Cavity-enhanced direct frequency comb
spectroscopybyMichael J. Thorpe
B.S., Western Washington University, 2001

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2009This thesis entitled:
Cavity-enhanced direct frequency comb spectroscopy
written by Michael J. Thorpe
has been approved for the Department of Physics

Jun YeEric A. Cornell

Date

The final copy of this thesis has been examined by the signatories, and we findthat both the content and the form meet acceptable presentation standards ofscholarly work in the above mentioned discipline.

Thorpe, Michael J. (Ph.D., Physics)
Cavity-enhanced direct frequency comb spectroscopy
Thesis directed by Professor Adjoint Jun Ye

Cavity-enhanced direct frequency comb spectroscopy (CE-DFCS) combines
broad spectral bandwidth, high spectral resolution, precise frequency calibration,and ultra-high detection sensitivity, all in one experimental platform based on
an optical frequency comb interacting with a high-finesse optical cavity. Precise
control of the optical frequency comb allows highly efficient, coherent couplingof individual comb components with corresponding resonant modes of the highfinesse cavity. The use of low dispersion mirrors permits almost the entire spectral
bandwidth of the frequency comb to be employed for detection, covering a range
of > 10% of the actual optical frequency. The long cavity lifetime dramaticallyenhances the effective interaction between the light field and intra-cavity matter,increasing the sensitivity for measurement of optical losses by a factor that is on
the order of the cavity finesse. Finally, the light transmitted from the cavity isspectrally resolved to provide a multitude of detection channels with spectral resolutions ranging from several gigahertz to hundreds of kilohertz.
We begin with a discussion of the principles of cavity-enhanced direct frequency comb spectroscopy including the properties of frequency combs and optical cavities that are relevant to CE-DFCS systems. Methods for characterizing
cavity-comb interactions and achieving simultaneous broad bandwidth and highresolution detection of the cavity transmitted beam are discussed. Measurements
are presented that fully characterize the detection sensitivity, resolution, and spectral coverage of several CE-DFCS systems. We also outline several types of UV,
optical, and IR frequency comb sources and optical cavity designs that can beiv
used for specific spectroscopic applications. Finally, we present a series of experimental measurements on trace gas detections, human breath analysis, andcharacterization of cold molecular beams. These results demonstrate clearly thatthe wide bandwidth and ultrasensitive nature of the femtosecond enhancement
cavity enables powerful real-time detection and identification of many molecularspecies including their quantum state distributions in a massively parallel fashion.

Dedication

To my parents and grandparents who always emphasized
the importance of education.
For Mariah Palmer, the love of my life.

Acknowledgements

If you would have asked me at the beginning of my freshman year of college,“where will you be in 12 years?”, I think that “preparing to defend a PhD
thesis in physics,” would have been an unlikely answer. When I entered Western
Washington University in the fall of 1996, I was very interested in philosophy andunder the impression that philosophers were somehow the best suited to tackle thecomplicated problems facing society. However, after my first quarter of philosophy classes, I found myself dissatisfied with the process of formulating arguments
about topics that clearly had no definitive answer. I also sensed that philosophers
purposely avoided topics where answers could be reached because such questions
had already been, or would likely be, answered by the various fields of science. It
was this thinking that led me to enroll in my first physics class freshman year. I
was immediately impressed by the rigorous development of the topics of physics,
and how a few well constructed fundamental principles could describe such a wide
variety of phenomena. Needless to say, I was hooked!
During my time at Western, I had many instructors to whom I’m greatly
indebted. Most notably were Richard Atneosen and Brad Johnson. Over thecourse of several optics, lasers, and holography courses, Rich Atneosen helped me
develop a love for the field of optics that remains at the heart of my research interests today. Brad Johnson provided me with my first glimpses of the incredible
breadth, complexity and beauty of physics. His courses on quantum mechanics,

vii
solids, and general relativity always left me dreaming about the solutions to great
problems.
After my undergraduate studies I had the pleasure of experiencing the commercial world for the first time as a technician working for a small optics start-up
company called Orca Photonic Systems. The company had nine employees, mostof which held masters degrees or higher, and two of the three founders received
their PhD’s from JILA. A bit of foreshadowing I guess! At Orca, I learned a great
deal of practical skills that are essential in any technical pursuit, academic orcommercial. Steve Moody and Doug Donaldson taught me about circuit design.
Matt Moore patiently helped me learn the basics of computer programming. Dave
Cunningham taught me how to design optical systems and more importantly, how
to troubleshoot them. Finally, Bill Pence introduced me to the art of machining,a skill that has been incredibly useful for designing experiments throughout my
PhD studies. After a couple of years at Orca, their guidance also helped me realizethat I wanted to pursue a PhD in physics.
During college and my time at Orca, my love for physics was always slightly
overshadowed by my love of the outdoors, in particular the mountains, skiing,and climbing. So when it was time to apply for graduate school, the Universityof Colorado was on a short list of schools that could satisfy both my interest in
physics and my love of the outdoors. At the time, I wasn’t aware that CU hada tendency to attract outdoor enthusiasts. However, after arriving, I quickly realized that many of the students shared my motivation for choosing CU. Overthe course of the last five years I was privileged to spend so many weekends withgreat friends like Dave Hume, Paul Parazzoli, Ben Egbers, Chris Ticknor, Patrick
Maletinski, David James, Darren Hudson, Eric Hudson, Jason Brietbarth, Paul
Kuntz, Ron Pepino, and Evan Salim, climbing, skiing, biking, rafting, and hiking
in the outdoor playgrounds of Colorado. These times were essential for rejuvenat-

viii
ing my spirits and recharging my interest and curiosity for the next week in thelab. Since most of these guys are also physicists, we spent much of these times
discussing physics ideas and experiments, which only heightened the outdoor experience.
I think that I was blessed with good timing when I applied to the graduate program at CU. In the spring of 2003, the Optical Science and Engineering
Program, started by Dana Anderson, was still young and growing rapidly. This
education program, funded by NSF/IGERT, was designed to train physics, chemistry, and engineering graduate students in the field of optics. It was the perfectopportunity for a student with my background to secure funding for my PhD
studies. I attribute a large portion of my success at CU to having received an
OSEP fellowship. In particular, I think Jun’s decision to hire me was in largepart due to my OSEP funding. I still remember the day that I asked Jun if I
could work in his lab. I could tell that he was reluctant, and since I didn’t havea research background to speak of, I could offer no good reason why he should
hire me. In the end he agreed to give me a chance, and I set out to prove to himthat he made the right choice. I hope that, over the course of the last five years,
I have succeeded in that goal. Jun has truly been an incredible advisor. He hastaught me how to be an experimental physicist, and has given me extraordinary
opportunities to explore my own creativity that will certainly prove invaluable in
my future endeavors.
It has been a great pleasure working in Jun’s lab. The atmosphere is one ofhard work, ingenuity, and collaboration. I am especially grateful to Kevin Moll,
Thomas Schibli, Jan Hall, Matt Stowe, Jason Jones, and Eric Hudson for theirsage-like advice and undying willingness to help me find answers to my questions.
I’m also indebted to the many people who helped develop the cavity-enhanced
frequency comb spectroscopy project: Kevin Moll, Jason Jones, David Balslev-

ix
Clausen, Darren Hudson, Ben Safdi, Matt Kirchner, Kevin Cossel, and Florian
Adler. Probably the best aspect of Jun’s lab is that there is such a wide variety
of different experiments being conducted in such a small lab space. The result isthat the collective Ye labs team has a huge amount of technical expertise covering
a wide range of topics. If you have a question, chances are that someone within
50 feet of you knows the answer. Aside from the scientific resource, the ability
to learn about, and track the progress of, so many different experiments was agreat experience. Fortunately people like Thomas Schibli, Brian Sawyer, Seth
Foreman, Marty Boyd, Dylan Yost, Matt Stowe, Gretchen Campbell, Sebastian
Blatt, and Dajun Wang were consistently willing to spend time telling me about
their experiments.
The high quality of the experiments at JILA is in large part due to the excellent technical support staff. I’d like to thank Terry Brown, James Fung-a-Fat,
Mike Whitmore, Carl Sauer, and Dave Tegart for their help with the many custom
electronics used in my experiments. Also, thanks to Blaine Horner, Tom Foote,Todd Asnicar, Hans Greene, and David Alchenberger for their help with the mechanical design and fabrication of experimental components. Finally, the people
in computing department, the scientific reports office, and the many administrative staff members all contributed to making my time at JILA both productive
and rewarding.
Of course, nothing good gets done at work without a whole bunch of support
at home. I am fortunate that I have inherited a strong work ethic and sense of
integrity that was first shown to me by my parents Jim and Brenda. During mytime growing up in Idaho they not only taught me how to carry my own weight
in society, but they also provided me with endless opportunities to explore all of
the richness that life has to offer. I can’t begin to express how grateful I am fortheir love and support. Similarly, my grandparents, Ruth, Cecil, Evelyn, and Billx
always took a very active role in my life and I deeply appreciate all they havegiven me. While I’ve been consumed with school for the last five years, it has
been a pleasure to watch my sister, Heather, start a family of her own. I hope tosoon follow in her footsteps. Finally, it was especially nice to have relatives in the
Boulder area during graduate school. A big thanks to John and Marsha for all of
the great meals and good times.
Over the years, many of my friends have helped me find my course in life.
The most notable influence came from Seth Rigles. During my high school years,
Seth helped me develop virtues such as patience and critical curiosity that have
served me well to this day. I deeply appreciate the time we spent discussing issues
and forming our adult perspectives of the world. I am also grateful for the freindship of Ben Egbers. By chance and good fortune, our paths have kept us in close
contact over the past decade. Ben’s good nature and easy going temperament
always remind me to try and be a better person.
Finally, I’d like to thank my lovely fiancée Mariah Palmer for her unwavering love, support, patience, and encouragement. Her many great qualities are
everything that I could hope for in a life partner. I am so excited about our future
together.

Contents

Chapter
1 Introduction1

1.1

Spectroscopy before the laser. . . . . . . . . . . . . . . . . . . .

1

1.2

The invention and development of laser spectroscopy . . . . . . .

3

1.3

Modern laser spectroscopy . . . . . . . . . . . . . . . . . . . . . .

3

1.4

The advent of the optical frequency comb

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

5

1.5

Direct frequency comb spectroscopy . . . . . . . . . . . . . . . . .

7

1.6

Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92 Optical frequency comb properties and sources
2.1

11

Optical frequency comb properties . . . . . . . . . . . . . . . . . .

11

2.1.1

Time and frequency domain properties of the frequency comb 12

2.1.2

Measurement and control of the comb frequencies . . . . .

15

2.2

Comb sources and their applications to spectroscopy. . . . . . .

15

2.3

The mode-locked erbium fiber laser . . . . . . . . . . . . . . . . .

19

2.3.1

Energy levels and mode-locking physics of the Er3+ fiber laser 20

2.3.2

Er3+ fiber laser design strategies . . . . . . . . . . . . . . .

24

2.3.3

Practical matters . . . . . . . . . . . . . . . . . . . . . . .

27

2.3.4

Laser performance . . . . . . . . . . . . . . . . . . . . . .

30

xii
3 Useful properties of passive optical cavities

33

3.1

Mirror reflectivity and spectral bandwidth . . . . . . . . . . . . .

33

3.2

Finesse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

3.3

Cavity mode frequencies and dispersion . . . . . . . . . . . . . . .

42

4 Cavity-comb coupling
4.1

49

Optimizing the coupling of a frequency comb into an optical cavity

50

4.1.1

Mode-matching of the laser beam to the cavity . . . . . . .

50

4.1.2

Alignment of the laser to the cavity modes . . . . . . . . .

53

4.1.3

Matching the laser fr and fo to the cavity modes . . . . .

54

4.2

Comb locked to cavity . . . . . . . . . . . . . . . . . . . . . . . .

56

4.3

Comb swept around cavity . . . . . . . . . . . . . . . . . . . . . .

59

4.4

Precision sweep of laser fr . . . . . . . . . . . . . . . . . . . . . .

63

5 Measurement schemes
5.1

68

Cavity-enhanced absorption spectroscopy . . . . . . . . . . . . . .

68

5.1.1

Steady-state CEAS . . . . . . . . . . . . . . . . . . . . . .

69

5.1.2

Trasient CEAS . . . . . . . . . . . . . . . . . . . . . . . .

70

5.2

Cavity ringdown spectroscopy . . . . . . . . . . . . . . . . . . . .

71

5.3

Sensitivity comparison . . . . . . . . . . . . . . . . . . . . . . . .

72

6 Detection schemes
6.1

6.2

6.3

75

The grating spectrometer . . . . . . . . . . . . . . . . . . . . . . .

75

6.1.1

Theory of the grating spectrometer . . . . . . . . . . . . .

76

Multi-channel cavity ringdown detection . . . . . . . . . . . . . .

79

6.2.1

Streak camera cavity ring-down . . . . . . . . . . . . . . .

79

6.2.2

Multichannel cavity ring-down sampling electronics . . . .

82

The virtually-imaged phased-array spectrometer . . . . . . . . . .

84

xiii

6.4

6.3.1

Theory and design considerations . . . . . . . . . . . . . .

86

6.3.2

VIPA spectrometer design example . . . . . . . . . . . . .

90

6.3.3

Construction, alignment, and calibration . . . . . . . . . .

94

6.3.4

VIPA image processing . . . . . . . . . . . . . . . . . . . .

99

Precision scan of fr and cavity filtration . . . . . . . . . . . . . . 101
6.4.1

Single detector element cavity filtered detection . . . . . . 101

6.4.2

Cavity filtration with multiple detection channels . . . . . 103

7 Applications

107

7.1

Initial demonstrations . . . . . . . . . . . . . . . . . . . . . . . . 108

7.2

Trace detection at 1.5 µm . . . . . . . . . . . . . . . . . . . . . . 110

7.3

7.2.1

Trace molecular concentration measurements . . . . . . . . 112

7.2.2

Isotope ratio measurements . . . . . . . . . . . . . . . . . 118

7.2.3

Human breath analysis . . . . . . . . . . . . . . . . . . . . 119

Spectroscopy of jet cooled molecules . . . . . . . . . . . . . . . . . 126
7.3.1

High-resolution spectroscopy of a supersonic jet of C2 H2 . 128

7.3.2

Tomography of a supersonic jet of C2 H2 . . . . . . . . . . 134

8 Conclusion

145

8.1

The state of the art . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8.2

Current research and future directions . . . . . . . . . . . . . . . 146

Bibliography

149

Appendix
A MATLAB genetic algorithm code for mode-matching

164

xiv
B Derivation of the cavity ringdown and cavity enhanced absorption equations

169

B.1 Cavity ringdown spectroscopy derivation . . . . . . . . . . . . . . 169
B.2 Steady-state cavity enhanced absorption spectroscopy derivation . 170
B.3 Transient cavity-enhanced absorption spectroscopy derivation. . 172

C Algorithm for finding the fringes in a VIPA interference image

174

D MATLAB code for a discrete inverse Abel transform

181

xv

Tables

Table
2.1

Dispersion of optical fibers used in Er3+ lasers . . . . . . . . . . .

29

2.2

Comparison of 100 MHz and 250 MHz fiber lasers . . . . . . . . .

29

6.1

VIPA spectrometer example design parameters . . . . . . . . . . .

90

7.1

Minimum detectable concentration of molecules measured with 1.5 µm
CE-DFCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Figures

Figure2.1

Properties of an optical frequency comb . . . . . . . . . . . . . . .

2.2Frequency comb spectra and the corresponding atomic and molecular transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

13

17

Energy level diagram of Er3+ and absorption spectrum for erbiumdoped fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.4

Schematic of the polarization additive-pulse mode-locking mechanism 23

2.5

Pulse energy, net cavity dispersion, and total fiber length for several
mode-locked erbium fiber lasers . . . . . . . . . . . . . . . . . . .

25

2.6

Schematic of an erbium-doped fiber comb . . . . . . . . . . . . . .

28

2.7

Spectra and autocorrelation of the 100MHz Er3+ fiber comb . . .

30

2.8

Spectra and autocorrelation of the 250MHz Er3+ fiber comb . . .

31

3.1

Reflectivity bandwidth of a quarter wave stack dielectric mirror .

34

3.2

Multi-beam interference in an optical cavity . . . . . . . . . . . .

39

3.3

Cavity transmission versus mirror reflectivity . . . . . . . . . . . .

41

3.4

Cavity finesse and mirror reflectivity . . . . . . . . . . . . . . . .

42

3.5

Phase accumulated in one round-trip of an optical cavity . . . . .

43

3.6

Frequency comb-based cavity dispersion measurement technique .

45

3.7

Cavity dispersion measurement . . . . . . . . . . . . . . . . . . .

47

xvii
4.1

Mode-matching an incident laser beam to an optical cavity . . . .

4.2

Optimizing the coupling between a frequency comb and an optical

51

cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

4.3

Locking a frequency comb to an optical cavity . . . . . . . . . . .

57

4.4

Incident and transmitted spectrum of comb locked to an opticalcavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5

59

Transmission of an optical cavity due to a frequency swept opticalfrequency comb . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

4.6

Servo for locking a frequency swept comb to an optical cavity . .

62

4.7

Cavity transmission for a precision sweep of the incident laser fr .

64

4.8

Frequency domain mismatch of cavity and comb modes . . . . . .

66

5.1

Absorption sensitivity and transmitted power for transient cavityenhanced absorption spectroscopy . . . . . . . . . . . . . . . . . .

73

6.1

Principles of a grating spectrometer . . . . . . . . . . . . . . . . .

77

6.2

Multiple channel ring-down measurement using the streak camera
method6.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

A circuit for parallel readout of cavity ringdown signals from a
photodiode array . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

6.4

A schematic of the VIPA spectrometer . . . . . . . . . . . . . . .

85

6.5

The spectrum of CO2 near 1.6 µm recorded using a VIPA spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

6.6

VIPA spectrometer alignment procedure . . . . . . . . . . . . . .

95

6.7

Aligning the VIPA imaging lens using the interference fringes . . .

98

6.8

The process of extracting absorption spectra from VIPA spectrometer images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.9

Schematic of the electronics for cavity filtered detection . . . . . . 102

xviii6.10 C2 H2 spectra recorded using cavity filtered detection . . . . . . . 104
6.11 Multi-channel cavity filtered detection . . . . . . . . . . . . . . . 105
7.1

Absorption spectra of H2 O, NH3 , and O2 near 800 nm . . . . . . . 110

7.2

Cavity ring-down spectrum of NH3 , C2 H2 , and CO near 1.5 µm . 111

7.3

Schematic of the 1.5 µm CE-DFCS trace detection system . . . . 113

7.4

Detection sensitivity of 1.5 µm trace detection system . . . . . . . 114

7.5

Trace spectra of C2 H4 , CO2 and CH4 near 1.6 µm . . . . . . . . . 115

7.6

Trace spectra of CO, C2 H2 and NH3 near 1.6 µm . . . . . . . . . 116

7.7

Spectra of CO and C2 H2 isotopes near 1.6 µm . . . . . . . . . . . 118

7.8

Human breath analysis measurement system . . . . . . . . . . . . 121

7.9

Human breath spectrum of CO2 isotopes near 1.63 µm . . . . . . 122

7.10 Human breath spectrum of CO near 1.565 µm . . . . . . . . . . . 123
7.11 Trace spectrum of NH3 in a forest of H2 O absorption lines near
1.515 µm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.12 Schematic of the cold molecular jet detection system . . . . . . . 129
7.13 Residual C2 H2 absorption versus time after firing the expansion valve131
7.14 High resolution C2 H2 pulsed jet absorption measurements . . . . . 132
7.15 C2 H2 pulsed jet characterization . . . . . . . . . . . . . . . . . . . 133
7.16 Molecular jet tomography via Inverse Abel transform . . . . . . . 135
7.17 Determining the rotational temperature and density distributionsof a pulsed jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.18 Rotational temperature and density maps of the C2 H2 pulsed jet . 140
7.19 Absorption line-shapes versus nozzle distance for the C2 H2 pulsed jet142

Chapter 1
Introduction

1.1

Spectroscopy before the laser
It has been known for more than one hundred years that electromagnetic

radiation is an excellent vehicle for studying the properties of matter. Alreadyin the early 1800s scientists such as Wollaston and Fraunhofer were observing
dark lines in the spectrum of light emitted from the sun that led future generations of scientist to determine the composition of the Sun’s gaseous outer layers.
Fraunhofer’s invention of the grating spectrograph, in 1814, allowed for precise
wavelength measurements of optical absorption and emission spectra with accuracies of up to seven significant digits. By the second half of the 19th century,
many scientists were using this device to record the spectra of several atomic
vapors, providing our first glimpses of atomic structure. This spectral data wasstudied extensively by Johannes Rydberg who, in 1888, presented an empiricalformula for describing the frequencies of spectral lines of alkali atoms
1
1
1
ν
=
= Z 2R 2 − 2 .
c
λvac
n1 n2

(1.1)

Here, c is the speed of light, λvac is the vacuum wavelength of light, Z is theatomic number of the element under study, R is the Rydberg constant for thatelement, and n1 and n2 are integers where n1 < n2 . In 1913, Niels Bohr used
Rydberg’s discovery that the frequencies of spectral lines could be described by

2
the difference of two quantities to formulate a quantum theory of the hydrogenatom. Perhaps the most important aspect of Bohr’s work was his postulate thatthe frequencies of atomic of spectral lines arose for transitions between discrete
electronic energy levels. This idea paved the way for modern quantum theory.
The development of modern quantum mechanics in the 1920s, by Schrodinger,
Heisenberg, and others, provided a theoretical framework for understanding andpredicting the frequencies of light absorbed and emitted by atoms and molecules
based on their internal energy level structure. During this time, spectroscopic observations played a crucial role in the development of the theory. The observationof the doublet fine structure in hydrogen led to the incorporation of relativistic
effects in atomic structure and the discovery of the electron spin. Pauli’s famous
exclusion principle stemmed from his observation of many atomic spectra whichindicated that no two electrons could occupy the same state. Also, the observation of atomic hyperfine structure led Pauli to postulate that the nucleus had asmall magnetic moment. The spectral observations of Stark and Zeeman provided
a framework for understanding how atomic energy levels are affected by external
electric and magnetic fields. Also, during this time, the work of C. V. Raman
led to the discovery of inelastic scattering in molecules resulting in a powerfulmechanism for the study of molecular vibrational and rotational structure. The
theoretical framework of quantum mechanics combined with the increased precision of spectroscopic measurement, enabled by the ingenuity of spectrometer
developers such as Fabry and Perot, led to an enormous growth in our understanding of atomic and molecular structure in the first half of the 20th century.

3
1.2

The invention and development of laser spectroscopyWith the invention of the laser in 1960, the field of spectroscopy was revo-

lutionized. Atomic and molecular absorption lines could be probed with precision
light sources characterized by high spectral brightness and precisely defined wavelengths. The invention of the dye laser, in 1966, provided a tunable, coherent lightsource that enabled access to a wide variety atomic and molecular transitions covering large spectral windows. New techniques were invented that dramatically improved the resolution and control of the light-matter interactions in spectroscopic
measurements. For example, saturation absorption spectroscopy led to improved
resolution and allowed the measurement of energy levels that had previously been
obscured by Doppler broadening such as the Hydrogen Lamb shift [1]. The development of laser cooling increased the resolution even further by suppressing
many transition broadening effects such as first and second order Doppler shifts
as well as transit time broadening [2]. Methods developed for frequency stabilizing lasers led to increased the precision of spectroscopic measurements and thecontrol of light-matter interactions [3] [4]. Finally, external optical cavities were
implemented to enhance the signals from weak or dilute resonant absorbers [5][6].
During this string of advances in spectroscopic techniques, a wide variety of laser
sources were developed including tunable solid-states lasers, tunable diode lasers,
and mode-locked lasers each with unique capabilities that have contributed to therapid growth of laser spectroscopy.

1.3

Modern laser spectroscopy
Today, the breadth and sophistication of the field laser spectroscopy is truly

amazing. In the physics community alone, laser spectroscopy is currently enabling

4
a large number of experiments that are at the forefront of scientific investigation.
Examples include: the creation and study of new forms of matter such as BoseEinstein condensates and degenerate Fermi gases [7], the measurement of theLamb shift as high precision test of quantum electrodynamics [8], the search forpermanent electric dipole moments and the variation of fundamental constantsfor table-top tests of the standard model [9], and the exploration of entanglement
as a resource for quantum information processing and quantum computing [10].
In chemistry, biology, and environmental sciences, lasers spectroscopy has beenused to study to structure and properties of a huge variety of molecules ranging
from the small few-atom-molecules to large macro molecules. Through these studies, modern society has developed a broad understanding of biological processes,
medical treatments, atmospheric chemistry, and chemical engineering processes
for industrial applications.
In the commercial world, use of laser spectroscopy is less widespread; however, the number and scope of these applications is rapidly increasing. Already,combustion spectroscopy is used for optimizing the efficiency of fossil fuel power
stations and car engines. The ‘dosage’ of anesthetics administered during surgery
can be controlled by real-time monitoring of the ratios of C02 , O2 and N2 exhaledby the patient using Raman spectroscopy [11]. Also, the detection of trace concentrations of molecules has emerged as a powerful tool in a wide range of commercial
and industrial settings. Applications include: the detection of contaminants forsemiconductor processing [12], human breath analysis for disease detection [13],air quality monitoring [14], and detection of biologically hazardous or explosive
materials for security purposes [15]. Each of these applications has been facilitatedby the development of new laser sources to access spectral domains of interest,
combined with cavity enhancement techniques for increased detection sensitivity.
For example, the advent of tunable laser diodes has led to sensitive measurements

5of impurity levels in gases such as phosphine used in semiconductor productions[12] and for monitoring ethylene in fruit storage facilities [16]. Subsequently, quantum cascade lasers and optical parametric oscillators operating in the mid-infrared
have been applied to human breath analysis for health screening and optical detection of explosives [17] [15]. These innovations represent remarkable advancesin our capability to monitor and understand our environment. However, thesetechnologies are currently based on single frequency laser sources that operate inrelatively narrow spectral regions and thus the number of different molecules thatcan be studied or detected with a single system is very small. This severely limits
our capability of performing surveyance on a global scale in real time.

1.4

The advent of the optical frequency comb
In the late 1990s, the previously disparate fields of mode-locked ultra-fastlasers and precision laser stabilization were merged with the invention of the
optical frequency comb [18][19]. By controlling the phase and repetition rateof femtosecond pulses emitted from a mode-locked laser, the frequency spectrum
corresponding to the pulse train is comprised of a well defined set of regularly
spaced frequencies
νn = nfr + fo ,(1.2)

where n is the mode number, fr is the rate that pulses are emitted from the laser,and fo is the rate that the phase of the laser evolves with respect to the centerof the pulse envelope. The key to producing the well defined set of frequencies
in Eq. 1.2 was the measurement of fo . While fr could easily be measured using
a fast photodiode, measuring fo required the use of nonlinear interferometry [18].
When both fr and fo are measured, servos acting on the mode-locked laser cavity

6length and dispersion can be used to stabilize the comb frequencies.
The potential applications of the optical frequency comb are considerable.
The first, and so far most developed, application is optical frequency metrology[20][21][22]. Before the optical frequency comb, long chains of nonlinear frequency
up-conversion were used to provide a coherent link between a microwave frequency,
which could be counted by conventional electronics, and the optical frequency thatwas being measured [23]. Due to the large size, complexity, and limited utility of
the frequency chains, only a few of these devices were constructed.
The first demonstration of optical frequency metrology using mode-locked
lasers measured the difference between a well known optical frequency (the 4th
harmonic of a methane stabilized HeNe laser) and ones that were not well known(the D1 lines of cesium) [24]. Because the difference frequency of two comb modes
does not depend on fo , only the laser fr needed to be stabilized for these measurements. After the development of fo stabilization, the absolute frequency of
any laser could be measured by its heterodyne beat with the nearest comb toothνCW = νm + fbeat ,(1.3)

Here, m is the mode number of the nearest comb tooth to the CW laser frequency.
Frequency-comb-based optical frequency metrology has lead to an explosion in thefield of optical atomic clocks and optical frequency standards [25] [26][27]. Already,
optical atomic clockes have enabled some of the highest precision measurements
fundamental constant variations and the coupling of the gravitational potential
to fundamental constants [28] [29].
Other applications of the optical frequency comb are less developed, butoffer equally exciting prospects for the role of frequencies combs in the future oflaser-based technologies [30]. Many of these applications require control over theamplitude and phase of individual comb components over broad spectral regions.

7
For instance, line-by-line pulse shaping can be used to generate arbitrary optical
waveforms with the potential for dramatic advances in spread-spectrum lightwave
communications, laser ranging and detection, and optical signal processing [31].
Another application is the coherent control of the quantum states of cold atomic
and molecular samples [32]. Phase and amplitude control of the comb has already been used to control quantum interference and quantum state populationsin atoms [32]. This same approach should also allow precise control of photoassociation and chemical reactions [33]. Finally, precise control of the phase of
few-cycle mode-locked lasers has already enabled control of ultra-fast processes
such as electron dynamics in atoms and molecules [34]. As a result, it is now
possible to generate phase stable attosecond laser pulses in the soft X-ray region
that will be used for the next generation of ultrafast experiments [35].

1.5

Direct frequency comb spectroscopy
Since optical frequency combs possess a very large number of precisely fre-quency calibrated laser channels, it is natural to think of using these lasers to
perform high precision and broadband spectroscopy over massively parallel detection channels. The first implementation of this idea, called ‘direct frequency
comb spectroscopy’ (DFCS), used a precision scan of fr to recover a serial detection of high resolution spectral features of 87 Rb across a broad spectral range [36].
More recent approaches to molecular detection that use broad bandwidth frequency combs have indeed created parallel detection schemes that in a single shotrecord large spectral bandwidths [37][38]. Such detection schemes have allowed
the simultaneous measurement of hundreds of transitions from several different
molecules.
Since frequency combs have high peak intensities due to their pulsed output,

8
nonlinear frequency conversion can be implemented for easy access to spectral regions spanning from the UV to the far infrared [39] [40]. Therefore, these lasers canprovide broad spectral coverage for probing a wide variety of atomic and molecular transitions. Another nice feature is that the periodic structure of the comb isideal for efficient coupling to the resonant modes of a high finesse optical cavity.
The subject of this thesis, ‘Cavity-enhanced direct frequency comb spectroscopy’,(CE-DFCS) takes advantage of this fact to combine the previously mentioned
features of DFCS with highly sensitive detections of weak or dilute atomic andmolecular absorbers that is made possible by cavity enhancement. Because of
its combination of high resolution, high detection sensitivity, and broad spectral
coverage, CE-DFCS has enormous potential for applications in the scientific andindustrial communities.
In the scientific community, sensitive and high resolution spectroscopy overa large spectral window is useful for many areas of research. For instance, studies
of atmospheric chemistry, climate change, and pollution can benefit a great deal
from the capability to monitor multiple molecules simultaneously [41][42]. Molecular physics investigations of potential energy surfaces, energy level structure, local
and eigen-mode dynamics, intramolecular energy redistributions, interaction dynamics, and chemical reactions all desire spectroscopic probes with high spectral
resolution and accuracy over a wide range of spectral coverage [43][44]. For the
emerging field of ultracold molecules, our ability to photoassociate cold moleculesfrom pairs of free atoms depends on extensive spectroscopic probes to find themost efficient photoassociation pathways [45]. To study a new class of collisionand reaction dynamics in the low energy limit [46][47], the need for higher resolution and sensitivity while under a broad spectral coverage is now of more urgent
importance [48][49].
In the industrial community, CE-DFCS has the potential to revolutionize

9
applications that are currently being performed with single frequency laser andopen the door to many others that were not previously possible. For example,
human breath contains more than 1000 different molecules in trace quantities [50].
Already, many of these molecules, so called ‘biomarkers’, have been correlated to
specific diseases and metabolic processes. While single frequency trace detection
systems can determine the presence and quantity of one or a few of these molecules,
CE-DFCS has the potential to rapidly determine the entire chemical compositionof a breath sample. Such an advance in real-time breath measurement would enable large-scale studies of human breath with the goal of low cost, non-invasive
health screening. In the semi-conductor industry, the manufacturing yield andperformance of integrated circuits depends strongly on the concentrations of trace
impurities in the gases used to make these chips. While single frequency systems
have been used to detect one of these impurities, CE-DFCS could be used tomonitor a wide variety of detrimental impurities simultaneously. These are only acouple of examples where CE-DFCS enables trace detection applications that were
not previously possible. Other potential applications include hazardous materials
and explosives detection and environmental quality and pollution monitoring.

1.6

Thesis outline
In this thesis we will explore the newly developed technique in laser spec-

troscopy called cavity-enhanced direct frequency comb spectroscopy (CE-DFCS).
Since this is a new technique, developed during the course of my Ph.D. studies,many of the following chapters are devoted to introducing an understanding of thebuilding blocks and the concepts that comprise CE-DFCS systems. In the firstchapter, we explore the properties of optical frequency combs. We also discuss thevarious types of mode-locked laser that can be used to generate frequency combs,

10and their potential applications to atomic and molecular spectroscopy. Particularattention will be paid to the erbium-doped fiber comb including guidelines fordesigning these lasers and the performance that can be achieved.
In Chapter 3, we turn our attention to optical enhancement cavities andthe properties that make them useful for CE-DFCS. We develop a mathematicaldescription of a cavity made from two quarter wave stack dielectric mirrors. We
then discuss techniques for measuring the dispersion and finesse of these typesof cavities. Chapter 4 focuses on methods for coupling a frequency comb to ahigh finesse optical cavity. This discussion will begin with the fundamental topics
of aligning and mode-matching a laser beam to an optical cavity. We will thenpresent several strategies for efficiently coupling the comb frequencies to the optical cavity modes and discuss advantages and disadvantages of each.
In Chapter 5 we introduce several different implementations of cavity-enhanced
spectroscopy including cavity ringdown spectroscopy (CRDS), cavity enhanced
absorption spectroscopy (CEAS), and transient cavity enhanced absorption spectroscopy (T-CEAS). Evaluations of each measurement technique are given fromboth a mathematical and practical implementation point of view. In Chapter 6,
we present a discussion of detection techniques for reading out the multitude of
frequency channels contained in the cavity transmitted beam. Chapter 7 discusses
the first set of applications of CE-DFCS. Here, we present a series of experiments
using CE-DFCS for trace detection of molecules, human breath analysis, and characterization of supersonically cooled molecular jets. Finally, Chapter 8 concludes
with a summary of the current state of the art of CE-DFCS including a discussionof ongoing research and the latest applications of this technique. This chapter
ends with an outlook of experiments that are on the horizon and a bit of dreaming about the future of CE-DFCS.

