Boltzmann distribution, ω1(2) is the angular frequency of the upward (downward)

214
laser, and νz is the trap secular frequency along the direction of the lasers (defined to be thez-axis). From this, we see that a transfer probability of at least 0.5 can be obtained as long asΩ212 > (ω1 − ω2 )νi .
Additional complications arise when considering multiple Landau-Zener transitions. The
net transfer probability after two crossings is an oscillating function P = 4PLZ (1 − PLZ ) sin2 φS ,where φS is the Stückelberg phase [459]. This phase consists of a term acquired during the time
between crossings and a term from the non-adiabatic transition. These two terms can cause either
constructive or destructive interference for the second crossing, leading to an effect called LandauZener-Stückelberg interferometry. For a period series of crossings, the overall behavior is more
complicated and is discussed in a variety of papers [459, 460, 461, 462, 463, 464]. In general, thebehavior depends on the phase and amplitude of modulation (in this case, the Doppler shift) as
well as the two-photon Rabi rate. Since we have a thermal distribution, we have assumed that theinterference effects are averaged away and just use the simple Landau-Zener treatment.
In order to figure out the expected transfer efficiency in our system, we need to determine allof the parameters in the Landau-Zener formula. The detuning and trap secular frequencies are wellknown, and we can estimate the Doppler shift, so all that remains is to determine the one-photon
Rabi rates (Ω12 and Ω23 ). We can measure Ω12 (i.e., the Rabi rate for the 3 Π0 ← 1 Σ+ transition) bymeasuring the depletion rate with an on-resonant drive. Because of the oscillating Doppler shifts,
this depletion process is treated as a repeated one-photon Landau-Zener transition to the 3 Π0 fromwhich the population is lost due to spontaneous emission. For a single sweep, the probability ofremaining in the ground state is given by Equation 6.29 with Ω13 replaced by Ω12 and δ̇ given by
the first part of Equation 6.30. The probability of remaining in the ground state after N cycles isNPgiven by P0 (N ) = 1 −
P (1 − P )n−1 , which results in an exponential loss with time constant τ .
n=1

Thus
"
P0 (T ) = exp(−T /τ ) = exp −2π

Ω12 2
2
2∆ω
T

 #
,

(6.31)

215
which can be solved to give
r
Ω12 =

8∆ω
.
τ

(6.32)

Measurements of Ω12 at different laser intensities are shown in Figure 6.14.

For these

points, the laser was enlarged (∼15-mm diameter) so that the intensity was uniform across thecenter. The electric field strength was calculated from the laser power measured over a 9.5q
2I
mm-diameter power meter: E =c0 . Since Ω12 = µ21 E/~, a linear fit to the data gives thetotal dipole moment µ21 = 0.070(4) ea0 . This is the total dipole moment, which includes angular momentum coupling (S21 ), statistical weight (g1 ), and vibrational wave function overlap:pµ21 = S21 /g1 hχ2 |χ1 i 3 Π0 |d|1 Σ+ . Here S21 is referred to as the Hönl-London factor and can beobtained from Clebsch-Gordon coefficients or Wigner 3j symbols. Note that there are some inconstancies among different published versions [358, 365, 465, 466], here we have used an expression
for a Case (a) to Case (a) rotational transition:



0
00
J
1
J


S21 = (1 + δΩ0 0 + δΩ00 0 − 2δΩ00 0 δΩ0 0 )(2J 0 + 1)(2J 00 + 1) .
−Ω0 Ω0 − Ω00 Ω00(6.33)For this transition S21 = 2. Using the Franck-Condon overlap hχ2 |χ1 i2 = 0.38 and g1 = 1 (ignoring
hyperfine),

3 Π |d|1 Σ+0= 0.080(5) ea0 .

We can measure Ω23 in a similar fashion, but we do not initially start with populationin the 3 ∆1 . Instead, we can transfer some population into 3 ∆1 J = 1 and then deplete thispopulation; however, we did not have the ability to shift the laser frequency to be directly on
resonance and also follow the same beam path. While it should be possible to measure the depletionat a given detuning, this only works if the lineshape is well known. Given the sidebands andbroadening of the one-photon lineshape observed in Figure 6.15, we do not trust this measurement.
Instead, we send the on-resonant light in from the radial direction; however, the micromotion in
the radial plane (which varies depending on the location in the trap) significantly complicates theinterpretation of the depletion time scale. To overcome this, we use the rotating electric field toimpose a constant oscillating Doppler shift. We measure a depletion time constant of τ = 0.61(8) ms

216

0.35

Rabi Frequency Ω/2π (MHz)

0.3
0.25
0.2

Transition Dipole Moment:
0.070(4) ea0

0.15
0.1
0.05
0
0

0.5

1

1.5

2

Electric Field [V/cm]

2.5

3

Figure 6.14: Transition dipole moment calibration for the 3 Π0+ ← 1 Σ+ (v = 1, J = 1) ← (v =0, J = 0) transition.

Fraction Hf+ Depleted

217

Erot = 16 V/cm
Erot = 0

1
0.8
0.6
0.4
0.2
0
-300 -250 -200 -150 -100 -50
0ν12 - 899.66653 nm (MHz)

50

100

Figure 6.15: One photon lineshape measured by depletion from 1 Σ+ J = 0 (denoted as laser S).
The center peak is saturated to show the structure in the sidebands. Dark points are recorded with
Erot = 16 V/cm, light points are for 0 V/cm. Blue and green points are for two slightly differentalignments of the transfer laser.

218
with a depletion laser intensity of 25 mW/cm2 and a rotating field of 16 V/cm. Using Equation
6.32, with ∆ω = vrot /c × 2πc/λ, we find Ω23 = 2π × 1.03(6) MHz, so µ23 = 0.18(2) ea0 . For
this transition, we use the Hönl-London factor S32 = 3/2, which still ignores hyperfine. Using theFranck-Condon overlap hχ2 |χ1 i2 = 0.31, we find thattheoretical predictions in Table 5.6,

3 Π |d|1 Σ+0

3 Π |d|3 ∆01≈ 0.27(3) ea0 . Compared to theis low by about a factor of 2, while

3 Π |d|3 ∆0
1

matches the theory.
We initially operated with a total single-photon detuning (∆) for each laser of 160 MHz.
However, we found that the broadened lineshapes shown in Figure 6.15. To reduce the spontaneousemission, the single-photon detuning was increased to ∼1.5 GHz. For the data shown below, we
had about 200 mW of S and 350 mW of T just before the trap chamber. This was later increased
to 500 and 630 mW, respectively, by removing an AOM and using mechanical shutters instead.
To maximize the Rabi rate, we focus the lasers to a beam waist of 1.5 mm at the center of thetrap and increase the radial trap confinement to a secular frequency of (12.5, 10) kHz in (x, y). To
improve the Landau-Zener probability, we decrease the axial trap frequency to about 1 kHz. Using
these parameters and the measured dipole moments gives Ω12 = 2π × 6 MHz, Ω12 = 2π × 19 MHz,and Ω13 = 2π × 210 kHz for the lower laser power. From Equation 6.29, we expect a transition
probability for a single crossing of 17% and a transfer time scale from Equation 6.32 of 17 ms.

6.3.4

Transfer efficiency and timescale

We can measure the transfer efficiency by measuring the 1 Σ+ J = 0 population while scanningthe duration for which the transfer laser shutters are open. An example of such a measurement isshown in Figure 6.16 with both lasers linearly polarized and perpendicular to each other (Figure
6.11(b)). Each time point in this figure is the average of separate measurements alternating between
both transfer lasers, only the 899 nm laser, and no transfer lasers to measure the J = 0 baseline
population. From this data we obtain a transfer efficiency of 63(2)%, with a time constant of
5.8(7) ms; one axial trap period is ∼1 ms. In addition, the loss due to one-photon spontaneousemission is negligible. Because the initial state is J = 0, F = 1/2 and e parity and the final state219
is J = 1, F = 3/2, we would expect a transfer of 66.7% after the populations fully equilibratein the absence of an electric field. In the presence of a bias electric field, the degeneracy between
mF = ±1/2 and mF = ±3/2 is lifted, thus reducing the expected efficiency to 50%. With a rotating
bias field of 16 V/cm, we observe an efficiency of 37(2)% and a time constant of 15(2) ms. The
transferred fraction at short times and the transfer time scale agree reasonably well with the roughestimates in the previous paragraph. In addition, we observe no transfer with the laser polarizations
parallel to each other, which is expected due to selection rules (Section 6.3.5).
The maximum expected efficiency of 50% of course assumes multiple transits through the
avoided crossing, with transfer for just one-half cycle (one sweep through the avoided crossing), weshould be able to increase this to 100% assuming enough laser power. If we decrease the singlephoton detuning to 160 MHz and use the same laser parameters, we should have a Landau-Zener
probability of about 99.9%. However, even with the higher laser power, the maximum transfer that
we have seen after one-half of a trap cycle is around 40%. Some of this may be due to the opening
and closing time of the shutters or due to poor overlap of the trap lasers and ions. We are still
investigating this and hope to be able to increase the transfer efficiency to above 50%.

6.3.5

Transfer spectrum

Scanning the second transfer laser (denoted T in Fig. 6.11) through two-photon resonancewith each of the hyperfine and Stark levels of 3 ∆1 J = 1 yields the transfer spectrum shown in
Fig. 6.17(a). Because the single-photon detuning is large compared to the single-photon transition width, we assume that the total rate of transfer Wf into a particular Stark level of 3 ∆1 is
proportional to the modulus squared of the two-photon transition dipole moment,
2

Wf ∝

X X
i,m pT ,pS

hf |Tp(1)|mihm|Sp(1)
|ii .
T
S

(6.34)The states |ii, |mi, and |f i correspond to the various sub-levels of 1 Σ+ , 3 Π0+ , and 3 ∆1 respectively.
The rank-1 spherical tensor operators S (1) and T (1) correspond to the first and second transfer
(1)

(1)

lasers, respectively. The spherical tensor components in the molecule-fixed frame SpS and TpT are

220

Fraction Remaining1
0.8

τ = 15(2) ms
A = 0.37(2)

0.6
0.4
0.2
0

τ = 5.8(7) ms
A = 0.63(2)Transfer (E = 0 V/cm)
One laser (E = 0 V/cm)
Transfer (E = 16 V/cm)
One laser (E = 16 V/cm)

0

10

20

30

40

50

60

70

80

Transfer Duration (ms)Figure 6.16: Fraction of population remaining in 1 Σ+ J = 0 versus transfer laser on duration. Blue
points correspond to Erot = 0 V/cm and green points are for Erot = 16.5 V/cm. In both cases, dark
points indicate both transfer lasers and light points indicate only S (the 3 Π0+ − 1 Σ+ laser). The
data with both transfer lasers are each fit to a single exponential decay (shown as dashed lines)
with a variable time constant (τ ) and transfer efficiency (A). In addition, the S-only data showthat spontaneous emission losses are negligible.

