High-Precision Spectroscopy of Molecular Iodine:
From Optical Frequency Standards to Global
Descriptions of Hyper…ne Interactions andAssociated Electronic Structure
by
LISHENG CHEN
B.A., Beijing Polytechnic University, 1993
M.S., University of Colorado, 2002

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial ful…llmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
LISHENG CHEN
2005

This thesis entitled:
High-Precision Spectroscopy of Molecular Iodine: From Optical Frequency
Standards to Global Descriptions of Hyper…ne Interactions and AssociatedElectronic Structure
written by LISHENG CHEN
has been approved for the Department of Physics

Dr. Jun Ye

Dr. John L. Hall

Dr. David M. Jonas

Date

The …nal copy of this thesis has been examined by the signatories, and we …nd thatboth the content and the form meet acceptable presentation standards of scholarly
work in the above mentioned discipline.

iii
LISHENG CHEN (Ph.D., Physics)
High-Precision Spectroscopy of Molecular Iodine: From Optical Frequency Standards
to Global Descriptions of Hyper…ne Interactions and Associated Electronic Structure
Thesis directed by Associate Professor Dr. Jun Ye

A widely tunable and high-resolution spectrometer based on a frequency-doubled
Ti:sapphire laser is used to explore sub-Doppler transitions of molecular iodine in the
wavelength range 523

498 nm. We investigate the natural width of the hyper…necomponents at various transitions and its wavelength dependence is mapped out inthis region. The narrowest natural width observed is

52 kHz near 508 nm. The

observed excellent signal-to-noise ratio should lead to high-quality optical frequency
standards that are better than those of the popular 532-nm system. In addition, we
employ a self-referenced femtosecond optical comb to measure the absolute frequencyof the length standard at 514.67 nm, which is based on the a3 hyper…ne componentof transition P(13) 43-0. This technique improves the precision of the frequency measurement by two orders of magnitude as compared with previous wavelength-based
results.
The hyper…ne spectra of B

X transitions in the wavelength range 500–517

nm are investigated systematically. Four e¤ective hyper…ne parameters, eqQB , CB ,dB , andB , are determined for an extensive number of rovibrational levels spanning

the intermediate region 42 <
els

0

= 57

60, the 1g (103< 70 in the B0+u(u ) state.

Near vibrational lev-

+ 3g ) electronic state strongly perturbs the B0u (u ) statethrough rotational coincidence, leading to e¤ects such as abnormal variations in thehyper…ne parameters and strong u-g mixing recorded at the transition P(84) 60-0.
3
Various perturbation e¤ects in the B0+u(u ) state identi…ed so far are summarized.

iv
We have also performed a high-resolution analysis of the six electronic states
3
that share the same dissociation limit with the excited electronic state B0+u(3
molecular iodine. These six states are coupled to the B0+u(u ) inu ) state via hyper…neinteractions. The four hyper…ne parameters are calculated using available potential
energy curves and wave functions constructed from the separated-atom basis set. We
obtain a maximum separation of the respective contributions from all six electronic
states and compare each individual contribution with high-precision spectroscopic
data, allowing an independent veri…cation of the relevant electronic structure.

Dedication

To my parents, for their inspiring and unconditional support.

Acknowledgements

This thesis is the outcome of several graduate projects that are related to precision laser spectroscopy. I wish to take this opportunity to acknowledge my debts
to all those who have contributed to the work described in this thesis and apologize
in advance to those whose names and e¤orts I failed to recognize.
First of all, my great gratitude goes to my thesis advisor Jun Ye, who ushered
me into precision measurement and provided me with many research opportunities.
He showed me how to search around in the laboratory to …nd the right optical elements, taught me how to lock diode lasers to cavities, and performed the magic infront of me that makes a scrambled Ti:sapphire laser work again in a few minutes.
I had the freedom to explore various approaches and think independently and creatively in my experiments. He constantly encouraged me to broaden my researchexperience in the laboratory, to interact with other researchers, and to poke my head
into other exciting experiments in the laboratory. His diligence, patience, exceptional
high e¢ ciency, and determination were the driving forces of my graduate study andwill be a strong impetus to my future research. Without his mentorship, my graduate study would have gotten nowhere, and for this guidance I am forever indebted tohim.
John Hall is more like my second advisor. On many occasions, he o¤ered me
help not merely for immediate "pain relief" but rather to stimulate a new roundof inquiry. He carefully read several manuscripts related to this work and provided
valuable comments. He shared with me his experience in the frequency control of

vii
Ti:sapphire lasers. Many locking electronics he built years ago are still doing a wonderful job in our experiment. It is his hand drawings of circuits, and not the dazzlingelectronics on the market, that rekindled my old interests in circuit design.
Special thanks also go to Wang-Yau Cheng. We worked together on iodine
spectroscopy for more than one year. Before I joined the project, Wang-Yau had already assembled the Ti:sapphire laser and the iodine spectrometer and had obtainedsome interesting hyper…ne spectra of molecular iodine near the dissociation limit.
He helped me quickly come up to speed with the necessary experimental skills on
saturation spectroscopy. Without his hard work on the iodine spectrometer, I cannot
imagine that the follow-up investigations on the hyper…ne interactions would havecome to fruition.
Contributions from Tai Hyun Yoon at an early stage of the iodine experimentare acknowledged. My understanding of the Franck-Condon principle and molecularsymmetry is largely attributed to a one-semester course on molecular spectroscopyo¤ered by David Jonas, whose wave-function picture of electronic transitions made astrong impression on me. I would also like to thank Wibe de Jong at Paci…c Northwest National Laboratory for kindly sharing with us his ab initio potential energy
curves of molecular iodine.
I owe my gratitude also to Mark Notcutt, from whom I have received considerable help and education. His …nesse on vibration isolation, mechanical engineering,and various data collection and analysis methods in frequency domain has profoundlychanged my way of doing research. The experiences I accumulated on laser frequency
stabilization when I worked with Mark later turned out to be extremely helpful forthe iodine spectroscopy.
I have bene…ted tremendously from doing research in the JILA laboratories,
which retain a longstanding and rigorous investigation on the frequency stabilization of continuous-wave lasers and has extended this expertise to the frequency andviii
phase control of femtosecond laser frequency combs. I learned valuable lessons andsharpened my experimental skills through interactions with graduate students, postdocs, and visiting scientists in the laboratory. The absolute frequency measurementof molecular hyper…ne transitions described in chapter 3 is the fruit of collaboration
with Jason Jones and Kevin Holman, both of whom are experts on femtosecond laser
frequency combs. In addition, Kevin helped me to understand the details of the femtosecond laser frequency comb, and Jason explained the mounting structure of theultra-stable cavity used in his research on locking femtosecond pulses to stable cavities. I would also like to thank Seth Foreman for introducing to me his research on thesynchronization of two femtosecond lasers, Eric Hudson and Heather Lewandowski
for explaining their experiment on the collision properties of cold molecules and therelated apparatus, and Long-Sheng Ma for sharing his experience in iodine spectroscopy.
I had also the pleasure of working with Joshua Zirbel and Kang-Kuen Ni from
Carl Wieman’s group on the photoassociation experiment of cold atoms. Josh generously o¤ered me many projects closely related to the experiment and put up with
the inconvenience caused by my busy research schedule.
During the period of my graduate study, I received valuable assistance fromthe supporting sta¤ at JILA. The highly e¢ cient and diligent work of Ed Holliness,
Maryly Dole, Brain Lynch, Randall Holliness, and others in the purchasing o¢ cegreatly accelerated the progress of the experiment. The numerous purchase orders
they processed for a single experiment, including rush orders and instrumental trials,
repairs, and returns, are a testimony to their indispensable contributions. Technicalsupport from electronic shop is also highly appreciated. Far beyond the call of duty,
Terry Brown taught me many informal lessons on circuit design with great patienceand thoughtful ideas. James Fung-A-Fat repaired many electronic components forthe experiment. Julie Phillips in the scienti…c report o¢ ce contributed her editing

ix
work, which not only expedited the publishing process but also improved the qualityof the manuscript. I would like to thank Caroline Frazee, who helped to arrange myconference travel and who found me a desk and a book shelf in a quiet o¢ ce despite
the pressure of increasing demand for the o¢ ce place. I had the fortune to be able toattend a course taught by Barb Tennis on spoken English. Barb also organized many
seminars from which I began to learn the importance of academic communications. I
would also like to thank everyone in the instrument and specialty shops. In particular, Hans Green taught me every detail about the operation of the tooling machines
in the instrument shop, and the tricks he shared with me will be always treasured.
Indeed, without help from so many warmhearted and hardworking people, the joy of
discovery might have been a dream that never came true.
Finally, I must express my deepest thank to my parents, who have always been
there for me. Their unconditional support and love are a constant inspiration formy personal development. I also thank my two brothers Linqi and Zongze for theirsupport and for tolerating my long absence from family activities.

xContents

Chapter
1 INTRODUCTION
1.1 Sub-Doppler Saturation Spectroscopy . . . . . . . . . . . . . . . . . .

1
2

1.1.1

Doppler Broadening . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.2

Sub-Doppler Techniques . . . . . . . . . . . . . . . . . . . . .

4

1.1.3

Saturation Spectroscopy . . . . . . . . . . . . . . . . . . . . .

5

1.2 Hyper…ne Structure of Molecular Iodine . . . . . . . . . . . . . . . .

8

1.2.1

Hyper…ne Structure as Frequency References . . . . . . . . . .

8

1.2.2

Systematic Study of Hyper…ne Interactions . . . . . . . . . . .

10

1.2.3

Probing the Electronic Structure Through Hyper…ne Interactions 11

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2 SUB-DOPPLER MOLECULAR-IODINE TRANSITIONS NEAR THE DISSOCIATION LIMIT (523–498 nm)

15

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.2 Iodine Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.3 Sub-Doppler Line Shape Analysis . . . . . . . . . . . . . . . . . . . .

23

2.4 Power and Pressure Broadening . . . . . . . . . . . . . . . . . . . . .

25

2.5 Hyper…ne-Transition Linewidth near the B-State Dissociation Limit .

31

2.6 Hyper…ne Spectra near the B-State Dissociation Limit . . . . . . . .

35

xi
2.7 System Performance . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

3 OPTICAL FREQUENCY MEASUREMENT OF MOLECULAR IODINE HYPERFINE TRANSITIONS

41

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

3.2 Optical Frequency Comb Based on Femtosecond Lasers . . . . . . . .

44

3.2.1

Kerr-Lens Mode-Locked Ti:Sapphire Laser . . . . . . . . . . .

44

3.2.2

Comb Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3.2.3

Frequency Stabilization of the Femtosecond Comb . . . . . . .

50

3.2.4

Self-Referencing Carrier-Envelope O¤set Frequency . . . . . .

54

3.3 Optical Frequency Measurement . . . . . . . . . . . . . . . . . . . . .

57

3.3.1

Iodine Spectrometer . . . . . . . . . . . . . . . . . . . . . . .

57

3.3.2

Self-Referenced Femtosecond Laser Frequency Comb . . . . .

59

3.3.3

Absolute Frequency of the Iodine-Based Length Standard at
514:67 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

Characterizing the Iodine Spectrometer . . . . . . . . . . . . .

65

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

3.3.4

4 HYPERFINE INTERACTIONS AND PERTURBATION EFFECTS IN THE
3
B0+
u(

u ) STATE OF

127

I2

69

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

4.2 E¤ective Hyper…ne Hamiltonian . . . . . . . . . . . . . . . . . . . . .

72

4.2.1

Hyper…ne Spectra of Molecular Iodine . . . . . . . . . . . . .

72

4.2.2

Matrix Elements of the Four E¤ective Hyper…ne Hamiltonian75

4.3 Experimental Setup and Data Collection . . . . . . . . . . . . . . . .

77

4.3.1

Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . .

77

xii
4.3.2

Extracting the E¤ective Hyper…ne Parameters from the Hyper…ne Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4 B-State E¤ective Hyper…ne Parameters in the Region 42 683

6 70 .

85

4.5 Vibrational Dependence of Hyper…ne Parameters in the B State . . .

90

4.6 Radial Dependence of Hyper…ne Interactions . . . . . . . . . . . . . .

92

0

4.7 Rovibrational Dependence of Hyper…ne Parameters as a Result of theSecond-Order E¤ect. . . . . . . . . . . . . . . . . . . . . . . . . . .

4.8 Strong Perturbation from the 1g (1g ) State

96. . . . . . . . . . . . . . 101

4.9 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 105
5 CHARACTERIZING MOLECULAR-IODINE ELECTRONIC STRUCTURE
THROUGH HYPERFINE INTERACTIONS

107

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2 E¤ective Hyper…ne Hamiltonian in Molecular Iodine . . . . . . . . . . 109
5.2.1

Matrix Elements of the Hyper…ne Hamiltonian . . . . . . . . . 112

5.2.2

Second-Order Contributions to the E¤ective Hyper…ne Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.3 Preparation of Potential Energy Curves and Electronic Wave Functions 117
5.3.1

Potential Energy Curves . . . . . . . . . . . . . . . . . . . . . 117

5.3.2

Electronic Wave Functions in the Separated-Atom Model . . . 118

5.3.3

Hyper…ne Matrix Elements in the Separated-Atom Model . . . 122

5.4 Computing the E¤ective Hyper…ne Parameters . . . . . . . . . . . . . 125
5.5 Numerical Analysis of E¤ective Hyper…ne Parameters and RelatedElectronic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.5.1

Spin-Rotation Parameter CB and the Admixture of Two 1u
States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.5.2

Separation of Contributions from 0g (3g ) and (3)0uStates . . 131

xiiiStrong Perturbation from the 1g (15.5.3g ) State and its Contribu-

tion to the E¤ective Hyper…ne Parameters

B , dB , and eqQB

. 136

5.6 First-Order Electric Quadrupole Interaction . . . . . . . . . . . . . . 137
5.7 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 141

143

Bibliography

Appendix
A Matrix Elements of the Hyper…ne Hamiltonian155

A.1 Tensorial Form of the Hyper…ne Hamiltonian . . . . . . . . . . . . . . 155
A.2 Reduced Nuclear Matrix Element . . . . . . . . . . . . . . . . . . . . 157
A.3 Reduced Electronic Matrix Element . . . . . . . . . . . . . . . . . . . 158
A.4 Matrix Elements of the Hyper…ne Hamiltonian . . . . . . . . . . . . . 161
B Hyper…ne Spectra and E¤ective Hyper…ne Parameters for 79 B-state levels( 0 = 42

70,

= 500

517 nm)162

B.1 P(10) 42-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
B.2 P(19) 42-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
B.3 R(30) 42-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
B.4 R(41) 42-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
B.5 R(49) 42-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
B.6 P(13) 43-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
B.7 P(25) 43-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
B.8 R(27) 43-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
B.9 P(38) 43-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
B.10 R(40) 43-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

xiv
B.11 R(63) 43-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
B.12 R(21) 45-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
B.13 R(52) 45-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
B.14 P(69) 45-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
B.15 P(91) 45-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
B.16 R(31) 47-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
B.17 R(51) 47-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
B.18 R(92) 47-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
B.19 R(111) 47-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
B.20 P(19) 49-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
B.21 P(41) 49-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
B.22 P(62) 49-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
B.23 R(93) 49-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
B.24 P(108) 49-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
B.25 R(55) 50-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
B.26 R(20) 51-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
B.27 P(30) 51-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
B.28 P(50) 51-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
B.29 P(73) 51-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
B.30 P(90) 51-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
B.31 R(32) 53-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
B.32 R(51) 53-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
B.33 P(70) 53-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
B.34 P(89) 53-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
B.35 R(21) 55-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
B.36 P(61) 55-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
B.37 P(83) 55-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

xv
B.38 P(102) 55-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
B.39 P(19) 57-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
B.40 P(33) 57-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
B.41 R(65) 57-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
B.42 R(74) 57-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
B.43 P(80) 57-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
B.44 R(83) 57-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
B.45 R(95) 57-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
B.46 P(69) 58-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
B.47 P(17) 59-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
B.48 R(18) 59-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
B.49 P(27) 59-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
B.50 R(28) 59-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
B.51 P(49) 59-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
B.52 R(81) 59-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
B.53 P(87) 59-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
B.54 P(21) 60-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
B.55 R(34) 60-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
B.56 R(45) 60-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
B.57 P(77) 60-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
B.58 P(84) 60-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
B.59 R(99) 60-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
B.60 P(17) 61-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
B.61 R(36) 61-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
B.62 P(53) 61-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
B.63 P(83) 61-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
B.64 P(23) 63-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

xvi
B.65 R(31) 63-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
B.66 P(43) 63-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
B.67 P(52) 63-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
B.68 P(19) 65-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
B.69 P(33) 65-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
B.70 R(43) 65-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
B.71 P(33) 69-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
B.72 P(39) 69-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
B.73 R(44) 69-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
B.74 R(49) 69-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
B.75 P(53) 69-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
B.76 P(35) 70-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
B.77 R(37) 70-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
B.78 R(45) 70-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
B.79 P(63) 70-0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

xvii

TablesTable
2.1 Estimation of various linewidth broadening e¤ects . . . . . . . . . . .

31

4.1 B-state e¤ective hyper…ne parameters derived from the measurementof 127 I2 rovibrational transitions in the wavelength range 500

517 nm 87

4.2 Vibration-removed interpolation function eqQB (R) . . . . . . . . . .

96

B.1 Hyper…ne spectrum recorded at P(10) 42-0 . . . . . . . . . . . . . . . 164
B.2 Hyper…ne spectrum recorded at P(19) 42-0 . . . . . . . . . . . . . . . 165
B.3 Hyper…ne spectrum recorded at R(30) 42-0 . . . . . . . . . . . . . . . 166
B.4 Hyper…ne spectrum recorded at R(41) 42-0 . . . . . . . . . . . . . . . 167
B.5 Hyper…ne spectrum recorded at R(49) 42-0 . . . . . . . . . . . . . . . 168
B.6 Hyper…ne spectrum recorded at P(13) 43-0 . . . . . . . . . . . . . . . 169
B.7 Hyper…ne spectrum recorded at P(25) 43-0 . . . . . . . . . . . . . . . 170
B.8 Hyper…ne spectrum recorded at R(27) 43-0 . . . . . . . . . . . . . . . 171
B.9 Hyper…ne spectrum recorded at P(38) 43-0 . . . . . . . . . . . . . . . 172
B.10 Hyper…ne spectrum recorded at R(40) 43-0 . . . . . . . . . . . . . . . 173
B.11 Hyper…ne spectrum recorded at R(63) 43-0 . . . . . . . . . . . . . . . 174
B.12 Hyper…ne spectrum recorded at R(21) 45-0 . . . . . . . . . . . . . . . 175
B.13 Hyper…ne spectrum recorded at R(52) 45-0 . . . . . . . . . . . . . . . 176
B.14 Hyper…ne spectrum recorded at R(69) 45-0 . . . . . . . . . . . . . . . 177

xviiiB.15 Hyper…ne spectrum recorded at P(91) 45-0 . . . . . . . . . . . . . . . 178
B.16 Hyper…ne spectrum recorded at R(31) 47-0 . . . . . . . . . . . . . . . 179
B.17 Hyper…ne spectrum recorded at R(51) 47-0 . . . . . . . . . . . . . . . 180
B.18 Hyper…ne spectrum recorded at R(92) 47-0 . . . . . . . . . . . . . . . 181
B.19 Hyper…ne spectrum recorded at R(111) 47-0 . . . . . . . . . . . . . . 182
B.20 Hyper…ne spectrum recorded at P(19) 49-0 . . . . . . . . . . . . . . . 183
B.21 Hyper…ne spectrum recorded at P(41) 49-0 . . . . . . . . . . . . . . . 184
B.22 Hyper…ne spectrum recorded at P(62) 49-0 . . . . . . . . . . . . . . . 185
B.23 Hyper…ne spectrum recorded at R(93) 49-0 . . . . . . . . . . . . . . . 186
B.24 Hyper…ne spectrum recorded at P(108) 49-0 . . . . . . . . . . . . . . 187
B.25 Hyper…ne spectrum recorded at R(55) 50-0 . . . . . . . . . . . . . . . 188
B.26 Hyper…ne spectrum recorded at R(20) 51-0 . . . . . . . . . . . . . . . 189
B.27 Hyper…ne spectrum recorded at P(30) 51-0 . . . . . . . . . . . . . . . 190
B.28 Hyper…ne spectrum recorded at P(50) 51-0 . . . . . . . . . . . . . . . 191
B.29 Hyper…ne spectrum recorded at P(73) 51-0 . . . . . . . . . . . . . . . 192
B.30 Hyper…ne spectrum recorded at P(90) 51-0 . . . . . . . . . . . . . . . 193
B.31 Hyper…ne spectrum recorded at R(32) 53-0 . . . . . . . . . . . . . . . 194
B.32 Hyper…ne spectrum recorded at R(51) 53-0 . . . . . . . . . . . . . . . 195
B.33 Hyper…ne spectrum recorded at P(70) 53-0 . . . . . . . . . . . . . . . 196
B.34 Hyper…ne spectrum recorded at P(89) 53-0 . . . . . . . . . . . . . . . 197
B.35 Hyper…ne spectrum recorded at R(21) 55-0 . . . . . . . . . . . . . . . 198
B.36 Hyper…ne spectrum recorded at P(61) 55-0 . . . . . . . . . . . . . . . 199
B.37 Hyper…ne spectrum recorded at P(83) 55-0 . . . . . . . . . . . . . . . 200
B.38 Hyper…ne spectrum recorded at P(102) 55-0 . . . . . . . . . . . . . . 201
B.39 Hyper…ne spectrum recorded at P(19) 57-0 . . . . . . . . . . . . . . . 202
B.40 Hyper…ne spectrum recorded at P(33) 57-0 . . . . . . . . . . . . . . . 203
B.41 Hyper…ne spectrum recorded at R(65) 57-0 . . . . . . . . . . . . . . . 204

xix
B.42 Hyper…ne spectrum recorded at R(74) 57-0 . . . . . . . . . . . . . . . 205
B.43 Hyper…ne spectrum recorded at P(80) 57-0 . . . . . . . . . . . . . . . 206
B.44 Hyper…ne spectrum recorded at R(83) 57-0 . . . . . . . . . . . . . . . 207
B.45 Hyper…ne spectrum recorded at R(95) 57-0 . . . . . . . . . . . . . . . 208
B.46 Hyper…ne spectrum recorded at P(69) 58-0 . . . . . . . . . . . . . . . 209
B.47 Hyper…ne spectrum recorded at P(17) 59-0 . . . . . . . . . . . . . . . 211
B.48 Hyper…ne spectrum recorded at R(18) 59-0 . . . . . . . . . . . . . . . 213
B.49 Hyper…ne spectrum recorded at P(27) 59-0 . . . . . . . . . . . . . . . 214
B.50 Hyper…ne spectrum recorded at R(28) 59-0 . . . . . . . . . . . . . . . 216
B.51 Hyper…ne spectrum recorded at P(49) 59-1 . . . . . . . . . . . . . . . 217
B.52 Hyper…ne spectrum recorded at R(81) 59-0 . . . . . . . . . . . . . . . 218
B.53 Hyper…ne spectrum recorded at P(87) 59-0 . . . . . . . . . . . . . . . 219
B.54 Hyper…ne spectrum recorded at P(21) 60-0 . . . . . . . . . . . . . . . 220
B.55 Hyper…ne spectrum recorded at R(34) 60-0 . . . . . . . . . . . . . . . 221
B.56 Hyper…ne spectrum recorded at R(45) 60-1 . . . . . . . . . . . . . . . 222
B.57 Hyper…ne spectrum recorded at P(77) 60-0 . . . . . . . . . . . . . . . 223
B.58 Hyper…ne spectrum recorded at P(84) 60-0 . . . . . . . . . . . . . . . 225
B.59 Hyper…ne spectrum recorded at R(99) 60-0 . . . . . . . . . . . . . . . 227
B.60 Hyper…ne spectrum recorded at P(17) 61-0 . . . . . . . . . . . . . . . 228
B.61 Hyper…ne spectrum recorded at R(36) 61-0 . . . . . . . . . . . . . . . 229
B.62 Hyper…ne spectrum recorded at P(53) 61-0 . . . . . . . . . . . . . . . 230
B.63 Hyper…ne spectrum recorded at P(83) 61-0 . . . . . . . . . . . . . . . 231
B.64 Hyper…ne spectrum recorded at P(23) 63-0 . . . . . . . . . . . . . . . 232
B.65 Hyper…ne spectrum recorded at R(31) 63-0 . . . . . . . . . . . . . . . 233
B.66 Hyper…ne spectrum recorded at P(43) 63-0 . . . . . . . . . . . . . . . 234
B.67 Hyper…ne spectrum recorded at P(52) 63-0 . . . . . . . . . . . . . . . 235
B.68 Hyper…ne spectrum recorded at P(19) 65-0 . . . . . . . . . . . . . . . 236

xx
B.69 Hyper…ne spectrum recorded at P(33) 65-0 . . . . . . . . . . . . . . . 237
B.70 Hyper…ne spectrum recorded at R(43) 65-0 . . . . . . . . . . . . . . . 238
B.71 Hyper…ne spectrum recorded at P(33) 69-0 . . . . . . . . . . . . . . . 239
B.72 Hyper…ne spectrum recorded at P(39) 69-0 . . . . . . . . . . . . . . . 240
B.73 Hyper…ne spectrum recorded at R(44) 69-0 . . . . . . . . . . . . . . . 241
B.74 Hyper…ne spectrum recorded at R(49) 69-0 . . . . . . . . . . . . . . . 242
B.75 Hyper…ne spectrum recorded at P(53) 69-0 . . . . . . . . . . . . . . . 243
B.76 Hyper…ne spectrum recorded at P(35) 70-0 . . . . . . . . . . . . . . . 244
B.77 Hyper…ne spectrum recorded at R(37) 70-0 . . . . . . . . . . . . . . . 245
B.78 Hyper…ne spectrum recorded at R(45) 70-0 . . . . . . . . . . . . . . . 246
B.79 Hyper…ne spectrum recorded at P(63) 70-0 . . . . . . . . . . . . . . . 247

xxiFiguresFigure1.1 Experimental setup for sub-Doppler saturation spectroscopy . . . . .

7

1.2 Lamb dip in the absorption pro…le . . . . . . . . . . . . . . . . . . . .

7

1.3 A simpli…ed energy level diagram of molecular iodine . . . . . . . . .

9

2.1 Experimental setup of I2 spectrometer . . . . . . . . . . . . . . . . .

18

2.2 Single-mode selection using a birefringent …lter and two solid etalons
for a ring Ti:sapphire laser . . . . . . . . . . . . . . . . . . . . . . . .

20

2.3 Beat signal between a cavity-stabilized Ti:sapphire laser and a freerunning Yb:YAG laser at 1301 nm . . . . . . . . . . . . . . . . . . . .

21

2.4 Temperature control of the iodine-cell cold …nger . . . . . . . . . . .

24

2.5 Sub-Doppler FM line shape recorded at the a3 hyper…ne componentof the P(13) 43-0 transition . . . . . . . . . . . . . . . . . . . . . . .

26

2.6 Power broadening at the a3 hyper…ne component of the P(13) 43-0transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.7 Pressure broadening at the a1 component of the P(17) 54-0 transition.

29

2.8 Half-width versus pressure of three iodine transitions . . . . . . . . .

30

2.9 Comparision of linewidth measurement at the a3 hyper…ne componentof the P(13) 43-0 transition . . . . . . . . . . . . . . . . . . . . . . .

32

2.10 Wavelength dependency of linewidth . . . . . . . . . . . . . . . . . .

36

2.11 Typical spectra of I2 hyper…ne transitions from 523 nm to 500 nm . .

37

xxii
2.12 Typical FM signal recorded for a hyper…ne component of R(86) 57-0
at 508.060 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

3.1 Schematics of two Kerr-lens mode-locked Ti:sapphire lasers . . . . . .

45

3.2 Frequency pulling e¤ect in a dispersive laser cavity . . . . . . . . . .

48

3.3 E¤ective dispersion relation of a mode-locked laser . . . . . . . . . . .

49

3.4 A pulse train from a mode-locked laser and its frequency-domain representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

3.5 A simpli…ed picture for the frequency control of mode-locked Ti:sapphirelasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3.6 Simpli…ed block diagrams for feedback control of comb frequencies . .

55

3.7 Self-referencing determination of the comb o¤set frequency fceo . . . .

56

3.8 A precision I2 spectrometer based on a widely tunable Ti:sapphire laser 58
3.9 Experimental setup for a self-referenced frequency comb . . . . . . . .

61

3.10 Frequency measurement at the a3 component of the P(13) 43-0 transition 63
3.11 Long-term frequency measurement of the Nd:YAG/I2 -532 nm system

64

3.12 Long-term frequency measurement at the a3 component of the P(13)43-0 transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

3.13 Pressure shift of the a3 component at P(13) 43-0 . . . . . . . . . . . .

68

4.1 A simpli…ed energy level diagram of molecular iodine . . . . . . . . .

73

4.2 Schematic diagram of saturation spectroscopy in the wavelength range
517 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

4.3 Hyper…ne spectrum recorded at R(41) 42-0 . . . . . . . . . . . . . . .

80

4.4 Hyper…ne spectrum recorded at P(13) 43-0 and R(15) 43-0 . . . . . .

81

4.5 Hyper…ne spectrum recorded at P(35) 70-0 . . . . . . . . . . . . . . .

82

500

4.6 Vibrational dependence of the B-state hyper…ne parameters eqQB , CB ,dB , andB .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

xxiii
4.7 R dependence of the hyper…ne operators eqQB (R), CB (R), dB (R), and. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B (R)

4.8 eqQB , CB , dB , and00
B versus R centroid h J 0 j R j J 0 i .

4.9 Rovibrational dependence of eqQB

94

. . . . . . . . .

95. . . . . . . . . . . . . . . . . . .

98

4.10 Semilog plot of rovibrational dependence of CB. . . . . . . . . . . .

4.11 Semilog plot of rovibrational dependence of dB andB

99. . . . . . . . 100

4.12 Strong u-g perturbation observed at transition P(84) 60-0 . . . . . . . 103
5.1 Potential energy curves (PECs) for ten electronic states converging to
the 2 P 3 + 2 P 1 dissociation limit over the range 2:5 Å < R < 6:5 Å . . 119
22

5.2 Long-range PECs for the ten electronic states . . . . . . . . . . . . . 120
5.3 Combining the short- and long-range potentials . . . . . . . . . . . . 121
5.4 The B-state rovibrational structure . . . . . . . . . . . . . . . . . . . 126
5.5 Two 1u PECs used in the calculation of CB . . . . . . . . . . . . . . . 127
5.6 Second-order calculations of the spin-rotation parameter CB . . . . . 130
5.7 Rotational dependence of CB around two vibrational levels

0

= 47

and 70 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.8 Separation of the contributions from the 0u and 0g states . . . . . . . 134
5.9 Semilog plot of the second-order calculation of the scalar spin-spinparameterB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.10 Semilog plot of the second-order calculation of the tensor spin-spinparameter dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.11 Second-order calculation of the electric quadrupole parameter eqQB . 140
B.1 Hyper…ne spectrum recorded at P(69) 58-0 and R(65) 57-0 . . . . . . 210
B.2 Hyper…ne spectrum recorded at P(17) 59-0 and R(18) 59-0 . . . . . . 212
B.3 Hyper…ne spectrum recorded at P(27) 59-0 and R(28) 59-0 . . . . . . 215
B.4 Hyper…ne spectrum recorded at P(77) 60-0 . . . . . . . . . . . . . . . 224

xxiv
B.5 Hyper…ne spectrum recorded at P(84) 60-0 . . . . . . . . . . . . . . . 226
B.6 Hyper…ne spectrum recorded at P(63) 70-0 . . . . . . . . . . . . . . . 248

Chapter 1
INTRODUCTION

Precision spectroscopy provides an e¤ective way to probe the internal structureof atoms and molecules. Moreover, rigorous spectroscopic investigations have in many
ways re…ned and deepened our understanding of the fundamental laws of physics.
The spectral measurement of black-body radiation in the late 19th century lifted
the curtain on the quantum revolution. Precision measurement of Lamb shift in
atomic hydrogen provided strong supporting evidence for quantum electrodynamics.
A number of ongoing precision measurements, such as the detections of the time
variation of the …ne structure constant [1, 2, 3] and the particle intrinsic electric
dipole moment (EDM) [4, 5, 6, 7], will put stringent constraints on various new
theories beyond the standard model.
The narrow spectral lines of atoms and molecules are obscured or totally buriedby three major broadening e¤ects: the spectral impurity of the interrogating light, thebroad Doppler pro…le due to the relentless thermal motions of atoms or molecules,
and the limited interrogating time. With the advent of lasers, a number of subDoppler techniques have emerged that can e¤ectively remove the Doppler broadening.
Advances in laser frequency control and stabilization have greatly accelerated theimprovement in spectral resolution. New methods and techniques in spectroscopy or
related …elds are continuously developed. Two distinct examples are the laser-coolingof atoms and ions [8, 9, 10, 11, 12] and the introduction of femtosecond laser frequency

2
combs [13, 14, 15] that have revolutionized the optical frequency measurement.
This thesis covers a systematic investigation of the hyper…ne structure in molecular iodine. By using high-precision saturation spectroscopy, the hyper…ne spectraof the B

X transitions in molecular iodine are measured for a large number oflevels in the second excited electronic state B. In the following sections I will explain several key ingredients of our spectroscopic investigations such as the Doppler
broadening that severely masks the hyper…ne structure, how this broadening e¤ect is
circumvented by the saturation spectroscopy, energy levels in molecular iodine, andan e¤ective hyper…ne Hamiltonian developed by Broyer et al. [16] that is especiallysuitable for describing the hyper…ne interactions in molecular iodine. The majortasks of this spectroscopic study are also discussed in this introduction. A thesis
outline is given at the end of this chapter.
1.1

Sub-Doppler Saturation Spectroscopy

1.1.1

Doppler Broadening

Atomic or molecular transitions can be probed by passing an interrogating lightthrough a gas sample. The spectral lines obtained in this way are usually much broadthan what are expected from the natural widths of these transitions. This broadeninge¤ect prevents accurate and high-precision determination of the center of the spectral
line. Furthermore, a broadened strong line can often skew or totally conceal theadjacent weak features from which valuable information might be extracted.
The spectral lines are broaden by the Doppler e¤ect because atoms and molecules in a gas have di¤erent thermal velocities [17, 18]. A moving particle feels an
increased or decreased frequency of the incoming light, depending on whether it ismoving toward or receding from the incoming light. The …rst-order Doppler shift
experienced by the particle isk v [18], where k is the wave vector of the inter-3
rogating light and v is the velocity of the particle. As a result, if the particle has
a sharp spectral feature at frequency !0 in its rest frame, then the frequency of theinterrogating light has to be tuned to! = !0 + k v= !0 1 +

vz
c

(1.1)

in order for the light to be in resonance with the the moving particle. Note thatin arriving at the second equality in equation 1.1, we have assumed that the lightpropagates in the +z direction.
Gas particles move in all directions with di¤erent speeds. At thermal equilibrium and far from the quantum degeneracy, the probability distribution for each of
the three velocity components is Maxwellian [19]:w(vx ) = p

1
vp

exp

vx2 =vp2 ;(1.2)

where vx is one of the three components of the velocity in Cartesian coordinate,pvp = 2kB T =m is the most probable velocity, m is the mass of the particle, kB
is the Boltzmann’s constant, and T is the absolute temperature of the gas sample.

Because of this velocity di¤usion, the sharp resonances of the particle are spread out
in the laboratory frame. The broadened line pro…le can be obtained by combining
equations 1.1 and 1.2 [18]:I (!) = I0 exp
where

"

#
4 ln 2 (! !0 ) 2
;
2
!D

(1.3)

p
!D = 2 ln 2!0 vcp is the full spectral width at half maximum (FWHM).

For iodine molecules at room temperature (300 K), the Doppler width at 514 nm
is

D

=

!D
2

= 454 MHz. In contrast to this broad pro…le, the narrow lines ofiodine molecules around 514 nm have natural widths of

100 kHz, and 15 or 21

such lines are clustered together with distance between two adjacent lines typically4ranging from 100 Hz to 100 MHz. These …ne features are masked by the Doppler
pro…le associated with each narrow line. In the case of detecting spectral lines with abroad natural width that is comparable to the Doppler width, the intensity pro…le for
an isolated line is no longer Gaussian. Instead, it is a convolution of the Lorentzianand Doppler pro…les, which is called a Voigt pro…le [18].
Since the Doppler width is scaled down in accordance with the square root of
temperature, Doppler broadening is still prominent at low temperatures reached bycryogenic cooling or laser cooling. For instance, the 51 S0

53 P1 transition (689 nm)

in 88 Sr has a natural width of 7:6 kHz, but the associated Doppler width remains
to be 33 kHz even if atoms are laser cooled to 1 K. In this case, the informationof the line center, the natural width, and the Doppler width can be extracted froma deconvolution of the experimentally determined Voigt pro…le [20]. However, thisseparation of the contributions of natural width and Doppler broadening becomes
more di¢ cult when the atomic transition is orders of magnitude narrower than thatof the Doppler pro…le. Besides, the Doppler broadening prevents a real-time accessto narrow atomic transitions in applications such as optical frequency standards.
Consequently, without special countermeasures such as laser cooling and trapping
atoms into the Lamb-Dicke regime [21, 17] the Doppler broadening can persist to bea limiting factor in the spectroscopy of some extremely narrow transitions in atoms
or ions.

1.1.2

Sub-Doppler Techniques

To circumvent the resolution limit imposed by Doppler broadening, a direct
approach is to reduce the term

k v by collimating the atoms or molecules into abeam and probing all the particles in a direction that is perpendicular to the beam.
Indeed, atomic or molecular beams were major tools for sub-Doppler spectroscopy
before the era of lasers, and they are still widely used in spectroscopic studies.

5
The advent of lasers has completely changed the landscape of high-precision
spectroscopy. With the development of various tunable lasers, such as dye lasers,
semiconductor diode lasers, Ti:sapphire lasers, and …ber lasers, the spectral coverageof the intense laser radiation has been extended from visible spectrum to infraredand ultraviolet. On the other hand, the spectral purity of these tunable lasers has
been greatly improved thanks to the advances in laser frequency stabilization [22, 23,
24, 25, 26]. By virtue of nonlinear e¤ect, the intense radiation from a laser can exert
strong in‡uence on the gas sample. Based on this capability, a number of sub-Doppler
techniques have emerged to remove the …rst-order Doppler e¤ect. These sub-Doppler
schemes can be classi…ed as two major categories, namely, the saturation spectroscopy
and the two-photon spectroscopy. In the …rst category, a subgroup of atoms or
molecules with a narrow range of velocities in a particular direction is selected tointeract with laser radiations, while in the second category all the atoms or moleculesinteract simultaneously with two counter-propagating laser beams via two-photontransition and the …rst-order Doppler shifts of the two beams are therefore cancelled.
The following section gives a brief introduction of the saturation spectroscopy on
which our investigation of the hyper…ne spectrum of molecular iodine relies.

1.1.3

Saturation Spectroscopy

Rather than cooling the atoms or molecules to tame their thermal motions,saturation spectroscopy is based on the velocity-selective saturation of atomic ormolecular transition [27]. The velocity-selective saturation was …rst observed [28, 29]
as a Lamb dip [30] at the center of the Doppler-broadened absorption pro…le inside
the cavity of a HeNe laser. The experimental implementation of saturated absorption
spectroscopy was later re…ned by introducing the lock-in detection to remove theDoppler background [31, 32, 33]. Frequency-modulation (FM) technique has also been
applied to the saturation spectroscopy [34, 35, 36]. This FM saturation spectroscopy,

6
combined with a high-…nesse optical cavity for signal enhancement, has achieved highsensitivities in the detection of extremely weak molecular overtones [37, 38, 39].
Figure 1.1 (a) sketches the basic setup of the saturated absorption spectroscopy.
The laser beam is divided by a beam splitter into two beams that counter propagate
inside the gas sample. The strong pump beam selectively saturates the absorptionof a subgroup of atoms or molecules with a very narrow velocity distribution in the
direction of the pump beam. This saturation e¤ect does not a¤ect the absorptionof the weak probe beam as long as the two beams are in resonance with di¤erent
velocity groups. However, when the laser is tuned within a narrow bandwidth around!0 , both beams are simultaneously in resonance with the same group of particles
whose axial-velocity distribution is sharply centered around zero. Now the probe
beam experiences an extra reduction of the loss because the available absorbers are
depleted by the strong pump beam. As a result, the narrow spectral line is recovered
as a sharp dip at the center of the Doppler-broadened absorption pro…le, as shownin …gure 1.2. To remove the background signal arising from the Doppler-broadened
absorption pro…le of the probe beam, the pump beam is chopped at frequencies
ranging from several to a few tens of kilohertz. The background-free signal is thenobtained with a lock-in ampli…er.
FM sideband technique can be used to detect the changes both in the absorption and the dispersion, and to suppress the low frequency instrumental noise. Figure
1.1 (b) shows the setup for this detection scheme. The probe beam …rst passes a phasemodulator to develop two FM sidebands with opposite phases. When the optical frequency is scanned across the resonance, changes in the absorption and the dispersion
are converted to relative phase and amplitude variations among the two sidebands
and the carrier. The line pro…le (either absorption or dispersion, or their combinede¤ect) is recovered by a phase-sensitive detection at the modulation frequency witha fast photodetector and a RF mixer. By using a high modulation frequency, this

7
La
se

r

Lock-in
amplifierBeamsplitter
Chopper

Pump
Beam

Probe
beam

Detector

I2 cell

(a)

La
se

r

Lock-in
amplifierBeamsplitter
ChopperPhase
modulationPhase
sensitive
detection

Pump
Beam

Probe
beam

Detector

I2 cell(b)
Figure 1.1: Experimental setup for sub-Doppler saturation spectroscopy. (a) Saturated absorption spectroscopy. (b) FM sideband spectroscopy.

α(ω)
ω, k

ω0

v

ω, -k

ω

Figure 1.2: Lamb dip in the absorption pro…le. The frequency of two counterpropagating beams is tuned across the Doppler pro…le. For ! = !0 both waves
are simultaneously in resonance with the same group of molecules that are moving
perpendicularly to the laser beams, producing a sharp change of absorption around!0 .

8signal recovery scheme can e¤ectively suppress the low-frequency instrumental noise.
The line shape of sub-Doppler resonances recovered by FM sideband technique has
been studied in reference [36].
1.2Hyper…ne Structure of Molecular Iodine
The saturation spectroscopy introduced in the previous section is particularly

suitable for probing the gas sample around room temperatures. Using this subDoppler technique, narrow hyper…ne lines of molecular iodine can be readily resolved.
We move on now to highlight the central tasks of our spectroscopic investigation on
the hyper…ne spectrum of molecular iodine.

1.2.1

Hyper…ne Structure as Frequency References

Figure 1.3 shows the simpli…ed energy level diagram of molecular iodine. Plotted in this …gure are several potential energy curves (PECs) converging to threedi¤erent dissociation limits. Among these states are the ground state X, the second
excited sate B, and a 1g state converging with the B state to the second dissociation
limit. Several rovibrational levels in the X and the B states are also shown in the…gure. The thick vertical line in the …gure indicates an electric dipole transition
between two rovibrational levels ( 00 ; J 00 ) and ( 0 ; J 0 ) in the ground state X and thesecond excited state B, respectively. Because of hyper…ne interactions, each rovibrational level is further split into hyper…ne levels. Thus a rovibrational transition
contains many hyper…ne components.
High-precision measurement of hyper…ne spectrum in molecular iodine over an
extended range of rovibrational levels is valuable for selecting frequency referencesin the visible and in the near infrared. These frequency references can be used for
frequency stabilization of lasers and for serving as wavelength standards. Among the
wavelength standards recommended by Comité International des Poids et Mesures930

0g+(1Σ+g)2

P1/2+2P1/2

Energy (x103 cm-1)1g(1Πg)2

20

P3/2+2P1/2( υ',J' )Energy scale expanded

Hyperfinelevels in
the B state3
B:0+u( Πu)

2

P3/2+2P3/2

10

1 +
X:0+
g( Σg)( υ'',J'' )Energy scale expanded

034Hyperfinelevels in
the X state

5Internuclear separation (A)Figure 1.3: A simpli…ed energy level diagram of molecular iodine. Several rovibrational levels in the X and the B states are shown. The thick vertical line on the left
indicates a B
X rovibrational transition between two rovibrational levels ( 00 ; J 00 )and ( 0 ; J 0 ) in the X and the B states, respectively. Each rovibrational level is further split into hyper…ne sublevels. Thus one rovibrational transition contains many
hyper…ne components. Such a hyper…ne component is indicated as a thin verticalline on the right.

10(CIPM), molecular iodine (I2 ) holds a unique position in that it o¤ers …ve referencelines that have been widely used for metrological calibration [40]. The 633-nm HeNelaser stabilized by an iodine line is till widely used in the length metrology. Nd:YAG
lasers stabilized by hyper…ne lines at 532 nm are preferred to their companions at
longer wavelengths because of their compact size, reliability, and high stability (witha fractional frequency instability of < 5

10 14 at 1 s) [41].

To reach higher frequency stability, it is useful to explore I2 transitions at
wavelengths below 532 nm, where the natural widths can decrease at a faster rate
than that for the line strengths. Thus, it is of great interest to monitor the variationof the transition linewidth when iodine molecules approach the dissociation limit. We
built a widely tunable and high-resolution I2 spectrometer and measured the natural
width of hyper…ne components within the wavelength range 523

498 nm. This

investigation is described in chapter 2. In addition, to evaluate the potential of these
hyper…ne lines for serving as future frequency standards, we performed frequency
measurements at 514:67 nm with the help of a femtosecond frequency comb andinvestigated the performance of our iodine-based laser system. Chapter 3 covers thedetails of this survey.