Chapter 2
Optical frequency comb properties and sources

In this chapter, we will explore mode-locked laser sources that can be used
to generate optical frequency combs for CE-DFCS systems. We begin with a discussion of the general properties of frequency combs by developing mathematicalrepresentations for the output of these lasers in both the time and frequency domains. We then discuss methods for measuring and controlling the frequencies of
the comb. In the following section, we present an overview of the different kinds ofmode-locked lasers that have been developed, and the corresponding atomic andmolecular transitions that they can be used to probe. The chapter concludes witha detailed discussion of mode-locked erbium fiber lasers including: the physicalmechanisms responsible for their operation, design guidelines, and the performance of these lasers.

2.1

Optical frequency comb properties
Many of the same properties that make the frequency comb an excellent

tool for counting optical frequencies and transferring coherence across vast spectral regions also make it an ideal light source for broadband, cavity enhanced
spectroscopy. Frequency combs are capable of covering very large spectral rangesof hundreds of terahertz, thus enabling investigations of global energy level struc-12ture for multiple atomic and molecular species. At the same time, the spectrally
sharp and precisely defined frequency structure of the comb components allow
high-resolution and frequency-calibrated spectroscopy across massively parallel
detection channels. For the same reason, the frequency comb is ideal for sensitive intra-cavity spectroscopy since it can be efficiently coupled to a corresponding
set of resonant modes of a high finesse optical cavity across a wide spectral region.

2.1.1

Time and frequency domain properties of the frequency comb

The time and frequency domain representations of a comb are illustrated
in Fig. 2.1. In the time domain, the laser emits a train of femtosecond pulses
described by the time varying electric field
E(t) =nfX

An (t)ei(ωc t−nωc trt +n∆φce +φo ) .

(2.1)

n=1

Here, n is the pulse number, nf is the total number of pulses in the train, An (t) is
the shape of the nth pulse, and trt is the time between consecutive pulses. ωc is thecenter frequency of the comb spectrum, ∆φce is the phase slip between the pulse
envelope and the carrier wave for adjacent pulses, and φo is a constant phase-offset
between the pulse envelope and the underlying carrier wave. In order to derive
specific connections between the time and frequency representations of the comb,
a Gaussian pulse shape is assumed
An (t) = Ep e−(t−ntrt )2
τ2

.

(2.2)

Here, Ep is the peak electric field and τ is the 1/e half-width of the pulse duration.
Furthermore, we assume that all frequencies contained in the pulse are perfectly
overlapped in time. This condition is equivalent to a ‘transform-limited’, or ‘chirpfree’ pulse. The unique property of a transform-limited pulse is that the pulse

13

Figure 2.1: The time and frequency domain properties of an optical frequency
comb. The top panel shows the time domain representation of a mode-locked pulse
train including the temporal spacing between pulses (trt ) and the evolution of thecarrier envelope phase (∆φce ). The bottom panel shows the frequency domain
picture where the evenly spaced comb modes with frequencies νn lie beneath thelaser’s spectral envelope.

duration is determined only by the width of the frequency comb spectrum. Whilethese assumptions do not affect the frequency comb structure, they are useful forunderstanding the relationship between the pulse duration and the extent of the
comb spectrum.
The frequency domain picture of the comb can be derived by taking theFourier transform of the pulse train.
iφE(ω) = Ep e

nf
X
n=1

e

i(n∆φce −nωc trt )

Z ∞
e
−∞

−(t−ntrt )2τ2

ei(ωc −ω)t dt.

(2.3)

14
Evaluating the Fourier integral yields an expression for the frequency spectrum of
the combnf√ X−(ω−ωc )2 τ 2
4e
E(ω) = Ep e τ πei(n∆φce −nωc trt ) eintrt (ωc −ω)
iφ

(2.4)

n=1
nf

√ X −(ω−ωc )2 τ 2 i(n∆φce −nωtrt )4e.
= Ep eiφ τ π
e

(2.5)

n=1

Analysis of this Eq. 2.5 reveals several properties of the frequency domain representation of the comb. First, the power spectral envelope (P ∝ |E|2 ) is a Gaussianwith a full width at half maximum (FWHM) that is inversely proportional to thepulse durationp
2 2 ln(2)δωF W HM =
.
τ

(2.6)

Also, the spectral envelope has a center frequency of ωc . From the oscillatory term,
we see that the broad spectral envelope is comprised of a comb of sharp peaks
occurring at frequencies for which the phase of consecutive pulses adds coherently.
This condition can be written as ∆φce − ωtrt = 2πm. Solving for ω results in an
equation for the comb frequencies [19]ωm =2πm ∆φce
+
.
trt
trt

(2.7)

Further analysis of Eq. 2.5 shows that as the total number of pulses nf becomeslarge, the individual comb components become narrow in frequency.
By defining the repetition frequency (fr = 1/trt ) and the carrier-envelopeoffset frequency (fo = ∆φce /2πtrt ) we arrive at the commonly used expression forthe comb frequencies νm = mfr + fo [51][52]. For optical frequency combs in thevisible and near IR, the comb mode index m takes on values ranging from 105 to
107 , depending on fr . Typical values for fr are constrained by the cavity length ofthe mode-locked laser. Practical lengths for these laser cavities range from 1 cm
to 10 m corresponding to an fr range of 10 MHz < fr < 10 GHz. fo can take on

15values of - f2r <fo < f2r determined by the difference between the group and phasevelocity of light inside the laser cavity.

2.1.2

Measurement and control of the comb frequencies

Precise control of fr and fo establishes the comb as an accurate frequency
tool for atomic and molecular spectroscopy and also permits efficient coupling of
the comb to the modes of a high finesse passive optical cavity. To implement
control, one first needs to measure fr and fo . fr can be determined directly fromthe pulse train incident on a fast photodiode while fo requires nonlinear interferometry [21][19][53]. The typical nonlinear interferometry involves generating anoctave of comb spectrum with the use of highly nonlinear fiber or photonic crystal
fiber. The long wavelength edge of the octave spectrum is then frequency doubled to form a heterodyne beat with the short wavelength edge of the spectrum.
The frequency of the resulting heterodyne beat provides a measure of the carrierenvelope-offset frequency of the comb. Two stable radio frequency references canbe used subsequently to stabilize fr and fo via servo transducers acting on thelaser cavity length and the intra-cavity dispersion. A frequency comb can also bestabilized directly to a passive optical cavity by detecting the cavity response to
the incident comb spectrum in the optical domain [54][55], as discussed later insection 4.2.

2.2

Comb sources and their applications to spectroscopy
Many different types of mode-locked laser sources exist to provide coverage

for a wide variety of spectral regions from blue to the mid IR spectral regions.
When nonlinear frequency conversion and spectral broadening techniques are im-16
plemented, the spectral coverage can be extended well into the VUV and the far
infrared. The diversity of spectral regions that can be accessed allows frequency
comb spectroscopy to be performed on a large variety of atomic and molecularsystems. In this section, we will provide an overview of several types of modelocked laser that can be used to generate frequency combs. Nonlinear frequency
conversion to other spectral regions based on these laser sources will also be discussed. A visual guide to the spectral coverage provided by the different kinds ofmode-locked lasers is shown in Fig. 2.2. This figure also shows spectral overlaps
between various frequency comb sources and relevant regions for useful atomic
and molecular spectroscopy.
As the first generation of precisely controlled optical frequency combs
were based on Ti:sapphire lasers, it is natural that they played important rolesin the first demonstrations of direct frequency comb spectroscopy [36]. Frequency
comb-based cavity ring-down spectroscopy (CRDS) was also first performed witha Ti:sapphire comb on molecular overtone transitions in the 750 nm to 850 nm
spectral region [37]. This system probed the high overtone transitions of C2 H2 ,H2 O and NH3 as well as a weak electronic transition in O2 .
Many types of nonlinear frequency conversion have been used to shift theTi:Sapphire comb from the VUV to far infrared spectral regions. Frequency down
conversion of Ti:sapphire combs has been used to produce infrared frequency
combs via difference frequency generation (DFG) and optical parametric oscillation (OPO) techniques [56][57] [58]. Such infrared combs have been used toprobe the strong fundamental vibration transition in molecules [59]. Also, highharmonic generation (HHG) has been used to generate frequency combs in thevacuum ultra-violet regions [60][61], and with improvement in the power [62] they
could be used to probe ground state transitions of noble gases [63]. Finally, while
the gain bandwidth of Ti:sapphire lasers cover only 700-950 nm, subsequent super-

17

Figure 2.2: Spectral coverage for several types of mode-locked lasers and corresponding atomic and molecular transitions to which they can be applied. 1,transitions from ground states of noble gas; 2 and 3, transitions from groundstates of alkaline earth or alkali atoms; 4 and 5, transitions between excited
atomic or molecular states; 6, transitions from ground states of molecules; 7,
overtones of molecular ro-vibrational transitions; 8 and 9, fundamental vibrationsof light-nuclei or heavy-nuclei molecules. a) Ti:sapphire mode-locked lasers, after nonlinear spectral broadening, covers the entire visible range from 400 to 1200
nm; b)Erbium-doped fiber lasers (centered around 1550 nm) can be made to cover1000 - 2400 nm via spectral broadening; c)Ytterbium-doped fiber lasers (centeredaround 1030 nm) can cover 600 - 1600 nm via spectral broadening. Each of theselasers can have their spectral coverage further extended via nonlinear frequency
conversion using techniques listed in (d), (e), (f), and (g). We thus have the possibility of performing comb spectroscopy for nearly all of the interested atomic
and molecular transitions.

continuum generations in photonic crystal and tapered telecommunications fibers[40][64] [18], and the later development of octave-spanning Ti:sapphire lasers via
intra-cavity nonlinear phase modulation allow simultaneous spectral coverage extending from the deep blue to the near Infrared (NIR) using a single laser source[65][66] [67]. Due to the spectral overlap of the Ti:Sapphire spectrum with somany atomic and molecular transitions, it is likely that future Ti:Sapphire based
CE-DFCS systems will find many important applications in atomic and molecular

18
physics.
Over the last decade, mode-locked erbium and ytterbium fiber lasers havebeen developed that generate highly stable frequency combs which are pumped byinexpensive laser diodes [68] [39] [69] [70]. These lasers can conveniently produce
super-continuum spectra via coupling to highly nonlinear fibers [71][72]. They
have been frequency converted to cover the spectral domain extending from the infrared to ultra-violet, using the same techniques described for Ti:sapphire [73][74].
The fiber lasers are compact, reliable, and robust, with turn-key operations. Thus
they have a very promising future for commercial applications. Recent work has
used an erbium fiber comb to perform CE-DFCS on low molecular overtones fortrace detection of a wide variety of molecules at concentrations approaching 1 partper billion [75][76]. These fiber combs are conveniently located in the 1.0 µm 2.3 µm spectral domain, overlapping with strong first-order vibration overtonesof many molecules containing O-H, C-H and N-H bonds. They are also ideal for
nonlinear frequency conversion to extend their spectral coverage to the mid andfar infrared; an important spectroscopic region where most molecules have uniqueabsorption signatures that can be detected using cavity enhanced spectroscopy atconcentrations at or below 1 part per trillion.
More recently, efforts have been made to develop several other types ofmode-locked lasers that operate in different spectral regions. The most matureof these efforts is the Cr:forsterite laser. Already these lasers have demonstrated
highly stable mode-locking with spectra covering 1.0 µm-2.1 µm [77][78]. Anotherfairly old technology that is receiving renewed attention due to its potential formolecular spectroscopy is the thulium-doped fiber laser [79]. These lasers canalready produce sub-200 femtosecond pulses with more than 100 mW of average
power and tunable soliton spectra covering the 1.9 µm-2.1 µm [80][81]. Thulium
fiber lasers are particularly attractive for detecting trace impurities in gases such

19
as arsine and phosphine used to manufacture semiconductor chips. Both arsine
and phosphine have low absorption in this spectral region, while the most common impurities in these gases (H2 O, CO2 , CH4 , SiH4 ), have strong absorption
signatures.
Probably the most exciting prospects are mid-infrared mode-locked lasers
based on Cr2+ :ZnSe and Cr2+ :CdSe gain media. Already, Cr2+ :ZnSe lasers havebeen passively mode-locked generating spectra tens of nanometers wide withpulse durations of several hundred femtoseconds [82]. With further development,
Cr2+ :ZnSe lasers could produce extremely broad spectra due to the 1.9-2.9 µm
gain bandwidth of this crystal [83]. Even more exciting is Cr2+ :CdSe gain medium[83]. This crystal has a gain spectrum that extends from 2.0 µm to 3.5 µm, ideal
for the observation of hundreds of molecules via strong fundamental vibrationaltransitions. To date, no one has succeeded in mode-locking a laser using thisgain medium. However, the wide range of applications that would be enabled by
mid-IR mode-locked lasers are very enticing, and will certainly motivate future
development of these types of lasers.

2.3

The mode-locked erbium fiber laser
Currently the most widely used mode-locked fiber laser, and also the eas-iest to design and construct, is the erbium-doped fiber laser. Due to the hugeamount development of erbium fiber components for the telecommunications industry, these lasers are relatively inexpensive and highly reliable. Furthermore,
the demonstrated ability to frequency convert their output to a variety of spectral
regions from the UV to mid-IR make them extremely useful for CE-DFCS applications.
In this section, we present an overview of mode-locked erbium-doped fiber

20
laser. We begin by describing physics that enables the operation of these lasers
including: the energy level structure of Er3+ , the absorption spectrum of erbiumdoped fibers, and the mechanism responsible for mode-locking. Next, we discuss
the crucial design criteria for constructing these lasers. We provide guidelines forchoosing values of the intra-cavity dispersion and pulse energy that lead to stablemode-locking. We also discuss how to achieve the correct gain ratio for light at
1530 nm and 1560 nm to initiate passive mode-locking. Finally, we present two
specific mode-locked erbium fiber lasers designs along with complete characterizations of the performance of each system.
The reader should note that there are two main types of mode-locked erbium fiber lasers, stretched-pulse lasers and soliton lasers. We will only discussdesign considerations for stretched-pulse lasers in this section. The reasons forthis are two-fold. First, sideband generation in soliton lasers due to periodic perturbations limit the pulse duration and lead to narrower spectral widths than canbe achieved with stretched-pulse lasers. Second, the net positive cavity dispersion
and longer average pulse durations in the stretched-pulse laser allow for higher
pulse energy and hence higher average laser power. For further information on
both types of erbium fiber lasers see references [84] [68].

2.3.1Energy levels and mode-locking physics of the Er3+ fiber laser

The Er3+ energy level diagram in Fig. 2.3a shows the relevant transitions
for the erbium-doped fiber laser. A diode laser at 980 nm is used to pump the
erbium from its ground state |1i, to excited state |3i. This state undergoes rapid,
nonradiative decay to metastable state |2i [85]. Lasing occurs on the |2i → |1itransition when the transition |1i → |3i is pumped sufficiently hard to create apopulation inversion between levels |2i and |1i. There have been several studies

21

Figure 2.3: a) Energy level diagram of Er3+ reproduced from [88]. b) The absorption spectrum for the Leikki Er110-4/125 erbium-doped fiber taken from theThorlabs website [89].

of the pumping threshold to achieve gain in erbium-doped fibers [86] [87][85]. A
particularly useful set of analytical equations for determining the optimal gainfiber length based on Er3+ doping concentrations and 980 nm pumping power
was presented by Peroni et al. [87]. When Er3+ is doped into a host glass, acomplicated combination of homogeneous and inhomogeneous broadening resultin a broad absorption band that arises from 56 unresolved Stark transitions [88].
The absorption bands for the Leikki Er110-4/125 erbium-doped fiber are shownin Fig. 2.3b. The broad emission/absorption spectrum for erbium-doped fibers
make them an excellent candidate for mode-locked operation.
Most Er3+ fiber lasers are mode-locked using a technique called polarization additive pulse mode locking (P-APM) [84][68]. Here, the optical Kerr effectinduces an intensity dependent rotation of the laser polarization that, in conjunction with polarization optics, creates different levels of intra-cavity loss for
CW and pulsed operation. The intensity dependent polarization rotation can be22understood by an examination of an elliptically polarized beam propagating in amaterial with a nonlinear index of refraction. As an optical field E travels throughthe fiber, it causes a polarization of the medium P = χe E. In the case of strong
optical fields, higher order terms of the material susceptibility must be consideredχe = χ(1) + χ(2) E + χ(3) E 2 + . . . .

(2.8)

The materials used to make optical fiber are typically amorphous which implies
inversion symmetry such that χ(2) = 0 [90]. In the SI unit system, the material
index of refraction is related to the susceptibility by n2 = 1 + χe [91]. The
intensity-dependent index of refraction takes the formn = no + n2 I,
where I = 2no |E|2 /Zo , Zo =

(2.9)

p
p
µo /o , no = 1 + χ(1) , and n2 = 3Zo χ(3) /(4n2o ).

The nonlinear index gives rise to an intensity-dependent phase shift for light propagating through the fiberφ(z, t) = (no + n2 I(t))kz.

(2.10)The phase shift can be viewed as a polarization rotation by writing the nonlinear
index of refraction in a circular polarization basis [92]n± ' no + 4no n2 (|E± |2 + 2|E∓ |2 ),(2.11)

where E− and E+ refer to right-hand and left-hand circular polarized fields. The
index difference experienced by fields E+ and E− leads to a polarization rotationof angleθ=ω2ωno n2(n+ − n− )z =(|E− |2 − |E+ |2 )z.
2cc(2.12)

Here, the tensor component of χ(3) responsible for the rotation is χxyyx [92]. From
Eq. 2.12 we can see that linearly polarized light experiences no rotation since23

Figure 2.4: Schematic of the polarization additive-pulse mode-locking mechanism.
The incident linearly polarized pulse is converted to an arbitrary polarization state
by intra-cavity wave plates. In the Kerr medium, the high-intensity pulse center
experiences a large rotation of its polarization state while the polarization of the
low intensity wings remain unchanged. The intra-cavity wave plates are used toalign the polarization of the center of the pulse to be transmitted through the
PBS for efficient recoupling to the laser cavity. Due to the intensity-dependent
polarization rotation, the recoupled pulse is shortened during each trip around thecavity resulting in passive mode-locking. The rejection port of the PBS is used as
the laser output.

|E+ |2 = |E− |2 . Also, rotations of circularly polarized light produce no measurable
effect. Finally, the rotation experienced by elliptically polarized light depend on
both the intensity and the ellipticity of the beam.
A schematic illustrating how nonlinear polarization rotation leads to polarization additive pulse mode locking is shown in Fig. 2.4. In this depiction, thelaser pulse is broken into three sections representing the low-intensity front andback edges of the pulse and the high-intensity pulse center. First, a linearly polarized pulse is incident from the left on a quarter and a half wave plate to create
an arbitrary elliptical polarization state. Next, the pulse enters the Kerr medium
where it experiences an intensity-dependent polarization rotation described by
Eq. 2.12. This is represented in Fig. 2.4 by a 90◦ rotation of the intense pulse
center while the polarization of the wings of the pulse remain unchanged. Finally,

24
a polarizing beam splitter (PBS) is used to recouple the center of the pulse into
the laser cavity while output-coupling the wings of the pulse. This configuration
leads to high intra-cavity loss for CW lasing and low loss for pulsed operation.
Therefore, by choosing the correct orientation of the wave plates with respect to
the PBS, the laser will passively mode lock. Of course, this is only a qualitativedescription of P-APM. A more rigourous derivation of the mode-locking dynamics
can be found in Ref. [68].

2.3.2Er3+ fiber laser design strategies

In the last section, we saw how nonlinear polarization rotation can give rise
to passive mode-locking of Er3+ fiber lasers. In this section, we will discuss design considerations for achieving an intra-cavity pulse intensity that leads to theappropriate the nonlinear polarization rotation for mode-locking. The first decision to make when designing any mode-locked laser is the repetition rate. Oncea repetition rate is chosen, target values for the average power and intra-cavity
dispersion can be specified that will lead to stable mode-locking. To date, modelocked erbium fiber lasers have been constructed with repetition rates ranging
from 10 MHz to 250 MHz, though higher repetition rates will likely be achievedin the near future [68][93]. For CE-DFCS applications, frequency combs withhigh repetition rates in the range of 100 MHz to 250 MHz are preferred sincethese lasers can be efficiently coupled into reasonably small (0.6 m-1.5 m) opticalcavities.
Figure 2.5 shows the dispersion, pulse energy, and total fiber length times
pulse energy for several mode-locked erbium fiber lasers. The most notable feature of Fig. 2.5a is the difference in laser performance between laser cavities withnet positive versus negative dispersion. As mentioned previously, lasers with neg-

25

Figure 2.5: a) The observed range of pulse energies (dots) and minimum pulse
duration (triangles) versus net cavity dispersion for several mode-locked erbium
fiber lasers (figure taken from [84]). The numbers next to the triangles give thetime-bandwidth product for each laser. b) The range of pulse energies (red) andpulse energy times fiber length (blue) versus total fiber length for four fiber lasers.
Data for the two 44 MHz lasers was from [84] while the 100 MHz and 250 MHz
lasers were constructed in Ye labs.

ative cavity dispersion have both lower pulse energy and smaller time-bandwidth
products than lasers with positive cavity dispersion. Also shown in Fig. 2.5a isthe fact that lasers with larger positive dispersion require higher pulse energies formode-locking. Generally speaking, the values for the net cavity dispersion shownin Fig. 2.5a provide good guidelines for designing a mode-locked erbium fiber
laser. For low repetition rate laser designs, it is wise to choose higher dispersion
to achieve longer intra-cavity pulses and hence higher output power. However,for designing a high repetition rate laser, it is better to choose lower dispersion tocompensate for the lower energy per pulse and the shorted fiber lengths associatedwith these lasers.
Figure 2.5b illustrates that the cavity round-trip nonlinear phase shift is
an important parameter for stable mode-locking. In Fig. 2.5b both the pulse energy (red) and the pulse energy times the total intra-cavity fiber length (blue) areplotted against the total cavity dispersion for four mode-locked lasers. If only the26pulse energy were plotted, there would be seemingly no trend linking the operation of these four lasers. However, when the pulse energy times the fiber lengthis plotted, a clear trend can be seen as a function of net cavity dispersion. This
indicates that the P-AMP process leads to mode-locking for a specific range ofvalues of the round-trip nonlinear phase shift [68]. In one cavity round-trip, a
pulse will experience a nonlinear phase shift given by ∆φnl = n2 IL, where n2 is
the nonlinear index of refraction of the fiber, I is the pulse intensity, and L is thetotal fiber length. The important point here is that pulse intensity is proportional
to the average intra-cavity pulse duration which is proportional to the net intracavity dispersion. Hence, I ∝ βEp , where β is the intra-cavity dispersion and Ep
is the pulse energy. So, if the intra-cavity dispersion is increased, either the pulse
energy, or the fiber length need to be increased to maintain the same nonlinear
phase shift.
Of course, there are many factors that complicate this general rule including
the fact that the pulse is changing duration as it propagates through the cavity so
it matters where in the cavity the pulse reaches its shortest duration. Also, different fibers have different mode field diameters which lead to different intensities
for the same pulse energy. Finally, different fibers are made with different typesof glass and sometimes include dopants which can lead to different values of thenonlinear index n2 . Nevertheless, the trend in Fig. 2.5b provides a good estimate
for the combinations of pulse energy, fiber length, and cavity dispersion that willlead to mode locking.
The final consideration for designing a mode-locked erbium fiber laser isgain. In the previous paragraphs, we have found that the proper balance of pulse
energy and cavity dispersion are required for mode-locking. Accordingly, we must
provide enough gain in the laser cavity to reach the required pulse energy. For
this, the gain equations from reference [87] and an estimate of the total loss in27
the laser cavity can be used to determine the intra-cavity pulse energy.
Besides providing sufficient gain to reach the desired pulse energy, there isalso an issue of gain competition. Experience with mode-locking erbium fiber
lasers reveals that a newly constructed fiber laser will be likely to mode-lock if
it will CW lase at 1560 nm. However, since the peak of the erbium emission
spectrum is at 1530 nm (see Fig. 2.3), a laser that is designed with the optimal
length of gain fiber will lase strongly at 1530 nm, but probably not at 1560 nm.
To remedy this problem, it is common practice to use a length of gain fiber thatis ∼20-40% longer than what is suggested by the gain equations. By doing this,the gain for light at 1530 nm will decrease due to reabsorption, while the gain
at 1560 nm will increase. The laser will be most likely to mode-lock when thegain fiber length is chosen such that CW lasing is equally likely at 1530 nm and1560 nm.
Of course, these are only guidelines for successfully designing an erbium
fiber mode-locked laser. By staying close to the parameter space outlined here,
chances are your design will turn out fine. But beware, even if you get all of thefiber parameters right and have a sound laser design, practical matters such as
excessive loss at a fiber splice, or back reflections of either the erbium or pumplaser wavelength can easily preclude mode-locking.

2.3.3

Practical matters

The basic schematic for a mode-locked erbium fiber laser and amplifier are
shown in Fig. 2.6. For the current discussion, we’ll focus our attention on only
the laser. For more information about erbium fiber amplifiers, see [39]. The laser
is pumped with a single commercially available 976-978 nm laser diode [94][95].
The pump laser is coupled into the cavity such that it is counter-propagating with

28

Figure 2.6: A schematic for the erbium-doped fiber comb.

respect to the laser beam. The cavity is constructed this way to extract the maximum gain from the erbium fiber. In this configuration, the laser pulse propagates
toward the region of greater gain medium inversion as the pulse energy is amplified, thereby maximizing the gain. Another nice side effect of counter-propagating
operation is that there is no pump light present in the laser output due to the freespace isolator and the polarizing beam splitter (PBS) used for output coupling.
The laser is constructed from three types of fiber and a free space region
that consists of an optical isolator and some polarization optics. The three different fiber types combine to provide the gain, dispersion, and nonlinearity required
for the mode-locking mechanism. The dispersions of five common fibers used toconstruct erbium fiber lasers are given in Table 2.1. Fiber manufacturers will often
times quote the fiber dispersion in units of (ps/nm/km). Equation 2.13 provides
the conversion to the more familiar units of (ps2 /m).
β(ps2 /m) = D(ps/nm/km) ×

λ(nm)λ(km)2πc(m/ps)(2.13)

Control of the repetition frequency of these lasers is addressed using two
intra-cavity actuators. For coarse, manual control of fr , one of the collimators

29
Table 2.1: Dispersion of fibers used to construct mode-locked erbium fiber lasers.
Fiber type

Nominal dispersion β(ps2 /m)

SMF28
Corning Flexcor-7/125
Lieki ER110-4/125
Lieki ER80-4/125
OFS EDF80-0.022-0.0115
0.01150.03360.075

can be mounted on a translation stage. Fine, albeit slow (< 500 Hz), control canthen be achieved by placing a piezo-electric transducer (PZT) in the translation
stage. For fast servoing of fr (upwards of 50 kHz), there are several options. If
a low profile collimator is chosen, the collimator can be mounted directly in afast ring PZT. Also, for lower repetition rate lasers (fr < 200 MHz), a PZT fiber
stretcher can be used. Finally, a small mirror mounted on a fast PZT can be usedin either a sigma cavity or a folded cavity configuration [96][97]. Slow control offo can be achieved by adjusting the angle of one of the intra-cavity waveplates.
Fast control of fo is achieved by modulating the power of the 980 nm pump laser.

Table 2.2: Comparison of the performance of the 100 MHz and 250 MHz modelocked fiber lasers.
Laser Parameter
Oscillator Pump Power
Oscillator Power
Amplifier Pump Power
Amplified Power
Pulse Duration
Cavity Dispersion
Fiber Length

100 MHz Laser

250 MHz Laser

500mW
35mW
1200 mW
300 mW
95 fs
0.022 ps2
183 cm

750 mW
150 mW
1500 mW
450 mW
81 fs
0.005 ps2
70 cm

30

Figure 2.7: a) The spectrum of the 100 MHz Er3+ mode-locked fiber laser directlyfrom the oscillator (red) and after the Raman shifting amplifier (blue). b) The
autocorrelation of the pulse after the laser showing an 95 fs FWHM pulse duration.

2.3.4

Laser performance

In this final section, we present a series of measurements that characterize
the performance of a 100 MHz and a 250 MHz erbium fiber laser built in Yelabs. The list of parameters for these two lasers, shown in Table 2.2, illustrate
many of the design issues mentioned in the previous sections. For instance, the100 MHz laser requires a lower pump power, and has a lower output power thanthe 250 MHz laser due to the higher intra-cavity pulse energy and longer fiber
lengths associated with lower repetition rate lasers. Similarly, the 250 MHz laser
does not even begin to mode-lock until the output power is above 100 mW due
to the short length of fiber available for accumulating nonlinear phase shift.
Figure 2.7a shows the laser spectrum (red) and the spectrum after a Raman
shifting amplifier (blue) for the 100 MHz laser. The spectrum from the amplifier
has been efficiently frequency shifted via a Raman process for CE-DFCS applications in the 1.6 µm to 1.7 µm spectral region. The frequency shift is achieved bypulse amplification followed by a temporal compression of the pulse in a negative
dispersion piece of fiber. A Raman soliton forms in the negative dispersion fiberthat shifts the spectrum of the pulse through a process known as Raman induced

31

Figure 2.8: a) The spectrum of the 250 MHz Er3+ mode-locked fiber laser directlyfrom the oscillator (red) and after the amplifier (blue). b) The autocorrelationof the pulse after the amplifier showing an 81 fs FWHM pulse duration. c) The
spectrum after the amplified pulse is passed through 10 cm of OFS highly nonlinear
fiber.

frequency shift (RIFS) [98]. An autocorrelation of the laser output, shown in
Fig. 2.7b, reveals a 95 fs pulse width.
Fig. 2.8a shows the laser spectrum (red) and the spectrum after the amplifier (blue) for the 250 MHz laser. The amplifier for this laser was designed
to compress the pulse to its minimum duration in the positive dispersion erbium
gain fiber, resulting spectral broadening due to self phase modulation (SPM) [99].
Due to SPM, the width of the spectrum after the amplifier is increased by roughly
50 nm. An autocorrelation of the pulse after the amplifier, shown in Fig. 2.7b,
reveals a pulse duration of 81 fs. Considering the width of the spectrum, it islikely that this pulse could be compressed to roughly 60 fs if a silicon prism compressor were used to compress the pulse rather than the piece of SMF 28 fiberthat we used to achieve 81 fs. However, for most CE-DFCS applications, the pulse
duration is not important, the spectral coverage is what really matters. When theamplified 81 fs pulse is propagated through 10 cm of OFS highly nonlinear fiber,

32
the resulting spectrum extends from 1.1 µm to 2.2 µm (Fig. 2.8c), providing avery broadband laser source for frequency comb spectroscopy.
In general, mode-locked erbium fibers lasers are highly useful both for theirwavelength agility and their ease of operation. Unlike Ti:Sapphire lasers, Er3+
lasers will routinely stay mode-locked for months at a time due robustness of PAPM and the lack of alignment issues in fiber based laser systems. These lasers
will be the primary frequency comb source used in the most of the CE-DFCS
measurements presented in this thesis.

Chapter 3
Useful properties of passive optical cavities

In this chapter, we outline the important properties for passive optical cavities used in CE-DFCS. These properties include the finesse for detection sensitivity, the intra-cavity dispersion for comb-cavity coupling, and the spectral
bandwidth for useable spectroscopy windows. This chapter is devoted to an analysis of these properties for a cavity constructed from two highly reflective mirrors
with quarter-wave stack dielectric coatings. Here, we will illustrate why high finesse, low dispersion, and large spectral bandwidth are desirable qualities for a
CE-DFCS cavity.

3.1

Mirror reflectivity and spectral bandwidth
The spectral bandwidth of a cavity refers to a spectral window for which

the cavity has a high finesse and nearly uniform cavity FSR that are important
for CE-DFCS applications. Both the cavity finesse and the spectral bandwidth
are determined by reflectors used to construct the cavity. In this section, we willdevelop the properties for a type of highly reflective mirror used in all of the CEDFCS experiments presented in this thesis. These mirrors, commonly referred toas quarter wave stack reflectors, are constructed from alternating layers of high(nh ) and low (nl ) index of refraction dielectric material (Fig. 3.1a). As we will

34

Figure 3.1: a) The quarter wave stack structure used to make ultra-high reflectivity mirrors. b) Schematic of the right and left propagating E-fields inside the ith
and the jth layer in the dielectric stack. c) A calculation of the wavelength dependent reflectivity for a quarter wave stack dielectric mirror with a peak reflectivityof 1.6 µm.

see, a condition of high reflectivity is produced from alternating dielectric layerswith an optical thickness of d = λ/4, where λ is the wavelength that the mirror
is designed to reflect.
The reflectivity of a quarter wave stack mirror will be developed following
the transfer matrix formalism found in Klein [100]. At normal incidence, thereflection coefficient at both the nh to nl and the nl to nh interfaces can be foundusing the Fresnel equations
nh − nl,
nh + nl
nl − nhrlh =.
nh + nl
rhl =

(3.1)
(3.2)

From these equations, we can see that rhl = −rlh . We will assume negligible
scattering loss in the dielectric material such that the transmission coefficient isdefined as t2 = 1 − r2 .
To simplify the mathematical description of the total interference from many
dielectric layers, the incident and reflected fields at each dielectric layer will be35
represented as column vectors Eli Ei ≡   ,
Eri
and



(3.3)



 Elj Ej ≡   .
Erj

(3.4)

Here, Er and El denote right and left propagating fields, as depicted in Fig. 3.1b.
Also, a prime is used to describe the fields on the right side of the layer while
unprimed denotes fields on the left side of the layer. Using these definitions, thefields on either side of an interface between the ith and j th can be determinedusing a transfer matrix
Hij ≡





11

t r

rij 
,
1

ij

(3.5)

such that
Ei0 = Hij Ej ,(3.6)

When propagating across dielectric layer j, the beams pick up a phase factor e−iβjthat depends on the layer thickness dj , the index of refraction nj , and the vacuum
wavelength of light λo according toβj =2πnj dj
.
λo

(3.7)

The phase shift due to propagation can also be represented as a transfer matrix


−iβ
0
1 e
Lj ≡ 
(3.8)
.
t
0 eiβ
For example, this matrix can be used to propagate the beam from the right side
to the left side of the j th
Ej = Lj Ej0 .