221a)Transfered signal (arbitrary units)23.75 V/cm

20.13 V/cm

16.17 V/cm

12.38 V/cm

8.25 V/cm

-40

-30

-20

-10

0

10

20

Detuning δ (MHz)

30

40

50

F = 3/2
F = 1/2b) mF = -3/2 mF = -1/2 mF = 1/2 mF = 3/2 mF = -1/2 mF = 1/2
25

Rotating field (V/cm)

20

15

10

5
dmf = 1.401(5) e a0
A|| = 62.2(1) MHz
0

-40

-30

-20

-10

0

10

20

Detuning δ (MHz)

30

40

50

Figure 6.17: Stark spectroscopy of 3 ∆1 J = 1. (a) Observed 3 ∆1 v = 0, J = 1 spectrum using
two-photon transfer, for a range of the rotating electric field Erot . The transfer duration was
decreased so that the peaks were mostly unsaturated. The solid lines are a fit of the simulated
spectrum, discussed in Section 6.3.5, dashed vertical lines indicate positions of the hyperfine peaks
for Erot = 0 V/cm. (b) Peak centers for the six Stark manifolds as a function of Erot . The centers
are fit to the expected energies (solid lines) to obtain the hyperfine constant, A|| , and the molecularframe dipole moment, dmf . The lines are color-coded according to the value of mJ in the high-field
limit.

222
related to the lab frame components according toSp(1) =

X
p0

Tp(1) =

X
p0

(1)

(1)

Dpp0 (ωrot t)∗ Sp0(6.35)(1)(1)
Dpp0 (ωrot t + θ)∗ Tp0

where D(1) is the J = 1 Wigner rotation matrix. In the lab frame, the first and second transfer
lasers are linearly polarized with a relative angle of θ. We find that the transfer rate Wf ' sin2 θand is independent of ωrot t. This confirms our experimental result that no transfer occurs when
θ = 0, and maximum transfer occurs when θ = π/2.
(1)Each of the matrix elements hf |Tp |ii in Eq. 6.34 can be expanded in terms of the quantum
numbers of each state using the Wigner-Eckart theorem. In zero electric field, the states 1 Σ+ ,3Π

0+ , and3∆

1 can all be written in the Hund’s case (c) basis as |η; I; J, Ω; F, mF i, where η areother quantum numbers specific to each state. The dipole matrix elements can be writtenhη 0 ; I; J 0 , Ω0 ; F 0 , m0F |Tp(1) |η; I; J, Ω; F, mF i =m1 (F 0 , m0F , p, F, mF ) m2 (I, F 0 , J 0 , F, J) m3 (J 0 , Ω0 , J, Ω) (6.36)
where


F0

0
0 
m1 (F 0 , m0F , p, F, mF ) = (−1)F −mF −m0F0

m2 (I, F 0 , J 0 , F, J) = (−1)F +J +I+1


1

F 

p mF

p
(2F 0 + 1)(2F + 1)



J

F



I

(6.37)


F 0 J 0 1



0
X
1 J  0 (1)00p
 J
m3 (J 0 , Ω0 , J, Ω) =(−1)J −Ω (2J 0 + 1)(2J + 1)  hη ||T ||ηi.
0q−Ω q Ω
In a nonzero electric field, the 3 ∆1 sub-levels are Stark states; linear combinations of the states|η; I; J, Ω; F, mF i. The eigenstates and energies are determined by diagonalizing an effective Hamiltonian that includes Stark, hyperfine, and Λ-doubling matrix elements. The hyperfine and Λdoubling terms are diagonal in the |η; I; J, Ω; F, mF i states with ∆EHF = 45.6(1) MHz and223∆EΩ = 740(40) kHz, and the Stark matrix elements are given by
HS = −dmf Erot m1 (F 0 , m0F , 0, F, mF ) m2 (1/2, F 0 , 1, F, 1) m3 (1, Ω0 , 1, Ω)(6.38)

where dmf is the molecule-frame dipole moment.
The relative energies of the Stark states obtained by diagonalizing the effective Hamiltonian,along with the relative transfer rates given by Eq. (6.34), allow us to simulate the transfer spectrum.
Figure 6.17 shows the observed and simulated spectra for a range of Erot , where the hyperfine
splitting, Λ-doubling, and molecule-frame dipole moment were fitted parameters. The full widthat half max of the lines was fixed at 6 MHz. The simulated spectra recreate the main featuresof the experimental results: The relative amplitude of each peak is qualitatively correct, withthe “stretched” transitions into |F = 3/2, mF Ω = ±3/2i appearing strongest. In addition, theobservation of suppressed transfer into the |3/2, −1/2i and |1/2, 1/2i states is corroborated by thesimulated spectra.
We can intuitively understand the amplitude of the transfer peaks by considering the casewith no nuclear spin, as shown in Figure 6.18. Since the light polarization is in the plane of
the quantization axis, we can only drive π or both σ ± transitions. From the 1 Σ+ state, the πtransition is forbidden (since ∆J = 0, ∆mJ = ±1 only), so we can only drive the blue transitions
to 3 Π0+ . From the 3 Π0+ , a second photon with π polarization (opposite of the first photon) candrive transitions to mJ = ±1 and a σ transition will go to mJ = 0; however, the two-σ pathways(all blue lines) cancel because of the signs of the Wigner 3j coefficients, so the transition to mJ = 0
is forbidden. This means that we would only expect transitions to mJ = ±1, and only transition
when the laser polarizations are opposite. If we project the |F, mF i states onto |J, mJ i, we findthat the relative amplitudes in the transfer spectra in Figure 6.17 correspond to the projection onto
|1, ±1i. In particular, the states that trend toward mJ = 0 (light blue in Figure 6.17(b)) decreasein amplitude as Erot increases, i.e., as the states approach states of good mJ .

224

3

Δ1

−

1/ 6

1/ 6

3

Π0+
1/ 3

1/ 3

1

Σ+

Figure 6.18: Possible two-photon transfer pathways from 1 Σ+ J = 0 through 3 Π0+ J = 1 to
3 ∆ J = 1, ignoring the nuclear spin of fluorine. Red lines indicate π transitions and blue lines
1
indicate σ transitions. The Wigner-3j coefficients for the σ transitions are also given.

225

6.4

Ramsey Spectroscopy
In the previous sections, we have demonstrated the ability to produce ions with 35% of the

population in 1 Σ+ J = 0, trap the produced ions without causing much heating, apply rotating
electric and magnetic fields, transfer up to 35% of the population from J = 0 to a single Stark
doublet in 3 ∆1 J = 1, and read out the population in single rotational levels with high efficiency.
We now consider how to perform Ramsey spectroscopy between the mF = ±3/2 levels in a single
doublet, which are the states used in the eEDM measurement as shown in Figure 6.19(a).
One complication arises due to the rotating quantization axis, which results in a coupling
between states with ∆mF = ±1 [467, 468, 62, 64]. This coupling can be understood by consideringa spin-1/2 system in a rotating magnetic field, which is similar to magnetic resonance [469]. Assume
that the the magnetic field is at an angle θr from the rotation axis, which defines the lab-fixed ẑ
~
axis. The Hamiltonian, H(t) = −~µ · B(t)can be written in the lab frame as−iωr t
cos(θ
)
sin(θ
)e
r
r
~gµB B H(t) =,2sin(θr )eiωr t− cos(θr )(6.39)

where the diagonal terms come from the projection onto ẑ and the off-diagonal terms are from therotating B⊥ in the xy plane. We can now apply a unitary transformation into the rotating frameby∂UH̃ = U † HU − iU †
∂t


−iω
/2t
r
0 
e
U =
.
0
eiωr t/2(6.40)

If we assume that the rotation is slow enough that the quantization axis tracks the field, we canalso rotate the ẑ axis by θr to lie along the field direction, defined as the 0 coordinate system. Then,the Hamiltonian in the field-fixed coordinates is~gµB Bωr
ωr− 2 sin(θr ) 2 + 2 cos(θr )H̃ 0 = .
~gµB Bωr
ωr− 2 sin(θr )− 2 − 2 cos(θr )(6.41)

The additional contributions on the diagonal are the standard energy contribution due to a Berry’s

226

(b)

1

u, upperStark
manifold

F

-

{

=-1

n̂

Hf
|

rot B

2+

|b

{ F
Stark

Hf

l
B - 2d e
3g F B

n̂

-

= +1

= -1

-1/2

1/2

= -1
Hf2+

eff

|c

|d

mF = -3/2

=+1

n̂

n̂
= -1

50
0
-50

-100

F
| = +1
-

-0.10

3/2

(c)

(d)

Strobed σ+

Strobed σ+/depletion REMPD| depletionrot

Spin precession
time τ

/2

Time

tdep

Frequency (Hz)

(e)

(f)

600

|g | = 0.00305(10)
l

400
200
0
-2

-1

0
B’ (G/cm)

1

2

-0.05

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0

Frequency (Hz)/2

20 V/cm
7 V/cm

100

-

2+

|

Hf2+

|

B+

F

B

= +1

l, lower

manifold

u

3g F

|a

eff
2d e

Fractional Population Difference, =0, J=1, F = 3/2

3

Frequency (Hz)

(a)

0.00

B’ (G/cm)

0.05

0.10

20 40 60 80 100 120 140 160 180
Time τ (ms)600

|gu| = 0.00306(10)

400
200
0
-2

-1

0
B’ (G/cm)

1

2

Figure 6.19: Ramsey spectroscopy between mF = ±3/2 levels in the 3 ∆1 . (a) Level structure of the
1 v = 0, J = 1, F = 3/2 manifold in electric and magnetic fields. (b) Avoided crossing between
mF = ±3/2 states due to rotation for two different electric fields with ωrot = 2π × 250 kHz. (c)Ramsey spectroscopy sequence. After the start of the trap, we ramp on Erot as shown by the grayline. After transfer (not shown), strobed depletion pulses prepare a single spin state. We thenreduce Erot for a period of time for the first π/2 pulse, wait a precession time τ , and then apply asecond π/2 pulse. After the second pulse, we repeat the strobed depletion process to remove either
the +3/2 or the −3/2 spin state, and finally read out the remaining population with REMPD.
(d) Example Ramsey fringe from the initial sets of measurements. (e) and (f) Measurements ofthe fringe frequency as a function of applied magnetic field gradient allow us to map the avoided
crossing and determine the magnitude of gF for the lower (e) and upper (f) doublets.
3∆