1.2.2

Systematic Study of Hyper…ne Interactions

In order to accurately describe the hyper…ne structure at wavelengths below
514 nm, where the perturbation e¤ect becomes prominent, it is important to obtaina thorough understanding of hyper…ne couplings between the B and the perturbing
electronic states. Despite the dense spectroscopic data at longer wavelengths [42],precision measurement of hyper…ne spectrum in the wavelength range 500
(42 <

0

517 nm< 70) is scarce, leaving the quantitative knowledge of the perturbatione¤ects rather vague. In fact, previous study [43, 44, 45] of hyper…ne interactions at
vibrational levels0= 71 82 in the B state has revealed a strong hyper…ne perturba-

11tion from a nearby electronic state, and the primary survey of the hyper…ne spectrum
described in chapter 2 shows that the hyper…ne patterns in a much broad range ofrovibrational levels are dramatically modi…ed. Motivated by these observations, we
performed a systematic investigation on the hyper…ne spectrum of B
tion in the wavelength range 500

X transi-

517 nm. We measured the hyper…ne spectra of alarge number of rovibrational levels spanning the intermediate region (42 <

0

< 70)

in the B state. This investigation, which is detailed in chapter 4, provides rich information about the rotational and vibrational dependence of hyper…ne interactions inmolecular iodine.
An important theoretical tool that is heavily used in our spectroscopic investigation is an e¤ective hyper…ne Hamiltonian developed by Broyer et al [16]. Thise¤ective Hamiltonian, which contains several e¤ective hyper…ne parameters, is used
to describe hyper…ne interactions that can couple di¤erent electronic states. Throughthis e¤ective hyper…ne Hamiltonian, the seemingly intriguing patterns of the hyper…ne spectrum, which can vary greatly even across neighboring rovibrational levels,
can be described by a small number of hyper…ne parameters whose variations acrossdi¤erent levels are more predictable. It will be shown in chapter 4 that the hyper…ne parameters determined from spectroscopic …ts at a large number of rovibrational
levels o¤er an e¤ective way to examine the global evolution of hyper…ne interactions
in molecular iodine.

1.2.3

Probing the Electronic Structure Through Hyper…ne Interactions

The hyper…ne parameters introduced in the previous section serve as a bridgethat connects spectroscopic data with the electronic structure of molecular iodine.
On the experimental side, the introduction of these hyper…ne parameters greatly simpli…es the spectroscopic analysis of the hyper…ne spectrum. On the other hand, these12
hyper…ne parameters can be dissected into pieces that correspond to the hyper…necouplings with various perturbing electronic states. Because these perturbing states
are very close to the high-lying vibrational levels of the B sate, the hyper…ne-couplingstrength depends very sensitively on the electronic structure of the associated electronic states. Accordingly, the extensive and high-precision spectroscopic data on
hyper…ne spectrum permit a detailed study of electronic structure of the moleculariodine. This is especially feasible for the B state because vibrational wave functionsin the B state samples a broad range of internuclear separations. In particular, experimentally determined hyper…ne parameters over a large range of rovibrational levels
allow a sensitive test of the relevant electronic wave functions and potential energy
curves (PECs).
Hyper…ne parameters have been calculated for several vibrational levels ( 04082) in the B state [45, 46, 47, 48, 43]. The detailed information about the

PECs of perturbing electronic states was largely unavailable when these calculations
were performed, and hence the properties of the molecular wave function of the corresponding electronic state could not be independently inferred from these analyses.
In our analysis, we take the advantage of the signi…cant improvements in both experimental data and theoretical PECs. Thus a detailed calculation of the hyper…neparameters enables either the determination of the admixture of the basis wave functions in the separated-atom model or the veri…cation of more-sophisticated molecularwave functions.
In chapter 5, we present a numerical analysis of the hyper…ne parameters,
using the related empirical/theoretical PECs and electronic wave functions derived
from the separated-atom basis set. We divide the perturbing electronic states into
several groups whose contributions to the hyper…ne parameters can be separated
and directly compared with the corresponding experimental data. With the helpof this numerical analysis several interesting issues can be quantitatively addressed,

13
such as (1) to what extent the separated-atom model can be extrapolated to smallinternuclear separations, (2) the admixture of the basis wave functions at various
internuclear separations, and (3) the susceptibilities of the hyper…ne interactions tovarious adjustments of the relevant PECs.
1.3

Thesis Outline
Chapter 2 describes our investigation of the sub-Doppler transitions of iodine

molecules in the wavelength range 523 498 nm. To explore various properties of thesetransitions, especially the variations of their natural widths, we developed a widely
tunable and high-resolution spectrometer based on a frequency-doubled Ti:sapphirelaser. The experimental details of the I2 spectrometer are presented. An FM line
shape analysis is used to extract the linewidth information from the sub-Doppler
resonance. The results of this investigation include the linewidth dependence of theB

X transitions in the wavelength range 523

498 nm, the strong modi…cation of

the hyper…ne-resolved patterns near the B-state dissociation limit, and the potential
performance of the iodine-cell-based frequency standard at wavelengths below 532
nm. The observed excellent signal-to-noise ration (S/N) at several transitions and aclear trend of the linewidth narrowing indicate that I2 transitions in the wavelength
range 532 501 nm hold great promise for the future development of optical frequency
standards.
Chapter 3 reports the frequency measurement of the 514:67-nm length standard
at a3 hyper…ne component of the P(13) 43-0 transition. The frequency measurementof such a transition in the interesting wavelength range 523

498 nm provides …rst-

hand information that can be used to evaluate the potential of iodine-cell-based
systems serving as future frequency standards. A femtosecond frequency comb spanning an octave bandwidth is employed to check systematics and measure the absolute
frequency of the optical transition. This chapter contains a brief introduction to fem-

14
tosecond lasers in the context of frequency metrology, the details of the experimental
setup, the results of the frequency measurement of the length standard, and a characterization of the iodine spectrometer by a femtosecond frequency comb.
Chapter 4 presents a systematic investigation of 127 I2 hyper…ne spectrum in the
wavelength range 500 517 nm. The precision measurement of hyper…ne spectra overan extended range of rovibrational levels is valuable both for studying the molecular
structure and for selecting references for laser frequency stabilization. This chapter
begins with a brief overview of the hyper…ne spectrum in molecular iodine. The
experimental setup and data collection are then described. The hyper…ne spectra ata large number of rovibrational levels are recorded and the four e¤ective hyper…neparameters, eqQB , CB , dB , andB , are extracted from the spectrum at each level.

Spectroscopic data are presented and discussed in detail. This chapter also discusses
the abnormal variations in eqQB , dB , andB observed at several vibrational levels.

Various perturbation e¤ects in the B state are summarized. In addition, the Rdependence of the four e¤ective hyper…ne parameters is obtained by removing the
average e¤ect of the molecular vibration.
Chapter 5 provides a detailed analysis of the excited electronic states converging to the second dissociation limit in molecular iodine. Comprehensive and highprecision measurements of the hyper…ne spectrum in the B

X system of moleculariodine provide a unique opportunity for detailed examination of the global electronic
structure relevant to the second dissociation limit, 2 P 3 + 2 P 1 . In this chapter, we
22extend previous calculations of hyper…ne parameters to low vibrational levels andto rotational dependence at each vibrational level and compare these calculations
with spectroscopic data. In doing so, we are able to perform a sensitive test of theassociated electronic wave functions and potential energy curves.

Chapter 2
SUB-DOPPLER MOLECULAR-IODINE TRANSITIONS NEAR THE
DISSOCIATION LIMIT (523–498 nm)

2.1

Introduction
Hyper…ne-resolved optical transitions in molecular iodine (I2 ) often provide sta-

ble references for precision spectroscopy [49, 50]. The narrow hyper…ne components
have been widely used in laser frequency stabilization, ever since the development ofan early HeNe/127 I2 system using intracavity saturated absorption in a red laser at
633 nm [51]. Other systems include HeNe/127 I2 at 640, 612, 576, and 543 nm andAr/127 I2 at 515 nm as well as the 129 I2 variants. Most of these systems have wavelengths recommended for realization of the SI meter [40]. The I2 lines near 532 nm arestronger than red transitions, have relatively narrower (200 300 kHz) linewidth, andare readily accessible by frequency-doubled solid-state Nd:YAG lasers [52, 53]. The
532-nm system has proved to be one of the best practical optical frequency standards
because of its compact size, reliability, and demonstrated high stability (< 510 14

at 1 s) [41]. Recent development of wide-bandwidth optical combs shows that such
I2 -based references can be useful not only for precision length metrology, but also forgeneration of highly stable time signals [54].
Our motivation for performing detailed studies of hyper…ne transitions of I2
in the wavelength range of 500

532 nm is two-fold. First, the search for the best

candidates of I2 -based optical frequency references requires a systematic study of the16linewidth and the strength of many transitions. To attain a higher frequency stability,
it is useful to explore I2 transitions at wavelengths below 532 nm where the naturallinewidth may decrease at a faster rate than the line strengths as molecular iodine
approaches its dissociation limit. In fact, coincidental overlaps between the I2 transitions and corresponding Ar+ laser lines at 515 and 501 nm have already revealedtransition linewidth as small as 50 100 kHz [half-width at half-maximum (HWHM)]
at these wavelengths [55, 56]. It will be interesting to monitor variations in the transition linewidth when the I2 molecules approach the dissociation limit. Second, awidely tunable laser system permits systematic studies of rotation-vibration dynamics and hyper…ne interactions near the dissociation limit, providing rich information
on molecular structure and dynamics. Speci…cally, a large range of rovibrational
levels can be accessed, allowing a detailed parametric study of transition strengths,
hyper…ne interactions, and collision physics. We built such a widely tunable, yethigh-resolution and high-sensitivity I2 spectrometer and measured the linewidths oftransitions within the range 523

498 nm. Signals are recovered with an excellent

signal-to-noise ratio (S/N). We observe a clear trend of linewidth narrowing with decreasing transition wavelength as molecular iodine approaches the dissociation limit.
However, this tendency is complicated by variations in linewidth among di¤erent rotational or hyper…ne components. The limit on lifetime imposed by predissociations
and the associated broadening of the linewidth is studied. We also discover that thehyper…ne patterns are dramatically in‡uenced by the predissociation e¤ect.
This chapter is outlined as follows. Section 2.2 covers the experimental detailsof the I2 spectrometer based on a frequency-doubled Ti:sapphire laser. Discussions
about single-mode selection in Ti:sapphire lasers, laser frequency stabilization andscanning, and saturated absorption spectroscopy are given in this section. Section
2.3 introduces the line shape analysis that is used to extract the linewidth fromthe sub-Doppler resonance in frequency modulation (FM) spectroscopy. Section 2.4

17
explains the processes used to remove the e¤ects of power and pressure broadening
from the measured linewidth. Section 2.5 gives experimental results on linewidth
dependence of the transitions in the wavelength range 523

498 nm, together witha discussion on the predissociation of the B state. The strong modi…cation of thehyper…ne spectrum near the dissociation limit is discussed in section 2.6. Section 2.7
discusses the potential performance of the iodine-cell based frequency standard atwavelengths below 532 nm. Section 2.8 summarizes the chapter.
2.2

Iodine Spectrometer
Figure 2.1 shows the experimental setup that implements precision scan andcontrol of the laser frequency. Our single-mode Ti:sapphire laser is constructed in aring-cavity con…guration with a set of long-wavelength (1- m) coating mirrors andis pumped by a commercial diode-pumped solid state laser at 532 nm. At a pump
power of 8 W, the Ti:sapphire laser can be tuned from 1080 to 953 nm and provides
300 mW of useful output power from 1000 to 1034 nm, with a moderate power dropat the long-wavelength end. The mirror set consists of two curved mirrors (folding
mirrors) and two ‡at mirrors. These four mirrors form the ring cavity with a beam
waist (

0:4 mm) at the middle of two ‡at mirrors and a small beam waist (0:03mm) in the center of the two curved mirrors where the Ti:sapphire gain medium isplaced. An optical diode is inserted into the beam path between the two ‡at mirrors
to enforce unidirectional propagation of the laser radiation.
Three additional …ltering components with decreasing bandwidth are placed
inside the cavity to obtain single-mode operation. A birefringent …lter with a bandwidth of

1:5 THz is used for coarse wavelength adjustment. Single-mode operationis then achieved with the combination of a thin and a thick solid etalon. The thicknesses of the two etalons used in this system are 0:5 mm [free spectral range (FSR)
200 GHz] and 10 mm (FSR

10 GHz), respectively. Because each etalon has re-

PD

PBS

Etalon 2 BW

OD

PD

PBS

Prism

Monitor25 MHzTi:sapphire
laser

BF

Etalon 1

Phase
delay

EOM

PBS
10 MHzEOM

λ/2

Lock-in
amplifierFabry-Perotcavity

AOM
1

VCO

VCO

AOM
2

MirrorSHG

Frequency-scansignalDouble-pass AOMPZT driverFrq.
ditherBS

PD

Figure 2.1: Experimental setup of I2 spectrometer. AOMs, acousto-optic modulators; BF, birefringent …lter; BW, Brewster window
plate; EOM, electro-optic modulator; OD, optical diode; PBS, polarization beam splitter; PD, photodiode; SHG, second-harmonic
generator; =4, quarter-wave plate.

PZT

PZT driver

Servo

λ/4

Phase
delay

Iodine cell

80 MHzSpatial filter

AOM
3

Temperature
controlSpatial filter

40 KHzZerodur cavity

Phase
delay

Lock-in
amplifier

Pump laser

Monitor

Signalrecord18

19
peating transmission peaks, the FSR of the thin etalon is chosen so that it can selecta single transmission peak of the thick etalon, which is narrower and in turn selects asingle laser mode. This mode selection scheme is illustrated in …gure 2.2. Because thelasing process is extremely sensitive to small variation of the net gain, a few percentof the loss provided by the etalon is su¢ cient to suppress the unwanted cavity modes
and hence the etalons used for intracavity mode selection are not coated. The anglesof the two solid etalons can be tuned by galvo scanners to select di¤erent laser modes.
To probe the narrow resonances of the 127 I2 transitions, we use a reference cavity with a …xed length to stabilize the frequency of the Ti:sapphire. A small portionof the fundamental power (15 mW) is sent to this reference cavity for laser fre-quency stabilization. The cavity uses two highly re‡ective broadband mirrors (99%;a ‡at and a concave mirror with R = 1 m) and its FSR and …nesse are approximately 234 MHz and 200, respectively. The Zerodur cavity spacer is suspended
inside an evacuated aluminum chamber that is temperature controlled to within 20
mK. To e¤ectively isolate the cavity from vibrations of di¤erent origins, we wrap
the aluminum chamber with lead foils sandwiched by thermal insulation, and threestages of damping structure are cascaded between the chamber and the optical table.
The laser is tightly locked to one of the cavity modes using the Pound-Drever-Hallmethod [22]. The laser frequency is corrected by two feedback transducers: One is apiezoelectric transducer (PZT) mounted on the back of a cavity mirror and the otheris an acousto-optic modulator (AOM 1 in …gure 2.1) that performs fast frequency
correction. The operational laser linewidth is limited by the vibration noise anddrift associated with the cavity. The linewidth of this cavity-stabilized Ti:sapphirelaser is examined by heterodyne beating its infrared radiation against a free running
Yb:YAG laser at 1031 nm. Figure 2.3 shows two beat notes obtained from the heterodyne measurement. The linewidth of the beat note is 23 kHz (FWHM) when the
baseplate of the Yb:YAG laser is not temperature controlled, and it falls to 16 kHz

20Thin etalon

Transmission

1

Birefringent filter

0.9

Thick etalon
0.8

1000

500

0

500

1000

Thin etalon

Transmission

1

Thick etalon
0.9

Selected mode

Laser cavity mode
0.8

15

10

5

0

5

10

15

Frequency offset (GHz)

Figure 2.2: Single-mode selection using a birefringent …lter and two solid etalons fora ring Ti:sapphire laser. The birefringent …lter consists of three quartz plates whose
thicknesses are related by integral multiples. The thicknesses of the thin and thicksolid etalons are 0.5 mm and 10 mm, respectively. The etalons are uncoated.

21(FWHM) once the laser is temperature controlled. This decrease in the linewidth
implies that these measurements are probably limited by the free-running linewidth30
20

FWHM: 23KHz

(a)

10
0
-10
-20
-30
3.7

3.8
3.9
4.0
Frequency offset (MHz)

4.1

Amplitude of beat note (a.u.)

Amplitude of beat note (dB)of the Yb:YAG laser.
1.0
0.8

FWHM: 16kHz

(b)

0.6
0.4
0.2
0.0
1.20
1.25
Frequency offset (MHz)1.30

Figure 2.3: Beat signal between a cavity-stabilized Ti:sapphire laser and a freerunning Yb:YAG laser at 1301 nm. (a) Yb:YAG laser without temperature control.
(b) Yb:YAG laser temperature controlled.

Frequency scanning of the Ti:sapphire laser is accomplished with a doublepassed AOM (AOM 2 in …gure 2.1) that is located at the cavity input. With a…xed cavity length (aside from drift), changing of the drive frequency of AOM 2 willthus tune the stabilized laser frequency continuously over a 150-MHz range without
losing the cavity lock. Because the cavity’s FSR (mode spacing) is 244 MHz (< 150
MHz 2), this AOM-based laser tuning allows for continuous frequency scan overconsecutive cavity modes, and is basically limited only by the single-mode tuning
range of the laser itself.
For second-harmonic generation (SHG), a KNbO3 crystal (353 mm; bcut) is placed inside a buildup cavity. The crystal temperature is varied from 172 C
to 71 C to achieve noncritical phase matching at wavelengths ranging from 523 to
498 nm. At 502 nm, SHG yields a roughly 30-mW output power with 190-mW
fundamental input. One of the transmission peaks of the buildup cavity is ditherlocked to the fundamental laser frequency (see …gure 2.1). In this locking scheme, thecavity length is dithered by applying a sinusoidal signal (10100 kHz) to the PZT22
mounted on the back of a ‡at cavity mirror. A photodiode receives the cavity leakage
at the back of a curved mirror. This signal is then sent to a lock-in servo ampli…er
that generates error signal by phase-sensitive detection at the dither frequency andits harmonics. The cavity-length correction signal is combined with the dither signal
at the high-voltage PZT driver.
Sub-Doppler saturated absorption spectroscopy of I2 hyper…ne components ismainly performed with the FM saturation spectroscopy [35]. The I2 spectrometershown in …gure 2.1 is similar to that described in reference [41] but has the added ‡exibility of using either FM spectroscopy (FMS) or modulation-transfer spectroscopy(MTS) for appropriate recovery of a saturated-absorption signal. The light from thefrequency-doubling cavity is divided by a polarization beam splitter into two beams
that counter propagate inside the iodine cell. The strong pump beam selectively
saturates a group of molecules with a very narrow velocity distribution along the direction of the pump beam, while the weak beam probes this group of molecules. The
probe beam is phase modulated by a EOM to develop two FM sidebands with opposite phases. When the optical frequency is scanned across the molecular resonance,changes in both absorption and dispersion are converted to the relative phase andamplitude variations among the two sidebands and the carrier. The line shape information is recovered by a phase-sensitive detection at the modulation frequency witha fast photodetector and a radio frequency (RF) mixer. By using high modulation
frequencies, this signal recovery scheme can e¤ectively suppress the low frequency instrumental noise. To allow for a lock-in detection of the resonance signal, the pumpbeam is chopped at 40 kHz by an 80-MHz AOM (AOM 3 in …gure 2.1).
The high-purity I2 cell was prepared by the Bureau International des Poids etMeasures and is 8-cm long, with a Brewster window at each end. The I2 pressure
is controlled by the temperature stabilization of the cell’s cold …nger. Figure 2.4
sketches the two-stage cooling of the cell cold …nger. With this two-stage cooling, the23
cold-…nger temperature can be varied from 15 C to

22 C to access vapor pressures

ranging from 18 to 0:37 Pa. The temperature of the copper tube enclosing the cold…nger is stabilized and is monitored by a thermistor inside the base of the copper tube.
A thermistor with a small bead size and of a di¤erent type is placed in contact withthe glass wall of the cold …nger to verify the accuracy of temperature measurement.
2.3

Sub-Doppler Line Shape Analysis
Using the I2 spectrometer described in section 2.2, we investigate sub-Dopplertransitions of iodine molecules in the wavelength range 523

498 nm. To search

for the best candidate for an I2 -based optical-frequency reference requires systematicstudies of the linewidth and strength of many transitions in this wavelength region.
For each hyper…ne component selected, the Ti:sapphire laser is scanned across thiscomponent with a typical range of

10 MHz (in green light) and the probe beam ismodulated at a frequency of 6 MHz. For a majority of the transitions investigated,the collimated pump and probe beams are of
powers are 1

3-mm diameter and typical pump

8 mW. The power of the probe beam is kept an order of magnitude

smaller than the pump power.
We …t the sub-Doppler resonance to the line shape derived by Hall et al. [36].
The linewidth and the center of the sub-Doppler resonance are extracted from the…t. With the introduction of the two pro…le functions
2

Lj =

2+( j

, and Dj =
2
0)

( j
2+( j

0)
2
0)

;

(2.1)

the signal obtained from phase-sensitive detection at modulation frequency fm canbe expressed as
IF M ( ) = [ LIn equations 2.1 and 2.2,

L+ sin + D+

2D0 + D

represents the optical frequency,of the optical resonance, superscripts j = +, 0, andcos ]:(2.2)

0 is the center frequency

label three FM components

24

Iodinecell

Brewster window
Enclosure
Thermistor

Thermalinsulation

Thermistor
Aluminum
plate

Copper tube
Cold finger

Thermistor
2nd stage TEC
1st stage TEC
Aluminium
baseplateFigure 2.4: Temperature control of the iodine-cell cold …nger. The small gap between
the glass wall of the cold …nger and the copper tube is …lled with thermal grease.
TEC, thermoelectric cooler.

25
j

=

+ f2m , , andfm 1, respectively, and2resonance. In equation 2.2,

is a scaling factor,

is the linewidth (HWHM) of theis a phase shift that can be adjustedwith a delay line inserted between the local oscillator and the mixer used in the phasesensitive detection. Note that in equation 2.2, we only consider the contributions fromthe carrier and the two …rst-order FM sidebands. The modulation spectrum contains
mainly these three components because the phase modulation index is always below
1 in our measurements.
Figure 2.5 shows a typical resonance signal recorded for the a3 hyper…ne component at rovibrational transition P(13) 43-0. The resonance signal is obtained with
1:425-Pa vapor pressure, 3:7-mW pump power, and 6-MHz modulation frequency.
The …t (solid line) gives a linewidth of 365:9(3:7) kHz. The residual error of the …t isplotted at the bottom of the …gure. To reduce the in‡uence of the slow and unidirectional drift of the external cavity, a scan across the resonance is followed immediately
by a reverse scan, and the averaged linewidth is adopted.
2.4Power and Pressure Broadening
To obtain the zero-power, zero-pressure linewidth of a particular line, we typ-ically vary the I2 cell’s pressure from 0:4 to 4 Pa. At each pressure we measure thelight-power dependence of the linewidth and determine its zero-power value. We then…t the zero-power linewidth versus pressure to …nd the zero-pressure linewidth. Neareach wavelength we usually measure and compare a few lines that either have different rovibrational quantum numbers in the excited state or are di¤erent hyper…necomponents within the same transition.
The light-power dependence of the linewidth is determined by measuring theresonance linewidth at several power levels. The data points are then …tted to the1

The frequencies of the pump and the probe beams are simultaneously tuned, but only the probe
beam is frequency modulated. This con…guration results in an extra factor of 21 for the modulation
frequency fm [36].

26

1.0
0.5
0.0
-0.5
-1.0
-1.5
0

2

4

6

8

10

12

14

Frequency (MHz)Figure 2.5: Sub-Doppler FM line shape recorded at the a3 hyper…ne component of
the P(13) 43-0 transition. The …t (solid line) gives a linewidth of 365:9(3:7) kHz. The
bottom curve is the residual error of the …t. The modulation frequency is 6 MHz,the vapor pressure is 1:425 Pa, and the pump power is 3:7 mW.

27
power-broadening relation(P; I) =p(P )1 + 1 + I=Is ;
2

(2.3)

where I is the power of the pump beam, P is the vapor pressure, and Is is a pressuredependent saturation power. Note that this relation applies to our experimental
condition in which only the strong pump bean can saturate the I2 molecules. Figure
2.6 shows the power …t at several vapor pressures for the a3 hyper…ne component at
the P(13) 43-0 transition. Each panel in …gure 2.6 is the power …t at a certain vapor
pressure. At each vapor pressure, power broadening is removed by extrapolating thelinewidth to zero power.
Once the e¤ect of the power broadening is removed, the zero-power linewidths
at di¤erent pressures are …tted to a quadratic pressure dependence(P ) =+ aP + bP 2 :(2.4)

Such a pressure …t at the a1 hyper…ne component of the P(17) 54-0 transition isshown in …gure 2.7. In this case, two …ts are performed. The …rst includes all thedata points. As can be seen in the …gure, a data point with a vapor of

2 Pa has arelatively large residual error. This point is excluded in the second …t, which produces
an 8-kHz di¤erence in the …tted zero-pressure linewidth. The 1- uncertainties of the…tted zero-pressure linewidth are also given in the …gure. As another example, …gure
2.8 shows three pressure …ts with the corresponding hyper…ne components indicatedin the …gure. We notice that, even at nearly the same wavelength, natural linewidthsof transitions with vastly di¤erent rotational quantum numbers can di¤er by as muchas 40 kHz and exhibit quite di¤erent collision broadening coe¢ cients.

28

Pressure: 0.518 Pa
Γ(P) = 192.7(2.7)

Resonance linewidth (HWHM) (kHz)

300

320

280

300

260

280

240

260

220

240
0.5

340

Pressure: 0.742 Pa
Γ(P) = 214.0(4.1)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

1.0
360

Pressure: 1.234 Pa
Γ(P) = 258.2(2.9)

350

1.5

2.0

2.5

3.0

3.5

4.0

3.0

3.5

Pressure: 1.661 Pa
Γ(P) = 301.2(3.2)

340

320

330

300

320
280
1.0
430
420

1.5

2.0

2.5

3.0

3.5

1.0
450

Pressure: 2.674 Pa
Γ(P) = 350.7(3.2)

440

410

430

400

420

390

410

380

400

370

390
1.0

1.5

2.0

2.5

3.0

3.5

4.0

1.5

2.0

2.5

Pressure: 3.122 Pa
Γ(P) = 379.0(7.6)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Pump power (mW)
Figure 2.6: Power broadening at the a3 hyper…ne component of the P(13) 43-0 transition. Each panel shows the power broadening at a certain vapor pressure. Solidlines are the …t to the power dependence describing two counter propagating beams
among which only the strong pump beam has a saturation e¤ect. At each pressure,
power broadening is removed by extrapolating the linewidth to zero power.

29

400

HWHM (KHz)

350
300
Dropped for thedashed line.

250
200

Γ(P)=94(11)+117.2P-9.24P2

150

All data points are used in the fit.
Γ(P)=85.5(9.6)+130.5P-12.7P2

One data point is dropped.

100
0

1

2

3

4

Vapor pressure (Pascal)
Figure 2.7: Pressure broadening at the a1 component of the P(17) 54-0 transition.
The power broadening at each pressure has already been removed.

30

400
360

HWHM (KHz)

320
280
240
200
514.673 nm, P(13) 43-0:a3
514.986 nm, R(107) 49-0:a1
508.077 nm, R(20) 50-0:b5

160
120
80
0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Vapor pressure (Pa)
Figure 2.8: Half-width versus pressure of three iodine transitions. The power broadening at each pressure has already been removed.

31
2.5

Hyper…ne-Transition Linewidth near the B-State Dissociation
Limit
In this section we investigate various broadening e¤ects encountered in the

linewidth measurement. Their estimated values and corresponding uncertainties are
summarized in table 2.1. At the a3 hyper…ne component of the P(13) 43-0 transition,
we compare our result with that obtained by two-photon resonant scattering technique in a previous experiment [57]. Figure 2.9 shows the comparison. In both cases,
various broadening e¤ects, such as transit-time, power, pressure, and laser linewidth,have been removed.
Table 2.1: Estimation of various linewidth broadening e¤ects.
Broadening e¤ect
Transit time

tt [18]Wave-front curvature [58]Dependence (HWHM)
1
2

p

Value

Uncertaintyq
kT 1
2 ln 2 2Mw(z)15 kHz

2:0 kHz

w(z)2
R(z)

1:8 kHz

1:0 kHztt

1 h 2
2 M c2

Recoil doublet [18]Laser linewidth
Pressure broadening
Total

2

beat (HWHM)

2(P ) =

3:0 kHz
8 kHz+ aP + bP 2

4:6 kHz
10 kHz28 kHz

12 kHzFigure 2.10 displays a number of measured zero-pressure linewidths. For comparison, the 543-nm (reference [60]) and 532-nm data points are also included. The
dashed line (28 kHz) at the bottom of the …gure represents the remediable technicallimit to the measured linewidth, including contributions from wave-front curvatureand transit time (17 kHz), laser linewidth (in the green, 8 kHz), and recoil doublet (3:0 kHz). The narrowest natural width that we have found so far is52

kHz (HWHM) for the b5 component at the R(20) 50-0 transition (508:077 nm). As

Linewidth (HWHM) (kHz)

32

140
130

1σ

120
1σ

110
This work

100

Tench, et al.

90

Figure 2.9: Comparision of linewidth measurement at the a3 hyper…ne componentof the P(13) 43-0 transition. The data on the right is from reference [59]. In bothresults, various broadening e¤ects, such as transit-time, power, pressure, and laser
linewidth, have been removed.

33
shown in …gure 2.10, we observe a clear trend of linewidth narrowing with decreasingtransition wavelength.
This linewidth narrowing at shorter wavelength can be explained by the decreasing B ! X transition probability when the levels in the B state approach thedissociation limit [61]. For a vibrational level close to the dissociation limit, a molecule spends most of its time around the outer classical turning point where it can beviewed as two separated atoms. With this separated-atom model in mind and considering that the B and the X sates dissociate respectively into 2 P 3 + 2 P 1 and 2 P 3 +
2

2

2

2

P 3 atomic states, a B ! X transition at large internuclear separations requires an2atomic 2 P 1 !2 P 3 transition, which is an electric dipole forbidden transition.
2

2

In an attempt to quantitatively analyze the linewidth narrowing, we performthe calculations of the radiative decay rate from B to X. The radiative decay rates
from a level ( 0 ; J 0 ) in the B state to all possible levels (including energy continuum)
in the X state amount to [62]A ( 0J 0) ==1 64 4 X X SJ 0 J 00 3 0 0
jh J j
4 "0 3hc3 00 J 00 =J 0 1 2J 0 + 1
1 64 4 X 3 0 0
jh J j
4 "0 3hc3 00

e (R) j00

e (R) j00

2

J 00 ij2

J 0 ij ;(2.5)

where SJ 0 J 00 is the Hönl-London rotational intensity factor (SJ 0 J 00 = J 0 for R branchand SJ 0 J 00 = J 0 +1 for P branch),

is the frequency of the transition ( 0 J 0 ) ! ( 00 J 00 ),R is the internuclear separation, ande (R) is the transition dipole moment.

Thesquared matrix element in equation 2.5 is often evaluated by employing the R-centroid
approximation:
jh 0 J 0 j

e (R) j00

2

J 00 ijj e (Rc )j2 jh 0 J 0 j00

2J 00 ij ;(2.6)

where the R-centroid is de…ned as
h 0 J 0 j R j 00 J 00 i:
Rc =h 0 J 0 j 00 J 00 i

(2.7)

34
The radiative decay rates of the B-state levels in the wavelength range 500

540 nmare calculated according to equation 2.5. In the …rst calculation we choose a transition
dipole moment obtained by Tellinghuisen [63] from an empirical …t of radiative decay
rates measured at various rovibrational levels in the B state [61]:2e (R) = c1 exp

c2 (Rc3 )2 R 2 ;(2.8)

where c1 = 21:5(2:1) Debye2 Å2 , c2 = 1:73(37) Å 2 , and c3 = 3:49(12) Å. The unit
Debye is de…ned as 1 Debye = (10 21 =c) = 3:33564095

10 30 Cm [64], where c isthe speed of light in m/s. In the second calculation we adopt the transition dipole
moment obtained by Lamrini et al. [62]. Lamrini et al. determined the transition
dipole moment by selectively exciting molecules to a B-state level and measuring theintensity spectrum of the ‡orescence. Because only relative values are available intheir measurement, the experimental determined radiative rate at0= 43 [65] wasused to calibrate the absolute scale of the transition dipole moment. This transition
dipole moment is given in the R-centroid approximation:8>>
134:0416 + 166:1989Rc 75:81764Rc2
>
>
>
>
>
>
2
< +15:12918R3 1:12537R4
j e (Rc )j
c
c
log10
2 = >
1 Debye
>
>
>
>
>
>
>
: 2:6642 1:1635Rc(2:633 Å 6 Rc 6 3:9 Å)(3:9 Å < Rc 6 6:035 Å)(2.9)

with a standard deviation8>< 0:150 j e (Rc )j2 (2:633 Å 6 Rc 6 3:9 Å)
2(j e j ) =>: 0:545 j e (Rc )j2 (3:9 Å < Rc 6 6:035 Å):(2.10)The radiative decay rates are calculated after a global shift of the right-hand side ofequation 2.9 by an amount of 0:30103 to allow agreement with the radiative decay

rate

rad = (31:4

1:8)

104 s 1 measured directly at0= 43 [65].

The calculated radiative decay rates are divided by a factor of 2 for HWHM
and then are plotted as solid curves in …gure 2.10. Both two calculations show a35decreasing trend consisting with the measurement. However, this tendency is complicated by variations in linewidth among various rotational or hyper…ne components
even at the same wavelength. Besides the radiative decay, the gyroscopic and thehyper…ne predissociations through the B 00 1u (1 ) state provide another depopulating
channel that increases the B-state natural width. The predissociation rates not only
depend on the rovibrational levels but are also di¤erent for hyper…ne sublevels within
one rovibrational level [66]. For example, the predissociations contribute 5

61 kHz(FWHM) [57] to the natural widths of various hyper…ne components belonging to
the P(13) 43-0 and the R(15) 43-0 transitions. As the B-state level approaches thedissociation threshold, the limit on lifetime imposed by predissociations will need tobe taken into consideration [66, 67, 61]. Last, we point out that both calculations
overestimate the radiative decay rates compared with the experimentally determined
natural widths, which are due to radiative decay and predissociations. The calculation can be improved by calibrating the absolute scale of the dipole moment, suchas that given in equation 2.9, with radiative decay rates measured at more hyper…nelevels in the wavelength range 500
2.6

550 nm.

Hyper…ne Spectra near the B-State Dissociation Limit
Figure 2.11 shows some typical saturation spectra of I2 transitions from 523 to

500 nm. These spectra were obtained with a vapor pressure of 4:12 Pa. The I2 transitions are identi…ed by the CNRS I2 spectral atlas [68]. However, spectra obtained
below 500:53 nm are out of the range of the I2 atlas. Our I2 spectrometer is foundto have a constant noise ‡oor throughout the wavelength range of interest. Strongtransitions (with strengths comparable to those of 532-nm lines) are relatively common in the range from 530 to 510 nm, as shown in …gure 2.11. Near the dissociation
limit (499:5 nm) [69], the line strengths and associated S/Ns decrease substantially.
With such a widely tunable spectrometer, a large range of rovibrational quan-HWHM at zero pressure (KHz)

36

350
300
250

Measurement
Calculation 1
Calculation 2 and its
1-σ uncertainy band

**

***

200
150

*

100
50
0
500

Contribution to linewidthof technical origin

510

520

530

Wavelength (nm)540

Figure 2.10: Wavelength dependency of linewidth. Solid circles are measurement
results with the collision-induced broadening removed. These results include thebroadening e¤ects such as transit time, laser linewidth, and recoil splitting, whose
accumulated contribution is represented by the dashed line. The solid curves with
open circles are calculated widths due to electric dipole transition B ! X. Calculations 1 and 2 adopt transition moment functions obtained by Tellinghuisen [63] andLamrini et al. [62], respectively. Gyroscopic and hyper…ne quenching of the B-statelevels through a B 00 1u (1 u ) state also broadens the upper levels, which may preventthe fast decreasing trend indicated by the calculation near and below 500 nm. *,wavelength standard P(13) 43-0, a3 at 514:673 nm [40]; **, measured by Nd:YAG
laser; ***, measured by HeNe laser [60].

37Line 2667, P(54)38-0, 522.474 nm

(a)

a15

a1

a21

Line 3749, P(61)43-0, 516.505 nm(b)a1Line 1275, P(61)53-0, 507.949 nm(c)a1

a15&a16a21

500.506 nm

(d)
a1

a15

Figure 2.11: Typical spectra of I2 hyper…ne transitions from 523 to 500 nm. (a), (b),(c) MTS, 273-kHz modulation frequency; (d) FMS, 6-MHz modulation frequency.
The quoted [68] line strengths are (a) 80, (b) 79, and (c) 55. The time constant is 5
ms for (a)–(c) and 100 ms for (d). Length of I2 cell, 8 cm; vapor pressure, 4:12 Pa.

38
tum numbers can be addressed, allowing a detailed parametric study of transition
strengths, hyper…ne interactions, and collision physics. For example, …gure 2.11 (a)shows that the spectrum of P(54) 38-0 at 522:474 nm (even rotational quantum
number J 00 ) follows the usual 15-line hyper…ne pattern (1–4–4–1–4–1), whereas the
spectrum of P(61) 43-0 at 516:505 nm (odd rotational number J 00 ), shown in …gure
2.11 (b), displays the typical 21 hyper…ne components (3–4–4–3–4–3). However, as
the transition wavelength approaches the predissociation region, e.g., line P(61) 53-0
at 507:949 nm […gure 2.11 (c)], the pattern of hyper…ne splittings starts to change.
When the wavelength approaches the dissociation limit, as at 500:506 nm, the 15-line
pattern is completely modi…ed […gure 2.11 (d)]. Similar observations are made forother lines near 500 nm. Clearly the hyper…ne interactions are dramatically in‡uenced by the dissociation e¤ect and other nearby electronic states converging with
the B state toward the second dissociation limit.
2.7

System Performance
In this section, we explore the potential performance of an iodine-cell-basedfrequency standard operating at wavelengths below 532 nm. The Ti:sapphire laser
is locked on a hyper…ne component of molecular iodine. We send the resonancesignal extracted from FM saturation spectroscopy to a JILA loop …lter to generate afrequency correction signal. This correction signal is dispatched to a voltage control
oscillator (VCO), which in turn adjusts the frequency shift of the double-passed AOM
between the external cavity and the Ti:sapphire laser (AOM 2 in …gure 2.1). Thissecond feedback loop basically suppresses the drift and low-frequency noise from theexternal cavity. The transition selected for this evaluation is one of the hyper…necomponents of R(86) 57-0 at 508:060 nm. Figure 2.12 shows the resonance signal of
the selected hyper…ne component recovered with FM saturation spectroscopy. The
modulation frequency and the time constant used in the FM saturation spectroscopy

39
are 6 MHz and 1 ms, respectively. The cold …nger temperature of the I2 cell is

15 C,

corresponding to a vapor pressure of 0:79 Pa.
The projected frequency noise of a laser locked to the center of a hyper…necomponent can be estimated from the discrimination slope and the rms noise at thebaseline of the FM resonance signal. For the signal shown in …gure 2.12, the estimatedfrequency noise is 3 kHz, corresponding to a fractional noise of 5
or 1:610 12 at 1 ms

10 13 at 1 s, with an 8-cm-long cell. The best (in)stability reported in an

I2 -stabilized laser system is 5

10 14 at 1 s with a 120-cm long I2 cell, also at

15 C[41]. Therefore the performance of the new system, if it is scaled to the same I2 cell
length, would have an (in)stability of 1

10 14 at 1 s. Placing the I2 cell inside a

Signal size (arb. unit)

multipass cavity will further improve system performance.

1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-5

0

5

Frequency (MHz)Figure 2.12: Typical FM signal recorded for a hyper…ne component of R(86) 57-0 at
508.060 nm. Laser frequency stabilization with the discrimination slope at the line
center will lead to a projected (in)stability of 1:6 10 13 at 1 s, or 1 10 14 at 1 s
for a 120-cm-long I2 cell.

40
2.8

Summary
A widely tunable and high-resolution spectrometer based on a frequency-doubled

Ti:sapphire laser is used to explore sub-Doppler transitions of iodine molecules in the
wavelength range 523

498 nm. The wavelength dependence of the hyper…ne tran-

sition linewidth of iodine is mapped out in this region, and the narrowest linewidthis

52 kHz near 508 nm. The observed excellent S/N for the recovered resonancesignal indicates that I2 transitions in the wavelength range 532

501 nm hold agreat promise for future development of optical frequency standards, especially whencoupled with the all-solid-state Yb:YAG laser. In addition, the hyper…ne-resolved
patterns are found to be largely modi…ed toward the dissociation limit.
Frequency-based measurements of hyper…ne spectra across a large number vibrational levels will reveal important information on the variations of charge distribution and molecular con…guration. Based on this observation, we perform an extensive
investigation of the hyper…ne spectrum in the wavelength range 517
details of this investigation will be covered in chapter 4.

500 nm. The

Chapter 3
OPTICAL FREQUENCY MEASUREMENT OF MOLECULARIODINE HYPERFINE TRANSITIONS

3.1

Introduction
Our knowledge of the precise values of physical quantities in many branches ofmetrology is ultimately tied to frequency measurement. The de…nition of the unit oflength and its practical realization are based on the adopted value for the speed oflight, c = 299 792 458 m/s, and the frequency of an optical transition. Thus, length
measurements are based on the unit of time. This notion is re‡ected by the fact thattwelve optical transitions in the infrared and visible regions are recommended by the
Comité International des Poids et Mesures (CIPM) [40] for the realization of meter,
the SI unit of length. Among the listed optical frequencies recommended by CIPM,
molecular iodine (I2 ) holds a unique position in that it o¤ers …ve reference lines thathave been widely used for metrological calibration.
Nd:YAG lasers, frequency doubled and stabilized by I2 hyper…ne transitions
near 532 nm, have demonstrated a remarkably high stability of 5

10 14 at 1 s[41]. Near 515 nm, frequency-doubled all-solid-state Yb:YAG lasers with compact
size and high output power also exhibit low intrinsic frequency noise. Moreover,
our work discussed in the previous chapter showed that below 532 nm, the natural
widths may decrease at a faster rate than line strengths. Therefore, to attain ahigher frequency stability, it is useful to explore the performance of I2 -stabilized

42
laser systems below 532 nm. Absolute frequency measurements performed in thiswavelength region are an e¤ective way to evaluate the potential of these systems forserving as future frequency standards. Although the system at 514:67 nm promises tobe one of the better-quality standards based on I2 , historically it has been probed onlywith cumbersome Ar+ laser systems and the transition frequency determined only
through wavelength interferometry. The recommended 514:67-nm length standard
was based on such measurements of the hyper…ne component a3 at transition P(13)43-0 of 127 I2 . The adopted value by the CIPM in 1997 is 582 490 603.37 MHz,
with a standard uncertainty of 0:15 MHz (2:5
temperature set at

10 10 ), for an iodine-cell cold-…nger

5 C [40].

The accepted primary frequency standard thus far is based on a hyper…ne transition in atomic cesium, with the transition frequency of 9:1926317700

109 Hz. Prior

to the introduction of femtosecond lasers, the optical frequency measurement wasdone by complex frequency chains (harmonic-synthesis frequency chains or opticalfrequency interval-divider chains) [70, 71, 72, 73, 74, 75, 76, 77] to bridge the enormous
frequency gap that separates the radio and optical frequencies. Since the introductionof Kerr-lens mode-locked femtosecond lasers for optical frequency metrology [13, 14],
wide bandwidth optical combs have revolutionized optical frequency measurements,
impacting a wide range of optical frequency standards based on transitions of atoms,
ions, or molecules [78, 79, 80, 81]. Femtosecond laser combs serve the same purpose
as frequency chains in measuring absolute optical frequencies, but do so in just one
step with access to a broad optical spectrum and with a greatly simpli…ed systemsetup. Basically, mode-locked lasers can generate a femtosecond-pulse train with precise periodicity. In the frequency domain, this pulse train corresponds to a comb ofnarrow and equidistant frequency components spanning a bandwidth ranging fromseveral tens of terahertz to about one hundred terahertz. The frequency structureof the comb can be characterized by a surprisingly small set of parameters [82, 15]:43m = mfrep + fceo , where m is the mode order (105106 ), frep is the repetition

rate of the pulse train or, equivalently, the mode spacing of the frequency comb,
and fceo is the carrier-envelope o¤set frequency [83]. Both frep and fceo are radio
frequencies (RF). Through this relationship, the optical frequencies are linked directly to the RF frequencies, and vice versa. In a typical frequency measurement,
the repetition rate, frep (or mode spacing of the comb), of the femtosecond laser isphase-coherently locked to the Cs microwave frequency standard. The other degreeof freedom associated with the femtosecond comb, i.e., the carrier-envelope o¤set
frequency, fceo , can also be measured or stabilized with the same RF reference using
a self-referencing approach [79] or another known optical standard. This drasticallysimpli…ed frequency-measurement scheme can be realized with a reliable and compact table-top system. The measurement reliability and accuracy of the optical comb
system have been carefully studied [80, 81]. The I2 -based 532-nm laser system has
been used to stabilize an entire octave-bandwidth-spanning optical frequency comb
based on a mode-locked Ti:sapphire laser, thus establishing an new concept of opticalatomic clock in which the RF signal is phase-coherently derived from the I2 opticaltransition [54].
This chapter reports the …rst absolute frequency determination of the 514:67nm length standard at the a3 hyper…ne component of the P(13) 43-0 transition in
127

I2 . Our absolute frequency measurement employs a phase-coherent optical fre-quency comb linked to the Cs primary clock, which is the current realization of theunit of time. This measurement produces a 100-fold improvement in the precision
over previous wavelength-based results. In section 3.2, I will give a brief introductionof the principles and operations of femtosecond laser frequency combs in the context
of frequency metrology. Section 3.3 covers the details of the experimental setup andmeasurement results. The frequency comb used in this experiment is described. The
experimental details of the iodine spectrometer have already been discussed in the44previous chapter. Therefore only the modi…cations made for this experiment are explained in this section. The remainder of the section presents the results of the shortand long- term optical-frequency measurements, along with a characterization of theiodine spectrometer by the femtosecond laser comb.
3.2

Optical Frequency Comb Based on Femtosecond Lasers

3.2.1

Kerr-Lens Mode-Locked Ti:Sapphire Laser

The Kerr-lens mode-locked Ti:sapphire laser [84, 85] plays a central role in our
optical frequency measurement. Figure 3.1 shows schematics of two typical Kerr-lensmode-locked Ti:Sapphire lasers with prism-based [86] and prismless ring-cavity [84]con…gurations. The Ti:sapphire crystal in the laser cavity is pumped by 532-nm green
light from a diode-pumped solid-state (DPSS) laser that provides a reliable single
mode operation. The Ti:sapphire crystal serves both as the gain medium and thesaturable absorber for passive mode locking. A pair of prisms is added to the cavity
to compensate for the group-velocity dispersion (GVD) of the Ti:sapphire crystal
in the standing-wave con…guration [87], while the same dispersion management isperformed by a set of chirped mirrors in the ring-cavity con…guration [88].
In sharp contrast to the singe-mode continuous-wave (CW) operation describedin previous chapter, a large number of cavity modes (Ti:sapphire gain bandwidth (700105106 ) within the1000 nm) are active and are synchronized inphase in a mode-locked Ti:sapphire laser, resulting in a train of femtosecond pulses
with precise periodicity and stable pulse shape. This soliton-like propagation of the
optical pulse in the cavity is a consequence of the nonlinear interaction between the
optical pulses and the Ti:sapphire gain medium. Most optical material exhibits a…eld-induced change in the refractive index, which is referred to as the optical Kerr

45

Output coupler &translating PZT(Mode spacing)
Prism pairPumpβ

Ti:sapphire
crystalHigh reflector& tilting PZT(Mode position)(a)
OutputcouplerM2

M3 & PZT(Mode spacing)

Pump power
control
M1 (Mode
position)
Ti:sapphire
crystal

AOM

Pump

(b)
Figure 3.1: Schematic of two Kerr-lens mode-locked Ti:sapphire lasers with (a)standing-wave con…guration and (b) ring-cavity con…guration. M1 M3 are chirped
mirrors.