(3.9)

36
Using these matrices, we can easily represent the multiple beam interference arising from a stack of N dielectric layers


S11 S12 
S1N ≡ H12 L2 . . . LN −1 HN −1,N = .
S21 S22(3.10)

There is one final boundary condition that is needed to make the transfer matrix
formalism useful for describing quarter wave stack mirrors. If we assume thatlight is incident on the stack from the left, then there can be no light incident
from the right on the final layer (the N th layer) of the stack and hence


 0 
EN = 
.
ErN(3.11)

Using this condition, the reflection and transmission coefficients for the entire
stack can be described according to
  

 

0
 El1  S11 S12   0  S12 ErN 
 =

=
,
0
Er1
S21 S22ErN
S22 ErN(3.12)

where
r=

0
El1
S12
=
0
Er1
S22

(3.13)

t=

ErN
1
=
.
0
Er1
S22

(3.14)

andThe peak reflectivity of the stack occurs whenβj =2πnj djπ= .
λo2(3.15)

for both the high and low index layers. Carrying out the matrix multiplications
in Eq. 3.10 leads to a reflection coefficient of
r=

n2N
− n2N
l
h
.
2N
n2N
+
n
l
h

(3.16)

37
Finally, the reflectivity of a stack of 2N layers or N layer pairs is
 !2
nh 2N
1
−
nl
R = |r|2 =
.

nh 2N
1 + nl(3.17)

As can be seen from Eq. 3.17, the reflectivity of a quarter wave stack depends on the number of layer pairs in the stack and the ratio nh to nl . Most of
the mirror coatings used for the work in this thesis were made from alternating
layers of fused silica (nl = 1.45) and niobium oxide (nh = 1.9). Fig. 3.1c shows
the reflectivity versus wavelength for a 20 layer pair mirror constructed from thesematerials with a center wavelength of 1.6 µm. The peak reflectivity for this mirror is Rpeak = 0.99992. The wavelength dependence of the reflectivity, shown in
Fig. 3.1, is found by changing the wavelength in the phase term β and repeating
the calculation in Eq. 3.17. From this calculation, we find that the spectral bandwidth of the mirror reflectivity is determined by the ratio nh to nl . Larger nh /nllead to broader spectral bandwidths of the mirror reflectivity. In Fig. 3.1c, we cansee that the combination of fused silica and niobium oxide provide high reflectivity over a spectral range covering 1.5 µm to 1.7 µm. Outside of this region, themirror reflectivity drops very rapidly. Therefore, the useful spectral bandwidthof these mirrors is about 200 nm or ∼13% of the center optical frequency of thecoating. However, the spectral coverage of modern optical frequency combs canbe very large (∼90% of the central frequency). As a result, mirror coating designs
currently limit the spectral coverage that can be achieved with CE-DFCS.
In response to this, efforts toward improved designs for low-loss and lowdispersion mirrors have been intense over the past few years including the useof different dielectric materials and specialized coating designs [101]. Recently,
new designs for high-finesse cavities that use prism retro-reflectors instead of mirrors have been under development. Since prism retro-reflectors are based on broad
bandwidth effects, such as Brewster’s angle and total internal reflection, these cav-

38
ities could provide spectral bandwidths of >80% of the center optical frequency
with a finesse of up to 50,000 [102][103]. Recently, Johnston et al. coupled aspatially coherent, broadband light source into a prism cavity demonstrating apeak finesse of better than 30,000 at 1.064 µm [104]. A research effort is currently
underway in Jun’s lab to couple an erbium fiber comb to a cavity made from fusedsilica prisms to cover the 1.0 µm to 2.0 µm spectral region with a single cavity.

3.2

FinesseSo far, we have only discussed the reflectors that are used to construct op-

tical cavities. Now, we turn our attention to the properties of the optical cavities
themselves. The finesse is the most important property of an optical cavity because it determines the enhancement factor for the intra-cavity absorption signalthat can be recovered from the cavity transmission. As the finesse increases, thecavity resonances become more sensitive to optical loss. As a result, small absorption signals become amplified by the cavity response. Precise knowledge of thecavity finesse is important for making accurate measurements of the intra-cavity
absorption. In this section, we will first develop the concept of the cavity finesse
and then present techniques for using an optical frequency comb to measure thefinesse.
A schematic of a two-mirror cavity with light incident from the left is shownin Fig. 3.2. The amount of light transmitted and reflected from the cavity depends
strongly on the frequency of the incident beam and the reflectivity and transmission coefficients of the mirrors. An expression for the transmitted field in Fig. 3.2
can be found by summing all of the intra-cavity fields
Eintra−cavity = E0 + E1 + E2 + . . . + En ,(3.18)

39

Figure 3.2: Multiple beam interference in an optical cavity with length L, mirror
reflection coefficients r and and transmission coefficients t.

where
E0 = tEo eiωt~

E1 = tr2 Eo ei(ωt−2L|k|)~

E2 = tr4 Eo ei(ωt−4L|k|)~

En = tr2n Eo ei(ωt−2nL|k|) .
Summing the geometric series yield the intra-cavity field


t
iωt
Eintra−cavity = E0 e,
1 − r2 e−iδ

(3.19)

where δ = 2L|~k|. Specifically, Eq. 3.19 describes the intra-cavity field at the input
mirror. The field transmitted from the cavity is found by applying the mirror
transmission coefficient to the intra-cavity field at the output mirror
 2 −iδ teiωt.
Et = E0 e
1 − r2 e−iδ(3.20)The ratio of transmitted to incident light intensity from the cavity is given by
It
1
=
.
Ii
1 + F sin2 (δ/2)(3.21)

where F is known as the coefficient of finesse and is related to the mirror reflection
coefficient according to

F =

2r
1 − r2

2.

(3.22)

40
The ratio It /Ii is plotted as a function of δ in Fig. 3.4 for three values of F .
As F increases, the window of frequencies transmitted from the cavity becomesnarrower. The most common measure of the width of the cavity resonance is called
the finesse F [105]. The finesse is defined as the ratio of the interval between cavity
modes (2π) to the full width at half maximum phase width of the cavity resonance(∆δ) such that

√
2π
π F
F =
=
.
∆δ
2

(3.23)The most common, and perhaps the most useful, expression for the finesse is
written in terms the cavity free spectral range (FSR) and linewidth (∆ν)
F=

F SR
.
∆ν

(3.24)In Chapter 5, we will quantify the relationship between the cavity finesse and theenhancement of detection sensitivity of intra-cavity absorption measurements. For
now, we will discuss how to measure the cavity finesse, an important precursor toquantitative intra-cavity absorption measurements.
For high finesse cavities (F > 1000), the cavity finesse is usually measured
via cavity ringdown spectroscopy (CRDS) [5] [6]. CRDS measurements are performed by injecting laser light into a resonant mode of the cavity and then rapidly
switching off the incident light. Without input, the intra-cavity power decays at a
rate that is determined by the ratio of the total cavity loss and the cavity roundtrip time (i.e., the linewidth of a cavity resonance mode). The cavity decay timeτcavity and finesse F are thus related by F = 2πτcavity F SR.
It is straightforward to use a single-frequency CW laser for CRDS measurements [106][107][108]. For fast switching of the light source, the laser frequency
can be swept quickly across the cavity resonance such that the time the laser
spends on resonance with the cavity is shorter than the cavity lifetime. Anotherway of switching the incident light is to use an acousto-optic or electro-optic mod-

41

Figure 3.3: The ratio of the incident and transmitted intensity as a function of δfor F values of 0.5, 2, and 200. As F increases phase interval corresponding tocavity transmission ∆δ decreases. When the phase axis is converted to units offrequency, the cavity FSR corresponds to the separation between cavity resonancesand ∆ν is the full width at half maximum linewidth.

ulator that can provide switching times of less than a few hundred nanoseconds,
sufficiently shorter than a typical high-finesse cavity lifetime. CRDS measurements can also be performed with an optical frequency comb if the mode spacingsof the cavity and the comb are roughly matched. For a more detailed discussionof techniques for coupling the comb frequencies to an optical cavity, see Chapter 4. The frequency comb components can then be frequency-scanned across thecorresponding cavity modes just as in the case for a CW laser. When the cavity
transmitted light is frequency resolved using a grating or some other forms of dispersion, the information about the cavity finesse, and thus the intra-cavity loss,
is obtained simultaneously across the entire spectrum of the comb [37].
A set of measurements using fast sweeping of the comb frequencies and a
grating disperser after the cavity are shown in Fig. 3.4. Here, a frequency combfrom a mode-locked erbium fiber laser was used to couple into a cavity constructed42

Figure 3.4: The finesse, and the corresponding cavity mirror reflectivity, of an
optical enhancement cavity spanning from 1.51 to 1.68 µm measured by a modelocked Er+3 fiber laser using frequency-resolved cavity ring-down spectroscopy.

from two mirrors with peak reflectivity at 1.6 µm [76]. A grating was used in cavity transmission to provide a 1-nm spectral resolution. A beam splitter was used
to direct a portion of the cavity transmission to an optical spectrum analyzer forwavelength calibration of the ring-down measurements. This cavity, used in the
trace detection experiments presented in Chapter 7, has a peak finesse of 28,000
and a spectral bandwidth of nearly 200 nm.

3.3

Cavity mode frequencies and dispersion
The dispersion of an optical cavity is important, especially for frequency

comb-based spectroscopy, as it determines the spectral bandwidth over which
the cavity modes and the comb components can be overlapped simultaneously.
While the comb modes are evenly spaced in the frequency domain, νn = nfr +fo , the cavity mode spacing is frequency-dependent. This is due to a variety of
frequency-dependent phase shifts inside an optical cavity. The cavity schematic in43

Figure 3.5: The total phase accumulated for a wave with frequency ω in one
round-trip of an optical cavity with contributions from the mirror coatings, theintra-cavity media, free space propagation, and diffraction.

Fig. 3.5 shows several sources of frequency-dependent phase shifts including themirror phase shift φr (ω), the diffraction phase shift φD (ω), and the phase shift
due to intra-cavity media φm (ω) [109]. To derive an expression for the frequencydependence of the cavity mode spacing, we begin by writing an expression for thephase shift accumulated during one round-trip inside the cavity
φrt (ω) =ωωnm(Lc − Lm ) +(Lm ) + φD (ω) + φmo (ω) + φr (ω).
cc

(3.25)

Here, the phase accumulation inside the intra-cavity medium has been separated
into a frequency-independent component, ωnm Lm /c, and a frequency dependent
component, φmo (ω). Cavity modes exist at frequencies where the round-trip phaseis equal to a multiple of 2π such that
2πN = φrt (ω),(3.26)

where N is the mode number. The change in mode number as a function of
frequency is found by taking the derivative of the mode number with respect to ω
1 Lc + Lm (nm − 1) ∂φD ∂φmo ∂φr
∂N
=
+
+
+
.
∂ω
2π
c
∂ω
∂ω
∂ω

(3.27)

44
The FSR is defined as the frequency interval between adjacent modes; therefore,setting ∆N = 1 yields an expression for the frequency dependent mode spacing.
F SR(ω) =c,∂φ
2L + c ∂ωω(3.28)

where L = Lc + Lm (nm − 1) and ∂φ/∂ω is the combined derivative of all of
the frequency-dependent phase terms inside the cavity. The degree to which thecavity FSR is wavelength (or optical frequency)-dependent is determined by the
intra-cavity dispersion term c(∂φ/∂ω)|ω [110][111][112][113].
Clearly, low intra-cavity dispersion is important for coupling a broad-bandwidth
comb spectrum to respective cavity modes. Furthermore, precise characterizationsof the intra-cavity dispersion are essential for the cavity filtered detection scheme
described in Section 6.4.1. For now we describe a method of using the frequency
comb itself to precisely measure the intra-cavity dispersion. We will discuss theapplication of these measurements in later sections.
A schematic of the comb-based cavity dispersion measurement is shown in
Fig. 3.6. The comb acts as a frequency ruler to determine the cavity mode structure and the wavelength-dependent FSR. First, fr of the laser is detuned awayfrom the cavity FSR at 800 nm as shown in Fig. 3.6b. A frequency discrimination
signal between the cavity modes and frequency combs is recovered and serves tolock the comb to the cavity near the edge of the cavity-comb spectrum (830 nm inthis case). We will present a detailed discussion of cavity-comb locking schemes in
Chapter 4. For now, we note that an electro-optic modulator was used to generate
sidebands on the comb and a Pound-Drever-Hall error signal was used to lock the
comb to the cavity [4]. The locking wavelength becomes a reference from which
the cavity FSR is compared to the laser fr . As one gradually moves away from
830 nm to other parts of the comb spectrum, the corresponding comb components
will spectrally walk away from their respective cavity resonance modes. However,45

Figure 3.6: A technique for using the frequency comb for measuring intra-cavity
dispersion. (a) The cavity transmission spectrum for two values of fr , showing thespectral regions where the comb modes and cavity modes overlap. While combcavity mode overlap is forced at the locking wavelength near 830 nm, the othercavity transmission peaks arise from the frequency dependence of the cavity FSR.
(b) The frequency dependent cavity FSR and values of fr that give rise to thetransmission spectra in (a). (c) A schematic of the cavity and comb modes for thecase of Spectrum 1 in (a), illustrating that cavity transmission peaks occur only
where there is overlap between the cavity and comb modes.

after a sufficiently large spectral gap, a set of comb lines will again come on resonance with cavity modes. Overlaps between the comb frequencies and cavity
modes that lead to cavity transmissions are schematically illustrated in Fig. 3.6c.
A spectrometer is used to measure the wavelength-resolved intensity of the light46
transmitted from the cavity, as shown in Fig. 3.6a for two different values of fr .
The frequency separation between the mth cavity mode and the mth comb
component from the locking point (830 nm) can be determined by the walk-off
integral [112],
Z np
∆νm =




F SR(nc ) − fr (np ) dnc .

(3.29)

nl

Here, FSR(nc ) is the frequency-dependent cavity FSR corresponding to cavity
mode number nc . ∆νm is the frequency gap between the mth comb component
and the mth cavity mode from the locking wavelength where m = nl − np . nl is
the mode number of the cavity that is locked to the comb. np is the mode numberof the comb tooth where the frequency difference between comb and the cavity
mode is ∆νm . Finally, fr (np ) is the repetition frequency of the comb that leads
to a frequency difference ∆νm between the comb and cavity at comb mode np .
When ∆νm = kfr (k = ..., -2, -1, 0, 1, 2, ...), the mth comb frequency overlaps
with the (m + k)th cavity mode. This comb component then appears as a peak inthe cavity transmitted spectrum.
Z np(np − nc )fr (np ) =F SR(nc ) − fr (np ) dnc .

(3.30)

nl

The walk-off integral can be changed into an integral over frequency by converting
the mode numbers into their corresponding frequency values.
Z νp

 dν
∆nfr (νp ) =F SR(ν) − fr (νp ).
fr
νl(3.31)

Here, ∆n refers to the mode number slippage between the cavity and comb overthe frequency interval νl to νp . Since fr is a constant, this part of the frequency
integral can be evaluated directly
Z νp
F SR(ν)dν = fr (νp )[νp − νl ] + ∆nfr2 (νp ).

(3.32)

νlFinally, we use the fundamental theorem of calculus F (b) − F (a) =Rbaf (x)dx

followed by a derivative with respect to νp to arrive at a differential equation that47
relates the frequency dependent cavity FSR to the frequencies of the peaks in thecavity transmission spectrum,
F SR(νp ) = fr (νp ) +∂fr (νp )∂fr (νp )[νp − νl ] + 2∆nfr (νp )
.
∂νp
∂νp

(3.33)

With the laser locked to the cavity at 830 nm, fr is scanned and the wavelength
values of the cavity transmission peaks are recorded for each value of fr . Using
Eq. 3.33, the values of fr that give rise to cavity transmission peaks at frequencies
νp are used to determine the frequency dependance of the cavity FSR. Finally, asimple equation
GDD(ν) =d
1dν F SR(ν)(3.34)

is used to calculate the intra-cavity group-delay-dispersion from the frequency
dependent cavity FSR. Fig. 3.7 shows a measurement of the intra-cavity dispersion for an evacuated two-mirror cavity centered at 800 nm using a Ti:Sapphirefrequency comb. With this level of dispersion, the comb frequencies and cavity
modes can be made to overlap over a spectral window of about 30 nm. However,this cavity has high finesse for a spectral window covering 100 nm. In the next

Figure 3.7: Measurement of the dispersion of a two-mirror cavity with peak reflectivity at 800 nm, using the comb-based cavity dispersion measurement technique.

48
chapter, we will discuss techniques for coupling the comb to the cavity that allow
us to take advantage of the entire 100 nm spectral bandwidth of the cavity finesse.
The precision of this dispersion measurement technique is directly related
to the cavity linewidth. Higher finesse leads to more sharply peaked features incavity transmission and reduced uncertainties in the determination of the center
wavelength of the transmission peak. In the example shown in Fig. 3.7, the cavity
under measurement had a finesse of 3000, an FSR of 380 MHz, and the achieved
experimental precision for intra-cavity dispersion measurement was better than 1
fs2 .

Chapter 4
Cavity-comb coupling

In this chapter we discuss techniques that are used to couple, and more
importantly stabilize the coupling of, the optical comb frequencies to the resonantmodes of an optical cavity. Three techniques will be discussed including (1) a tightlock of the comb components to the cavity modes, (2) a low-amplitude, fast sweepof the comb components over the cavity modes, and (3) precision scan between
initially mismatched FSR and fr such that one comb component leaks out of thecavity at a time. Each of these techniques offers different benefits and limitations
for performing CE-DFCS. Practical applications will dictate the choice for theright approach.
For instance, a tight lock is preferred when using a modulation spectroscopy
technique such as NICE-OHMS [114] or velocity modulation spectroscopy [115][116] in conjunction with CE-DFCS. This is because the continuous presence ofthe cavity transmitted beam leads to long measurement times while the modulation acts to separate the intensity noise of the cavity transmitted beam from theabsorption signal. So far, there have been no efforts to marry the CE-DFCS technique with NICE-OHMS, however, a CE-DFCS and velocity modulation hybrid
system is currently under development in Eric Cornell’s lab to map the transitionsin the HaF+ molecular ion. In cases where the is no modulation applied to theabsorption measurements, the sweep method is preferred because it reduces inten-

50sity noise, thereby yielding a higher signal to noise ratio (S/N) for measurementsof the cavity transmitted beam. This technique was used for all of the CE-DFCS
measurements presented in Chapter 7. Finally, a precision scan of mismatched
FSR and fr can be used to achieve high spectral resolution of the absorption
measurements without the use of a high resolution spectrometer in cavity transmission. An example of this type of measurement is presented in Section 6.4.1.

4.1

Optimizing the coupling of a frequency comb into an opticalcavityBefore discussing the details of the various techniques available for couplinga frequency comb to an optical cavity, we will spend a moment reviewing the basics of optical cavity alignment. This section will cover mode-matching, alignmentof the incident laser beam to the cavity mode, and finding the values of the laser fr
and fo that lead to overlap of the comb frequencies with the resonant cavity modes.

4.1.1

Mode-matching of the laser beam to the cavity

The first thing to consider when coupling a laser beam (frequency comb orotherwise) to an optical cavity is mode matching. Mode-matching refers to thepractice of using one or several lenses to match both the radius and wavefront
curvature of the incident laser beam to a transverse eigenmode of the opticalcavity (Fig. 4.1). The transverse cavity eigenmode used for this discussion, andin all other experiments described in this thesis, is the TEM00 mode.
To decide which lenses should be used and where to place them relative tothe laser and optical cavity, it is first necessary to know the beam parameters forthe both the laser beam and the cavity eigenmode. The beam parameters consist

51

Figure 4.1: Mode-matching schematic showing how two lenses are used to matchthe size and wavefront curvature of an incident laser beam to a cavity eigenmode,
thereby efficiently coupling the laser beam into the cavity. The laser is aligned suchthat the axis of the laser beam is collinear with the axis of the cavity eigenmode.
A photodiode and/or CCD camera can be used in cavity transmission to observeand optimize both the mode-matching and alignment of the laser beam to thecavity.

of the position of the beam focus and the Rayleigh range of that focus. The TEM00
cavity eigenmode parameters are calculated using the so-called ABCD matrices[117][118]. For the cavity shown in Fig. 4.1, the ABCD matrix for a complete
cavity round-trip starting from the intra-cavity focus is






 
L
L
A B  1 2   1 0 1 L  1 0 1 2 






=
0 1
− R2 1
0 1− R2 1
0 1
C D



=

2L2 −4LR+R2
R2
4(L−R)R2

L(L2 −3LR+2R2 )
R2
2L2 −4LR+R2
R2

.

(4.1)

(4.2)

Here, R is the radius of curvature of both cavity mirrors and L is the cavity length.
The q-parameter of the cavity, contains information about both the position of thebeam focus (the real part of q) and the Rayleigh range of the focus (the imaginary
part of q) according toq(z) = z − iz0 .

(4.3)

The Rayleigh range is related to the wavelength (λ) and the radius (wo ) of the
focused beam by
z0 =

πwo2
.
λ

(4.4)

52
The q-parameter for the TEM00 eigenmode of the cavity is found from the roundtrip ABCD matrix using the self consistency relationshipq=

Aq + B
Cq + D(4.5)

Using Eq. 4.5 one can quickly determine if a given cavity geometry will have astable TEM00 mode and what the beam parameters of that mode will be [117].
The laser beam parameters are found by measuring the laser mode diameterat two different distances from laser. This can be done using a beam profiler, ormore primitively, a razor blade on a translation stage and a power meter. Oncethe beam diameters at the two positions have been determined, the position of
the laser beam waist and the Rayleigh range are found by solving the following
system of equationsr
W (z1 ) =r
W (z2 ) =

"

λzo
1+
π



x + z1zo

2 #−1/2
,

(4.6)

"

2 #−1/2
λzo
x + z21+.
πzoHere, W (z1 ) and W (z2 ) are the 1/e electric field radii of the laser beam measured
at distances z1 and z2 from an arbitrary reference point. x is the distance fromthe reference point to the laser beam waist, and zo is the Rayleigh range of thebeam waist.
Once the laser and cavity eigenmode beam parameters are known, the laser
beam can be mode-matched to the cavity mode as shown in Fig. 4.1. Unfortunately, there is no closed-form solution for selecting the focal lengths and thepositions of the mode-matching lenses. This should make intuitive sense as one
would expect that several combinations focal lengths and lens positions will yield
successful mode-matching. To find a suitable set of lenses and positions, one typically uses some sort of computer search algorithm. One particularly nice tool for53
this purpose is the MATLAB genetic algorithm tool [119]. Example code for using
this tool to mode-match a laser beam with a known q-parameter to a two mirror
symmetric cavity is provided in Appendix A. Since lenses are only available in alimited number of focal lengths, it is useful to specify the available focal lengths
in the search program. It is also useful to set limits on the distances between thelenses and between the laser and the cavity based on the area available on the
optical table.
Once a set of focal lengths and positions for the mode-matching lenses has
been found, the solution should be verified on the optical table. With lenses inposition, the laser beam should come to a focus at the position of the intra-cavity
focus. This can be checked by removing the cavity mirrors and observing thebeam as it passes through the region of the intra-cavity focus. If there is someproblem, and the beam does not come to a focus in this region, double-check yourcalculations of the cavity eigenmode and the laser beam parameters. For instance,
when measuring the laser beam width, it is quite easy to forget that the resultof the measurement is the 1/e point of the beam intensity and not the electric field.

4.1.2

Alignment of the laser to the cavity modes

With the mode-matching telescope successfully installed, it is time to alignthe propagation axis of the laser beam to the axis of the cavity mode. For this task,
it is useful to have a CCD camera in cavity transmission as shown in Fig. 4.1. At
the very least, a photo-detector is required to observe the resonant cavity modes.
At this point a slow (< 100 Hz) modulation should be applied to either the cavity
length or the laser fr so that when the cavity-comb coupling becomes close to
optimal, it will be possible to observe the resonances in cavity transmission. It
is best to apply a length modulation that is larger than λ/2 so that at least one

54
entire FSR will be observed in cavity transmission. The laser beam should bealigned to the detector in cavity transmission with the cavity mirrors removed
from their mounts. Once the beam is aligned, the rear cavity mirror (mirror 2 inFig. 4.1) is placed in its mount and aligned such that the laser beam is perfectly
retro-reflected. Next, mirror 1 is placed in its mount and is also aligned such thatthe laser beam is perfectly retro-reflected. At this point, it should be possible tosee faint cavity modes using the CCD camera. If so, the image from the CCD canbe used to align the laser beam to the cavity such that the TEM00 is the mostprominent mode in cavity transmission. If a CCD is not used, or no image canbe seen, it may be necessary to adjust the cavity length or the laser fr such thatfr ≈ FSR before continuing with the laser beam alignment.

4.1.3

Matching the laser fr and fo to the cavity modes

One of the biggest differences between coupling a frequency comb versus a
CW laser to an optical cavity is that efficient coupling of a frequency comb requires that mfr ≈ FSR, where m is an integer. Unfortunately, this requirement
is a bit troublesome since there is nearly no signal in cavity transmission until
mfr ≈ F SR. Fortunately, there are a couple of tricks that work pretty reliably
for achieving coupling. The first trick involves a bit of blind faith: if you believethat the laser beam is reasonably well aligned to the cavity then you can try toscan either fr or the cavity FSR and look for resonances in cavity transmission.
If the cavity finesse is not too high (< 10, 000) and the laser power is larger thana few milliwatts this approach will work most of the time. However, if the laser
power is too low, the finesse is too high, the mirrors are dirty, or you’re justhaving an off day and you can’t find the modes, you can try knocking the laser
out of mode-locking. With the laser operating in CW mode, it will be possible

55
to see resonance in cavity transmission regardless of the cavity length. Using the
CW resonances as a guide, the alignment of the laser beam to the cavity can be
optimized. When the laser is once again mode-locked you can scan the laser fr
and be certain that the laser beam has a good spatial overlap with the cavity
mode. Another trick that is sometimes used to make aligning the beam easier isspectral broadening. Highly nonlinear fiber or a Raman shifting medium can beused to generate wavelengths that are outside of the reflectivity bandwidth of themirrors. The wavelengths corresponding to low mirror reflectivity will pass nearly
unattenuated through the cavity making it easy to see the low-spectral-resolution
fringes in cavity transmission when the incident beam is aligned with the cavity
mode.
Once the cavity transmission resonances have been found, the laser beamalignment, mode-matching, fr and fo can be tuned to optimize the coupling between the comb and the cavity. Guidelines for how to tune these quantities basedon the structure of the resonances observed in cavity transmission are shown in
Fig. 4.2. The situation of optimal coupling of the comb to the cavity is shown in
Fig. 4.2a. In this case, only the TEM00 appears in cavity transmission. Also, thevalue of fo is set such that the height of the fr = FSR resonance is maximized.
This occurs when the fr = (1 ± 1/n)FSR resonances have equal amplitude, where
n is the comb mode number at the center of the cavity transmitted spectrum. Figures 4.2 b) and c) provide examples of non-optimal coupling: b) shows the cavity
transmission when fo is not optimized resulting in a lower signal for the fr = FSR
resonance and unequal amplitudes for the fr = (1 ± 1/n)FSR resonances, whilec) shows the laser beam misaligned and/or poorly mode-matched to the cavity
resulting in the appearance of higher order transverse modes [120].

56

Figure 4.2: a) Optimized coupling of a frequency comb to an optical cavity. The
prominent fr = FSR mode is surrounded by smaller and equally sized fr = (1 ±1/n)FSR modes indicating that the laser fo optimally matches the evolution of
the incident laser pulse phase to the phase evolution of the intra-cavity pulse.
The inset shows that good mode-matching and the laser beam alignment resultin higher order modes that are suppressed by a factor of 200 compared to the
TEM00 . b) and c) show examples of non-optimal coupling. In b) the value of fo
is set incorrectly resulting in fr = (1 ± 1/n)FSR modes that differ in height anda less prominent fr = FSR mode. In c) the laser beam is poorly aligned and/or
mode-matched to the cavity resulting in prominent higher order mode structure
and a reduced TEM00 signal.

4.2

Comb locked to cavity
With the frequency comb optimally aligned and mode-matched to the cav-ity, it is natural to think of ways to stabilize the coupling of the comb to thecavity. The most intuitive method of stabilization is to maintain a precise andsimultaneous overlap of all of the comb frequencies with corresponding cavity
modes (Fig. 4.3). To perform this type of coupling, the laser cavity must main-

57

Figure 4.3: A frequency comb can be locked to an optical cavity by generating
spectrally resolved cavity-locking error signals. An error signal at the center frequency is used to lock a collection of comb modes at ωc to be on resonance with
their respective cavity modes. Error signals at the high (ωh ) and low (ωl ) ends ofthe comb spectrum can be used to maintain a correct value of fo . Adjustment of
fo and fr (via the optical lock at ωc ) ensures an optimal overlap of the comb andcavity modes over a wide spectral bandwidth.

tain an integer multiple of the intra-cavity optical path length (i.e. FSR = mfr ).
Furthermore, the value of fo must be correct such that the comb frequencies areoverlapped with the cavity modes. Finally, the dispersion of the optical cavity
must be sufficiently low to ensure a large number of comb modes overlap with
their corresponding cavity modes.
For a broadband frequency comb, two servo loops are needed to lock the
comb to a cavity [53][54][55], one for fr and the other for fo . The two servo error
signals are generated through spectrally resolved detection of the cavity-comb interaction as shown in Fig. 4.3. For a robust lock, there are several schemes that58
can be used to derive error signals including: a dither-lock modulation scheme, aPound-Drever-Hall (PDH) modulation scheme [4], or a Hänsch-Couillaud polarization scheme[3]. Typically, we use a modulation/lock-in detection approach. For
example, an EOM can be used to place modulation sidebands on the frequency
comb to implement the PDH scheme, or the cavity length can be dithered rapidly
to derive a dither lock error signal. A collection of comb modes at the center of thespectrum is detected, after their interaction with the corresponding cavity modes.
This generates an error signal that is used to lock the optical comb frequencies tothe cavity modes by acting on fr . The interaction of the high and low frequency
ends of the comb spectrum with the cavity can be used to lock fo to the value that
results in the maximum spectral bandwidth coupled into the optical cavity. These
two loops act together to optimize the coupling between the comb and cavity.
Figure 4.4 shows the spectrum of an incident frequency comb and the spectrum transmitted from the optical cavity (F = 2500) when the comb frequencies
are tightly locked to the cavity modes. This cavity contained a 600 µm intracavity Brewster plate made of sapphire that was dispersion compensated with a-40 fs2 negative dispersion mirror. Due to the dispersion compensation, 30 nmof spectrum are efficiently coupled into the cavity with minimal filtering of theincident comb spectrum due to frequency mismatch of the comb frequencies andcavity modes.
The main benefit of locking the comb to the cavity is that the opticalpower transmitted through the cavity is nearly that of the incident beam for thecase of a well impedance-matched cavity. Furthermore, the cavity transmission iscontinuously present. These two features result in a high transmitted power andfast averaging of the intensity noise. The largest drawback is that the spectral
bandwidth of the comb that is coupled into the cavity is typically limited to < 10%of the center frequency, due to mirror dispersion. Also, the continuous coupling

59

Figure 4.4: The spectrum of a Ti:Sapphire laser frequency comb spectrum (red)incident on an optical cavity. The spectrum transmitted from the cavity when the
comb frequencies are locked to the cavity modes (black). The cavity transmission
shows that 30 nm of comb spectrum centered at 800 nm are coupled into thecavity with a minimal amount of filtering due to frequency mismatch of the cavity
modes and comb frequencies.

of the comb to the cavity is very sensitive to acoustic and vibration-induced frequency noise, which gets converted to a substantial amount of intensity noise on
the cavity transmission if the servo loops do not have sufficient feedback bandwidth.

4.3

Comb swept around cavity
The second method for stabilizing the coupling of the comb frequencies tocavity modes involves a periodic injection of the comb into the cavity. This isaccomplished by introducing a low-amplitude, high-frequency sweep between thecorresponding modes of the cavity and the comb [121] [112]. This type of cou-

60
pling has several advantages and some drawbacks compared to directly locking
the comb to the cavity.
The first advantage is that the requirement on the intra-cavity dispersion
is relaxed. Since the comb and cavity modes are being swept over each other,
it is no longer required that all of the comb and cavity modes are exactly overlapped simultaneously. Instead, different modes are allowed to come on resonanceat slightly different times, thus increasing the useful spectral bandwidth that isrecovered in cavity transmission. With a reasonably low dispersion, high finesse,
and fast frequency sweeping, the time it takes for different comb modes to comeon resonance with the cavity can be shorter than the cavity lifetime. For theexample in Fig. 4.5, 100 nm of comb spectrum centered at 1.55 µm is swept overthe cavity modes. Due to low intra-cavity dispersion and optimal matching of the
comb to the cavity, all comb frequencies come onto resonance with the cavity atnearly the same time. Under these conditions, the cavity transmission signal is
sharply peaked. The amplitude of the frequency sweep can be made very small
such that the comb is on resonance with the cavity for a large fraction of thesweep time. By using the smallest possible sweep amplitude, the power containedin the cavity transmission is maximized.
Another advantage of sweeping over the resonance is that it largely avoids
intensity noise on the cavity transmitted beam. Under the locked condition, mechanical vibration-induced cavity length fluctuations cause amplitude noise on
cavity transmission, which can place a limit on the detection S/N of the transmitted beam. However, by sweeping at rates larger than the Fourier frequencies
of dominant mechanical noise processes, the intensity noise of cavity transmission
is drastically reduced.
The final advantage of sweeping is the ease of maintaining the desired coupling between comb and cavity modes. In this case, the servo only has to stabilize

61

Figure 4.5: Cavity transmission for a 100 nm input spectrum coupled to a F= 28,000 cavity. Due to low dispersion and optimal cavity-comb matching, theentire 100 nm incident spectrum becomes resonant with the cavity at nearly thesame time during a frequency sweep.

the time at which the comb modes come onto resonance with the cavity modes
during the sweep cycle. As a result, a low bandwidth servo acting on the cavity
length or laser length is sufficient for maintaining a constant sweep-based combcavity coupling. An example of this type of servo is given in Fig. 4.6. The cavity
transmission is compared against a preset DC threshold in a comparator, generating a train of pulses that indicate when the comb is resonant with the cavity.
These pulses become the ‘clock’ input for a subsequent D-flip flop. The ‘D’ inputof the flip flop is provided by a TTL synchronization signal for the sawtooth sweep
waveform. The output is high when sawtooth is increasing in amplitude and low
when it is decreasing. The output of the flip flop can then be integrated to indicate the separation between successive cavity transmission peaks. A DC offset
voltage can then be used to lock the separation of successive cavity transmission
peaks to any desired value. Using this servoing scheme, fo is measured either by
the interaction of the comb with the cavity, or by f - 2f interferometry. Typically,

62

Figure 4.6: A schematic of the servo used to maintain the position at which the
comb and the cavity come onto resonance during each sweep cycle.

the value of fo drifts quite slowly such that infrequent measurements, once every
several minutes, can be made to maintain the value that optimizes the couplingof the comb to the cavity.
The primary disadvantage of sweep-based coupling is the loss of duty cycle
for cavity transmission measurements. Making a faster sweep reduces the intensity
noise, but the power transmitted through the cavity is also reduced. Therefore, itis necessary to choose an optimum sweep rate that maximizes the overall S/N of
the cavity transmission by making a compromise between the intensity noise andthe transmission power. For systems implemented in our lab, the optimal sweep
rate is typically 1-2 kHz, such that the cavity is allowed to build-up intensity for∼1/10 τcavity . Of course, the optimal settings depend on the bandwidth of thesweep actuator, the vibrational mode frequencies of the laser and cavity length,

63
and the degree of isolation of the cavity and laser from mechanical vibrations of
their environment. Less mechanical noise allows for a longer intra-cavity build-up
time and higher optical power in the cavity transmitted beam.