227
phase [470, 471, 472] of ±A/2, where A is the solid-angle enclosed by one revolution of the quantization axis. It can be shown that, more generally, the Berry’s phase contribution is mF A for a statewith total angular momentum F and projection mF [473, 467, 474]. If we set θr = π/2 – that is,
the rotating field is perpendicular to the rotation axis as it is in the ion trap – the diagonal Berry’s
phase contribution vanishes. We can then re-write the diagonal terms as ωr mF sin α ≈ ωr mF α,
where α is the tilt angle of the quantization axis from the plane perpendicular to the rotation axis.
This tilt could be caused by an axial electric field; however, because the time averaged axial electricfield must be zero for the ion to remain trapped, we expect that hαi = 0 and so there should be noenergy shift due to Berry’s phase. We discuss other sources of a Berry’s phase shift in Section 6.6.
The magnetic-resonance picture also provides an intuitive explanation of what causes thecoupling between spin states [475]. A classical dipole µ~ = g J~ will precess about the static magnetic
along ẑ at the Larmor frequency, due to the torque µ~ ×B~k . If we now transform into a rotating frame,
the precession rate appears either faster or slower, which means that the effective magnetic field
has changed by an amount ωr /g. Now, if we include the effect of a rotating field rotating at ωr , theq
2 + B − ω /g 2 .
total magnetic field is tilted away from the ẑ axis and has a magnitude B = B⊥rkThis changes the net project onto the ẑ axis, which implies that neighboring mJ values are coupled.
The rotation-induced coupling Hamiltonian, combined with the Λ-doubling Hamiltonian(which connects terms with ∆Ω = ±1), results in a perturbation that couples |mF , Ωi with|−mF , −Ωi. Thus, states in the upper Stark manifold (states |ai and |bi in Figure 6.19(a)) arecoupled as are states in the lower Stark manifold (|ci and |di). This results in an avoided crossing
between these states, as shown in Figure 6.19(b) as a function of magnetic field.
The first step in the Ramsey sequence shown in Figure 6.19(c) is to start with a pure spinstate far outside of the avoided crossing. Since the transfer process creates an incoherent mixture of
mF = ±3/2 level, we must remove all of the population from a single level. Since we want to addressa single spin state and the Zeeman splitting between the two spin states is unresolvable (several
hundred Hz at most), we must use a laser with circular polarization and propagation direction
along the quantization axis (set by the rotating electric field). This laser drives the 3 Π0+ ← 3 ∆1

228
(v = 1, J = 1) ← (v = 0, J = 1) transition and follows the same beam path through the trap as
the ionization and dissociation lasers (i.e., it is in the radial plane of the trap). Population excited
to the 3 Π0+ will then spontaneously decay to the 1 Σ+ or to other vibrational and rotational statesin 3 ∆1 and (ideally) will not appear in the measurement. We use light from the second transfer
laser (the first order of an AOM, see Figure 6.13) for this depletion light; however, we need to drive
the depletion transition on resonance, so we rapidly jump the frequency of laser by stepping thegrating position. The jump is adjusted so that the laser frequency changes by 1 FSR and thus
we do not lose lock to the cavity. After the AOM and fiber coupling, the depletion laser power istypically about 400 mW. The polarization of the depletion light is set using a combination of a λ/2and λ/4 waveplate to be circular measured before and after the chamber. Typically, we measure amax −Pmin
contrast ( PPmax+Pmin ) of under 5% when rotating a linear polarizer and measuring the power after

the polarizer.
Since the electric field (and thus the quantization axis) is rotating, the depletion light needs
to be pulsed on using the AOM only when the electric field is aligned (or anti aligned) with the
propagation direction of the depletion laser. We sync the timing of the AOM with the trap frequency
and adjust the phase so that we deplete either the mF = +3/2 or the mF = −3/2 depending on the
direction of the electric field. As discussed in Section 6.2.3, the rotating micromotion results in alarger micromotion velocity, which in turn results in large Doppler shifts (of about ±600 MHz) for
the depletion laser. Thus, the laser is only on resonance for a short time, so the depletion efficiency
for a single pulse is low. To alleviate this, we pulse the depletion laser at frot for about 5 ms witha duty cycle of 10%.
Figure 6.20 shows the fraction of the population removed from 3 ∆1 J = 1 after two setsof depletion pulses versus depletion duration (tdep ). The light-blue data points show the fraction
removed with two sets of depletion pulses of the same polarization (i.e., same phase of Erot , referred
to as inphase/inphase) while the dark-blue points correspond to the opposite polarization between
the two sets of depletion pulses (antiphase/inphase). The difference between these two chops iscalled the depletion contrast, as shown by the purple data points.

229

Fractional signal

1
0.8
0.6

Antiphase/Inphase
Inphase/Inphase
Contrast

0.4
0.2
0
0

2

4

6

8

10

12

Depletion duration tdep (ms)Figure 6.20: Fraction of the population depleted from 3 ∆1 (light- and dark-blue points) and depletion contrast (purple points) as a function of the depletion duration (tdep ). The light-blue data
points show the fraction removed with two sets of depletion pulses of the same polarization (i.e.,same phase of Erot ) while the dark-blue points correspond to the opposite polarization between
the two sets of depletion pulses. The difference between the two depletion chops is the depletion
contrast, shown in purple.

230
Ideally, for the inphase/inphase chop, we would expect to remove all population from asingle mF level only, resulting in 50% depletion. The antiphase/antiphase chop should remove all
population from 3 ∆1 , so we should see 100% depletion and 50% contrast. Instead, we see morethan 50% depletion for inphase/inphase and less than 100% for antiphase/inphase for tdep = 4 ms
and a total contrast of 20%.
We have tried to understand this reduced depletion contrast, but so far we have not beenable to determine the cause. Interestingly, sometimes the total depletion fraction is more than
90%, but the contrast is not better in these cases. One possible cause of poor depletion contrast
is impure circular polarization: we can measure the polarization before and after the chamber but
cannot independently measure the polarization at the ions. Stress-induced birefringence in thevacuum windows could lead to the polarization at the ions being different. The linear polarization
rotation of a single window outside of vacuum was under 1◦ , but mounting the window of thechamber and evacuating the chamber will likely result in more birefringence. We have measured
the polarization before and after the chamber, and see the same purity of circular polarization.
With linear polarization, we observe minimal change of contrast (i.e., minimal added ellipticity)
and a rotation of less than 2◦ . It is possible that the two windows happen to cancel each other;
however, the observed rotation is consistent with more careful measurements of a similar window[476]. They noticed significant variability across the window, which we did not consider. Our
measurements, combined with theirs, suggest that the window contributes less than 1% to theobserved circular polarization impurity, which would not cause the loss of contrast. In addition,
we see in Figure 6.20 (as well as in other data sets) that the depletion fraction for inphase/inphaserises rapidly at first and then slowly rises. We attribute this slow rise to polarization ellipticity;
comparing the time scale for the fast rise with the slow rise also suggests a 10% ellipticity.
Another explanation for above 50% depletion for inphase/inphase is poor selectivity in transfer. Because of the large, rapidly changing Doppler shifts from the rotating micromotion, any
population in all the mF = ±1/2 levels will also be depleted. We have used microwave spectroscopy between J = 1 and J = 2 to attempt to measure population in the other Stark pairs. We

231have some indication of population in the closest Stark pair to the target pair, but the poor S/N in
the microwave spectroscopy prohibited a quantitative comparison. In addition, while transferring
for a shorter duration seemed to decrease the population in the other Stark pair when measured
with microwave spectroscopy, it did not seem to improve the contrast, so more work is necessary
to understand this. Based on the transfer spectrum in Figure 6.17, the Stark pairs should be wellresolved at 16 V/cm, so we do not think that directly populating one of the other Stark pairs could
account for the added population. Instead, it is possible that mF level changing collision could
occur when the trap is tight in the radial direction for transfer or there are additional coupling
effects that we have neglected.
There are also several possible reasons for the < 100% total depletion. To address concernsof dark states or other coherent effects, we have measured depletion with the pulse rate 2×frot (i.e.,
depleting both +3/2 and −3/2), with linear polarization, and with a pulse rate frot /10 all withsimilar results. One possible scenario is that population excited to the 3 Π0+ is decaying back to
3∆

1 and not being depleted. Based on the approximate electronic dipole matrix elements, we would

expected ∼90% of the excited 3 Π0+ population to decay back to 3 ∆1 . This population could decay
to various vibrational states, with about 30% (the Franck-Condon factor) going to v = 0. Selection
rules only allow decay from 3 Π0+ J = 1 to 3 ∆1 J = 1 or J = 2; ignoring degeneracy, we would
expect something like 15% of the depleted population to return to each rotational level. Because
most of the population that returns to J = 1 should be depleted again (except for population that
returns to the opposite m − F = ±3/2 levels), this should not cause a problem. However, at highdissociation laser power (for the first dissociation step), we lose selectivity between J = 1 andJ = 2 for readout, so population that returns to J = 2 may be detected, depending on power. If
we are dissociating all of the population in J = 2, this would explain the undepleted fraction that
we observe, and if the dissociation power is not constant, this could explain why the undepleted
fraction seems to drift.
After preparing a single spin state, we pulse Erot to preform a π/2 pulse. Because the avoided
π/2crossing splitting (∆) increases with decreasing Erot , we can choose a value of Erot so that the

232
eigenstates are not the spin states (e.g., |ai and |bi) but are instead nearly pure superpositions:
π/2|a0 i = √12 (|ai + |bi) and |b0 i = √12 (|ai − |bi). If we start with |ψi = |ai, |ψi → √12 (|a0 i + |b0 i) at Erot .
After a dwell time of Tπ/2 ≈ 4∆1u/l , |ψi = √12 (|a0 i + e−iπ/2 |b0 i). Ramping back to the higher Erotgives |ψi = eiπ/4 |ai + e−iπ/4 |bi. We then allow the states |ai and |bi to evolve for a time τ overwhich they accumulate an additional phase shift, φ = ∆Eτ /~. After this, we perform a second
π/2 pulse, which maps the phase shift onto a population difference. One thing to note is thatour π/2-pulses are not perfect because the magnetic field results in some detuning from the centerof the avoided crossing which results in the eigenstates not being perfect superpositions. This isdiscussed in more detail in Section 6.6.4.
Finally, we measure the population in either mF = ±3/2 after the second π/2 pulse using
a second set of pulsed depletion either inphase or antiphase compared to the initial preparation
followed by REMPD. We repeat the entire experimental cycle many times and chop between inphaseand antiphase for the second depletion as well as no depletion (to measure the total transferred
population in 3 ∆1 ) and no transfer or depletion (to measure the background). By measuring asa function of time T between, we obtain a Ramsey fringe such as shown in Figure 6.19(d). The
fractional population difference plotted on the y-axis is the difference between antiphase depletion
and inphase depletion normalized to the difference between transfer and background.
We can measure the fringe frequency as a function of B 0 , as shown in Figure 6.19(e) and(f) for the lower and upper doublet, respectively. These data clearly show the avoided crossing
illustrated in Figure 6.19(b). We can extract the values of gF for both doublets, which are indeed
small as expected, as well as measure ∆ for both doublets as a function of Erot [64, 65].
6.4.1

Sensitivity Estimates

The fringe shown in Figure 6.19(d) forms the basis for the eEDM measurement. We canestimate statistical sensitivity of the first generation of experiments using HfF+ from Equation 6.6.
Relevant parameters are given in Table 6.2 (updated with newer results compared to the fringe in
Figure 6.19(d)). From this table, we project a shot-noise limited statistical sensitivity of 1 × 10−27

233√
e cm/ day for the first generation measurement. This is of course only a rough estimate. In fact,√using measurements in both doublets and for both signs of B should give a 4 improvement sinceeach measurement contributes linearly to the eEDM signal (i.e., the total signal is 8de Eeff ) and√only increases the noise by 4. However, we have also neglected the time that would be necessary
for measurements that don’t relate directly to the eEDM signal (e.g., the transfer and dissociation
background chops, monitoring fringe contrast, initial phase, etc.).
This table provides a good reference for considering how to further optimize the experiment.
Many of the key inefficiencies (transfer efficiency, dissociation efficiency, fast loss, and depletion
contrast) have been discussed above. A discussion of possible improvements towards a 10−29 e cm
can be found in Section 6.8, first though, we turn our attention to optimizing τ .