46
e¤ect and can be written as [89](3.1)

n = n0 + n2 I;

where I is the optical intensity and n2 > 0 for Ti:sapphire gain medium. When alight beam passes through the Kerr medium, it creates an index pro…le that resembles
the beam’s transverse intensity distribution. Because this index pro…le is equivalent
to a convex lens (Kerr lens), the beam is focused toward the center and its intensity
is ampli…ed.
The nonlinear Kerr-lens mechanism initiates and enforces the passive modelocking process. When one of the folding mirrors1 is translated away from its optimized position for the CW-laser operation, the lasing mode in the cavity can nolonger saturate the Ti:sapphire gain medium. However, the spatially focused pulses
with enhanced peak intensity can still saturate the gain medium and pass throughthe small gain region. Noise spikes in the cavity provide the seed from which the
Kerr-lens mode-locking (KLM) [90] starts. In addition, the KLM tends to temporally compress the pulse because only the central portion of the pulse has enoughintensity to burn its way through the saturable absorber, whereas the pulse tails
experience a net loss.
As circulating pulses are continuously compressed by the KLM process, theself-phase modulation (SPM) plays an important role in pulse shaping [89]. The
Ti:sapphire crystal exhibits positive dispersion2 within its gain bandwidth. To compensate for this positive dispersion, a prism pair or chirped mirrors are introduced
into the cavity, resulting in a negative net dispersion. A nonzero dispersion leads toa rapid breakup of the pulse as it travels through a dispersive medium. However,this “cold”cavity picture is completely modi…ed by the SPM process. An intensity1

One of the curved mirrors away from the pump laser.
2dk(!)
A positive dispersion refers to d d!
> 0, which gives a negative GVD: d!g =2
k is the propagation constant.
22

2 d k(!)
g d! 2 < 0.

47dependent refractive index means that when a pulse passes through the Kerr medium
it sees a time-varying phase shift. For positive optical Kerr coe¢ cients (n2 > 0), theSPM lowers the frequency at the leading edge of the pulse, while the frequency at thetrailing edge of the pulse is blue chirped3 . Because the group velocity

g increases

with increasing frequency for a negative dispersion, the leading edge of the chirped
pulse will begin to slow down and fall back to the pulse center. Similarly, the trailingedge of the pulse will catch up with the main pulse. Furthermore, the SPM acts
together with the KLM in a mutual-enforced manner. As a result of this delicate
balance between the SPM and the negative dispersion, di¤erent groups of spectral
components are merged into the pulse spectrum and move at a single group velocity,
producing an ultrashort and stable pulse with an enormous bandwidth.

3.2.2

Comb Spectrum

To appreciate the strictly uniform distribution of the comb spectrum, it is
helpful to …rst have an understanding of the cavity-mode structure [85]. The cavity
supports longitudial modes with discrete frequencies !m (in radians/second) suchthat the phase accumulated through a round trip inside the cavity is an integral
multiple of 2 :k(!m ) =

2
m;
L

(3.2)

where k is the propagation constant and L is the round-trip length. The discrete
frequencies !m are determined by equation 3.2 and the dispersion relationk(!) =
3

!
n(!);
c

(3.3)

This phenomenon bears an analogy to the Doppler e¤ect. A receiver experiences a increasingor decreasing frequency depending on whether the source is approaching to or receding from thereceiver. Similarly, a time-varying refractive index stretches or compresses the optical path, leading
to chirped frequencies.

48
where n is the averaged refractive index of the cavity and is de…ned through
nL =

Z L

(3.4)

n(z)dz:0

In a cavity without dispersion, i.e., n does not depend on the frequency, the dispersion
relation 3.3 is strictly linear, meaning that all the modes are separated by an equal
distance 2 c=nL from their neighboring components. However, for a CW laser withintracavity dispersion, these frequencies are pulled away from the uniform distributionof an empty cavity, as illustrated in …gure 3.2. Furthermore, in CW operations, thek(ω)
Total dispersion
2π
(m+2)
L

(m+1)

2π
L

m

2π
L

(m-1)

2π
L

Empty cavity
dispersion

Cavitymode

Pulling
effect

ωm-1

ωm ωm+1 ωm+2

ωFigure 3.2: Frequency pulling e¤ect in a dispersive laser cavity. The cavity modes
are pulled away from an equal-spaced distribution by a small amount because of
intracavity dispersions.

lasing modes are only restricted to a few longitudinal and transverse modes becauseof the mode competition. This severely limits the bandwidth of the laser output.
The spectrum of a mode-locked laser di¤ers essentially from that of a CW
laser. As discussed in section 3.2.1, the ultrashort pulses emerging from the KLM
process experience zero e¤ective GVD because of a balance between the negative

49
dispersion of the “cold” cavity and the nonlinear SPM process. Accordingly, thee¤ective dispersion relation within the pulse bandwidth is strictly linear:k(!) = k(!c ) +

@k
(!
@!

(3.5)

!c );

@k
where !c is the carrier frequency and @!
= 1g holds in the mode-locked bandwidth(see …gure 3.3). This linear dispersion relation again gives uniformly distributed

k(ω)
dk 1
=
dω υg
k(ωc)

k(0)

ω

ωc

0

Figure 3.3: E¤ective dispersion relation of a mode-locked laser. The shaded region
represents the mode-locked bandwidth.

modes with a global shift associated with k(0), the propagation constant extrapolated
to zero frequency. Combining the above dispersion relation with equation 3.2, thefrequencies of these modes can be written as
!m =
where

2

g

L

m + !c 1gp (!c )

(3.6);

p (!c ) is the phase velocity at which the wavefront of the carrier advances.

These frequencies can be rewritten in the units of hertz as(3.7)

m = mfrep + fceo ;where frep = Lg is the repetition frequency and fceo = 2!c (1o¤set frequency.

gp) is the carrier-envelope

50
That the frequency comb is globally shifted by an amount of fceo relative tothe empty cavity modes can also be understood in the time domain. The pulse
envelopes propagate with group velocity while the wavefront of the carrier advanceswith phase velocity. The di¤erence in these two velocities results in a pulse-to-pulse
phase slippage

ce between the envelope and the carrier, as illustrated in …gure 3.4.

The phase slippagece accumulated through a round trip is

ce = k(!c )[ngnp (!c )]L,

(3.8)

where ng and np are averaged group and phase refractive indices of the cavity, respectively. In the frequency domain, this phase shift is corresponding to a frequency
ce

shift

2

frep , which gives the same result for fceo .

Applying femtosecond mode-locked lasers to optical frequency measurements
relies on the uniformity of the comb components across the whole mode-locked bandwidth. A number of experiments have been devoted to the veri…cation of this uniformity. In one of these experiments [14], a frequency comb spanning about 20.12 THz(

267; 200 modes) was tested. The test was based on the fact that if an odd total

number of modes is chosen and if these modes are uniformly distributed, then there isa mode located exactly at the center of the spectrum. In this experiment, the opticalfrequency that bisects the lowest and highest modes of the selected comb spectrum
was independently obtained using a frequency-interval divider [91, 76]. By comparing
this optical frequency with the expected center mode in the comb spectrum under
test, the uniform distribution of the modes was veri…ed within the experimental uncertainty of 3:0

3.2.3

10 17 .

Frequency Stabilization of the Femtosecond Comb

To use mode-locked lasers for optical frequency metrology, their comb frequencies need to be stabilized and referenced to some existing frequency standard. With-

51(a) Time domain
∆φce

E(t)

2πfc

t

1/frep(b) Frequency domain
I(ν)
fceo

0

2πfceo = ∆φcefrep
frep

νm = mfrep + fceoν

Figure 3.4: A pulse train from a mode-locked laser and its frequency-domain representation. The pulse envelopes propagate at group velocity while the wavefront of
the carrier advances with phase velocity. The di¤erence in these two velocities causesa pulse-to-pulse phase shift
ce between the carrier and the envelop that shifts theentire comb by an amount of fceo =ce frep =2 .

52
out active stabilization, frep and fceo are perturbed by various noise sources such as
cavity length changes and pulse intensity ‡uctuations. The techniques developed for
CW laser stabilization can be applied to this task. However, two degrees of freedom,
namely frep and fceo , will need to be controlled independently.
The mode spacing frep can be controlled and servo-locked by adjusting the cavity length L, which in practice is performed by translating a piezoelectric transducer
mounted on one of the cavity mirrors. This method is used for both prism-based
and prismless mode-locked lasers such as those illustrated in …gure 3.1. For these
two systems, controlling of the carrier-envelope o¤set frequency fceo is implementeddi¤erently. With prism-based lasers, fceo can be adjusted by rotating the back mirror
located after the prism pair [92, 13, 80], as indicated in …gure 3.1 (a). The rotation
produces an extra amount of cavity dispersion because the comb components are
spatially spread across the back mirror. The change of linear dispersion results ina shift of fceo , while other high-order dispersions incurred during the rotation will
be balanced by the nonlinear SPM in the mode-locking process. In prismless lasers,
where the spatially-separated modes are not available, an acousto-optic modulator(AOM) is inserted into the pump beam path to control the pump power [93, 94],
which a¤ects

g and

p in a di¤erent way. Figure 3.5 gives a simpli…ed interpretation

for the comb frequency control.
There are cross-couplings between the two control channels for frep and fceo
[95, 96]. Cavity-length changes used to control frep also modify the cavity-…lling
factor (the length of the dispersive medium divided by the overall cavity length) andhence a¤ect fceo , though this e¤ect is small compared with the direct response [95].
Control of fceo a¤ects frep too. Rotating the back mirror in prism-based systems has
a very small cross-coupling to frep compared to the direct channel, while responses of
fceo and frep to pump power in prismless systems can have comparable magnitudes
and opposite signs [95, 97].

53

I(ν)
0

0

1

2

m-3

m-2

m-1

m

m+1

ν

∆f

fceo

∆f = m∆frep

(a)
I(ν)
0

0

1

2

m-3

m-2

m-1

m

m+1

ν

|∆fceo|∆fceo = -m∆frep

νm = mfrep+fceo

(b)
I(ν)
Figure 3.5: A simpli…ed picture for the frequency control of mode-locked Ti:sapphire1
m-3 modi…es
m-2 them-1
m
m+1
lasers. (a) A0small change
in2cavity length
mode spacing
frep , giving rise
to a large frequency shift at optical frequencies because of their large mode orders(m 105 106 ). Note that the changes of mode frequencies near the zero frequency
are greatly exaggerated. (b) With the anchor position moved to a mode in opticalνregion,0a small change in frep has a large e¤ect on fceo . In practice, this is achievedby rotating the end mirror in prism-based femtosecond lasers.
lasers, theνm=mfIn+fceo
repprismlesse¤ect of the pump-power change can be viewed as a linear combination of (a) and(b).

A simplified picture for comb frequency control. (a) A small change in cavity length
modifies the mode spacing frep, giving rise to a large frequency shift at
optical frequencies due to their large mode orders (m~10^5 - 10^6). Note thatthe changes of mode frequencies near the zero frequency are greatly exaggerated.
(b) With the anchor position moved to a mode in optical region, a small change in fre
has a large effect on fceo. In practice, this is achieved by rotating the end mirror inprism-based femtosecond lasers. In prismless lasers, the effect of the pump-powerchange can be viewed as a linear combination of (a) and (b).

54
To summarize the above discussion, …gure 3.6 shows simpli…ed block diagrams[98] for the comb frequency stabilization, where G11 and G22 are plant transfer functions for frep and fceo , respectively. G12 and G21 represent the cross-couplings between
the two channels, and D1 and D2 are transfer functions of two loop …lters. This simpli…ed model of the comb frequency stabilization provides some general guidance forservo-locking both frep and fceo . For example, in prism-based systems, a qualitative
analysis of cross-couplings resulting from cavity-length and mirror-angle adjustments
21
12shows that G> 0 and G< 0. Thus the transfer function characterizing the relG11G22

G12 G21
ative strength of the cross-coupling, de…ned as S = G, is negative, implying a
11 G22potential oscillatory instability when both feedback loops are closed [98, 99]. In thiscase, decoupling of the two channels can be introduced in the loop …lters to improve
the locking performance.

3.2.4

Self-Referencing Carrier-Envelope O¤set Frequency

The detection of frep is straightforward. A fast photodetector is used to measure
the heterodyne beats among adjacent comb components. The S/N is enhanced ina sense that a large number of modes contribute to the beat notes at frep and its
harmonics. On the other hand, the carrier-envelope o¤set frequency fceo cannot be
detected in this manner because it is a common variable for all the comb components.
Several methods have been developed to detect this value such as the cross-correlation
measurement of the pulse-to-pulse phase slippagece [93], a heterodyne beat witha stabilized CW optical frequency, and self-referencing techniques [78, 100]. A -to2 detection scheme, which falls into the category of self-referencing techniques, isparticularly suitable for the absolute frequency measurement because it allows fceo
to be directly phase-locked to an RF reference.
Figure 3.7 illustrates the basic idea of the -to-2 detection scheme. A comb
component

m at the “red” end is frequency doubled by sum-frequency generation

55

r1

y1
+

Σ

D1

u1

G11

+

Σ

+

y1
(frep)

G21
G12
r2

+
-

Σ

D2

u2

G22

+
+Σ

y2

y2
(fceo)

(a)

r1

r2

y1
+

Σ

+
-

Σ

+

D1

D2

u1

Σ

G11

+
K21

G21

K12

G12
+

y2

+
Σ

u2

G22

+

Σ

+

+
+Σ

y1
(frep)

y2(fceo)(b)
Figure 3.6: Simpli…ed block diagrams for feedback control of comb frequencies. G11
and G22 are plant transfer functions for frep and fceo , respectively. G12 , and G21
represent the cross-couplings between two channels. D1 and D2 are transfer functionsof two loop …lters. In (b), decouplers K12 and K21 are introduced to reduce the crosscouplings through G21 and G12 .

56

I(ν)

Fundamental spectrum
SFG spectrum

fceo

frep
ν

νm = mfrep + fceo

ν2m = 2mfrep + fceo2νm = 2mfrep + 2fceo

fceo

Figure 3.7: Self-referencing determination of the comb o¤set frequency fceo . The fundamental comb spectrum spanning an octave is further expanded using sum-frequency
generation (SFG). In the overlapping region of the fundamental and SFG spectra,a heterodyne beat between 2m and frequency-doubled m gives fceo . The extracted
fceo can be referenced to an RF frequency standard.

57(SFG) in a nonlinear crystal and is heterodyne beat against the component

2m at

the “blue”end. The beat frequency gives the carrier-envelope o¤set frequency fceo :2 m

2m = 2(mfrep + fceo )(2mfrep + fceo ) = fceo

(3.9)

This scheme requires that the comb spectrum spans an octave4 such that between thefundamental and SFG spectra there is an overlap in which the heterodyne beat canbe performed. An octave-spanning frequency comb is usually obtained by externallybroadening the spectrum from a mode-locked laser, using a segment of speciallydesigned microstructure …bre [101, 102, 103]. Alternatively, an octave-spanning comb
can be directly generated inside the cavity of a femtosecond laser with exquisite
intracavity dispersion management [104, 105, 106].
3.3

Optical Frequency Measurement

3.3.1

Iodine Spectrometer

Figure 3.8 shows the experimental scheme for precision scanning and controlof the laser frequency based on a highly stable optical cavity augmented by a subDoppler I2 resonance recovered from the spectrometer. The femtosecond comb systemused for frequency measurement is also shown. The experimental details of the iodine
spectrometer have already been discussed in the previous chapter. Some modi…cations are made to suit the optical frequency measurement of a particular I2 hyper…necomponent.
The single-mode Ti:sapphire laser employed in the I2 spectrometer can be tuned
from 953 to 1080 nm. It provides about 300 mW of useful output power at 1030 nm.
A small portion of the fundamental power is used for laser frequency prestabilization
to an evacuated, vibration-isolated, and thermally stabilized optical cavity. The operational laser linewidth is limited by the vibration noise and drift associated with the
4

The interval between any two frequencies having a ratio of 2 to 1.

58

Iodine spectrometer
Frequency stabilization& precision tuning

Highly stable
optical cavity< 1 mW

Widely tunable,
single mode Ti:s

SHG

750 MHz
mode-locked Ti:s

Optical octave
B W generation~ mW

Self-referencing
Cs clock

Heterodyne beatmeasurement

Figure 3.8: A precision I2 spectrometer based on a widely tunable Ti:sapphire laser.
The femtosecond comb system used for frequency measurement is also shown.

59
cavity. Most of the Ti:sapphire laser power is used for second harmonic generation(SHG) to probe I2 transitions. The I2 spectrometer is con…gured for frequency modulation (FM) spectroscopy, with additional chopping of the pump beam to further
reduce the in‡uence of the Doppler background. The size of both pump and probe
beams is 3 mm in diameter. We typically maintain < 100 W for the probe beam
while the pump-beam power can be varied from 1 to 6 mW to study power-related
frequency shift and broadening of I2 transitions. The high-purity I2 cell was preparedby the Bureau International des Poids et Mesures (BIPM) and it has an 8-cm usefullength with Brewster windows at both ends. For studies of pressure-related e¤ects,
the I2 vapor pressure is controlled by varying the temperature of the cell cold …nger.
For frequency measurement of the a3 , P(13) 43-0 transition, we use a cold …nger
temperature of

5 C (vapor pressure = 2:38 Pa), in compliance with the CIPM

recommendation.
In addition to locking the single-mode Ti:sapphire laser to the external cavity,
the long-term stabilization of the laser is obtained by referencing its frequency to an
I2 transition. The double-passed AOM, shown as AOM 2 in …gure 2.1, ful…ls this task.
Drift and low-frequency noise on the laser stemming from the external cavity can besuppressed by feeding the error signal derived from the I2 resonance into a voltage
control oscillator (VCO), which in turn determines the optical frequency shift of the
double-passed AOM. The laser light, now carries the information of I2 resonance, isthen directed to a femtosecond laser comb for frequency measurement. Since there
is more power available in the fundamental laser light near 1030 nm, it is used for
frequency measurement.

3.3.2

Self-Referenced Femtosecond Laser Frequency Comb

Figure 3.9 shows the details of the experimental setup for the frequency comb
system. In our experiment, the Kerr-lens mode-locked femtosecond laser used for

60
the frequency comb generation has a repetition rate of 750 MHz and produces anoptical spectrum spanning

30 nm, centered at 800 nm [84]. An octave bandwidthof the comb is generated by coupling 100 mW of laser light through a 20 cm long
microstructure …ber. The input to the -to-2 interferometer is divided into infrared(IR) and visible wavelengths by a dichroic beam splitter. After the IR light is frequency doubled by a BBO ( -BaB2 O4 ) crystal, the beams in the two arms of theinterferometer are recombined and detected with a photodiode. A prism is inserted
into one of the beam path to match the lengths of the two arms to achieve temporal
overlap of the pulses.
Both the repetition frequency frep and the carrier-envelope o¤set frequency
fceo are stabilized with reference to the Cs standard, thus establishing an opticalfrequency grid of 750 MHz spacing, extending from 500 to 1100 nm. In practice, an
intracavity piezo-activated mirror is used to control frep while the pump power for the
femtosecond laser is used to control fceo . The frequency of the mth comb component
is then given by fm = mfrep + fceo . The absolute frequency of the I2 -stabilized
CW laser can thus be determined by counting its heterodyne beat frequency against
a corresponding comb line. The comb order, m, is determined from a mediumresolution (500 MHz) wavemeter.

3.3.3

Absolute Frequency of the Iodine-Based Length Standard at
514:67 nm.

A remarkable feature of the femtosecond comb system is its reliability. It is
now possible to acquire hours of measurement data without interruption. This will
be important for the future development of an optical clock [107, 54]. Furthermore,
the S/N of the beat signal between the CW laser and the comb exceeds 45 dB within
a 100-kHz bandwidth, permitting a direct frequency count without further signal
processing. Figure 3.10 (a) shows the line shape of the a3 , P(13) 43-0 transition

61

OC

M2

Modelocked
Ti:Sapphirering laser
frep = 750 MHz

M3 & PZT

M1
AOM
F1

30 mm

frep

Pump

HW

F2

frep servo

fceo servo

Cs clock

Digitalφ detector

PD
Microstructure
optical fiber
Synthesizer

Dichroic
BS

IR fn

BBO

2fn

Synthesizer
f/N

Filter Polarizer
PD

Visible f2n

Prism

PD
Beat noteIR from I2
spectrometer Optical
Fiber

Grating
Polarizer

Figure 3.9: Experimental setup for a self-referenced frequency comb. The system
employs a mirror-dispersion-controlled Ti:sapphire laser with a ring-cavity con…guration and a repetition frequency of 750 MHz. Both frep and fceo are locked to the Cs
clock. The absolute frequency of the I2 -stabilized CW Ti:sapphire laser (not shown)at IR is determined by counting its heterodyne beat frequency against a nearby comb
component with predetermined mode order. Note that several half wave plates used
to rotate polarizations are not shown. F1, F2, chirped folding mirrors with f = 15mm; M1–M3, ‡at chirped mirrors; OC, output coupler; PZT, piezoelectric transducer; AOM, acousto-optic modulator; HW, half wave plate; PD, photo detector;
BS, beam splitter; BBO, -BaB2 O4 crystal.

62
recovered using FM saturation spectroscopy with a modulation frequency of 6 MHz
and a 5-ms time constant. Figure 3.10 (b) shows a record of the frequency count of
the length-standard transition, i.e., a3 , P(13) 43-0, 127 I2 . The corresponding Allan
deviation determined from the measurement is presented in Fig. 3.10 (c). It is clearthat the measurement noise is dominated by the instability of the Cs clock as well,
just as that for the 532-nm system.
To check the reliability of the 750-MHz femtosecond comb, we use a frequencydoubled Nd:YAG laser stabilized on a hyper…ne component of 127 I2 transition [a10 ,R(56) 32-0] at 532 nm as a calibration tool [108, 41]. The absolute frequency of the532-nm system has been measured for over two years with a standard deviation of lessthan 120 Hz at 1064 nm [54]. Previous measurements were carried out using a wellcharacterized 100-MHz femtosecond laser system referenced to a Cs clock. Figure 3.11
shows the augmented record of the absolute frequency measurements performed at
a10 , R(56) 32-0, accumulated during a two-year period. The last four data points
represent measurements of the same I2 transition using the current 750-MHz comb
system. The agreement between the two systems is within 20 Hz. At the present
stage, the frequency noise of the 532-nm system is limited by the instability of theCs clock (510 12 at 1 s). A H-maser-referenced optical comb system is also usedin the system.
Figure 3.12 shows a two-week record of the frequency-measurement performed
at a3 , P(13) 43-0. Our measurement gives an average value of 71:8
than the CIPM-recommended value of 582 490 603 307

0:3 kHz higher

150 kHz [40]. To include

other unforeseeable e¤ects on the transition frequency, we quote a 5-

value of

1.5 kHz for the uncertainty of the measurement. To be more speci…c, further evaluations should include studies of beam size and wavefront curvature, signal-recovery
techniques other than FM spectroscopy, and a large collection of various I2 cells.
However, we emphasize that even for the present work, we have used three di¤erent

8000

(b)

(a)

4000
0
-4000

1-s gate time
-8000
0

500

1000

1500

2000

Time (s)

2500

3000

4

Allan varianceI2 signal (Arb. unit)

Optical freq. count (Hz)

63

(c)

2
-12

10

6
4
2
-13

10

6

1

2

3

4 5 6

10

2

3

4 5 6

100

Averaging time (s)

2

3

4 5 6

600 kHz

Scan frequency

Figure 3.10: Frequency measurement at the a3 component of the P(13) 43-0 transition. (a) Line shape recovered from FM spectroscopy. Cell cold-…nger temperaturenear -5 C, 5-ms averaging time. (b) Direct frequency measurement of a3 at P(13)43-0 by the Cs-referenced fs comb. Gate time 1 s. Standard deviation is limited by
the Cs clock. (c) Allan variance is determined from the time record in (b).

64f1064 - CIPM frequency (kHz)19

Measured by a 750 MHz fs comb
and a 20-cm Danish microstructure fiber
Mean value: 17.221 kHz, standard deviation: 54 Hz

18

17

Measured by 100-MHz fs combs and an 8-cm Lucent fiber-13
Mean value: 17.240 kHz, Standard deviation: 118 Hz (4 x 10 )16

(CIPM frequency 281 630 111.74 MHz)15
02 70 72 74 430

435

440

445

506 508 510

724

730

Days
Figure 3.11: Long-term frequency measurement of the Nd:YAG/I2 -532 nm system.
Agreement is excellent with previous measurements based on 100-MHz systems withdi¤erent microstructure …bers.

65
I2 cells. Two of the cells were made in the same run while a third one was made morethan eight years ago, also at BIPM. We observe no statistically signi…cant di¤erence
among these three cells, as shown in Fig. 3.12. While we congratulate our BIPM
colleagues for making cells on a consistent basis, it is clear that the capability of absolute frequency measurement referenced to the Cs standard gives us a new approach
to characterize reference cells and track their long-term variations.

3.3.4

Characterizing the Iodine Spectrometer

The capability of absolute frequency measurement greatly facilitates the characterization of systematic e¤ects of the spectrometer. While traditionally one employsa heterodyne beat between two similar systems to characterize optical standards,
we can now evaluate the performance of a single system and calibrate it using Cs
frequency standard. We have carefully studied frequency shifts depending upon thefollowing experimental parameters: optical power, modulation frequency and amplitude, vapor pressure, and optical alignment. To minimize subjective in‡uences in
the frequency measurement, we typically have a few experimenters align the spectrometer independently, with the only common objective being maximization of thesignal size. Residual amplitude modulation (RAM) in FM saturation spectroscopy is
a common source of systematic errors. The RAM on the frequency-modulated probe
beam is minimized by careful adjustment of the polarization optics throughout theI2 spectrometer. When RAM is minimized, we can choose di¤erent values of themodulation frequency (4, 6, 8, and 10 MHz, for example) with various modulation
amplitudes and still obtain consistent and reproducible results for frequency measurement. Within this optimized operation regime, optical-power-related frequency
shifts are small, usually less than the measurement standard deviation, in the normal
operation range of 1 mW to 4 mW for a 3-mm diameter pump beam. By a controlled
variation of the cold-…nger temperature of the iodine cell, we change the sample den-f514.67 nm - CIPM Frequency (kHz)66

BIPM I2 cell a (#338S)
BIPM I2 cell b (#337S)
BIPM I2 cell c

CIPM recomendation:

Weighted mean:

Days
Figure 3.12: Long-term frequency measurement at the a3 component of the P(13)43-0 transition. Three di¤erent I2 cells are used in the measurement, as indicated bysquares, triangles, and circles. The BIPM-designated numbers for cells a and b are
shown in parenthesis.

67
sity in the cell. The related pressure-dependent frequency shift is measured for thislength standard transition, with a linear shift of

2:5

0:5 kHz/Pa, as shown in

Fig. 3.13.
3.4

Summary
A femtosecond laser based optical comb spanning an octave bandwidth is em-

ployed conveniently to check systematic errors and measure absolute frequencies of
selected optical standards. The transition frequency of the length standard at 514:67
nm, i.e. the a3 hyper…ne component at transition P(13) 43-0, 127 I2 , has been determined to be 71:8 kHz

1:5 kHz above the recommended value of 582 490 603 307

kHz. The measurement was performed with a pump power of 1
size of 3 mm in diameter, and a cold-…nger temperature of

4 mW, a beam

5 C for the iodine cell

(as speci…ed in reference [40]). The average value of the frequency measurements is
obtained over a 15-day measurement period. This result is well within the 1- uncertainty (150 kHz) of the recommended value. The coe¢ cient of the pressure shift
is found to be

2:5(5) kHz/Pa.

f514.67 nm - CIPM Frequency (KHz)

68

Presure shift: -2.5

0.5 kHz/Pa

Pressure (Pa)
Figure 3.13: Pressure shift of the a3 component at P(13) 43-0.

Chapter 4
HYPERFINE INTERACTIONS AND PERTURBATION EFFECTS IN
3
THE B0+u(4.1u ) STATE OF

127

I2

Introduction
High-precision measurement of hyper…ne spectrum in molecular iodine over an

extended range of rovibrational levels is valuable both for studying the molecular
structure and selecting references for laser frequency stabilization. The hyper…nestructure in atoms and molecules originates from the electromagnetic interaction between the nuclear and surrounding charge distributions. This seemingly weak e¤ect
has a subtle but noticeable in‡uence on molecular structure. For instance, the hyper…ne coupling between the B state and the repulsive B 00 1u (1u ) state is responsible

for the predissociation of the molecules [66, 67], an e¤ect that results in a variation of
the lifetime of the hyper…ne levels belonging to a single rovibrational level. Another3
example of this e¤ect is the hyper…ne-induced u-g mixing of the B0+
u(

1g (1
els

u ) and theg ) states that dramatically modi…es the hyper…ne structure at vibrational lev0

= 76

378 in the B0+u(u ) state [44, 45]. Considering the broad scope within

which the hyper…ne e¤ect operates, it is of great interest to examine the evolution ofthis e¤ect with the progression of the vibrational levels toward the dissociation limit
or, equivalently, with the stretch of the molecular bond from its equilibrium length
to large internuclear separations, where a separated-atom model represents a good
approximation to the molecular electronic wave function. Hyper…ne structures in the

70
1
X0+
g(

+
+ 3
g ) and the B0u (the prominent Bu ) states have received extensive investigations throughX transition manifold in the visible and near-IR regions. By

use of sub-Doppler detection schemes such as saturation spectroscopy and supersonic molecular beams, hyper…ne spectra of B

X transitions were measured fora large number of rovibrational levels with a precision that reached the kilohertz
level for each hyper…ne component. Broyer et al. developed an e¤ective Hamiltonian
to describe the hyper…ne interaction that can couple di¤erent electronic states [16].
Derived from this Hamiltonian are e¤ective hyper…ne parameters that characterize
hyper…ne coupling strength. Among these parameters, electric quadrupole (eqQ),
spin-rotation (C), tensorial spin-spin (d), and scalar spin-spin ( ) represent the four
dominant parameters that have been widely used to analyze experimental spectrum.
For the B state in 127 I2 , perturbations from other electronic states can contribute signi…cantly to various terms in the hyper…ne Hamiltonian. Located above
the B state on the molecular energy scale are a total of nine weakly bounded electronic states (0+g , 0g , 0u , two 1g , two 1u , 2g , and 2u ) [109, 59, 110], which sharethe same 2 P 3 + 2 P 1 dissociation limit with the B state. Furthermore, rovibrational
22levels in the B state are not far removed in energy from this cluster of states. As a result, these electronic states perturb the B state to various degrees through hyper…necoupling. Indeed, this qualitative estimate of the perturbation strength is con…rmedby the observed global trend of a monotonic increase of the magnitude of the effective hyper…ne parameters CB , dB , andB

along with the ascending vibrational

level, a trend especially remarkable for those levels approaching the dissociation limit[111, 50, 46, 112]. More quantitatively, for levels with vibrational quantum numbers0< 43 the four-term e¤ective Hamiltonian describes satisfactorily the hyper…nespectrum with small residual errors ranging from a few tens to several kilohertz oreven less [42], indicating that in this region second-order perturbation is su¢ cient toaccount for the contributions from states other than the B state.

71
Moreover, hyper…ne interaction in the region 71 <0

< 82 has been systemat-

ically investigated by Pique et al [44, 45, 43]. In addition to the second-order e¤ect,strong hyper…ne perturbation from the 1g (1g ) state was found at levels

0

= 76,

77, 78 owing to small energy di¤erences and large Franck-Condon overlaps between
the rovibrational levels in the B and the 1g (1the strong perturbations at
1g (1

0

= 77 and0g ) states.

In particular, to deal with= 78 requires that basis functions of theg ) state be included in the construction of the Hamiltonian matrix:

A directdiagonalization of this new matrix gives the correct hyper…ne spectrum. The resultant strong u-g mixing was directly con…rmed by the extra hyper…ne lines recordedat the 78-0 band head.
Despite the fact that ample data were collected in the regions summarized
above, the intermediate levels that spread from0

= 44 to0= 70 will require

more information. This is an important region for study of the molecular structureof 127 I2 because vibrational wave functions of these levels sample a relatively large
range of internuclear separations. Direct hyper…ne-spectrum measurements in thisregion have been made only for a limited number of transitions to which access isgained primarily by isolated laser lines such as 514:5- and 501:7-nm argon-ion laser
wavelengths [55, 48, 113].
Using a widely tunable and high-resolution spectrometer based on a frequencydoubled CW Ti:sapphire laser, we carried out systematic investigations of the hyper…ne spectra of B

X transitions in the wavelength range 500

517 nm anddetermined the detailed rovibrational dependence of the four e¤ective hyper…ne parameters in the region 42 <0

< 70. In addition to the usual monotonic trend that

results from the second-order e¤ect, we observed strong perturbations at

0= 57 60.

The source of this perturbation was con…rmed, and its properties were studied. We
also established a systematic dependence of the hyper…ne interactions on internuclear
separation R.

72
This chapter describes a systematical investigation of 127 I2 hyper…ne spectrum
in the wavelength range 500

517 nm. After a brief overview of the hyper…nespectrum in molecular iodine, the experimental setup and data collection are covered
in section 4.3. Section 4.4 gives the four e¤ective hyper…ne parameters extracted
from hyper…ne spectra of nearly 80 rovibrational transitions. These results are thenexamined and discussed in the following sections. First, these date are combinedwith previous measurements to give a global picture of the vibrational dependenceof the hyper…ne parameters in the B sate. Second, this vibrational dependence isconverted to the radial dependence. Third, the rotational dependence of the four
hyper…ne parameters are examined in detail. Finally, a strong perturbation observed
at vibrational levels

0

= 57

60 is discussed in section 4.8. Section 4.9 summarizes

the chapter.
4.2

E¤ective Hyper…ne Hamiltonian

4.2.1

Hyper…ne Spectra of Molecular Iodine

Figure 4.1 shows the simpli…ed energy level diagram of the molecular iodine.
There are totally 23 valence states that converge to three atomic states (2 PJ + 2 PJ ,J= 32 ; 21 ) at large internuclear separations. Shown in this simpli…ed energy level diagram are the potential energy curves (PECs) of four electronic states: the groundstate X, the second excited sate B, the 1g (1g ) state converging with the B state to1
the second dissociation limit, and the 0+g(+g ) state converging to the third dissoci-ation limit. Several rovibrational levels in the X and the B states are also shown inthe diagram.
A rovibrational transition of B

X manifold occurs between two rovibrational

levels ( 00 ; J 00 ) and ( 0 ; J 0 ) in the ground state X and the second excited state B,
respectively. The selection rule for an electric dipole transition is

J = J0J 00 =

173

30

0g+(1Σ+g)2

P1/2+2P1/2

Energy (x103 cm-1)1g(1Πg)2

20

P3/2+2P1/2( υ',J' )Energy scale expanded

Hyperfinelevels in
the B state3
B:0+u( Πu)

2

P3/2+2P3/2

10

1 +
X:0+
g( Σg)( υ'',J'' )Energy scale expanded

034Hyperfinelevels in
the X state

5Internuclear separation (A)Figure 4.1: A simpli…ed energy level diagram of molecular iodine. Several rovibrational levels in the X and the B states are shown. The thick vertical line on the left
indicates a B
X rovibrational transition between two rovibrational levels ( 00 ; J 00 )00
and ( ; J ) in the X and the B states, respectively. The selection rule involved is
J = J 0 J 00 = 1 ( J = 1: P branch, J = 1: R branch). A rovibrationaltransition is labeled as P(J 00 ) 0 - 00 or R(J 00 ) 0 - 00 . Each rovibrational level is further
split into hyper…ne sublevels. Thus one rovibrational transition contains many hyper…ne components. Such a hyper…ne component is indicated as a thin vertical lineon the right.

74( J = 1: P branch,as P(J 00 )0

J=

- 00 or R(J 00 )0

1: R branch)1 . Each rovibrational transition is labeled- 00 according to the value ofJ.

Because of the hyper…ne interaction, each rovibrational level is further split
into hyper…ne sublevels. The iodine nucleus has a nuclear spin of Ia = 25 . Thus I,
the total nuclear spin of a iodine molecule, can take on the possible values of 0, 1, 2,
3, 4, and 5. The coupling of the nuclear spin I and the molecular angular momentum
J gives the total angular momentum F = J + I. For each I and J, the quantum
number F can take on the values of jJ + Ij ; jJ + I

1j ;

jJ

I + 1j ; jJ

Ij. Thehyper…ne interaction between the nucleus and the surrounding charge distributions
breaks the degeneracy on I and F , resulting in many hyper…ne sublevels.
The total number of hyper…ne sublevels at a rovibrational level is further limitedby the nuclear statistics. An iodine molecule contains two identical femionic nuclei,
which demands that the total nuclear wave function must be antisymmetric underthe exchange of the two nuclei. This requirement imposes additional restrictions
on the possible values of the nuclear spin. In particular, for the ground state X,
rovibrational levels with even (odd) J 00 can have hyper…ne sublevels with even (odd)
I 00 . Consequently, levels with even (odd) J 00 are split into 15 (21) hyper…ne sublevels.
The restriction is reversed for the B state: even (odd) J 0 can have hyper…ne sublevels
with odd (even) I 0 . Figure 4.1 illustrates two sets of hyper…ne sublevels for the X
and the B states.
The selection rule for the electric dipole transition between two hyper…ne sublevels in two di¤erent electronic states is

F = F0F 00 =

J. Thus each B

X

rovibrational transition contains either 15 (even J 00 ) or 21 (odd J 00 ) hyper…ne components. Such a hyper…ne component is illustrated as a thin vertical line on the rightof …gure 4.1. Transitions with1

F = 0 orF =J do occur, but they are usually

B
X transitions do not possess Q branch ( J = 0) because there is no angular momentum
parallel to the internuclear axis for the B and the X states.

75
much weaker than the main lines ( F =4.2.2

J) and therefore are not considered here.

Matrix Elements of the Four E¤ective Hyper…ne Hamiltonian

The hyper…ne Hamiltonian in a homonuclear diatomic molecule with two nuclei,
labeled a and b, can be formally written as
Hhf = Hhf (a) + Hhf (b) + Hhf (a; b);(4.1)

where Hhf (a) [Hhf (b)] and Hhf (a; b) represent the nucleus-electron and nucleusnucleus interactions, respectively. The fact that Hhf (a) and Hhf (b) can couple different electronic states was studied through the introduction of an e¤ective hyper…neHamiltonian [16]. This e¤ective Hamiltonian acts only on one electronic state but
has its matrix elements modi…ed by second-order perturbation to take into accounthyper…ne couplings with external electronic states. The advantage of this treatment
is that, despite their di¤erent origins, various tensorial terms can be incorporated
into a small set of e¤ective hyper…ne parameters according to their overall geometrical dependences evaluated with irreducible tensorial algebra, thereby simplifying
the experimental analysis of hyper…ne structure. The e¤ective Hamiltonian can besummarized as
Hhf;ef f = HeqQ + HSR + HT SS + HSSS + He + Hf + Hh + HhH ;(4.2)

where the four leading terms on the right-hand side represent major contributions
from nuclear electric quadrupole, spin-rotation, tensorial spin-spin, and scalar spinspin interactions, respectively. The last four terms arise primarily from second-ordercontributions not included in the four major terms and have been examined experimentally only for a few transitions without con…rming evidence of their existence[55]. The matrix elements of all the terms in equation 4.2 have be derived and canbe found in the literature [114, 16]. For the four major terms, their matrix elements

76
are
1
hF IJj HeqQ jF I 0 J 0 i = ( 1)F +J+2Ia [(2I + 1)(2I 00 + 1)]1=2 eQq(J; J 0 )
2
20
10
13 1
0
J 2 C7
6B Ia Ia 2 C B J
4@
A@
A5
Ia
Ia 0
J
J 0
8
98
9
>
>
>
< Ia Ia I = < F J I >=;>: 2 I 0 Ia >;>: 2 I0 J0 >;q(J; J + 2) = 3(J + 1)1=2 q=[(2J + 3)(2J + 5)]1=2 ;q(J; J) =hF IJj HSR jF I 0 J 0 i =JJ 0 II 0(4.3)

Jq=(2J + 3);

1
C [F (F + 1)
2

I (I + 1)

J (J + 1)] ;0

1)F +I +1 d(2J + 1)[Ia (Ia + 1) (2Ia + 1)]
10p
B J 2 J C
30 (2I + 1) (2I 0 + 1) @
A
0 0 0
9
8
9>
8
>
>
Ia Ia 1 >
>
>
>
>
>
>
=
< F J I =<Ia Ia 1 > ;>>;>>: 2 I0 J0 >>>>;: I I0 2 >hF IJj HT SS jF I 0 J 0 i =JJ 0 (hF IJj HSSS jF I 0 J 0 i =(4.4)

JJ 0 II 0

1[I (I + 1)
2

2Ia (Ia + 1)] ;

(4.5)

(4.6)

where I, J, and F are the quantum numbers of the total nuclear spin, total angularmomentum, and total angular momentum including nuclear spin, respectively, large
parentheses are 3-j symbols, large curly braces are 6-j and 9-j symbols, and eqQ, C,d, and , which characterize the hyper…ne-coupling strength, are electric quadrupole,spin-rotation, tensorial spin-spin, and scalar spin-spin parameters, respectively. Eachmatrix element in equations 4.3 - 4.6 is expressed as a product of a hyper…ne parameter and a geometrical function G depending only on the quantum numbers I, J,

77
and F . This is true also for the last four terms in equation 4.2. Thus the the matrix
element of the hyper…ne Hamiltonian in a particular electronic state(either the X

or the B state) can be expressed as
h ; I 0 J 0 F jHhf;ef f j ; IJF i = eqQ GeqQ + C GSR + d GT SS +e Ge + f Gf + h Gh + hH GhH ;GSSS +(4.7)

The hyper…ne parameters, i.e., eqQ, C, d, etc., can be determined by …tting thise¤ective Hamiltonian to the hyper…ne spectrum of a rovibrational transition. A
detailed investigation of this e¤ective Hamiltonian and the second-order calculationof its hyper…ne parameters will be given in chapter 5.
4.3

Experimental Setup and Data Collection

4.3.1

Experimental Setup

A schematic of the experimental setup is depicted in …gure 4.2. The system
consists of a widely tunable frequency-doubled Ti:sapphire laser, a cell-based 127 I2
spectrometer for saturation spectroscopy with the frequency modulation (FM) technique, and a self-referenced optical frequency comb [115] that makes the absolute
frequency measurement. The experimental details of the Ti:sapphire laser and the I2
spectrometer can be found in chapter 2.
The details of the cell-based 127 I2 saturation spectroscopy have been describedin chapter 2. Some aspects of the system are summarized brie‡y as follows. The
127

I2 cell was prepared at the Bureau International des Poids et Mesures, and it has

an 8-cm useful length and Brewster windows at both ends. The 127 I2 vapor pressure
is maintained at 3:3 Pa (22.3 C ) by temperature control of the cold …nger of thecell. Whereas for a majority of the transitions investigated the collimated pump andprobe beams are of

3-mm diameter and typical pump powers are 2

some weak transitions with

0

4 mW, for> 60 the beams are focused and pump powers are

78

127
I2
spectrometer

Hyperfine spectrum;
frequency stabilization

Reference cavity

l/4

PBS
EOM

PD

f

Slow loop

Pump laser@ 532 nm

Prism

Double-pass AOM
AOM
2

24.8 MHz

Servo
MixtureFreq. synth.