4.4

Precision sweep of laser fr
The final method of coupling the comb to cavity involves purposely mis-matching the cavity FSR and the laser fr . The comb modes are then swept
precisely through successive cavity modes such that the cavity acts as a frequency
filter in transmission, allowing one or a handful of comb components to be transmitted at a time. This allows serial detection of individual frequency comb components across the comb spectrum [122][123].
To illustrate this approach, we demonstrate the measurement of the spectrum of our mode-locked fiber laser using two distinctly different methods. Figure 4.7a displays the spectrum of the mode-locked laser recorded by a traditional
optical spectrum analyzer. In comparison, Fig. 4.7b shows the cavity transmission signal when the laser fr is detuned from its optimal value for coupling to thecavity and then scanned with respect to the cavity modes. For this measurement,
the cavity had an FSR of 396 MHz and the laser had an fr of 99 MHz, such that4 × fr ≈ F SR. The scan of the laser fr in Fig. 4.7b lasted about 20 ms, andthe resulting cavity transmission signal recorded over this time interval closely
resembles the optical spectrum of the mode-locked laser. The optical frequency
resolution and spectral bandwidth provided by the cavity filtering scheme are determined by the the detuning of the laser fr from its optimal value for coupling
to the cavity (δf = mfr − F SR), the cavity FSR, and the cavity linewidth (∆ν).

64

Figure 4.7: a) The optical spectrum of a mode-locked Er+3 fiber laser measured
with an optical spectrum analyzer. b) The cavity transmission signal as a functionof the laser detuning δf form the cavity FSR. The resulting cavity transmission
signal recorded over time is not sharply peaked, but rather resembles the original
spectrum of the mode-locked laser source.

The resolution is given byresolution = ∆νfor ∆ν < |δf |,

(4.7)

or
resoltuion =∆ν
F SR for ∆ν > |δf |.
|δf |

(4.8)

The first scenario corresponds to the case where only one comb mode is resonantwith the cavity at a time while the second case gives the resolution when several
adjacent comb modes are simultaneously resonant with the cavity.
The spectral bandwidth that can be achieved in a single scan is determined65
by δf , FSR, m, and fr . Figure 4.8 illustrates that if the comb is on resonancewith the cavity at a particular frequency, δf will initially cause the comb modes
to become detuned from the cavity modes in the neighboring spectral region.
However, after a frequency interval of
bandwidth =

fr
mfr2
mfr =
|δf |
|δf |

(4.9)

the comb will once again become resonant with the cavity modes. This relation
can be easily derived by noting that
N mfr − fr = N × F SR,(4.10)

where N is the number of cavity modes between the two spectral regions where thecomb is resonant with the cavity. Rewriting this expression yields an expression
for N in terms of fr and δf
N=

fr
fr
=
.
mfr − F SR|δf |(4.11)

Equations 4.7, 4.8 and 4.9 can be used to determine the resolution and spectral bandwidth for the cavity transmitted signal in Fig. 4.7b. The 396 MHz FSR
and the average finesse of 6300 in the 1.5 µm to 1.6 µm spectral region resultin a cavity linewidth of ∆ν = 63 kHz. Since the repetition frequency detuning(δf = 1620 Hz), was much smaller than ∆ν, the resolution of the cavity transmitted beam, given by Eq. 4.8, is 15.4 GHz. The maximum spectral bandwidth thatcan be achieved using these parameters is 207 nm. The scan in Fig. 4.7b shows
130 nm or 63% of the available scan range.
Equations 4.7 through 4.11 describe the case where fr < FSR. In cases where
FSR > fr , such that mFSR ≈ fr , the substitutions FSR → fr and fr → FSR must
be made for Eqs. 4.7-4.11 to be valid. In this case, δf = fr − mFSR, the spectral
bandwidth is mFSR2 /|δf |, and the resolution is ∆νfr /|δf | when δf < ∆ν.

66

Figure 4.8: An example of the resolution (300 MHz) and spectral bandwidth(fr2 /δf = 10 THz) provided by cavity filtering for fr = 100 MHz, δf = 1 kHz,and ∆ν = 3 kHz.

When more than one spectral region is transmitted from the cavity at atime, there is no longer a unique mapping between scan time and the frequency
in cavity transmission. To avoid ambiguity in such a case, a dispersive elementwith a coarse resolution (better than mfr2 /δf ) should be placed between the cavity transmission and the detector. To obtain a precise calibration of the cavity
transmitted wavelength during a scan of fr requires that the frequencies of thecavity modes be known to high precision. For this, the dispersion measurement
technique described in Section 3.3 can be used to provide a high precision measurement of the wavelength dependence of the cavity mode frequencies. Figure 4.8
illustrates the frequency domain picture of cavity filtering of a comb where fr =100 MHz, ∆ν = 3 kHz, and δf = 1 kHz.
To achieve the actual cavity-linewidth-limited spectral resolution for ∆ν <
δf , the frequency difference between the cavity and comb modes needs to be stabilized to within the cavity linewidth. If the jitter of the cavity or comb modes
is larger than the cavity linewidth, then the value of ∆ν used for determining theresolution should be replaced with the frequency jitter. Since cavities used forsensitive detection can have linewidths in the kHz range, stabilizing the jitter be-

67
low this level is straightforward in a research laboratory but could be challenging
in field applications.
Compared to the frequency sweep technique for coupling the comb to thecavity, the approach of precision fr scan typically uses slower sweep rates of the
comb modes. It is therefore more susceptible to cavity-comb coupling noise arisingfrom mechanical vibrations. Furthermore, this approach takes measurements ina serial manner, prolonging the measurement time. However, this technique does
have the benefit of spectrally resolving the cavity transmitted signal without theneed for an external spectrometer, a unique feature not shared by the two othercoupling techniques.

Chapter 5
Measurement schemes

This chapter describes three measurement schemes that can be used to recover intra-cavity absorption information from the cavity transmitted beams. The
two common methods are cavity-enhanced absorption spectroscopy (CEAS) andcavity ringdown spectroscopy (CRDS). In addition, CEAS can be implemented
in two forms: steady-state or transient. The type of measurement that will bemost convenient for a given CE-DFCS application depends on both the type of
comb-cavity coupling that is implemented and the scheme for detecting the cavity
transmitted beam that is used (see Chapter 6).
5.1

Cavity-enhanced absorption spectroscopy
CEAS refers to measurements of the intra-cavity absorption that are madeby comparing the power transmitted through the cavity with and without absorption present. These measurements can be performed by either locking the comb
to the cavity or by sweeping the comb frequencies over the cavity modes. For the
purposes of this section, tightly-locked measurements will be referred to as steadystate CEAS while measurements involving frequency sweep will be referred to astransient CEAS. As will be seen in the following section, the signal to noise ratio achieved in cavity transmission and thus the absorption sensitivity depends
slightly on the type of CEAS that is implemented.

69

5.1.1

Steady-state CEAS

For steady state CAES measurments, the comb is locked to the cavity, allowing the intra-cavity power in each mode to reach steady state after several
cavity lifetimes. For a single comb component that is on resonance with a cavity
mode, the effect of absorption on the power of the light transmitted through thecavity is given byT1 T2 e−α(ν)LPt (ν)=√
2.
Pinc
1 − R1 R2 e−α(ν)L(5.1)

Here Pt (ν) is the transmitted power, Pinc is the power incident on the cavity,
and R1,2 and T1,2 are the reflectivity and transmission of the cavity mirrors at
frequency ν. L is the cavity length and α(ν) is the frequency-dependent intracavity absorption [124][125]. In the limit that α(ν)L  (1 − R1,2 ) and R1,2 ≈ 1,
the change in transmitted power due to absorption α(ν) can be written as
2Fα(ν)LδPt (ν)
=
.
Pinc
π

(5.2)

A detailed derivation of these expressions can be found in Appendix B.2. Equation 5.2 illustrates that in the limit of low absorption, steady-state CEAS results in an enhancement of the single pass absorption signal by a factor of 2F/π.
Steady-state CEAS has the benefit that it is the most sensitive of the intra-cavity
absorption measurement techniques. However, as discussed in the previous section, there are several drawbacks to this technique. The first is the complexity of
the comb-cavity servo system necessary to control both the laser fr and fo to highprecision. Secondly, intra-cavity dispersion due to the mirrors and sample gas results in a limited spectral bandwidth over which the comb frequencies and cavity
modes are overlapped. Finally, when locking a comb to a high finesse cavity, it is

70
difficult to avoid frequency to amplitude noise conversion, unless a technique similar to NICE-OHMS is employed [114][126]. Consequently, the sensitivity benefitof steady state CEAS is often compromised by technical noise.

5.1.2

Trasient CEAS

Transient CEAS, performed by a frequency sweep between the comb andcavity modes, overcomes some of the limitations of steady-state CEAS. This
scheme is relatively easier to implement due to the relaxed locking requirements
and it solves the problem of limited spectral bandwidth encountered when locking
the comb to the cavity. The cost of transient CEAS is a reduced sensitivity for
absorption measurements.
To perform transient CEAS, the comb modes are made to overlap the cavity modes like they would be for steady state CEAS. However, instead of locking
the comb to the cavity, a low modulation amplitude (<10% FSR), fast (>1 kHz)sweep is applied to either the laser fr , fo , or the cavity FSR. For a single comb
component, the effect of intra-cavity absorption on the transmitted power is givenby∞X
Pt (ν)
2= T1 T2
r1n−1 r2n−1 e−(2n−1)αL/2 eiφn (t) .
Pinc
n=1

(5.3)

Here n is the number of cavity round trips and r1,2 are the electric field reflection
coefficients for the cavity mirrors. The phase term is given byν(t)L2φn (t) = 2π(2n − 1)(L− 2(n − 1)β 2 ),
c
c

(5.4)

where ν(t) is the instantaneous frequency of the comb component, and β is its frequency sweep rate [127] [128]. A derivation of the transient CEAS equations canbe found in Appendix B.3. Besides the lower detection sensitivity, transient CEAS

71
also has the disadvantage that quantitative measurements of the intra-cavity absorption require careful calibration. The calibration accuracy is contingent on
precise control of the frequency sweep. Maintaining a constant sweep rate is important. A common culprit causing non-constant sweep rates is hysteresis foundin transducers such as piezo electrics. It is therefore necessary to either make measurements only in one sweep direction or determine the actual frequency sweep
function and take measurements when the opposing sweep rates are equal.

5.2

Cavity ringdown spectroscopy
Another method for making intra-cavity absorption measurements is cavity

ringdown spectroscopy (CRDS) [6][106] [129][130][107]. To perform CRDS, light isfirst injected into the resonant modes of the optical cavity. When the intra-cavity
power in each cavity mode reaches a sufficient level, the light incident on thecavity is switched off. The cavity decay signal is then recorded by a photodiodewith a detection bandwidth exceeding that of the cavity decay rate. The powerin each cavity mode decays exponentially with a decay constant given byτcavity =1
1.
=
F SR(ν)Loss
F SR(ν)(1 − R1 R2 e−2α(ν)L )(5.5)

Depending on the length and finesse of the optical cavity, this decay time can takeon values ranging from tens of nanoseconds to hundreds of microseconds. In thelimit that α(ν)L  (1 − R1,2 ) and R1,2 ≈ 1, the change in cavity decay time due
to absorption α(ν) can be written asδτcavity (ν)Fα(ν)L=.
τcavityπ(5.6)

A derivation of the cavity ringdown equations is provided in Appendix B.1. For
CRDS, the enhancement of the absorption signal in the low absorption limit is72
exactly half of the enhancement in steady-state CEAS. This is due to the fact thatfor steady-state CEAS, the presence of the incident beam suppresses decay of theintra-cavity field from the input mirror. While in CRDS, the field decays throughboth the input and output mirrors. In the case that the reflectivity of the input
and output mirrors are exactly equal, CEAS is twice as sensitive to intra-cavity
absorption as CRDS. In the case of unequal reflectivities, Eqs. 5.1 and 5.5 can beused to determine the relative sensitivities.

CRDS has a couple of advantages over the CEAS methods for measuring
intra-cavity absorption. First, since the laser is effectively switched off during themeasurement time, cavity ringdown measurements are free of laser intensity noise.
Also, since the finesse of an optical cavity is typically a very stable quantity thatcan be measured to high precision, background measurements of the cavity decayrate without any intra-cavity absorption can be performed relatively infrequently.
Hence, more time can be devoted to making absorption measurements rather thancalibrating them. The primary drawback of cavity ringdown spectroscopy is thatit requires fast sampling to record the cavity decay rate. This becomes challenging
in broadband applications where hundreds or even thousands of channels must be
simultaneously sampled to record an intra-cavity absorption spectrum.
5.3

Sensitivity comparison
To conclude this section we present a comparison of the absorption signal

enhancement for each of the three measurement schemes. This comparison will beillustrated by the enhancement of absorption signals for a transient CEAS measurement as a function of the speed at which a single comb ‘tooth’ is swept overa cavity mode. In the limit of an infinitely slow sweep, the signal enhancement oftransient CEAS approaches that of steady-state CEAS. In the limit of a very fast

73

Figure 5.1: Absorption sensitivity and integrated power of cavity transmission
versus sweep speed (β) for transient CEAS. Cavity FSR = 400 MHz and mirror
reflectivity R = 0.9995 resulting in a cavity linewidth ∆ν = 63.7 kHz. A time
window of 100 µs was used for integrating the cavity transmitted signal.

sweep speed, the sensitivity of transient CEAS approaches that of CRDS. Fig. 5.1
shows both the low absorption limit detection sensitivity and the time-integrated
fractional transmitted power in transient CEAS as a function of the sweeping
speed of the incident laser frequency.
For the calculations displayed in Fig. 5.1, FSR = 400 MHz and R1,2 = |r1,2 |2= 0.9995, resulting in a cavity lifetime of 2.5 µs. A time window of 100 µs was chosen for integrating the cavity transmitted signal. The center wavelength was 1.525µm and the intra-cavity absorption was 10−8 cm−1 . At slow sweep speeds, it takes
a comb frequency many cavity lifetimes to scan across the cavity linewidth. Under these conditions, the intra-cavity power nearly reaches the steady state value
and cavity transmitted beam power is Pα = Po (1 − 2F αL/π). Here, Po is thecavity transmitted power with no intra-cavity absorption. For a fast sweep speed,
the comb frequency scans across the cavity resonance in much less than a cavity
lifetime. In this case, the peak intra-cavity power is much smaller than the steady74
state value, and the majority of the cavity transmitted signal takes the form ofan exponential decay. Since the time integral of an exponentially decaying signal
is proportional to its decay time, the change in the integrated cavity transmission
due to absorption is just proportional to the change in the cavity ringdown time,thus linking transient CEAS to CRDS. When the frequency sweeping speed (β)
is such that the comb frequency traverses the cavity line-width in one or a few
cavity lifetimes (β ≈ ∆ν 2 ), changes in the cavity transmission due to intra-cavity
absorption arise from a combination of steady state CEAS and CRDS effects.
Unless a modulation spectroscopy technique is used to separate intensity
noise for the absorption signal, it is usually advantageous to sacrifice the factorof 2 enhancement associated with steady state CEAS and work in the regime offast sweeping. Doing so allows us to avoid relatively low frequency mechanicalnoise present in both the laser and the passive optical cavity. The gain in S/N bymaking detection at Fourier frequencies where mechanical noise is low is typicallymuch greater than the factor of 2 gained via steady-state detection.

Chapter 6
Detection schemes

In the previous chapter we outlined a few strategies for making intra-cavity
absorption measurements. In this chapter we focus on broad-bandwidth detection schemes that can be implemented to provide frequency-resolved and massively
parallel detection channels for the intra-cavity absorption. Depending on the particular application, different detection schemes can provide uniquely optimized
performances.

6.1

The grating spectrometer
The simplest way to perform parallel detection of a broadband cavity trans-mission signal is to disperse the beam using a diffraction grating. The dispersed
beam can then be imaged onto a multi-channel detector such as a photodiode array or a CCD camera. Since nearly all of the multi-channel detection techniquespresented in this chapter involve the use of a diffraction grating, we will devote
the first section to developing the theory of the grating spectrometer. We willthen highlight the capabilities of the grating spectrometer for parallel detections
before presenting schemes for implementation in the following sections.

76
6.1.1

Theory of the grating spectrometer

When used for parallel detection of many frequency channels, a grating
spectrometer consists of a diffraction grating, an imaging lens, and a detector
array, as shown in Fig. 6.1. In the following paragraphs, we derive equations forseveral properties of a grating spectrometer including the spatial dispersion, thespectral resolution, and intensity distribution of light at the detection plane.
As shown in Fig. 6.1, the cavity transmitted beam, with beam radius w, isincident on the diffraction grating at an angle θig . The diffracted angle of the lightleaving the grating θdg is determined by the grating line-spacing d, the wavelengthof the incident light λ, and the incident angle of the beam according to the familiar
grating equation [131]mλ = d(sin(θdg ) − sin(θig )),(6.1)

where m is the order of the diffracted light. Typically the first diffraction order(m = 1) is used to maximize the diffraction efficiency. The angular separation
between two different wavelengths as a function of diffracted angle can be foundby solving equation 6.1 for θdg−1θdg = sinmλ + d sin(θig )
.
d

(6.2)

Next, Eq. 6.2 is differentiated with respect to wavelength and then we substituteEq. 6.1 into the denominator yielding
dθdg
=q
dλ
1−

1

m
1
m
p
=
.
mλ+d sin(θig ) 2 d
1 − sin2 (θdg ) d

(6.3)

d

Finally, we use the identity 1 = sin2 (θ) + cos2 (θ) to obtain a compact formula forthe angular dispersion of light diffracted from the grating∆θdg =m∆λ.
d cos(θdg )

(6.4)

77

Figure 6.1: Grating spectrometer schematic. A cavity transmitted beam with 1/eelectric field radius w is incident on a diffraction grating with line-spacing d atangle θig . Light is diffracted from the grating at a wavelength dependent angle
θdg . The dispersion of grating is characterized by the angular separation ∆θdg
between two wavelengths separated by ∆λ. The dispersed light is imaged onto a
detector array with pixel pitch l using a lens of focal length f . The detector arrayis placed at the focal plane where the focused light has a diffraction limited spot
size wo .

Once the grating dispersion is known, an imaging relationship from Fourier optics,∆x = ∆θf , can be used to determine the spatial separation (∆x) of the opticalfrequencies at the detector array (the focal plane) [132][133].
α=

λ2 f
dx
=
.
dω
2πcd cos(θdg )(6.5)

Here, the diffraction order m = 1 has been assumed, and the differential relationship∆λ = −

λ2
∆ω
2πc

(6.6)

has been used to convert from wavelength to angular frequency. The radius ofa focused, single wavelength, beam on the detector array is related to w by the

78
imaging lens focal length, and the incident and diffracted angles from the grating[134][133]wo =cos(θig ) f λ.
cos(θdg ) πw

(6.7)

Combining equations 6.5 and 6.7 with the Fourier relation results in an equation
for the frequency dependent spatial intensity distribution of light on the detector
array
Idet (x, ω) ∝ Iin e−2(x−αω̃)2
2
wo

.

(6.8)

Here ω̃ = ω − ωo and ωo is the frequency at the center of the detector array. The
maximum resolution that can be achieved by a grating spectrometer depends onthe diameter of the incident beam w, the wavelength of the incident light, the linespacing of the grating, and the incident and diffracted angles [135]. If we choose
a fairly conservative resolution condition (e.g. two spots must be separated by a
1/e2 beam diameter to be resolvable) we find that the maximum resolution is∆νmin =

2cd
.
πwλ

(6.9)

Two conditions must be met to achieve the resolution limit of Eq. 6.9. First,
the lens used to image the diffracted light onto the detector must have low aberrations (chromatic, spherical, etc.) and must be sufficiently well aligned to reachthe diffraction limit. Second, the diffraction limited spot size must be much larger
than the pixel pitch l such that the spectrometer resolution is not limited by spatial sampling of the detector array.
Grating spectrometers typically provide resolutions on the order of a few tomany tens of gigahertz. The overall spectral bandwidth that can be recorded by
the detector in a single shot is limited by both the size of the detector array andthe area over which a flat, or abberation free, field of imaging can be maintained.
In general, single shot spectral bandwidths range from hundreds of gigahertz toseveral terahertz. For a fixed detector size with n resolvable detection channels

79
and a resolution of ∆ν, the spectral bandwidth that can be recorded in a single
shot is n∆ν. This relationship illustrates the compromise that must be madebetween single shot bandwidth and resolution that is encountered when designinga broadband detection system.

6.2

Multi-channel cavity ringdown detection
A serious issue that arises from multichannel detectors is the rate at which

the array can be read out. Depending on the detector size and the type of amultiplexer used for sampling, read-out rates in the tens of Hz to tens of kHz
range can be achieved. While the later provides fast frame acquisitions for signal
averaging, it is still too slow to directly observe cavity ring-down decays. Most
cavities require a sampling rate of around 1 MHz to perform accurate ring-down
measurements. In the following subsections two techniques are described for simultaneously recording multiple channels of cavity ring-down signals.

6.2.1

Streak camera cavity ring-down

The first method, originally referred to as ‘Ring-down spectral photography’,uses a rotating mirror to streak the cavity transmitted light across a CCD camera[136]. Using this technique the cavity ringdown decay signal is recorded by the
CCD as a as a function of position on the detector array [137]. During our firstexperiment involving CE-DFCS, we implemented this strategy. Images of theposition-dependent decay from these experiments are shown in Fig. 6.2a, b, & c.
Here, the cavity transmitted beam was dispersed in the horizontal dimension bya grating and a rotating mirror was positioned between the imaging lens and thecamera to streak the beam across the camera in the vertical dimension [37]. For

80this experiment, we used a resonantly driven (355 Hz) gold mirror mounted on a
torsion spring as our rotating mirror. The mirror had a total deflection angle of40◦ (20◦ in each direction from its rest position) and was placed 10 cm from the
CCD. The mirror was aligned so that the cavity transmitted beam was incident
on the center of the camera when the mirror was at rest. Accordingly, the beam
deflection as a function of time at the CCD plane is given byx(t) = xo cos(ωt),(6.10)

where xo = 10(cm) × tan(20◦ ) and ω = 2π×355 Hz. The velocity of the beamacross the center of the CCD is found by differentiating Eq. 6.10 with respect totimev(t) = ωxo sin(ωt).

(6.11)The grating spectrometer used for these measurements produced a 2 cm diameterbeam that was focused onto the CCD with a 25 cm focal length lens. The neardiffraction limited spot size measured at the CCD had a 1/e width of a = 12.5
µm. At the center of the CCD, the focused beam traverses each pixel in time
δt =

a
vmax

=

12.5µm
= 0.15µs.
ωxo(6.12)The detection cavity for this experiment had a finesse of 4500 and an FSR of
380 MHz, resulting in an empty cavity ring-down time of 1.88 µs. Therefore, thestreak camera spectrometer sampled the cavity transmitted intensity more than
12 times during each 1/e-decay interval of the intra-cavity power. Equation 6.11
is used to precisely convert the decay-versus-position signal to decay versus time.
The cavity ring-down signal is then fit with a decaying exponential to extract
the ring-down time, which is related to the intra-cavity absorption at the corresponding detection channel. Each column of the CCD simultaneously records a
decay signal at a different wavelength, resulting in a single-shot, broad bandwidth

81

Figure 6.2: Streak camera images recorded near 820 nm with a) 5 Torr and b)
10 Torr of H2 O inside the detection cavity. c) A 3-D image of the 10 torr streak
image reveals the lower build-up and faster ring-down of the wavelengths that
correspond to strong H2 O absorptions. d) The mirror rotation rate and mirror to
CCD distance are used to convert the streak images into ring-down times which
are then converted to absorptions.

ring-down spectrum shown in Fig. 6.2d.
To successfully record broadband ring-down images like those in Fig. 6.2,
two conditions must be met. First, the cavity decay events must be synchronizedwith the rotating mirror such that the ring-down signal hits the same area on the
CCD for every measurement, thus preserving the position-time calibration. Thiscan be accomplished by sweeping the laser fr with a sawtooth waveform that istriggered by the rotating mirror driver electronics. A convenient way to derive
this signal is to integrate the TTL synchronization signal from the rotating mirror
driver electronics. Secondly, fo and fr of the comb must be well matched to thecavity as described in Section 4.1.3 such that all wavelengths are simultaneously

82
resonant with the optical cavity.
Since the scanning mirror in this experiment operated at 355 Hz, we wereable to acquire a ring-down spectrum every 1.4 ms. However, the CCD we used
is capable of exporting a ring-down image to the data acquisition computer every
300 µs. Therefore, the use of a faster scanner could result in a faster spectral acquisition rate. Eventually, the camera frame rate will be the ultimate limitation
for the streak camera detection scheme. In the next section, we discuss an electronics architecture that could be used to acquire ring-down spectra much fasterthan is possible using the streak camera method.

6.2.2

Multichannel cavity ring-down sampling electronics

The main disadvantage of the streak camera method is that it requires a 2D
detector array to record 1D spectral data. To directly sample ringdown signals
from a large-numbered 1D photodiode array, we have developed an electronics
architecture that is capable of sampling and recording ringdown events from anarbitrarily large photodiode array at rates that can easily exceed 1 MHz. A block
diagram of the electronics architecture is shown in Fig. 6.3. A large photodiodearray, consisting of hundreds to thousands of detector elements, is first divided
into sub-arrays of 10 to 50 elements that detect adjacent portions of the optical
spectrum. The photodiode array in Fig. 6.3 represents one of these sub-arrays.
The current from each element in the sub array is converted to a voltage by a bankof transimpedance amplifiers. Next, a bank of comparators is used to compare
these photo-voltages against a known reference voltage that is set by a digitalto-analog converter (DAC). The reference voltage is chosen to be a fraction of
the peak voltage recorded by one of the detector elements in the sub-array. As
the intra-cavity field decays, each photo-voltage drops below the reference voltage

83

Figure 6.3: A circuit for parallel readout of cavity ringdown signals from a photodiode array.

and the corresponding comparator emits a pulse that is recorded by a clocked
memory chip. The comparator pulse is also sent to the input of an AND gate.
When each signal drops below the DAC voltage, the output of the AND gate
switches, instructing the DAC to set a new reference voltage. The subsequent
reference voltages are chosen to be a power law series such that an exponentiallydecaying ring-down signal will produce a series of comparator pulses that are
equally spaced in time. This sequence continues until all signals drop below thelowest reference voltage. During this process, a field programmable gate array(FPGA) performs a number of operations to ensure that the reference voltages
switch at the appropriate times and that the ringdown signal for each channel
is recorded correctly. When the comparator pulse arrival data is retrieved fromthe memory chip, a simple linear fit of the arrival time of the comparator pulses
versus pulse number reveals the ringdown time of that channel. So far, only a small

84
prototype of this circuit consisting of four channels has been built and tested, but
the scalability of this architecture will allow for hundreds to thousands of channels
to be simultaneously sampled with a small additional effort.
6.3

The virtually-imaged phased-array spectrometer
One way to overcome the compromise between spectral bandwidth and res-olution is to use a virtually-imaged phased-array (VIPA) spectrometer shown in
Fig. 6.4 [138][38][76]. This spectrometer contains a solid etalon (VIPA etalon),similar to a Fabry-Perot, whose spectral dispersion is oriented orthogonally to thatof a diffraction grating. The etalon is tilted slightly (∼2◦ ) from normal incidence
such that the beam transmitted from the VIPA resembles an elongated stripe oflight. The name and dispersing capability of the VIPA come from the fact thatas light bounces back and forth up the etalon, the transmitted light appears tooriginate from a phased-array of sources (see inset in Fig. 6.4). These sources
are termed ‘virtual’ because their apparent locations are on a line that extends
from the back side of the VIPA normal to the surface. The interference of thesource array gives rise to a large angular dispersion of light transmitted from theetalon that is roughly 20 times greater than the dispersion of an optical grating[139]. A diffraction grating is placed after the etalon to disperse the transmitted
beam into a 2D pattern, with each spatial element corresponding to a unique
comb frequency. The 2D spatial pattern of the comb is then imaged onto a CCD
camera to record the frequency-resolved image of the cavity transmitted power.
While light dispersed along the tilt of etalon provides a high spectral resolution,light dispersed along the orthogonal direction by the grating acts to resolve thedifferent mode orders transmitted through the VIPA.
Like a Fabry-Perot etalon, each mode order of a VIPA etalon covers a frequency range that is equal to the FSR of the etalon. Unlike a Fabry-Perot, the85

Figure 6.4: A schematic of the VIPA spectrometer showing the incident beamfocused into the VIPA etalon via a cylindrical lens. The transmission of the VIPA
etalon is dispersed in the horizontal dimension using a diffraction grating and thenimaged onto the focal plane of a detector array. The expanded VIPA schematicshows the incident angle of the focused beam θiv and the propagation angle inside
the VIPA θv . The array of virtual sources that generate the interference patterntransmitted from the VIPA are shown originating from a line perpendicular tothe surface of the etalon.

surface of the VIPA is designed with three regions of different reflectivity. The
bottom of the front side of the etalon has an anti-reflective (AR) coating and isused to couple light into the VIPA. The top of the front side has a highly reflective (HR) coating (R∼100%) that directs light inside the VIPA toward thetransmission direction. Finally, the back side has a partially-reflective (PR) coating (r∼96%) that transmits a small portion of the light from inside the VIPA ateach reflection point. The sum of all of the transmitted waves from the partiallyreflective surface creates the phased-array interference effect and hence the angular dispersion of the transmitted beam.

86
6.3.1

Theory and design considerations

There have been several theoretical treatments of the VIPA etalon [139][140][138][141][142][143]. These theories vary in complexity and in agreement withexperimental observations. The discussion presented here will be based on amaster equation for the propagation of a beam through a VIPA spectrometer
developed by Xiao et al. [138]−2fc2 y 2

Idet (x, y, ω) ∝ Iin e f 2 w2−2(x−αω̃)2
1
2
wo
e
.
2 k∆
2
(1 − Rr) + 4Rr sin ( 2 )(6.13)

This equation describes the 2D intensity distribution of the frequency-resolved
interference image at the detector plane. fc and f are the focal lengths of thecylindrical and imaging lenses, w is the incident beam radius, R and r are thehigh and partial reflectivities, and k = 2π/λ is the wavenumber of the beam. Since
the grating and VIPA dispersions are orthogonal, Eq. 6.13 is a simple product of
the grating intensity distribution (x-direction) and the VIPA intensity distribution(y-direction) [133][141]. Therefore, the term for the grating dispersion intensity
distribution is identical to Eq. 6.8. Most of the physics that determines the VIPA
dispersion is contained in the parameter∆ = 2tnr cos(θv ) − 2ttan(θv ) cos(θiv )y t cos(θv )y 2−.
f
nr f 2(6.14)

Here, t is the etalon thickness, nr is the index of refraction, θiv is the incident
angle of the focused beam on the VIPA, and θv is the propagation angle of thebeam inside the VIPA. The incident and internal propagation angles of beam are
related according to Snell’s law (nr sin(θv ) = sin(θiv )). A thorough derivationof the VIPA intensity distribution using paraxial wave theory can be found inreference [141]. In the following paragraphs, Eq. 6.13 will be used to determine
most of the important properties of the VIPA spectrometer.
The first important property we will examine is the spatial dispersion of

87
light at the detection plane. The intensity pattern in Eq. 6.13 has maxima whenk∆ = 2mπ,(6.15)

where m is the mode order of the VIPA transmission. The approximate mode
number is given by m ∼ 2tnr cos(θv )/λc where λc is the center wavelength of thedispersed spectrum. Using this condition, the wavelength can be related to the
y-position in the detection plane bymλ = 2tnr cos(θv ) − 2ttan(θv ) cos(θvn )y t cos(θv )y 2−.
f
nr f 2(6.16)Taking the derivative of this expression with respect to y yields an equation for
the VIPA etalon dispersion


2t
∆λ
cos(θv )y=−tan(θv ) cos(θiv ) +
.
∆y
mf
nr f

(6.17)

Of course, this equation is important for mapping wavelengths to positions in thedispersed interference pattern. It is also useful during the spectrometer design
process for determining if one FSR of the VIPA transmission will fit within thearea of the detector array.
Another important property of the VIPA spectrometer is the spectral resolution. The resolution can be determined from the second term in Eq. 6.13. We
know from Eq. 6.15 that the transmitted intensity is maximum when1
(1 − Rr)2 + 4Rr sin2

k∆
2

=

1
.
(1 − Rr)2(6.18)As the wave number k changes, the transmitted intensity is reduced from Imax to
Imax
2

when
2

(1 − Rr) = 4Rr sin

2




k∆
, or
2

1 − Rr
√
= sin
2 Rr




k∆
.
2

(6.19)

Since Rr ∼ 1, Eqs. 6.19 indicate that sin(k∆/2)  1 when the transmitted
intensity drops to Imax
. Accordingly, we can use the paraxial approximation2

88(sin(2πmx + δx) ≈ δx) by definingk∆ko ∆= 2πmx + δx, and= 2πmx,22(6.20)

where ko is a wavenumber that corresponds to Imax . Combining the two equationsin 6.20, we find thatδx =∆ko δλ.
2λ(6.21)

Substituting this result into Eq. 6.19 yields an expression for the wavelength interval δλ between the positions corresponding to Imax and Imax /2 in the detection
plane. The ‘full width at half maximum’ (FWHM) linewidth is defined as twice
this interval such that
δλF W HM ≈λ2o (1 − Rr)
√
.
π∆ Rr

(6.22)

Using the familiar relation δν = −(c/λ2 )δλ we can convert Eq. 6.22 to the frequency domain.
δνF W HM ≈c(1 − Rr)
√
.
π∆ Rr

(6.23)

Equation 6.23 describes the position dependent resolution of the VIPA spectrometer in the y-direction. An examination of ∆ reveals that the resolution is only
weakly position dependent, so for most cases it is sufficient to write the VIPA
spectrometer resolution using only the first term in ∆
δνF W HM ≈c(1 − Rr)
√ .
2πtnr cos(θv ) Rr

(6.24)

A rigorous determination of the spectrometer resolution also requires consideration of the pixel size of the detector array. In particular, if the diffraction limited
spot size of the focused beam on the detector is of similar size or smaller thanthe detector pixels, the pixel size must be considered in the resolution calculation.
In this case, Eq. 6.17 is used to find the frequency interval that corresponds to
one pixel width. The pixel frequency interval is then convolved with the VIPA

89
resolution function, which results in the overall spectrometer resolution.
Another important consideration regarding the VIPA resolution is the lineshape of a single frequency at the detector plane. The lineshape function can bederived by substituting the approximations in Eqs. 6.20 and 6.21 into the second
term in Eq. 6.13g(λ) ≈

12 k2o(1 − Rr)2 + Rr∆(λ − λo )2
λ2.