6.5

Coherence Time
The primary motivation for using ions for a precision measurement is the possibility of a long

free-evolution time, τ , because of the long possible coherence time. Of course here, like usual inexperimental physics, it is not possible to escape Murphy’s law. In fact, there are many potential
sources of decoherence: in particular the lifetime of the states used in the measurement, collisions,and inhomogeneous fields.

6.5.1

Lifetime

The fundamental limit to the time of any measurement is the lifetime of the state or statesused in the measurement, which can be set by several different factors. If both states participatingin the Ramsey measurement decay to a third state at the same rate independent of each other anddo not decay to each other, then the fringe contrast (normalized to the total number) does notdecay; however, the signal-to-noise will decrease due to the loss of population. In the case of themetastable 3 ∆1 J = 1, v = 0 state used in the eEDM measurement, the lifetime could be limitedby spontaneous electronic decay or blackbody excitation of either rotation or vibration. Since theonly electronic state below the 3 ∆1 state electronic state is the 1 Σ+ state, we can write the decay

234

Table 6.2: Estimated statistical sensitivity of the first generation
measurement using HfF+
Parameter

Estimate

Eeff

23.3 GV/cm

Total trapped HfF+

3000

Detector open-area-ratio

63%

Percent in single J, mJ

35%

Fast loss

50%

Transfer efficiency to 3 ∆1

35%

Depletion contrast

20%

Photodissociation detection efficiency

13%

Maximum fringe counts at τ = 0

3Free-evolution time (τ )250 msCounts per shot (N ) after τ

1Background counts (N0 )4Contrast (N/N0 )0.25

Duty cycle

50%

Rep rate

0.5 Hz

Fourier-limited linewidth (1/2πτ )640 mHzδde (assuming a 10-hour day)√1 × 10−27 e cm/ day235
rate of the lowest ro-vibrational state in the 3 ∆1 manifold as [378]:
X
2e2 X ∆E 3
ψv0 i |d|ψv0
Γel = τel−1 =A1i =30 hc3 0
~
i

≈

vi

2e2

µ2
3 12

X

30 hc

vi0

∆E
~

3

2

(6.42)
χ0vi |χv02

where the A1i is the Einstein A coefficient for each possible transition, the prime indicates 1 Σ+ states,∆E is the energy difference, and d is the dipole moment operator. In the approximation, µ21 is theelectronic transition dipole moment andχ0vi |χv02

is the vibrational wave function overlap (i.e.,

the Franck-Condon factor). Using the measured energy separation and calculated dipole moments(Table 5.6), we estimate the lifetime of 3 ∆1 v = 0 to be about 2 s.
To estimate the blackbody excitation rate, we start with the number of photons-per-mode ata frequency ν for a blackbody spectrum at a temperature T
1

n̄ =
e

hν
kB T

−1

≈

kB T
,
hν

(6.43)

where the approximation is valid for hν >> kb T . For a two-level system, the blackbody excitation
rate from the ground to excited state, Γexc , is related to the spontaneous emission rate from theexcited state, Γem : Γex = n̄Γem . Γem can be obtained from Equation 6.42 with the sum over
1 Σ+ states replaced by 3 ∆1 states.

For rotational excitation from J = 1 → J = 2, µ21 ≈ dmf =1.4 ea0 and the energy difference is 4B = 1.2 cm−1 , so n̄ ≈ 174 at 300 K and Γex ≈ 0.0025 s−1 .
Thus the expected blackbody rotational excitation timescale at 300 K is ∼800 s and so can beneglected.
For vibrational excitation from v = 0 → v = 1, the energy difference is about 760 cm−1 son̄ ≈ 0.027 (here the approximation in Equation 6.43 is not valid). The dipole moment is a bit more
complicated to approximate however. The transition dipole moment for a vibrational transition
arises from the change in the molecular-frame dipole moment, dmf (R), with internuclear distanceR. We can write dmf (R) ≈ dmf (R0 ) + d0mf (R0 )R + · · · , where R0 is the internuclear distance at
the minimum of the potential. Then µ12 = hχv1 |dmf (R)|χv0 i ≈ d0mf (R0 ) hχv1 |R|χv0 i. The value ofd0mf (R0 ) can be approximated from [378] to be 1.2 (ea0 )/a0 . We can do the integral using harmonic

236
q
√
~
oscillator wavefunctions and obtain hχv1 |R|χv0 i = 1/ 2α = 2µω≈ 3.6 × 10−12 m. This gives alifetime of v = 1 of 280 ms (which matches reasonably well with the previously calculated lifetimeof 184 ms from [378]) and a blackbody vibrational excitation timescale of 10.4 s (6.8 ms using the
calculated lifetime).
We have used state-resolved photodissociation to measure the 1 Σ+ J = 0, v = 0 lifetime asshown in Figure 6.9(a). The measured lifetime (ignoring the initial, fast collisional loss) of 7.1(8) sis shorter than the trap lifetime (i.e., total ion number decay) and matches well with the estimatedblackbody vibrational excitation rate. By using coherent transfer and photodissociation, we canmeasure the lifetime of the 3 ∆1 J = 1, v = 0 state to be 2.1(2) s as shown in Figure 6.9(b). Thisagrees with the predicted electronic lifetime of 2 s.

6.5.2

Field inhomogeneity

An field inhomogeneity that results in different fringe frequencies for different ions will resultin a reduced coherence time. The fringe frequency in the eEDM is set, to first order, by the Zeeman
~ Since Brot (R)~ ∼ ∇r B(~ R)~~ , any spatial variability in Erot (R)~splitting, 3gF µB Brot (R).
Erot (R)~ R)~ can potentially lead to decoherence. Since the average position of any ion over theor in ∇r B(
course of one measurement (i.e., many secular trap oscillations) is very close to the trap center,
D E~ = 0 to high precision, linear gradients in either Erot (R)~ or in ∇r B(~ R)~ will be averaged
that is R
over to give hErot i = Erot (0) (similarly h∇r Bi = ∇r B(0)) for for every ion. Second-order gradients,
however, will result in ions that have a larger R2 (e.g., hotter ions) experiencing a different valueof hErot i or h∇r Bi than ions that are, on average, closer to the trap center.
The coherence time limited by field inhomogeneity can be approximated as the time necessary
for different ions to acquire 1 rad of phase difference. Since the field inhomogeneity results in a
fractional frequency difference, the phase difference per period remains the same regardless of
frequency, thus, the coherence time should be a fixed number of periods. For a fringe of frequency
1
f , the coherence time can be estimated to be τinhom = 2πf ∆f/f , where ∆f /f is the fractional

inhomogeneity. In general, the fringe would not necessarily be expected to decay exponentially,237
but instead would have a decay profile obtained from the time-averaged distribution of fringe
22

frequencies. Frequently, this will result in a Gaussian decay profile ∼ e−t /τ [477]. We tried fitting
the decay profile with a Gaussian form, but did not find any noticeable improvement in the qualityof fit over a fit with a simple exponential decay. Since the decay constant from an exponential fit
versus a Gaussian fit is typically similar [478, 479], we used exponential fits for all presented data.
Figure 6.21 shows several different measurements of the coherence time (from a singe exponential fit) as a function of fringe frequency. All data points are measured with less than 2000 ionsand in a loose trap to reduce the effect of collisions. The dashed line shows the 1/f scaling due toinhomogeneity; the point at the lowest fringe frequency deviates significantly from the dashed line
indicating that the coherence time is at least partially limited by collisions, see Section 6.5.3. The
coherence time measured with a fringe frequency of 780 Hz is 48(7) ms, which implies a fractional
inhomogeneity of about 0.5% over the size of the cloud.
The applied magnetic field gradient, combined with trapping fields, contributes to a decoherence similar to field inhomogeneity. This can be seen from the expression for Brot from Equation
6.26. The second term in this equation gives the largest contribution to δBrot . To estimate thedecoherence, we assume a thermal distribution and replace Ex,y,z with kB T ≈ 10 K. This gives
δBrot /Brot ≈ 0.006, which approximately matches the observed decoherence. Decoherence due tothis effect can be reduced by decreasing ωrot . This will result in a larger rrot , so we can reduce B 0
for the same value of Brot .

6.5.3

Collisions

Because of the long-range nature of Coulomb interactions, ion-ion collisions can be a major
source of decoherence even in low density ion clouds. The primary concern with ion-ion collisions
is the Berry’s phase caused by the electric field of one ion tilting the quantization axis of the otherion. In this section we review the basics of ion-ion collisions and use this to obtain estimates forthe coherence time.

238

Coherence time (ms)

103

102

101
101

102
Fringe frequency (Hz)103

Figure 6.21: Coherence time versus fringe frequency. The dashed line is an approximate fit to a 1/ffunctional form, not including the lowest f point since this point is most likely limited by collisions,
see Section 6.5.3.

239
6.5.4

Collision backgroundAn ion-ion collision, often called a Rutherford collision, is usually treated as a one-bodyproblem in the center-of-mass frame. In this setup, shown generically in Figure 6.22(a), a single
particle with mass µ = m1 m2 /(m1 + m2 ) is incident on a central Coulomb potential located at
the center-of-mass of the two-body collision. The incident particle is offset from the collision axis
by a distance b, called the impact parameter. After the collision, the particle is deflected by the
Rutherford scattering angle, ∆θ. We can treat the collision in polar coordinates, (r, θ), and defineθ̂ = 0 as the direction opposite of the outgoing particle. With this definition, θ runs from (∆θ, π).
The incoming particle has angular momentum L = µr2 θ̇ = µvb and kinetic energy Ukin =1L2
2
2 µv = 2µb2 .

2e
The total energy is E = 21 µṙ2 + 12 µr2 θ̇2 + Vc (r), where Vc (r) = 4π
. We can use
0r

conservation of energy and angular momentum to obtainbθ̇ = v 2
r(6.44)

andsṙ = ±vb2
Vc (r)
1− 2 −
= ±v
r
Ukin

r
1−

b2
− A.
r2

(6.45)

Since ṙ|rmin = 0, we can solve forrmin = A +p
A2 + b2 .