Fast loop
Integ.

Scaling
BW OD

EtalonPZT

SHG

EtalonAOM
1

BF
Ti: sapphire Monitor
Laser

750 MHz mode-locked
Ti:sapphire laser

Optical octave
BW generation

Self-referencing
Cs clock

Frequency
measurement

Heterodyne
beat

Figure 4.2: Schematic diagram of saturation spectroscopy in the wavelength range
500 517 nm. A frequency-doubled Ti:sapphire laser is stabilized by a reference
cavity and scanned by a RF synthesizer. Hyper…ne spectra are extracted by the FM
sideband technique. Introducing a self-referenced optical frequency comb into the system signi…cantly simpli…es the absolute frequency calibration. AOMs, acousto-optic
modulators; BF, birefringent …lter; BW, Brewster window plate; EOM, electro-optic
modulator; OD, optical diode; PBS, polarization beam splitter; PD, photodiode;
SHG, second-harmonic generator; =4, quarter-wave plate; , phase delay.

79
increased to 7

10 mW to compensate for the large saturation intensity. The probe

beam has 1-MHz frequency modulation sidebands and the pump beam is amplitude
modulated by an in-line AOM.
Crucial to this high-resolution spectroscopy is precise and continuous scanningof the laser frequency over 1 GHz (a typical span of the hyper…ne spectrum for one
transition) without mode hop. The laser frequency, which is locked to the reference
cavity, is scanned by a frequency synthesizer that drives a double-pass AOM (AOM
2 in …gure 4.2) inserted between the laser and the reference cavity. The feedback
signal applied to the PZT is integrated and then supplied to an intracavity Brewster
plate, providing a large tuning range that is limited mostly by the RF bandwidth ofthe double-pass AOM. During the course of a frequency scan, the angle of the thicketalon is controlled in a feed-forward manner to synchronize the transmission peak of
the thick etalon with the laser mode. We obtain continuous tuning in the green overa 1-GHz range, which we can enlarge greatly by locking the laser to di¤erent modesof the reference cavity. Figures 4.3 - 4.5 show several hyper…ne spectra scanned by
the Ti:sapphire laser.
Hyper…ne splittings are measured in a pairwise manner. For each rovibrationaltransition measured, we typically select an isolated hyper…ne component, roughly
in the middle of the hyper…ne spectrum whenever possible, as a reference line toform a pair with another component. The laser, locked to the reference cavity, isscanned across this pair of hyper…ne components. The direction of the frequency
scan is then reversed to take the laser frequency back to the initial position. For eachscan direction, the corresponding resonance structure is recorded and a line-shapeanalysis of the saturation resonance determines the frequency di¤erence between thetwo components. The bidirectional average of the frequency di¤erence is then adopted
to circumvent the slow and quasi-linear cavity drift (< 20 Hz/s) that dominates in ashort period. This procedure is followed for all hyper…ne components that are isolated

Strength (a.u.)

-10

-5

0

5

10

-10

-5

0

5

10

0

0

Frequency (MHz)

400

400

Figure 4.3: Hyper…ne spectrum recorded at R(41) 42-0

200

200

600

600

800

800

80

-10

-5

0

5

10

-10

-5

0

5

10

-10

-5

0

0

0

0

b1

a1

a5
b2

a6
b3

200

200

a7

a2

a8
a9

200

a3

a10 a11 a12b4

a4

a14/b6b7 a15

400

a55

b10

b3

b21
600

600

a8
a9

a20

a18/a19b8b
a16 b a17 b11 12
9

b2

a7

Frequency (MHz)

400

400

b5 a13

b1

a6

b14

b13

a21

600

a10
b4 a11

b16

800

800

b15

a12
b5

b17

a13

b18

a14/b6

b19 b

b7

20

1000

1000

800

a15

Figure 4.4: Hyper…ne spectrum recorded at P(13) 43-0 and R(15) 43-0. Hyper…ne components a1 - a21 belong to P(13) 43-0, b1 -b21
belong to R(15) 43-0.

Strength (a.u.)

5

81Strength (a.u.)

-4

-2

0

2

4

-2

-1

0

1

2

0

0

100

a2

200

200

a3

a16

500

a6

a8

a17 a
18

a7

600

a19

a20

a9

600

Frequency (MHz)

400

a5

a21

700

Figure 4.5: Hyper…ne spectrum recorded at P(35) 70-0.

300

400

a4

a10 a11

a13

800

800

a12

900

a14 a15

a16

1000

1000

82

83
and hence can be well resolved by the FM sideband technique with a modulation
frequency of 1 MHz.
The performance of the system can be evaluated by comparisons made at transitions where the hyper…ne splittings obtained from high-precision beat experiments
are available. For example, the frequency scan of the a1 and a3 components at transition P(13) 43-0 gives a frequency di¤erence of 131:764(8) MHz between these twocomponents, consistent with the published value of 131:7701(2) MHz [55]. Absolute
frequency measurement of any desired 127 I2 transition is facilitated by a self-referenced
optical frequency comb based on a mode-locked femtosecond laser, which has been
described in chapter 3. Various system parameters, such as frequency and pressure
shift, optical alignment, and residual amplitude modulation [116], are characterizedor optimized with the help of the optical frequency comb. Whereas some of the hyper…ne components have their absolute frequencies measured by the comb, most of
the hyper…ne-splitting measurements are carried out by the cavity-based laser scanwith its intrinsic frequency-calibrated scale coming from the frequency synthesizer
that drives the AOM tuning element.

4.3.2

Extracting the E¤ective Hyper…ne Parameters from the Hyper…ne Spectrum

Based on Bordé’s FORTRAN code, a computer program that performs thenonlinear least-squares …t is developed to extract the e¤ective hyper…ne parametersfrom the hyper…ne spectrum. We select a rotational basis set that spans

J = 0,

2,

4 to construct the Hamiltonian matrix with vibrational and rotational constants
E , B , D , H , L , and M taken from the I2 atlas [68]. Although two neighboring
vibrational levels with di¤erent J values can have a nonvanishing Franck-Condon
overlap, their large energy separation ensures that the hyper…ne coupling to di¤erent
vibrational levels inside the B state is negligible. A practical issue inherent in the84measurement of main lines (transitions withF =J) is that the …t does not permit

simultaneous determination of the hyper…ne parameters both for the B and the X
states. Without additional information, only the di¤erence between the hyper…neparameters in the upper and the ground levels can be reliably obtained. To separate
the B-state parameters from those of the X state we selectively excite the transitions
with low vibrational level00= 0 [transitions P(49) 59-1 and R(45) 60-1 are two

exceptions] and relate the J dependence of the ground-level parameters eqQX and
CX to that predetermined by interpolation formulas [42, 117]. The other two major
parameters for the ground state, i.e., the tensorial and scalar spin-spin parameters
dX andX , are …xed in the …t to the values of 1:524 and 3:705 kHz [42], respectively,

which are good approximations for low-lying vibrational levels in the ground state X[42].
The nonlinear least-squares …t has been explained in great detail elsewhere[118, 119] and hence will not be discussed here. However, it is worth mentioning a
special case of the …t. It was observed that at transition P(121) 35-0, the a8 and a12
components could not be …tted by the program used at the time [120, 121], thoughthese two components are isolated and their transition frequencies were measured
with about the same S/N as that of other components.
This problem arises from the matching of the upper and lower hyper…ne levels after the diagonalization of the associated Hamiltonians in the …tting procedure.
Keeping track of I 0 and I 00 before and after the diagonalization of the Hamiltonians for the B and the X states works for most levels in that it can generate thecorrect frequency of a hyper…ne component by identifying the upper and lower levels involved. However, it occasionally gives erroneous pairing of the two levels andconsequently one sees interchanged frequencies for two hyper…ne components. Here
we show explicitly the matching of the relevant hyper…ne levels for the a8 and a12
components of transition P(121) 35-0. After switching on the hyper…ne Hamiltonian,

85
the total nuclear wave functions for the two relevant hyper…ne levels in the B stateevolve as
jI 0 = 3i ! j"0 (1)i = 0:714 jI 0 = 3i + 0:701 jI 0 = 5i ;jI 0 = 5i ! j"0 (2)i =

0:701 jI 0 = 3i + 0:714 jI 0 = 5i ;(4.8)

and likewise for the two levels in the X state:
jI 00 = 3i ! j"00 (1)i = 0:709 jI 00 = 3i

0:706 jI 00 = 5i ;jI 00 = 5i ! j"00 (2)i = 0:706 jI 00 = 3i + 0:709 jI 00 = 5i :(4.9)

Note that an overall phase factor is dropped for these wave functions. Because thedipole transition operator acts only on electronic degrees of freedom, the matching
criterion is jh"0 j"00 ij = 1, which indicates that j"0 (1)i [j"0 (2)i] is matched to j"00 (2)i[j"00 (1)i]. Thus if the …tting procedure only tracks the total nuclear spins, then it
will attempt to use the pairing j"00 (1)i ! j"0 (1)i [j"00 (2)i ! j"0 (2)i] to calculate thehyper…ne splittings of the a8 (a12 ) component with respect to a reference position. The
matching criterion jh"0 j"00 ij = 1 can be readily incorporated into the …tting procedureby using the eigenvectors obtained from the diagonalization of the Hamiltonians forthe upper and lower electronic states.
B-State E¤ective Hyper…ne Parameters in the Region 42 6

4.4

In the wavelength range 500

0

6 70

517 nm we measured hyper…ne spectra of 79

X rovibrational transitions that cover a large set of rotational and vibrationalB

quantum numbers in the B state. These transitions were chosen to include uppervibrational levels starting at
00= 42; 43 and thence progressing in increments of 2 to

= 63: For each of these vibrational levels, we measured four to seven rotational

levels that spread roughly from J 0 = 10 to J 0 = 100. Abovelevels,00= 63, three vibrational= 65, 69, 70, were accessed, and the rotational levels measured for each

86
vibrational level are restricted to the range J 0 = 30 60 because of the weak transition
moment associated with this region. In addition to the above data set, six rotational
levels at

0= 60 and two rotational levels at

0= 50, 58 were also measured. Most of

the hyper…ne spectra recorded have been …tted by the four-term e¤ective Hamiltonian
described in section 4.2.2, and the extracted hyper…ne parameters, together with thecorresponding standard deviations, are listed in table 4.1. The hyper…ne splittings
for all the transitions measured are listed in appendix B. The list of table at the
beginning of this thesis can be used to locate the spectrum for a speci…c transition.

87
Table 4.1: B-state e¤ective hyper…ne parameters derived from the measurement of
I2 rovibrational transitions in the wavelength range 500 517 nma;b;c

127

eqQB
CB
dB
S. D.
B
Transition
(MHz)
(MHz)
(MHz)
(MHz)
(KHz)
P(10) 42-0 -557.119(20)d 0.176138(96) -0.08916(86) -0.00520(85)7.6
P(19) 42-0-556.988(66)0.17666(17)-0.0921(43) -0.0016(39)33
R(30) 42-0 -557.184(44) 0.177751(85) -0.0946(22)0.0099(33)22
R(41) 42-0 -557.553(30) 0.179178(24) -0.0929(20) -0.0003(16)17
R(49) 42-0 -557.612(50) 0.180487(46) -0.0915(25)0.0051(23)30
P(13) 43-0-558.613(18) 0.190361(78) -0.09899(62) -0.00083(56)10
P(25) 43-0-558.823(15) 0.191161(39) -0.1009(12)0.0025(11)9.4
R(27) 43-0 -558.667(23) 0.191593(38) -0.0988(16)0.0015(12)11
P(38) 43-0-558.565(12) 0.192839(21) -0.10109(69) -0.00001(81)5.9
R(40) 43-0 -558.604(17) 0.193497(14) -0.10303(86) 0.00034(86)6.4
R(63) 43-0 -559.456(16) 0.198545(12) -0.10629(90) 0.00234(85)11
R(21) 45-0 -560.838(13) 0.223669(23) -0.12073(80) 0.00623(57)7.6
R(52) 45-0 -561.356(22) 0.229843(18) -0.1239(10)0.0095(15)10
P(69) 45-0-561.804(11) 0.2349204(71) -0.12617(80) 0.00739(59)6.8
P(91) 45-0-562.546(13) 0.2452897(55) -0.1324(10) 0.00975(82)7.9
R(31) 47-0 -562.816(27) 0.264762(35) -0.1390(16)0.0172(12)12R(51) 47-0 -563.294(12) 0.270529(10) -0.14741(84) 0.01588(64)7R(92) 47-0 -564.742(23) 0.293399(10) -0.1598(10)0.0200(13)9.4
R(111) 47-0 -565.651(56) 0.310759(29) -0.1626(44)0.0254(35)37
P(19) 49-0 -564.6327(72) 0.309250(21) -0.16923(43) 0.02631(31)4.6
P(41) 49-0-564.886(26) 0.314896(28) -0.1706(18)0.0262(13)16
P(62) 49-0-565.295(39) 0.324667(25) -0.1794(17)0.0326(22)16
R(93) 49-0 -566.485(19) 0.3509271(87) -0.1940(13)0.0403(10)11P(108) 49-0 -567.273(24) 0.3662298(84) -0.1996(10)0.0427(13)9.7
R(55) 50-0 -566.0544(80) 0.3504090(55) -0.19199(56) 0.03910(46)4.7
R(20) 51-0 -566.275(21) 0.366031(35) -0.20074(96) 0.0469(11)7.1
P(30) 51-0-566.415(29) 0.368223(40) -0.2005(13)0.0401(19)13
P(50) 51-0-566.763(14) 0.377297(11) -0.20747(63) 0.05024(66)5.6
P(73) 51-0
-567.132(23) 0.394644(14) -0.2134(16)0.0552(12)12P(90) 51-0
-567.872(28) 0.413339(14) -0.2258(13)0.0628(19)13
R(32) 53-0 -567.752(27) 0.437830(32) -0.2383(12)0.0785(16)11
R(51) 53-0 -568.171(11) 0.450037(11) -0.24241(82) 0.08388(55)6.6
P(70) 53-0
-568.623(34) 0.466803(19) -0.2535(15)0.0937(19)14P(89) 53-0-569.274(40) 0.493524(21) -0.2664(27)0.1073(19)21
R(21) 55-0 -568.807(18) 0.513082(32) -0.2718(12) 0.12206(91)10
P(61) 55-0
-569.444(22) 0.544230(21) -0.2854(16)0.1444(13)15P(83) 55-0
-570.208(27) 0.579914(17) -0.3006(16)0.1701(13)15
P(102) 55-0 -571.145(42) 0.626595(27) -0.3212(21)0.2092(28)22(Table Continued)

Table 4.1: Continued
eqQB
Transition(MHz)
P(19) 57-0 -569.534(27)P(33) 57-0 -569.335(44)R(65) 57-0 -571.765(86)R(74) 57-0 -570.210(94)P(80) 57-0 -570.38(12)R(83) 57-0 -570.632(90)R(95) 57-0 -570.701(83)P(17) 59-0 -568.58(15)P(27) 59-0 -573.49(28)R(28) 59-0 -573.32(48)P(49) 59-1 -571.695(39)R(81) 59-0 -572.12(11)P(87) 59-0 -572.555(90)P(21) 60-0 -569.937(20)R(34) 60-0 -569.964(45)R(45) 60-1 -570.446(92)R(99) 60-0 -573.28(23)P(17) 61-0 -569.550(24)R(36) 61-0 -569.603(42)P(53) 61-0 -570.037(79)P(83) 61-0 -570.14(14)P(23) 63-0 -568.464(78)R(31) 63-0 -568.62(11)P(43) 63-0 -568.854(83)P(52) 63-0 -569.08(11)P(19) 65-0 -566.342(63)P(33) 65-0 -566.504(73)R(43) 65-0 -567.21(11)P(33) 69-0 -558.63(12)P(39) 69-0 -560.05(23)R(44) 69-0 -561.17(27)R(49) 69-0 -559.39(26)P(53) 69-0 -562.09(50)P(35) 70-0 -557.32(24)R(37) 70-0 -557.47(35)R(45) 70-0 -558.80(44)
a

88
CB
(MHz)
0.605935(82)0.614577(46)0.660831(90)0.678716(54)0.688579(74)0.702332(76)0.741636(48)0.72125(42)0.71536(54)0.72088(55)0.750322(35)0.840592(60)0.856503(46)0.783323(37)0.799015(53)0.816760(82)1.03267(11)0.850067(59)0.875111(89)0.906087(75)1.02283(10)1.01785(15)1.03395(15)1.05691(12)1.084930(90)1.21056(14)1.24010(14)1.28195(20)1.85049(22)1.89425(41)1.96255(38)2.02075(33)2.04973(57)2.08735(37)2.12619(51)2.22994(62)

dB
B
(MHz)
(MHz)
-0.3091(19) 0.1993(11)-0.3103(26) 0.2058(20)-0.4260(68) 0.3647(59)-0.3092(44) 0.2500(65)-0.298(14)0.240(21)-0.311(12)0.280(11)-0.3270(59) 0.3323(50)-0.290(13)0.141(11)-0.569(14)0.382(14)-0.476(24)0.316(18)-0.4326(26) 0.3335(20)-0.4355(78) 0.4337(59)-0.4360(65) 0.4444(55)-0.3773(13) 0.34612(86)-0.3820(20) 0.3619(30)-0.3800(62) 0.3793(37)-0.499(17)0.688(14)-0.3944(14) 0.4414(10)-0.4056(19) 0.4638(27)-0.4162(42) 0.5154(42)-0.422(10) 0.6224(83)-0.4219(56) 0.6974(44)-0.4281(76) 0.7225(58)-0.4436(43) 0.7560(36)-0.4532(49) 0.7987(73)-0.4648(37) 1.0373(32)-0.4893(33) 1.0966(34)-0.4901(68) 1.1712(49)-0.6800(72) 2.3114(69)-0.696(12)2.422(10)-0.732(12) 2.5258(96)-0.854(14)2.559(14)-0.793(28)2.691(22)-0.809(12) 2.8335(88)-0.867(22)2.886(14)-0.937(24)3.045(17)

S. D.
(KHz)
15
23
60
44
37
46
51
73
131
93
22
65
55
11
21
43
121
13
16
44
87
46
70
40
51
43
46
57
44
119
76
116
252
111
173
196

A four-term e¤ective Hamiltonian is used in the …t. Transitions P(69) 58-0, R(18) 59-0,
P(84) 60-0, P(77) 60-0, and P(63) 70-0 are not included in this table because their upperlevel hyper…ne spectra are altered considerably by the hyper…ne perturbation from a
1g (1 g ) state.
b
eqQX and CX are determined by equations (14) and (10), respectively, in reference [42] andare held …xed in the …t.
cdX and X are …xed in the …t to 1.524 and 3.705 kHz [42], respectively.
d
Quoted uncertainties (1 ) are estimated from the standard deviations of the …ts.

89First, for the transition P(13) 43-0 the four e¤ective parameters obtained by our
measurement are in good agreement with previous results within the measurement
uncertainties. The standard deviation of the …t is typically 10 20 kHz for levels below
0= 55 and is caused primarily by various experimental sources such as backgroundinterference fringes superimposed upon the saturation resonance, residual cavity drift,
variations of the optical power and alignment arising from the frequency scan, andresidual amplitude modulation in the saturation spectroscopy.
At rovibrational levels above0= 55, transitions R(65) and R(74) of the 57-

0 band and P(17), P(27), and R(28) of the 59-0 band distinguish themselves fromadjacent transitions by large standard deviations that cannot be explained by theexperimental sources mentioned above. Furthermore, …ts for transitions P(69) 580, R(18)59-0, P(77) 60-0, and P(63) 70-0 produce very large standard deviations(0:3

1 MHz), and the hyper…ne spectrum of the P(84) 60-0 transition cannot be…tted at all by the e¤ective hyper…ne Hamiltonian2 . The spectra of these transitions
are provided in appendix B. We attribute this abnormality to the in‡uence of thesame 1g (1g ) state that strongly perturbs the0= 76, 77, 78 vibrational levels close

to the dissociation limit. Because of the strong perturbation, hyper…ne spectra atthese transitions cannot be fully described by the e¤ective Hamiltonian based on
second-order perturbation theory. The source of this perturbation is con…rmed andits speci…c properties in the intermediate region that we investigate are discussed
further in section 4.8. We have also attempted to introduce the last four terms inequation 4.7. At levels of

0

= 57

65, these four additional terms do not improve

the standard deviations of the …ts. Besides, e¤ective hyper…ne parameters e, f , g,and hH derived from the new …ts all have large uncertainties that are comparable to
the …tted values, and contradict the smooth trend of the second-order contribution
2

With a2 and a10 components excluded, the spectrum can be …tted to the four-term e¤ective
hyper…ne Hamiltonian with large residual error (1:1 MHz). See appendix B.58 for details.

90
by altering their signs across adjacent rotational levels. Although vibrational levels
with0= 65 would be better candidates for testing these additional terms becauseof the increasing perturbation, the present large uncertainties associated with theseweak transitions preclude any accurate assessment. Also, as pointed out in reference[16], the total number of hyper…ne parameters that can be …tted reliably is restrictedby the use of a maximum of 15 or 21 main lines in the hyper…ne spectrum.
Vibrational Dependence of Hyper…ne Parameters in the B State

4.5

Combining the data from the study reported here and from the literature[45, 42, 55, 117, 40, 120, 122, 123, 124, 125, 126, 127, 128, 129] has facilitated investigations of the hyper…ne spectra that now cover a majority of the vibrational levels(3 <0< 82) in the B state. Therefore it is now possible to explore the global

trend of these hyper…ne parameters in the B state and uncover systematic variationsof the corresponding hyper…ne interactions. With the rotational dependence being
suppressed, the hyper…ne parameters as functions of pure vibrational energy E( 0 )are illustrated in …gure 4.6. As shown in the …gure, the values of these hyper…ne parameters increase rapidly when the molecule approaches the dissociation limit, which
is a result of the perturbations from the cluster of electronic states that share thesame dissociation limit with the B state. While the variation of CB is essentiallysmooth over the whole range of vibrational energies, eqQB , dB , andcal irregularities at several regions. First, in the region 3 << E( 0 ) < 17 700 cm 1 ], both dB andB0< 17 [16 000 cm 1have visible ‡uctuations because of themagnetic dipole coupling between the B and the B 00 1u (1potential crossover near level0

B all have lo-u ) states that arises from a= 5 [67]. Meanwhile, the absence of such ‡uctuationin CB at these levels is due to the fact that the accompanying gyroscopic coupling,which is involved only in CB , is a weak e¤ect compared with the magnetic-dipole
coupling [66, 67, 42, 61]. Second, irregularities in hyper…ne parameters are found

91

-350

(a)

This work
Previous data
Experimental fit

(b)

10

CB (MHz)eqQB (MHz)

-300

-400
-450
-500

1

0.1

-550
-600

0.01
100

(c)

(d)
δB+0.02 (MHz)-dB (MHz)

10

1

0.1

10
1
0.1

0.01

0.01
16

17

18

E(υ') [cm-1]

19

20x10

3

16

17

18

19

20x10

3

E(υ') [cm-1]

Figure 4.6: Vibrational dependence of the B-state hyper…ne parameters eqQB , CB ,dB , and B . Note that a semilog plot is used for CB , dB , and B . The steep trend
near the dissociation energy is due to the perturbations from the electronic states
converging with the B state to the second dissociation limit: 2 P3=2 + 2 P1=2 .

92
for levels neardB , and0= 57

60 [19 863 cm 1 < E( 0 ) < 19 915 cm 1 ] at which eqQB ,B deviate from the smooth trend for a small range of rotational levels in acertain vibrational level. As will be discussed in section 4.8, this resonance feature
results from the hyper…ne coupling between the B and the 1g (1the rotational coincidence. Finally, the same 1g (1from0= 76 to0= 78 [E( 0 )g ) states throughg ) state strongly perturbs levels

20 034 cm 1 ] with the favorable Franck-Condon

overlaps and the small energy separations [45, 43]. In last two perturbing regions,
CB also keeps its smooth trend because the gyroscopic Hamiltonian cannot couple
the B state with the 1g (1Theg ) state [16].

dependence of eqQB reverses its trend after

cm 1 ]; hence its values for higher

00= 60 [E( 0 )levels overlap those of lower

0

19 915levels, a conse-

quence of the two e¤ects that dominate at di¤erent regions: a …rst-order term— theinteraction between nuclear quadrupole moment Q and electric …eld gradient q—
increases with decreasing vibrational energy but approaches a constant value at large
energies at which the second-order term from perturbing electronic states moves in
gradually and causes a steep trend near the dissociation energy.
4.6

Radial Dependence of Hyper…ne Interactions
In this section we examine the variation of the hyper-…ne interaction with inter-

nuclear separation R. Because hyper…ne parameters already involve averaging overthe molecular rotation, the electronic motion in the molecular frame, and the nuclear
vibration, it is desirable to remove the last vibrational average in these hyper…ne parameters such that a straightforward relation between the hyper…ne interaction andthe internuclear separation can be obtained.
We remove the vibrational average in the hyper…ne parameters by inverting theexpression
O( 0 ; J 0 ) = h J0 0 jO(R)j J0 0 i ;(4.10)

93
where O( 0 ; J 0 ) denotes one of the four hyper…ne parameters for a particular level( 0 ; J 0 ) in the B state and j J0 0 i is the corresponding vibrational wave function (properly normalized). The inversion is performed by expanding the function O(R) as apolynomial with its coe¢ cients determined from an experimental …t [130, 131]. Le
Roy’s program LEVEL [132] is used in the process to calculate the vibrational wave
functions from the B-state potential energy curve [133]. Figure 4.7 plots eqQB (R),CB (R), dB (R), andB (R), along with the corresponding residual errors of the inter-

polation. Consistent with the smooth variation in CB , interpolation function CB (R)has small residual errors (within

0:02, relative) for the entire range

0=3

70, asshown in the inset of …gure 4.7(b). Conversely, the large residual errors in the interpolation of eqQB , dB , and
near

0

B

for

0

> 56 re‡ect the abnormal variations observed= 57, 59 [see the insets in …gure 4.7], restricting a reliable interpolation only

to levels of

0

< 56.

The hyper…ne parameters extracted from the spectra are directly associatedwith speci…c quantum numbers ( 0 ; J 0 ) but not internuclear separation R. Alternatively, these hyper…ne parameters can be plotted against R-centroid h J0 0 jRj J0 0 i,
which is the expectation value of the internuclear separations sampled by the nuclear
vibration and evaluated by use of the B-state potential [133]. Figure 4.8 shows suchplots for eqQB , CB , dB , and B , where solid curves are calculated from h J0 0 jO(R)j J0 0 i
and symbols are experimental data. Again, the visible scatter of data points in eqQB ,dB , and1
B near 5 Å is due to the strong perturbation from the 1g (g ) state.

In the region R < 5 Å valuable information can readily be extracted from eqQB
to assist the investigation of the electronic structure of 127 I2 . Unlike the other three
hyper…ne parameters whose major parts originate from perturbations at nearly all
possible values of R, a signi…cant part of eqQB is due to the interaction between
the nuclear quadrupole moment Q and the electric …eld gradient q(R) generated
by the surrounding charge distribution of a largely B-state character [16, 47]. This

-450
-500

2.5

υ′≤55

0

20

40
υ′

60

-550

2.0
1.5

0.0 2
0.0 0
-0. 02-0. 04
20

1.0

40
υ′

60

0.5

(a)

-600

1.0

0 .05
0 .00

δ B (R) [MHz]dcal - dfit (MH z)

1.5

υ′≤55

-0. 05-0. 10
0

20

40
υ′

60

0.5

(c)

0.0
3

4

5

R (Å)

6

7

(b)

0.0
4

-d B (R) [MHz]

0.0 4

0

3

2

δcal - δmea (MH z)

eqQ B (R) [MHz]

-400

3.0
3
2
1
0
-1

C B (R) [MHz]eqQ c al - eqQ mea (MH z)-350(C c al - C mea )/C mea

94

0.1 0
0.0 5
0.0 0
-0. 05-0. 10

υ′≤55
0

20

1

40
υ′

60

0

(d)
3

4

5

6

7

R (Å)

Figure 4.7: R dependence of the hyper…ne operators eqQB (R), CB (R), dB (R), andB (R). These hyper…ne operators are obtained by inverting the expression O 0 J 0 =h J0 0 j O(R) j J0 0 i, where O represents one of the hyper…ne parameters. Insets areresidual errors of the interpolation. (a), (c), (d) Solid and dashed curves are …ts up
to 0 = 55 and 0 = 70, respectively. The …t in (b) is up to 0 = 70. The large
residual errors for eqQB , dB , and B in the region 55 < 0 6 70 are due to the strong
perturbation from the 1g (1 g ) state.

95

3

(a)

eqQ B (MHz)

-500

(b)

This work
Previous data
Experimental fit

C B (MHz)

-520
-540

2

1

-560
0
10

(c)

(d)
δ B +0.02 (MHz)-d B (MHz)

1

0.1

1

0.1

0.01
0.01
3

4

5

6

R-centroid (Å)

7

3

4

5

6

7

R-centroid (Å)

Figure 4.8: eqQB , CB , dB , and B versus R centroid h J0 0 j R j J0 0 i. Solid curves arecalculated from the interpolation functions for these hyper…ne parameters. The data
presented here cover a relatively large range of internuclear separation and join theprevious results at both small and large R-centroids. The visible scatter of the data
points in eqQB , CB , and B near 5 Å is due to the strong perturbation from the1g (1 g ) state.

96
fact is more evident if we focus on the region R < 5 Å where perturbations fromother electronic states are negligible by comparison. Moreover, when they movefrom a separated-atom model to small internuclear separation, the two atomic wave
functions are increasingly distorted, their overlapping also increases, and hence eqQB
deviates gradually from the value calculated from a separated-atom basis set or linear
combinations of atomic orbitals. Therefore, when R < 5 Å the vibration-removed
interpolation function eqQB (R), coupled with a prior information on q(R), can beused to determine iodine’s nuclear quadrupole moment or to serve as a benchmark
for molecular ab initio calculations of the electronic structure at various values of R.
For this purpose the coe¢ cients of the interpolation function for eqQB (R) over therange R = 3

5 Å is given in table 4.2.

Table 4.2: Vibration-removed interpolation function eqQB (R)a
i
0
1
2
3
4
5
6
Standard deviation (MHz)ai2487.80(17)b
2236(11)112(26)163(28)2210(18)
87.8(6.5)
212.73(94)
0.21

P6

Re )i , 3 Å < R < 5 Å, and Re = 3:02708Å.
The …t includes 69 levels with 0 < 57, which are taken from this studyaeqQB (R) =ii=0 a (Rand from published data summarized in [42, 117].
b
Quoted uncertainties (1 ) are estimated from the standard deviationsof the …t.

4.7

Rovibrational Dependence of Hyper…ne Parameters as a Resultof the Second-Order E¤ect
Four hyper…ne parameters, eqQB , CB , dB , andB , all exhibit systematic rovi-

brational dependence in the measured vibrational levels from0

= 42 to

0

= 70.

97
Figures 4.9–4.11 display variations of these parameters. In the …gures each solid line
is a …t of experimental data that belong to a single vibrational level0( 0 is indi-cated as a number attached to each line in the …gures). In general, these parameters
have monotonic dependence on both rotational and vibrational quantum numbers,
except for the levels in the vicinity of
the 1g (10= 57

59, where strong perturbation fromg ) state produces irregularities that superpose upon the smooth second-order background. Except for the variation of eqQB at vibrational levels lower than0

= 60, the smooth and monotonic trend of these e¤ective hyper…ne parameters iswell characterized by second-order perturbation theory. For instance, spin-rotation
parameter CB can be expressed as [45, 16, 46, 134]"#
Z
X X hV 0 i hV 1 i
hV 0 i hV 1 i
CB ( ; J) = CD ( ; J)+
(E 0 J )dE 0 J :E0+u J E 0 0 J
E0+u J E 0 J
00(4.11)The …rst-order contribution CD is less than 0:15 kHz [16], which is negligible comparedwith the second-order contribution. The summation on0

and the integral in brackets

in equation 4.11 together account for the second-order contribution from both discretelevels ( 0 ; J) and energy continuum EJ . In the numerators hV 0 i and hV 1 i denotegyroscopic and magnetic dipole couplings, respectively, which can be approximatedas a product of a Franck-Condon integral and an electronic matrix element with theassumption that the latter has a weak R dependence [16]. The other three e¤ective
parameters can be expressed in the same manner with di¤erent types of hyper…necoupling [45, 16].
As the B-state rovibrational levels ( ; J) progress toward the dissociation limit,
their overall Franck-Condon overlaps with the perturbing levels continue to increase
while the corresponding energy denominators decrease, thereby creating monotonic
trends in hyper…ne parameters, with large slopes near the dissociation limit. Morespeci…cally, in a given vibrational level , as the rotational quantum number J in-

98

42

-558

eqQB (MHz)

-562

45

-564

47

-566

50
51

-568

49

53

-570

55

57

-572

60

59

-558

70

-560

eqQB (MHz)

( a)

43

-560

( b)

69

-562
-564
-566

65
63

-568

61

-570
-572

60

0

2

4

6

8

12x10

3

J'(J'+1)
Figure 4.9: Rovibrational dependence of eqQB . Each solid line is the linear …t in
J 0 (J 0 + 1) of experimental data that belong to one vibrational level, 0 ( 0 is indicatedas a number attached to each line). Experimental data (open squares and circles)
show abnormal variations near 0 = 57, 59. The vibrational dependence of eqQB
reverses its trend after 0 = 60 and overlaps that of lower 0 levels. Also, in theregion 0 = 57 60 several rovibrational levels experience strong perturbation fromthe 1g (1 g ) state; thus eqQB for these levels are not shown in the …gure.

99

2.5
2.0

70

65

69

CB (MHz)

1.5

63
61
60
59

1.0

57
55
53
51
49

0.5
50

47

45
43
42

0

2

4

6

8

12x10

3

J'(J'+1)
Figure 4.10: Semilog plot of rovibrational dependence of CB . Each solid line, calculated from interpolation function CB (R) [see …gure 4.7 (b)], is the J dependence forone vibrational level 0 ( 0 is indicated as a number attached to each line). Unlike
eqQB , dB , and B , CB does not have abnormal variations near 0 = 57, 59 because
the gyroscopic Hamiltonian, which is involved only in CB , cannot couple the B and
the 1g (1 g ) states.

100

1

70

8
7
6

65

-dB (MHz)

5

( a)

69
59

63

60

4

61

57

3
2

50

47

55
53
51
49

45

0.1

43

42

8

70

69

65

δB (MHz)

1

63
59

0.1
50

( b)
61
57

60
55
53
51
49
47
45

0.01

42
42

0

2

4

6

J'(J'+1)

8

12x10

3

Figure 4.11: Semilog plot of rovibrational dependence of dB and B . Note that thevertical scale has been inverted for dB . Each solid line is the linear …t in J 0 (J 0 + 1) ofexperimental data that belong to one vibrational level, 0 ( 0 is indicated as a number
attached to each line). Experimental data (open squares and circles) show abnormal
variations near 0 = 57, 59. In the region 0 = 57 60 several rovibrational levels
experience strong perturbation from the 1g (1 g ) state; thus hyper…ne parameters forthese levels are not shown in the …gure.

101
creases the total number of discrete vibrational levels inside each perturbing electronic state drops, whereas the contribution from energy continuum increases. The
combined contribution from discrete levels and continuous spectrum gives a smoothvariation of the hyper…ne parameters even for large J values.
The systematic data from this study and from the literature o¤er a means for acomplementary approach to analyzing the electronic structure of the states converging
to the second dissociation limit, 2 P 3 + 2 P 1 . Previous calculations [45, 46, 134] of2

2

hyper…ne parameters can be extended to low vibrational levels as well as to therotational dependence at each vibrational level, using the available information on theelectronic wave functions and the related potential energy curves (PEC) determined
either by experiment [133, 135] or by theoretical calculation [110]. Several interesting
issues can be addressed with the help of these calculations, such as (1) to what
extent the separated-atom wave functions can be extrapolated to short internuclear
separations, (2) an estimation of the admixtures of the basis wave functions in theseparated-atom model, and (3) how sensitively the hyper…ne parameters depend on
the choice of PECs. This line of investigation is beyond the scope of current chapter
and will be detailed in chapter 5.
4.8Strong Perturbation from the 1g (1g ) State

Apart from second-order e¤ect, we observed strong perturbation in the intermediate region that spans

0

= 42

70. Hyper…ne parameters eqQB , dB , andall exhibit abnormal J dependence for levels

0B= 57, 59, which are clearly present

in …gures 4.9 and 4.11. Moreover, as pointed out in section 4.4, hyper…ne spectraof several transitions have exceptionally large residual errors in the …t or cannot be…tted by the e¤ective hyper…ne Hamiltonian in the region

0

= 57

60.

The nature of this perturbation is analogous to that of the rotational andvibrational perturbations studied in a variety of molecules [136]. Among the cluster

102
of electronic states converging with the B state, the 1g (1g ) state has a potential

whose outer limb is closest to that of the B state. It is possible that an approximate
energy coincidence and a moderate Franck-Condon overlap between the B and the
1g (1g ) rovibrational levels arise simultaneously as two sets of rovibrational levels

progress respectively inside the B and the 1g (1g ) states. This perturbation depends

sensitively on the energy di¤erences among the levels involved and hence cannot betreated by the e¤ective Hamiltonian based on second-order perturbation theory.
The strong u-g mixing that we recorded at transition P(84) 60-0 permits a morequantitative analysis. The hyper…ne spectrum of this transition is shown in …gure
4.12, where vertical bars labeled a1 a15 above the spectrum are simulated by the fourterm e¤ective Hamiltonian with the hyper…ne parameters obtained by interpolationof the parameters of neighboring rovibrational levels. This simulation produces the
spectrum in the absence of strong perturbation. As shown in …gure 4.12, the a10
component does not show up at the position predicted by the simulation; instead, atleast four new components, namely, b1b4 , emerge nearby: Apparently the upper

level of a10 is strongly perturbed by hyper…ne levels of a perturbing electronic state
and the mixed levels appear as b1 and b3 , with the crossover b2 appearing in the
middle.
The upper level of a10 (F = 87, J = 83, and I = 4) is isolated from otherhyper…ne levels inside the B state because it is coupled to only two hyper…ne levels
with large energy separations (1g (1111 GHz). However, calculation that uses the B andg ) empirical PECs [133, 135] shows that the perturbing levels could be the two

hyper-…ne levels (resulting from linear combinations of I 0 = 3; 5) at rovibrational level( 0 = 15, J 0 = 84) inside the 1g (1g ) state. Nevertheless, judging from the relatively

large strength of the b1 and b3 components, an e¤ective two-level model, characterized
by a energy di¤erence

E and an coupling matrix element H12 , is adequate for

estimating the hyper…ne-coupling strength between the B and the 1g (1g ) states.

0

2

4

700

C7

a7

a8

a10

740

57.2 MHza9200

C9 C10 C C
11
12

720

b1

C8

a1 a2

760

C13

C14

b2

a6

820

a12

600

b3

Frequency (MHz)800

57.2 MHz

C15

a11

a5

780

400

a3 a4

800

b2
b3
b4

840

860

880

b4

1000

a14 a15
d1

900

a13

Fitted by four-term effective
hyperfine Hamiltonian(a2 and a10 exclueded.)b1

a7 - a10 a11 a12

920

Figure 4.12: Strong u-g perturbation observed at transition P(84) 60–0. (a) The whole spectrum; the part of interest (700 920 MHz)is expanded and replotted in (b), where b1 –b4 arise from u-g mixing between the B and the 1g (1 g ) states and b2 is the crossover ofb1 and b3 . In (a) c7 –c15 are identi…ed as lines of a neighboring transition R(30) 64-1, and d1 , left of a14 , belongs to another transition,
P(81) 59-0. The vertical bars labeled a1 –a15 indicate the hyper…ne spectrum that the P(84) 60-0 transition would assume if there
were no such strong perturbation from the 1g (1 g ) state. Note that the 1g (1 g ) state is also designated as 1g or c1g in the literature.

(b)

-4

-2

0

2

4

6

-4

-2

(a)

Strength (a.u.)Strength (a.u.)

6

103

104
Accordingly, the two mixed states can be written as
j1i =
j2i =

jB; JIF i + j1g ; 0 J 0 "0 F i
jB; JIF i +(4.12)

j1g ; 0 J 0 "0 F i(4.13)

where "0 labels the nuclear spin wave function of the perturbing level. Simple calculation that uses the frequency di¤erences among b1 , b3 , and simulated a10 gives theenergy di¤erence
parameters

E = 39 MHz, the hyper…ne coupling H12 = 54 MHz, and mixing= 0:82 and= 0:57. With the PECs of the B and the 1g (1g ) states

the Franck-Condon overlap j h 0 j i j is calculated to be 0:1 and then the electronic
1
matrix element for the hyper…ne coupling is estimated as j hB0+
u j Hhf j1g (g )i j = 540

MHz.
Electronic states other than the 1g (1g ) state are unlikely to produce suchstrong perturbation because the associated Franck-Condon overlaps for levels at
which the energy coincidence could happen are at least two orders of magnitude
smaller. Even if the perturbation from such a state can shift a particular hyper…nelevel on the kilohertz level, it cannot produce a noticeable in‡uence on the whole hyper…ne structure at a certain rovibrational level in this intermediate region because
the strict energy equality can be satis…ed only for a small number of hyper…ne levels.
In this intermediate region, owing to the relatively large di¤erences in the vibrational and rotational constants and the moderate Franck-Condon overlap between the
1g (1g ) and the B states, these resonances each spans only a small range of rotational

levels, a property that is clearly identi…ed in …gures 4.9 and 4.11.
These abnormal variations directly con…rm the previously observed u-g mixing
in the optical-optical double-resonance (OODR) experiments [137, 138, 139, 135].
In these OODR experiments it was found that high-lying ion-pair states with usymmetry can be accessed by a two-photon excitation starting from the ground stateX0+
g . This seems to contradict the symmetry requirement of a two-photon transition,

105
which requires that the initial and the …nal states must have the same u(g) symmetry.
However, if the two-photon transition takes a two-step pathway, namely, X0+
!
g
jB0+
ui +

j1g (1g )i ! ju i, then the symmetry requirement is satis…ed in eachof the two steps. In one of these OODR experiments [135], a strong hyper…ne u-g
mixing was inferred from the rovibrational spectrum that exhibits a resonance feature
at vibrational level0= 59 and around rotational levels centered on J 0 = 22. Clearly,

this resonance feature is consistent with the abnormal variations of eqQB , dB , andB in …gures 4.9 and 4.11.

4.9

Summary and Conclusions
3
In this chapter hyper…ne interactions in the B0+u(u ) state of127

I2 have beensystematically studied. By saturation spectroscopy of hyper…ne spectra of the B
X transitions in the wavelength range 500

517 nm, four e¤ective hyper…ne parame-ters, eqQB , CB , dB , and B , have been determined for a large number of rovibrational
levels spanning the intermediate region (42 <0< 70), yielding rich information

about the rovibrational dependence of the hyper…ne parameters. In addition to thesecond-order e¤ect that is responsible for the smooth trend of these hyper…ne parameters, abnormal variations in eqQB , dB , andB at vibrational levels

0

= 57

60 havebeen observed, and it was found that some hyper…ne spectra in this region cannot
be fully described by the e¤ective hyper…ne Hamiltonian. By examining the clusterof electronic states converging with the B state and the strong u-g mixing recorded
at transition P(84) 60-0, we have concluded that these anomalies are caused by thehyper…ne perturbation from the 1g (1g ) electronic state through the rotational coin-cidence, which con…rms the previous observation of u-g mixing at level0= 59 in the

optical-optical double-resonance excitation of the ion-pair states with u symmetry.
Various perturbation e¤ects in the B state so far identi…ed have been summarizedbrie‡y. We also examined the radial dependence of the hyper…ne interactions by

106
removing the vibrational average in the hyper…ne parameters.
The systematic determination of hyper…ne parameters in the intermediate region provides useful information for applications of optical frequency stabilizationthat explore the transitions with wavelengths below 517 nm. Also, extensions of thecalculation of the hyper…ne parameters to low-lying vibrational levels as well as totheir rotational dependence permits a detailed analysis of the molecular electronic
structure from the perspective of hyper…ne interactions. Such a follow-up investigation is given in chapter 5.