(6.25)

Here we see that while the grating spectrometer lineshape is Gaussian, the VIPA
lineshape is Lorentzian. This arises because the light intensity transmitted fromthe VIPA decreases exponentially as a fucntion of height along the etalon [142][141].
Since the Lorentzian function decays slowly, the frequency channels in the VIPA
spectrometer have high levels of cross-talk. This fact must be considered when
making quantitative spectroscopic measurements of closely spaced absorption features. There have been several proposals to alter the VIPA lineshape including
graded reflectivity coatings, but so far none of the solutions are practical to manufacture [142].
The final VIPA spectrometer property that we will discuss is the physicalsize of the interference pattern at the detection plane. The first term in Eq. 6.13
determines the y-direction extent of the interference pattern. Specifically, thedistance between the 1/e intensity points in the detection plane is
√
2f w
.
Wy =
fc

(6.26)

Accordingly, the y-direction size of the interference pattern depends on the widthof the cavity transmitted beam and the ratio of the focal lengths of the imaging
lens and the cylindrical lens. Equation 6.26 is useful for matching the size of
the VIPA interference pattern to the size of the detector array. When designing
a VIPA spectrometer, it can happen that you wish to change Wy to match the90
Table 6.1: VIPA spectrometer example design parameters.
VIPA and grating parametersnr
2t
1.5 mm
R
1.0
r
0.96
θv
1.25◦
θiv
2.5 ◦
d
0.909 µm
θig
- 75 ◦
θdg
52.6◦

Camera parametersl
25 µm
pixels320×256
Lenses
f
20 cm
fc
6 cm
Incident beamw
1.5 mmλ
1.6 µmsize of the detector, but it is not desirable to change either f or fc . In this case,a telescope can be used to expand or contract the cavity transmitted beam toproduce the desired value of Wy .

6.3.2

VIPA spectrometer design example

In this section, we present an example design of a VIPA spectrometer. Using
the parameters contained in Table 6.1 and the equations in Section 6.3.1, we willcalculate some of the spectrometer properties and illustrate some of the importantdesign considerations.
We begin with a calculation of the how efficiently light is coupled into the
VIPA etalon. The coupling efficiency is determined by how of much the incident
beam is clipped by the interface between the HR and AR coatings as it entersthe etalon. The two important quantities for this calculation are the size of the
focused beam as it enters the etalon wv and the length interval δh, shown in
Fig. 6.4. Using the parameters in Table 6.1, the 1/e radius of the beam power isgiven byfc λ
wv = √= 14.4 µm.
2πw(6.27)

91
The length interval δh can be found from the etalon thickness and the beam
propagation angle inside the VIPAδh = 2t tan(θv ) = 65.4 µm.

(6.28)

The clipping loss is minimized when the center of the incident beam is positioned
δh/2 below the line separating the HR and AR coatings. The beam experiences
clipping loss twice as it enters the etalon. Once as it enters the front surface of the
VIPA and a second time when it makes its first reflection from the HR coating.
According to the paramters in Table. 6.1, the portion of the beam coupled into
the VIPA is−37.82
Pcoupled= (1 − e 14.42 )2 = 0.9885 = 98.85%.
Pincident(6.29)To understand how sensitive the coupling is to the spectrometer parameters wewill make the same calculation, but this time we’ll reduce the incident beam radiusby half−37.82
Pcoupled= (1 − e 28.82 )2 = 0.525 = 52.5%.
Pincident(6.30)

This small change in parameters has increased the clipping loss by a factor of 40
from 1.2% to almost 50%!
For the next calculation, we’ll determine the vertical size of the VIPA interference pattern at the detector plane. To make efficient use of the light transmitted
from the spectrometer, it is desirable to match the size of the interference pattern
to the size of the detector array. In the y-direction, the detector has 256 pixels
with a pixel spacing l of 25 µm for a total height of 6.4 mm. The size of theinterference pattern can be calculated using Eq. 6.26,
√
2f w
Wy =
= 7.1 mm.
fc

(6.31)

According to Eq. 6.31, we can calculate that at the top and bottom of the image,2

2

the intensity of the VIPA fringes will have dropped to e−3.2 /3.55 ≈ 44% of their

92
value at the center of the image.
The next question we’ll address is, “what is the size of one FSR of the VIPA
transmission at the detector plane?” To make sure that most of the detector area
contains unique spectroscopic information, one FSR should cover more than halfof the y-direction size of the detector array. The VIPA etalon used here has an
FSR of approximately c/(2nr t) = 50 GHz, and the mode number of the VIPA
transmission at 1.6 µm is m ≈ 2tnr cos(θv )/λ = 3749. With this information,
Eq. 6.17 can be used to determine the spatial extent of one FSR∆y ≈F SRλ2 mf
 = 4.2 mm.
cos(θv )yave
2tc tan(θv ) cos(θiv ) +nr f(6.32)

Here the average value of the y-coordinate (yave ) in the detector plane is 1.6 mm.
The result of this calculation is that one FSR of the VIPA transmission occupies(4.2 mm)/(6.4 mm) = 65% of the detector area.
The final calculation in this section will determine the resolution of our
example VIPA spectrometer. According to Eq. 6.23, the VIPA resolution isδνF W HM ≈c(1 − Rr)√
= 650 MHz.
2πtnr cos(θv ) Rr

(6.33)To check if the pixel size affects the spectrometer resolution we can calculate thefrequency interval corresponing to one pixel widthδνpixel ≈2ltctan(θv ) cos(θi v) = 256 MHz.
λ2 mf

(6.34)A convolution of these two frequencies results in a total resolution of 697 MHz,
which shows that the contribution of the pixel width is minimal.
We conclude this section by presenting a CE-DFCS absoprtion image recorded
by a VIPA spectrometer with paramters very similar to those in Table 6.1. Figure 6.5 shows an absorption image of the rovibrational spectrum of CO2 centeredat 1.609 µm. Each vertical fringe contains roughly 1.7 FSR mode orders of the93

Figure 6.5: An image of a rotational spectrum of a CO2 vibration overtone centered at 1.609 µm recorded with the VIPA spectrometer. The image contains 25
nm of spectrum with 833 MHz resolution. The panel to the right shows the P(13)line of the (011) to (311) vibration overtone band, illustrating the resolution.

etalon transmission, which has been dispersed and resolved in the horizontal direction by the grating. The vertical fringes in the interference pattern are wellresolved due to the 25 GHz resolution of the grating spectrometer compared to
the 50 GHz FSR of the VIPA etalon. Probably the most impressive aspect of
this image is that it contains more than 3500 spectral channels that were acquired simultaneously. This is far more than what could be achieved using a
one-dimensional grating spectrometer, and with significantly higher resolution.
To create this image the cavity was filled with such a large concentrationof CO2 that the cavity response has been completely saturated. As a result, thecavity transmitted light is extinguished where there are CO2 absorption features.
This has been done for the purpose of visualizing the frequency resolved absorption patterns in the VIPA interference image; however, during typical operationof the spectrometer, the cavity is operated in the linear regime and absorption
features are not be visible with a naked eye. Instead, a differential measurement

94
is made that subtracts the VIPA image with intra-cavity absorptions present from
one that was taken when the cavity contains no absorption. This procedure will
be discussed further in Section 6.3.4. However, before one can make absorption
measurements, the VIPA spectrometer must first be constructed, aligned and calibrated. This procedure will be the subject of the next section.

6.3.3

Construction, alignment, and calibration

Before beginning to construct the VIPA spectrometer, there are few experimental conditions that must be met to ensure a successful experience. First, theaverage power of the beam used to align the VIPA must be sufficiently high tobe easily detected. Ideally, the beam could be visibly detected via an IR card ora thermally sensitive liquid crystal sheet. Often times, the average power of thecavity transmitted beam is quite low and so it may be necessary to use a second
laser beam that has been aligned collinear to the cavity transmission for aligning
the VIPA. Second, the cavity transmitted beam should be aligned to propagate
parallel to the optical table prior to aligning the VIPA spectrometer. Finally, thecavity transmitted beam should be well collimated. If the detection cavity has agentle intra-cavity focus, this condition should already be met. Otherwise, a lensshould be used to collimate the beam.
Figure 6.6 shows a four step procedure for constructing and aligning a VIPA
spectrometer. In the first step, the VIPA etalon is positioned such that the division between the HR and AR coatings perfectly bisects the beam. Also, thebisecting line should be horizontal, or parallel with the optical table. The beam
bisection can be optimized by adjusting the etalon position while watching either
the transmitted or reflected beam from the VIPA. Next, the horizontal angle of thebeam on the VIPA should be aligned to normal incidence. This is accomplished95

Figure 6.6: Procedure for constructing and aligning a VIPA spectrometer.

by adjusting the VIPA so that the reflected beam is directly above the incident
beam. Finally, the vertical angle of incidence for the beam should be aligned tothe desired value (typically 1.5◦ -3.5◦ ). For this, a ruler can be placed at a known
distance (∆z) from the etalon to measure the vertical displacement between theincident and reflected beams (∆y). These distances can then be used to calculate
the vertical angle of incidence.
In the second step, the cylindrical lens is placed roughly one focal lengthin front of the VIPA etalon. This lens should create a horizontal focused line at
the front surface of the etalon. The axis of the focus should be parallel to theline dividing the HR and AR coatings. The line-focus should intersect the front
surface of the etalon approximately δh/2 below the HR/AR division as shown in
Fig. 6.4. To adequately align the cylindrical lens, it is useful to mount it on a
two-axis translation stage for fine control of the y and z position. Also, the lens

96
can be mounted on a goniometer for adjusting the angle of the line focus relative
to the line dividing the HR and AR regions. The alignment of the cylindrical lenscan be optimized by maximizing the power transmitted from the VIPA. When the
beam is optimally coupled into the VIPA etalon, the transmitted beam should resemble a single vertical stripe that has the same width as the input beam. If the
stripe is not vertical, or if it appears split into multiple stripes, the alignment is
not optimal.
When the transmission of the VIPA looks optimal, it is time to introduce
the grating. The grating should be mounted on a rotation stage such that the center of the grating’s front surface is along the axis of rotation. The mount should
also allow for control of the vertical tilt of the grating, which can be achieved
by fixing the grating to a mirror mount. The grating should be positioned suchthat the VIPA transmission is centered on the grating as shown in Fig. 6.6. The
zeroth-order reflection from the grating and the vertical tilt control can be used
to make sure that the beam reflected from the grating is level with the optical
table. Finally, the camera should be positioned in preparation for introducing
the imaging lens. The distance between the camera and the grating should be atleast 1.5 times the focal length of the imaging lens. The height of the detector
array should be the same height of the beam above the optical table, and it should
be positioned along the anticipated propagation direction of the beam diffracted
from the grating. Finally, the plane of the detector array should be oriented perpendicular to the incident beam. This can be accomplished by positioning one
arm of a carpenter’s square along the line from the center of the grating to thecenter of the detector array. The other arm is then perpendicular to the incident
beam and can be used as a reference for orienting the detector.
The final, and most difficult, step in aligning the VIPA spectrometer is optimizing the interference pattern at the detection plane. This discussion will first

97
address the procedure for coarse alignment of the imaging lens before moving on
to fine-tuning the alignment. Five control axes are required to achieve optimalalignment of the imaging lens: a three-axis translation stage allows control of the(x, y, & z) positions of lens while a mirror mount provides tilt control in the vertical and horizontal directions.
For coarse alignment, place the imaging lens roughly one focal length fromthe detector plane. Position the lens so that the incident beam passes throughthe center of the lens. Again, a carpenter’s square can be used to orient the lensperpendicular to the incident beam. At this point, an interference pattern similar
to the one in Fig. 6.5 should be recorded by the camera. This image will be used
to fine tune the alignment of the imaging lens. A particularly useful diagnostic isthe line that transects the interference pattern in Fig. 6.7a, which corresponds toan intensity cross-section of the interference fringes on the detector. The camera
images should be streamed via computer so that intensity cross-sections can beobserved in real-time. Fig. 6.7 shows some of the ways that the fringe curve canbe used to optimize the alignment of the imaging lens.
The first step in fine-tuning the imaging lens alignment is to optimize thefocus of the fringe pattern. To do this, move the lens toward and away from thedetector plane. You should observe the fringe contrast changing as you move thelens. At the lens position that produces the maximal fringe contrast, record thefringe contrast (Imax − Imin )/Imax . Next, move the lens toward the detector plane
until the fringe contrast has been reduced by half. Record the lens position. Similarly, move the lens away from the detector plan, past the optimal position, until
the contrast is again reduce by half and record the lens position again. The lensshould be placed at the average of the two recorded positions for optimal focusingof the fringe pattern.
While moving the lens toward and away from the detector plane it is worth

98

Figure 6.7: a) The fringe intensity cross-section. b) The focus of the imaging
lens can be aligned by maximizing the fringe contrast. Curve 1 shows the fringe
contrast when the imaging lens is either too close to (or far from) the detector.
Curve 2 shows the fringes when the lens is optimally positioned. c) The tiltof the lens and centering of the diffracted beam on the lens can be optimizedby comparing the fringe contrast at the left versus right side interference image.
Similarly, the y-direction tilt can be optimized by comparing the fringe contrast
between the top and bottom of the image.

noting if the contrast of the fringe pattern is changing uniformly across the image. If the fringe contrast changes are not spatially uniform, the imaging lens tilt
and/or the centering of the incident beam on the imaging lens must be optimized.
This is an iterative process by which the lens is tilted and/or translated until thefringe pattern is in sharp focus everywhere in the recorded interference pattern.
The final steps in preparing the VIPA spectrometer for making quantitative absorption measurements are wavelength and resolution calibrations. These
calibrations can be done by filling the detection cavity with a low pressure, dilute sample of an appropriate calibration gas. The calibration gas should haveabsorption lines in the spectral region where the VIPA spectrometer will be op-

99
erated. The absorption lines should have well-known transition frequencies forwavelegnth calibration and well-known pressure broadening coefficients for resolution calibration of the spectrometer. The interference image in Fig. 6.5 shows
the spectrum of CO2 , a good candidate for a calibration gas at 1.6 µm. Using thisimage, one can quickly determine the VIPA FSR interval. The many absorption
features, each with a well-known transition frequency, allow for a complete mapping of wavelength to position within the interference image. Since the linewidthof the absorption feature is known (i.e. the Doppler width convolved with pressure broadened width), the resolution of the spectrometer can be determined bydeconvolving the known linewidth from the linewidth measured by the VIPA spectrometer. An accurate measurement of the VIPA spectrometer resolution requiresthat the intra-cavity absorption be sufficiently weak such that it doesn’t saturate
the cavity response. Also, the intra-cavity pressure must be low enough thatpressure broadening is negligible and the VIPA resolution is the dominant contribution to the measured linewidth.

6.3.4

VIPA image processing

Once the VIPA spectrometer is well-aligned and calibrated, it can be used torecord spectra of the intra-cavity absorption. To extract a traditional absorption
spectrum from the recorded VIPA images requires a modest amount of image processing. A flow diagram of the spectral recovery process is presented in Fig. 6.8,
where a pair of 256 x 320 pixel images are converted into a 25 nm absorption
spectrum. The first step is to locate the nearly vertical fringes produced within
the picture. This is done using the reference image, which is recorded without
intra-cavity absorption. Fringes are located by an algorithm that begins at thepeak of each fringe at the middle of the picture (i.e. row 128) and ‘walks’ along

100

Figure 6.8: Generation of an absorption spectrum from VIPA images. a) absorption and reference images are recorded. b) The nearly vertical fringes in both thereference and absorption images are collected into column vectors and processed
according to (Pr − Pa )/Pr to form a relative power array. The relative powerarray is used to find the VIPA FSR, indicated by the area between the solid lines.
c) The fringe intervals between the solid lines are then collected and arranged
endwise to form a traditional absorption spectrum.

each fringe up and down to find the crest of each fringe within the entire picture
frame. The intensity of each fringe is converted into a column vector forming
an array that contains frequency resolved cavity transmitted power information.
A MATLAB routine for finding the fringes with and calculating the normalized
cavity transmitted power at each frequency within the VIPA image is provided inAppendix C.
To correct for possible drifts in the cavity transmitted power between theacquisition of the absorption and reference pictures, an area of low absorption in
the absorption fringe array is equalized in power to the reference fringe array. The
two fringe arrays are subtracted and then divided by the reference fringe array,
resulting in a power-normalized fringe array shown in Fig. 6.8b. The FSR interval of the VIPA, shown as the interval between the two solid horizontal lines inFig. 6.8b, is determined from the recognition of the repetitive patterns of absorp-

101tion peaks in the power-normalized fringe array. All unique information about
the molecular absorption is contained within this interval. Once this interval isdetermined, the area of the normalized fringe array between the horizontal lines is
unwrapped into a one dimensional spectrum of relative power versus wavelength.
Here the Eq. 6.13 can be used to obtain a high precision mapping of the positionof each wavelength on the normalized fringe array. Finally, the cavity enhancement equations from Chapter 5 are used to convert changes in cavity transmitted
power to their corresponding values of intracavity absorption.

6.4

Precision scan of fr and cavity filtration
Another approach for achieving high spectral resolution over a large spectral

window is cavity-filtered comb detection. Unlike previously mentioned approaches
that rely completely on optical dispersers to analyze the cavity transmission, thecavity-filtered detection uses the frequency response of the cavity itself to only allow specific spectral components of the frequency comb to be transmitted throughthe cavity at a time. Cavity filtered detection can be performed using a single
detector element in cavity transmission, or in conjunction with a low resolution
spectrometer and a detector array. The relationship between the fr of the comb
and the FSR of the cavity determine which detection scheme should be used incavity transmission.

6.4.1

Single detector element cavity filtered detection

In this section, we present an example of cavity-filtered detection that uses
only a single detector element in cavity transmission. A detailed discussion ofcavity frequency filtration is presented in Section 4.4. The results presented here
are based on a fr = 99 MHz mode-locked erbium fiber laser comb, a detection

102

Figure 6.9: Schematic for the cavity filtration measurement setup. An iodine
stabilized Nd:YAG laser (1064 nm) is used to stabilize the cavity frequencies via
a piezo electric actuator that controls the cavity length. The 10th harmonic of
the laser fr is locked to a microwave signal that is the mixing product of a 1 GHz
Wenzel oscillator and a 10 MHz DDS. To scan a spectrum, the DDS frequency
is set such that the laser fr is detuned from the cavity FSR. The DDS frequency
is then linearly scanned, resulting in a time-dependent output frequency for thecavity transmission.

cavity with an FSR of 396 MHz (FSR ≈ 4 ×fr ), and a finesse of 6300 at thedetection wavelength 1.525 µm. According to the relationship ∆ν = FSR/finesse,
the cavity has an intrinsic linewidth of 63.5 kHz.
To use the cavity-filtration approach for frequency-resolved detection of cavity transmission, it is necessary to precisely control the frequencies of both thecavity and comb modes. For the measurements presented here, the frequency
stabilization scheme is shown in Fig. 6.9. The cavity is stabilized by locking aparticular cavity resonance (and thus the cavity length) to a frequency-stabilized
Nd:YAG laser at 1064 nm, resulting in a λ/3200 stabilization of the cavity length.
For an FSR ≈400 MHz cavity, this corresponds to stabilization of a cavity resonance frequency to within 250 kHz. To stabilize the relative frequency between thecavity and the comb, the tenth harmonic of the laser fr is locked to the frequency

103
difference of a 1 GHz low-phase-noise crystal oscillator and a 10 MHz auxiliary
RF signal from a direct digital synthesizer (DDS). The time base of the DDS isslaved to the same low-phase-noise crystal oscillator. The DDS is used to generatea high precision linear frequency ramp that sweeps the comb frequencies over thecavity modes. The DDS makes a 500 Hz ramp in steps of 23 mHz, each lasting
10.2 µs for a total sweeping time of 0.22 s.
To illustrate the utility of this scheme for frequency-resolved detection ofcavity transmission, the cavity was backfilled with 100 mTorr of C2 H2 and 500 Hz
scans were taken at detunings of δf = 10 kHz and δf = 20 kHz, where δf = 4fr −FSR (Fig. 6.10). Since the cavity modes were stabilized to within 250 kHz (effective ∆ν), the δf = 10 kHz scan resulted in a resolution of (∆ν/δf )×FSR = 10GHz. The available spectral bandwidth for this scan is mfr2 /δf = 3.92 THz, or 30
nm at 1.525 µm. Figure 6.10a shows a 25 nm spectrum, or 83% of the available
scan range, with the C2 H2 absorption spectrum clearly extracted in Fig. 6.10b.
For the 20 kHz scan, δf was increased by a factor of 2 and consequently the resolution increased and the bandwidth decreased by the same factor of 2. This leads to
the 12 nm spectrum with 5 GHz-resolution shown in Fig. 6.10c. The variations in
the baselines of these two scans arise from a combination of cavity-comb coupling
noise and absorptions due to unresolved hot bands of C2 H2 that occupy the same
spectral region as the first overtone of the asymmetric C-H stretch.

6.4.2

Cavity filtration with multiple detection channels

A couple of options exist to use cavity filtration to achieve higher resolution
detection. The first and the most intuitive approach is to increase the detuning δf and perform scans of higher resolution with narrower spectral bandwidth.
The second and more elegant method was recently demonstrated by Gohle et al.

104

Figure 6.10: a) The cavity transmission signal with and without 100 mTorr ofC2 H2 inside the cavity for a δf = 10 kHz detuning of 4 × fr from the cavity F SR.
b) The C2 H2 absorption spectrum extracted from a) showing 25 nm of spectrumwith 10 GHz resolution. c) A second sweep, where δf = 20 kHz, resulting in a
12.5 nm spectrum with 5 GHz resolution.

[122]. This method combines cavity filtering with a grating spectrometer in cavity
transmission to achieve high resolution and large spectral bandwidth in a single
system. In a sense this scheme shares the same spirit as the VIPA spectrometer,

105

Figure 6.11: An example of how the comb-cavity detuning (δf ) can be selected tomatch the cavity-filtered spectral bandwidth to the corresponding spatial periodof a multichannel spectrometer in cavity transmission.

where a high resolution element (etalon) is combined with a low resolution grating. Whereas the VIPA scheme uses a second spatial dimension to provide highresolution, cavity filtration uses a temporal scan of the comb frequencies.
Figure 6.11 shows a schematic of how the cavity filtration and spatial dispersion can work together to provide high resolution and large spectral bandwidth.
First, the detuning δf is chosen such that the spectral bandwidth of cavity filtration is equal to or larger than the resolution of the grating spectrometer. In thecase shown in Fig. 6.11, FSR = fr = 100 MHz and δf = 400 kHz resulting ina spectral bandwidth of 25 GHz between frequencies where the comb is resonantwith the cavity. Since the cavity linewidth, ∆ν = 50 kHz, is much less than δf ,only one comb component is transmitted from the cavity in each 25 GHz frequency
interval. Figure 6.11 shows a grating spectrometer with a resolution of 25 GHz
can then be used to record the transmitted power of each comb component. Using this technique, the single-shot bandwidth of the grating spectrometer limits

106
the spectral bandwith of the measurement, while the resolution is limited by thecavity linewidth. Finally, each scan covers 250 cavity modes requiring a scan timeof at least 250 cavity lifetimes for single comb mode resolution. For typical cavities, 250 cavity lifetimes corresponds to a scan time of one to tens of milliseconds.

Chapter 7
Applications

In the final chapter, we present a series of cavity-enhanced direct frequency
comb spectroscopy measurements, beginning with initial demonstrations of thesetechniques followed by more advanced and recent measurements. This overview
is meant to demonstrate the general capabilities of these systems and highlightrecent technological advances that have allowed significant simplifications and expanded capabilities of the experimental setups. The initial demonstrations of
CE-DFCS used Ti:Sapphire lasers at 800 nm and relatively low resolution grating
spectrometers to probe weak high molecular overtone transitions. While thesesystems were adequate to develop the CE-DFCS measurement techniques, they
were not well suited for demanding applications such as trace molecular detection.
As a first step toward addressing this problem we used an Er3+ fiber laser
at 1.5 µm to probe the much stronger first overtones of molecules containing C-H,
O-H and N-H bonds. More recent work using the Er3+ fiber laser and the high resolution VIPA spectrometer have allowed CE-DFCS to emerge as a powerful spectroscopic tool that combines high resolution, high sensitivity, and broad spectral
bandwidth. The majority of this chapter is devoted to CE-DFCS measurements
made using an Er3+ fiber laser source and VIPA spectrometer detection. These
measurements include trace detection of room temperature gas samples, molecular isotope analysis, human breath analysis, and detection and characterization

108
of supersonically cooled molecular jets.

7.1

Initial demonstrations
There were several demonstrations of cavity-enhanced absorption spectroscopyusing short pulses with broad spectral coverage that provided the foundation forour eventual development of CE-DFCS. The first of these, performed by Crosson
et al., was referred to as ‘Pulse-stacked cavity ring-down spectroscopy’ and used a
mid-infrared free electron laser (FEL) to coherently add picosecond pulses insideof an optical enhancement cavity [144]. This approach increased the S/N of cavity ring-down measurements by several orders of magnitude compared to a single
pulse measurement. Using the pulse stacking scheme they were able to achieve anabsorption sensitivity of 2 × 10−9 cm−1 at 10 seconds averaging time for an H2 O
absorption line at 5.38 µm. Crosson et al. realized that efficient coupling of pulses
into the cavity required that the pulse injection rate be equal to a harmonic of
the cavity FSR and that the phase of the injected pulse must match the phaseof the pulse inside to cavity. However, since the FEL source had a large amountof phase noise and no means by which to control the phase of the incident pulse
train, it could not be coupled optimally to the cavity.
Shortly thereafter, work by Scherer et al. made progress toward broad bandwidth detection by developing a spectrometer for the parallel acquisition of manycavity ringdown signals [137]. Their technique, known as ‘Broadband ring-down
spectral photography’, used a broad-bandwidth dye laser and a rotating mirror
in cavity transmission to streak the spectrally dispersed cavity-transmitted light
across a CCD camera. This initial demonstration probed the rotational spectrumof the b1 Σg −X 3 Σg transition in O2 and achieved the detection of a 8.5 nm spectral
bandwidth with a 45 GHz resolution in a single measurement. This corresponds

109
to the parallel acquisition of 120 ring-down measurement channels. The primarydrawback of this work was the extremely inefficient coupling of the dye laser spectrum into the optical enhancement cavity.
The final precursor to CE-DFCS was performed by Gherman and Romanini
using a mode-locked Ti:sapphire laser to probe the high overtone spectrum ofC2 H2 at 860nm [121]. In this work, Gherman et al. came close to realizing thefull set of requirements for efficient coupling of a frequency comb to an optical cavity. They determined that fr = FSR was a necessary condition, and also observedthat even though fr = FSR they still had not achieved optimal coupling. They
guessed that the non-optimal coupling must be due to fo . Unfortunately, they
had no means to control fo and therefore were not able verify their suspicion, nor
could they achieve optimal coupling of the comb frequencies to the cavity modes.
Their experiment used transient CEAS measurements and a cavity with a finesseof 420 for an absorption sensitivity of 2 × 10−7 cm−1 Hz−1/2 . A high resolution
grating spectrometer and a CCD camera were implemented in cavity transmission
to provide 14 nm of single shot bandwidth with a resolution of 6 GHz.
The first demonstration of CE-DFCS was made using a Ti:Sapphire laserbased frequency comb at 800 nm to probe the high overtone spectra of C2 H2 ,H2 O, NH3 , and a weak forbidden electronic transition in O2 [37]. This work, forthe first time, exploited control of the fo to maximize the coupling efficiency of
the frequency comb to the resonant modes of the optical cavity. This capability
was coupled with parallel detection of cavity ringdown signals resulting in greatly
improved S/N and sensitivity of absorption measurements. An example of thesemeasurements is provided in Fig. 7.1. This system used a F = 3300 cavity andcavity ringdown detection to achieve an integrated absorption sensitivity of 10−8
Hz−1/2 (2.5×10−10 cm−1 Hz−1/2 ). In cavity transmission a grating spectrometerand CCD were implemented to record 20 nm of single shot spectra with 25 GHz

110

Figure 7.1: The overtone spectrum of H2 O (red), the P-branch of the b1 Σg −X 3 Σg (0,0) vibrational transition of the electronic transition in O2 (blue), and theovertone spectrum of NH3 near 790 nm (green) demonstrate the broadband, highresolution, and high sensitivity capabilities of this spectroscopic technique. For
each molecule, 1000 spectral scans were averaged to produce the traces shown.
The inset graph show that the individual rotational lines of each spectrum havebeen resolved revealing the characteristic doublet structure of the O2 rotational
spectrum.

resolution. To achieve adequate time resolution of the ringdown signals, a rotating
mirror was placed between the imaging lens and the CCD and the streak camera
technique described in Section 6.2.1 was used. This combination of spectral bandwidth and resolution allowed for the parallel detection of 375 ring-down channels.

7.2

Trace detection at 1.5 µm
While the Ti:Sapphire-based CE-DFCS system provides good sensitivity

for the study of high molecular overtones, the weak absorption associated withthese transitions make them not very desirable for investigating low concentration

111

Figure 7.2: Absorption spectrum measurements for 2 Torr CO, 10 mTorr NH3 ,and 1.5 mTorr C2 H2 , showing 150 nm of spectral information. The inset of the Pbranch of the CO overtone spectrum shows individually resolved rotational lines.

molecular samples. Furthermore, the resolution of this system is too low to resolve
the Doppler line-widths of room temperature gases, further limiting the sensitivity of molecular concentration measurements. In response to these limitations, webegan constructing CE-DFCS systems based on Er3+ fiber laser frequency combs
at 1.5 µm. In addition, we implemented a high resolution VIPA spectrometer incavity transmission to dramatically increase the detection sensitivity and spectral
resolution without diminishing the spectral bandwidth available with CE-DFCS
systems [76].
The first demonstration of CE-DFCS using an erbium fiber laser probed thefirst overtone transitions of NH3 and C2 H2 , and the second overtone of CO [75].
This system still used a low resolution grating spectrometer in cavity transmission,and therefore was not well suited for performing trace detection measurements.
However, the advantages of probing lower vibrational overtones for trace detection
can already be seen in the measurements, shown in Fig. 7.2. Here the absorption
spectra of 2 torr CO, 10 mTorr NH3 , and 1.5 mTorr C2 H2 demonstrate that athousand times greater concentration of CO is required to produce an absorption

112
signal that is similar to the signals from NH3 and C2 H2 . The factor of 1000 inconcentration is due to the fact that both NH3 and C2 H2 are detected using afirst overtone transition while CO is detected via its second overtone. Because ofthis, the 1.5 µm spectral region is especially useful for detecting molecules with
N-H, C-H, and O-H bonds.
The final step toward constructing a suitable trace gas detection system at
1.5 µm was the implementation of a high resolution VIPA spectrometer in cavity
transmission. The complete 1.5 µm trace detection system was first reported in
the breath analysis measurements in Ref. [76], and a schematic of this system
is shown in Fig. 7.3. The move to high resolution detection allows for the optimization of the molecular absorption signal by matching the resolution of thespectrometer to the absorption line-width. This point will be addressed in moredetail in the next section. For now, consider the simple case where the molecularabsorption is much narrower than the spectrometer resolution. In this case, mostof the light in a given detection channel will not interact with the molecular absorption. The result is that the molecular absorption signal is diluted by a factor
equal to the ratio of the spectrometer resolution to the absorption line-width.

7.2.1

Trace molecular concentration measurements

In this section, we present a series of trace molecular concentration measurements in the 1.5-1.7 µm spectral region made using the CE-DFCS system shownin Fig. 7.3a. These measurements are meant to demonstrate the capabilities of
our trace detection system and provide examples of real-world applications of CEDFCS technique. The frequency comb used for this system was a 35 mW average
power Er3+ fiber laser with fr = 99 MHz. This laser generates a 100 nm spectrum
from 1.5 to 1.6 µm. For measurements in the 1.6-1.7 µm spectral region, a Raman

113

Figure 7.3: a) Schematic of the 1.5 µm CE-DFCS trace detection system. b) Spectrum directly from the laser (right) and the spectrum from the Raman-shifting
amplifier (left). c) The wavelength dependent finesse of the detection cavity.

shifting amplifier is used to shift the spectrum and increase the average power to
300 mW (Fig. 7.3b) [98]. The detection cavity had an FSR of 99 MHz with apeak finesse of 28,000 at 1.61 µm (Fig. 7.3c). The comb was coupled to the cavity
via swept frequency coupling and transient CEAS measurements were performed.
A VIPA spectrometer was used in cavity transmission to achieve a 25-nm single
shot bandwidth and 800 MHz resolution.
To characterize the absorption sensitivity of the system we performed measurements of the cavity transmitted light with the VIPA spectrometer centeredat 1.61 µm and only dry nitrogen in the detection cavity. We recorded two consecutive VIPA images that were converted into an absorption spectrum using theprocedure outlined in Section 6.3.4. The residual noise from these measurements

114

Figure 7.4: The absorption noise versus number of measurements at 1.61 µm.

are plotted in Fig. 7.4 against the number of pictures that were averaged to makethe measurement. The upper data set (red) shows the residual noise when thecavity transmitted light was incident on the camera and the lower data set (blue)shows the noise when the cavity transmitted light was blocked from the camera.
The camera frame rate for these measurements was 60 Hz. Therefore, ∼30 s wererequired to average 2000 pictures at which point the residual absorption noise was
8×10−10 cm−1 per detection channel.
For these measurements there are several effects to consider in order to optimize the spectrometer performance. The equation that relates the absorption
coefficient measured by the spectrometer to the gas concentration is [145]α(ν) = ρSi g(ν).