(6.46)The scattering angle can be related to the impact factor by relating the momentum change
in y to the integral of the force:Z ∞δpy = p0 sin (∆θ) =−∞Fy dt = −Z πF sin θ
∆θ

dt
dθ =
dθ

π
e2 r2
e2
=
−
cos
θ
(1 + cos ∆θ).
4π0 r2 vb
4π0 vb∆θ

We can solve for b to obtainb=e2
1 + cos ∆θA=.
4π0 p0 v sin ∆θ
tan ∆θ
2(6.47)

Don’t believe the last step? Brush up on your half-angle formulas here2 since you will need themmore later. Equation 6.47 can now be used to determine the differential scattering cross section. As
2

http://mathworld.wolfram.com/Half-AngleFormulas.html

240

(a)

(b)
μ, Ukin, p, Lbr = rmin δ
Δθ

^

θ

center of mass
(origin of Vc(r))

dϕ

ϕ
db

dΩ

Figure 6.22: (a) Schematic of ion-ion collision treated as a single particle with mass equal to the
reduced mass µ incident on a central (Coulomb) potential. The collision coordinates are (r, θ),
where ~r is the position vector from the center of mass to the incoming particle and θ is the angle
between θ̂ and r̂, where θ̂ points opposite of the outgoing particle velocity. (b) Differential solid
angle.

241
shown in Figure 6.22(b), the differential cross section relates the number of particles, dN 0 , exiting
in a differential solid angle dΩ to the total number of incoming particles, N, by
dN 0
dσ
=N
.
dΩ
dΩ
Since dN 0 = N 2πbdb and dΩ = 2π sin ϕdϕ we can solve for the differential cross section (using bfrom Equation 6.47)
db
dσ
b
=
=
dϕ
sin ϕ dϕ 2
 2 2
A
1
1
e
.
ϕ =
4
2
2
4π0
sin 2
16Ukin sin4 ϕ2(6.48)The trajectory of the particle can be described by a hyperbolar(θ) =
Here ε =

6.5.5

1
b2
.
A ε cos (θ + δ) − 1(6.49)q2
1 + Ab 2 is the eccentricity and δ is the angle at which r = rmin .
Coherence time from collisions

Now that we have the basic collision theory under our belts, we can go on to look at how
collisions limit the coherence time. As mentioned above, the effect of an ion-ion collision is tointroduce an electric field that will add with the rotating electric field and tilt the quantization
axis. Over the course of the collision, the electric field vector traces a closed path on the surface
of a sphere. For example, the collision trajectory shown in Figure 6.23(a) with Erot perpendicular
to the collision plane results in the path illustrated in 6.23(b). This trajectory was generated by
Equation 6.49 and then the net electric field vector was generated for points along the trajectoryby summing Erot and the electric field from the ion-ion interaction:
Eion−ion =e2.
24π0 rmin

The solid angle, A, enclosed by this path results in a Berry’s phase from the collision of ∆φb =∆mF A, where ∆mF = 3 for ions that are in a superposition of mF = ±3/2 levels. This phasethen leads to decoherence, either from a single collision if the phase is large enough or from phasediffusion after multiple collisions.

242

(a)

(b)

y (μm)
6

z

4
2
2

4

6 x (μm)

Figure 6.23: (a) Collision trajectory for two particles in the xy plane. (b) The path of Etot is shown
as a dark curve for the collision trajectory shown in (a). Erot is assumed to point along z. The
area of this closed path is the solid angle, A, which leads to a Berry’s phase ∆φb = ∆mF A for thiscollision.

243
The rate of collisions, Z, for a collisional process is given by Z = nσv, where n is the density,
σ is the collision cross section, and v is the velocity. Since σ can be thought of as an area for a
hard-sphere collision σv is a volume per unit time that a single particle covers; thus the value of Z
gives the number of collisions per unit time. The rate of decoherence, Γ, then is given by Z/Nd ,where Nd is the number of collisions required for decoherence. A reasonable assumption would bethat decoherence occurs when the total accumulated phase is equal to π, i.e., Nd = π 2 /(∆φb )2 fora random-walk phase diffusion.
We now have all of the tools necessary to numerically estimate the collisionally limited coherence time. Initially we assume that all collisions lie in the plane perpendicular to Erot since thisshould represent the worst case scenario. We first calculate the solid angle for a given scattering
angle by numerically integrating to find the area enclosed by the path of the unit electric field
vector:A(∆θ, Ukin ) =ZπZZsin φdφdθ =
A

∆θ

dφ
sin φ(ϕ) dϕ
dϕ

θmax
Z (φ)

dθ,

(6.50)

θmin (φ)

where φ(ϕ) and θ(ϕ) are the polar angles of Êtot as a function of angle during the collision. Figure
6.24 shows plots of A(∆θ) for different values of Ukin . Putting everything together, we have thatΓ(Ukin ) =∆φ2b (∆θ, Ukin )2Z(∆φb )2=Z=nσ(∆θ,
U
)v(U
)
.
kin
kin
Nd
π2
π2

(6.51)

We can now use the differential cross section from Equation 6.48, substitute ∆φb = ∆mF A, assumepthat v = 2 kb T /m, and integrate over possible scattering angles to obtain the decoherence rate
as a function of collision energy. The coherence time (inverse of Equation 6.51) is shown in Figure6.25 for 20,000 ions with our measured trap parameters and different values of Erot .
In addition to the numerical results, it is instructive to obtain some general scaling rules. In
particular, the “big paper” [62] has some approximate formulas that appear out of no-where, can we
reproduce the scaling in these? First, we will need to find the maximum electric field experienced
during the collision, Eion−ion r=rmin. After substituting rmin from Equation 6.46 and then b from

244

0

10

−2

10

−4

Solid angle A (rad)

10

−6

10

−8

10

−10

10

−12

10

−14

10

−16

10

−4

10

−3

10

−2

Δθ/π (rad)

10

−1

10

0

10

Figure 6.24: Log-log plot of A as a function of scattering angle for different values of Ukin /kB
4 .
ranging from 0.3 K to 10 K with Erot = 10 V/cm. For small scatting angles, A ∼ Ukin245

10 V/cm
20 V/cm2

10

1

Coherence time (s)

10

0

10

−1

10

−2

10

−3

10

−4

10

1

10

2

Ukin/kb (K)10

Figure 6.25: Log-log plot of estimated coherence time as a function of collision energy (in temperature units) for Erot = 20 V/cm (red) and 10 V/cm (blue). These calculations are expected to give
the worst-case coherence time as discussed in the text. The calculated curves are for 20,000 ionsin the normal trap given in Table 6.1.

246
Equation 6.47 and finally using seemingly random combinations of half-angle formulas we find that!2sin ∆θ
4π0 2max
2∼ T 2.
(6.52)
Eion−ion (∆θ) = 3 4Ukin∆θe
1 + sin 2
max
For small values of Eion−ion
the solid angle can be approximated by

A≈

2
πE⊥
2 ,
Erot

(6.53)

where E⊥ is the component of the electric field perpedicular to Erot . Another approximation thatworks in more limits is


A≈



4
E2
Erot
π − ∆θ  ≈ π − ∆θ ion−ion ∼ T ,
1− q
2
2
2
2
2Erot
Erot
2 + E2Erot
ion−ion(6.54)

where in the second step we have assumed Eion−ion < Erot . From Equation 6.48, we see that the−2dσ−2 . Since v ∼ T 1/2 we see thatdϕ ∼ Ukin ∼ TΓ ∼ nT

−2



T4
2
Erot

2

√

−4
T ∼ nT 13/2 Erot

(6.55)

for low collision energies.
In the high collision energy limit, there is a high probability for a single collision to result ina Berry’s phase ∼ π, which would lead to decoherence. For a single collision to result in enoughphase shift, we need to have Eion−ion ∼ Erot . This implies that the distance of closest approach
must be
2
rmin
≈

e2
.
4π0 Erot(6.56)−1
2
We can use the standard hard-sphere collision rate, Γ = nσv, with σ = πr2 and r2 = rmin
∼ Erotto arrive at the high-energy scaling√−1Γ ∼ nErot
T.

(6.57)

These two limits can be seen in Figure 6.25.
We have measured the coherence time as a function of ion number and as a function of Erotas shown in Figure 6.26. Taken together, these measurements provide strong evidence for ion-ion
collisions. Figure 6.26(a) shows the measured coherence time as a function of ion number as well

247

(a)

Coherence time (ms)

103

102

101
102

103
Ion number

104

Coherence time (ms)

(b)

2

10

1

10

0

10

10

Erot (V/cm)

Figure 6.26: Coherence time versus ion number and Erot . (a) Log-log plot of coherence time as afunction of total ion number. The red dashed line shows the expected 1/n scaling (assuming thatthe density changes linearly with ion number, i.e., that the trap volume is constant). The blue
dashed line is the expected coherence time due to field inhomogeneity for a 40 Hz fringe frequency.
The black dashed line is the combined contribution from both sources: 1/τ = 1/τcoll + 1/τinhom .
(b) Log-log plot of various coherence time measurements as a function of Erot . The coherence timepwould be expected to scale like Erot
, where p is between 1 and 4 (see Equations 6.56 and 6.57). The
blue line is a fit to this scaling, which gives p = 2.3.

248
as the expected 1/n dependence (red dashed line), the inhomogeneity limit for a 40 Hz fringe
frequency (blue dashed line), as well as the combined limit (black dashed line). The ion number
was changed primarily by adjusting the power of the second ionization laser and has been correctedby the open-area ratio of the MCP to reflect the total ion number in the trap. We assume thatthe density scales linearly with the ion number; that is, we assume that the trap volume does
not change when the ion number changes. This assumption is reasonable as long as mean-field
effects do not get too large and as long as the ionization laser power do not significantly affect thetemperature. The 1/n dependence to the coherence time shown by the red dashed line is manually
adjusted to the match the highest ion number point.
As the ion number is decreased, the coherence time begins to deviate from the 1/n dependence. The measurements with low ion number were all taken at a fringe frequency of 40 Hz, which
contributes to the coherence time limit as shown by the blue dashed line. The black dashed line
shows the coherence time limit estimated by combining these two sources of decoherence. In general,
this combination is done by multiplying the decay from each source of decoherence. If each sourceis treated as a simple exponential decay (this is not true in general for inhomogeneity but is a good
approximation, see Section 6.5.2), the decoherence rates add linearly, so 1/τtot = 1/τcoll + 1/τinhom ,if we just consider inhomogeneity and collisional decoherence. This gives the black dashed line in
Figure 6.26, which matches well with the data.
Figure 6.26(b) shows the dependence of the coherence time on Erot . These measurements
were taken under a variety of conditions and have been corrected for ion number changes. All ofpthe data are fit to a power lower, Erot, yielding p = 2.3 (blue line). This is consistent with the

expected range of p = 1 − 4 from Equations 6.56 and 6.57.
We can compare the numerical results shown in 6.25 with the data in 6.26. With 20,000 ionsin the trap, we measure a coherence time of 48(7) ms. Using the normal trap parameters given in
Table 6.1, this corresponds to about 1 × 107 ions/cm3 . If we use a trap temperatures of 20 K and
Erot = 20 V/cm, the numerical estimates give a worst-case coherence time of 60 ms and p ≈ 3. If
instead we use the a temperature of 30 K, we would expect a worst-case coherence time of 10 ms

249
and p ≈ 2.3. Overall, the numerical estimates are in reasonable agreement with the data. We havealso decreased the trap temperature as discussed in Section 6.2.1 (“relaxed + ramp” in Table 6.1)
and measured an improved coherence time of 270(80) ms with 20,000 ions and 390(90) ms with
10,000 ions, which is an improvement by a factor of roughly 5.