Chapter 5
CHARACTERIZING MOLECULAR-IODINE ELECTRONICSTRUCTURE THROUGH HYPERFINE INTERACTIONS

5.1

Introduction
Comprehensive and high-precision measurements on hyper…ne spectra of mole-cular iodine provide a unique opportunity for a detailed examination of the molecule’s
electronic structure. Molecular iodine is a rare case in that its hyper…ne spectra in
the B

X system have been recorded not only with high precision at the kilo-

hertz level, but also for a large set of rovibrational levels, extending fromjust below the dissociation limit (03= 82) in the excited B0+
u(0= 2 tou ) electronic state[55, 42, 45, 117, 140, 141]. These vibrational levels cover a broad range of internuclear
separation, with R-centroids (Rc , the mean of internuclear separations weighted by
the B-state vibrational wave function) ranging from about 3 Å to 12 Å. Because hyper…ne interactions can exert large in‡uences on electronic structure, experimentally
determined hyper…ne parameters over a large range of Rc allow a sensitive test of therelevant electronic wave functions and potential energy curves (PECs).
Hyper…ne parameters from precision measurements have been used to test several relevant electronic states, i.e., X, B, and E in molecular iodine [45, 46, 47, 134,
3
142, 48, 112] and in other diatomic molecules [131]. In the case of the B0+u(u ) state

in molecular iodine, second-order contributions to the four hyper…ne parameters, theelectric quadrupole parameter eqQB , spin-rotation parameter CB , tensorial spin-spin

108parameter dB , and scalar spin-spin parametervibrational levels ( 040

B , have been calculated for several

82) in the B state [45, 46, 47, 48, 43]. In these analyses,

the molecular wave functions involved were constructed from a separated-atom basis
set because the molecule spends much of its time at large internuclear separations
for high-lying vibrational levels close to the dissociation limit. The good agreementwith the experimental data achieved in these calculations justi…es the separated-atommodel as a simple and e¤ective tool in characterizing the electronic structure at large
internuclear separations. Nonetheless, there is room to improve this approach. First,
the same calculation can be carried out with improved resolution for vibrational levels
lower than0

40 as well as for the detailed rotational dependence in each vibrational

level. This approach has become particularly relevant given the latest spectroscopic
data with an extensive coverage of rovibrational dependence of the hyper…ne interactions as described in the previous chapter and in references [140, 141]. In addition,
detailed information on the short- and intermediate-range (R < 7 Å) PEC of theperturbing electronic state, obtained either empirically or theoretically, was largely
unavailable when these calculations were performed, and therefore the properties ofthe molecular wave functions of the corresponding electronic state could not be independently inferred from these analyses. In contrast, with adequate constraints on
the PECs, a correct account of the hyper…ne parameters enables either the determination of the admixture of the basis wave functions in the separated-atom model orthe veri…cation of more-sophisticated molecular wave functions.
In this chapter we extend calculations of hyper…ne parameters to low vibrational levels and to rotational dependence at each vibrational level, using electronic
wave functions derived from the separated-atom basis set and the related empirical/theoretical PECs. In doing so, we are able to quantitatively address several issues
such as (1) to what extent the separated-atom wave functions can be extrapolated
to small internuclear separations, (2) the estimation of the admixture of the basis

109
wave functions in the separated-atom model, and (3) how sensitively the hyper…neparameters depend on the choice of PECs. For the spin-rotation parameter CB , theresults of both vibrational and rotational dependence are in good agreement with theexperimental data for0> 42 and the admixture of the two 1u states is independently

determined. Moreover, calculations of the other three parameters, eqQB , dB , andB,

place a stringent constraint on the long-range PECs of the relevant electronic states.
Here we consider the smooth second-order contributions. Thus the calculation does
not cover the abnormal variations of the hyper…ne parameters around
and

0

= 76

0

= 57

60

78 because of the strong hyper…ne coupling, which has already been

analyzed in chapter 4 and in references [45, 140].
This chapter is outlined as follows. Section 5.2 gives a close examination of
the e¤ective hyper…ne Hamiltonian developed by Broyer et al., which is the basisof our calculations of the e¤ective hyper…ne parameters. This section also lists theformulae of second-order contributions to the four hyper…ne parameters. Section 5.3
prepares the PECs and the electronic wave functions that are used to calculate therovibrational levels and coupling matrix elements. Section 5.5 presents the results
for the four e¤ective hyper…ne parameters at various internuclear separations. The
calculation is performed and discussed in three steps to demonstrate a maximum
separation of the contributions from di¤erent perturbing electronic states. In thissection we also make comparisons between the results and the experimental data andthe information of the electronic structure is deduced from these comparisons. The
conclusions are given in section 5.7.
5.2

E¤ective Hyper…ne Hamiltonian in Molecular Iodine
In chapter 4 an e¤ective hyper…ne Hamiltonian is introduced to describe thehyper…ne spectra in molecular iodine. In this section we begin with an introductionof the origins of various terms in this e¤ective hyper…ne Hamiltonian, which provides

110
the theoretical background for the calculations of the e¤ective hyper…ne parameters
performed in this chapter.
The hyper…ne interaction, which is not included in the Born-Oppenheimer
molecular Hamiltonian, may couple the B state to several electronic states sharing
the same 2 P 3 + 2 P 1 dissociation limit [111, 16]. Consequently, the hyper…ne spectra2of the X

2

B transitions are altered to various degrees depending on the couplingstrength and the energy di¤erences between the B and the perturbing states.
A direct approach to address these perturbing electronic states is to deal witha hyper…ne-Hamiltonian matrix that spans not only the B electronic state but also
all the perturbing states involved. In principle, a diagonalization of this Hamiltonian
matrix gives the hyper…ne spectra of all the electronic states spanned by the matrix.
However, this approach is not feasible in the spectroscopic analysis. First, to construct such a Hamiltonian matrix, one needs detailed information on the perturbing
electronic states, such as their rovibrational constants, which are largely not available in spectroscopic date. Second, this Hamiltonian matrix contains a large numberof hyper…ne parameters. Similar to the B state, each perturbing state has its own
hyper…ne parameters. In addition, extra parameters have to be used to characterize
the couplings between the B and the perturbing states. Such a large number of parameters cannot be determined from the spectroscopic …t that uses at most 15 or 21
hyper…ne lines for each rovibrational transitions.
An e¤ective Hamiltonian based on second-order perturbation theory was developed by Broyer et al. [16] to treat hyper…ne couplings with the external electronic
states. Reference [143] explains the notion of an e¤ective Hamiltonian in a much more
broad context. This e¤ective-Hamiltonian matrix spans only one electronic state (Bstate in this case) but has its matrix elements modi…ed by second-order perturbation theory to take into account indirect hyper…ne couplings with external electronic
states. The bene…t of this treatment is that despite their di¤erent origins, various per-111
turbation terms from di¤erent perturbing electronic states can be incorporated into asmall set of parameters according to their overall geometrical dependences evaluatedwith irreducible tensorial algebra, thereby simplifying the experimental analysis of
hyper…ne structure. This e¤ective Hamiltonian, which includes four major terms,
can be summarized as(5.1)

Hhf;ef f = HeqQ + HSR + HT SS + HSSS ;where the four terms on the right-hand side stand respectively for nuclear electric
quadrupole, spin-rotation, tensorial spin-spin, and scalar spin-spin interactions. The
matrix elements of this e¤ective Hamiltonian can be written as
h ; I 0 J 0 F jHhf;ef f j ; IJF i = eqQ GeqQ + C GSR + d GT SS +where eqQ, C, d, andGSSS ;(5.2)

are e¤ective hyper…ne parameters and G’s are geometricalfunctions depending only on the quantum numbers I, J, and F , whose explicit forms
are given by equations 4.3 - 4.6. Except for levels where strong coupling is present,
perturbed hyper…ne spectra can generally be described by this e¤ective hyper…neHamiltonian with high precision using the four e¤ective hyper…ne parameters, which
are extracted from …tting the spectroscopic data to the e¤ective Hamiltonian.
Second-order calculations of these parameters relate their experimental values
to the relevant PECs and the admixture of the basis wave functions, imposing stringent constraints on the associated electronic structure. References [44] and [16] …rst
give the matrix elements of the hyper…ne Hamiltonian that can couple di¤erent electronic states. The evaluation of these matrix elements is detailed in appendix A.
Based on these matrix elements, the e¤ective hyper…ne parameters, which encompass the contributions from the perturbing states, are then derived in reference [16]with tensorial algebra and second-order perturbation theory. For clarity of the notation and discussion, we will provide an overview of this derivation before we list the
explicit expressions for the e¤ective hyper…ne parameters.

112
5.2.1

Matrix Elements of the Hyper…ne Hamiltonian

The hyper…ne Hamiltonian of the iodine molecule can be formally written as
Hhf = Hhf (a) + Hhf (b) + Hhf (a; b);(5.3)

where a and b label the two iodine nuclei, respectively, and Hhf (a) [Hhf (b)] andHhf (a; b) represent the nucleus-electron and nucleus-nucleus hyper…ne interactions,
respectively. The last term in the above Hamiltonian is left out in the following discussion because it only contains nuclear coordinates and thus cannot couple di¤erent
electronic states. With the last term dropped, this Hamiltonian can be expressed intensorial form:
Hhf =

k
XXX

( 1)q Qkq (I ) V kq (e ) ;(5.4)=a;b k q= kwhere the rank-k spherical tensor operators Qk (I ) and V k (e ) act on the nuclear
spin I and the electronic degrees of freedom, respectively.
Considering the strong spin-orbit interaction in this heavy molecule, a Hund’scase c coupling scheme at large internuclear separations is appropriate for the molecular basis set, which we denote as
j i=j
where

JIF MF i ;(5.5)

is the projection of the total angular momentum J on to the molecular axis

connecting the two nuclei,

the vibrational quantum number, I the total nuclear

spin, F = I + J, and MF the projection of F onto a quantization axis. Note that in
the energy continuum, the discrete level index

is replaced by a continuous energy

spectrum E, while J is always a good quantum number.
Appendix A provides a detailed evaluation of the matrix elements of the hyper…ne Hamiltonian in a Hund’s case c coupling scheme. Here we brie‡y summarize
the result. Averaging explicitly over the molecular rotation in the laboratory frame

113
gives the matrix elements of the hyper…ne Hamiltonian in terms of 3-j (bracket) and
6-j (curly bracket) symbols and electronic matrix elements:kh 0 j Hhf(a) j i = F F 0 MF MF0 ( 1)F +k+2Ia + I+ Jp
(2J 0 + 1) (2J + 1) (2I 0 + 1) (2I + 1)
98
9
0
18
>
>
>
>
0
0
k J C < J k J = < k Ia Ia =
B J
@
A
>
0
;>
: Ia I 0 I >;: I F I0 >fk (a;

where

J = J0J,

I = I00; J0 0 ; ; J ) ;
I,

0

=

, and fk (a;

0

(5.6)

; J0 0 ; ; J ) encapsulates

the averaging over the electronic motion in the molecular frame and the molecularvibration in the laboratory frame, which is written as
fk (a;

Here C1 =0; J0 0 ; ; J ) = ( 1)

0

0

Ck

B k Ia
@
0 Ia

1

Ia C
A
Ia

h

0 0
k
J0 j Vn gIa is the nuclear magnetic dipole moment with

(a) j

Ji :

(5.7)

n being the nuclear

magneton and g the Landé factor for the iodine nucleus and C2 = 21 eQIa is the nuclear
electric quadrupole moment with e being the proton charge and QIa de…ned as
QIa = e hIa ; MIa = Ia j

X

3zp2

p

rp2 jIa ; MIa = Ia i ;(5.8)

where zp and rp refer to the coordinates of the protons p in the iodine nucleus.
For

6= 0, the projection of the angular momentum J onto the molecularaxis can take either the positive or negative direction, a degeneracy that is lifted by
the hyper…ne interaction or other couplings with external states. In this case themolecular wave function is represented by [44]j "i =

1p
2

j"j[j ; JIF MF i + " j; JIF MF i] ;

(5.9)

114
where " can take on the values of 0 (= 0) and

1(> 0). The net e¤ect of this-doubling on the matrix elements of the hyper…ne Hamiltonian is to replace the0

fk (a;; J0 0 ; ; J ) de…ned in equation 5.7 by
j"j+j"0 j

1
p
2
C
0

where C (

+

h

+

J

B
B
B
B
@

)= 0
B
B
B
B
@

0

0

" + " ( 1)
k

0

J

h
i
f 1 + ""0 ( 1) J fk (a;

J

+

0

k
0

J

J

1
C
C
C
C
A

1

i

fk (a;0; J0 0 ;

+

=

0

and

0

; J0 0 ; ; J ) +; J )g;(5.10)+ . Expression 5.10 is equivalent

C
C
C
C
A

to equation (6) in reference [44] when " 6= 0. For the case of the B state, expression
5.10 is reduced to1
p2

j"0 j hi
1 + "0 ( 1) J fk a;0; J0 0 ; 0+u; J :(5.11)The matrix elements of the hyper…ne Hamiltonian for the second nucleus b isrelated to that of the …rst nucleus a because of symmetry considerations [44, 16]:
0
u;I 0

k
Hhf
(b) j u;I i = ( 1) I

0
u;I 0

k
Hhf
(b) j g;I i = ( 1) I+1

0
u;I 0

k
Hhf
(a) j u;I i ;

0
u;I 0

k
Hhf
(a) j g;I i ;(5.12)

where the u(g) symmetry refers to the odd (even) electronic wave functions underre‡ection with respect to the center of the molecule.
Apart from the hyper…ne Hamiltonian, the o¤-diagonal part of the rotational
Hamiltonian, V 0 , can also couple the B state to other electronic states:
V0 =
where

~2
[J+ (L + S ) + J (L+ + S+ )] ;2 R2

(5.13)

is the reduced mass of two nuclei and R is the internuclear separation. V 0

can only couple electronic states withhJ;

1j J jJ; i =

=
p
(J

1 and
)(J

J = 0 because of [144]+ 1):(5.14)

115
In addition, V 0 cannot couple u and g states. As a result of these restrictions, V 0 can
only couple the B state to 1u states. Using equation 5.14 and notice that the wave
1) state is j1u;"0 = 1 i = p12 [j1u (function of the 1u ("0 =

0

j1u (

= 1)i

the matrix element of V 0 between the B state and a 1u state with

0

=

1)i],

-doubling "0 = 1

is evaluated aspp
2J(J + 1) 2f0 1u ; 0+u ;(5.15)~2= 1) ; J0 0 j p(L+ + S+ ) 0+
u J :
2
2 2 R

(5.16)

h1u;"0 =1 j V 0 0+
u =
where
f0 = h1u (The extra factor of

5.2.2

p

0

2 in equation 5.15 arises from the-doubling of the 1u state.

Second-Order Contributions to the E¤ective Hyper…ne Parameters

With the matrix elements of hyper…ne Hamiltonian given in the section 5.2.1,
the hyper…ne parameters are calculated by second-order perturbation theory to include the contributions of all rovibrational levels allowed by the selection rules ineach of the six perturbing states [C:1u , 1u , 1g (13g ), 1g (3g ), 0g (g ), and (3)0u]. In the derivation of these parameters (see Appendix I in reference [16]), a weakJ 0 dependence of fk (a;0; J0 0 ; ; J ) across neighboring rotational levels is assumed

when the summation on J 0 is performed. However, to account for thefk (a;

0-doubling,; J0 0 ; ; J ) de…ned in equation 5.7 has to be replaced by expression 5.10. In

fact, this modi…cation does not interfere with the just-mentioned assumption because
the coupling selection rules for

J = J0J are di¤erent for "0 = +1 andthe combined e¤ect adds no extra J 0 dependence to the modi…ed fk (a;01; hence; J0 0 ; ; J ).

After summations on I 0 and J 0 and reorganizing various terms according to theirangular momentum dependence, the explicit expressions for the four e¤ective para-

116
meters are obtained and are listed below [16].
CB = CD"p4
Ia (Ia + 1) (2Ia + 1)

X X f0 (1u ; 0+ ) f1 (a; 0+ ; 1u ) 2uu+
E0+u J E1u 0 J
0
1u

r

#
+
f0 (1u ; 0+u ) f1 (a; 0u ; 1u )dE1u J p;
2~2 E0
Eu E0+u J E1u J
Z 1(5.17)

2
3Ia (Ia + 1) (2Ia + 1)"
X
X jf1 (a;0i (1 + j" j)
E0+u J
0
0

B =

0

2

2
; 0+
u )j
+
E 0 0Jr

Z 1

2

jf1 (a; 0 ; 0+
u )j
0J p
dE
2
2~ E0
Eu E0+u J E 0 J#;(5.18)dB =1
3Ia (Ia + 1) (2Ia + 1)

X02

eqQB = eqQ0

i

r

3

02

Z 1

2~2 E00

2 (1 + j" j)"X jf1 (a;0

E0+u J2dE

0J

jf1 (a; 0 ; 0+u )jp
Eu E0+u J E 0 J

2

0

; 0+
u )j
+
E 0 0J#;(5.19)"+
p X X 4f0 (1u ; 0+u ) f2 (a; 0u ; 1u )+4 2
E0+u J E1u 0 J
0
1u
#
r
Z 1
+
2
;
1
)
4f0 (1u ; 0+
)
f
(a;
0
u
2
u
u
+
dE1u J p
2~2 E0
Eu E0+u J E1u J"
X X
X 2f (a; 0 ; 0+ ) fk0 (a; 0 ; 0+ )
0
u
u
k
10
( 1)2Ia + (1 + j"0 j)
+
0 0J
+
E
E
0
0
0
0
J
u
k ;k=1;2
#
r
Z 1
0 +
0
2
2fk (a; 0 ; 0+
)
f
(a;
;
0
)
k
u
u
p
dE 0 J
2~2 E0
Eu E0+u J E 0 J
98
9
0
18
>
>
>
>
0
k
2 C < Ia 2 Ia = < 2 Ia Ia =B k:
(5.20)
@
A
>
>
>
0 >
:
;
:
;
0
k Ia k0 Ia
Ia

CD and eqQ0 are the …rst-order contributions to the corresponding parametersin equations 5.17 and 5.20, respectively. In equations 5.18 – 5.20,
can take on values of 0 ( 0 = 0) and0> 0 and "0

1 ( 0 > 0). For each perturbing electronic

117
0

state

,

1) if

i = 1(

0has g(u) symmetry and its contribution to the hyper…neparameters comes from both discrete levels ( 0 , J) and the energy continuum EJ ,q
whose density of state is 2 2~2 Eu , where Eu is the radial kinetic energy. E0 , thelower limit of the integrals for the energy continuum, is the energy just above the last
discrete rovibrational level that an attractive potential can support. In the equations
for0
B , dB , and eqQB , the extra factor (1 + j" j) arises from the0

perturbing state
5.3-doubling of the.

Preparation of Potential Energy Curves and Electronic Wave
Functions
We move on now to several computational issues concerning the preparation of

the PECs, the electronic wave functions constructed from the separated-atom basis
set, and the electronic matrix elements. The information is used in the calculationof the various electronic matrix elements in equations 5.17 - 5.20.

5.3.1

Potential Energy Curves

We use the available empirical and theoretical I2 PECs to construct the rovibrational structure and the associated wave functions inside each electronic state. The
B-state empirical PEC is adopted from the analysis performed by Gerstenkorn et al.
[133]. This empirical PEC was constructed from Fourier spectroscopic data by using
an “inverted perturbation approach” (IPA) method. For the six states that perturb
the B state, we adopt the empirically improved relativistic PECs obtained by deJong et al. [110], and these PECs are used in the short and intermediate internuclear
separations (R

6:5 Å). Figure 5.1 plots these PECs over the range 2:5 Å < R < 6:5

Å, along with the B-state empirical PEC and the other three PECs belonging to thesame dissociation limit. For the large internuclear separations (R > 7 Å), the PECs
C5
C6
take the inverse-power form, V (R) = R
5 + R6 + Ediss , which includes quadrupolar

118
electrostatic and dispersion energy terms, with the corresponding coe¢ cients C5 andC6 determined by Saute et al. [109]. Figure 5.2 shows these long-rang PECs for all
ten electronic states that share the same dissociation limit, 2 P 3 +2 P 1 . To make a2

2smooth transition between the two segments of the PEC, each short-range PEC isshifted vertically to match the corresponding long-rang PEC at around R = 7 Å.
The matched short- and long-range PECs are illustrated in …gure 5.3.

5.3.2

Electronic Wave Functions in the Separated-Atom ModelAt large internuclear separations (R >7 Å) the iodine molecule is treatedas two separated atoms with their charge distributions barely distorted. In thisseparated-atom model the molecular basis functions can be constructed from symmetrized products of two atomic wave functions:
j i=

X
l

Cl jjl ml ia jjl0 m0l ib(5.21)

where a and b denote two iodine atoms. A subset of these basis functions relevant to
the B and six perturbing states [C:1u , 1u , 1g (13g ), 1g (3g ), 0g (g ), and (3)0u ] is

1
2

1 1
;
2 2

+ 21 ; 12

3
;
2

1
2

];

listed below [109].
1 3 1
0+
u = 2[ 2; 2

1
;
2

1
2

+ 12 ;

j10u i = p12 [ 32 ; 21

1 1
;
2 2

+ 12 ; 21

j100u i = p12 [ 32 ; 231;2

10g = p12 [ 32 ; 21

1 1
;
2 2

100g = p12 [ 32 ; 23

1
;
2

1
2

1
2

+ 21 ;
1 1
;
2 2

1
2

1
;
2

3 1
;
2 2
3 1
;
2 2

];

1
2

3 3
;
2 2

3 1
;
2 2

];

1
2

3 3
;
2 2

+ 32 ;];];= 21 [ 23 ; 21

1
;
2

1
2

1
;
2

1
2

3 1
;
2 2

3
;
2

1
2

1 1
;
2 2

+ 12 ; 12

3
;
2

1
2

];

0u = 21 [ 23 ; 21

1
;
2

1
2

+ 12 ;

1
2

3 1
;
2 2

3
;
2

1
2

1 1
;
2 2

1 1
;
2 2

3
;
2

1
2

]:

0g

(5.22)

Each electronic wave function is a linear combination of the basis functions given
in equation 5.22. For example, the wave functions of two 1u states, 1u (3+u ) and119

1u( 3 S +u )4800

C:1u(3 S+u )Energy (cm -1)

4600

-

(3)0u
2 g ( 1 ∆g )

4400
4200

2 u ( 3 ∆u )-1 g ( 3 Σg )4000

1g (1Pg )

3800
3600

0 g- (3 Pg )(3)0+g3400

B:0 +u (3 Pu )

3200
3

4

R (A)

5

6

Figure 5.1: Potential energy curves (PECs) for ten electronic states converging to
the 2 P 3 + 2 P 1 dissociation limit over the range 2:5 Å < R < 6:5 Å. These PECs
2
2
are adopted from the relativistic calculations by de Jong et al [110] except for the B
state, for which an empirical PEC [133] is used.

120(3)0+g4380

2g

Energy (cm -1)

0g- ; 0u-

4370

«1u′ » [1u(3Σ+u)]2u«1u" »«1g′ » [1g(1Πg)]«1g" »

4360

B:0+u

4350

7

8

9

R (A)

10

11

12

Figure 5.2: Long-range PECs for the ten electronic states. These PECs are adopted
from reference [109]. The dashed line indicates the 2 P 3 + 2 P 1 dissociation limit. Note223that the 100u and the 100g states refer to the C:1u(3 +u ) and the 1g(g ) states,
respectively.

121

4400

«1u' »(3)0+g1u( 3 S +u )Energy (cm -1 )4350

«1g' »
0 g- (3 Pg );
(3)0u

4300

-

1g(3Σg); C:1u(3 S+u )1g (1Pg )

4250
4200

B:0 +u (3 Pu )

4150
4

5

6

R (A)

7

8

9

Figure 5.3: Combining the short- and long-range potentials. The solid lines are theshort-range (R < 7 Å) PECs from the ab initio calculation [110], and the dashedlines are the ab initio long-range PECs from reference [109]. To make a smoothconnection between two segments of the PEC, each short-range (R < 7 Å) PEC has
been shifted vertically to match the corresponding long-range one. The unchanged
B-state empirical PEC [133] and its ab initio long-range PEC are also included.

122
C:1u (3+u ), also designated as

10u

100u

and

respectively, are linear combinationsof two basis functions:

with

p

j 10u

i=

j10u ij 100ui=

j100u i +p

1

2 j100 iu

1

2 j10 iu(5.23)

being the admixture of the two basis functions. Likewise, the wave functionsof two 1g states, 1g (1g)

and 1g (3g ),are also linear combinations of two basis

functions:

where

10g

and

100g

10g

=

10g +

100g

=

100g

refer to the 1g (1

p
1
p
1

2

100g

2

10g ;

(5.24)

3g ) and the 1g (g ) states, respectively.

The R dependence of the mixing parameters

andwill be examined in section 5.5.

3Each of the wave functions of the other three electronic states, B0+u(3u ), 0g (g ),and (3)0u , consists of only one basis function.
5.3.3

Hyper…ne Matrix Elements in the Separated-Atom Model

With molecular wave functions derived from the basis functions in the separatedatomic model, the hyper…ne matrix elements, hV 0 i, hV 1 i, and hV 2 i ; can be readily
determined from the experimental values of atomic hyper…ne constants [46, 44]. In
this section two 1u basis wave functions are used to demonstrate the evaluation ofvarious hyper…ne matrix elements. A complete list of values of those matrix elements
between the B and six perturbing electronic states are given in reference [44].

5.3.3.1

Gyroscopic Matrix Elements hV 0 i

The matrix element of V 0 between the B state and a 1u state with-doubling"0 = 1 is given by equations 5.15 and 5.16. In Born-Oppenheimer approximation, f0

123
in equation 5.16 be rewritten as,1~2
p=
h J0 0 j 2 h1u (f0 1u ; 0+uR
2 2where 1u (0= 1) can be either theelectronic matrix element h1u (

0

10u

0

= 1)j (L+ + S+ ) 0+u

[1u (3+u )] orj J0 0 i ;[C:1u (3100u(5.25)+u )] state. The= 1)j (L+ + S+ ) j0+
u i in equation 5.25 is in general

R dependent and can be evaluated by rewriting the operator L+ + S+ as(5.26)

L+ + S+ = j+ = j+ (a) + j+ (b);where j is the total angular momentum of the electrons in the molecular frame and it
is decomposed to j(a) and j(b), which are the electronic angular momenta of the two
separated iodine atoms a and b, respectively. Then it is straightforward to write down
the matrix elements hV 0 i for the two 1u basis wave functions in the separated-atommodel:
h10u j V 0 0+u =h100u j V 0 0+
u =

p
~2
1
3 2J(J + 1) p h J0 0 j 2 j J i ;
R
2 2
2
p p
~
1
3 2J(J + 1) p h J0 0 j 2 j J i :R
2 2

According to equation 5.23, the matrix elements hV 0 i for the(5.27)

10u

100u

and

states are simply the linear combinations of the two matrix elements in equation
5.27:
h 10u

h 100uj V 0 0+
u =j V 0 0+u =p
~2
2J(J + 1) p
2 2

3 h J0 0 jp

3 h J0 0 j~2
2J(J + 1) p
2 2

R

j Ji
2

p

3 h J0 0 j

1 p
1
R2

2j

Ji

(5.28)
1 p
1
R2

2j

Ji +

p

3 h J0 0 j

R2j Ji :(5.29)

Hyper…ne Magnetic Dipolar hV 1 i and Electric Quadrupolar

5.3.3.2

Matrix Elements hV 2 i
The hyper…ne magnetic dipolar matrix element h
in f1 (a;

0

;

0 0
1
J0 j V(a) j

J i contained; J0 0 ; ; J ) can be approximated as a product of the Franck-Condon over-124
lap and an electronic matrix element, namely,h1
0 0
J0 j V(a) j

Ji

h J0 0 j0Ji hj V 1 (a) j i ;(5.30)

assuming that the latter has a weak R dependence over the range sampled by thevibrational wave function. Here we use the 10u basis state and the B state to show
how the electronic matrix element h10u j V11 (a) j0+
u i is related to the experimentally
determined atomic hyper…ne constants. In the separated-atom model, h10u j V11 (a) j0+
ui
is evaluated as
1
1
3
3 1
p
h10u j V11 (a) 0+
u = 2 2 [ 2 ; 2 V1 (a) 2 ;

1
2

1 1
;
2 2

+

V11 (a) 12 ;

1
2

];

(5.31)

while the atomic magnetic hyper…ne constant Aj is de…ned as [145]1
n gV1 (a)Aj = hj; m = jjjjj; m = ji :(5.32)To relate hV 1 i to the atomic hyper…ne constant Aj , we apply the Winger-Eckart
theorem to both equations 5.31 and 5.32. Equation 5.31 is then expressed in termsof reduced matrix elements:
1 p1
h10u j V11 (a) 0+
u = 2 [ 1532V 1 (a)

3
2

+ p16

1
2

V 1 (a)1
2];(5.33)and Aj is also related to the same set of reduced matrix elements:01
B j 1 j C
1
Aj = j @A hjk
n gV (a) kji ;j 0 j
A 3 = p115
2q
A 1 = 23
2

3
2

n gV

1

(a)

3
2

;

1
2

n gV

1

(a)

1
2

:

(5.34)

By comparing equations 5.33 and 5.34 the matrix element h10u j V11 (a) j0+
u i is determined by,
h10u j1
n gV1 (a)

0+
u =

1
(2A 3 + A 1 );2
2
4

(5.35)

125
where A 3 = 827:127 MHz and A 1 = 6587:34 MHz [146]. Similarly, the matrix22element h100u j V11 (a) j0+
u i is found to be
h100u j1
n gV1 (a)

0+
u

=

p

3
A3 :
4 2

(5.36)The matrix elements hV 1 i for other basis wave functions can be evaluated in the sameway. With the same method, the electric quadrupolar matrix elements hV 2 i can be
related to the atomic electric quadrupolar parameter jej q 3 QIa = 1146:356 MHz [147].
2

5.4

Computing the E¤ective Hyper…ne Parameters
FORTRAN codes LEVEL [132] and BCONT [148] developed by Le Roy et al.

are used to calculate the Franck-Condon integrals h J0 0 j

1
0J i and h J 0 j R2 j J i for thediscrete levels and energy continua, respectively. The rovibrational energies obtained
from LEVEL are also used to compute the energy denominators in equations 5.17
- 5.20. As an example, a portion of the B-state rovibrational structure is sketchedin …gure 5.4, and the vibrational levels inside a perturbing state C:1u are shown in…gure 5.5. A special case arises when a potential can support rotational levels above
its asymptotic energy because of a potential barrier located at the PEC outer limb.
Wave functions of such levels can tunnel through the potential barrier. As a result,
their energy spectra can be broadened signi…cantly and can no longer be treated as
discrete or quasi-discrete levels. We isolate these levels from the rest of the discretelevels and treat them instead as part of the energy continuum.
5.5

Numerical Analysis of E¤ective Hyper…ne Parameters and Related Electronic Structure
In this section we present results for the calculated e¤ective hyper…ne para-

meters, eqQB , CB , dB , andB , together with a detailed analysis of the electronic

structure of the associated states. The calculation covers nearly all the vibrational

126

J'=83

-1

Energy (cm )

4350
4300

u '=60, J'=83

4250

u '=60, J'=0
u '=58u '=57

4200

J'=0

3

4

5
R (A)

6

7

Figure 5.4: The B-state rovibrational structure. The thick solid curves are theempirical B-state rotationless potential [133]. Vibrational levels 0 = 57 68 areindicated as dotted lines. The rotational levels belonging to several vibrational levels
are also drawn as short bars. The line of long dashes is the e¤ective potential (asum of the rotationless and centrifugal potentials) associated with a rotational level( 0 = 60, J 0 = 83). The probability distributions of the molecular vibration forlevels ( 0 = 60, J 0 = 0) and ( 0 = 60, J 0 = 83) are plotted as thin solid curves forcomparison of the internuclear separations sampled by the two levels.

127«1u' »

4400-1

Energy (cm )

4350
4300

Long-range PECs

4250
4200
4150

«1u" » ( C:1u )

4

5

+

B:0u

6

R (A)

7

8

Figure 5.5: Two 1u PECs used in the calculation of CB . Also plotted is the B state
empirical potential [133], which is used for all internuclear separations. C:1u and 1u
PECs (R < 7 Å) are constructed from theoretical [110] (R < 7 Å) and long-range
PECs [109] (R > 7 Å). Those from ab initio calculation are shifted vertically to matchthe corresponding long-range one. The dotted lines indicate the vibrational levels of
C:1u state. In the literature, the 1u and C:1u states are also denoted as 10u and100u , respectively.

128
levels (3 606 82) in the B state with Rc extending from

3:5 Å to12:5 Å.

These results are discussed in three steps to demonstrate the maximum separation of the contributions from di¤erent perturbing electronic states. First, adetailed analysis of CB is required for the determination of the admixture of thetwo 1u states. Then, through the calculation of

B , contributions from the othertwo states (0u and 0g ) are separated from the rest (two 1g states), and the corresponding long-range PECs are tested. Finally, we present results for eqQB and dB ,which are in part based upon the information derived in previous two steps. Additionally, to demonstrate the numerical analysis on a …ner energy scale, we givethe calculated rotational dependence of the hyper…ne parameter CB for several vibrational levels. New information of electronic structure is inferred at each step of
this numerical analysis by comparing the result of the calculation with the highprecision spectroscopic data. In these comparisons we incorporated, to the bestof our knowledge, all available experimental data reported in previous publications[55, 42, 45, 117, 40, 120, 122, 123, 124, 125, 126, 127, 128, 129, 140, 141].

5.5.1

Spin-Rotation Parameter CB and the Admixture of Two 1u
States

The states that contribute to CB are restricted to the 1u states by the coupling
selection rule associated with the o¤-diagonal rotational Hamiltonian V 0 . Thus thecalculation of CB automatically isolates the second-order contributions of the two
1u states from those of the other four perturbing states. Furthermore, this allows adetermination of the mixing parameter

between the two 1u states that in turn canserve as a consistency check of the ab initio calculation of the PECs based on theseparated-atom model. The details of the PECs and molecular wave functions usedin equation 5.17 have already been described in section 5.3. Here we stress that in
the calculation the same value for mixing parametercan be used at a wide range

129
of internuclear separations, especially in the intermediate and long range, and thisvalue is adjusted until a global agreement with experimental data is achieved.
Figure 5.6 summarizes the comparison between the calculation and the experiment. The R-centroid used for the horizontal axis is the averaged internuclear separations weighted by the B-state vibrational wave function (not by the wave functionof the perturbing electronic state). In the …gure, open squares, circles, and diamonds
are calculations performed with the mixing parameter= 0:9, 0:9, and 1, respec-tively. The experimental data are also plotted as solid circles. First, the result showsa sensitive dependence on the mixing parameterR >at large internuclear separations

7 Å, as shown in …gure 5.6 (a). In contrast, we verify that the uncertainty

in the two 1u PECs has merely a small in‡uence on the calculation for large R. For
instance, we intentionally decrease the potential depths of the two 1u states by 50
cm 1 and 30 cm 1 , respectively, and repeat the calculation. While this modi…cation
results in a noticeable reduction of CB between 4:0 Å and 5:5 Å as indicated bytriangles in …gure 5.6 (b), it induces less than a 2% change of CB at Rc >7 Å.

Additional diagnostic tests on the di¤erent parts of the two 1u PECs produce similar
outcomes. Consequently, the mixing parameteris essentially constant and takes

the value of 0:99 for internuclear separation R > 7 Å. Moreover, a calculation using
the same

at even shorter Rc is in good agreement with the experiment until Rc

reaches 4 Å, as shown in panels (b) and (c) in …gure 5.6. On the other hand, a close
examination of CB at Rc < 4 Å in …gure 5.6 (c) shows that the calculated results
gradually deviate from the experiment, quantitatively illustrating a transition fromthe separated-atom model to the overlapped and distorted charge distribution that
calls for a description employing more sophisticated molecular wave functions.
Figure 5.7 plots the calculated rotational dependence for two vibrational levels,0= 70 and0= 47, with the mixing parameter= 0:99. Given a large energydi¤erence between the two levels and a …xed mixing parameter, the agreement with

130

CB (MHz)

20

Experimental data
Calculation (a = 0.99)
Calculation (a = 0.9)

15
10
5

(a)

0
4
2.0

CB (MHz)

1.5

6

8

Experimental data
Calculation
Calculation. 1u potential
depthes reduced by 50
and 30 cm -1, respectively.

10

12

a = 0.9
a=1

1.0

a = 0.99
0.5

4.0

CB (MHz)0.3(b)a = 0.99

4.5

5.0

Experimental data
Calculation
Calculation. 1u potential
depthes reduced by 50
and 30 cm -1, respectively.

5.5

6.0

a = 0.9

a = 0.99

0.2

0.1

(c)
0.0
3.2

3.4

3.6
3.8
R-centroid (Å)

4.0

4.2

Figure 5.6: Second-order calculations of the spin-rotation parameter CB . (a) Witha mixing parameter = 0:99 for two 1u states, the calculations agree well with theexperimental data, while changing to 0.9 results in a large global deviation fromthe experimental data. (b) A deliberate modi…cation of the two 1u PECs produces
a noticeable discrepancy at Rc = 4.2 – 5.5 Å. (c) At Rc <4 Å, the calculation
begins to depart from the experimental data.

131
the experimental data is reasonably good on both vibrational levels. Similar to theglobal trend exhibited in the …gure 5.6, The rotational dependence of CB is due
to smooth variations of the Franck-Condon overlap and the energy denominator inequation 5.17, which is also responsible for the perturbation-induced rovibrational
dependence in the other three hyper…ne parameters.
When dealing with high-lying levels of the B state, we estimate the possible
contributions beyond the second-order perturbation. The strength of the perturbation is characterized by the parameter, whereis the coupling between

the two discrete levels inside the B and the 1u states, andE is their energy di¤er-=

ence. Numerical estimation shows thatE

is well below 1 for these high-lying levels.

Moreover, when there is a possibility of energy coincidence, the coupling

betweenthe two levels involved is veri…ed to be below a few megahertz. Given this couplingstrength,

E has to be less than a few tens of megahertz for a strong perturbation to

occur, which is unlikely because of the large mismatch in the rovibrational constantsof the B and the 1u states. Thus second-order perturbation theory is su¢ cient forthe calculation of CB and the analysis of the related 1u states. However, for theother three hyper…ne parameters precautions should be taken because the B state
can be strongly coupled to a discrete level in a particular electronic state. This strong
coupling will be discussed in section 5.5.3.

5.5.2

Separation of Contributions from 0g (3g ) and (3)0uStates

While only two 1u states contribute to CB , the other three parameters,

B,dB , and eqQB , contain perturbations from all six perturbing states. In section 5.5.1
we determined the mixing parameterassociated with the two 1u states, and thesensitivity of the CB to the related PECs was investigated. With the help of thisinformation, we decompose the hyper…ne parameter

B to contributions from several

groups of perturbing states, or even from individual perturbing states when it isCB (MHz)

132

Experimental data
Calculation

2.4

u '=70

2.2

u '=69
2.0

0

2000

3000

u '=490.32

CB (MHz)

1000

0.30

u '=470.28
u '=45

0.26
0

2

4

6

8

J'(J'+1)

10

3

14x10

Figure 5.7: Rotational dependence of CB around two vibrational levels 0 = 47 and70. Solid circles and lines are experimental data and …t, respectively. Empty circles
are calculation by second-order perturbation theory. The dotted lines are …ts to thecomputed points.

133
possible. This decomposition allows a sensitive characterization of the 0g and the 0u
states and their long-range potentials.
We divide the six states into four groups: two 1u states, the 0g state, the 0u
state, and two 1g states. Each of the three hyper…ne parameters, namely eqQB , dB ,and

B ; can be decomposed into terms that correspond to the contributions fromthese four groups of electronic states. For instance, the scalar spin-spin parameter
B can be written as

B =(1u ) + (1g ) +0g +From equations 5.18 and 5.19, it follows that dB is closely related to
dB =

1
1
(1u ) +
(1g )
2
2

0g

(5.37)

0u :

0u :

B by

(5.38)

Using equations 5.37 and 5.38, we can isolate the contributions of the 0g and the 0u
states from those of the other four states by
0g +

0u

=

1
( B
3

2dB ) ;

(5.39)

where the right-hand side can be determined experimentally by using the valuesofBand dB extracted from the spectroscopic …t. The left-hand side of equation

5.39 is calculated and plotted (open squares on dashed curve) in …gure 5.8 (a). For
comparison we also plot the experimental values (solid circles) according to the righthand side of equation 5.39. Recall that in the calculation there is no adjustableparameter for the electronic wave functions of the 0g and the 0u states. As can beseen in …gure 5.8 (a), there is a noticeable discrepancy between the experiment andthe calculation, which increases with the increasing Rc .
To resolve the source of this discrepancy, the contributions from 0g and 0u need
to be further distinguished from each other. We notice that the last term in equation
5.20 for eqQB dominates when k and k 0 equal 1. Thus one can keep only these terms

134

Experimental data
Calculation
Calculation with
0u long-range PEC
replaced by that of 2g

-

15
10

-

d (0g ) + d(0 u ) [MHz]

20

5

(a)

0

d (1u ) - 2 d(0 u- ) [MHz]0-5Experimental data
Calculation
Calculation with
0u- long-range PEC
replaced by that of 2g

-10
-15
-20
-25

(b)

d (1u ) + 2 d(0 g- ) [MHz]20

Experimental data
Calculation

15
10
5

(c)

0
4

6

8

10

R-centroid (A)Figure 5.8: Separation of the contributions from the 0u and 0g states. The overall
contribution of the 0u and 0g states to B is plotted in (a), while the two separated
contributions are plotted in (b) and (c), respectively. Solid circles are values deduced
from the experimental data on B , dB , and eqQB [see equation 5.39, 5.41, and text
for detail]. Open squares on dashed line represent the calculation; open circles onsolid line are the calculation with the 0u state long-range PEC replaced by that of2g state (see …gure 5.2 for the two long-rang PECs). In this calculation, there is no
adjustable parameter for the electronic wave functions.

135
and compare them with equation 5.18. This leads to the following relation between
eqQB and

B:

1
[eqQB
20

1
1
(1u ) +
(1g )2
2

eqQ0 ] t

0g +

0u ;

(5.40)

where eqQ0 is the …rst-order contribution, which is calculated in the separated-atombasis set at large internuclear separations. From equations 5.37, 5.38, and 5.40, thecontributions from 0u and 0g are isolated from each other by
(1u )

2

0u

= dB

(1u ) + 2

0g

=

2
B
3

1
(eqQB eqQ0 ) ;
20
1
1
dB
(eqQB eqQ0 ) ;3
20(5.41)

where the right-hand side again can be determined experimentally by using the values ofB , dB , and eqQB extracted from the spectroscopic …t.

we have determined the mixing parameterIn the section 5.5.1,of two 1u states. This information isincorporated into the calculation of the left-hand side of equation 5.41 that also involves the contribution from two 1u states. In …gure 5.8, panels (b) and (c) show
our results for the 0g (open squares on dashed curve) and 0u (open circles on solid
curve) states, respectively, along with the experimental data (solid circles) plotted inboth panels according to the right-hand side of equation 5.41. While the calculation
for 0g agrees with experimental data, a discrepancy exists for the 0u state that is
consistent with the discrepancy observed in …gure 5.8 (a). Evidently, the calculationof 0u is the source of this discrepancy.
We perform additional diagnostic calculations of (0u ). We verify that moderate modi…cations to the short- and intermediate-range potential of the 0u statecannot be responsible for the large discrepancy at Rc >8 Å, as shown in …gure 5.8(b). Conversely, (0u ) is sensitive to the C5 and C6 parameters of the 0u long-range
potential. In fact, at large Rc , the magnitude of (0u ) has a steep increase that is
dominated by the perturbations from the last few discrete levels in the 0u state whose136
energies depend sensitively on the shape of long-range potential. We therefore adjust
the C5 and C6 parameters of the 0u long-range potential to compensate for the error
in the calculation of (0u ). The modi…ed long-rang potential is very close to the 2glong-range potential as shown in …gure 5.2. In light of this observation, we repeat thecalculation of (0u ) with the 0u long-range potential replaced by that of the 2g state.
These new results (open circles on solid curve) are also presented in …gures 5.8 (a)and (b). The discrepancy is greatly reduced in comparison with the dashed curves
using the original 0u long-range potential. We also estimate the coupling-strength
introduced in section 5.5.1 for each discrete level in the 0u state to con…rm thevalidity of the above analysis using second-order perturbation theory.

5.5.3

Strong Perturbation from the 1g (1g ) State and its Contribution

to the E¤ective Hyper…ne Parameters

B , dB , and eqQB

In section 5.5.2 we divided six perturbing states into four groups, and thecontributions of three groups to the e¤ective hyper…ne parameters were analyzed
in detail. Once the contributions from the last two 1g states are determined, thethree hyper…ne parameters

B,dB , and eqQB can be calculated by summing up

contributions from the four groups of perturbing states.
Among the two 1g states, the outer branch of the 1g (1g ) PEC is close to thatof the B state (see …gures 5.1 and 5.2), resulting in a strong hyper…ne coupling tohigh-lying levels (70 60B

6 82) of the B state through favorable Franck-Condon

overlaps [45]. Furthermore, this perturbation is enhanced because for each high-lying
level in the B state, there is a nearby discrete level in the 1g (1g ) state whose energy

separation with the perturbed level is typically only a few gigahertz. In calculating
the contribution of the 1g states, we monitor the coupling strength between the B
and the 1g (1g ) states for these high-lying levels. For0
B 6 78, the parametercoupling strength is veri…ed to be < 0:1, whereas at the last four levelsof the0
B = 7982,

137
can be as large as 0:4. Therefore, the second-order calculation of the contribution
from the 1g (1g ) state is primarily restricted to levels below

0
B = 78.

Despite the inadequacy of the second-order calculation at levels

0
B > 78, thestrong perturbation allows a high-resolution examination of the 1g (1g ) long-rang

potential. We …nd that only a minor adjustment on the 1g (1…gure 5.2) is needed for the calculation ofBg ) long-range PEC (seeto achieve good agreement with theexperimental data, as shown in …gure 5.9. In the calculation, the value of the mixing
parameter

for the two 1g states is taken from reference [45] and kept …xed.