(7.1)

Here, α is the absorption per unit length (cm−1 ), ρ is the gas concentration (cm−3 ),Si is the transition strength (cm), and g(ν) is the absorption line-shape (cm). The
values for Si can be measured, found, or inferred from a variety of references [146][147]. Although the values for Si have a strong temperature dependence, all trace
concentration measurements presented in this section were made at a temperature of 296 K and so the values of Si will be treated as constants for the present

115

Figure 7.5: The rovibrational overtone spectra of a) 330 parts per billion (ppb)
C2 H4 and 3.3 ppm CO2 and b) 1 ppm CH4 .

discussion [148]. The temperature dependence of Si will be discussed in detail in
Section 7.3.2. The absorption line-shape depends on both the temperature andpressure conditions of the measurement. In particular, choosing the appropriate
intra-cavity gas pressure is important for maximizing the molecular absorption
while not diminishing spectral resolution via collision broadening. The gases thatwere measured and analyzed in our 1.5 µm CE-DFCS system include: CO, CO2 ,C2 H2 , C2 H4 , CH4 , and NH3 . Parts of the recorded spectra for each of these gases
are shown in Figs. 7.5 and 7.6. Since these trace gases are measured in a host gasof N2 , the air broadening coefficient adequately describes the observed collision
broadening. For the measurements presented in this section, an intra-cavity gaspressure of 250 Torr was chosen such that the average line-width for a gas under

116

Figure 7.6: The rovibrational overtone spectra of a) 32 ppm CO, b) 78 ppb C2 H2 ,and c) 4.4 ppm NH3 .

measurement was 1.8 GHz, resolvable by the 800 MHz resolution of our system.
For the molecules studied by this system, the spectral window between
1.5 µm and 1.7 µm contains more than 10,000 individual lines for measurements
at 296 K. To handle this large amount of information, we created a database containing the center frequency, line intensity, and the air-broadening coefficient of
more than 60% of the existing lines. This database was compiled from a variety
of sources [146][147][16]. Using this database, gas concentrations were determinedfor each molecule by performing modified Voigt fits for several absorption fea-

117
Table 7.1: Spectral location and minimum detectable concentration (MDC) of
measured molecules using the 1.5 µm CE-DFCS system.
Molecule

Wavelength (µm)

MDC

CO
CO2
CH4
C2 H2
C2 H4
NH3

1.56-1.62
1.56-1.62
1.63-1.69
1.51-1.54
1.61-1.65
1.50-1.55

900 ppb
770 ppb
10 ppb3 ppb
33 ppb15 ppbtures that also included the resolution of the VIPA spectrometer. By finding thevalue of the instrument resolution that provided the best fit to the experimentally
recorded line-shapes, an instrument resolution of 800 MHz was experimentally
verified.
Calibration of the spectrometer was performed using three gas mixturescontaining a total of four target gases consisting of 10 parts per million (ppm)
CO2 , 1 ppm CH4 , 1 ppm C2 H4 , and 1 ppm NH3 , all mixed in high purity N2 .
The accuracy of trace gas concentrations was specified to within 1% of the total
concentration by Airgas, the supplier of our calibration gases. Since the spectral
lines from these four molecules cover the entire spectral region from 1.5 µm to
1.7 µm, we were able to verify the calibration of the absorption loss provided by
the cavity transmission equation (Eq. 5.3) for the entire spectral region. Finally,
we made mixtures of CO and C2 H2 by mixing pure samples of these gases with
N2 to determine the minimum detectable concentrations of these gases. These
results, along with their detection wavelengths, are listed in Table 7.1.

118

Figure 7.7: a) The isotope spectrum of CO near 1.605 µm at a CO concentrationof 4.5 parts per thousand. b) The isotope spectrum of C2 H2 near 1.524 µm at aC2 H2 concentration of 2.2 ppm. The unlabeled lines in the C2 H2 spectrum are
12
C2 H2 while the labeled lines are H12 C13 CH.

7.2.2

Isotope ratio measurements

A powerful demonstration of the sensitivity, dynamic range, and resolutionof our spectrometer can be made via isotope measurements. During the course ofmeasuring the spectra of CO, CO2 , C2 H2 , H2 O, and CH4 , we recorded spectral
regions that contain multiple isotopes of all of these molecules. Figure 7.7 shows
the spectra of CO and C2 H2 containing multiple isotopes. For both CO and C2 H2 ,the relative abundances of the isotopes measured by our system matched thosefound in the HITRAN database to within 0.4%. We also estimated the minimum
concentration of CO and C2 H2 that would allow detection of the second mostabundant isotope to be 200 ppm for CO and 200 ppb for C2 H2 .
The ability to accurately measure molecular isotope ratios has found impor-119tant applications in many fields of science. For instance, doctors use stable isotope
ratios of 13 CO2 on the breath of patients to detect Heliobacter Pylori for diagnosing ulcers [149]. Geologists use stable isotopes of H2 O to investigate the transportof groundwater sources [150]. Climatologists and paleontologists use stable isotopes of O2 to determine the past temperature of the Earth’s atmosphere fromice cores [151]. These examples represent just a few of the useful applications ofstable isotope ratio measurements.

7.2.3

Human breath analysis

Another emerging application of trace gas detection is human breath analysis. To date, researchers have identified over 1000 different compounds containedin human breath. These molecules have both endogenous and exogenous origins
and provide information about physiological processes occurring in the body aswell as environment-related ingestion or absorption of contaminants [50] [152].
While the presence and concentration of many of these molecules are poorly understood, many ‘biomarker’ molecules have been correlated to specific diseases
and metabolic processes. Such correlations can result in non-invasive methodsof health screening for a wide variety of medical conditions. Several methods oftrace molecular detection have been applied to the problem of breath analysis
including optical detection [149][17], mass spectrometry [153][154], and electronic
noses [154] [155].
To understand the choice of optical detection as the preferred technique forbreath analysis, we first evaluate the available techniques in the context of thesystem criteria. For instance, mass spectrometry (MS) is extremely sensitive andthus capable of detecting very small quantities of the analyte molecule. However,when several molecules are present, MS has difficulty identifying a single component of the mixture. To remedy this problem, MS is often used in conjunction

120
with gas chromatography to separate out various components of a mixture prior
to measurement. While these hybrid systems are highly sensitive and accurate fora large number of biomarkers, they are also large, complex, and require a long period of time to perform a measurement [156]. Conversely, electronic nose devices
are typically inexpensive and perform measurements rapidly. However, these devices, which are designed to measure volatile organic compounds, have difficulty
distinguishing and accurately measuring concentrations of individual moleculesfrom the group they are designed to detect [157]. Optical detection provides agood compromise as a general approach that can be applied to many molecules.
The unique absorption spectrum of each molecule allows accurate identificationand concentration measurements of a single molecule in the presence of manyothers. Furthermore, cavity enhancement techniques permit highly sensitive detection in a matter of seconds [130][158]. CE-DFCS is particularly well suitedfor breath analysis because of its wide spectral coverage allowing for detection ofmany different biomarkers simultaneously.
In the 1.5 µm to 1.7 µm spectral region, there are more than 10 biomarkers
that exist in human breath in detectable quantities. The measurements presented
in this section explore three of these biomarkers: CO, CO2 , and NH3 . These
measurements were performed using the 1.5 µm trace detection system describedin the previous section.
Our protocol for collecting breath samples to be measured requires thatthe test subject first take one deep breath. Next, the subject inhales normally andthe breath is held for five seconds. Finally, the first half of the breath is released
into the air before the second half of the breath is exhaled into the 1 liter Tedlar
sample bag shown in Fig. 7.8. The use of only the second half of the breath sample
increases the concentration of alveolar breath that is measured.
The first set of our breath measurements involves the analysis of stable iso-

121

Figure 7.8: Schematic of the cavity-enhanced direct-frequency-comb spectrometer,with a modified gas handling system for breath analysis.

topes. Here we focus on measurements of CO2 isotopes, but other isotopes such as
H2 O and CH4 also exist in measurable quantities within our spectral region. Exhaled CO2 is of a medical interest because the ratio of 13 C16 O2 to 12 C16 O2 can beused to determine if a patient is infected with Heliobacter pylori, a common causeof ulcers. This breath test is conducted by having the patient ingest a 13 C-labeledurea and then measuring the ratio of 13 C16 O2 to 12 C16 O2 on the patients breath[159]. Increased levels of 13 C16 O2 indicate the presence of H. pylori, a bacteriathat helps to convert the 13 C-labeled urea into 13 CO2 before being exhaled in thepatients breath.
Two spectral windows centered at 1.59 µm and 1.63 µm contain spectroscopic features of three isotopes of CO2 , all with roughly equal absorption
strengths. Figure 7.9 shows the breath spectrum of a healthy graduate student
in the window centered at 1.63 µm and illustrates a unique feature of broadband
detection. In the spectral region shown in Fig. 7.9, there are 78 12 CO2 absorption
lines and 29 13 CO2 lines. The two zoomed-in panels (a) and (b) both contain anumber of spectral features for each of the three isotopes. By measuring many
absorption lines and computing a concentration for each line, the overall accuracyof the isotope ratio measurement is enhanced by the square root of the number

122

Figure 7.9: a)Breath spectrum of CO2 isotopes near 1.63 µm. The inset shows azoomed-in portion of the CO2 landscape. The spectral region between 1.622-1.638µm contains 78 CO2 lines, 29 13 CO2 lines, and 62 18 O12 C16 O lines making it an
ideal spectral region for determining isotope ratios.

of lines measured. To calculate the isotope ratios from the spectrum shown in
Fig. 7.9c, a modest selection of 5 lines is chosen for each isotope. Typically, isotope measurements are expressed in parts per thousand such that δ = (Rsample- Rstandard )/Rstandard x 1000, where Rstandard is the natural abundance of the isotope and Rsample is the abundance found in the breath sample [149]. For thismeasurement, we used the natural isotopic abundances reported in the HITRAN
database to calculate the values of Rstandard . Using this analysis we find that δ 18 O
= -9.1 ± 4.2 and δ 13 C = -28.8 ± 4.1. The δ 13 C ratio is currently the only breathbiomarker that is used for widespread clinical diagnosis. However, the ability tomeasure other stable isotopes of CO2 at increased accuracy, and the capability ofmeasuring isotopes of other molecules such as H2 O (1.55 µm) or CH4 (1.65 µm),
could significantly broaden the scope of future breath research.
The second breath measurement shown in Fig. 7.10 illustrates how environ-123

Figure 7.10: a) The CO2 and CO absorption spectra of student 1 (smoker) andstudent 2 (non-smoker) in the 1.564 µm spectral region, along with line intensities
from the HITRAN spectral database. The smoker’s obvious increase in CO concentration is clearly detected. b) The breath concentration of CO2 in parts per
thousand (ppt) and CO in parts per million (ppm) as a function of time during
which the test subject holds their breath prior to exhaling into the sample bag.

mental and behavioral factors that determine health can be monitored via breath
analysis. Although CO has more than 170 spectral features between 1.55 µm and1.61 µm, only one absorption line at 1.56474 µm does not overlap with interfering
absorptions from H2 O and CO2 . The measurement shown in Fig. 7.10a utilizes
this spectral feature to detect the concentration of CO contained in the breathof two students. Student 1 is a smoker who had a cigarette 15 minutes prior to
the beginning of the measurements while student 2 is a non-smoker. Both students provided five breath samples, each separated by 10 minutes. Both students
exhibited constant levels of CO in their breath above the minimum detectableconcentration of CO at 1.564 µm, which is 900 parts per billion (ppb). The average CO concentration measured for the smoker (6.5 ppm) was five times that of
the non-smoker (1.3 ppm). While analyzing the data for the CO concentration,
we noticed a small difference in the concentration of CO2 exhaled by the two
students (Fig. 7.10a). This observation led us to examine the sensitivity of our124concentration measurements to possible variations in our measurement protocol.
In particular, we investigated both the CO and CO2 breath concentrations as
a function of time the test subjects held their breath prior to exhaling into thesample bag. Figure 7.10b shows these dynamic results. For CO2 , two separate
measurements were taken at each hold time and five spectral features were usedfrom each measurement for a total of ten samples to determine the concentration
at each hold time. For CO, only a single measurement was taken at each holdtime and consequently the error bars shown for CO correspond to the minimum
detectable concentration of 900 ppb. This test shows that while the CO2 concentration is quite sensitive to how long the test subject holds their breath, theCO concentration is not to within the statistical uncertainty of the measurement.
Finally, a previous study of CO contained in the breath of smokers finds concentrations ranging from 1-68 ppm [160], indicating our system is capable of measuring
CO breath concentrations for this entire population. While smoking provides anexample of the behaviorial effect on the CO level in breath, environmental conditions can also lead to elevated CO breath levels. For instance, individuals who
spend a large portion of their day in congested traffic or people live with elevatedlevels of CO at home are also prone to elevated CO levels in their breath [160].
The final breath-related measurement was the determination of the minimum detectable concentration of NH3 , a biomarker for renal failure that can bedetected within a large concentration of H2 O. The absorption spectrum of NH3
overlaps heavily with H2 O in the spectral region from 1.5 µm to 1.55 µm. Two
spectral windows, one at 1.513 µm and the other at 1.517 µm, were found tocontain at least two strong NH3 absorption lines that have minimal overlap with
H2 O absorptions (Fig. 7.11). To find these spectral windows and determine theminimum detectable NH3 concentration, we measured the absorption spectrumof a calibrated sample of 4.4 ppm NH3 in high purity N2 and compared this125

Figure 7.11: a)The absorption spectrum of lab air superimposed over the spectrumof 4.4 ppm of NH3 . HITRAN line intensities for H2 O are included to identify theabsorption features of the lab air spectrum. The dashed grey line represents the
typical concentration of NH3 contained in the breath of a patient at final stagesof renal failure.

spectrum to both the lab air spectrum and the spectra generated from breath
samples. The feature with the greatest NH3 /H2 O absorption ratio was centeredat 1.512231 µm. The minimum detectable concentration of NH3 is considered tohave a corresponding absorption feature that is twice the standard deviation of
the measurement noise. According to this criterion, the minimum detectable concentration of NH3 in the lab air at a 25% relative humidity was 18 ppb. Since ourbreath measurements nominally contain 3 to 4 times higher H2 O concentration,
small overlaps with H2 O absorptions lead to slightly higher minimum detectableconcentrations of ammonia in breath. Although this limit is still higher thandetectable NH3 present in the breath of most healthy individuals, it is sufficient
to detect increased NH3 present in the breath of people at early stages of renal
failure (Fig. 7.11) [161].
The breath measurements presented here provide a modest glimpse of the

126health information that can be acquired by analyzing a patient’s breath. In the
1.5 µm spectral region alone there are also strong absorptions from methylamines
which can be used to detect kidney failure and liver disease [162], acetone which isan indicator for diabetes [48], and ethane and ethylene for detecting lung cancer[13]. Currently the biggest obstacle to clinical application of breath analysis isthe variation of breath samples from patient to patient, and the lack of a complete picture regarding how molecules contained in breath relate to disease andmetabolic processes. The large number of biomarkers that can be measured using the CE-DFCS system, along with sub-minute measurement times, presents a
cost effective way to accelerate research and future clinical applications of humanbreath analysis. With the rapid progress of frequency comb technology it most
likely will not be long until commercial CE-DFCS instrumentations are available
for this purpose.
7.3

Spectroscopy of jet cooled molecules
The measurements presented so far in this chapter have been on room

temperature gas samples at relatively high pressures and therefore do not require higher spectral resolution than what is provided by the VIPA spectrometer. In this section, we discuss a technique that uses precise control of the
comb and cavity frequencies in conjunction with a VIPA spectrometer to extend the spectral resolution far beyond the limitations of the VIPA spectrometer.
We then show how the high resolution capability allows CE-DFCS systems tostudy molecular ensembles involving complex velocity and internal state dynamics. In particular, correlations between the internal energy states and externalmotions of molecular ensembles can be observed due to the simultaneous highresolution and broad spectral coverage the observation. This demonstration suggests that CE-DFCS will be an ideal tool for the emerging field of cold molecule

127
physics and cold chemistry, where experimental conditions involve small numbersof molecules, narrow resonances, and potentially many internal energy states fordifferent species [49][47][163][164][165][166]. Indeed, femtosecond pulse trains andfrequency combs are already being used for population transfer and coherent control of cold molecules [167][168][33] [169].
As a first effort in making a complete characterization of cold molecularensembles with CE-DFCS, we have performed a tomographic reconstruction ofabsorption measurements from a supersonically cooled jet of C2 H2 molecules. Tomographic reconstructions are enabled by the use of a movable jet nozzle, allowing
us to measure the integrated absorption through the supersonic jet at a variety
of distances from the nozzle and heights from the jet axis. The precise frequency
control of the comb allows us to record the absorption line-shapes at each measurement position, while the broad spectral coverage of the comb allows us to record
the entire ν1 + ν3 rovibrational band of C2 H2 (both P and R branches) in a single
measurement. As a result, the transverse velocity distributions of the molecules in
the jet and their corresponding internal energy states are determined at the sametime. Due to these capabilities, the use of CE-DFCS provides unprecedented
advantages over previous supersonic jet studies that used Raman spectroscopy
or contiunuous-wave cavity ringdown spectroscopy for supersonic beam investigations [170][171] [172][130]. The ability to simultaneously observe the internal andexternal energies of the molecular jet allows us to rapidly generate spatial maps ofthe density, velocity, and internal state distributions via tomographic reconstruction [173], essentially providing a thermodynamic snapshot of the entire jet.
The tomographic reconstructions presented in this section reveal several interesting features of the pulsed supersonic jet. First, we find spatially varying
anomalous line-shapes that, to our knowledge, had not been previously reported.
We also observe an anomalous rotational temperature map that suggests the cold-

128
est temperatures in the jet are not at the center, but rather form a ring around theaxis of the jet. We also infer that the radial velocity is strongly correlated withthe radial positions of the molecules in the beam. Finally, we present evidence
indicating a transverse velocity dependence of the rotational cooling rate of themolecules within the expansion.

7.3.1

High-resolution spectroscopy of a supersonic jet of C2 H2For the this experiment, a cavity FSR of 700 MHz is chosen to be seven times
that of fr (100 MHz) such that every seventh comb mode is coupled to the cavity.
This choice of FSR acts to filter the comb modes as shown in Fig. 7.12 so thateach molecular absorption line, having a nominal Doppler line-width of 350 MHz,
only interacts with one comb mode. The comb teeth are rapidly swept over thecavity modes by modulating the laser fr , and absorption measurements are performed using the technique of transient cavity-enhanced absorption spectroscopy[123][128]. The cavity is constructed from two 2-meter radius of curvature mirrors
with a reflectivity of 99.95%, resulting in a finesse of 6300. The TEM00 cavity
mode has a 1/e electric field diameter of 1 mm which sets the spatial resolution
limit of the tomographic measurements presented here. The Rayleigh range ofthe cavity mode is 50 cm such that its diameter is nearly constant over the lengthof the cavity. To avoid degrading the finesse by scattering loss from the supersonic expansion nozzle, the cavity mode must be positioned at least 2.5 mm fromthe nozzle. This requirement sets the minimum distance from the nozzle thatmeasurements can be made with this system. Of course, engineering a smaller
diameter cavity mode would allow for both higher spatial resolution and measurements closer to the expansion nozzle. One of the cavity mirrors is mounted on a
1 inch diameter ring PZT for servoing of the cavity length at bandwidths of up

129

Figure 7.12: a) Schematic of the basic CE-DFCS system. A mode-locked laser
generates a pulse train that is coupled into a high finesse detection cavity. The
light transmitted from the cavity is analyzed using a VIPA spectrometer producing
an interference image (bottom right). The parabolic patterns of dark spots are
due to CO2 absorption at 1.6 µm. The frequency diagram (bottom left) shows
how the comb modes are coupled into the high finesse cavity. The molecularabsorption line is measured by scanning both the comb and cavity modes over 1
FSR of the cavity.

to 1 kHz. Large changes to the cavity length are made by a resistive heater that
holds the cavity at a nominal temperature of 35 ◦ C.
The VIPA spectrometer is aligned to capture the cavity transmission within
a 25 nm spectral range from 1.515 µm to 1.54 µm. The etalon used in this experiment has an FSR of approximately 50 GHz. A grating with 1100 lines/mm is usedwith a 20 cm focal length imaging lens to achieve a resolution of 25 GHz in thehorizontal direction. As a result, the interference image in Fig. 7.12 shows dark
stripes separating the bright vertical fringes, indicating that the mode orders of the
VIPA etalon are well-resolved. We performed careful measurements of the VIPA
FSR and position-dependent resolution of the spectrometer prior to the C2 H2 jet130
experiment. These measurements were made by observing the cavity transmitted
absorption spectrum of a dilute sample of C2 H2 . Using the well-known transition
frequencies of the ν1 + ν3 rovibrational band of C2 H2 , the VIPA FSR was determined to be 49.52 GHz [174][146]. The spatially varying spectrometer resolution
was determined by comparing the spatial extent of absorption transitions thatappeared at different heights on the interference image to the VIPA FSR interval.
The VIPA spectrometer configuration used in this experiment had a resolutionthat varies from 1.0 GHz at the bottom of the FSR interval to 1.3 GHz at the
top of the interval. Due to the 700 MHz spacing of the cavity transmitted comb
modes and the Lorentzian transfer-function of the VIPA etalon transmission, eachresolvable frequency channel recorded by the InGaAs camera samples an averageof 2.5 comb modes. This acts to dilute the signal carried by each comb mode thatinteracts with the intra-cavity absorption. However, since only one comb mode
is interacting with the molecular absorption within each resolvable channel, thefrequency resolution of the measurement is not limited by the VIPA spectrometer,
rather by the stability of individual comb modes.
To demonstrate high resolution and stable frequency measurements, thecavity mode frequencies are stabilized to within 700 kHz using a frequency stabilized Nd:YAG laser [175]. The comb frequencies are slaved to the cavity modes
using a servo scheme described in Section 4.3. Frequency scans of the comb andcavity modes are performed by stepping the frequency of the 1.064 µm Nd:YAG
laser in 25 MHz steps over 1 GHz, resulting in a 700 MHz (1 FSR) total scan of
the cavity mode frequency at 1.525 µm made in 17.5 MHz steps. The 650 m/s
average forward velocity of molecules moving through the 1 mm diameter of cavity mode results in a transit-time broadened line-width of 200 kHz. Accordingly,
resolutions of better than 1 MHz can be achieved, which is more than sufficient torecord the nominal Doppler limited line-width for molecules in the jet. Meanwhile,

131

Figure 7.13: Residual C2 H2 absorption of the R(1) transition versus time after
firing the expansion valve.

the VIPA spectrometer easily captures the entire cooled rotational spectrum of
the C2 H2 molecules in a single measurement.
To create the pulsed supersonic expansion, a mixture of 2.5% C2 H2 and97.5% Ar at 650 torr is allowed to expand through a 1 mm diameter circularvalve into an evacuated chamber with pressure < 10−5 Torr. The nozzle wasconstructed with a pseudo-trumpet geometry shown in Fig. 7.18 as a compromise between ease of manufacturing and optimal cooling and forward collimationof the molecular beam [176]. A 2 Hz valve rate was chosen to allow adequate
evacuation of the residual C2 H2 inside the detection cavity before the again firing the valve. The data in Fig. 7.13 shows that the decay constant for residual
C2 H2 absorption on the R(1) transition inside the detection chamber is 68.7 ms.
Therefore, after 500 ms, the residual R(1) absorption signal will be on the order of
αmax e500/68.7 = 7 × 10−7 αmax . Opening and closing of the value is actuated using
a piezo-electric disc. The expanding jet of molecules is oriented perpendicularly
to the mode of an optical cavity where CE-DFCS measurements are performed.
The valve is mounted in a bellows and on translation stages, allowing control of132

Figure 7.14: a) Schematic of the supersonic jet measurement via CE-DFCS. b) A
snapshot of the rotational spectrum of the C2 H2 beam at z = 6 mm, y = 1 mm.
c) The peak absorption values for the R-branch transitions at z = 6 mm and y= 1 mm as the cavity length is scanned through one FSR in 17.5 MHz steps.
The rotational spectrum snapshot in b) was captured at a frequency of 600 MHz
during the scan in c).

the distance from the valve to the cavity mode (z-coordinate) and the height ofthe valve with respect to the cavity mode (y-coordinate) (Fig. 7.14a). Specifically,
the z-coordinate is measured from the tip of the expansion nozzle (Fig. 7.18), andthe y-coordinate is measured from the jet axis. Due to the mobility of the valve,line integrals along the cavity mode (x-coordinate) of the absorption through thebeam can be performed at a variety of positions, allowing for tomographic reconstruction. The tomographic reconstructions are used to determine the exact
position of the jet axis (y = 0 mm).
To characterize the time-dependent shape and forward velocity of our133

Figure 7.15: .a) The C2 H2 pulse shape measured by scanning the delay between
firing the valve and measuring the intra-cavity absorption of the R(1) transition.
b) The delay time corresponding to the center of the R(1) absorption as a functionof nozzle distance. Fitting the data points yields an average forward velocity of
647 ± 21 m/s.

molecular packet, we measured the pulse shape for y = 0 mm at four different
distances from the nozzle (z = 3.5, 9.85, 16.2 & 22.55 mm). At each z-position wevaried the time delay between opening the valve and performing the absorption
measurement to map out both the pulse shape and the arrival time of the frontedge of the pulse. The molecular pulse shape resembled a plateau with sharp front
and back edges and a flat top at all measurement positions (Fig. 7.15a). Fromthese measurements, we determined that the average forward velocity of our jet
was 647 ± 21 m/s. The repeatability of these measurements indicated that ourcontrol of the valve opening time was better than ±5 µs.
Figure 7.14b shows the cooled C2 H2 rotational spectrum at a distance ofz = 6 mm from the nozzle. This spectrum is an average of 10 measurements
for a total acquisition time of 5 seconds. Frequency scans of the cavity over one
FSR are made by successively changing the cavity length to map out the entire
frequency spectrum within the 25 nm bandwidth of the VIPA spectrometer. Figure 7.14c shows a one FSR frequency scan of the peak absorption wavelengths of
the 0 through 5 transitions of the R-branch. This scan was performed in approximately 30 minutes, and yielded line-shape measurements for each of the P and R

134
branch transitions. For the frequency scans in Fig. 7.14c, the valve height was y= 1 mm such that the cavity mode was approximately 1 mm below the axis of themolecular beam. The minimum detectable integrated absorption of the system is
1 × 10−6 at 1 ms of integration time. For the typical temperature and diameterof our beam (T = 10 K and d = 1 cm), the minimum detectable concentration ofC2 H2 is 1.7 × 1011 cm−3 .

7.3.2

Tomography of a supersonic jet of C2 H2Our measurements in this experiment were restricted to parallel integrated
absorption measurements as a function of height from the jet axis due to our
setup. Since our scheme provides measurements at only one angle through the supersonic jet, our tomographic reconstructions are limited to distributions that areaxis-symmetric [173]. In this case, an inverse Abel transform is used to convert thesingle set of parallel integrated absorption measurements into a radial absorption
distribution α(r), where r is the radial position [173]. We justify our assumptionof a radially symmetric jet based on our use of the cylindrically symmetric nozzle shown in Fig. 7.18. Furthermore, we have performed frequency scans of theintegrated absorption both above and below the jet axis which yield very similar
absorption lines, both in shape and intensity. Evidence of the agreement between
above and below axis measurements can be seen by comparing the y = -1.1 mm
and y = 0.9 mm measurements in Fig. 7.16a. For jets that lack axial symmetry,
our general approach can still be implemented; however, the number and type ofmeasurements required to reconstruct the beam are significantly more extensive.
In the asymmetric case, sets of parallel height-dependent measurements must be
performed at a variety of angles covering 180◦ through the jet, and the more complicated Radon transform must be implemented to reconstruct the asymmetric135

Figure 7.16: a) Integrated absorption frequency scans of the P(2) transition at
a distance of z = 11 mm from the nozzle for several heights of the cavity modewith respect to the axis of the expansion. b) The data from a) are frequencyintegrated and plotted versus the measurement height. A cubic spline of this datais performed in preparation for the inverse Abel transform. c) The inverse Abeltransform of the cubic spline curve in b) results in a radial absorption profile.

absorption distribution α(r, φ) [177].
The steps for performing tomographic reconstructions of the radial absorption distribution are shown in Fig. 7.16 using data from the P(2) transition at a
nozzle distance of z = 11 mm. First, each of the integrated absorption line-shapes
is fit to a sum of three Gaussians (Fig. 7.16a). This fitting function was chosen because it provided an excellent fit to every absorption line-shape that we observed.
The triple Gaussian line-shape implies that the distribution of rotational groundstates are populated differently in the center of the beam than in surrounding
region. This is a point that will be discussed in more detail in conjunction with
Fig. 7.19. Next, the fitted curves at each height were integrated along the frequency axis yielding net absorption versus height (Fig. 7.16b). The data points

136
in Fig. 7.16b were interpolated using a cubic spline. Finally, the spline of the integrated absorption data was converted to a radial absorption distribution using
the inverse Abel transform:
1α(r) = −
π

Z ∞
r

dy
dα(y)
p
.
dy
y 2 − r2

(7.2)

Here, α(y) represents height-dependent integrated absorption measurements (integrated along the cavity mode (x) and the optical frequency (ν)), and α(r) is
the radial absorption distribution. Since this integral has a singularity at y = r,direct integration of this transform results in inaccurate reconstructions of radial
distribution. Consequently, there is an extensive literature concerning various algorithms that can be used to overcome the singularity issue. We have selecteda discrete inverse Abel transform algorithm based on an eigenvalue approach toprovide rapid and accurate tomographic reconstructions of the radial absorption
profile [178]. An example MATLAB code for executing this algorithm is provided
in Appendix D.
Using the above procedure, the radial absorption distributions were calculated for each R-branch rotational transition at the nozzle distance z = 11 mm asshown in Fig. 7.17a. The rotational absorption distribution at each radial position can be used to determine the molecular rotational temperature and density
distributions via the relation α(ν) = ρSi g(ν) introduced in section 7.2. Whenthe molecular ensemble is in thermal equilibrium, the line intensities (Si ) of therotational transitions can be described by a Boltzmann distribution [148][179]−Eje kB TR.
Si = gI Li νCsi , where si =
Qr

(7.3)

Here, j is the ground state rotational quantum number, Ej is the ground stateenergy of the jth rotational level, gI is the nuclear spin state degeneracy (3 forodd and 1 for even rotational ground states), kB is the Boltzmann constant,

137

Figure 7.17: a) Tomographically reconstructed radial absorption profiles for the
R-branch transitions at a distance of z = 11 mm from the nozzle. b) The density
and rotational temperature of the jet are determined by fitting of the rotational
absorption profile to a Boltzmann distribution at several radii from the beamaxis. c) The radial density and rotational temperature distributions from thefits in b) show that the C2 H2 density is largest on the beam axis with a steadydecrease toward the edge of the beam. Counterintuitively, the radial temperature
distribution reveals a warmer temperature of 9 K at the center of the beam that
reduces to 5.5 K at r = 4 mm before warming to 12 K near the edge of the jet.

and TR is the rotational temperature of the molecular ensemble. Li is the HölnLondon factor, ν is the rovibrational transition frequency, and C is a constantthat includes the vibrational transition dipole moment and a collection of otherphysical constants [148]. The quantity Qr in the denominator is the rotational

138
partition function and has the form
Qr =

X

−

Ej

gI (2j + 1)e kB TR .