6.6

Systematic Errors
Of course, you might be wondering,“what about Berry’s phase?” or perhaps more generally,“don’t you expect some killer systematic error?” Well, let me ease your concerns a bit and let you
know that, while crazy, we are not actually crazy enough to do an experiment where we expect amajor systematic above the 10−28 e cm level. If this is enough to convince you, then you can skip
the rest of this section since we will now discuss in detail potential sources of systematic errors.
The first step in determining possible sources of systematic errors is to identify all of thedependences of the measurement signal. The eEDM experiment basically amounts to measuring
energy splitting (in our case, actually the unsigned frequency difference) between either the upperor lower pair of mF = ±3/2 states. We can write the Hamiltonian for these two states in the upper(u) or lower (l) doublet as

3 u/l
3
− 2 gf µB Brot ± de Eeff + 2 αωrot
H = ~
∆u/l
2

∆u/l
2
3 u/l
3
2 gf µB Brot ∓ de Eeff − 2 αωrot.

(6.58)The diagonal elements of this matrix contain the Zeeman splitting, eEDM contribution (where
the upper (lower) sign refer to the upper (lower) doublet), and the energy shift due to a periodic
Berry’s phase caused by a tilt of the quantization by an angle α from the horizontal plane. Recall
eErot
that, because of the way we generate the rotating magnetic field, Brot ≈ B 0 rrot ≈ B 0 mω
2 . Therot

off-diagonal components are caused by the coupling in the rotating field, as discussed in Section¯ ± δ∆ [62, 64]:6.4. Since ∆ is different in the upper and lower doublet, we can define ∆u/l = ∆~ωrot 3∆u + ∆ l¯≈ 27ωef∆=;2
dmf Erot(6.59) ∆u − ∆l81~ωrot 3 dmf Erot 2 1 dmf Erot 2 ¯
δ∆ =
≈ ωef
≈
∆.
2
8
dmf Erot
EHF
3
EHF

250
To get the frequency difference f u/l , we diagonalize the Hamiltonian and take the difference between√b2
the eigenvalues, making use of the approximation a2 + b2 ≈ a + 2a
for b << a:
2πf u/l (Erot , Brot , ωrot , α) =(∆u/l )2u/l

3gF µB Brot ± 2de Eeff − 3αωrot +u/l

6gF µB Brot+

6αωrot (∆u/l )2u/l(6gF µB Brot )2(6.60),

where the Zeeman splitting is assumed to be much larger than any of the other terms. In addition,
we assume that trap motion results in hαi ≈ 0. A very rough eEDM measurement could then be
made by measuring hf u (E, B)i − f l (E, −B) ≈ 4de Eeff ; however this will be ruined by an imperfect
B reversal, as well as other deviations from our initial assumptions. So our goal is to find out how
much of an effect deviations have and how we can minimize the impact: in particular, we consider
the effects due to non-reversing B (e.g., from stray fields), magnetic fields perpendicular to thequantization axis B⊥ , shifts in Erot , and field inhomogeneities. We also briefly discuss any potential
systematics arising from state preparation or readout issues. A summary of systematic effects isgiven in Table 6.3.

6.6.1

Non-reversing magnetic field

A non-reversing magnetic field, B nr , will cause a systematic in the simple B reversal:
f u (E, B + B nr ) − f u (E, −B + B nr ) ≈ 4de Eeff + 6gFu µB B nr ,(6.61)

where we have removed the time averaging for simplicity. To reduce this systematic, we can combinea B switch with a doublet switch, which givesf u (E, ±B + B nr ) − f l (E, ±B + B nr ) ≈ ±4de Eeff + 6δgF µB (B ± B nr ),(6.62)

where δgF = gFu − gFl . This final four-way difference
fedm = [f u (B) − f l (B)] − [f u (−B) − f l (−B)] = 8de Eeff + 12δgF µB B nr ,(6.63)

so the effect of B nr is suppressed by the value of δgF /gF . The difference in gF arises from twocontributions [62, 468]:

251

Table 6.3: Systematic error summary. The first column lists different contributions to the four-way
difference, fedm . The next three columns give the sign of the contribution for each of the doublet,
applied B 0 , and rotation direction switch. Here + indicates that the contribution does not change
sign and − indicates a sign reversal under the switch. Columns 4-6 give the power-law dependence ofeach effect as a function of the values of |B 0 |, ωrot , and |Erot |. Finally, the final estimated systematic
contribution is given in column 7. These assume a fringe frequency (3gF µB Brot )of 40 Hz. See text
for more details.
SwitchDependence

Est. Contribution

DoubletB0

Rot.

|B 0 |

ωrot|Erot |

to fedmde (< 3 × 10−29 e cm)

-

-

+

0

0

0

1 mHz

B nr (switch)

+

-

+

0

0

0

0.6 mHzB nr (leakage)

+

-

-

0

1

1

0.6 mHz

∂Erot
∂z

+

-

-

0

1

-1

1 mHz

Motional Fields

+

-

-

0

-3

1

4 µHz

δ∆

-

+

+

0

6

-4

∼0

δ hB·Ei
|E|

+

+

+

1

0

-1

40 µHz

B⊥

+

+

-

0

1

-1

2 µHz

Effect252
δg(1) mixing of the Stark levels in J = 1 with levels in J = 2, g F =3/2 ≈F =3/29dmf Erot
40hcBe ;(2) rotation-induced mixing of ∆mF = ±1 levels that is different for the upper and lower Stark
δg

√

2
6ωrot
.
d
E E
F =3/2 mf rot hf

pair because the separation to the mF Ω = ±1/2 pairs is different, g F =3/2 ≈ γ 2
F =3/2

These two contributions are expected to be (1) 9.1 × 10−4 and (2) 8.6 × 10−4 for a maximum total
contribution of 1.8 × 10−3 . These assume that 3gF µB B >> ∆, if this is not true then δ∆ will also
δgcontribute to an apparent δgF . Our current best measurement is g F =3/2 = 0.003(6).
F =3/2

There are several potential sources of B nr , the most obvious of which is poor reversal of
the applied B-field gradient. By using current monitors on the anti-Helmholtz coils and/or several
magnetometers, we should be able to reverse B 0 to 1%. This systematic is at most 12δgF µB B nr out
nr

δgF B
−4
of a fringe frequency of 3gF µB Brot , i.e., δff ≈ 4 gF Brot ≈ 1×10 . This corresponds to a systematicof 2 mHz for a 20 Hz fringe, which gives an error of 3 × 10−29 e cm. Another source of B nr arises
from leakage current in the trap electrodes that oscillates at ωrot . Such a leakage current arises
max =due to capacitance C in the electrodes, which results in an oscillating current IleakVpp
2 Cωrot

for a peak-peak voltage Vpp . We have measured the capacitance of a single electrode plus SHV
feedthroughs to be about 40 pF (the electrode itself is estimated to be ∼2 pF); if we assume
that this capacitance all contributes to a current that flows through the electrode (worst case),
this gives a maximum current of 8 mA for Vpp = 250 V (corresponding to Erot = 20 V/cm) and0I
ωrot = 2π × 250 kHz. The magnetic field at a distance r = 4 cm from a wire with current I is µ2πr,

so each electrode produces a peak contribution of 400 µG at the trap center. The total contribution
nr < 1.2 mG,from all electrodes is three times the peak contribution from a single electrode, so Bleak

which contributes at most 15 Hz to the Zeeman splitting and a systematic of 0.2 Hz to the four-way
difference (Equation 6.63). This contribution assumes that all the electrodes are driven from one
direction only and assumes the worst case capacitance. We have switched to driving the electrodes
symmetrically from the top and bottom, which suppresses the leakage currents dramatically. A
reasonable estimate in this case is that the currents from the top drive balance those from thebottom drive to better than 10%, which reduces the systematic to < 15 mHz or 3 × 10−28 e cm.

253
This effect will also depend on the value of Erot , so should be detectable. A final contribution to
B nr would be a stray magnetic field gradient. This would originate most likely from magnetism ina material, such as the welds in the vacuum chamber or from the optical table: since all material,
especially welds, is located a large distance from the trap, stray gradients are expected to be small.
If we assume a stray gradient of 2 mG/cm, the resulting systematic is around 12 mHz.
All of the estimates in the previous paragraph assume that we do no active shimming of
B nr . A simple potential shim procedure would be to apply a compensation value of B nr to makef u (B) − f u (−B) ≈ 6gFu µB B nr = 0. Then a measurement in the lower doublet would provide ameasurement of 4de Eeff with a systematic of 6δgF µB B nr . A shim of f u (B) − f u (−B) < 100 mHz
would make B nr < 4 µG, which then contributes a systematic of 0.6 mHz or 3 × 10−29 e cm. A
100 mHz precision will hopefully only require perhaps 10 minutes of measurement time (assumingthat we are at the desired statistical sensitivity of 10−29 in a day!), so such a shim would not
be too hard to implement. There are several potential issues however with this simple shimming
procedure. One concern is a time dependence of B nr ; since we can monitor the reversal of theapplied B-field gradient at this level (see previous paragraph), the only concern is changes in theleakage current contributions. Other than possible slow temperature-dependent changes, thesecurrents should be static between a shim measurement in one doublet and an eEDM measurementin the other doublet. Another potential issue is a value of B nr that correlates with the doublet.
Since the doublet switch is set by a laser detuning, it is hard to come up with a likely source ofcorrelation.