Considering the linear independence of equations 5.37, 5.38, and 5.40, the calculations of dB and eqQB provide a cross-check for the modi…ed 1g (1PEC and the mixing parametersand . Based on this observation, we proceed toperform the calculations of dB and eqQB using the modi…ed 1g (1and the mixing parametersg ) long-rangeg ) long-range PEC

and . The hyper…ne parameter dB is calculated using

equation 5.38. In the case of eqQB , our calculation includes both major contributions
in equation 5.40 and other minor terms in equation 5.20. The results for dB , andeqQB agree reasonably well with experimental data, as shown respectively in …gures
5.10 and 5.11. Compared with dB , our calculation of eqQB has a relatively large
discrepancy because the residual error of (1g ) is ampli…ed by a factor of 20 in thecase of eqQB according to equations 5.38 and 5.40. Once the mixing parameterof the two 1g states is independently determined, the calculations of dB and eqQB
can be further improved by a global …t of the 1g (1hyper…ne parameters
5.6g ) long-range PEC to all three

B , dB , and eqQB .

First-Order Electric Quadrupole Interaction
Unlike CB , B , and dB , which result solely from perturbations given by external

electronic states over the whole range of internuclear separation, eqQB has both …rstorder and second-order components that are important in di¤erent regions. Our

138

100

dB +0.02 (MHz)

10

1

0.1

Experimental data
Calculation (a = 0.99, b = 0.84)

0.01
4

6

8
R-centroid (A)1012Figure 5.9: Semilog plot of the second-order calculation of the scalar spin-spin parameter B . Solid circles are experimental data and open circles represent calculations.
In the calculation the mixing parameters for the two 1u and the two 1g states are …xed
to = 0:99 and = 0:84, respectively. Because the contribution from the 1g (1 g )state is extremely sensitive to the corresponding long-range potential, the 1g (1 g )long-range potential is …ne-tuned to obtain good agreement with the experimental
data.

139

-dB (MHz)

10

1

0.1

Experimental data
Calculation (a = 0.99, b = 0.84)

0.01
4

6

8
R-centroid (A)1012Figure 5.10: Semilog plot of the second-order calculation of the tensor spin-spin parameter dB . Solid circles are experimental data and open circles represent calculations.
The calculation combines the contributions from all six states (see equation 5.38)
that are determined in the calculations of B . The calculation provides a cross-checkof the 1g long-rang PEC and the mixing parameters and .

140

-300

eqQB (MHz)-350

Experimental data
Experimental fit
Calculation (a = 0.99, b = 0.84)

-400
-450
-500
-550
4

6

8
R-centroid (A)1012Figure 5.11: Second-order calculation of the electric quadrupole parameter eqQB .
Solid circles are experimental data and open circles represent calculations. The calculation combines the contributions from the same six states used for …gures 5.9 and5.10. The calculation provides a cross-check of the 1g long-rang PEC and the mixing
parameters and . The relatively large discrepancy is due to the fact that theresidual error of (1g ) (the contribution of the 1g states to B ) has been ampli…ed bya factor of 20 in the case of eqQB (see equations 5.38 and 5.40).

141
numerical analysis makes it possible to locate more accurately the regions at which
each component dominates. As shown in …gure 5.11, at short internuclear separations(Rc < 5 Å), the perturbation from the external states is negligible. The perturbation
sets in at Rc

5 Å and increases along with the internuclear separation. Finally, it isworth noting that at Rc < 5 Å the …rst-order component of eqQB departs from theconstant value predicated by both SAB (separated-atom basis) and LCAO (linear
combination of atomic orbitals) models with the decreasing internuclear separation
because of a combined e¤ect of (1) the nuclear quadrupole-quadrupole interactionand (2) the distortion and overlapping of the electric charge distributions of two
iodine atoms.
5.7

Summary and Conclusions
We have performed second-order calculations of four hyper…ne parameters CB ,B , dB , and eqQB , using the available molecular PECs and electronic wave functions

derived from the separated-atom basis set. We have shown that by dividing thesix perturbing electronic states into four groups, their contributions to the hyper…neparameters can be separated and directly compared with the corresponding experimental data. For the spin-rotation parameter CB , the results for both vibrationaland rotational dependence agree well with the experimental data for a wide range ofinternuclear separations (Rc >4 Å), and the mixing parameter of the two 1u statesis independently determined as= 0:99. However, calculations depart from experi-mental data at short internuclear separations (Rc <4 Å), indicating the breakdownof the separated-atom model. Moreover, calculations of the other three hyper…ne parameters verify the mixing parameter of two 1g states and place a stringent constraint
on the long-range PECs of the relevant electronic states.
The numerical analysis of e¤ective hyper…ne parameters based on the separatedatom model is in remarkably good agreement with the experimental data over a broad

142
range of internuclear separations. This implies that the electronic structure of a large
number of B-state rovibrational levels, not limited to those high-lying ones ( 0 > 70),
preserves a long-range character.
The analysis presented in this chapter can also be applied to the gyroscopic
and hyper…ne predissociation of molecular iodine in the B state [66, 67, 61]. In
this decay channel, molecules in the B sate are predissociated through the repulsive
B 00 1u (1u ) state that converges with the ground state X

to the …rst dissociation

limit, 2 P 3 +2 P 3 . In particular, this new analysis may help to elucidate the source of22

the large discrepancy found in the calculation of the predissociation matrix element
C [149], which is related to the gyroscopic predissociation.

Bibliography[1] E. Peik, B. Lipphardt, H. Schnatz, T. Schneider, and Chr. Tamm, “Limit on
the present temporal variation of the …ne structure constant”, Phys. Rev. Lett.,
vol. 93, pp. 170801, 2004.
[2] H. Marion, F. Pereira Dos Santos, M. Abgrall, S. Zhang, Y. Sortais,
S. Bize, I. Maksimovic, D. Calonico, J. Grünert, C. Mandache, P. Lemonde,
G. Santarelli, Ph. Laurent, and A. Clairon, “Search for variations of fundamental constants using atomic fountain clocks”, Phys. Rev. Lett., vol. 90, pp.
150801, 2003.
[3] S. Bize, S. A. Diddams, U. Tanaka, C. E. Tanner, W. H. Oskay, R. E. Drullinger,
T. E. Paker, T. P. Heavner, S. R. Je¤erts, L. Hollberg, W. M. Itano, and J. C.
Bergquist, “Testing the stability of fundamental constants with the 199 H+gsingle-ion opttical clock”, Phys. Rev. Lett., vol. 90, pp. 150802, 2003.
[4] B. C. Regan, E. D. Commins, C. J. Schmidt, and D. DeMille, “New limit on
the electron electric dipole moment”, Phys. Rev. Lett., vol. 88, pp. 071805,
2002.
[5] J. J. Hudson, B. E. Sauer, M. R. Tarbutt, and E. A. Hinds, “Measurement of
the electron electric dipole moment using YbF molecules”, Phys. Rev. Lett.,
vol. 89, pp. 023003, 2002.
[6] D. Wilkening, N. Ramsey, and D. Larson, “Search for P and T violations in
the hyper…ne structure of thallium ‡uoride”, Phys. Rev. A, vol. 29, pp. 425,
1984.
[7] D. Cho, K. Sangster, and E. E. Hinds, “Tenfold improvement of limits on T
violation in thallium ‡uoride”, Phys. Rev. Lett., vol. 63, pp. 2559, 1989.
[8] T. W Hänsch and A. L. Schawlow, “Cooling of gases by laser radiation”, Opt.
Commun., vol. 13, pp. 68, 1975.
[9] E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. E. Pritchard, “Trapping of
neutral sodium atoms with radiation pressure”, Phys. Rev. Lett., vol. 59, pp.
2631, 1987.

144[10] Dalibard J. and C. Cohen-Tannoudji, “Laser cooling below the Doppler limit
by polarization gradients: simple theoretical models”, J. Opt. Soc. Am. B, vol.
6, pp. 2023, 1989.
[11] P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I.
Westbrook, “Optical molasses”, J. Opt. Soc. Am. B, vol. 6, pp. 2084, 1989.
[12] D. S. Weiss, E. Riis, Y. Shevy, P. J. Ungar, and S. Chu, “Optical molasses andmultilevel atoms: experiment”, J. Opt. Soc. Am. B, vol. 6, pp. 2072, 1989.
[13] T. Udem, J. Reichert, R. Holzwarth, and T. W. Hänsch, “Absolute opticalfrequency measurement of the cesium D1 line with a mode-locked laser”, Phys.
Rev. Lett., vol. 82, no. 18, pp. 3568, 1999.
[14] T. Udem, J. Reichert, R. Holzwarth, and T. W. Hänsch, “Accurate measurement of large optical frequency di¤erences with a mode-locked laser”, Opt.
Lett., vol. 24, no. 13, pp. 881, 1999.
[15] H. R. Telle, G. Steinmeyer, A. E. Dunlop, J. Stenger, D. H. Sutter, and
U. Keller, “Carrier-envelope o¤set phase control: A novel concept for absolute
optical frequency measurement and ultrashort pulse generation”, Appl. Phys.
B, vol. 69, no. 4, pp. 327, 1999.
[16] M. Broyer, J. Vigué, and J. C. Lehmann, “E¤ective hyper…ne hamiltonian
in homonuclear diatomic molecules. Application to the B state of moleculariodine”, J. Phys. (Paris), vol. 39, pp. 591, 1978.
[17] J. C. Bergquist, “Doppler-free spectroscopy”, in Atom, molecular, and opticalphysics: electromagnetic radiation, F. B. Dunning and R. G. Hulet, Eds., vol.
29B. Academic Press, San Diego, 1996.
[18] W. Demtröder, Laser spectroscopy Basic concepts and instrumentation,
Springer, Berlin, 1998.
[19] L. D. Landau and E. M. Lifshitz, Statistical physics, vol. 5 of Course oftheoretical physics, Butterworth-Heinemann, Oxford, 1980.
[20] T. Ido, T. H. Loftus, M. M. Boyd, A. D. Ludlow, K. W. Holman, and J. Ye,“Precision spectroscopy and density-dependent frequency shifts in ultracold
Sr”, arXiv:physics/0410110 v1, 2004.
[21] R.H. Dicke, “The e¤ect of collisions upon the doppler width of spectral lines”,
Phys. Rev., vol. 89, pp. 472, 1953.
[22] R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an opticalresonator”, Appl. Phys. B, vol. 31, no. 2, pp. 97, 1983.
[23] B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, “Visible lasers
with subhertz linewidths”, Phys. Rev. Lett., vol. 82, no. 19, pp. 3799, 1999.

145[24] C. Salomon, D. Hils, and J. L. Hall, “Laser stabilization at the millihertz level”,
J. Opt. Soc. Am. B, vol. 5, no. 8, pp. 1576, 1988.
[25] D. Hils and J. L. Hall, “Frequency standards and metrology”, in Frequency
standards and metrology, A. De Marchi, Ed. Springer-Verlag Berlin, Heidelberg, 1989.
[26] M. Zhu and J. L. Hall, “Stabilization of optical-phase frequency of a laser
system - application to a commercial dye-laser with an external stabilizer”, J.
Opt. Soc. Am. B, vol. 10, no. 5, pp. 802, 1993.
[27] W. R. bennet, Jr., “Hole-burining e¤ects in a He-Ne-optical maser”, Phys.
Rev., vol. 126, pp. 580, 1962.
[28] A. Szöke and A. Javan, “Isotope shift and saturation behavior of the 1.15transition of Ne”, Phys. Rev. Lett., vol. 10, pp. 521, 1963.
[29] R. A. MacFarlane, W. R. Bennett, Jr., and W. E. Lamb, Jr., “Single modetuning dip in the power output of an He-Ne maser”, Appl. Phys. Lett., vol. 2,
pp. 189, 1963.
[30] W. E. Lamb, Jr., “Theory of an optical maser”, Phys. Rev., vol. 134, pp.
A1429, 1964.
[31] C. Bordé, “Saturated absorption spectroscopy of various molecules using lasers
with carbonic gas and nitrogen protoxide”, Cr. Acad. Sci. B Phys., vol. 271,
no. 6, pp. 371, 1970.
[32] T. W. Hänsch, M. D. Levenson, and A. L. Schawlow, “Complete hyper…nestructure of a molecular iodine line”, Phys. Rev. Lett., vol. 26, no. 16, pp. 946,
1971.
[33] T. W. Hänsch, “High resolution laser spectroscopy”, in Advances in laser
spectroscopy, F. T. Arecchi, F. Strumia, and H. Walther, Eds., vol. 95. Plenum
Press, New York, 1981.
[34] G. C. Bjorklund, “Frequency-modulation spectroscopy - new method for measuring weak absorptions and dispersions”, Opt. Lett., vol. 5, no. 1, pp. 15,
1980.
[35] J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, “Optical heterodyne
saturation spectroscopy”, Appl. Phys. Lett., vol. 39, no. 9, pp. 680, 1981.
[36] J. L. Hall, H. G. Robinson, T. Baer, and L. Hollberg, “The lineshapes of subDoppler resonances observable with FM side-band (optical heterodyne) laser
techniques”, in Advances in laser spectroscopy, F. T. Arecchi, F. Strumia, andH. Walther, Eds., vol. 95. Plenum Press, New York, 1981.

146[37] J. Ye, L. S. Ma, and J. L. Hall, “Sub-Doppler optical frequency reference
at 1.064 m by means of ultrasensitive cavity-enhanced frequency modulation
spectroscopy of a C2 HD overtone transition”, Opt. Lett., vol. 21, no. 13, pp.
1000, 1996.
[38] J. Ye, L. S. Ma, and J. L. Hall, “Ultrastable optical frequency reference at
1.064 m using a C2 HD molecular overtone transition”, IEEE Trans. Instrum.
Meas., vol. 46, no. 2, pp. 178, 1997.
[39] J. Ye, L. S. Ma, and J. L. Hall, “Ultrasensitive detections in atomic andmolecular physics: demonstration in molecular overtone spectroscopy”, J. Opt.
Soc. Am. B, vol. 15, no. 1, pp. 6, 1998.
[40] T. J. Quinn, “Practical realization of the de…nition of the metre (1997)”,
Metrologia, vol. 36, pp. 211, 1999.
[41] J. Ye, L. Robertsson, S. Picard, L.-S. Ma, and J. L. Hall, “Absolute frequency
atlas of molecular I2 lines at 532 nm”, IEEE Trans. Instrum. Meas., vol. 48,
pp. 544, 1999.
[42] B. Bodermann, H. Knöckel, and E. Tiemann, “Widely usable interpolation
formulae for hyper…ne splittings in the 127 I2 spectrum”, Eur. Phys. J. D, vol.
19, pp. 31, 2002.
[43] J. P. Pique, F. Hartmann, R. Bacis, S. Churassy, and J. B. Ko¤end, “Hyper…neinduced ungerade-gerade symmetry breaking in a homonuclear diatomic molecule near a dissociation limit: 127 I2 at the 2 P3=2 - 2 P1=2 limit”, Phys. Rev. Lett.,
vol. 52, pp. 267, 1984.
[44] J. P. Pique, F. Hartmann, S. Churassy, and R. Bacis, “Hyper…ne interactions
in homonuclear diatomic molecules and u-g perturbations. I. theory”, J. Phys.
(Paris), vol. 47, pp. 1909, 1986.
[45] J. P. Pique, F. Hartmann, S. Churassy, and R. Bacis, “Hyper…ne interactions
in homonuclear diatomic molecules and u-g perturbations. II. experiments on
I2 ”, J. Phys. (Paris), vol. 47, pp. 1917, 1986.
[46] J. Vigué, M. Broyer, and J. C. Lehmann, “Ab initio calculation of hyper…neand magnetic parameters in the I2 B state”, Phys. Rev. Lett., vol. 42, pp. 883,
1979.
[47] R. Bacis, M. Broyer, S. Churassy, J. Vergès, and J. Vigué, “eQq measurements in the X, 1g , 0+g and B state of I2 : A test of the electronic moleculareigenfunctions”, J. Chem. Phys., vol. 73, pp. 2641, 1980.
[48] J. P. Pique, F. Hartmann, R. Bacis, and S. Churassy, “Hyper…ne structure ofhigher rovibrational levels in the iodine B state studied by Ar+ laser induced‡uorescence”, Opt. Commun., vol. 36, pp. 354, 1981.

147[49] S. Ezekiel and R. Weiss, “Laser-induced ‡uorescence in a molecular beam ofiodine”, Phys. Rev. Lett., vol. 20, pp. 91, 1968.
[50] M. D. Levenson and A. L. Schawlow, “Hyper…ne interactions in moleculariodine”, Phys. Rev. A, vol. 6, pp. 10, 1972.
[51] G. R. Hanes and C. E. Dahlstrom, “Iodine hyper…ne structure observed insaturated absorption at 633 nm”, Appl. Phys. Lett., vol. 14, pp. 362, 1969.
[52] A. Arie, S. Schiller, E. K. Gustafson, and R. L. Byer, “Absolute frequency
stabilization of diode-laser-pumped NdYAG lasers to hyper…ne transitions inmolecular iodine”, Opt. Lett., vol. 17, no. 17, pp. 1204, 1992.
[53] M. L. Eickho¤ and J. L. Hall, “Optical frequency standard at 532 nm”, IEEE
Trans. Instrum. Meas., vol. 44, no. 2, pp. 155, 1995.
[54] J. Ye, L. S. Ma, and J. L. Hall, “Molecular iodine clock”, Phys. Rev. Lett.,
vol. 8727, no. 27, art. no.-270801, 2001.
[55] Ch. J. Bordé, G. Camy, B. Decomps, and J.-P. Descoubes, “High precision
saturation spectroscopy of 127 I2 with argon lasers at 5 145 Å and 5 017 Å: main resonances”, J. Phys. (Paris), vol. 42, pp. 1393, 1981.
[56] A. N. Goncharov, M. N. Skvortsov, and V. P. Chebotayev, “Study of relaxationof molecular-iodine by the method of saturated absorption-spectroscopy”, Appl.
Phys. B, vol. 51, no. 2, pp. 108, 1990.
[57] R. E. Tench and S. Ezekiel, “Precision measurements of hyper…ne predissociation in I2 vapor using a two-photon resonant scattering technique”, Chem.
Phys. Lett., vol. 96, pp. 253, 1983.
[58] K. Shimoda, “Line broadening and narrowing e¤ects”, in High-resolution
spectroscopy, K. Shimoda, Ed. Springer, Berlin, Heidelberg, 1976.
[59] C. Teichteil and M. Pelissier, “Relativistic calculations of excited states ofmolecular iodine”, Chem. Phys., vol. 180, pp. 1, 1994.
[60] W. Y. Cheng and J. T. Shy, “Wavelength standard at 543 nm and the corresponding 127 I2 hyper…ne transitions”, J. Opt. Soc. Am. B, vol. 18, no. 3, pp.
363, 2001.
[61] J. Vigué, M. Broyer, and J. C. Lehmann, “Natural hyper…ne and magentic
predissociation of the I2 B state III. - experiments on magnetic predissociation”,
J. Phys. ( Paris), vol. 42, pp. 961, 1981.
[62] M. Lamrini, R. Bacis, D. Cerny, S. Churassy, P. Crozet, and A. J. Ross, “The+electronic transition dipole moment of the B0+
u ! X0g transition in iodine”,
J. Chem. Phys., vol. 100, no. 12, pp. 8780, 1994.

148[63] J. Tellinghuisen,“The electronic transition moment function for the
3
1 +
B0+
(
)
$X
transitionin I2 ”, J. Chem. Phys., vol. 106, no. 4, pp. 1305,
u
g
1997.
[64] P. Dubé and M. Trinczek, “Hyper…ne-structure splittings and absorption
strengths of molecular-iodine transitions near the trapping frequencies of francium”, J. Opt. Soc. Am. B, vol. 21, no. 6, pp. 1113, 2004.
[65] J. P. Pique, R. Bacis, F. Hartmann, N. Sadeghi, and S. Churassy, “Hyper…nepredissociation in the B state of iodine investigated through lifetime measurements of individual hyper…ne sublevels”, J. Phys. (Paris), vol. 44, pp. 347,
1983.
[66] J. Vigué, M. Broyer, and J. C. Lehmann, “Natural hyper…ne and magnetic
predissociation of the I2 B state I. - Theory”, J. Phys. ( Paris), vol. 42, pp.
937, 1981.
[67] J. Vigué, M. Broyer, and J. C. Lehmann, “Natural hyper…ne and magnetic
predissociation of the I2 B state II. - Experiments on natural and hyper…nepredissociation”, J. Phys. ( Paris), vol. 42, pp. 949, 1981.
[68] S. Gerstenkorn and P. Luc, Atlas du Spectre d’Absorption de la Moleculed’Iode 14 800 cm 1 - 20 000 cm 1 , Editions du CNRS, Paris, 1985.
[69] J. I. Steinfeld, Molecules and radiation, MIT Press, New York, 1986.
[70] K. M. Evenson, J. S. Wells, F. R. Petersen, B. L. Danielson, and G. W. Day,
“Accurate frequencies of molecular transitions used in laser stabilization 3.39m transition in CH4 and 9.33- and 10.18- m transitions in CO2 ”, Appl. Phys.
Lett., vol. 22, no. 4, pp. 192, 1973.
[71] H. Schnatz, B. Lipphardt, J. Helmcke, F. Riehle, and G. Zinner, “First phasecoherent frequency measurement of visible radiation”, Phys. Rev. lett., vol. 76,
no. 1, pp. 18, 1996.
[72] B. G. Whitford, “Phase-locked frequency chains to 130 THz at NRC”, inFrequency standards and metrology, A. De Marchi, Ed. Springer-Verlag Berlin,
Heidelberg, 1989.
[73] J. E. Bernard, A. A. Madej, L. Marmet, B. G. Whitford, K. J. Siemsen, andS. Cundy, “Cs-based frequency measurement of a single, trapped ion transitionin the visible region of the spectrum”, Phys. Rev. Lett., vol. 82, no. 16, pp.
3228, 1999.
[74] T. Udem, A. Huber, B. Gross, J. Reichert, M. Prevedelli, M. Weitz, and T. W.
Hänsch, “Phase-coherent measurement of the hydrogen 1S-2S transition frequency with an optical frequency interval divider chain”, Phys. Rev. Lett., vol.
79, no. 14, pp. 2646, 1997.

149[75] T. Udem, A. Huber, M. Weitz, D. Leibfried, W. Konig, M. Prevedelli, A. Dimitriev, H. Geiger, and T. W. Hänsch, “Phase-coherent measurement of thehydrogen 1S-2S frequency with an optical frequency interval divider chain”,
IEEE Trans. Instrum. Meas., vol. 46, no. 2, pp. 166, 1997.
[76] H. R. Telle, D. Meschede, and T. W. Hänsch, “Realization of a new concept
for visible frequency division: phase locking of harmonic and sum frequencies”,
Opt. Lett., vol. 15, no. 10, pp. 532, 1990.
[77] D. A. Jennings, K. M. Evenson, and D. J. E. Knight, “Optical frequency
measurements”, Proceedings of the IEEE, vol. 74, no. 1, pp. 168, 1986.
[78] S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundi¤, J. L. Hall, J. K. Ranka, R. S.
Windeler, R. Holzwarth, T. Udem, and T. W. Hänsch, “Direct link between
microwave and optical frequencies with a 300 THz femtosecond laser comb”,
Phys. Rev. Lett., vol. 84, no. 22, pp. 5102, 2000.
[79] D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall,and S. T. Cundi¤, “Carrier-envelope phase control of femtosecond mode-locked
lasers and direct optical frequency synthesis”, Science, vol. 288, no. 5466, pp.
635, 2000.
[80] J. Reichert, M. Niering, R. Holzwarth, M. Weitz, T. Udem, and T. W. Hänsch, “Phase coherent vacuum-ultraviolet to radio frequency comparison with amode-locked laser”, Phys. Rev. Lett., vol. 84, no. 15, pp. 3232, 2000.
[81] J. Ye, T. H. Yoon, J. L. Hall, A. A. Madej, J. E. Bernard, K. J. Siemsen, and
L. Marmet, “Accuracy comparison of absolute optical frequency measurement
between harmonic-generation synthesis and a frequency-division femtosecondcomb”, Phys. Rev. Lett., vol. 85, no. 18, pp. 3797, 2000.
[82] J. Reichert, R. Holzwarth, T. Udem, and T. W. Hänsch, “Measuring thefrequency of light with mode-locked lasers”, Opt. Commun., vol. 172, no. 1-6,
pp. 59, 1999.
[83] J. Ye and S. T. Cundi¤, Eds., Femtosecond optical frequency comb: principle,
operation, and applications, Kluwer Academic Publishers / Springer, Norwell,
2004.
[84] A. Bartels, T. Dekorsy, and H. Kurz, “Femtosecond Ti:sapphire ring laser witha 2-GHz repetition rate and its application in time-resolved spectroscopy”, Opt.
Lett., vol. 24, no. 14, pp. 996, 1999.
[85] S. T. Cundi¤, “Phase stabilization of ultrashort optical pulses”, J. Phys. D,
vol. 35, no. 8, pp. R43, 2002.
[86] M. T. Asaki, C. P. Huang, D. Garvey, J. P. Zhou, H. C. Kapteyn, and M. M.
Murnane, “Generation of 11-fs pulses from a self-mode-locked Ti-sapphirelaser”, Opt. Lett., vol. 18, no. 12, pp. 977, 1993.

150[87] R. L. Fork, O. E. Martinez, and J. P. Gordon, “Negative dispersion using pairsof prisms”, Opt. Lett., vol. 9, no. 5, pp. 150, 1984.
[88] R. Szipöcs, K. Ferencz, C. Spielmann, and F. Krausz, “Chirped multilayer coatings for broad-band dispersion control in femtosecond lasers”, Optics Letters,
vol. 19, no. 3, pp. 201, 1994.
[89] A. E. Siegman, Lasers, University Science Books, Mill Valley, 1986.
[90] D. E. Spence, P. N. Kean, and W. Sibbett, “60-fsec pulse generation from aself-mode-locked Ti-sapphire laser”, Opt. Lett., vol. 16, no. 1, pp. 42, 1991.
[91] T. W. Hänsch, “High resolution spectroscopy of hydrogen”, in The hydrogenatom, G. F. Bussani, M. Inguscio, and T. W. Hänsch, Eds. Springer, Berlin,
Heidelberg, 1989.
[92] K. F. Kwong, D. Yankelevich, K. C. Chu, J. P. Heritage, and A. Dienes, “400Hz mechanical scanning optical delay-line”, Opt. Lett., vol. 18, no. 7, pp. 558,
1993.
[93] L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch,“Route to phase control of ultrashort light pulses”, Opt. Lett., vol. 21, no. 24,
pp. 2008, 1996.
[94] A. Poppe, R. Holzwarth, A. Apolonski, G. Tempea, C. Spielmann, T. W. Hänsch, and F. Krausz, “Few-cycle optical waveform synthesis”, Appl. Phys. B,
vol. 72, no. 8, pp. 977, 2001.
[95] K. W. Holman, R. J. Jones, A. Marian, S. T. Cundi¤, and J. Ye, “Detailed studies and control of intensity-related dynamics of femtosecond frequency combs
from mode-locked Ti:sapphire lasers”, IEEE J. Sel. Top. Quant., vol. 9, no. 4,
pp. 1018, 2003.
[96] K. W. Holman, R. J. Jones, A. Marian, S. T. Cundi¤, and J. Ye, “Intensityrelated dynamics of femtosecond frequency combs”, Opt. Lett., vol. 28, no. 10,
pp. 851, 2003.
[97] K. W. Holman, D. J. Jones, J. Ye, and E. P. Ippen, “Orthogonal control of thefrequency comb dynamics of a mode-locked laser diode”, Optics Letters, vol.
28, no. 23, pp. 2405, 2003.
[98] A. H. Glattfelder and W. Schaufelberger, Control systems with input andoutput constraints, Springer, London, 2003.
[99] G. F. Franklin, J. D. Powell, and A. Emami-Naeini,
dynamic systems, Addison-Wesley, Reading, 1994.

Feedback control of[100] R. Holzwarth, T. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, andP. S. J. Russell, “Optical frequency synthesizer for precision spectroscopy”,
Phys. Rev. Lett., vol. 85, no. 11, pp. 2264, 2000.

151[101] B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler, and
G. L. Burdge, “Cladding-mode-resonances in air-silica microstructure optical…bers”, J. Lightwave Technol., vol. 18, no. 8, pp. 1084, 2000.
[102] J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generationin air-silica microstructure optical …bers with anomalous dispersion at 800 nm”,
Opt. Lett., vol. 25, no. 1, pp. 25, 2000.
[103] J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Optical properties of highdelta air-silica microstructure optical …bers”, Opt. Lett., vol. 25, no. 11, pp.
796, 2000.
[104] R. Ell, U. Morgner, F. X. Kartner, J. G. Fujimoto, E. P. Ippen, V. Scheuer,
G. Angelow, T. Tschudi, M. J. Lederer, A. Boiko, and B. Luther-Davies, “Generation of 5-fs pulses and octave-spanning spectra directly from a Ti:sapphirelaser”, Opt. Lett., vol. 26, no. 6, pp. 373, 2001.
[105] T. M. Fortier, D. J. Jones, and S. T. Cundi¤, “Phase stabilization of an octavespanning Ti:sapphire laser”, Opt. Lett., vol. 28, no. 22, pp. 2198, 2003.
[106] L. Matos, D. Kleppner, O. Kuzucu, T. R. Schibli, J. Kim, E. P. Ippen, andF. X. Kaertner, “Direct frequency comb generation from an octave-spanning,prismless Ti:sapphire laser”, Opt. Lett., vol. 29, no. 14, pp. 1683, 2004.
[107] S. A. Diddams, T. Udem, J. C. Bergquist, E. A. Curtis, R. E. Drullinger,
L. Hollberg, W. M. Itano, W. D. Lee, C. W. Oates, K. R. Vogel, and D. J.
Wineland, “An optical clock based on a single trapped 199 Hg+ ion”, Science,
vol. 293, no. 5531, pp. 825, 2001.
[108] J. L. Hall, L. S. Ma, M. Taubman, B. Tiemann, F. L. Hong, O. P…ster, andJ. Ye, “Stabilization and frequency measurement of the I2 -stabilized Nd:YAG
laser”, IEEE Trans. Instrum. Meas., vol. 48, no. 2, pp. 583, 1999.
[109] M. Saute and M. Aubert-Frécon, “Calculated long-range potential-energy
curves for the 23 molecular states of I2 ”, J. Chem. Phys., vol. 77, pp. 5639,
1982.
[110] W. A. de Jong, L. Visscher, and W. C. Nieuwpoort, “Relativistic and correlated
calculations on the ground, excited, and ionized states of iodine”, J. Chem.
Phys., vol. 107, pp. 9046, 1997.
[111] N. F. Ramsey and E. M. Purcell, “Interactions between nuclear spins in molecules”, Phys. Rev., vol. 85, pp. 143L, 1952.
[112] A. Morinaga, “Hyper…ne structure and hyper…ne coupling constant of molecular iodine”, Jpn. J. Appl. Phys., vol. 23, pp. 774, 1984.
[113] H. J. Foth and F Spieweck, “Hyper…ne structure of the R(98), 58-1 line of 127 I2
at 514.5 nm”, Chem. Phys. Lett., vol. 65, pp. 347, 1979.

152[114] G. R. Hanes, J. Lapierre, P. R. Bunker, and K. C. Shotton, “Nuclear hyper…ne structure in the electronic spectrum of 127 I2 by saturated absorption
spectroscopy, and comparison with theory”, J. Mol. Spectrosc., vol. 39, pp.
506, 1971.
[115] R. J. Jones, W. Y. Cheng, K. W. Holman, L. Chen, J. L. Hall, and J. Ye,“Absolute-frequency measurement of the iodine-based length standard at
514.67 nm”, Appl. Phys. B, vol. 74, pp. 597, 2002.
[116] C. Ishibashi, J. Ye, and J. L. Hall, “Issues and applications in ultra-sensitive
molecular spectroscopy”, in Methods for Ultrasensitive Detection II, C. W.
Wilkerson, Jr., ed., Proc. SPIE, vol. 4634, pp. 58, 2002.
[117] F.-L. Hong, J. Ye, L.-S. Ma, S. Picard, Ch. J. Bordé, and J. L. Hall, “Rotation
dependence of electric quadrupole hyper…ne interaction in the ground state ofmolecular iodine by high-resolution laser spectroscopy”, J. Opt. Soc. Am. B,
vol. 18, pp. 379, 2001.
[118] S. Picard-Fredin, “On the hyper…ne structure of iodine: 1. how to calculate
hyper…ne transition energies”, Tech. Rep. BIPM-90/5.
[119] S. Picard-Fredin and A. Razet, “On the hyper…ne structure of iodine: 2. tocalculate hyper…ne constants on the basis of experimental data”, Tech. Rep.
BIPM-91/2.
[120] F. L. Hong and J. Ishikawa, “Hyper…ne structures of the R(122) 35-0 and P(84)33-0 transitions of 127 I2 near 532 nm”, Opt. Commun., vol. 183, pp. 101, 2000.
[121] Y. R. Lin, S. Picard, T. L. Huang, and J. T. Shy, “New hyper…ne structuresof iodine molecule near 532 nm”.
[122] C. S. Edwards, G. P. Barwood, P. Gill, F. RodriguezLlorente, and W. R. C.
Rowley, “Frequency-stabilised diode lasers in the visible region using dopplerfree iodine spectra”, Opt. Commun., vol. 132, no. 1-2, pp. 94, 1996.
[123] C. S. Edwards, G. P. Barwood, P. Gill, and W. R. C. Rowley, “A 633 nm
iodine-stabilized diode-laser frequency standard”, Metrologia, vol. 36, no. 1,
pp. 41, 1999.
[124] A. Razet, J. Gagniere, and P. Juncar, “Hyper…ne-structure analysis of the33P (6-3) line of 127 I2 at 633 nm using a continuous-wave tunable dye-laser”,
Metrologia, vol. 30, no. 2, pp. 61, 1993.
[125] H. R. Simonsen, “Iodine-stabilized extended cavity diode laser at lambda=633
nm”, IEEE Trans. Instrum. Meas., vol. 46, no. 2, pp. 141, 1997.
[126] A. Morinaga, K. Sugiyama, and N. Ito, “Hyper…ne-structure of low-lying vibrational levels in the B-electronic state of molecular-iodine”, J. Opt. Soc. Am.
B, vol. 6, no. 9, pp. 1656, 1989.

153[127] P. Gill and J. A. Clancy, “A microprocessor-controlled iodine-stabilized ion
laser”, J. Phys. E, vol. 21, no. 2, pp. 213, 1988.
[128] F. Bertinetto, P. Cordiale, S. Fontana, and G. B. Picotto, “Helium-neon lasers
stabilized to iodine at 605-nm”, IEEE Trans. Instrum. Meas., vol. 36, no. 2,
pp. 609, 1987.
[129] W. Y. Cheng, J. T. Shy, and T. Lin, “A compact iodine-stabilized HeNe laserand crossover resonances at 543 nm”, Opt. Commun., vol. 156, no. 1-3, pp.
170, 1998.
[130] V. Špirko and J. Blabla, “Nuclear-quadrupole coupling functions of the 1 +gand 3 +
statesofmolecular-iodine”
,
J.
Mol.
Spectrosc.,
vol.
129,
no.
1,
pp.
0u
59, 1988.
[131] W. S. Barney, C. M. Western, and K. C. Janda, “Measurement of the electronic
wave function: Separated atom wave function analysis of the R-dependent hyper…ne constants of the iodine monochloride A state”, J. Chem. Phys., vol.
113, no. 17, pp. 7211, 2000.
[132] R. J. Le Roy, “LEVEL 7.5: A computer program for solving the radial
Schrödinger equation for bound and quasibound levels”, University of Waterloo
Chemical Physics Research Report CP-655 (2002). The source code and manual
for this program may be obtained from the "Computer Programs" link on the
www site http://leroy.uwaterloo.ca.
[133] S. Gerstenkorn, P. Luc, and C. Amiot, “Analysis of the long range potential ofiodine in the B 3 +0u state”, J. Phys. (Paris), vol. 46, pp. 355, 1985.
[134] J. P. Pique, R. Bacis, M. Broyer, S. Churassy, and J. B. Ko¤end, “Calculationof the magnetic hyper…ne interaction in the E and X state of iodine with theseparated-atom theory”, J. Chem. Phys., vol. 80, pp. 1390, 1984.
[135] P. J. Jewsbury, T. Ridley, K. P. Lawley, and R. J. Donovan, “Parity mixing in
the valence states of I2 probed by optical-optical double-resonance excitationof ion-pair states - characterization of a new ion-pair state, H1u (3 P1 ), and avalence state, c1g ”, J. Mol. Spectrosc., vol. 157, pp. 33, 1993.
[136] G. Herzberg, Molecular spectra and molecular structure, vol. I, Krieger, Malabar, 1989.
[137] M. D. Danyluk and G. W. King, “2 photon sequential absorption spectroscopyof iodine molecule”, Chem. Phys., vol. 22, no. 1, pp. 59, 1977.
[138] T. Ishiwata, H. Ohtoshi, M. Sakaki, and I. Tanaka, “Spectroscopic study on
the iodine molecule by a sequential 3-photon excitation”, J. Chem. Phys., vol.
80, no. 4, pp. 1411, 1984.

154[139] T. Ishiwata, A. Tokunaga, T. Shinzawa, and I. Tanaka, “An analysis of
the F’(0+u ) ion-pair state of I2 by optical optical double-resonance”, J. Mol.
Spectrosc., vol. 117, no. 1, pp. 89, 1986.
[140] L. Chen, W. Y. Cheng, and J. Ye, “Hyper…ne interactions and perturbation
3
127
I2 ”, J. Opt. Soc. Am. B, vol. 21, no. 4, pp.
e¤ects in the B0+u ) state ofu(820, 2004.
[141] L. S. Chen and J. Ye, “Extensive, high-resolution measurement of hyper…neinteractions: precise investigations of molecular potentials and wave functions”,
Chem. Phys. Lett., vol. 381, no. 5-6, pp. 777, 2003.
[142] F. Martin, S. Churassy, R. Bacis, R. W. Field, and J. Vergès, “Long range
behavior of the gerade states near the 2 p3=2 + 2 p3=2 iodine dissociation limit
by laser-induced-‡uorescence fourier-transform spectroscopy”, J. Chem. Phys.,vol. 79, no. 8, pp. 3725, 1983.
Atom-photon[143] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg,
interactions Basic process and applications, John Wiley Sones, New York,
1992.
[144] Jon T. Hougen, The calculation of rotational energy levels and rotational line
intensities in diatomic moledcules, NBS Monograph 115. Nat. Bur. Stand., U.S.
Department of Commerce, 1970.
[145] H. G. Kuhn, Atomic spectra, Academic Press, New York, 1969.
[146] R. Engleman, R. A. Keller, Jr., and B. A. Palmer, “Hyper…ne structure andisotope shift of the 1.3- m transition of 129 I”, Appl. opt., vol. 19, pp. 2767,
1980.
[147] V. Jaccarino, J. G. King, R. A. Satten, and H. H. Stroke, “Hper…ne structureof 127 I. Nuclear magentic octupole moment”, Phys. Rev., vol. 94, pp. 1798L,
1954.
[148] R. J. Le Roy and G.T. Kraemer, “BCONT 2.0. computer program for calculating absorption coe¢ cients, emission intensities or (golden rule) predissociation rates”, University of Waterloo Chemical Physics Research Report CP-650(2001). The source code and manual for this program may be obtained fromthe www site http://leroy.uwaterloo.ca.
[149] E. A. Pazyuk, A. V. Stolyarov, V. I. Pupyshev, N. F. Stepanov, S. Ya. Umanskii,
and A. A. Buchachenko, “The I2 (B) predissociation by solving an inverse atomsin-molecule problem”, Molec. Phys., vol. 99, pp. 91, 2001.
[150] A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, Princeton, New Jersey, 2nd edition, 1960.
[151] J. Y. Zeng, Quantum mechanics, vol. II, Science Press, Beijing, 1993.

Appendix A
Matrix Elements of the Hyper…ne Hamiltonian

A.1

Tensorial Form of the Hyper…ne Hamiltonian
The hyper…ne Hamiltonian of iodine molecule can be formally written as
Hhf = Hhf (a) + Hhf (b) + Hhf (a; b);(A.1)

where a and b label the two iodine nuclei, respectively, and Hhf (a) [Hhf (b)] andHhf (a; b) represent the nucleus-electron and nucleus-nucleus hyper…ne interactions,
respectively. The last term acts only on nuclear coordinates and thus cannot coupledi¤erent electronic states. This last term is left out in the following discussion because
only terms that can couple di¤erent electronic states are relevant to the second-order
calculation of e¤ective hyper…ne parameters [16]. With the last term dropped, various
multipole interactions in this Hamiltonian can be expressed in tensorial form [16, 44]:
Hhf =

XX

Qk (I ) Vk eL=a;b k=kXXX( 1)q Qkq (I ) V kq eL ;(A.2)

=a;b k q= kwhere the rank-k spherical tensor operators Qk (I ) and V k (eL ) act on the nuclear spin
I and the electronic degrees of freedom, respectively. By labeling e , the coordinatesof the electrons, with a superscript L, we emphasize that the tensor operator Vk eL
acts on the coordinates of electrons expressed in the laboratory frame. Later in section
A.3 we will explain the signi…cance of making this seemingly redundant clari…cation.

156
In evaluating the matrix elements of Hhf , we choose a Hund’s case c coupling
scheme that is suitable for the dominance of a strong spin-orbit coupling in the iodine
molecule at intermediate and large internuclear separations. At short internuclear
separations such as R <3 Å for the electronic state B, a Hund’s case a coupling

scheme is more appropriate. As long as these short internuclear separations are barely
sampled by the vibrational wave functions, the Hund’s case c coupling scheme canstill be used in the evaluation.
The basis wave function in the Hund’s case c coupling scheme is denoted as
j i=j
where

JIF MF i ;(A.3)

is the projection of the total angular momentum J (nuclear spin excluded)

onto the molecular axis connecting the two nuclei,

the vibrational quantum number,

I the total nuclear spin, F = I + J, and MF the projection of F onto a quantization
axis.
Each rank-k component of Hhf is a scalar product of two commuting tensor
operators:kHhf(a) = Qk (Ia ) Vk eLa :(A.4)

A direct application of (7.1.6) in reference [150] gives the matrix element of thiscomponent in terms of the reduced matrix elements of Qk and Vk in the uncoupled
scheme1 :
h

0 0

k
(a) j
J 0 I 0 ; F 0 MF0 j Hhf

JI; F MF i

J 0 I 0 ; F 0 MF0 j Qk (Ia ) Vk eLa j JI; F MF i
89>
< F I0 J0 >=J+I 0 +F
= F F 0 MF MF0 ( 1)hI 0 k Qk (Ia ) kIi h>>: k J I ;

=h

1

0 0

(A.5)
0 0

J 0 k V k eLa k

Ji ;
(A.6)

Here the uncoupled scheme means that the nuclear spin I is decoupled from the angularmomentum J.