(7.4)

j

Equation 7.3 and the expression for α(ν) are used in a fitting procedure to determine first the rotational temperature and then the density of the molecularensemble as a function of r. The temperature is determined by fitting the exponential decay of each rotational transition as a function of the rotational groundstate energy (Fig. 7.17b). To generate the data points in Fig. 7.17b the absorption
values are divided by both the pre-factors in Equation (7.3) and the absorption
line-shape g(ν), leaving only ρsi . We then take the natural logarithm of the dataso the changes in rotational temperature and density can be more easily visualized
as linear relationships rather than decaying exponentials. The rotational temperature at each r is obtained from the slope of the linear fits. Once the rotationaltemperature of the ensemble has been determined, the rotational partition function can be calculated. Using the partition function, the molecular density canbe determined from the the y-intercept of the fits. The fit results for seven radial
positions are shown in Fig. 7.17c. As expected, the molecular density is largest on
the jet axis and decreases steadily with increasing radius. More interesting is theradial dependence of the rotational temperature of the supersonic jet. The coldest
rotational temperature observed does not appear at the center of jet. Instead, atemperature of 9 ± 0.9 K is observed on the jet axis, the temperature then drops
to 5.5 ± 0.7 K at r = 4 mm before rising to 12 ± 1.6 K at the edge of the jet(r=10 mm).
To gain a better understanding of how the jet evolves as it expands from thevalve, we have performed height and frequency scans at five distances from thenozzle (z = 3.5, 6, 8.5, 11, & 13.5 mm). The procedures described in Figures 7.16
and 7.17 were used to convert the integrated absorption measurements into den-139
sity and rotational temperature distributions at each measurement distance. The
radial distributions calculated for each z-position were then combined into a 2D
‘map’ describing the rotational temperature and density distributions (z, r) within
the measurement volume. To generate a smooth plot of the spatial-varying rotational temperature and density, a 2D spline was used to interpolate between thefive radial distribution curves. The resulting 2D density and rotational temperature maps are shown in Fig. 7.18. The data used to construct this map, acquired
at a 2 Hz valve rate and without an automated system for moving the nozzle, wasrecorded in 28 hours of measurement time.
The density map on the right side of Fig. 7.18 shows a well collimated andhigh molecular density region centered on the jet axis. The collimated packet issurrounded by a lower density cloud of molecules that expands as a function of distance from the nozzle. The most prominent feature of the rotational temperature
map is the well understood cooling of the rotational temperature as a functionof distance from the nozzle [180]. However, the temperature map also shows amore subtle effect. The coldest rotational temperatures are not located on the
jet axis. Rather, this temperature minimum exists in a ring around the jet axis.
Also, the radius of this ring appears to increase with increasing distance form thenozzle. To verify the results of our tomographically reconstructed temperature
map, we performed a couple of confirmation checks to make sure that what we
observe does not arise from a systematic error in either our absorption measurements or our procedure for tomographic reconstruction. First, we analyzed the
P-branch data and found that the tomographic reconstructions for both the density and temperature maps were in good agreement with the maps derived fromthe R-branch measurements. To rule out systematic error in our tomographic
reconstruction procedure, we performed a two-exponential (or two-temperature)
fit of the rotational distributions for the height-dependent integrated absorption

140

Figure 7.18: A scaled drawing of the supersonic expansion nozzle is shown above
tomographically reconstructed 2D (z, r) maps of the rotational temperature (left)and density (right). The temperature map reveals the typical rotational coolingas a function of distance from the nozzle. The radial temperature distribution
shows that the coldest molecules in the jet reside at a radial position that increases with increasing distance from the nozzle. The diagonal black line on thetemperature map defines a molecular trajectory corresponding to molecules witha transverse velocity of 230 m/s. The significance of this trajectory will be discussed in conjunction with the absorption line-shapes shown in Fig. 7.19. The
density distribution shows a well collimated jet centered at r = 0 mm surroundedby an expanding cloud of lower density molecules.

measurements. These fits showed a higher rotational temperature for the y = 0
mm measurements, which then decreased as the values of y increased. Finally,
the temperature indicated by the fits increased toward the edge of the jet. In
short, we confirmed that the temperature map in Fig. 7.18 faithfully represents
the position-dependent rotational temperature of the beam.
To gain some insight into the cause of the observed rotational temperature
distribution, we turn our attention to position and rotation dependance of theintegrated absorption line-shapes. The absorption line-shapes for the R-branch141transitions 0-5 measured at y = 0 mm are shown in Fig. 7.19a for nozzle distancesof z = 3.5 mm to z = 16 mm. To aid in the discussion of the line-shapes inFig. 7.19a, we have divided the frequency axis of each graph into three regions (I,II, and III) as denoted in the R(0) graph.
One interesting feature of these line-shapes is shown in Fig. 7.19b. Here
we have plotted the frequency-integrated absorption signals of the R(0) and R(5)transitions as a function of distance from the nozzle. To provide some context forthese curves, consider what the frequency-integrated absorption signal would look
like for a collisionless expanding jet with a Gaussian velocity distribution. As thejet begins to expand, the signal changes very little until the point where the jet
diameter becomes larger than the optical detection beam (cavity mode). Beyond
this point, the amplitude of the absorption signal will decrease as a function of
distance from the nozzle at the asymptotic rate of z−1 . Also, as the amplitudedecreases, the line-shape remains constant. In contrast, the signal for R(0) decreases at the much slower rate of z−0.26 while the signal for R(5) decreases at themuch faster rate of z−2.1 . The reason for the slow reduction of the R(0) signal andthe fast reduction of the R(5) signal is due to rotational cooling of the moleculesin the supersonic expansion.
Another interesting feature of the absorption lines are the anomalous shapesthat differ considerably from the expected Gaussian line-shape. While anomalous
line-shapes have been observed in a continuous beam of C2 H2 /Ar, those anomalies
manifest as a dip in the center of the absorption line for the lowest rotations (j= 0, 1, 2), and were attributed to C2 H2 clustering occurring at regions of highdensity within the beam [171]. In contrast, the low rotational state molecules inour measurements always exhibit peaks at the line-center. Furthermore, due tothe large nozzle aperture and low stagnation pressure in our experiment comparedto [171], we suspect that the primary cause of the line-shape anomalies observed

142

Figure 7.19: a) The absorption line-shapes for the R-branch transitions 0-5 measured through the jet axis at y = 0 mm for nozzle distances ranging from z = 3.5mm to z = 16 mm. Each panel has been labeled with three regions of interest. The
light area between the two shaded regions (I) shows the low transverse velocity
behavior, the shaded regions (II) show the medium transverse velocity behavior,and the outer light regions (III) show the high transverse velocity behavior. b)
The frequency-integrated absorption signals for R(0) and R(5) versus nozzle distance showing a signal decay that is slower than z−1 for R(0) and faster for R(5).
c) Frequency dependent integrated absorption signals for R(0), R(2), and R(4) attransverse velocities of 230 m/s and at 0 m/s. d) Absorption ratios of the vt =230 m/s versus vt = 230 m/s molecules for transitions R(0), R(2), and R(4) show
a faster decay of vt = 230 m/s molecules for the higher rotational levels.

here are not due to clustering [181].
The anomalies in our observations tend to occur at frequencies within thelightly shaded region II between 100 MHz and 200 MHz. In this frequency range
we observe a depletion of the absorption signal for higher rotation molecules (j143= 2, 3, 4, 5) and an excess of signal for the low rotations (j = 0 and 1). The
data in Fig 7.19c shows the absorption trends as a function of nozzle distance forthe transitions R(0), R(2) and R(4) for the Doppler shift frequencies of 0 MHz(vt = 0 m/s) and 150 MHz (vt = 230 m/s). Since the nuclear-spin states are
not changed by the rotational cooling of the supersonic expansion, the parity of
the rotational ground states does not change [182]. Hence rotational cooling in
the expansion proceeds in steps of ∆j = 2n where n is an integer. The data in
Fig. 7.19c illustrates the transverse velocity dependence of the rotational coolingof molecules with even rotational ground states. At zero transverse velocity, thepower law decay exponents of the absorption signals are -0.16, -0.2, and -1.5 for
R0, R2, and R4 respectively, while at vt = 230 m/s the decay exponents are -0.14,-1, and -2.4. Comparing the two sets of decay exponents shows that the absorption signals for R(2) and R(4) decay much slower at vt = 0 m/s than at vt =230 m/s. In contrast, the exponents for R(0) show a comparable decay of the vt= 0 m/s and vt = 230 m/s signals. This can be seen clearly in the ratios of theabsorption at vt = 230 m/s versus vt = 0 m/s for each rotation (Fig. 7.19d). A
further analysis of these decay exponents could be used to determine the population transfer rates between the different rotations. However, for the present work
we only wish to point out that the population transfer seems to depend on thetransverse velocity of the molecules within the jet.
The line-shape measurements and subsequent analysis shown in Fig. 7.19
can offer some insight into the interesting rotational temperature map of Fig. 7.18.
Once again we consider the center frequency of the darkly shaded ‘anomalous’ regions in Fig. 7.19a (f = 150 MHz), which has a transverse velocity of 230 m/s.
This velocity and the average forward velocity of the jet (vf = 650 m/s), resultin the trajectory illustrated by the diagonal black line in the rotational temperature map of Fig. 7.18. The interesting feature of this trajectory is that it closely

144
follows the minimum rotational temperature as the jet expands. Since the anomalous characteristic of this frequency range is a depletion of higher rotations andan excess of lower rotations, molecules at these velocities will exhibit a lower rotational temperature than other frequency regions. Furthermore, since this same
temperature dependence is manifested in the position of the ‘rotationally coldest’molecules in Fig. 7.18, it indicates that the transverse velocity of molecules in
the jet are strongly correlated to their position in the jet. Again, further analysis
could be implemented to determine the level of correlation between position andvelocities of molecules within the jet.
The measurements presented in this section provide a general example of
how CE-DFCS systems can be used to rapidly obtain highly detailed information about complex molecular ensembles. In particular, we have demonstrated
how CE-DFCS is used to obtain simultaneous high sensitivity, high resolution,and broad-bandwidth spectroscopic measurements. In the jet cooled C2 H2 study,
the ability to rapidly record broad-bandwidth and high resolution spectra has allowed us to perform tomographic reconstructions of the density and temperature
distributions of a C2 H2 /Ar supersonic expansion. These reconstructions reveal
an interesting behavior of the position dependance of the rotational temperature
distribution. Our ability to observe the absorption line-shapes of each rotationalline as a function of distance from the nozzle allows us to interpret the anomalous
rotational temperature distribution as resulting from a transverse-velocity dependent rotational cooling rate. Furthermore, comparing the position and line-shapedata allows us to infer strong correlations between the radial velocity and theradial position of molecules in the supersonic jet.

Chapter 8
Conclusion

8.1

The state of the art
The techniques and measurements described in this thesis represent the cur-rent state of the art for cavity-enhanced direct frequency comb spectroscopy. As
this is a relatively new field, the capabilities of these systems and the enabling
technologies are improving rapidly. Already CE-DFCS has demonstrated detection of molecular concentrations in the low ppb range for many molecules andsub megahertz spectral resolutions. Furthermore, these detections have been performed rapidly and covering spectral ranges extending hundreds of nanometers.
Due to the preliminary successes and great potential of CE-DFCS for a wide
variety of applications, we have submitted a patent application for this technology.
We are currently in the final stages of the application process. After our demonstration of human breath analysis using CE-DFCS, we were approached by several
companies who wanted to license our pending patent and develop CE-DFCS for
applications ranging from breath analysis, to industrial process applications, toexplosive and hazardous material detection. The breath analysis work has also
excited many in the medical field due to the prospect of low cost and rapid medical screening. As a result, we have started several collaborations with medical
researchers at institutes across the country to further advance the field of breath
research. We hope that through commercial development and collaboration with

146
medical experts, we can eventually help CE-DFCS technology find its way into
clinical applications.
The demonstration of broad bandwidth, high sensitivity, and high resolution spectroscopy of jet cooled molecules presents a different, but equally exciting
direction for CE-DFCS. Here, the high resolution and broad spectral coverage areshown to beautifully tackle the challenges facing cold molecule community (i.e.
the simultaneous high resolution investigation of many molecular transitions).

8.2

Current research and future directions
Currently, graduate student Kevin Cossel and postdoc Florian Adler arecontinuing to expand the trace detection applications of CE-DFCS. With myhelp, they have constructed a trace detection system that operates in the 1.75 µm
to 2.05 µm spectral region. This system was designed to detect trace contaminants in gases that are used for producing semiconductors [12]. In particular, theirsystem will look for traces of H2 O, CO2 , CH4 , C2 H6 , SiH4 , and potentially othercontaminants in an atmosphere of pure AsH3 . If this system reaches the part perbillion (ppb) level of detection for these contaminants, it would greatly benefit
the semiconductor industry, since ppb level contamination of these compounds
can drastically reduce the semiconductor performance.
The CE-DFCS technique has already began to spread to other groups within
JILA. Eric Cornell and his student Laura Sinclair are currently designing a system
to detect transitions in the poorly understood HaF+ molecular ion. They havedevised a clever combination of CE-DFCS and velocity modulation spectroscopy
to separate the neutral molecule absorption signals from those that are due tothe ions. Also, with the help of Jun and myself they will implement a scheme
for coupling the comb to the cavity that combines principles of both the cavity

147
filtration and tight cavity locking schemes presented in Chapter 4. The goal ofthis spectroscopy is to identify transitions in HaF+ that can be used to search forthe electron electric dipole moment.
As applications of CE-DFCS systems become more demanding, the current
limits of CE-DFCS systems will certainly be overcome. Directions for improving
CE-DFCS performance include new broadband cavity designs such as prism cavities mentioned in Chapter 3. Such cavities would greatly increase the numberof molecules that can be probed with a single CE-DFCS system by providing
detection cavities with high finesse over very large spectral bandwidths. Current
work with fused silica retro-reflecting prism suggests that a spectral bandwidth of
1 µm with a center wavelength of 1.5 µm can be achieved. This corresponds to a
fractional frequency coverage of 75% of the center frequency.
Extending high-precision and high average power frequency comb sources
to the mid-IR spectral region will also greatly enhance the capabilities of CEDFCS systems. Mid-IR detection will enable extremely high sensitivity absorption measurements, due to the strong fundamental molecular vibration transitions,
reducing the minimum detectable concentration of many molecules to the parts
per trillion level. Also, hundreds of molecules have absorption signatures in the
mid-IR spectral region greatly expanding the number of molecules that can bedetected compared to the 1.0 to 2.0 µm spectral region. Toward this goal, we are
currently developing an optical parametric oscillator (OPO) to generate up to 1
Watt of average power in the spectral range between 2.8 and 4.5 µm. The OPO
will be pumped by a 5 Watt mode-locked Yb fiber laser and will provide a tunable
spectrum using a fan-out type periodically poled lithium niobate crystal.
Aside from trace detection, new applications for CE-DFCS include detection
and characterization of cold molecules, including their dynamics and interactions.
High-resolution coherent control of atomic and molecular systems similar to work

148
already performed using free-space DFCS [32][183] represents another exciting
possibility. Considering the progress up to this point, it is likely that future experiments and technological advances will allow CE-DFCS to be applied to many
new areas of scientific inquiry and commercial application. Fortunately, the next
generation of students working on these projects, Kevin and Florian, are already
dreaming up the things to come.

Bibliography[1] T. W. Hänsch, I. S. Shahin, and A. L. Schowlow, Optical resolution of theLamb shift in atomic hydrogen by laser saturation spectroscopy, Nature 235,
63 (1972).
[2] D. J. Wineland, R. E. Drullinger, and F. L. Walls, Radiation-pressure cooling of bound resonant absorbers, Phys. Rev. Lett. 40, 1639 (1978).
[3] T. W. Hänsch and B. Couillaud, Laser Frequency Stabilization By Polarization Spectroscopy of a Reflecting Reference Cavity, Opt. Commun. 35, 441(1980).
[4] 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
Optical Resonator, Appl. Phys. B 31, 97 (1983).
[5] D. Z. Anderson, J. C. Frisch, and C. S. Masser, Mirror reflectometer basedon optical cavity decay time, Appl. Opt. 23, 1238 (1984).
[6] A. O’Keefe and D. D. A. G., Cavity ring-down optical spectrometer for
absorption measurements using pulsed laser sources, Rev. Sci. Instrum. 59,2544 (1988).
[7] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A.
Cornell, Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor, Science 269, 198 (1995).
[8] M. Niering et al., Measurement of the Hydrogen 1S-2S Transition Frequency
by Phase Coherent Comparison with a Microwave Cesium Fountain Clock,Phys. Rev. Lett. 84, 5496 (2000).
[9] M. V. Romalis, W. C. Griffith, J. P. Jacobs, and E. N. Fortson, New Limiton the Permanent Electric Dipole Moment of 199 Hg, Phys. Rev. Lett. 86,
2505 (2001).

150[10] D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, andD. M. Meekhof, Experimental Issues in Coherent Quantum-State Manipulation of Trapped Atomic Ions, J. Res. Natl. Inst. Stand. Technol. 103, 259(1998).
[11] W. Demtröder, Laser spectroscopy, 3 ed. (Springer-Verlag, Berlin, 2003), pp.
851–892.
[12] S. Y. Lehman, K. A. Bertness, and J. T. Hodges, Detection of trace waterin phosphine with cavity ring-down spectroscopy, J. Cryst. Growth 250, 262(2003).
[13] C. S. Patterson, L. C. McMillan, C. Longbottom, G. M. Gibson, M. J. Padgett, and K. D. Skelton, Portable optical spectroscopy for accurate analysis
of ethane in exhaled breath, Meas. Sci. Technol. 18, 1495 (2007).
[14] S. Spuler, M. Linne, A. Sappey, and S. Snyder, Development of a cavity ringdown laser absorption spectrometer for detection of trace levels of mercury,Appl. Opt. 39, 2480 (2000).
[15] M. W. Todd, R. A. Provencal, T. Owano, B. A. Paldus, A. Kachanov, K. L.
Vodoyanov, M. Hunter, S. L. Coy, J. I. Steinfeld, and J. T. Arnold, Application of mid-infrared cavity-ringdown spectroscopy to trace explosives vapor
detection using a broadly tunable (6-8 µm) optical parametric oscillator,
Appl. Phys. B 75, 367 (2002).
[16] E. H. Wahl, S. M. Tan, S. Koulikov, B. Kharlamov, C. R. Rella, and E. R.
Crosson, Ultra-sensitive ethylene post-harvest monitor based on cavity ringdown spectroscopy, Opt. Express 14, 1673 (2006).
[17] A. A. Kosterev, A. L. Malinovsky, F. K. Tittel, C. Gmachl, F. Capasso, D. L.
Sivco, J. N. Baollargeon, A. L. Hutchinson, and A. Y. Cho, Cavity ringdown
spectroscopic detection of nitric oxide with a continuous-wave quantumcascade laser, Appl. Opt. 40, 5522 (2001).
[18] D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L.
Hall, and S. T. Cundiff, Carrier-Envelope Phase Control of Femtosecond
Mode-Locked Lasers and Direct Optical Frequency Synthesis, Science 288,635 (2000).
[19] S. T. Cundiff, J. Ye, and J. L. Hall, Optical frequency synthesis based onmode-locked lasers, Rev. Sci. Instrum. 72, 3749 (2001).
[20] T. Udem, J. Reichert, R. Holzwarth, and T. W. Hänsch, Accurate measurement of large optical frequency differences with a mode-locked laser, Opt.
Lett. 24, 881 (1999).

151[21] S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, 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. 84, 5102 (2000).
[22] S. T. Cundiff and J. Ye, Colloquium: Femtosecond optical frequency combs,
Rev. Mod. Phys. 75, 325 (2003).
[23] T. Udem, A. Huber, B. Gross, J. Reichert, M. Prevedilli, M. Weitz, andT. W. Hänsch, Phase-Coherent Measurement of the Hydrogen 1S-2S Transition Frequency with an Optical Frequency Interval Divider Chain, Phys.
Rev. Lett. 79, 2646 (1997).
[24] T. Udem, J. Reichert, R. Holzwarth, and T. W. Hänsch, Absolute Optical
Frequency Measurement of the Cesium D1 Line with a Mode-Locked Laser,
Phys. Rev. Lett. 82, 3568 (1999).
[25] S. A. Diddams et al., An Optical Clock Based on a Single Trapped 199 Hg+
Ion, Science 293, 825 (2001).
[26] H. Katori, M. Takamoto, V. G. Pal’chikov, and V. D. Ovsiannikov, Ultrastable Optical Clock with Neutral Atoms in an Engineered Light Shift Trap,
Phys. Rev. Lett. 91, 173005 (2003).
[27] A. D. Ludlow et al., Sr Lattice Clock at 1 x 1016 Fractional Uncertainty by
Remote Optical Evaluation with a Ca Clock, Science 319, 1805 (2008).
[28] T. Rosenband et al., Frequency Ratio of Al+ and Hg+ Single-Ion Optical
Clocks; Metrology at the 17th Decimal Place, Science 319, 1808 (2008).
[29] S. Blatt et al., New Limits on Coupling of Fundamental Constants to Gravity
Using 87 Sr Optical Lattice Clocks, Phys. Rev. Lett. 100, 140801 (2008).
[30] J. Ye, Everything under control, Nat. Photonics 1, 447 (2007).
[31] Z. Jiang, C.-B. Huang, D. E. Leiard, and A. M. Weiner, Optical arbitrary
waveform processing of more than 100 spectral comb lines, Nat. Photonics
1, 463 (2007).
[32] M. C. Stowe, F. C. Cruz, A. Marina, and J. Ye, High Resolution Atomic
Coherent Control via Spectral Phase Manipulation of an Optical Frequency
Comb, Phys. Rev. Lett. 96, 153001 (2006).
[33] A. Pe’er, E. A. Shapiro, M. C. Stowe, M. Shapiro, and J. Ye, Precise Controlof Molecular Dynamics with a Femtosecond Frequency Comb, Phys. Rev.
Lett. 98, 113004 (2007).

152[34] M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T.
Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, Attosecond
metrology, Nature 414, 509 (2001).
[35] A. Baltuska, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C.
Gohle, R. Holzwarth, V. S. Yakovlev, T. W. Scrinzi, A. Hänsch, and F.
Kraus, Attosecond control of electronic processes by intense light fields,
Nature 421, 611 (2003).
[36] A. Marian, M. C. Stowe, J. R. Lawall, D. Felinto, and J. Ye, United TimeFrequency Spectroscopy for Dynamics and Global Structure, Science 306,
2063 (2004).
[37] M. J. Thorpe, K. D. Moll, R. J. Jones, B. Safdi, and J. Ye, Broadband
Cavity Ringdown Spectroscopy for Sensitive and Rapid Molecular Detection,
Science 311, 1595 (2006).
[38] S. A. Diddams, L. Hollberg, and V. Mbele, Molecular fingerprinting withthe resolved modes of a femtosecond laser frequency comb, Nature 445, 627(2007).
[39] F. Tauser, A. Leitenstorfer, and W. Zinth, Amplified femtosecond pulses
from an Er:fiber system: Nonlinear pulse shortening and self-referencing detection of the carrier-envelope phase evolution, Opt. Express 11, 594 (2003).
[40] J. K. Ranka, R. S. Windeler, and A. J. Stentz, Visible continuum generationin air-silica microstructure optical fibers with anomalous dispersion at 800
nm, Opt. Lett. 25, 25 (2000).
[41] K. W. Jucks, D. G. Johnson, K. V. Chance, W. A. Traub, J. J. Margitan,
G. B. Osterman, R. J. Salawitch, and Y. Sasano, Observations of OH, HO2 ,H2 O, and O3 in the upper stratosphere: implications for HOx photochemistry, Geophys. Res. Lett. 25, 3935 (1998).
[42] J. N. Begum, Modeling of chemical tracer transport in the atmospheric environment and its impact on the global climate, J. Quant. Spectrosc. Radiat.
Transfer 95, 423 (2005).
[43] K. K. Lehmann, B. H. Pate, and G. Scoles, On the measurement of therate of intermolecular vibrational energy redistribution via high resolution
molecular beam optothermal spectroscopy, Laser Chem. 11, 237 (1991).
[44] J. E. Gambogi, E. R. Th. Kerstel, K. K. Lehmann, and G. Scoles, Eigenstate
resolved infrared/infrared double resonance spectroscopy of the 3ν1 overtone
band of I-propyne: Intramolecular vibrational energy redistribution into a
Coriolis-coupled bath, J. Chem. Phys. 100, 2612 (1994).

153[45] H. R. Thorsheim, J. Weiner, and P. S. Julienne, Laser-induced photoassociation of ultracold sodium atoms, Phys. Rev. Lett. 58, 2420 (1987).
[46] E. R. Hudson, C. Ticknor, B. C. Sawyer, C. A. Taatjes, H. J. Lewandowski,
J. R. Bochinski, J. L. Bohn, and J. Ye, Production of cold formaldehyde
molecules for study and control of chemical reaction dynamics with hydroxyl
radicals, Phys. Rev. A 73, 063404 (2006).
[47] J. Doyle, B. Friedrich, R. V. Krems, and F. Masnou-Seeuws, Quo vadis, cold
molecules?, Eur. J. Phys. D. 31, 149 (2004).
[48] C. Wang, S. T. Scherer, and D. Hossain, Measurements of Cavity Ringdown
Spectroscopy of Acetone in the Ultraviolet and Near-Infrared Spectral Regions: Potential for Development of a Breath Analyzer, Appl. Spectrosc. 58,
784 (2004).
[49] K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Peer, J. J. Neyenhuis,B. Zirbel, S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye, A highphase-space-density gas of ground-state polar molecules, Science 322, 231(2008).
[50] T. H. Risby and S. F. Solga, Current status of clinical breath analysis, Appl.
Phys. B 85, 421 (2006).
[51] H. R. Telle, G. Steinmeyer, A. E. Dunlop, J. Stenger, D. H. Sutter, and U.
Keller, Carrier-envelope offset phase control: A novel concept for absolute
optical frequency measurement and ultrashort pulse generation, Appl. Phys.
B 69, 327 (1999).
[52] T. Udem, R. Holzwarth, and T. W. Hänsch, Optical frequency metrology,
Nature 416, 233 (2002).
[53] J. Ye and S. T. Cundiff, Femtosecond optical frequency comb technology:
Principle, operation, and application. (Springer, New York, 2005).
[54] R. J. Jones and J. C. Diels, Stabilization of Femtosecond Lasers for Optical Frequency Metrology and Direct Optical to Radio Frequency Synthesis,
Phys. Rev. Lett. 86, 3288 (2001).
[55] R. J. Jones, I. Thomann, and J. Ye, Precision stabilization of femtosecond
lasers to high-finesse optical cavities, Phys. Rev. A 69, 051803 (2004).
[56] S. M. Foreman, D. J. Jones, and J. Ye, Flexible and rapidly configurable
femtosecond pulse generation in the mid-IR, Opt. Lett. 28, 370 (2003).
[57] E. Penman, C. McGowan, P. Loza-Alvarez, D. T. Reid, M. Ebrahimzadeh,and W. Sibbett, Femtosecond optical parametric oscillators based on periodically poled lithium niobate, J. Mod. Opt. 45, 1285 (1998).

154[58] D. Mazzotti, P. Cancio, G. Giusfredi, P. De Natale, and M. Prevedelli,Frequency-comb-based absolute frequency measurements in the midinfrared with a difference-frequency spectrometer, Opt. Lett. 30, 997 (2005).
[59] A. Schliesser, M. Brehm, F. Keilmann, and D. W. van der Weide, Frequencycomb infrared spectrometer for rapid, remote chemical sensing, Opt. Express
13, 9029 (2003).
[60] R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye, Phase-Coherent Frequency
Combs in the Vacuum Ultraviolet via High-Harmonic Generation inside a
Femtosecond Enhancement Cavity, Phys. Rev. Lett. 94, 193201 (2005).
[61] C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, A.
Schuessler, F. Kraus, and T. W. Hänsch, A frequency comb in the extreme
ultraviolet, Nature 436, 234 (2005).
[62] D. C. Yost, T. R. Schibli, and J. Ye, Efficient output coupling of intracavity
high-harmonic generation, Opt. Lett. 33, 1099 (2008).
[63] E. E. Eyler, D. E. Chieda, M. C. Stowe, M. J. Thorpe, T. R. Schibli, andJ. Ye, Prospects for precision measurements of atomic helium using direct
frequency comb spectroscopy, Eur. Phys. J. D. 48, 1 (2008).
[64] T. A. Birks, W. J. Wadsworth, and P. S. J. Russel, Supercontinuum generation in tapered fibers, Opt. Lett. 25, 1415 (2000).
[65] T. M. Fortier, D. J. Jones, and S. T. Cundiff, Phase stabilization of an
octave-spanning Ti:sapphire laser, Opt. Lett. 28, 2198 (2003).
[66] R. Ell et al., Generation of 5-fs pulses and octave-spanning spectra directlyfrom a Ti:sapphire laser, Opt. Lett. 26, 373 (2001).
[67] T. M. Fortier, A. Bartels, and S. A. Diddams, Octave-spanning Ti:sapphirelaser with a repetition rate > 1 GHz for optical frequency measurements
and comparisons, Opt. Lett. 31, 1011 (2006).
[68] L. E. Nelson, D. J. Jones, K. Tamura, H. A. Haus, and E. P. Ippen,
Ultrashort-pulse fiber ring lasers, Appl. Phys. B 65, 277 (1997).
[69] P. Pal, W. H. Knox, I. Hartl, and M. E. Fermann, Self referenced Yb-fiberlaser frequency comb using a dispersion micromanaged tapered holey fiber,
Opt. Express 15, 12161 (2007).
[70] P. Balling, M. Fischer, P. Kubina, and R. Holzwarth, Absolute frequency
measurement of wavelength standard at 1542nm: acetylene stabilized DFB
laser, Opt. Express 13, 9196 (2005).

155[71] B. R. Washburn, S. A. Diddams, N. R. Newbury, J. W. Nicholson, M. F.
Yan, and C. G. Jorgensen, Phase-locked, erbium-fiber-laser-based frequency
comb in the near infrared, Opt. Lett. 29, 250 (2004).
[72] T. R. Schibli, D. C. Yost, M. J. Martin, J. Ye, I. Hartl, A. Marcinkevicius,and M. E. Fermann, Optical frequency comb with submillihertz linewidthand more than 10 W average power, Nat. Photonics 2, 355 (2008).
[73] K. Moutzouris, F. Adler, F. Sotier, D. Träutlein, and A. Leitenstorfer, Multimilliwatt ultrashort pulses continuously tunable in the visible from a compact fiber source, Opt. Lett. 31, 1148 (2006).
[74] C. Erny, K. Moutzouris, J. Biegert, D. Kühlke, F. Adler, and A. Leitenstorfer, Mid-infrared difference-frequency generation of ultrashort pulses tunable
between 3.2 and 4.8 µm from a compact fiber source, Opt. Lett. 32, 1138
(2007).
[75] M. J. Thorpe, D. D. Hudson, K. D. Moll, J. Lasri, and J. Ye, Cavityringdown molecular spectroscopy based on an optical frequency comb at
1.451.65 µm, Opt. Lett. 32, 307 (2007).
[76] M. J. Thorpe, D. Balslev-Clausen, M. S. Kirchner, and J. Ye, Cavityenhanced optical frequency comb spectroscopy: application to human breath
analysis, Opt. Express 16, 2387 (2008).
[77] I. Thomann, A. Bartels, K. L. Corwin, N. R. Newbury, L. Hollberg, S. A.
Diddams, J. W. Nicholson, and M. F. Yan, 420-MHz Cr:forsterite femtosecond ring laser and continuum generation in the 12 µm range, Opt. Lett. 28,
1368 (2003).
[78] A. Bartels, N. R. Newbury, I. Thomann, L. Hollberg, and S. A. Diddams,
Broadband phase-coherent optical frequency synthesis with actively linked
Ti:sapphire and Cr:forsterite femtosecond lasers, Opt. Lett. 29, 403 (2004).
[79] R. C. Sharp, D. E. Spock, N. Pan, and J. Elliot, 190-fs passively mode-locked
thulium fiber laser with a low threshold, Opt. Lett. 12, 881 (1996).
[80] S. Kivistö, T. Hakulinen, M. Guina, and O. Okhotnikov, Tunable Raman
Soliton Source Using Mode-Locked TmHo Fiber Laser, IEEE Photon. Technol. Lett. 19, 934 (2007).
[81] M. A. Solodyankin, E. D. Obraztsova, A. S. Lobach, A. I. Cherno, A. V.
Tausenev, V. I. Konov, and E. M. Dianov, Mode-locked 1.93 µm thulium
fiber laser with a carbon nanotube absorber, Opt. Lett. 33, 1336 (2008).
[82] I. S. Moskalev, V. V. Fedorov, and S. B. Mirov, Conference on Lasers andElectro-Optics, 2008. (OSA, San Jose, CA, 2008), pp. 1–2.

156[83] E. Sorokin, S. Naumov, and I. T. Sorokina, Ultrabroadband Infrared SolidState Lasers, IEEE J. of selected topics in Quantum Electronics 11, 690(2005).
[84] K. Tamura, L. E. Nelson, H. A. Haus, and E. P. Ippen, Soliton versus
nonsoliton operation of fiber ring lasers, Appl. Phys. Lett. 64, 149 (1994).
[85] C. R. Giles and E. Desurvire, Modeling Erbium-Doped Fiber Amplifiers, J.
Lightwave Technol. 9, 271 (1991).
[86] E. Desurvire, J. R. Simpson, and P. C. Becker, High-gain erbium-doped
traveling-wave fiber amplifier, Opt. Lett. 12, 888 (1987).
[87] M. Peroni and M. Tamburrini, Gain in erbium-doped fiber amplifiers: a
simple analytical solution for the rate equations, Opt. Lett. 15, 842 (1990).
[88] Miniscalco, Erbium-Doped Glasses for Fiber Amplifiers at 1500 nm, J.
Lightwave Technol. 9, 234 (1991).
[89] Thorlabs, http://thorlabs.com.
[90] R. W. Boyd, Nonlinear optics, 2 ed. (Academic Press, San Diego, 2003), pp.
1–65.
[91] J. D. Jackson, Classical electrodynamics, 3 ed. (Wiley, Hoboken, NJ, 1999),
pp. 151–165.
[92] R. W. Boyd, Nonlinear optics, 2 ed. (Academic Press, San Diego, 2003), pp.
189–209.
[93] Menlo, http://www.menlosystems.com/.
[94] EM4, http://www.em4inc.com/.
[95] Bookham, http://www.bookham.com/.
[96] J. W. Lou, T. F. Carruthers, and M. Currie, Mode-Locked MultipleWavelength Erbium-Doped Fiber Laser in a Sigma Configuration, IEEE
Photon. Technol. Lett. 14, 281 (2002).
[97] D. D. Hudson, K. W. Holman, R. J. Jones, S. T. Cundiff, J. Ye, and
D. J. Jones, Mode-locked fiber laser frequency-controlled with an intracavity
electro-optic modulator, Opt. Lett. 30, 2948 (2005).
[98] G. P. Agrawal, Nonlinear fiber optics, 4th ed. (Academic Press, New York,
2007), pp. 162–168.
[99] R. W. Boyd, Nonlinear optics, 2 ed. (Academic Press, San Diego, 2003), pp.
356–363.

157[100] M. V. Klein and T. E. Furtak, in Optics, 2 ed., edited by J. W. Goodman(Wiley, New York, 1986), pp. 295–324.
[101] The work at JILA has greatly benefited from collborative coating runs from
R. Lalezari of Advanced Thin Films.
[102] K. K. Lehmann and P. Rabinowitz, High-Finesse Optical Resonator for Cavity Ring-Down Spectroscopy based upon Brewsters Angle Prism Retroreflectors,, U. S. Patent 5973864, 1999.
[103] V. N. Kuryatov and A. L. Sokolov, Polarisation losses in a ring prism cavity,
Quantum Electron. 30, 125 (2000).
[104] P. S. Johnston and K. K. Lehmann, Cavity enhanced absorption spectroscopy using a broadband prism cavity and a supercontinuum source, Opt.
Express 16, 15013 (2008).
[105] E. Hecht, Optics, 4th ed. (Addison-Wesley, New York, 2002), pp. 416–421.
[106] D. Romanini and K. K. Lehmann, Ring-down cavity absorption spectroscopy of the very weak HCN overtone bands with six, seven, and eightstretching quanta, J. Chem. Phys 99, 6287 (1993).
[107] J. Ye and J. L. Hall, Cavity ringdown heterodyne spectroscopy: High sensitivity with microwatt light power, Phys. Rev. A 61, 061802 (2000).
[108] Y. He and B. Orr, Ringdown and cavity-enhanced absorption spectroscopyusing a continuous-wave tunable diode laser and a rapidly swept opticalcavity, Chem. Phys. Lett. 319, 131 (2000).
[109] A. E. Siegman, in Lasers, edited by A. Kelly (University Science Books,
Sausalito, Ca, 1986), pp. 573–576.
[110] R. G. DeVoe, C. Fabre, K. Jungmann, J. Hoffnagle, and R. G. Brewer,
Precision optical-frequency-difference measurements, Phys. Rev. A 37, 1802
(1988).
[111] C. J. Hood, H. J. Kimble, and J. Ye, Characterization of high-finesse mirrors:
Loss, phase shifts, and mode structure in an optical cavity, Phys. Rev. A
64, 033804 (2002).
[112] M. J. Thorpe, R. J. Jones, K. D. Moll, J. Ye, and R. Lalezari, Precise
measurements of optical cavity dispersion and mirror coating properties via
femtosecond combs, Opt. Express 13, 882 (2005).
[113] A. Schliesser, C. Gohle, T. Udem, and T. W. Hänsch, Complete characterization of a broadband high-finesse cavity using an optical frequency comb,
Opt. Express 14, 5975 (2006).