6.6.2

Perpendicular magnetic fields

Because the quantization axis rotates rapidly, static magnetic fields in the xy plane quickly
average to zero. However, a magnetic field that is perpendicular to the quantization axis – in thiscase an axially oriented magnetic field – will not completely average to zero. The effect of such a

254
B⊥ is to tilt the quantization axis, which will cause a non-zero Berry’s phase. The tilt angle,
α≈

3 u/l
2 gF µB B⊥
,
3
3 u/l
γ
d
E
+
g
µ
B
rot
rot
F
B
mf
2
2 F

(6.64)

contributes 3ωrot α to f u/l (Brot ). A value of B⊥ = 0.5 G contributes 130 Hz to a measurement off u/l (Brot ). This is mostly removed in a B switch:
"
#
u/l
u/l
gF µB B⊥gF µB B⊥
ωrot
u/l
u/l
−
f (Brot ) − f (−Brot ) ≈ 3
2π γF dmf Erot + g u/l µB Brot γF dmf Erot − g u/l µB Brot
FF
u/l

(6.65)

ωrot (gF )2 µ2B B⊥ Brot≈6.
2π (γF dmf Erot )2

The contribution in the four-way difference is further reduced by the difference in g factors, soωrot 2ḡF δgF µ2B B⊥ Brot
ωrot (gFu )2 − (gFl )2 µ2B B⊥ Brot
B⊥=
6
≈ 2µHz.
(6.66)
fedm = 62π(γF dmf Erot )22π(γF dmf Erot )2
6.6.3

Field inhomogeneity

As mentioned earlier, trap motion results in hαi ≈ 0, where α is the tilt angle of the quantization axis from the plane perpendicular to the rotation axis. This is because the average axial
electric field experienced by an ion oscillating in the trap is zero since the ion is assumed to bestably trapped. Thus, the Berry’s phase acquired in one-half of an axial cycle (e.g., for z > 0)
cancels the phase from the other half-cycle (z < 0). However, this is not true if there is an axial
gradient to Erot such that Erot (z > 0) 6= Erot (z < 0). Since α ≈ Ez /(Erot + δErot ), where δErot is the(average) added contribution to Erot for z > 0, we can write the energy shift as mF ωrot EE2z 2δErot .
rot

2
zz
At a distance z = 0.5 cm, Ez = mω4e ≈ 1 V/cm.

If we assume δErot /Erot ∼ 0.5% (the worst caselimit from the measured inhomogeneity) and a typical value of Erot = 20 V/cm, the energy shift
would be expected to be around 2π × 0.2 Hz. This almost entirely cancels in a doublet switch,f u (B) − f l (B), except for a term that goes as(∆u )2 −(∆l )2
u/l(6gF µB Brot )2∼ 0.01, which would give a system-

atic of 2 mHz or 5 × 10−29 e cm. If we do a B switch first, then the shift does not cancel, andf u (B) − f u (−B) ∼ 0.4 Hz. This would then be shimmed out, leading to a systematic in the fourway difference of δgF /gF × 0.4 Hz or about 1 mHz (3 × 10−29 e cm). We can change the sign of
the Berry’s phase contribution by changing the rotation direction. In addition, the Berry’s phase255
should scale with Erot as well as ωrot , so should be noticeable. Finally, we could attempt to measure
this shift by kicking the ions and looking for a frequency shift.
The magnetic field gradient also causes a shift in Brot according to Equation 6.26. This same
effect contributes to decoherence as discussed in Section 6.5.2. All terms in Equation 6.26 depend
linearly on B 0 , so the frequency shift will be the same sign for both signs of Brot . Therefore, thisshift should cancel to a high degree in the B switch. The largest residual effect arises due to thepresence of B nr . In particular, if the current in the electrodes causes a B nr that we shim with a
0 , then this systematic shift will be different for the two signs of B . We
deliberately applied Bnrrot

can estimate the leading contribution of this systematic by replacing Ex,y,z in Equation 6.26 with
Tx,y,z . Assuming that the temperatures are all about the same, we would expect a contribution
u/l

6kB T
0 δB , where δBof 6gF µB Bnrrotrot ≈ eErot . This is the same sign in the doublet switch, so thecontribution to the four-way difference isδBrot0= 6δgF µB Bnr
fedm

6kB T
≈ 40µHz,eErot(6.67)

assuming a temperature of 10 K. Other radial inhomogeneities can be treated in a similar fashionand should also cancel in a B switch.

6.6.4

Other effects

Motion in electric and magnetic fields can lead to induced (relativistic) fields. The largest
contribution to the ion velocity comes from the rotating electric field, which yields a micromotion
eErotvelocity of mω
≈ 600 m/s for Erot of 20 V/cm. This motion in a magnetic field will produce
rot~ For a static axial magnetic field, the motional electric field
an electric field E~motional = ~v × B.
will always be parallel (or antiparallel) to Erot and so could cause a systematic. The largest axial
magnetic field will probably be due to Earth’s magnetic field currently (as there is no shielding orcancellation): a 0.5 Gauss axial field will result in a δErot = 0.3 mV/cm, which is suppressed by
10−5 from the applied electric field. This electric field will result in a small B nr , which will result

256
in a contribution to the four-way difference of
motional
fedm= 6δgF µB B 0

e
2
mωrot



eErot
B⊥
mωrot


≈ 4µHz.

(6.68)

Furthermore, Emotional will change sign with a rotation direction switch. Similarly, motion in an~ motional = 12 ~v × E.
~ The two dominant electric field
electric field will create a magnetic field: B
ccontributions are from Erot and from trapping fields. With Erot = 20 V/cm, Bmotional ≈ 0.1µG andis oriented axially. Such a small B⊥ will have no effect. Radial trapping fields are smaller than Erotand so will generate even smaller magnetic fields. Axial trapping fields will lead to a magnetic fieldthat could contribute to the applied magnetic field; however, even an anomalously large trapping
field of 1 V/cm will cause Bmotional < 0.01µG, which is less than 10−7 of Brot .
According to Equation 6.59, ∆ differs between the upper and lower doublet by 2δ∆. Because
the Zeeman shift is not infinitely large compared to ∆, there will be a difference in the fringe
frequency the depends on the doublet. This is difference is expected to be
2π(f u (B) − f l (B)) =(∆u )2 − (∆l )2u/l

6gF µB Brot

=

¯
4∆δ∆
u/l

6gF µB Brot

≈

4
3



dmf Erot
EHF
u/l

2

¯2
∆

6gF µB Brot

≈ 0.5Hz(6.69)

for a 40 Hz fringe with Erot = 20 V/cm. While this initially is concerning, it should be removed in
the B switch and so should not cause a problem.
Other potential effects could arise from imperfect state preparation or from phase shifts
during the Ramsey sequence. Previous eEDM experiments have observed differential Stark shiftsof the upper and lower Stark manifolds from various issues due to state preparation [420, 31], so
it is worth considering these effects. We can shutter all lasers during the free evolution period, so
Stark shifts due to laser light should not be a problem. In order to look for other effects, we cansimulate our Ramsey sequence by numerically integrating the time-dependent Schrödinger equation
using the Hamiltonian given in Equation 6.58. The electric field during the Ramsey sequence isshown in Figure 6.27(a). Figure 6.27(b) shows | h+3/2|ψi |2 (blue) and | h−3/2|ψi |2 (red) during
the Ramsey sequence. The large population oscillation after the second π/2 pulse is due to to theproximity to the avoided crossing, which means that the |±3/2i states are not perfect eigenstates.

257This oscillation means that the measurement of the phase accumulated during the free-evolution
period by measuring population after the second π/2 pulse will depend on the time the population
measurement is done. As long as the time after the second π/2 pulse is constant, this should notcause a problem, although since our depletion pulses are finite duration we will have to carefully
check from potential systematics by measuring with different delay times.
The presence of a finite magnetic field during the π/2 pulse means that our pulses are actuallyslightly “off-resonant”, i.e., it is equivalent to driving a microwave π/2 pulse with the microwave
frequency detuned from resonance. Assuming a two-level system and that all of the population
starts in |+3/2i = |zi, we can write the population difference during as the standard Rabi result:
Pz =where Ω =

2 − Ω20ΩtΩt
sin2 ( ) + cos2 ( ),2Ω2
2

(6.70)pΩ20 + 2 is the generalized Rabi frequency with on-resonance Rabi frequency Ω0 and

detuning . In this case, the π/2-pulse duration is ∼ π/(2Ω). For our system, Ω0 = ∆u/l and
u/l

eE

π/2

 = 3gF µB B 0 mωrot2 . From this, we can see that the π/2-pulse duration will be different for therotupper and lower doublets, which we must account for in the experiment. In addition, the offresonant pulses can lead to small phase shifts. We can use the detailed theory for a Ramsey
sequence using off-resonant pulses found in [480] to estimate that potential differential phase shifts
should be < 0.01 rad. These phase shifts can also be measured by measuring a fringe at shortfree-evolution times, so they should not cause a problem.

6.7

eEDM measurement
To perform a basic eEDM measurement, we need to evaluate the four-way difference from

Equation 6.63. This can be accomplished by measuring a Ramsey fringe in each combination of
doublet and sign of B 0 . Of course, measuring a full fringe in one configuration (switch state) takes
a long time, which increases the susceptibility to experimental drift between fringes. So, a betterdata taking strategy is needed.
The above method can be immediately improved by alternating between switch configurations

258

(a)

20

Erot (V/cm)

15
10
5
0
0

State probability

(b)

0.02

0.04

0.06

0.08

0.10

0.02

0.04

0.06

0.08

0.10

Time (s)

1.0
0.8
0.6
0.4
0.2
0
0

Time (s)Figure 6.27: Simulation of Ramsey sequence. (a) Erot as a function of time (in seconds). (b)Simulation of the population in |+3/2i (red) and |−3/2i (blue) as a function of time during theRamsey sequence. The delay for the second π/2 pulse delay is adjusted so that there is a bit of an
offset between the average population in |+3/2i and |−3/2i.

259
at one time point in the fringe, thus recording the four fringes simultaneously. An example data set
is shown in Figure 6.28, where each column corresponds to the data for one specific switch state.
At each time step and in each switch state, we measure Hf+ in four different chops as shown in
the third row: dissociation only (light green); transfer + dissociation (dark green); transfer, σ +depletion, wait T, σ + depletion, dissociation (light blue); and transfer, σ + depletion, wait T, σ −depletion, dissociation (dark blue). The two depletion chops are converted to a fractional depletionby subtracting the dissociation background and then dividing by the transfer - dissociation, as
shown in the second row. Finally, the difference of the two depletion chops gives the Ramsey fringe(fractional population difference) shown in the first row. We fit each fringe to an exponentiallydecaying sine wave with a decay time fixed to 300 ms. The frequency, amplitude, initial phase,and offset were fit individually for each switch state as shown in Figure 6.28. Taking the four-way
difference gives 0.34(33) Hz, so de = 7.5 ± 7.3 × 10−27 e cm. Not bad for the initial test with a few
hours of data!
We have now repeated this measurement multiple times, as shown in Figure 6.29. These
measurements include one at a 20 Hz fringe frequency as well as with different values of B nr
(sometimes removed with a B 0 shim, sometimes not, see Section 6.6.1). The combined eEDM
measurement is now de = −1.5 ± 3.0 × 10−27 e cm, as shown by the solid red line and dashed lines.
For a better measurement, we will want to further improve the data taking efficiency. In
principle, we only care about measuring a phase shift after a long wait time, which corresponds
to measuring a frequency difference. This phase could be measured by measuring near the zerocrossing of the fringe and looking for a change in measured population difference for different switch
states. However, to convert a change in the population difference to a phase difference it is necessary
to also know the slope of the fringe, which is related to the frequency and the fringe contrast. One
option for monitoring the slope is to measure two points at the side of a fringe, one below and one
above the zero crossing. This measurement could then be repeated on the opposite slope of thefringe, similar to what is done in the YbF experiment [420]. Such a procedure provides an inherent
calibration for the fringe contrast, which is necessary for determining the slope of the fringe andContrast

Depletion

5

10

1
0.8
0.6
0.4
0.2
0

-0.5

0

0.5

0
4000
2000
0

Hf+

HfF+

0

50

T (ms)

100 150 200

φ: (2π)0.581(11);offset: 0.0198(58); Contrast: 0.374(20);f: 39.26(12) Hz

State = [- l]050

T (ms)

100 150 200

φ: (2π)0.530(13);offset: 0.0125(76); Contrast: 0.402(26);

f: 38.72(14) Hz

State = [- u]050

T (ms)

100 150 200

φ: (2π)0.639(16);offset: 0.0264(80); Contrast: 0.323(31);f: 39.79(17) Hz

State = [+ l]050

T (ms)

100 150 200

φ: (2π)0.568(20);

offset: 0.016(11); Contrast: 0.408(41);f: 39.59(22) Hz

State = [+ u]260

Figure 6.28: Ramsey fringe measured for each switch state [±B 0 , u/l doublet].