157
where hI 0 k Qk (Ia ) kIi and h

0 0

J 0 k V k eLa k

Ji are reduced nuclear and electronicmatrix elements, respectively. In the following two sections we will show the evaluations of these two reduced matrix elements.
A.2

Reduced Nuclear Matrix Element
The reduced nuclear matrix element hI 0 k Qk (Ia ) kIi in equation A.6 are ex-pressed in the coupled scheme I = Ia + Ib . We rewrite this matrix element in theuncoupled scheme Ia so that it can be related to the nuclear multipole moments suchas the magnetic dipole and electrical quadrupole moments, which in turn are linked
to the experimentally determined atomic hyper…ne constants. By using (7.1.7) inreference [150], hI 0 k Qk (Ia ) kIi is rewritten in the uncoupled scheme:
hI 0 k Qk (Ia ) kIi = hIa Ia ; I 0 k Qk (Ia ) kIa Ia ; Ii
= ( 1)2Ia +I+kp

(2I + 1) (2I 0 + 1)

89>< Ia I 0 Ia >
=
>
: I

Ia

>
k ;

hIa k Qk (Ia ) kIa i :(A.7)

Note that in above equation we have put Ia = Ib . Using the Winger-Eckart theorem,
the reduced matrix element hIa k Qk (Ia ) kIa i for each nucleus can be related to the matrix element of Qk0 (Ia ) evaluated with a speci…c nuclear wave function jIa MIa = Ia i,i.e.,hIa k Qk (Ia ) kIa i = 0

Ck1;

B Ia k Ia C@
A
Ia 0 Ia

Ck = hIa Ia j Qk0 (Ia ) jIa Ia i ;(A.8)

where the atomic matrix element Ck = hIa Ia j Qk0 (Ia ) jIa Ia i can be related to thecorresponding atomic hyper…ne constants. For k = 1, C1 =n gIa

is the nuclear

158
magnetic dipole moment withn being the nuclear magneton and g the Landé factor

for the iodine nucleus. For k = 2, C2 = 21 eQIa is the nuclear electric quadrupole
moment with e being the proton charge and QIa de…ned as
QIa = e hIa ; MIa = Ia j

X

rp2 jIa ; MIa = Ia i ;

3zp2

p

(A.9)

where zp and rp refer to the coordinates of the protons p in the iodine nucleus.
Plugging equation A.8 into equation A.7, the reduced matrix element hI 0 k Qk (Ia ) kIi
in equation A.6 is evaluated as
hI 0 k Qk (Ia ) kIi = ( 1)2Ia +I+kp

(2I + 1) (2I 0 + 1)
89>< Ia I 0 Ia >
=
Ck
0
1
:
>
>
:
;
I Ia k
B Ia k Ia C@
A
Ia 0 Ia

A.3(A.10)

Reduced Electronic Matrix Element
We proceed to evaluate the reduced matrix element h

0 0

J 0 k V k eLa k

Ji inequation A.6. Similar to the procedure used in the evaluation of the reduced matrix
element hIa k Qk (Ia ) kIa i, we …rst relate h
0 0element h

J 0 MJ0 j V kq eLa j

J 0 MJ0 j V kq eLa j
0
J0
0
0 B
= ( 1)J MJ @
MJ0

h

0 0

0 0

J 0 k V k eLa k

Ji to a speci…c matrix

JMJ i by applying the Winger-Eckart theorem:
JMJ ik

The evaluation of matrix element h

1

J C
Ah
q MJ
0 0

0 0

J 0 k V k eLa k

J 0 MJ0 j V kq eLa j

Ji(A.11)JMJ i involves the averag-ing of V kq eLa over the electronic motion, the molecular rotation, and the molecularvibration. Note that the tensor operator Vk eLa acts on the coordinates of electrons
expressed in the laboratory frame, which ensures that the hyper…ne Hamiltonian can

159
directly describe the hyper…ne spectrum observed in the laboratory frame. This explicit choice of the coordinate system poses no di¢ culty for the averaging over themolecular rotation and vibration. However, the averaging over the electronic motion
demands a special treatment because the electronic wave function obtained in the
Born-Oppenheimer approximation is expressed in the molecular frame (body …xed
coordinate system). To work with the Born-Oppenheimer electronic wave function,
one can …rst use the tensor operator Vk (ea ) expressed in the molecular frame, andthen adopts an active rotation to transform this operator to the laboratory frame,i.e.,
V kq eLa =X(k)

D qq0 (!) Vqk0 (ea ) ;(A.12)

q0(k)

where D qq0 (!) is the rotation operator (Wigner D function), ! represents the three
Eular angles , , and , and ea represents the electronic coordinates in the molecular
frame. The operator Vqk0 (ea ) and the electronic wave function j i, both expressed in
the molecular frame, are used to perform the averaging over the electronic motion in
the molecular frame. The rotational wave function can also be written in terms of
the Wigner D function [151]:
jJ MJ i =r

2J + 1 (J)DMJ (!) :8 2(A.13)

Using equations A.12 and A.13, we can explicitly perform the averaging over themolecular rotation:
J 0 MJ0 j V kq eLa j JMJ ip
X(2J + 1) (2J 0 + 1)
q+q 0
( 1)
=
h
8 2
q0
Z
(J)
(k)
(J 0 )
d!DMJ (!) Dq q0 (!) DM 0 0 (!) :
h

0 0

J

0 0
k
J 0 j Vq 0 (ea ) j

Ji(A.14)

160
The integral of the product of three Wigner D functions has been evaluated in reference [151]:
Z(J)(J 0 )(k)

d!DMJ (!) Dq q0 (!) DM 0 0 (!)
J=

q;MJ MJ0 q 0 ;J = J0

where

0

B k
( 1) +k+ J+MJ 8 2 @
q0

0

J,

=

0

, and

J

0
0

MJ = M J

10

J CB
A@

J

0

k

1

J C
A;
q MJ

MJ0

(A.15)

MJ0 . Inserting integral A.15

into equation A.14 gives
J 0 MJ0 j V kq eLa j JMJ i00 p= ( 1)k+ + J+MJ (2J + 1) (2J 0 + 1)
0
10
1
0
0
J
J CB J
k
J C
B k
@
A@
Ah
0
MJ0
q MJ

h

0 0

The matrix element h

0 0k
J0 j V(ea ) j

0 0
k
J0 j V(ea ) jJi :(A.16)

J i encapsulates the averaging over the vibra-tion and the electronic motion. Combining equations A.11 and A.16, the reduced
matrix element h
h

0 0

0 0

J 0 k V k eLa k

J 0 k V k eLa k= ( 1)k+0 +Jp

Ji in equation A.6 is evaluated as

Ji
0

B k
(2J + 1) (2J 0 + 1) @

J

0
0

1

J C
Ah

0 0
k
J0 j V(ea ) jJi :(A.17)

161
A.4

Matrix Elements of the Hyper…ne Hamiltonian
The …nal form of the matrix element h

0 0k
J 0 I 0 ; F MF j Hhf(a) jJI; F MF i is

obtained by substituting equations A.10 and A.17 into equation A.6, which givesk(a) j
J 0 I 0 ; F MF j Hhf

h

0 0

h

0 0
k
J0 j VJI; F MF i
0 p
= ( 1)F +k+2Ia + I+ J+(2J 0 + 1) (2J + 1) (2I 0 + 1) (2I + 1)
8
90
9
18
>
>
>
>
< k Ia Ia =J0
k J C < J0 k J =
Ck
B
0
1
@
A
>
>
0
: Ia I 0 I >;: I F I0 >
;
k
I
I
a
a C
B
@
A
0 Ia
Ia
(a) j

Ji ;

where Ck is given in equation A.8. As reference [16], we de…ne a function fk (a;(A.18)0; J0 0 ; ; J )that encapsulates the averaging over the electronic motion in the molecular frame andthe molecular vibration in the laboratory frame:fk (a;

00; J0 0 ; ; J ) = ( 1)Then the matrix element h

0 0

0

Ck

B k Ia
@
0 Ia

1

Ia C
A
Ia

k
J 0 I 0 ; F MF j Hhf
(a) j

h

0 0
k
J0 j V

(a) j

Ji :

JI; F MF i can be expressed ask
J 0 I 0 ; F MF j Hhf(a) j JI; F MF ip
= ( 1)F +k+2Ia + I+ J (2J 0 + 1) (2J + 1) (2I 0 + 1) (2I + 1)
98
9
0
18
>
>
>
>
0
0
<
=
<
k J C J k J
k Ia Ia =
B J
@
A
>
0
: I F I0 >;>: Ia I 0 I >;h(A.19)0 0

fk (a;0; J0 0 ; ; J ) :(A.20)

This expression and equation A.19 can be directly compared with those given inreference [16] for a Hund’s case a coupling scheme.

Appendix B
Hyper…ne Spectra and E¤ective Hyper…ne Parameters for 79 B-statelevels ( 0 = 42

70,

= 500

517 nm)This appendix supplies spectroscopic data for all measured I2 rovibrationaltransitions in the wavelength range 500

517 nm. For each transition, the hyper…nesplittings and experimental conditions are summarized in a table, together with the…tted B-state e¤ective hyper…ne parameters. The list of table at the beginning of
this thesis can be used to locate a speci…c transition.
The hyper…ne spectrum of each rovibrational transition is …tted to a four-terme¤ective Hamiltonian [16]Hhf;ef f = HeqQ + HSR + HT SS + HSSS ;(B.1)

where the four leading terms on the right-hand side represent major contributions
from nuclear electric quadrupole, spin-rotation, tensorial spin-spin, and scalar spinspin interactions, respectively. The four B-state e¤ective hyper…ne parameters, eqQB ,CB , dB , andB are derived from the …t. In the …t, the four hyper…ne parameters for

the X ground state are determined by [42]1
0:542( 00 + )2
1+ 0:4535 10 1 ( 00 + )2
2
1
0:1927 10 3 ( 00 + )2 J 00 (J 00 + 1)
2
1
+ 0:694 10 5 ( 00 + )J 00 (J 00 + 1)
2

eqQX [MHz] =2452:2916

(B.2)

163
and
1
CX [MHz] = 1:9245 + 0:01356( 00 + )2respectively, and the values for dB and

15098
E(

00 )[cm 1 ]

12340

;

(B.3)

B are …xed to 1:524 and 3:705 kHz, respec-tively.
For transitions P(69) 58-0, P(84) 60-0, P(77) 60-0, and P(63) 70-0, their hyper…ne levels in the B state are altered considerably by the strong perturbation fromthe 1g (1g ) state.

Therefore hyper…ne spectra at these transitions cannot be …tted

to the above-mentioned e¤ective Hamiltonian. In this case, the spectrograms for the
relevant transitions are also given in addition to their hyper…ne splittings.

164
B.1

P(10) 42-0

Table B.1: Hyper…ne spectrum recorded at P(10) 42-0. Pump power: 2.4 mW, probe
power: 0.22 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger temperature:-2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
13
14
15

F0 Measurement (MHz) Calculation (MHz)9
484:613
484:6215
244:156
244:15210
228:548
228:5368
178:269
178:27713
168:798
168:800
6
135:256
135:257
7
87:017
87:018
11
61:017
61:01112
0
9
71:654
71:664
10
246:617
246:60911
263:601
263:6059359:347
359:343

Mea.-Cal. (MHz)
0:007
0:004
0:012
0:008
0:002
0:001
0:001
0:006
0:010
0:008
0:005
0:004

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)
0:0076
eqQB (MHz)557:119(20)z
CB (MHz)
0:176138(96)dB (MHz)
0:08916(86)(MHz)
0:00520(85)BHyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:57
CX (MHz)
0:00315478
ydX (MHz)0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

165
B.2

P(19) 42-0

Table B.2: Hyper…ne spectrum recorded at P(19) 42-0. Pump power: 2.7 mW, probe
power: 0.2 mW, time constant of lock-in ampli…er: 30 ms, cold-…nger temperature:-2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13

F0 Measurement (MHz) Calculation (MHz)13
621:512
621:45218
564:180
564:16623
513:899
513:89114
387:729
387:77319
292:759
292:79217
269:423
269:45815
208:705
208:71822
184:469
184:47516
155:325
155:347
20
131:746
131:75421
109:967
109:98615
58:38958:41918
0

Mea.-Cal. (MHz)
0:059
0:014
0:008
0:044
0:033
0:034
0:013
0:007
0:022
0:008
0:019
0:030

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:033
eqQB (MHz)556:988(66)z
CB (MHz)
0:1766(17)dB (MHz)
0:0921(43)(MHz)
0:0016(39)B
Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:62
CX (MHz)
0:00315478
ydX (MHz)0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

166
B.3

R(30) 42-0

Table B.3: Hyper…ne spectrum recorded at R(30) 42-0. Time constant of lock-in
ampli…er: 30 ms, cold-…nger temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
13
14
15

F0 Measurement (MHz) Calculation (MHz)31
566:231
566:19727
316:504
316:49232
287:001
287:01830
278:416
278:44335
250:426
250:44328
180:707
180:70729
156:209
156:21933
128:855
128:86134
103:198
103:17531
032
159:272
159:30033
167:739
167:73631
284:698
284:706

Mea.-Cal. (MHz)
0:034
0:012
0:016
0:027
0:017
0:001
0:010
0:006
0:024
0:029
0:003
0:008

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:021
eqQB (MHz)557:184(44)z
CB (MHz)
0:177751(85)dB (MHz)
0:0946(22)0:0099(33)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:73
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

167
B.4

R(41) 42-0

Table B.4: Hyper…ne spectrum recorded at R(41) 42-0. Pump power: 2.2 mW, probe
power: 0.26 mW, time constant of lock-in ampli…er: 30 ms, cold-…nger temperature:-2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)37
437:984
438:00742
380:721
380:73447
326:979
326:97138
171:674
171:72146
26:633
26:63139
040
26:674
26:67044
60:643
60:65345
79:27079:277
39
134:667
134:66142
186:006
186:00145
242:928
242:92140
273:458
273:44741
309:117309:08943
342:946
342:93744
377:681
377:67141
447:497
447:49642
466:190466:20243
490:582
490:571

Mea.-Cal. (MHz)
0:023
0:013
0:008
0:046
0:002
0:005
0:010
0:007
0:005
0:004
0:006
0:011
0:028
0:009
0:010
0:001
0:012
0:011

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:0173732
eqQB (MHz)557:553(30)z
CB (MHz)
0:179178(24)dB (MHz)
0:0929(20)0:0003(16)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:88
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

168
B.5

R(49) 42-0

Table B.5: Hyper…ne spectrum recorded at R(49) 42-0. Pump power: 2.6 mW, probe
power: 0.26 mW, time constant of lock-in ampli…er: 30 ms, cold-…nger temperature:-2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
7
8
9
10
11
12
13
14
15
16
17
18
20
21

F0 Measurement (MHz) Calculation (MHz)45
576:746
576:75050
515:333
515:29155
456:864
456:85646
304:832
304:83954
161:791
161:81247
134:469
134:51348
110:056
110:10552
70:926
70:960153
52:322
52:321347
050
51:75051:721253
107:133
107:15148
142:691
142:72149
175:505
175:50751
209:070
209:08852
240:809240:84750
332:847
332:83751
355:802355:772

Mea.-Cal. (MHz)
0:003
0:042
0:007
0:007
0:021
0:045
0:049
0:034
0:001
0:029
0:018
0:030
0:001
0:018
0:038
0:011
0:029

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:030
eqQB (MHz)557:612(50)z
CB (MHz)
0:180487(46)dB (MHz)
0:0915(25)(MHz)
0:0051(23)BHyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:01
CX (MHz)
0:00315478
ydX (MHz)0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

169
B.6

P(13) 43-0

Table B.6: Hyper…ne spectrum recorded at P(13) 43-0. Pump power: 2.2 mW, probe
power: 0.23 mW, time constant of lock-in ampli…er: 30 ms, cold-…nger temperature:-2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
11
12
13
15
16
17
20
21

F0 Measurement (MHz) Calculation (MHz)7
567:371
567:38012
495:511
495:50917
435:604
435:6038
359:570
359:56113
232:354
232:37311
194:828
194:8299
180:603
180:60510
96:926
96:907
16
85:893
85:895
15
41:65041:6349
0
12
64:104
64:10811
151:780
151:78815
181:156
181:14513
225:322
225:33112
322:024
322:02213
381:739
381:731

Mea.-Cal. (MHz)
0:009
0:003
0:001
0:008
0:019
0:001
0:002
0:020
0:002
0:016
0:004
0:008
0:011
0:009
0:003
0:008

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:011
eqQB (MHz)558:613(18)z
CB (MHz)
0:190361(78)dB (MHz)
0:09899(62)0:00083(56)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:59
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

170
B.7

P(25) 43-0

Table B.7: Hyper…ne spectrum recorded at P(25) 43-0. Pump power: 2.6 - 3.1
mW, probe power: 0.2 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
9
10
11
12
13
16
17
20

F0 Measurement (MHz) Calculation (MHz)19
736:133
736:12824
681:548
681:56329
632:257
632:25920
488:441
488:44925
405:369
405:36123
391:711
391:69722
272:136
272:126
26
246:431
246:428
27
226:166
226:16321
171:678
171:67724
116:559
116:55923
025
40:811
40:80424
158:852
158:863

Mea.-Cal. (MHz)
0:005
0:015
0:002
0:008
0:008
0:014
0:010
0:003
0:003
0:001
0:000
0:007
0:011

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:0094
eqQB (MHz)558:823(15)z
CB (MHz)
0:191161(39)dB (MHz)
0:1009(12)0:0025(11)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:67
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

171
B.8

R(27) 43-0

Table B.8: Hyper…ne spectrum recorded at R(27) 43-0: 2.6 - 3.1 mW, probe power:
0.2 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger temperature: -2.28 C(vapor pressure: 3.3 Pa).
a
1
3
4
5
6
7
8
10
13
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)23
033
108:536
108:53424
250:629
250:63829
334:396334:38327
345:334
345:333
32
422:632
422:64425
427:231
427:24930
493:499
493:50728
622:000
622:02127
739:288739:27929
779:734
779:72430
826:140
826:124
27
878:055
878:05828
898:204
898:20329
930:144
930:145

Mea.-Cal. (MHz)
0:002
0:009
0:013
0:002
0:013
0:018
0:008
0:021
0:009
0:011
0:016
0:003
0:002
0:000

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:012
eqQB (MHz)558:662(26)z
CB (MHz)
0:191579(37)dB (MHz)
0:0980(17)0:0006(13)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:69
CX (MHz)
0:00315478
dX (MHz)y0:001524
y(MHz)
0:003705
X**eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

172
B.9

P(38) 43-0

Table B.9: Hyper…ne spectrum recorded at P(38) 43-0. Pump power: 2 - 3 mW, probe
power: 0.18 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger temperature:-2.28 C (vapor pressure: 3.3 Pa).
a
1
2
5
7
8
9
10
11
12
13
14
15

F0 Measurement (MHz) Calculation (MHz)37
033
245:633
245:63541
319:675
319:67335
407:707
407:69839
440:260
440:25640
462:861
462:86437
566:241
566:24335
684:788
684:79436
691:039
691:04038
724:433
724:43639
733:674
733:67937
850:653
850:644

Mea.-Cal. (MHz)
0:001
0:003
0:008
0:004
0:003
0:002
0:006
0:001
0:003
0:005
0:009

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:0059eqQB (MHz)558:565(12)z
CB (MHz)
0:192839(21)dB (MHz)
0:10109(69)0:00001(81)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:83
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

173
B.10

R(40) 43-0

Table B.10: Hyper…ne spectrum recorded at R(40) 43-0. Pump power: 2 - 3 mW,probe power: 0.18 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
5
6
7
8
9
10
11
12
13
14
15

F0 Measurement (MHz) Calculation (MHz)41
383:822
383:82345
61:284
61:28638
039
22:442
22:44043
58:002
58:00444
80:82880:83041
182:464
182:46839
299:737
299:72940
306:832
306:82342
341:128
341:12643
351:060
351:05741
466:834
466:845

Mea.-Cal. (MHz)
0:001
0:002
0:002
0:002
0:002
0:004
0:009
0:009
0:002
0:002
0:011

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:0064
eqQB (MHz)558:604(17)z
CB (MHz)
0:193497(14)dB (MHz)
0:10303(86)0:00034(86)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:86
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

174
B.11

R(63) 43-0

Table B.11: Hyper…ne spectrum recorded at R(63) 43-0. Pump power: 2 - 3 mW,probe power: 0.18 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)59
585:205
585:20064
508:804
508:795
69
435:562
435:56560
305:174
305:19263
233:362
233:36965
217:508
217:52268
148:841
148:83161
134:901
134:89562
110:741
110:71966
57:767
57:762
67
37:348
37:35461
064
57:826
57:823
67
117:403
117:40262
148:618
148:61463
180:107
180:09065
217:406
217:40066
247:556
247:55063
319:064
319:06364
339:379
339:38865
362:791362:803

Mea.-Cal. (MHz)
0:004
0:009
0:002
0:018
0:007
0:014
0:010
0:006
0:022
0:005
0:006
0:003
0:001
0:004
0:016
0:006
0:006
0:001
0:010
0:012

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:011
eqQB (MHz)559:456(16)z
CB (MHz)
0:198545(12)dB (MHz)
0:10629(90)(MHz)
0:00234(85)BHyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:31
CX (MHz)
0:00315478
ydX (MHz)0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

175
B.12

R(21) 45-0

Table B.12: Hyper…ne spectrum recorded at R(21) 45-0. Pump power: 1.4 - 1.7
mW, probe power: 0.14 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19

F0 Measurement (MHz) Calculation (MHz)17
238:548
238:55322
175:128
175:125
27
118:645
118:640
18
0
23
98:41098:41021
116:180
116:18119
179:723
179:71126
205:151
205:14520
230:346
230:33924
259:859
259:85725
282:533
282:52919
326:388
326:37922
388:084
388:09120
442:094
442:09125
472:740
472:73921
497:713
497:70723
547:330
547:324
24
604:238
604:24221
638:454
638:472

Mea.-Cal. (MHz)
0:005
0:003
0:005
0:000
0:001
0:012
0:005
0:007
0:002
0:004
0:008
0:008
0:003
0:002
0:006
0:006
0:005
0:018

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)
0:0076
eqQB (MHz)560:838(13)z
CB (MHz)
0:223669(23)dB (MHz)
0:12073(80)0:00623(57)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:64
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

176
B.13

R(52) 45-0

Table B.13: Hyper…ne spectrum recorded at R(52) 45-0. Pump power: 2.2- 2.9
mW, probe power: 0.2 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

F0 Measurement (MHz) Calculation (MHz)53
337:180
337:16449
109:749
109:76052
60:715
60:73054
47:772
47:78057
050
37:28937:30051
60:792
60:79655
112:364
112:34956
135:798
135:79153
228:501
228:49851
339:039
339:04252
350:586
350:59354
389:084
389:09155
402:748
402:75153
512:392
512:388

Mea.-Cal. (MHz)
0:016
0:011
0:014
0:007
0:011
0:003
0:014
0:006
0:003
0:003
0:008
0:007
0:003
0:004

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:010
eqQB (MHz)561:356(22)z
CB (MHz)
0:229843(18)dB (MHz)
0:1239(10)0:0095(15)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:07
CX (MHz)
0:00315478
dX (MHz)y0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

177
B.14

P(69) 45-0

Table B.14: Hyper…ne spectrum recorded at R(69) 45-0. Pump power: 2.8 - 3.4
mW, probe power: 0.3 mW, time constant of lock-in ampli…er: 30 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)63
548:614
548:61368
456:865
456:85073
368:565
368:55364
264:184
264:19467
185:571
185:569
69
162:293
162:28665
92:046
92:04772
86:986
86:98366
65:786
65:78970
071
22:99922:99865
43:00043:002
68
108:963
108:96371
176:045
176:04666
195:318
195:31767
228:449
228:457
69
271:141
271:15070
302:706
302:70667
367:416
367:42568
390:220
390:22469
415:931
415:934

Mea.-Cal. (MHz)
0:001
0:015
0:012
0:010
0:001
0:007
0:002
0:002
0:004
0:002
0:002
0:000
0:001
0:000
0:008
0:009
0:000
0:009
0:004
0:003

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:0068eqQB (MHz)561:804(11)z
CB (MHz)
0:2349204(71)dB (MHz)
0:12617(80)(MHz)
0:00739(59)BHyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:47
CX (MHz)
0:00315478
ydX (MHz)0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

178
B.15

P(91) 45-0

Table B.15: Hyper…ne spectrum recorded at P(91) 45-0. Pump power: 3.8 - 4.0
mW, probe power: 0.39 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)85
469:110
469:10890
349:744
349:74395
233:796
233:78986
175:156
175:14289
85:54085:532
91
48:142
48:14687
088
29:615
29:61194
38:333
38:32892
118:993
118:98287
136:025
136:02093
146:439
146:43390
216:099
216:10489
331:122331:13791
383:234
383:24592416:858
416:85889
470:571
470:57390
497:738
497:73791
527:179
527:176

Mea.-Cal. (MHz)
0:001
0:001
0:008
0:014
0:008
0:003
0:003
0:005
0:011
0:005
0:006
0:005
0:015
0:012
0:000
0:002
0:002
0:003

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)
0:0079eqQB (MHz)562:546(13)z
CB (MHz)
0:2452897(55)dB (MHz)
0:1325(10)0:00975(82)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2454:14
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

179
B.16

R(31) 47-0

Table B.16: Hyper…ne spectrum recorded at R(31) 47-0. Pump power: 2.7- 3.0
mW, probe power: 0.33 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
9
10
11
12
13
14
16
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)27
470:966
470:98232
402:022402:11937
338:264
338:25428
212:885
212:88630
034
39:748
39:72535
61:885
61:886
29
100:491
100:49432
161:983
161:96435
233:666
233:54731
278:810
278:80534
365:406
365:40531
416:858
416:85532
438:774
438:77433
470:182
470:187

Mea.-Cal. (MHz)
0:016
0:097
0:010
0:001
0:023
0:001
0:003
0:019
0:119
0:005
0:001
0:003
0:000
0:005

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:012
eqQB (MHz)562:816(27)z
CB (MHz)
0:264762(35)dB (MHz)
0:1390(16)0:0172(12)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:74
CX (MHz)
0:00315478
dX (MHz)y0:001524
y(MHz)
0:003705
X* Not included in the …t.
**eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

180
B.17

R(51) 47-0

Table B.17: Hyper…ne spectrum recorded at R(51) 47-0. Pump power: 2.0 - 2.25
mW, probe power: 0.17 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)47
480:121480:13852
393:354
393:36057
310:700
310:68348
203:309203:30951
119:163
119:16153
102:583
102:585
49
28:885
28:89156
22:577
22:58250
054
59:405
59:37655
82:684
82:683
49
105:259
105:254
52
171:461
171:46055
240:605
240:61350
252:819
252:82451
289:990289:99353
333:136
333:12954
368:690368:68351
428:227
428:22252
451:387
451:38753
478:967
478:966

Mea.-Cal. (MHz)
0:016
0:006
0:017
0:001
0:002
0:002
0:006
0:005
0:028
0:001
0:006
0:002
0:008
0:005
0:003
0:006
0:006
0:005
0:000
0:001

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:010
eqQB (MHz)563:295(17)z
CB (MHz)
0:270541(14)dB (MHz)
0:1473(12)(MHz)
0:01565(88)BHyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:05
CX (MHz)
0:00315478
ydX (MHz)0:001524
y(MHz)
0:003705
X**eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

181
B.18

R(92) 47-0

Table B.18: Hyper…ne spectrum recorded at R(92) 47-0. Pump power: 1.7 - 2.3
mW, probe power: 0.2 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

F0 Measurement (MHz) Calculation (MHz)93
478:463
478:46989
308:818
308:83292
219:986
219:97394
172:081
172:08190
144:124
144:11491
110:288
110:28097
84:522
84:51595
096
33:266
33:27693
86:463
86:472
91
169:791
169:79192
196:460
196:45594
258:805
258:79395
286:950
286:94993
369:907369:912

Mea.-Cal. (MHz)
0:006
0:014
0:012
0:000
0:010
0:008
0:006
0:010
0:009
0:000
0:005
0:012
0:001
0:005

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:0094
eqQB (MHz)564:742(23)z
CB (MHz)
0:293399(10)dB (MHz)
0:1598(10)0:0200(13)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2454:17
CX (MHz)
0:00315478
dX (MHz)y0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

182
B.19

R(111) 47-0

Table B.19: Hyper…ne spectrum recorded at R(111) 47-0. Pump power: 2.4 mW,
probe power: 0.18 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
8
9
10
11
12
13
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)107
432:018
432:018112
250:387
250:437
108
123:928
123:925
117
72:598
72:562111
0110
102:973
102:942116
184:443
184:433
109
198:522
198:51
114
242:838
242:808115
282:012
282:005112
314:466
314:439110
371:409
371:366111
417:403
417:362115
429:972
429:934
113
493:450
493:416114
537:349
537:338111
556:820
556:848112
595:553
595:592113
636:524
636:600

Mea.-Cal. (MHz)
0:001
0:050
0:004
0:036
0:031
0:010
0:012
0:030
0:007
0:026
0:042
0:041
0:038
0:034
0:011
0:028
0:039
0:075

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:037
eqQB (MHz)565:651(56)z
CB (MHz)
0:310759(29)dB (MHz)
0:1626(44)0:0254(35)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2454:9CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

183
B.20

P(19) 49-0

Table B.20: Hyper…ne spectrum recorded at P(19) 49-0. Pump power: 4.6 - 5 mW,
probe power: 0.3 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)13
631:145
631:150
18
561:147
561:14823
499:371
499:36914
396:296396:29119
288:728
288:72817
270:560
270:55715
215:591
215:58722
174:032
174:033
16
159:690
159:68820
126:110
126:10821
102:016102:01415
65:517
65:52118
0
16
46:623
46:61921
91:701
91:698
17
105:592
105:59219
160:216
160:20920
220:933
220:92717
247:383
247:38618
268:959
268:960
19
312:608
312:619

Mea.-Cal. (MHz)
0:005
0:000
0:002
0:005
0:000
0:002
0:004
0:001
0:002
0:002
0:002
0:004
0:004
0:003
0:000
0:006
0:007
0:003
0:002
0:011

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:0046
eqQB (MHz)564:6327(72)z
CB (MHz)
0:309250(21)dB (MHz)
0:16923(43)(MHz)
0:02631(31)B
Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:62
CX (MHz)
0:00315478
ydX (MHz)0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

184
B.21

P(41) 49-0

Table B.21: Hyper…ne spectrum recorded at P(41) 49-0. Pump power: 3.4 - 3.8
mW, probe power: 0.38 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)35
532:653
532:66440
449:385
449:39045
370:892
370:87536
262:237
262:22439
172:967
172:93641
161:763
161:74637
86:186
86:19144
77:488
77:49438
54:55554:56042
043
23:542
23:53837
48:011
47:99940
114:420
114:42043
185:772
185:79338
191:181191:18139
231:748
231:77041
275:656
275:68542
314:638
314:63839
369:603
369:60340
392:932
392:92241
422:354
422:341

Mea.-Cal. (MHz)
0:012
0:005
0:017
0:012
0:031
0:017
0:005
0:006
0:006
0:004
0:012
0:000
0:021
0:001
0:022
0:029
0:000
0:000
0:010
0:012

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:016
eqQB (MHz)564:886(26)z
CB (MHz)
0:314896(28)dB (MHz)
0:1706(18)(MHz)
0:0262(13)BHyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:88
CX (MHz)
0:00315478
ydX (MHz)0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

185
B.22

P(62) 49-0

Table B.22: Hyper…ne spectrum recorded at P(62) 49-0. Pump power: 3.0 - 3.5
mW, probe power: 0.26 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

F0 Measurement (MHz) Calculation (MHz)61
463:477
463:46157
266:405
266:42760
196:429
196:44062
166:266
166:26158
111:003
110:99465
97:743
97:71659
81:646
81:62863
064
28:966
28:977
61
100:699
100:72359
196:833
196:83760
215:995
216:00062
267:353
267:35963
288:514
288:510
61
383:956
383:955

Mea.-Cal. (MHz)
0:016
0:022
0:011
0:005
0:008
0:027
0:018
0:011
0:024
0:004
0:005
0:006
0:003
0:001

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:016
eqQB (MHz)565:295(39)z
CB (MHz)
0:324667(25)dB (MHz)
0:1794(17)0:0326(22)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:29
CX (MHz)
0:00315478
dX (MHz)y0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

186
B.23

R(93) 49-0

Table B.23: Hyper…ne spectrum recorded at R(93) 49-0. Pump power: 2.1 - 2.5
mW, probe power: 0.2 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
8
9
10
11
12
13
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)89
530:500
530:50594
356:084
356:10790
226:206
226:22299
185:887
185:88793
103:786
103:78592
098
74:256
74:26891
95:18995:18696
133:201
133:197
97
171:449
171:45794
208:029
208:03992
265:469
265:46893
311:768
311:75197
320:691
320:68795
385:677
385:66096
429:766
429:76493
450:802450:795
94
488:630
488:63495
529:072
529:067

Mea.-Cal. (MHz)
0:005
0:023
0:015
0:001
0:001
0:012
0:004
0:005
0:008
0:010
0:000
0:017
0:005
0:017
0:002
0:007
0:004
0:005

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:011
eqQB (MHz)566:485(19)z
CB (MHz)
0:3509271(87)dB (MHz)
0:1940(13)0:0403(10)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2454:21
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

187
B.24

P(108) 49-0

Table B.24: Hyper…ne spectrum recorded at P(108) 49-0. Pump power: 2.5 - 2.7
mW, probe power: 0.25 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

F0 Measurement (MHz) Calculation (MHz)107
343:852
343:860103
221:718
221:729106
98:224
98:214104
44:79044:803108
25:55325:558105
0111
96:563
96:578109
157:592
157:579110
201:694
201:685107
220:194
220:188105
280:207280:201106
318:668
318:666108
403:545
403:544109
443:583
443:582107
503:277
503:280

Mea.-Cal. (MHz)
0:008
0:011
0:010
0:013
0:005
0:016
0:013
0:008
0:006
0:006
0:002
0:001
0:001
0:003

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:0097
eqQB (MHz)567:273(24)z
CB (MHz)
0:3662298(84)dB (MHz)
0:1996(10)0:0427(13)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2454:78
CX (MHz)
0:00315478
dX (MHz)y0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

188
B.25

R(55) 50-0

Table B.25: Hyper…ne spectrum recorded at R(55) 50-0. Pump power: 3.2 - 3.7
mW, probe power: 0.3 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
10
11
12
13
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)51
461:997
462:00756
348:169
348:17061
238:876
238:87052
178:699
178:69755
80:59080:58857
51:977
51:97853
054
33:276
33:283
58
114:773
114:76653
134:040
134:03959
143:050
143:04556
215:510215:51354
287:561
287:56459
299:143
299:14955
328:873
328:87657
382:073
382:07158
421:377
421:37155
466:964
466:96356
494:948
494:94757
527:203
527:200

Mea.-Cal. (MHz)
0:009
0:001
0:006
0:001
0:002
0:001
0:007
0:006
0:000
0:005
0:003
0:003
0:006
0:002
0:002
0:006
0:001
0:001
0:003

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:0047
eqQB (MHz)566:0544(80)z
CB (MHz)
0:3504090(55)dB (MHz)
0:19199(56)0:03910(46)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:13CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

189
B.26

R(20) 51-0

Table B.26: Hyper…ne spectrum recorded at R(20) 51-0. Pump power: 2.8 - 3.2
mW, probe power: 0.27 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
13
14

F0 Measurement (MHz) Calculation (MHz)21
476:456
476:44917
244:318
244:32022
201:904
201:91220
188:646
188:65025
149:119149:11018
113:232
113:234
19
77:998
77:99423
38:140
38:14524
021
84:216
84:22022
250:370
250:38123
263:803
263:798

Mea.-Cal. (MHz)
0:007
0:001
0:008
0:004
0:009
0:002
0:004
0:004
0:004
0:011
0:005

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)
0:0071
eqQB (MHz)566:275(21)z
CB (MHz)
0:366031(35)dB (MHz)
0:20074(96)0:0469(11)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:63
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

190
B.27

P(30) 51-0

Table B.27: Hyper…ne spectrum recorded at P(30) 51-0. Pump power: 3 mW, probe
power: 0.2 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger temperature:-2.28 C (vapor pressure: 3.3 Pa).
a
1
2
5
6
7
8
9
10
11
12
13
14
15

F0 Measurement (MHz) Calculation (MHz)29
334:316
334:323
25
107:632
107:643
33
026
32:685
32:687
27
62:13062:14431
110:609
110:62032
140:840
140:830
29
228:095
228:09727
336:702
336:71728
345:822
345:82130
391:399
391:39031
405:258
405:26929
511:372
511:351

Mea.-Cal. (MHz)
0:007
0:011
0:002
0:014
0:012
0:010
0:002
0:016
0:002
0:009
0:011
0:021

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:013
eqQB (MHz)566:415(29)z
CB (MHz)
0:368223(40)dB (MHz)
0:2005(13)(MHz)
0:0401(19)B
Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:73
CX (MHz)
0:00315478
ydX (MHz)0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

191
B.28

P(50) 51-0

Table B.28: Hyper…ne spectrum recorded at P(50) 51-0. Pump power: 3 mW, probe
power: 0.2 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger temperature:-2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

F0 Measurement (MHz) Calculation (MHz)49
490:442
490:44545
290:131
290:13548
221:508
221:50750
196:015
196:01346
137:487
137:48853
128:980
128:96847
107:257
107:25351
29:993
29:99452
049
72:79072:79847
169:996
169:99748
187:690
187:69850
239:328
239:32651
259:589259:59049
355:790355:783

Mea.-Cal. (MHz)
0:003
0:004
0:000
0:001
0:001
0:012
0:003
0:000
0:008
0:001
0:007
0:002
0:001
0:007

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:0056
eqQB (MHz)566:763(14)z
CB (MHz)
0:377297(11)dB (MHz)
0:20747(63)0:05024(66)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:03
CX (MHz)
0:00315478
dX (MHz)y0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

192
B.29

P(73) 51-0

Table B.29: Hyper…ne spectrum recorded at P(73) 51-0. Pump power: 3.8 - 4.2
mW, probe power: 0.36 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
5
6
7
8
9
10
11
12
13
14
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)67
480:189
480:19772
326:030
326:02971
68:69968:70873
19:894
19:89469
070
38:243
38:24876
90:816
90:823
69
134:791
134:77774
153:996
153:98375
189:061
189:06472
237:443
237:45270
299:126
299:15073
411:323
411:33474
454:085
454:07771
482:720
482:71372
517:349
517:32973
555:230
555:232

Mea.-Cal. (MHz)
0:008
0:001
0:009
0:000
0:005
0:007
0:014
0:013
0:003
0:009
0:024
0:011
0:007
0:007
0:020
0:003

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:012
eqQB (MHz)567:132(23)z
CB (MHz)
0:394644(14)dB (MHz)
0:2134(16)0:0552(12)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:57
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

193
B.30

P(90) 51-0

Table B.30: Hyper…ne spectrum recorded at P(90) 51-0. Pump power: 1.7 - 2.1
mW, probe power: 0.15 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

F0 Measurement (MHz) Calculation (MHz)89
431:434
431:42985
300:951
300:955
88
183:264
183:24686
127:129
127:14390
116:265
116:28587
83:606
83:60493
091
65:01965:00492
107:884
107:87089
131:971
131:977
87
195:850
195:85688
231:921231:93590
313:511
313:52291351:392
351:38889
414:856
414:845

Mea.-Cal. (MHz)
0:005
0:004
0:017
0:014
0:020
0:002
0:015
0:014
0:006
0:006
0:014
0:011
0:004
0:012

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:013
eqQB (MHz)567:872(28)z
CB (MHz)
0:413339(14)dB (MHz)
0:2258(13)0:0628(19)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2454:1CX (MHz)
0:00315478
dX (MHz)y0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

194
B.31

R(32) 53-0

Table B.31: Hyper…ne spectrum recorded at R(32) 53-0. Pump power: 2.7 - 3.1
mW, probe power: 0.25 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

F0 Measurement (MHz) Calculation (MHz)33
452:076452:07529
240:316
240:335
32
176:361
176:35534
165:370
165:36937
103:313
103:29630
95:91895:90431
63:436
63:42335
036
33:09833:096
33
109:781
109:79531
211:158
211:15332
224:113
224:11234
276:222
276:21635
293:751
293:75633
392:736
392:738

Mea.-Cal. (MHz)
0:001
0:019
0:005
0:001
0:017
0:014
0:013
0:002
0:014
0:006
0:001
0:006
0:006
0:002

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:011
eqQB (MHz)567:752(27)z
CB (MHz)
0:437830(32)dB (MHz)
0:2383(12)0:0785(16)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:75
CX (MHz)
0:00315478
dX (MHz)y0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

195
B.32

R(51) 53-0

Table B.32: Hyper…ne spectrum recorded at R(51) 53-0. Pump power: 3 - 3.5
mW, probe power: 0.3 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
12
13
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)47
696:728
696:73252
562:532
562:53457
433:120
433:10848
411:628
411:63251
298:949
298:95653
263:445
263:452
49
228:228
228:22350
189:830
189:82456
155:643
155:64555
59:98760:00052
050
61:70861:71055
97:171
97:16951
108:513
108:51253
170:551
170:54654
215:124
215:11851
246:245
246:23752
278:494
278:49753
315:689
315:690

Mea.-Cal. (MHz)
0:004
0:002
0:012
0:004
0:007
0:007
0:006
0:006
0:002
0:013
0:002
0:002
0:001
0:005
0:007
0:008
0:003
0:001

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:0066
eqQB (MHz)568:171(11)z
CB (MHz)0:450037(11)dB (MHz)
0:24241(82)0:08388(55)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:05
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

196
B.33

P(70) 53-0

Table B.33: Hyper…ne spectrum recorded at P(70) 53-0. Pump power: 3.7 - 4.2
mW, probe power: 0.2 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

F0 Measurement (MHz) Calculation (MHz)69
487:144
487:13565
340:022340:03768
233:902
233:88670
177:705
177:70566
171:985
171:98467
131:193
131:17873
72:96072:93771
072
40:253
40:26769
75:55075:562
67
147:197
147:19068
178:672
178:68170
253:581
253:59371
287:196287:20669
358:195
358:190

Mea.-Cal. (MHz)
0:009
0:016
0:016
0:000
0:000
0:015
0:024
0:013
0:012
0:007
0:009
0:012
0:010
0:005

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:014
eqQB (MHz)568:623(34)z
CB (MHz)
0:466803(19)dB (MHz)
0:2535(15)0:0937(19)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:49
CX (MHz)
0:00315478
dX (MHz)y0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

197
B.34

P(89) 53-0

Table B.34: Hyper…ne spectrum recorded at P(89) 53-0. Pump power: 3.9 - 4.1
mW, probe power: 0.2 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
2
3
4
5
6
7
8
9
10
11
14
15
16
17
18
19
20

F0 Measurement (MHz) Calculation (MHz)88
472:286472:28084
386:075
386:08893
249:234
249:21287
231:148
231:12485
189:769
189:78389
150:824
150:84586
138:414
138:40785
54:50454:51692
090
37:024
36:996
86
125:571
125:60087
182:974
183:00891
235:089
235:11089
278:320
278:31087
321:650
321:63990
333:204
333:19788
370:022370:018

Mea.-Cal. (MHz)
0:006
0:013
0:022
0:024
0:014
0:022
0:006
0:012
0:027
0:028
0:034
0:021
0:011
0:011
0:007
0:004

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:021
eqQB (MHz)569:287(39)z
CB (MHz)
0:493519(20)dB (MHz)
0:2670(27)0:1078(18)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2454:07
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

198
B.35

R(21) 55-0

Table B.35: Hyper…ne spectrum recorded at R(21) 55-0. Pump power: 4.5 - 5.0
mW, probe power: 0.37 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
7
8
9
10
13
14
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)17
428:467
428:48022
332:184
332:18527
243:711
243:70318
185:247
185:227
19
0
20
57:271
57:27826
71:524
71:51924
113:118
113:11222
227:333
227:34220
267:258
267:27423
392:239392:23824
454:918
454:91021
469:676
469:67322
496:595
496:595
23
543:361
543:361

Mea.-Cal. (MHz)
0:013
0:001
0:008
0:020
0:007
0:005
0:006
0:009
0:016
0:002
0:008
0:004
0:001
0:001

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:010
eqQB (MHz)568:807(18)z
CB (MHz)
0:513082(32)dB (MHz)
0:2718(12)0:12206(91)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:64
CX (MHz)
0:00315478
dX (MHz)y0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

199
B.36

P(61) 55-0

Table B.36: Hyper…ne spectrum recorded at P(61) 55-0. Pump power: 2.8 - 3.1
mW, probe power: 0.2 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
8
9
10
11
12
13
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)55
430:752
430:76960
253:022
253:047
56
133:332
133:34265
79:462
79:44759
0
58
101:179
101:19064
185:056
185:06857
190:447
190:45862
234:852
234:82963
275:478
275:45860
309:023
309:01858
357:424
357:435
59
409:802409:811
63
429:101
429:11261
487:740
487:73662
537:737
537:72459
547:717
547:71460
587:742
587:74061
631:900
631:889

Mea.-Cal. (MHz)
0:017
0:025
0:010
0:015
0:010
0:012
0:011
0:022
0:020
0:004
0:011
0:008
0:011
0:004
0:012
0:004
0:002
0:012

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:015
eqQB (MHz)569:444(22)z
CB (MHz)
0:544230(21)dB (MHz)
0:2854(16)0:1444(13)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:27
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

200
B.37

P(83) 55-0

Table B.37: Hyper…ne spectrum recorded at P(83) 55-0. Pump power: 2.9 - 3.3
mW, probe power: 0.2 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
6
9
10
11
12
13
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)77652:372
652:39882
403:835
403:82878
335:857
335:86879
134:909
134:89979
086
86:447
86:473
84
113:515
113:49482
158:238
158:24485
167:113
167:10480
183:712
183:72281
246:188
246:19385
316:773316:78183
350:503
350:48581
384:586
384:57384
410:482
410:47382
437:482
437:47283
493:596493:591

Mea.-Cal. (MHz)
0:027
0:007
0:010
0:010
0:026
0:021
0:006
0:009
0:010
0:005
0:007
0:018
0:013
0:009
0:009
0:005

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:015
eqQB (MHz)570:208(27)z
CB (MHz)
0:579914(17)dB (MHz)
0:3006(16)0:1701(13)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:87
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

201
B.38

P(102) 55-0

Table B.38: Hyper…ne spectrum recorded at P(102) 55-0. Pump power: 3.8 - 4.3
mW, probe power: 0.34 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
5
6
7
8
9
10
11
12
13
14
15

F0 Measurement (MHz) Calculation (MHz)101
635:470
635:47197
611:087
611:111
99
341:713
341:666102
294:079
294:082105
98:506
98:483103
86:937
86:969101
73:527
73:52999
62:505
62:499104
18:193
18:201100
0102
133:752
133:762103
198:269
198:265101
208:666
208:664

Mea.-Cal. (MHz)
0:000
0:023
0:047
0:003
0:023
0:031
0:002
0:007
0:009
0:011
0:004
0:002

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:022
eqQB (MHz)571:145(42)z
CB (MHz)
0:626595(27)dB (MHz)
0:3212(21)(MHz)
0:2092(28)BHyper…ne parameters for the ground state X used in th …teqQX (MHz)2454:54
CX (MHz)
0:00315478
ydX (MHz)0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

202
B.39

P(19) 57-0

Table B.39: Hyper…ne spectrum recorded at P(19) 57-0. Pump power: 2.2 - 2.2
mW, probe power: 0.14 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
2
3
4
7
8
9
10
13
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)18
558:632
558:62223
469:088
469:07414
416:678
416:69115
231:451
231:42816
170:097
170:10822
150:824
150:84220
114:286
114:30118
0
16
35:121
35:114
17
99:434
99:431
21
108:059
108:05519
165:132
165:14820
230:981230:99717
240:902
240:91118
267:519
267:52919
316:778316:764

Mea.-Cal. (MHz)
0:010
0:014
0:013
0:024
0:011
0:018
0:015
0:007
0:002
0:004
0:017
0:016
0:009
0:009
0:014

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:015
eqQB (MHz)569:534(27)z
CB (MHz)
0:605935(82)dB (MHz)
0:3091(19)(MHz)
0:1993(11)B
Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:62
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

203
B.40

P(33) 57-0

Table B.40: Hyper…ne spectrum recorded at P(33) 57-0. Pump power: 4.4 - 4.9
mW, probe power: 0.31 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
18
19
20

F0 Measurement (MHz) Calculation (MHz)27
456:274
456:28632
332:844
332:85237
213:728
213:71128
186:110186:06631
63:368
63:33633
40:79840:80229
030
44:532
44:55536
76:30876:30734
129:422
129:40529
135:102
135:10435
162:236
162:21732
228:139
228:15230
280:455
280:49434
449:815
449:81131
471:647
471:65932
503:762
503:748

Mea.-Cal. (MHz)
0:012
0:008
0:018
0:045
0:032
0:004
0:024
0:001
0:017
0:002
0:018
0:013
0:038
0:005
0:011
0:014

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:023
eqQB (MHz)569:335(44)z
CB (MHz)
0:614577(46)dB (MHz)
0:3103(26)0:2058(20)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:76
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

204
B.41

R(65) 57-0

Table B.41: Hyper…ne spectrum recorded at R(65) 57-0. Pump power: 2.5 - 3
mW, probe power: 0.2 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
2
3
4
5
6
7
8
9
10
11
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)66
062
77:885
77:86771
229:054
229:02765
241:469
241:41663
277:207277:21067
319:838
319:97964
331:932
331:89863
410:792
410:86470
481:387
481:38768
509:301
509:40864
589:635
589:67465
651:382
651:38269
713:708
713:69467
750:519
750:48765
789:097
789:06368
811:485
811:48366
840:531
840:530
67
894:544
894:478

Mea.-Cal. (MHz)
0:018
0:027
0:053
0:003
0:141
0:034
0:072
0:000
0:108
0:039
0:000
0:013
0:032
0:034
0:002
0:001
0:066

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:060
eqQB (MHz)571:756(86)z
CB (MHz)
0:660831(90)dB (MHz)
0:4260(68)(MHz)
0:3647(59)BHyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:36
CX (MHz)
0:00315478
ydX (MHz)0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

205
B.42

R(74) 57-0

Table B.42: Hyper…ne spectrum recorded at R(74) 57-0. Pump power: 5.2 - 5.4
mW, probe power: 0.16 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

F0 Measurement (MHz) Calculation (MHz)75
489:552
489:56171
416:884
416:94374
255:753
255:69172
230:555
230:52973
171:628
171:56876
161:499
161:52879
077
34:745
34:67375
72:286
72:27278
93:62093:567
73
106:680
106:66174
156:671
156:68176
268:664
268:677
77
321:271
321:28375
354:572
354:571

Mea.-Cal. (MHz)
0:009
0:059
0:062
0:026
0:059
0:029
0:071
0:014
0:054
0:019
0:010
0:013
0:012
0:001

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:044
eqQB (MHz)570:210(94)z
CB (MHz)
0:678716(54)dB (MHz)
0:3092(44)0:2500(65)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:6CX (MHz)
0:00315478
dX (MHz)y0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

206
B.43

P(80) 57-0

Table B.43: Hyper…ne spectrum recorded at P(80) 57-0. Pump power: 5.7 - 6.0
mW, probe power: 0.19 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
9
12
13

F0 Measurement (MHz) Calculation (MHz)79
561:844
561:84375
502:683
502:70178
331:833
331:78076
312:486
312:49977
250:721250:69180
229:954
230:01483
58:66058:64279
078
81:331
81:33880
199:481
199:476

Mea.-Cal. (MHz)
0:001
0:019
0:053
0:013
0:030
0:060
0:018
0:007
0:005

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:037
eqQB (MHz)570:38(12)z
CB (MHz)
0:688579(74)dB (MHz)
0:298(14)0:240(21)
B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:78
CX (MHz)
0:00315478
dX (MHz)y0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

207
B.44

R(83) 57-0

Table B.44: Hyper…ne spectrum recorded at R(83) 57-0. Pump power: 4.2 - 4.8
mW, probe power: 0.2 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
8
9
14
15
16
17
18
21

F0 Measurement (MHz) Calculation (MHz)79
641:420
641:455
84
336:607
336:54280
314:070
314:07983
111:242
111:16481
101:945
101:91385
081
32:891
32:905
87
269:197269:22083
301:691
301:71387
418:846
418:90885
429:085
429:07283
440:033
440:02185
572:169
572:135

Mea.-Cal. (MHz)
0:035
0:066
0:009
0:078
0:031
0:014
0:023
0:022
0:062
0:013
0:011
0:035

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:046
eqQB (MHz)570:632(90)z
CB (MHz)
0:702332(76)dB (MHz)
0:311(12)(MHz)
0:280(11)B
Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:87
CX (MHz)
0:00315478
ydX (MHz)0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

208
B.45

R(95) 57-0

Table B.45: Hyper…ne spectrum recorded at R(95) 57-0. Pump power: 4.6 - 5
mW, probe power: 0.18 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a F0 Measurement (MHz) Calculation (MHz)2 92
387:112
387:192
3 96
364:004
364:0194 93
164:377
164:353
5 95
151:276
151:193
6 94
86:081
86:0137 93
29:022
29:054
8 97
14:711
14:779
9 101
0
10 94
178:608
178:632
13 100219:665
219:643
14 95
262:914
262:92115 99
277:825
277:728
16 95
401:440
401:469
17 97
413:479
413:46118 99
426:148
426:143
19 96
477:337
477:35420 98
495:774
495:771
21 97
556:222556:223

Mea.-Cal. (MHz)
0:080
0:016
0:024
0:083
0:068
0:032
0:068
0:024
0:022
0:007
0:097
0:029
0:017
0:005
0:017
0:003
0:001

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:051
eqQB (MHz)570:699(83)z
CB (MHz)
0:741638(48)dB (MHz)
0:3274(59)(MHz)
0:3327(50)BHyper…ne parameters for the ground state X used in th …teqQX (MHz)2454:28
CX (MHz)
0:00315478
ydX (MHz)0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

209
B.46

P(69) 58-0

Table B.46: Hyper…ne spectrum recorded at P(69) 58-0. Pump power: 2.5 - 3 mW,probe power: 0.2 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
3
4
9
10
11
12
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)63
064
314:464314:57467
483:552
483:83965
653:996
653:96772
760:887
761:10670
774:748
774:01868
814:304
814:40967
902:211
902:52471
988:523
988:77669
1012:634
1012:631
67
1043:160
1043:09770
1079:411
1079:31868
1098:391
1098:21369
1158:781
1158:773

Mea.-Cal. (MHz)
0:110
0:287
0:029
0:219
0:730
0:105
0:313
0:253
0:003
0:063
0:093
0:178
0:009

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:30
eqQB (MHz)553:38(69)z
CB (MHz)
0:72696(53)dB (MHz)
0:589(49)
0:726(32)
B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:47
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0.003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

Strength (a.u.)