158[114] J. Ye, L. S. Ma, and J. L. Hall, Sub-Doppler optical frequency reference at
1.064 mm by means of ultrasensitive cavity-enhanced frequency modulation
spectroscopy of a C2HD overtone transition, Opt. Lett. 21, 1000 (1996).
[115] C. S. Gudeman, M. H. Begemann, J. Pfaff, and R. J. Saykally, Velocitymodulated infrared laser spectroscopy of molecular ions: The ν1 band of
HCO+ , Phys. Rev. Lett. 50, 727 (1983).
[116] D. J. Nesbitt, H. Petek, C. S. Gudeman, C. B. Moore, and R. J. Saykally, A
study of the ν1 fundamental and bend-excited hot band of DNN+ by velocity
modulation absorption spectroscopy with an infrared difference frequency
laser, J. Chem. Phys. 81, 5281 (1984).
[117] J. T. Verdeyen, in Laser Electronics, 3rd ed., edited by N. J. Holonyak(Prentice Hall, Upper Saddle River, NJ, 1995), pp. 76–136.
[118] A. E. Siegman, in Lasers, edited by A. Kelly (University Science Books,
Sausalito, Ca, 1986), pp. 786–797.
[119] MATLAB, Genetic Algorithm and Direct Search Toolbox, version R2007b,function name ‘ga( )’.
[120] A. E. Siegman, in Lasers, edited by A. Kelly (University Science Books,
Sausalito, Ca, 1986), pp. 700–770.
[121] T. Gherman and D. Romanini, Mode-locked cavity-enhanced absorption
spectroscopy, Opt. Express 10, 1033 (2002).
[122] C. Gohle, B. Stein, A. Schliesser, T. Udem, and T. W. Hänsch, Frequency Comb Vernier Spectroscopy for Broadband, High-Resolution, HighSensitivity Absorption and Dispersion Spectra, Phys. Rev. Lett. 99, 263902(2007).
[123] M. J. Thorpe and J. Ye, Cavity-enhanced direct frequency comb spectroscopy, Appl. Phys. B. 91, 397414 (2008).
[124] J. Ye and W. Lynn, Applications of Optical Cavities in Modern Atomic,
Molecular, and Optical Physics, Adv. At. Mol. Opt. Phys. 49, 1 (2003).
[125] L. Gianfrani, R. W. Fox, and L. Hollberg, Cavity-enhanced absorption spectroscopy of molecular oxygen, J. Opt. Soc. Am. B 16, 2247 (1999).
[126] L. S. Ma, J. Ye, P. Dube, and J. L. Hall, Ultrasensitive frequency-modulation
spectroscopy enhanced by a high-finesse optical cavity: theory and application to overtone transitions of C2H2 and C2HD, J. Opt. Soc. Am. B 16,2255 (1999).

159[127] J. Poirson, F. Bretenaker, M. Vallet, and A. Le Floch, Analytical and experimental study of ringing effects in a Fabry-Perot cavity. Application to
the measurement of high finesses, J. Opt. Soc. Am. B 14, 2811 (1997).
[128] R. Bakowski, L. Corner, G. Hancock, R. Kotchie, R. Peverall, and G. A. D.
Ritchie, Cavity-enhanced absorption spectroscopy with a rapidly swept
diode laser, Appl. Phys. B 75, 745 (2002).
[129] M. D. Levenson, B. A. Paldus, T. G. Spence, C. C. Harb, J. Harris, J. S., and
R. N. Zare, Optical heterodyne detection in cavity ring-down spectroscopy,
Chem. Phys Lett. 290, 335 (1998).
[130] J. J. Scherer, J. B. Paul, A. O’Keefe, and R. J. Saykally, Cavity Ringdown
Laser Absorption Spectroscopy: History, Development, and Application to
Pulsed Molecular Beams, Chem. Rev. 97, 25 (1997).
[131] M. V. Klein and T. E. Furtak, in Optics, 2 ed., edited by J. W. Goodman(Wiley, New York, 1986), pp. 280–284.
[132] M. V. Klein and T. E. Furtak, in Optics, 2 ed., edited by J. W. Goodman(Wiley, New York, 1986), pp. 457–465.
[133] A. M. Weiner, Femtosecond pulse shaping using spatial light modulators,
Rev. Sci. Instrum. 71, 1929 (2000).
[134] E. Hecht, Optics, 4th ed. (Addison-Wesley, New York, 2002), pp. 471–474.
[135] E. Hecht, Optics, 4th ed. (Addison-Wesley, New York, 2002), pp. 476–481.
[136] J. J. Scherer, Ringdown spectral photography, Chem. Phys. Lett. 292, 143(1998).
[137] J. J. Scherer, J. B. Paul, H. Jiao, and A. OKeefe, Broadband ringdown
spectral photography, Appl. Opt. 40, 6725 (2001).
[138] S. Xiao and A. M. Weiner, 2-D wavelength demultiplexer with potential for≥ 1000 channels in the C-band, Opt. Express 12, 2895 (2004).
[139] M. Shirasaki, Large angular dispersion by a virtually imaged phased array
and its application to a wavelength demultiplexer, Opt. Lett. 21, 366 (1996).
[140] A. Vega, A. M. Weiner, and C. Lin, Generalized grating equation forvirtually-imaged phased-array spectral dispersers, Appl. Opt. 42, 4152
(2003).
[141] S. Xiao, A. M. Weiner, and C. Lin, A Dispersion Law for Virtually Imaged
Phased-Array Spectral Dispersers Based on Paraxial Wave Theory, IEEE J.
Quantum Electron. 40, 420 (2004).

160[142] M. Shirasaki, A. N. Akhter, and C. Lin, Virtually Imaged Phased Array
with Graded Reflectivity, IEEE Photon. Technol. Lett. 11, 1443 (1999).
[143] A. Mokhtari and A. A. Shichegar, Optical Fiber Communication and Optoelectronics Conference, 2007 Asia (IEEE, Shanghai, 2007), pp. 514–516.
[144] E. R. Crosson, P. Haar, G. A. Marcus, H. A. Schwettman, B. A. Paldus,
T. G. Spence, and R. N. Zare, Pulse-stacked cavity ring-down spectroscopy,
Rev. Sci. Instrum. 70, 4 (1999).
[145] A. E. Siegman, in Lasers, edited by A. Kelly (University Science Books,
Sausalito, Ca, 1986), pp. 286–291.
[146] L. S. Rothman et al., The HITRAN 2004 molecular spectroscopic database,J. Quant. Spectrosc. Radiat. Transfer 96, 139 (2005).
[147] L. Lundsberg-Nielsen, F. Hegelund, and F. M. Nicolaisen, Analysis of theHigh-Resolution Spectrum of Ammonia (14 NH3 ) in the Near-Infrared Region, 6400-6900 cm−1 , J. Mol. Spectrosc. 162, 230 (1993).
[148] G. Herzberg, Molecular spectra and Molecular structure, 2 ed. (Krieger, Malabar, Florida, 1950), Vol. I, pp. 122–129.
[149] E. R. Crosson et al., Stable Isotope Ratios Using Cavity Ring-Down Spectroscopy: Determination of 13 C/12 C for Carbon Dioxide in Human Breath,Anal. Chem. 74, 2003 (2002).
[150] L. N. Plummer, E. Busenber, J. K. Bohlke, D. L. Nelms, R. L. Michel, and
P. Schlosser, Groundwater residence times in Shenandoah National Park,
Chem. Geol. 179, 93 (2001).
[151] E. Mosely-Thompson, L. G. Thompson, and P. N. Lin, A multi-century
ice-core perspective on 20th-century climate change with new contributions
from high-Arctic and Greenland (PARCA) cores, Ann. Glaciology 43, 42(2006).
[152] W. Q. Cao and Y. X. Duan, Breath analysis: Potential for Clinical Diagnosis
and Exposure Assessment, Clin. Chem. 52, 800 (2006).
[153] S. Dill, J. J. Payne-James, J. J. Misiewicz, G. K. Grimble, K. McSwiggan,
K. Pathak, A. J. Wood, C. M. Scrimgeour, and M. J. Rennie, Evaluation of
13
C-urea breath test in the detection of Heliobacter pylori and in monitoring
the effect of tripotassium dicitratobismuthate in non-ulcer dyspepsia, Gut.
31, 1237 (1990).
[154] F. Di Francesco, R. Fuoco, M. G. Trivella, and A. Ceccarini, Breath analysis:trends in techniques and clinical applications, Microchem. J. 79, 405 (2005).

161[155] R. F. Machado et al., Detection of Lung Cancer by Sensor Array Analysesof Exhaled Breath, Am. J. Respir. Crit. Care Med. 171, 1286 (2005).
[156] W. L. Wood, D. J. Higbee, M. Gooldy, S. Glogowski, R. Fitzpatrick, R. J.
Karalus, T. D. Wood, and D. J. Mangino, Analysis of Volatile Metabolites
by Gas Chromatography-Mass Spectrometry, Spectroscopy 21, 20 (2006).
[157] R. Polikar, R. Shinar, V. Honavar, L. Udpa, and M. D. Porter, Detection
and Identification of Odorants Using an Electronic Nose, in Proceedings ofIEEE 26th International Conference Acoustics, Speech and Signal Processing
5, 3137 (2001).
[158] 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 15, 6 (1998).
[159] P. D. Klein and D. Y. Graham, Minimum analysis requirements for thedetection of Helicobacter pylori infection by the 13C-urea breath test, Am.
J. Gastroenterol. 88, 1865 (1993).
[160] A. J. Cunnington and P. Hombrey, Breath analysis to detect recent exposure
to carbon monoxide, Postgrad. Med. J. 78, 233 (2002).
[161] J. Manne, O. Sukhorukov, W. Jäger, and J. Tulip, Pulsed quantum cascadelaser-based cavity ring-down spectroscopy for ammonia detection in breath,Appl. Opt. 45, 9230 (2006).
[162] D. Marinoy, J. M. Rey, M. G. Muller, and M. W. Sigrist, Spectroscopic investigation of methylated amines by a cavity-ringdown-based spectrometer,
Appl. Opt. 46, 3981 (2007).
[163] A. André, D. Demille, J. Doyle, M. D. Lukin, S. E. Maxwell, P. Rabl, R. J.
Schoelkopf, and P. Zoller, A coherent all-electrical interface between polar
molecules and mesoscopic superconducting resonators, Nat. Phys. 2, 636(2006).
[164] B. C. Sawyer, B. L. Lev, E. R. Hudson, B. K. Stuhl, M. Lara, J. L. Bohn,
and J. Ye, Magneto-electrostatic trapping of ground state OH molecules,
Phys. Rev. Lett. 98, 253002 (2007).
[165] S. Y. T. Van De Meerakker, H. L. Bethlem, and G. Meijer, Taming molecularbeams, Nat. Phys. 4, 595 (2008).
[166] J. G. Danzl, E. Haller, M. Gustavsson, M. J. Mark, R. Hart, N. Bouloufa, O.
Dulieu, H. Ritch, and H.-C. Nägerl, Quantum gas of deeply bound groundstate molecules, Science 321, 1062 (2008).

162[167] M. Viteau, A. Chotia, M. Allegrini, N. Bouloufa, O. Dulieu, D. Comparat,
and P. Pillet, Optical Pumping and Vibrational Cooling of Molecules, Science 321, 232 (2008).
[168] W. Salzmann et al., Coherent Transients in the Femtosecond Photoassociation of Ultracold Molecules, Phys. Rev. Lett. 100, 233003 (2008).
[169] E. A. Shapiro, A. Pe’er, J. Ye, and M. Shapiro, Piecewise Adiabatic Population Transfer in a Molecule via aWave Packet, Phys. Rev. Lett. 101, 023601(2008).
[170] G. Tejeda, B. Maté, J. M. Fernández-Sánchez, and S. Montero, Temperature
and Density Mapping of Supersonic Jet Expansions Using Linear Raman
Spectroscopy, Phys. Rev. Lett. 76, 34 (1996).
[171] K. Didriche, C. Lauzin, P. Macko, W. J. Lafferty, R. J. Saykally, and M.
Herman, On the role of clustering in an acetylene supersonic expansion,
Chem. Phys. Lett. 463, 345 (2007).
[172] M. Herman, K. Didriche, D. Hurtmans, B. Kizil, P. Macko, A. Rizopoulos,
and P. Van Poucke, FANTASIO: a versatile experimental set-up to investigate jet-cooled molecules, Mol. Phys. 105, 815 (2007).
[173] R. N. Brancewell, Fourier transform and it’s applications, 3 ed. (McGrawHill, New York, 2000), pp. 351–358.
[174] A. A. Madej, A. L. Alcock, A. Czajkowski, J. E. Bernard, and S. Chepurov, Accurate absolute reference frequencies from 1511 to 1545 nm of theν1 +ν3 band of 12 C2 H2 determined with laser frequency comb interval measurements, J. Opt. Soc. Am. B. 23, 2200 (2006).
[175] J. L. Hall, M. S. Taubman, and J. Ye, in Handbook of Optics IV, edited by
M. Bass, J. M. Enoch, E. Van Stryland, and W. L. Wolfe (McGraw-Hill,
New York, 2000).
[176] U. Valbusa, Atomic and Molecular Beam Methods, Atomic and Molecular
Beam Methods 1, 443 (1988).
[177] S. R. Deans, The Radon Transform and Some of Its Applications (Wiley,
New York, 1983).
[178] E. W. Hansen and A. Jablokow, State Variable Representation of a Classof Linear Shift-Variant Systems, IEEE. Trans. Acous. Speech Sig. Proc. 30,
874 (1982).
[179] G. Herzberg, Molecular spectra and Molecular structure (Krieger, Malabar,
Florida, 1945), Vol. II, pp. 18, 382.

163[180] D. R. Miller, Atomic and Molecular Beam Methods, Atomic and Molecular
Beam Methods 1, 14 (1988).
[181] W. R. Gentry, Atomic and Molecular Beam Methods, Atomic and Molecular
Beam Methods 1, 70 (1988).
[182] M. D. Duncan, P. Osterlin, and R. L. Byer, Pulsed supersonic molecularbeam coherent anti-Stokes Raman spectroscopy of C2 H2 , Opt. Lett. 6, 90(1981).
[183] M. C. Stowe, A. Pe’er, and J. Ye, Control of Four-Level Quantum Coherence
via Discrete Spectral Shaping of an Optical Frequency Comb, Phys. Rev.
Lett. 100, 203001 (2008).

Appendix A
MATLAB genetic algorithm code for mode-matching

The following MATLAB code uses a genetic algorithm to find a suitable
two lens telescope for mode-matching a laser beam having q-parameter (ql) at
location (zl) to a cavity with eigenmode q-parameter (qc) at location (zc). For
simplicity, this notebook assumes that both ql and qc correspond to a beam waist
and are therefore purely imaginary quantities. Note, all distances are expressedin millimeters. The output vector (x) specifies the focal lengths and positions of
the telescope lenses with respect to the position of the laser beam waist according
to x = [f1 f2 z1 z2]. For best performance, use a high resolution discretization of
the lens positions (i.e. ≤ 100 µm steps) and no more than 5 cm steps for the focal
length discretization. Since this is a genetic algorithm, it does not arrive at thesame answer each time you run the program. Therefore, it can be useful to run
the program several times in hopes of finding a telescope that can be made fromlenses that you already have in the lab. If increased precision of (x) is desired,
the population size and number of generations for the genetic algorithm can beincreased. The precision of the telescope (x) for matching the laser beam modeto the cavity mode is given by output variable (fval).

%------------Input parameters------------%
ql = 200;%Rayleigh range of laser beam waist

165
zl = 0;%location of laser beam waist

qc = 500;%Rayleigh range of cavity focus

zc = 2000;%location of cavity beam waist%If coupling to the cavity through a curved mirror, set flag == 1, and%specify input mirror location and radius of curvature.

Otherwise,%set flag == 0.
flag = 0;
Rm = 0;%Radius of curvature of the input mirror

nm = 1.5;%Index of refraction of input mirror

fm = -Rm/(nm-1); %transmission focal length of input mirror
zm = 1700;%location of the input mirror

zmin = 300;

%minimum z-value where a lens can be positioned

zmax = 1600;%maximum z-value where a lens can be positioned

fmax = 2000;%maximum available focal length

fmin = 100;%minimum available focal length

pd = 10;%position discretization (10 = 100um, 1 = 1mm, etc.)

fd = 0.02;%available focal length discetization%(0.1 = 1 cm, 0.02 = 5cm, etc.)%--------Define input vectors for search algorithm-----------%Z = [zl zc zm zmin zmax fm]*pd; % Vector of locations
q = [ql qc];

% Vector of q-parametes and input mirror ROC

dv = [fd fd pd pd];

% Discretization vector%--------Use genetic algorithm to find f1, f2, z1, z2---------%%Set intital population bounds
NumVar = 4;

%Number of variables for the genetic algorithm

LB = [fmin*fd fmin*fd Z(4) Z(4)];UB = [fmax*fd fmax*fd Z(5) Z(5)];%Bounds on lens focal lengths and locations

166
IniValBounds = [LB; UB];
A = [0 0 1 -1];

%Constraint requiring lens 2 be locatedb = 0;%after lens 1 (i.e. zl2 > zl1)

% Execute the Genetic algorithm
FitnessFunction = @(x) fitFunction(x,q,Z,dv,flag);
options = gaoptimset(’CreationFcn’, @int_pop,’MutationFcn’,...
@int_mutation, ’Generations’,500,’PopulationSize’,250,’TolFun’,1e-2,...
’PopInitRange’,IniValBounds,’CrossoverFraction’,.8,’EliteCount’,7);[x fval] = ga(FitnessFunction, NumVar, options);x = x./dv;x% focal lengths and locations for telescope lenses [f1 f2 z1 z2]

fval%value of the fitness function at x
The fitness function used by the genetic algorithm is given below.

function scores = fitFunction(x,q,Z,dv,flag)
if flag == 1
Mc = [[1 0]’ [(Z(2)-Z(3))/dv(3) 1]’]*[[1 -dv(3)/Z(5)]’ [0 1]’]*...
[[1 0]’ [(Z(3)-x(4))/dv(3) 1]’]*[[1 -dv(2)/x(2)]’ [0 1]’]*...
[[1 0]’ [(x(4)-x(3))/dv(3) 1]’]*[[1 -dv(2)/x(1)]’ [0 1]’]*...
[[1 0]’ [(x(3)-Z(1))/dv(3) 1]’];else
Mc = [[1 0]’ [(Z(2)-x(4))/dv(3) 1]’]*[[1 -dv(2)/x(2)]’ [0 1]’]*...
[[1 0]’ [(x(4)-x(3))/dv(3) 1]’]*[[1 -dv(2)/x(1)]’ [0 1]’]*...
[[1 0]’ [(x(3)-Z(1))/dv(3) 1]’];end
AA = Mc(1,1);

167
BB = Mc(1,2);
CC = Mc(2,1);
DD = Mc(2,2);
qprop = (AA*1i*q(1)+BB)/(CC*1i*q(1)+DD);
if x(3) > x(4)scores = 100*((real(qprop-1i*q(2)))^2+imag(qprop-1i*q(2))^2)^(1/2);
elsescores = ((real(qprop-1i*q(2)))^2+imag(qprop-1i*q(2))^2)^(1/2);
end
The functions that define the populations and mutations of the populations usedin the genetic algorithm are written such that the vector variable (x) is an elementof a discrete rather than continuous number set. The benefit of this approach isthat the answers given for the lens focal lengths correspond to values that canbe purchased from actual companies like Thorlabs. The functions that define thepopulations and mutations based on integer variables are given below.

function Population = int_pop(GenomeLength,FitnessFcn,options)

totalpopulation = sum(options.PopulationSize);
range = options.PopInitRange;
lower= range(1,:);span = range(2,:) - lower;% The use of ROUND function will make sure that individuals are integers.
Population = round(repmat(lower,totalpopulation,1) +...

repmat(span,totalpopulation,1) .* rand(totalpopulation,GenomeLength));% End of creation function

168% Mutation function to generate children satisfying the range and integer% constraints on decision variables.
function mutationChildren = int_mutation(parents,options,GenomeLength, ...
FitnessFcn,state,thisScore,thisPopulation)
shrink = .01;
scale = 1;
scale = scale - shrink * scale * state.Generation/options.Generations;range = options.PopInitRange;
lower = range(1,:);upper = range(2,:);scale = scale * (upper - lower);mutationPop =length(parents);% The use of ROUND function will make sure that children are integers.
mutationChildren =round(repmat(lower,mutationPop,1) +...

repmat(scale,mutationPop,1) .* rand(mutationPop,GenomeLength));% End of mutation function
All of these functions should be saved a separate m-files with names that are
identical to their function names. For instance, the population generation function
should be named int pop.m. Finally, all files should be saved in the same folder sothat they can be found by MATLAB when you run the genetic algorithm. Goodluck!

Appendix B
Derivation of the cavity ringdown and cavity enhanced absorption
equations

B.1

Cavity ringdown spectroscopy derivation
Begin with a two mirror cavity with length L, mirror reflectivity R, intra-

cavity power Po . Let the cavity contain a uniformly distributed and frequency
dependent intra-cavity absorber α(ω). The time rate of change of power inside
this cavity can be written ascdP (ω)= −Po (1 − R2 e−2α(ω)L ).
dt
2L

(B.1)

This rate equation can be integrated to yield the power in the cavity as a functionof time.
t

P (ω, t) = Po e(− τ (ω) ) + C(B.2)

where the decay constant τ takes the form
τ (ω) =2L1=.
2−2α(ω)Lc(1 − R e)
F SR(1 − R2 e−2α(ω)L )(B.3)

In the low absorption limit, where the approximation e−2α(ω)L ≈ (1 − 2αL) is
valid, the change in the ringdown due to absorption is
1 − R21δτ (ω)=1−=1−.
2α(ω)L2τ (ω)(1 − R (1 − 2α(ω)L))
1+
2
1−R

(B.4)

170
We can then Taylor expand this result for the case of 2α(ω)L  1 − R2 whichgivesδτ (ω)
2α(ω)L≈.
τ (ω)
1 − R2(B.5)

Finally, we use the approximations 1 − R2 ≈ 2(1 − R) and F ≈ π/(1 − R) to
arrive at the final resultδτ (ω)
Fα(ω)L
≈
.
τ (ω)
π

(B.6)

Equation B.6 shows the enhancement of the intra-cavity absorption signal provided by cavity ringdown spectroscopy in the low absorption limit.
B.2

Steady-state cavity enhanced absorption spectroscopy derivation
Once again we’ll begin with a two mirror cavity of length L with mirror

transmission T = |t|2 and reflectivity R = |r|2 , where t and r are the electric fieldtransmission and reflectivity coefficients. However, this time there will initiallybe no intra-cavity power. Instead, we imagine a Gaussian beam that is perfectly
mode-matched to the cavity mode incident on one of the cavity mirrors. Let thisbeam have electric field amplitude Eo and propagation vector ~k. As before, let
the cavity contain a uniformly distributed and frequency dependent intra-cavity
absorber α(ω) with electric field absorption coefficient κ = α2 . At the instant thatthe incident field is turned on, a portion of the field is coupled into the cavity.
After each round trip that the Gaussian beam makes inside the intra-cavity electricfield is given by
E0 = tEo~

E1 = tEo + tr2 Eo e2L(i|k|−κ)~~E2 = tEo + tr2 Eo e2L(i|k|−κ) + tr4 Eo e4L(i|k|−κ)~~

En = tEo + tr2 Eo e2L(i|k|−κ) + . . . + tr2n Eo e2nL(i|k|−κ) .

171Note that though we have not included any phase terms associated with the cavity
mirrors it is easy to add a wavelength dependent phase to the mirror reflectivity
if necessary. The steady state intracavity field is found by letting n → ∞.
∞X

Ei = tEo

tEo~

r2n e2nL(i|k|−κ) =n=0

1 − r2 e2L(i|~k|−κ)(B.7)

From Eq. B.7 it is easy to calculate the cavity transmitted field.
~

Et =t2 Eo eL(i|k|−κ)(B.8)

1 − r2 e2L(i|~k|−κ)The intracavity absorption will be detected by measuring the cavity transmitted
intensity
It =T 2 Io eα(ω)L
1 − Re(2i|~k|L−α(ω)L)

2.

(B.9)

For the remainder of this derivation, we’ll assume that the incident electric fieldis resonant with the cavity such that the phase factor in the denominator of the
Eq. B.9 is a multiple of 2π.
It =T 2 Io eα(ω)L
1 − Re−α(ω)L2.

(B.10)

The change in cavity transmitted intensity due to absorption in the low absorption
limit (2αL  1 − R) isδIt
1 − α(ω)L2α(ω)L 
=1−
≈
1
−
1
−
.
2
It
1−R
1 + α(ω)LR(B.11)

δI
2Fα(ω)L≈.
It
π(B.12)

1−R

Equation B.12 shows that in the low absorption limit, steady-state cavity-enhanced
absorption spectroscopy provides twice the signal enhancement compared to cavity ringdown spectroscopy.

172
B.3

Transient cavity-enhanced absorption spectroscopy derivation
The transient CEAS derivation is carried out using the same procedure asin the steady-state case with the exception that the for the transient case, a linear
frequency sweep is applied to the incident laser frequency. The time-dependent
frequency of one of the frequency comb teeth is given by
ν(t) = νo + βt.

(B.13)The time variable in this equation can be discretized according to the cavity
round-trip time such thatLν(tn ) = νo + 2nβ .
c(B.14)

Using this definition for the frequency of the incident beam, the intra-cavity electric field measured at the output coupler after n round-trips is given byL

αL

Eout (tn ) = t2 Eo e2πiν(tn ) c e− 2 +
L 3L

3αL

L 5L

5αL

t2 r2 Eo e2πi(ν(tn )−2β c ) c e− 2 +t2 r4 Eo e2πi(ν(tn )−4β c ) c e− 2 + . . .
L (2n−1)Lc+ t2 r2n Eo e2πi(ν(tn )−2(n−1)β c )e−(2n−1)αL
2.

Here, the variable t in ν(t) is defined as the time at the moment that the measurement is made. Once again, we let n → ∞ the and sum the geometric series. Also,
we note that since the cavity transmitted field is the one that will be measured,Leach term in the geometric sum has been multiplied by a factor of e2πiβ c . The
resulting expression for the cavity transmitted field is
Et = T Eo

nf
XLRn−1 e−(2n−1)α 2 e2πiφ(tnf ) ,(B.15)

n=1

where the phase term is given byφ(tnf ) = (2n − 1)(ν(tnf )L
L2− 2(n − 1)β 2 ).
cc

(B.16)

173
Unfortunately, the complexity of the arguments of the exponentials in the Eq. B.15
make direct summation of this expression impossible. Instead, this sum can be
carried out numerically to reveal that the enhancement of absorption signals fortransient CEAS depends on the sweep speed β and the cavity linewidth ∆ν asshown in Fig.5.1.

Appendix C
Algorithm for finding the fringes in a VIPA interference image

The following MATLAB function locates the fringe pattern on a VIPA interference image, and converts intensity of each fringe into a column vector to createa matrix that contains the normalized change in cavity transmitted intensity ateach frequency. This matrix has dimensions n × m, where n is the number ofpixels in the vertical direction along the detector array, and n is the number offringes. After this step, the reference and absorption intensity matrices are combined according to (Iref − Iabs )/Iref and then unwrapped yielding a traditional 1D
absorption spectrum. The low-pass filtering and background subtraction routines
in this code are used to improve the accuracy of the absorption measurements
extracted from the VIPA images and reduce the noise contained in the resulting
spectrum. Care must be taken when setting the parameters for these routines
such that absorption information is not ‘subtracted’ from or ‘filtered’ out of theabsorption spectrum.

function collectFringesCM%----- input from LabVIEW ------%
global pic;

%Absorption picture (pic)global bckg;%Reference picture

global freqCut;%LowPass filter frequency(bckg)

175
global thresh;%Threshold for fringe finderglobal ba;%Ratio of ref to abs image intensities

global reflambda;%Wavelenght at bottom left of%VIPA image%----- output to LabVIEW -----%
global out;

% Find baseline intensity of the (bckg) and (pic) images using a%’min_finder’ algorithm.
baseline = bckg;
bckgSize = size(bckg);
number_of_iterations=100;
ssize = length(baseline(:));
temp1 = baseline(:);
temp2 = baseline(:);
for k = 1:number_of_iterations
for l = (1+k):(ssize-k)
temp2(l) = min(temp1(l),(temp1(l+k)+temp1(l-k))/2);
end
temp1 = temp2;
endbaseline = reshape(temp1,bckgSize);baseline1 = pic;
picSize = size(pic);
number_of_iterations=100;
ssize = length(baseline1(:));

176
temp1 = baseline1(:);
temp2 = baseline1(:);
for k = 1:number_of_iterations
for l = (1+k):(ssize-k)
temp2(l) = min(temp1(l),(temp1(l+k)+temp1(l-k))/2);
end
temp1 = temp2;
endbaseline1 = reshape(temp1,picSize);%This routine determines the number of fringes and their locations. It%collects the (bckg) and (pic) fringes into column vectors.
%iidx will automaticly become the second minimum.
rmax = size(pic,1);
rmid = floor(rmax/2);
fringeProfile =sum(Ref((rmid-5):(rmid+5),:),1)/11;

if istest, plot(fringeProfile),pause;close; end
fringeDiff = fringeProfile - thresh;poss = find(fringeDiff >= 0);
idx = 1;
fringeN = 1;
while idx < length(poss)-10
for k = 1:2
while poss(idx)+1 == poss(idx+1)
idx = idx + 1;
end
idx = idx + 1;

Note that

177
if k == 1; iidx = poss(idx); oldidx = idx; end
end
fidx = poss(idx-1);[maxI,midx] = max(Ref(129, iidx:fidx));
w = floor((fidx-iidx)/3);
midx = iidx + midx+1;
for ridx = 129:256;[maxI,midx] = max(Ref(ridx, (midx-w):(midx+w)));midx = iidx + midx+1;
picBasev(ridx,fringeN) = sum(baseline1(ridx, ...
(midx-w):(midx+w)),2)/(2*w+1);
bckgv(ridx,fringeN) = sum(bckg(ridx, ...
(midx-w):(midx+w)),2)/(2*w+1);
picv(ridx,fringeN) = sum(pic(ridx, ...
(midx-w):(midx+w)),2)/(2*w+1);
bckgBasev(ridx,fringeN) = sum(baseline(ridx, ...
(midx-w):(midx+w)),2)/(2*w+1);
end[maxI,midx] = max(Ref(128,iidx:fidx));
midx = iidx + midx+1;
for ridx = 128:-1:1;[maxI,midx] = max(Ref(ridx, (midx-w):(midx+w)));midx = iidx + midx+1;
picBasev(ridx,fringeN) = sum(baseline1(ridx, ...
(midx-w):(midx+w)),2)/(2*w+1);
bckgv(ridx,fringeN) = sum(bckg(ridx, ...
(midx-w):(midx+w)),2)/(2*w+1);

178
picv(ridx,fringeN) = sum(pic(ridx, ...
(midx-w):(midx+w)),2)/(2*w+1);
bckgBasev(ridx,fringeN) = sum(baseline(ridx, ...
(midx-w):(midx+w)),2)/(2*w+1);
end
fringeN = fringeN + 1;
idx = oldidx;
end%Lowpass filter picv, bckgv and basev before calculating the normalized%changes in the cavity transmitted light due to absorption.
fringeSize=size(picv);[b,a]=butter(2,freqCut);butter_pic=filtfilt(b,a,picv(:));
butter_bckg=filtfilt(b,a,bckgv(:));
butter_picBasev=filtfilt(b,a,picBasev(:));
butter_bckgBasev=filtfilt(b,a,bckgBasev(:));
picv = reshape(butter_pic,fringeSize);bckgv = reshape(butter_bckg,fringeSize);
pBasev = reshape(butter_picBasev,fringeSize);bBasev = reshape(butter_bckgBasev,fringeSize);%Calculate the normalized change in cavity transmission due to%absorption at all points along the fringes of the VIPA image.
fringes = (bckgv-picv/ba)./(bckgv-bBasev);
fringes = fringes(:,end:-1:1);

179%Find values closest to zero near top and bottom of adjacent fringes%and connects them with a straight line.

This step is in preparation%the background subtraction that occurs in the next code block.
for index = 2:fringeN-1
aFringes = abs(fringes);[minTop,indTop] = min(aFringes(10:40,index-1));[minBot,indBot] = min(aFringes(216:246,index));
iTop = 10+indTop;
iBot = 210+indBot;
deltaI = (256-210+indBot)+(10+indTop);deltaV = fringes(iTop,index-1)-fringes(iBot,index);
fringes(iBot:256,index) = fringes(iBot,index)+...
deltaV/deltaI*(1:(256-iBot+1));
fringes(iTop:-1:1,index-1) = fringes(iTop,index-1)...
-deltaV/deltaI*(1:(iTop));end% Remove non-zero relative intensity baselines that arise from%differences in the reference and absorption images intensities.
%This should only be done for spectra containing only sharp%absorption features.
fringeSize = size(fringes);for n = 1:fringeN-1
spectrum = fringes(:,n);number_of_iterations=19;
ssize = length(spectrum(:));
temp1 = spectrum(:);

180
temp2 = spectrum(:);
for k = 1:number_of_iterations
for l = (1+k):(ssize-k)
temp2(l) = min(temp1(l),(temp1(l+k)+temp1(l-k))/2);
end
temp1 = temp2;
endfringes(:,n) = spectrum(:)-temp1(:);
end%Reshape the normalized changes in cavity transmission array and write%this array to the output variable.
[fringeL,fringeN] = size(fringes);out = fringes(fringeL:-1:1,:);

Appendix D
MATLAB code for a discrete inverse Abel transform

The following MATLAB function calculates the inverse Abel transform fora data vector ‘V ’. Each value of the vector is the line-integral of a density distribution measured at different distances ‘y’ from the axis of the radial distribution.
The distance interval between adjacent line-integral measurements is ‘dy’. Thisalgorithm was taken from Ref. [178].

function out = AbelTransform(V, dy)%--------------Initialize the variables-----------------%
la = length[V];gPrime = zeros(1,la-1);
xk = zeros(9,la);f = zeros(1,la);%----------Compute spatial derivatives of input Matrix-----------%
gPrime(1,:) = diff(V(1,:))/dy;%------EigenVectors for computing the inverse Abel transform-------%
h = [.318 .19 .35 .82 1.8 3.9 8.3 19.6 48.3];
lam = [0 -2.1 -6.2 -22.4 -92.5 -414.5 -1889.4 -8990.9 -47391.1];
C = [1 1 1 1 1 1 1 1 1];%-----------Execute the inverse Abel transform alogorithm-----------%

182
for j = la-1:-1:2
Phi(1,:) = (j/(j-1)).^lam(1,:);Gamma(1,1) = h(1,1)*(-log(j/(j-1)));Gamma(1,2:9) = h(1,2:9)./lam(1,2:9).*(1-(j/(j-1)).^lam(1,2:9));for i = 1:9a(1,i) = Phi(1,i).*xk(i,j);end
xk(:,j-1) = a(1,:) + Gamma(1,:)*gPrime(1,j);f(1,j-1) = dot(C,xk(:,j-1));
end
end
out = f;