261
thus the phase shift. In addition, since we really care about phase shifts that occur at long time due
to frequency differences, we need to remove any differences in initial phase. One nice advantage
of our experiment is that we can easily measure the phase and contrast at short times by onlychanging the delay time between π/2 pulses, as demonstrated in Figure 6.28. This means that we
can calibrate the initial phase by alternating between measurements at short time and long time.

6.8

Outlook√
The estimates from Table 6.2 give an initial statistical sensitivity of around 1×10−27 e cm/ daywith the current parameters. We also have several areas to improve upon these numbers withoutmajor upgrades to the experiment. In particular:(1) by upgrading the photodissociation laser to a higher pulse rate, we should be able to increase
the dissociation efficiency by a factor of at least 2;(2) we should be able to increase the free evolution time to 500 ms by fully understanding thedecoherence issues;(3) we should be able to increase the depletion contrast by a factor of 2.
This would increase N to 4 counts-per-shot (also probably increase the background, N0 , to some√thing like 6) and would increase τ , which would give a statistical sensitivity of around 2×10−28 e cm/ day.
This sensitivity will enable initial tests of systematic errors at the few 10−28 e cm level, which will
provide vital information for planning the next generation experiment.
On a longer timescale, we could hope to increase the free evolution time up to perhaps 1 s
while increasing the total ion number by lowering the temperature of the ions. This free evolution
time is still below the 3 ∆1 lifetime, but we will also need to improve the inhomogeneity limit shownin Figure 6.21 while keeping the fringe frequency significantly larger than ∆. We can accomplish
this by decreasing ωrot , which will both increase Brot for a given B 0 and decrease ∆. Effects due tothe ponderomotive potential from the rotating electric field will increase with decreasing ωrot , but

262

20

0.8

15

0.6
0.4

5

0.2

0

0

-5

-0.2

-10

-0.4

Hz

10-27 e cm

10

-0.6

-15

-0.8

-20

14/06/15 14/06/22 14/06/29 14/07/06 14/07/13 14/07/20 14/07/27

Date
Figure 6.29: All eEDM measurements to date. The red line and dashed line show the mean andstandard error, respectively, corresponding to de = −1.5 ± 3.0 × 10−27 e cm.

263
we should be able to manage this by optimizing the field homogeneity, for example, by increasing
the distance between the trap electrodes. In addition to increasing the free-evolution time, a larger
trap chamber with a higher pump-out speed (by removing LIF mirrors that limit the conductanceto the center of the trap) could also reduce the fast initial loss caused by collisional heating withbackground Ar. Several further possibilities exist for improving the signal-to-noise:(1) It might be possible to increase the fraction of population in 1 Σ+ J = 0 by a factor of
2. This could be accomplished by using microwave/laser STIRAP to excite the neutral
precursor to a non-core penetrating (high angular momentum) Rydberg state. Because theelectron in a non-core penetrating state does not interact with the ion core, it can thenbe removed without influencing the quantum state of the ion core. We could also consider
implementing a sympathetic cooling scheme using an atomic magneto-optical trap [447].
(2) Based on the transfer time scales that we have observed, we should be able to transfer more
population in a half-trap cycle if we can increase the transfer laser power and decrease theone-photon detuning. This should be possible without causing significant spontaneousemission by eliminating scattered light, which causes the sidebands observed in the onephoton linewidth shown in Figure 6.15. Removing the LIF mirrors and possibly following
the “paint it black” philosophy should significantly reduce the scattered light. This could
provide another factor of 2-3.
(3) By reducing the Hf+ loss between photodissociation shots and increasing the MCP efficiency
with a combination of a return bias voltage and coating with CsI or similar material, we
could increase the REMPD detection efficiency by a factor of 2.
√
With these improvements, we should be able to reach 1 × 10−29 e cm/ day.
Instead of a trapped ion cloud, it is intriguing to consider the use of a single trapped ion
as in optical clocks using atomic ions [481, 482, 483]. Quantum-logic spectroscopy [484] is usedin atomic ion clocks for state detection with extremely high fidelity, so the noise is limited byquantum-projection noise (i.e., shot noise on one count). Typically, quantum-logic spectroscopy

264
relies on the ability to perform clean π-pulses between motional states, which can be challenging
with molecules without cycling transitions; however, several options exist for extending quantumlogic spectroscopy to molecules [485, 486, 487]. One of motivations for considering a single trapped
ion is the potential for a very long coherence time, since ion-ion collisions are eliminated and field
inhomogeneity is significantly reduced. With HfF+ , our free-evolution time will be limited by the3∆+
1 lifetime to 2 s. However, ThF should have a significantly longer lifetime, currently estimatedto be greater than 30 s, so this would not be a limit. Other possible ions include TaO+ and WN+ ,which both might be expected to have 3 ∆1 ground states. WN+ has been proposed for an eEDM
measurement in a storage ring due to it’s expected 3 ∆1 ground state [488]. This can be explained byligand-field theory (what, you don’t remember this?, see Section 5.1.1): typically more ionic bonds
due to the O2− or N3− ligands will stabilize an sd configuration to below the s2 configuration,
see for example ScF vs TiO [489]. In addition, WC – which is isoelectronic with WN+ – has a
3∆

1 ground state [490]. Assuming then that state lifetime is not a problem, how long of a coherencetime can we expect? A coherence time of around 5 s has been demonstrated between hyperfine
states in trapped 43 Ca+ [491], so a coherence time of perhaps 10 s seems reasonable. Because weuse states which are first-order Zeeman sensitive, a coherence time of of 10 s requires better than
0.25% magnetic and electric field stability given the current Zeeman splitting. At this point, we
would also have to worry about the coherence time due to blackbody radiation, which is around 7 sfor HfF+ at room temperature. Assuming we reach 10 s, this would result in a sensitivity of around√1 × 10−29 e cm/ day, which is comparable to the projected sensitivity of the current experimental
design. However, many systematic errors will be very different using a single ion confined to a welldefined location. For example, Berry’s phase due to axial motion should be suppressed. On theother hand, stray electric and magnetic field gradients would not be averaged over and would haveto be shimmed away. Depending on what systematics are discovered in the current experiment, it
may be worthwhile considering a single-ion experiment, but, for statistical sensitivity, there is notmuch of a gain.

265
It is worth noting that ground state 3 ∆1 could be very beneficial for pushing below the
10−29 e cm level with a trapped ion cloud. Efficient ionization combined with optical pumping andpossibly sympathetic cooling could produce perhaps 25% of the ions in a single mF level, which is1000 ions at a modest ion number of 4000! In addition, precision spectroscopy of molecular ionsis currently in an infancy, so the techniques that we have developed and will continue to develop
should be beneficial for other experiments as well.

6.9

Useful numbers
In this section, we provide several tables of useful numbers.

266

Table 6.4: Useful numbers
Parameter

SymbolEstimate

Rotating electric field

Erot

20 V/cm

Electric field rotation angular frequency

ωrot

2π × 253.7912 kHz

Trap rfωrf

2π × 50 kHz

Magnetic field gradient

B0

0.08 G/cm

Molecular frame dipole momentdmf

1.401(5) e a0 ≈ 2π × 1.8 MHz/(V/cm)

Stark energy

dmf Erot /h

36.6 MHzElectric field coupling factor

γF

1/3 for F = 3/2, 2/3 for F = 1/2

Stark shift of mF = ±3/2

γF mF dmf Erot /h

18.3 MHz

Magnetic g-factor in 3 ∆1

gF

0.00305(10)Difference in g-factorδgF /gF

0.003(6) measured; < 0.002 theory

Radius of micromotion

rrot

0.4 mmRotating micromotion velocity

vrot

620 m/sZeeman splitting between ±3/2

3gF µB B

2π × 40 HzΩ-doublet splitting

ωef

2π × 740(40) kHz

Hyperfine splitting

Ehf

h × 45.6(1) MHz

267¯ δ∆, and Stark shift for different
Table 6.5: ∆,values of Erot , from theory
Erot

γF mF dmf Erot /h

¯
∆

δ∆

5

4.6 MHz434.5 Hz31.4 Hz6

5.5

251.4 Hz21.8 Hz7

6.4

158.3 Hz16.0 Hz8

7.3

106.1 Hz12.3 Hz

10

9.1

54.3 Hz

7.8 Hz12

11.0

31.4 Hz

5.4 Hz16

14.6

13.3 Hz

3.1 Hz20

18.3

6.8 Hz2.0 Hz25

22.8

3.5 Hz†1.3 Hz†

30

27.4

2 Hz†

0.9 Hz††Due to a level crossing between the upper Stark
manifold and the F = 1/2, mF = ±1/2 levels,
this might deviate from the theory.

Chapter 7

Conclusion

In this thesis, we have presented a variety of experiments covering a range of techniques
and applications of molecular spectroscopy. Direct frequency comb spectroscopy – with or withoutcavity enhancement – provides a unique combination of bandwidth, resolution, and sensitivity thatis not easily available in other systems. We have shown how this combination can be beneficial
for trace species detection by enabling simultaneous, multi-species detection even with competing background absorption or Rayleigh scattering. Spectroscopy systems using the demonstrated
technology or extensions farther into the mid-infrared have many potential applications including
industrial gas purity monitoring, pollution control, breath analysis, and atmospheric chemistry. In
addition, the newly-developed technique of time-resolved frequency-comb spectroscopy has excitingprospects for the study of chemical reactions including characterization of reaction intermediates
and precise measurements of branching ratios to different products. Frequency comb velocity modulation spectroscopy extends this combination of bandwidth, resolution, and sensitivity for the firsttime to the study of molecular ions with applications to astrochemistry, chemical physics, as well
as fundamental physics. Precision spectroscopy of molecular ions is an exciting direction for testsof fundamental physics because of the long coherence times and thus narrow linewidths that canbe obtained. Currently, we have achieved a Fourier-limited linewidth of 400 mHz in trapped HfF+
and used this to perform a preliminary measurement of the electron electric dipole moment. By
further increasing the signal and coherence time, this system will be competitive with the current
best measurement of the eEDM.

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