-2

-1

0

1

2

-2

-1

0

1

0

0

a1
100

100

200

200

b7 b8
b9

a3

300

a15

400

a4

500

600

a7&8

b16

700

a21

b3

600

a5&6

b15

a16 a17 a18 a19 a20

b10 b11 b12&13 b14

500

Frequency (MHz)400

a10 a11 a12 a13&14

b6

300

a2

b2

800

b17 b18 b19 b20

700

a9
800

900

b21

b4 b5

Figure B.1: Hyper…ne spectrum recorded at P(69) 58-0 and R(65) 57-0. Hyper…ne components a1 - a21 belong to P(69) 58-0, b1 -b21
belong to R(65) 57-0. Vertical bars are …tting results using a four-term e¤ective hyper…ne Hamiltonian.

Strength (a.u.)

b1

210

211
B.47

P(17) 59-0

Table B.47: Hyper…ne spectrum recorded at P(17) 59-0. Pump power: 6.3 - 6.7
mW, probe power: 0.36 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
13
14
15
16
21

F0 Measurement (MHz) Calculation (MHz)11
0
16
105:642
105:66421
201:806
201:90212
232:772
232:80517
381:348
381:216
15
389:629
389:87313
419:638
419:50714
489:586
489:84620
524:174
524:13018
551:001
550:86516
662:991
663:05314
686:773
686:80815
756:124
756:19919
781:578781:537
17
982:114982:092

Mea.-Cal. (MHz)
0:021
0:096
0:034
0:132
0:244
0:131
0:260
0:044
0:136
0:062
0:035
0:076
0:041
0:021

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:080
eqQB (MHz)568:61(21)z
CB (MHz)
0:72138(46)dB (MHz)
0:287(19)0:139(16)
B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:61
CX (MHz)
0:00315478
dX (MHz)y0:001524
y(MHz)
0:003705
X* Not included in the …t.
** eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

Strength (a.u.)

-3

-2

-1

0

1

2

3

4

-4

-2

0

0

0

100

100

200

200

a1

300

300

400

400

a3

a15

a17

600

500

600

700

b15

a20

a18&19

700

a21
800

800

900

900

a6
a14
a8
b3 b4 a5
b6 b7
a9 a10b a
a11&12 10 13
b5 a7

b11&12 a16 b13 b14

500

b2

Frequency (MHz)

b1 a2

a4

b8

1000

1000

Figure B.2: Hyper…ne spectrum recorded at P(17) 59-0 and R(18) 59-0. Hyper…ne components a1 - a21 belong to P(17) 59-0, b1 -b15
belong to R(18) 59-0.

Strength (a.u.)

2

212

213
B.48

R(18) 59-0

Table B.48: Hyper…ne spectrum recorded at R(18) 59-0. Pump power: 6.3 - 6.7
mW, probe power: 0.36 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
5
6
7
8
10
11
12
13
14

F0 Measurement (MHz) Calculation (MHz)19015
204:841
204:62916
338:729
338:84323
354:575
354:76017
382:732
382:97421
450:367
450:12719
558:746
558:80517
655:972
656:08018
663:964
664:03020
732:906
732:81921
755:079
754:957

Mea.-Cal. (MHz)
0:212
0:114
0:185
0:242
0:240
0:059
0:108
0:067
0:086
0:122

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:20
eqQB (MHz)564:99(44)z
CB (MHz)
0:7332(14)dB (MHz)
0:197(26)(MHz)
0:030(37)B
Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:62
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* Not included in the …t.
** eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

214
B.49

P(27) 59-0

Table B.49: Hyper…ne spectrum recorded at P(27) 59-0. Pump power: 2.5 - 2.7
mW, probe power: 0.27 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
2
3
4
5
6
8
12
13
14
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)26
491:743
491:82131
373:759
374:29922
354:209354:16825
220:600
220:72227
203:946
203:67324
115:496
115:654
29
0
26
66:577
66:632
24
110:338
110:43027
235:758
235:80828
295:120
295:03125
306:135
306:16026
340:070
340:10727
384:520
384:435

Mea.-Cal. (MHz)
0:078
0:540
0:041
0:122
0:273
0:159
0:056
0:092
0:050
0:088
0:025
0:038
0:085

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:13
eqQB (MHz)573:49(28)z
CB (MHz)
0:71536(54)dB (MHz)
0:569(14)0:382(14)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:69
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* Not included in the …t.
** eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

Strength (a.u.)

-4

-2

0

2

4

6

-6

-4

-2

0

2

4

0

0

a1

100

100

b3 b4

b5

a2

200

200

b6 b7
300

b9

b10

a6

400

a17 a18

400

a5

a19

b11

a7

Frequency (MHz)300

a15 a16

a4

b8

a14

a3

a20

500

500

b12

a8

a21

b13

a9

600

600

b14

a10 a11

b15

a12

700

700

a13

a14

800

800

Figure B.3: Hyper…ne spectrum recorded at P(27) 59-0 and R(28) 59-0. Hyper…ne components a1 - a21 belong to P(27) 59-0, b3 -b15
belong to R(28) 59-0. Vertical bars are …tting results using a four-term e¤ective hyper…ne Hamiltonian.

Strength (a.u.)

6

215

216
B.50

R(28) 59-0

Table B.50: Hyper…ne spectrum recorded at R(28) 59-0. Pump power: 2.5 - 2.7
mW, probe power: 0.27 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
3
4
5
8
9
10
12
14
15

F0 Measurement (MHz) Calculation (MHz)28
030
21:725
21:876
26
62:10062:17431
194:318
194:38132
236:910
236:82729
289:840
289:89028
395:205
395:17631
488:189
488:16429
571:196571:211

Mea.-Cal. (MHz)
0:151
0:074
0:063
0:083
0:049
0:029
0:026
0:016

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:093
eqQB (MHz)573:32(48)z
CB (MHz)
0:72088(55)dB (MHz)
0:476(24)0:316(18)
B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:7CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

217
B.51

P(49) 59-1

Table B.51: Hyper…ne spectrum recorded at P(49) 59-1. Pump power: 4.6 mW, probe
power: 0.09 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger temperature:-2.28 C (vapor pressure: 3.3 Pa).
a
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)48
345:734
345:72344
247:286
247:26653
151:343
151:32747
95:746
95:73645
51:473
51:46949
34:934
34:90746
045
81:782
81:797
52
112:833
112:80550
148:415
148:41851
194:168
194:16348
214:852
214:87946
249:948
249:98547
309:851
309:89051
351:467
351:47849
397:873
397:87747
447:114
447:10350
456:179
456:14348
492:516
492:52649
542:739
542:738

Mea.-Cal. (MHz)
0:011
0:019
0:015
0:010
0:004
0:027
0:015
0:028
0:003
0:005
0:026
0:037
0:040
0:010
0:003
0:011
0:036
0:010
0:000

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:022
eqQB (MHz)571:695(39)z
CB (MHz)
0:750322(35)dB (MHz)
0:4326(26)0:3335(20)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:45
CX (MHz)
0:00318986
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

218
B.52

R(81) 59-0

Table B.52: Hyper…ne spectrum recorded at R(81) 59-0. Time constant of lock-in
ampli…er: 100 ms, cold-…nger temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
13
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)77
636:036
636:16778
299:459
299:48482
281:279
281:36179
77:525
77:537
81
66:353
66:35980
079
56:951
56:90783
64:794
64:68187
72:902
73:001
80
261:676
261:68686
296:240
296:28281
345:647
345:63485
354:050
353:98181
483:678483:66983
493:246
493:18785
503:852
503:82982
558:535
558:52584
575:676
575:65583
636:260
636:202

Mea.-Cal. (MHz)
0:131
0:025
0:082
0:012
0:006
0:043
0:112
0:099
0:010
0:041
0:013
0:069
0:008
0:059
0:022
0:010
0:020
0:058

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:065
eqQB (MHz)572:12(11)z
CB (MHz)
0:840592(60)dB (MHz)
0:4355(78)(MHz)
0:4337(59)BHyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:81
CX (MHz)
0:00315478
ydX (MHz)0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

219
B.53

P(87) 59-0

Table B.53: Hyper…ne spectrum recorded at P(87) 59-0. Pump power: 5.7 - 6 mW,
probe power: 0.19 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
7
8
9
10
13
14
15
16
17
18
19
20

F0 Measurement (MHz) Calculation (MHz)81
082
342:209342:24986
377:556
377:57383
567:983
568:03385
587:498
587:54783
702:696702:75087
728:626
728:56891
754:889
754:99384
912:514
912:56190
972:537
972:53685
1000:174
1000:18389
1026:670
1026:58685
1138:301
1138:335
87
1156:506
1156:469
89
1175:584
1175:566
86
1217:443
1217:430
881242:645
1242:583

Mea.-Cal. (MHz)
0:041
0:017
0:050
0:048
0:054
0:058
0:103
0:047
0:000
0:009
0:084
0:033
0:036
0:017
0:013
0:062

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:055
eqQB (MHz)572:555(90)z
CB (MHz)
0:856503(46)dB (MHz)
0:4360(65)0:4444(55)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2454:0CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

220
B.54

P(21) 60-0

Table B.54: Hyper…ne spectrum recorded at P(21) 60-0. Pump power: 4.7 - 5.3
mW, probe power: 0.32 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
7
8
9
10
11
12
13
14
15
16
17
20
21

F0 Measurement (MHz) Calculation (MHz)15
536:534
536:54020
420:233
420:23225
308:604
360:21216
288:380
288:378
17
99:785
99:787
18
39:968
39:96124
0
22
33:774
33:75817
45:51945:522
23
67:34067:33120
138:500
138:50318
171:639
171:63919
236:848
236:86123
254:163
254:18121
307:497
307:50620
407:927
407:91521
458:147
458:132

Mea.-Cal. (MHz)
0:006
0:001
51:608
0:002
0:002
0:007
0:016
0:003
0:009
0:003
0:001
0:013
0:018
0:009
0:012
0:015

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:011
eqQB (MHz)569:937(20)z
CB (MHz)
0:783323(37)dB (MHz)
0:3773(13)0:34612(86)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:64
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* Not included in the …t.
** eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

221
B.55

R(34) 60-0

Table B.55: Hyper…ne spectrum recorded at R(34) 60-0. Pump power: 6.6 - 6.8
mW, probe power: 0.4 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

F0 Measurement (MHz) Calculation (MHz)35
403:333
403:34331
245:273245:30734
142:603
142:57836
103:025
103:02732
87:467
87:46333
42:67842:65839
037
75:35475:321
38
121:291
121:27735
157:115
157:12633
231:602
231:62034
258:132
258:14436
336:298
336:29037
367:531
367:50935
439:296439:292

Mea.-Cal. (MHz)
0:011
0:033
0:025
0:002
0:004
0:020
0:033
0:014
0:011
0:018
0:012
0:009
0:023
0:004

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:021
eqQB (MHz)569:964(45)z
CB (MHz)
0:799015(53)dB (MHz)
0:3820(20)0:3619(30)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:78
CX (MHz)
0:00315478
dX (MHz)y0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

222
B.56

R(45) 60-1

Table B.56: Hyper…ne spectrum recorded at R(45) 60-1. Pump power: 8 mW, probe
power: 0.16 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger temperature:-2.28 C (vapor pressure: 3.3 Pa).
a
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)42
408:631
408:68451
293:039
292:87045
248:752
248:67343
209:932
209:92147
185:893
185:85344
155:234
155:18643
76:442
76:44850
28:760
28:69048
0:000
49
48:41048:41346
63:01963:041
44
91:795
91:77145
155:502
155:52149
207:805
207:86647
248:369
248:39545
292:724
292:74548
310:467
310:48246
340:319
340:28947
393:809
393:795

Mea.-Cal. (MHz)
0:053
0:168
0:079
0:011
0:041
0:049
0:006
0:070
0:004
0:021
0:024
0:019
0:061
0:026
0:021
0:015
0:029
0:013

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:043
eqQB (MHz)570:446(92)z
CB (MHz)
0:816760(82)dB (MHz)
0:3800(0:0062)
0:3793(37)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:38
CX (MHz)
0:00318986
dX (MHz)y0:001524
y0:003705
X (MHz)* Not included in the …t.
** eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

223
B.57

P(77) 60-0

Table B.57: Hyper…ne spectrum recorded at P(77) 60-0. Pump power: 7.5 mW, probe
power: 0.16 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger temperature:-2.28 C (vapor pressure: 3.3 Pa).
0

a

F

1
2
3
4
5
6
7
8
9
10
13
14
15
16
17
18
19
20
21

71
72
76
73
75
74
73
77
81
74
80
75
79
75
77
79
76
78
77

Fit 1: a1 -a10 , a13 -a21
Fit 2: a1 -a10 , a13
Measurement Calculation Mea.-Cal. Calculation Mea.-Cal.
(MHz)
(MHz)
(MHz)
(MHz)
(MHz)
0:000
335:373
334:984
0:388
335:5140:141
353:666
352:257
1:408
354:035
0:369
559:358
562:147
2:789
559:578
0:220
570:643
571:304
0:661
570:609
0:034
637:558
639:582
2:024
637:5180:040
694:979
698:194
3:216
695:096
0:118
702:798
703:467
0:669
702:235
0:563
714:630
718:434
3:804
714:9630:334
897:526
898:3960:870
897:636
0:110
937:374
939:232
1:858
937:075
0:299
987:660
990:276
2:616
984:829
2:831
1004:637
1001:267
3:370
995:625
9:012
1126:132
1127:383
1:251
1123:250
2:882
1143:943
1140:909
3:034
1134:240
9:703
1152:239
1149:468
2:770
1145:480
6:759
1197:484
1197:532
0:047
1196:070
1:414
1217:930
1216:899
1:031
1215:020
2:910
1283:635
1280:999
2:636
1276:850
6:785

Fitted hyper…ne parameters for the excited state B
Fit 1: a1 -a10 , a13 -a21
standard dev. (MHz)2:4
eqQB (MHz)550:7(3:8)z
CB (MHz)
0:9176(22)dB (MHz)
0:88(24)
0:30(18)
B (MHz)Fit 2: a1 -a10 , a13
0:33
563:68(89)z
0:91296(36)0:119(64)0:175(54)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:69
CX (MHz)
0:00315478
dX (MHz)y0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

0

a9

100

-1.0

-0.5

0.0

0.5

1.0

0

100

200

a10

a11&12

200

300

400

a16

500

a19

a18

500

a20

Frequency (MHz)

a15

400

a3

600

a21

600

700

700

Figure B.4: Hyper…ne spectrum recorded at P(77) 60-0.

a13

a14

300

Fit to four-term Hamiltonian: a1-a10, a13

-0.8

a2

Fit to four-term Hamiltonian: a1-a10, a13-a21

a1Strength (a.u.)Strength (a.u.)

-0.4

0.0

0.4

0.8

a4

800

800

a5

a6

900

a7

a9

224

225
B.58

P(84) 60-0

Table B.58: Hyper…ne spectrum recorded at P(84) 60-0. Pump power: 6.4 - 6.6
mW, probe power: 0.19 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

F0 Measurement (MHz) Calculation (MHz)79
383:291384:200
83
351:72680
170:299
170:52382
145:383
144:58881
86:608
86:10484
081
189:807
191:00983
207:952
207:79285
219:298
218:69387
228:97182
266:932
267:19186
297:494
299:53584
428:737
427:383
83
487:868
488:40885
504:357
503:047

Mea.-Cal. (MHz)
0:909
0:224
0:795
0:504
1:202
0:160
0:605
0:258
2:040
1:353
0:540
1:309

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)1:1
eqQB (MHz)566:1(3:2)z
CB (MHz)
0:9093(21)dB (MHz)
1:00(16)0:25(17)
B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:9CX (MHz)
0:00315478
dX (MHz)y0:001524
y(MHz)
0:003705
X* Not included in the …t.
** eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

Strength (a.u.)

-4

-2

0

2

4

6

-4

-2

0

2

4

0

0

a6

C7

100

100

C8

a1

C9

200

b1

a9

300

C13

b3

a12

a3
a4

400

400

C14

b4

a13

C15

a5

500

d1

a14

500

Frequency (MHz)

a11

300

b2

a10

C11 C12

a7 a8

200

C10

a2
a6

a15

700

600

700

Fit to four-term effective
hyperfine Hamiltonian(a2 and a10 exclueded.)

600

a7

800

800

Figure B.5: Hyper…ne spectrum recorded at P(84) 60-0. The vertical bars labeled a1 - a15 are the …tted spectrum using a four-terme¤ective Hamiltonian. Note that a2 and a10 components are excluded from the …t. b1 - b4 arise from u - g mixing between the B and
the 1g (1 g ) states and b2 is the crossover of b1 and b3 . c7 - c15 are identi…ed as lines of a neighboring transition R(30) 64-1, and d1 ,left of a14 , belongs to another transition, P(81) 59-0.

Strength (a.u.)

6

226

227
B.59

R(99) 60-0

Table B.59: Hyper…ne spectrum recorded at R(99) 60-0. Pump power: 5 - 5.7
mW, probe power: 0.2 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
17
18

F0 Measurement (MHz) Calculation (MHz)95
702:474
702:60896
329:889
329:961100
179:831
179:99197
76:59876:55699
0
98
33:094
33:14497
58:442
58:488101
201:312201:10598
298:417
298:442105
346:878
347:024100
380:897
380:83799
413:600
413:599102
448:910
448:698104
533:008
533:022101
628:219628:130100
660:375
660:380

Mea.-Cal. (MHz)
0:134
0:072
0:161
0:042
0:050
0:046
0:206
0:025
0:146
0:060
0:001
0:212
0:013
0:089
0:005

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:12
eqQB (MHz)573:28(23)z
CB (MHz)
1:03267(11)dB (MHz)
0:499(17)(MHz)
0:688(14)B
Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2454:42
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

228
B.60

P(17) 61-0

Table B.60: Hyper…ne spectrum recorded at P(17) 61-0. Pump power: 7.7 - 8.7
mW, probe power: 0.33 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
7
8
9
10
11
12
13
14
15
16
17
19
20
21

F0 Measurement (MHz) Calculation (MHz)11
797:493
797:51416
683:672
683:67121
574:382
574:35812
562:972
562:95513
374:641
374:64714
303:201
303:18920
254:512
254:53318
233:833
233:848
13
218:245
218:25019
199:944
199:94416
126:262
126:26214
105:785
105:774
15
34:674
34:66519
0
17
42:70842:71118
117:606
117:61216
138:459
138:455
17
195:324
195:324

Mea.-Cal. (MHz)
0:021
0:001
0:024
0:017
0:005
0:012
0:021
0:015
0:005
0:000
0:001
0:010
0:008
0:003
0:006
0:004
0:001

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:013
eqQB (MHz)569:550(24)z
CB (MHz)
0:850067(59)dB (MHz)
0:3944(14)(MHz)
0:4414(10)BHyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:61
CX (MHz)
0:00315478
ydX (MHz)0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

229
B.61

R(36) 61-0

Table B.61: Hyper…ne spectrum recorded at R(36) 61-0. Pump power: 6.2 - 6.9
mW, probe power: 0.38 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
3
6
7
8
9
10
11
12
13
14
15

F0 Measurement (MHz) Calculation (MHz)37
421:236
421:23836
165:277
165:26335
68:883
68:89241
039
66:597
66:57540
116:201
116:17537
139:458
139:47335
205:630
205:61336
236:722
236:73738
322:588322:595
39
358:049
358:04937
421:502421:504

Mea.-Cal. (MHz)
0:002
0:015
0:009
0:022
0:026
0:014
0:017
0:015
0:007
0:000
0:002

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:016
eqQB (MHz)569:603(42)z
CB (MHz)
0:875111(89)dB (MHz)
0:4056(19)0:4638(27)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:8CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

230
B.62

P(53) 61-0

Table B.62: Hyper…ne spectrum recorded at P(53) 61-0. Pump power: 3.6 - 3.8
mW, probe power: 0.27 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
2
3
4
5
6
9
10
11
12
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)52
397:073
397:10448
338:646
338:69351
158:392
158:31957
145:488
145:45349
133:464
133:43949
0
56
105:502
105:49854
120:218
120:14252
164:111
164:13351
248:943
249:00755
332:913
332:92553
358:141
358:10351
386:464
386:46354
426:238
426:23252
442:416
442:42953
502:797502:773

Mea.-Cal. (MHz)
0:031
0:047
0:073
0:035
0:025
0:004
0:076
0:022
0:064
0:012
0:038
0:001
0:006
0:013
0:023

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:044
eqQB (MHz)570:037(79)z
CB (MHz)
0:906087(75)dB (MHz)
0:4162(42)(MHz)
0:5154(42)BHyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:09
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

231
B.63

P(83) 61-0

Table B.63: Hyper…ne spectrum recorded at P(83) 61-0. Pump power: 2.9 - 3.3
mW, probe power: 0.3 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)78
507:178
507:35782
431:730
431:75479
271:267
271:25081
231:558
231:44180
179:580
179:47979
136:509
136:51183
70:062
70:12387
080
83:07083:11682
129:832
129:88884
159:110
158:99981
181:304
181:36486
207:358
207:32485
250:076
249:91481
319:585
319:63283
358:725
358:68685
399:644
399:58882
409:168
409:22984
456:054
456:05283
501:810
501:779

Mea.-Cal. (MHz)
0:180
0:023
0:017
0:117
0:101
0:002
0:061
0:046
0:055
0:110
0:060
0:034
0:163
0:048
0:040
0:057
0:061
0:002
0:030

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:087
eqQB (MHz)570:14(14)z
CB (MHz)
1:02283(10)dB (MHz)
0:422(10)0:6224(83)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:87
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

232
B.64

P(23) 63-0

Table B.64: Hyper…ne spectrum recorded at P(23) 63-0. Pump power: 10 - 12
mW, probe power: 0.56 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
11
12
13
14
15
16
17
18
19
21

F0 Measurement (MHz) Calculation (MHz)17
512:525
512:61022
369:072
369:10718
254:068
254:128
27
222:042
221:966
21
98:97599:00823
77:568
77:62119
60:721
60:72120
024
99:857
99:80825
139:832
139:81222
190:842
190:84320
217:894
217:900
21
285:940
285:92225
322:664
322:66823
366:131
366:08221
424:485
424:46224
436:181436:15923
516:474
516:446

Mea.-Cal. (MHz)
0:085
0:034
0:060
0:076
0:033
0:053
0:000
0:048
0:021
0:001
0:006
0:018
0:003
0:050
0:024
0:022
0:028

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:046
eqQB (MHz)568:464(78)z
CB (MHz)
1:01785(15)dB (MHz)
0:4219(56)(MHz)
0:6974(44)BHyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:66
CX (MHz)
0:00315478
ydX (MHz)0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

233
B.65

R(31) 63-0

Table B.65: Hyper…ne spectrum recorded at R(31) 63-0. Pump power: 11.5 - 12
mW, probe power: 0.56 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 0.5 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)27
538:951
539:09132
350:428
350:51328
259:187
259:239
37
156:654
156:52331
93:728
93:72329
59:582
59:59633
45:24045:29430
029
75:88975:86136
120:878
120:88734
139:986
139:89935
187:926
187:87432
210:440
210:42230
232:792
232:79031
301:834
301:79335358:922358:87733
394:247
394:12631
438:926
438:95034
464:144464:17632
486:095
486:05033
542:322542:321

Mea.-Cal. (MHz)
0:140
0:084
0:052
0:132
0:005
0:014
0:054
0:028
0:009
0:086
0:052
0:019
0:001
0:042
0:045
0:120
0:024
0:032
0:045
0:001

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:070
eqQB (MHz)568:62(11)z
CB (MHz)
1:03395(15)dB (MHz)
0:4281(76)(MHz)
0:7225(58)BHyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:74
CX (MHz)
0:00315478
ydX (MHz)0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

234
B.66

P(43) 63-0

Table B.66: Hyper…ne spectrum recorded at P(43) 63-0. Pump power: 2.9 - 3.1
mW, probe power: 0.31 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 0.5 Pa).
a
2
3
6
7
8
9
10
11
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)42
401:739
401:72838
338:308
338:360
39
133:354
133:29943
82:40882:43440
71:93971:89239
046
99:72999:76044
111:260
111:20141
245:329
245:37645
328:154
328:18643
352:523
352:51641
382:527
382:535
44
423:699
423:67942
438:014438:00343
498:539
498:517

Mea.-Cal. (MHz)
0:011
0:053
0:055
0:026
0:047
0:030
0:059
0:047
0:032
0:007
0:009
0:021
0:011
0:022

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:040
eqQB (MHz)568:854(82)z
CB (MHz)
1:05691(12)dB (MHz)
0:4436(43)0:7560(36)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:91
CX (MHz)
0:00315478
dX (MHz)y0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

235
B.67

P(52) 63-0

Table B.67: Hyper…ne spectrum recorded at P(52) 63-0. Pump power: 3.2 - 3.5
mW, probe power: 0.11 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 0.5 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

F0 Measurement (MHz) Calculation (MHz)51
512:816
512:83947
458:053
458:12250
281:919281:86548
269:828
269:80949
204:146
204:07152
181:445
181:43055
053
21:992
21:88951
48:952
48:93549
72:012
72:00654
89:917
89:86950
126:788
126:80552
252:280
252:28653
309:774
309:76251
330:435
330:441

Mea.-Cal. (MHz)
0:024
0:069
0:054
0:019
0:075
0:014
0:103
0:017
0:006
0:048
0:017
0:007
0:011
0:005

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:051
eqQB (MHz)569:08(11)z
CB (MHz)
1:084930(90)dB (MHz)
0:4532(49)0:7987(73)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:07
CX (MHz)
0:00315478
dX (MHz)y0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

236
B.68

P(19) 65-0

Table B.68: Hyper…ne spectrum recorded at P(19) 65-0. Pump power: 7 mW,time constant of lock-in ampli…er: 100 ms, cold-…nger temperature: -2.28 C (vapor
pressure: 0.5 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)13
298:340
298:40818
154:044
154:03914
50:181
50:20223
0
17
119:391
119:42919
134:999
134:98015
144:428
144:432
16
214:591
214:62115
292:968
292:97722
305:584
305:52720
315:048
314:98421
357:001
356:91318
405:858
405:90316
421:004
421:06017
495:405
495:42121551:325
551:29219
582:592
582:57217
635:772
635:800
20
662:449
662:43418
675:359
675:38519
735:217735:206

Mea.-Cal. (MHz)
0:068
0:006
0:021
0:039
0:019
0:004
0:030
0:009
0:056
0:064
0:089
0:045
0:056
0:016
0:033
0:019
0:028
0:015
0:026
0:011

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:043
eqQB (MHz)566:342(63)z
CB (MHz)
1:21056(14)dB (MHz)
0:4648(37)(MHz)
1:0373(32)BHyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:62
CX (MHz)
0:00315478
ydX (MHz)0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

237
B.69

P(33) 65-0

Table B.69: Hyper…ne spectrum recorded at P(33) 65-0. Pump power: 11 - 12
mW, probe power: 0.57 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)27
626:827
626:89432
410:684
410:69128
338:119
338:16437
181:026
180:98531
160:612
160:57729
134:159
134:135
33
97:27997:32830
71:597
71:54429
030
166:034
166:09731
238:347
238:35835
318:010318:00733
341:678341:63431
375:236
375:20834
417:943
417:15432
429:299
429:29433
489:868
489:808

Mea.-Cal. (MHz)
0:067
0:006
0:045
0:041
0:035
0:024
0:049
0:053
0:063
0:012
0:003
0:044
0:028
0:789
0:004
0:060

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:046
eqQB (MHz)566:504(73)z
CB (MHz)
1:24010(14)dB (MHz)
0:4893(33)1:0966(34)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:76
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* Not included in the …t.
** eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

238
B.70

R(43) 65-0

Table B.70: Hyper…ne spectrum recorded at R(43) 65-0. Pump power: 10.5 mW,
probe power: 0.41 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger temperature: -2.28 C (vapor pressure: 0.5 Pa).
a
1
2
3
8
9
10
14
15
19
20
21

F0 Measurement (MHz) Calculation (MHz)39
661:109
661:15144
367:446
367:40940
349:125
349:14345
35:32835:37341
046
171:306
171:28447
240:460
240:48643
269:130
269:08844
474:777
474:78346
485:587
485:66045
547:114547:020

Mea.-Cal. (MHz)
0:042
0:037
0:017
0:045
0:022
0:026
0:041
0:007
0:072
0:094

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:056
eqQB (MHz)567:21(11)z
CB (MHz)
1:28195(20)dB (MHz)
0:4901(68)(MHz)
1:1712(49)B
Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:91
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

239
B.71

P(33) 69-0

Table B.71: Hyper…ne spectrum recorded at P(33) 69-0. Pump power: 11 - 12.2
mW, probe power: 0.31 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
6
9
12
13
14
15
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)27
606:777
607:56930
029
53:172
53:17032
261:448
261:495
36
284:401
284:44635
321:367
321:29331
331:113
331:10435
492:871
492:88432
544:099
544:09034
572:866
572:835
33
622:070622:099

Mea.-Cal. (MHz)
0:792
0:002
0:047
0:045
0:073
0:008
0:013
0:009
0:031
0:029

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:044
eqQB (MHz)558:63(12)z
CB (MHz)
1:85049(22)dB (MHz)
0:6800(72)(MHz)
2:3114(69)BHyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:76
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* Not included in the …t.
** eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

240
B.72

P(39) 69-0

Table B.72: Hyper…ne spectrum recorded at P(39) 69-0. Pump power: 12.2 - 12.9
mW, probe power: 0.33 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
9
10
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)33
580:517580:62934
255:867
255:89638
218:638
218:82035
25:31425:31637
036
61:36861:432
35
105:586
105:69039
131:165
130:98743
183:857
184:84536
309:903
309:93641
444:440
444:50637
542:312542:23939
570:913
570:75841
610:643
610:78838
631:723
631:68140
678:759
678:78739
718:777
718:641

Mea.-Cal. (MHz)
0:112
0:029
0:183
0:003
0:063
0:104
0:178
0:988
0:033
0:066
0:073
0:155
0:145
0:042
0:028
0:135

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:12
eqQB (MHz)560:05(23)z
CB (MHz)
1:89425(41)dB (MHz)
0:696(12)2:422(10)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:85
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* Not included in the …t.
** eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

241
B.73

R(44) 69-0

Table B.73: Hyper…ne spectrum recorded at R(44) 69-0. Pump power: 16 - 17
mW, probe power: 0.5 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
1
2
3
4
5
6
7
8
11
13
14
15

F0 Measurement (MHz) Calculation (MHz)41
589:777
589:82745
520:813
520:85342
370:917
370:94144
319:268
319:20743
272:997
272:97046
154:066
154:05543
045
45:463
45:43649
137:248
138:27146
285:342
285:33445
324:711
324:55147
374:804
374:868

Mea.-Cal. (MHz)
0:050
0:040
0:025
0:062
0:027
0:011
0:027
1:023
0:007
0:160
0:065

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:076
eqQB (MHz)561:17(27)z
CB (MHz)
1:96255(38)dB (MHz)
0:732(12)2:5258(96)B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:93
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* Not included in the …t.
** eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

242
B.74

R(49) 69-0

Table B.74: Hyper…ne spectrum recorded at R(49) 69-0. Pump power: 14.3 - 15.2
mW, probe power: 0.36 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 3.3 Pa).
a
2
3
4
5
6
7
8
9
10
11
12
13
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)46
523:882
524:34350
380:255
380:08447
268:727
268:74249
192:278
191:88748
157:767
157:45347
137:994
137:93151
048
98:58998:70455
165:183
165:48150
186:022
186:21249
218:982
219:15052
253:243
253:09251
437:234
437:16350
471:497
471:60253
529:132
528:99252
569:054
569:04751
583:396583:252

Mea.-Cal. (MHz)
0:461
0:171
0:016
0:390
0:313
0:063
0:115
0:298
0:190
0:169
0:151
0:071
0:105
0:139
0:008
0:144

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:24
eqQB (MHz)560:28(43)z
CB (MHz)
2:01971(48)dB (MHz)
0:862(28)2:597(25)
B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:01
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

243
B.75

P(53) 69-0

Table B.75: Hyper…ne spectrum recorded at P(53) 69-0. Pump power: 11.8 - 12.3
mW, probe power: 0.32 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 0.5 Pa).
a
1
2
3
4
5
6
7
8
9
12
13
14
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)47
739:507
739:79748
373:953
374:12252
211:757
212:06349
114:853
114:91351
29:775
29:86050
049
15:948
15:98053
174:497
174:23150
259:066
258:93651
382:908
382:70154
432:661
432:59851
518:257
518:00353
610:770
610:52052
640:376
640:18855
711:531
711:98554
746:133
746:29953
756:360
756:079

Mea.-Cal. (MHz)
0:290
0:169
0:306
0:060
0:085
0:032
0:267
0:130
0:207
0:062
0:254
0:250
0:188
0:454
0:167
0:281

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:25
eqQB (MHz)562:09(50)z
CB (MHz)
2:04973(57)dB (MHz)
0:793(28)2:691(22)
B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:09
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

244
B.76

P(35) 70-0

Table B.76: Hyper…ne spectrum recorded at P(35) 70-0. Pump power: 8.7 - 9.6
mW, probe power: 0.24 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 0.5 Pa).
a
2
3
4
5
6
7
8
9
10
13
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)30
563:491
563:88734
528:460
528:51331
332:665
332:77833
307:378
307:19932
245:465
245:31531
202:640
202:59035
179:632
179:71539
124:601
124:33732
033
98:944
98:95838
109:664
109:66837
135:050
134:85933
232:623
232:66235
262:622
262:52837
305:238
305:13134
323:973
323:97336
374:275
374:17335
411:178
411:038

Mea.-Cal. (MHz)
0:396
0:053
0:113
0:179
0:150
0:050
0:083
0:264
0:015
0:004
0:192
0:039
0:094
0:106
0:001
0:102
0:140

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:17
eqQB (MHz)556:68(29)z
CB (MHz)2:08631(41)dB (MHz)
0:832(17)(MHz)
2:815(12)B
Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:79
CX (MHz)
0:00315478
ydX (MHz)0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

245
B.77

R(37) 70-0

Table B.77: Hyper…ne spectrum recorded at R(37) 70-0. Pump power: 12 - 13.3
mW, probe power: 0.34 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 0.5 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)33
762:560
762:98734
431:176
431:30438
359:002358:82335
191:524
191:41437
148:157
147:76236
95:586
95:25335
61:622
61:40539
043
93:693
94:097
36
151:684
151:81738
208:463
208:60040
235:942
235:78237
259:070259:16142
316:988
317:07941
333:885
333:69937
392:762
392:83439
441:726
441:60938
493:749
493:79841
502:559
502:42740
562:065
561:96939
589:733
589:542

Mea.-Cal. (MHz)
0:427
0:128
0:179
0:111
0:395
0:333
0:217
0:404
0:133
0:137
0:160
0:091
0:091
0:186
0:073
0:117
0:049
0:132
0:096
0:191

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:23
eqQB (MHz)557:55(32)z
CB (MHz)
2:12504(50)dB (MHz)
0:873(21)(MHz)
2:884(18)B
Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:82
CX (MHz)
0:00315478
ydX (MHz)0:001524
y(MHz)
0:003705
X* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

246
B.78

R(45) 70-0

Table B.78: Hyper…ne spectrum recorded at R(45) 70-0. Pump power: 14.5 - 15.5
mW, probe power: 0.37 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 0.5 Pa).
a
2
3
4
5
6
7
8
9
10
11
12
13
14
17
18
19
20
21

F0 Measurement (MHz) Calculation (MHz)42
531:974
532:26446
382:131
382:00143
274:590274:57745
194:590
194:25344
161:667
161:37243
145:104
144:97147
044
93:215
93:22751
176:439
178:04146
185:292
185:37245
216:692
216:695
48
256:183
256:20545
350:577
350:48947
439:247
439:14946
471:892
471:88849
537:401
537:67648
576:083
576:19047
585:542
585:354

Mea.-Cal. (MHz)
0:290
0:130
0:012
0:337
0:294
0:133
0:012
1:602
0:080
0:003
0:022
0:088
0:098
0:004
0:275
0:106
0:187

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0.20
eqQB (MHz)558:80(44)z
CB (MHz)
2:22994(62)dB (MHz)
0:937(24)(MHz)
3:045(17)B
Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2452:94
CX (MHz)
0:00315478
ydX (MHz)0:001524
y(MHz)
0:003705
X* Not included in the …t.
** eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

247
B.79

P(63) 70-0

Table B.79: Hyper…ne spectrum recorded at P(63) 70-0. Pump power: 9.2 - 10.2
mW, probe power: 0.39 mW, time constant of lock-in ampli…er: 100 ms, cold-…nger
temperature: -2.28 C (vapor pressure: 0.5 Pa).
a
1
2
3
4
5
6
7
8
9
10
11
12
13
16
17
18
19

F0 Measurement (MHz) Calculation (MHz)57
857:211
857:99858
434:684
434:92059
124:932
124:81762
51:34950:91159
060
42:332
42:58961
71:218
69:66960
302:823
303:00963
383:840
384:29761
477:670
477:77862
503:611
503:87861
605:952
605:51264
691:569
691:69263
811:490811:28665
863:653
863:53766
913:131
913:46863
950:210
949:238

Mea.-Cal. (MHz)
0:787
0:235
0:115
0:437
0:257
1:548
0:186
0:457
0:108
0:268
0:440
0:123
0:204
0:116
0:336
0:972

Fitted hyper…ne parameters for the excited state B
standard dev. (MHz)0:62
eqQB (MHz)584:3(1:2)z
CB (MHz)
2:5889(11)dB (MHz)
2:408(50)3:456(43)
B (MHz)Hyper…ne parameters for the ground state X used in th …teqQX (MHz)2453:31
CX (MHz)
0:00315478
dX (MHz)y0:001524
y0:003705
X (MHz)* eqQX and CX are determined by equations B.2 and B.3, respectively.
ydX and X are …xed in the …t to 1.524 and 3.705 kHz, respectively.
zQuoted uncertainties (1 ) are estimated from the standard deviation of the …t.

Strength (a.u.) Strength (a.u.) Strength (a.u.) Strength (a.u.)

0

0

a5

0

0

a6

100

100

a7

100

a2

100

a1

200

200

200

200

a3

300

a14 a15 a16

300

a8

300

300

a4

a19

500

a20

500

a10

500

a5

500

a21

a11

a6

600

600

a12

600

a7

600

Frequency (MHz)400

a17 a18

400

a9

400

400

a2

700

700

a13

700

700

800

800

a14 a15 a16

800

a8

800

a3

900

900

a17

900

a9

900

a10

1000

1000

a5

1000

1000

a18 a19a4

Figure B.6: Hyper…ne spectrum recorded at P(63) 70-0. Vertical bars are …tting results using a four-term e¤ective hyper…ne Hamiltonian.

-1.0

0.0

1.0

-1.0

0.0

1.0

-1.0

0.0

1.0

-1.0

0.0

1.0

248