Development of Frequency Comb Velocity-Modulation
Spectroscopy, Spectroscopy of HfF+ and the JILA eEDM
Experimentby
Laura C. Sinclair
B.S. California Institute of Technology, 2004

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
2012This thesis entitled:
Development of Frequency Comb Velocity-Modulation Spectroscopy, Spectroscopy of HfF+
and the JILA eEDM Experiment
written by Laura C. Sinclair
has been approved for the Department of Physics

Eric A. Cornell

Prof. Jun Ye

Date

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

iii
Sinclair, Laura C. (Ph.D. Physics)
Development of Frequency Comb Velocity-Modulation Spectroscopy, Spectroscopy of HfF+
and the JILA eEDM Experiment
Thesis directed by Prof. Eric A. Cornell

Broad bandwidth, precision spectroscopy of the molecular ions of interest to the JILA
electron electric dipole moment experiment, HfF+ and ThF+ , is necessary due to the limited amount of spectroscopic information available and the large theoretical uncertainties
in the energy level structure (thousands of wavenumbers). This thesis covers the development of a novel spectroscopic technique, frequency comb velocity-modulation spectroscopy,that provides high resolution, broad spectral bandwidth, ion discrimination and high sensitivity simultaneously. Frequency comb velocity-modulation spectroscopy as well as singlefrequency velocity-modulation spectroscopy have been used to identify five rotational bandsof HfF+ . This work discusses the first spectroscopic information for HfF+ which camefrom our measurement of the 1 Π1 −1 Σ+ (0,0) band recorded with single-frequency velocitymodulation spectroscopy with a sensitivity of 3x10−7 Hz−1/2 . The development of frequency
comb velocity-modulation spectroscopy allowed us to cover a thousand wavenumbers of spectral bandwidth and to identify an additional four HfF+ bands. The achieved sensitivity forfrequency-comb velocity-modulation spectroscopy was 4x10−8 Hz−1/2 (spectral element)−1/2
with 1500 simultaneous detection channels spanning 150 cm−1 of bandwidth. For a 30 minuteacquisition time using 30 interleaved images to densely sample the whole spectrum, this corresponded to a 3x10−7 single-pass fractional absorption sensitivity for each of the 45,000
measurement channels. The spectroscopic information from all five HfF+ rotational bands
is presented and molecular constants for the 1 Σ+ , 3 Π1 , and 1 Π1 states were extracted.

To Dr. Jack Blumenthal, mentor and softball coachand to my siblings, Betsy and Andy, for reminding me to “keep calm and carry on”

Acknowledgements

I would like to thank my advisor, Eric Cornell, for sharing his joy of science, for his
advice and ability to refocus our efforts when needed, and for his patience and kindness
when things really didn’t work. I would also like to thank Jun Ye for co-advising all of thefrequency comb work. To have Jun step in with his expertise in frequency combs was criticalto our success. Working with Jun and Eric was a fantastic experience although at times aterrifying experience due to the brain power in the room, and I’m very grateful for it.
I would like to thank experimental JILA eEDM-ers, past and present, Tyler Coffey,
Kevin Cossel, Matt Grau, Dan Gresh, Aaron Leanhardt, Huanqian Loh, Patrick Maletinsky,
Russ Stutz and Tyler Yahn. Special thanks go to Russ and Aaron for the early days in B238
and to Kevin for the late nights making frequency comb velocity-modulation spectroscopy
work. John Bohn and his student, Ed Meyer, provided theoretical guidance from the earlydays onward and many corny physics jokes.
Without the larger JILA scientific community, this work would not have been possible.
I’m grateful to Carl Wieman for stepping in as our advisor for those bad weeks of 2004.
Florian Adler and Mike Thorpe provided a great deal of help in the initial design of frequency
comb velocity-modulation spectroscopy for which I am grateful. Without the expertise of
the JILA machine and electronic shops this work would have been much more challenging.
Thank you, Todd Ascinar, for saving us multiple times with speedy repairs of our glass
work and, Dave Alchenberger, for re-polishing our ti:sapph crystal. The guidance of Debbie
Jin in the Cornell-Jin meetings as well as discussions with the members of the Jin, Cornell

vi
and Wieman groups were very helpful. Likewise, the cold molecule meetings with the Yemolecule folks and the Lewandowski group provided many fruitful discussions.
Finally, thanks are due to my parents, siblings, siblings-in-law, and husband for theirunwavering support.

viiContents

Chapter
1 Introduction

1

1.1

Electron EDM as a probe of physics beyond the Standard Model . . . . . . .

1

1.2

The Advantage of Molecular Ions and the JILA eEDM experiment . . . . . .

3

1.3

The Trouble with Molecular Ions: The Need for Spectroscopic Information .

5

1.4

Overview of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62 Spectroscopy with a Hollow Cathode Lamp

83 Velocity-Modulation Spectroscopy with a Diode Laser

14

3.1

Experiment Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

3.2

Characterizing the System . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

3.3

First Experimental Data of HfF+ Spectra . . . . . . . . . . . . . . . . . . . .

22

3.4

Extending Velocity-Modulation Spectroscopy . . . . . . . . . . . . . . . . . .

26

4 Development of Frequency Comb Velocity-Modulation Spectroscopy

30

4.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

4.2

Experiment Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

4.2.1

Cavity Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

4.2.2

Imaging System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

4.2.3

Generating a Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . .

45

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4.2.4

Control of the Frequency Comb . . . . . . . . . . . . . . . . . . . . .

52

4.3

Noise and the Sensitivity of the System . . . . . . . . . . . . . . . . . . . . .

53

4.4

A Thousand of Wavenumbers of HfF+

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

56

4.5

Coherent Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

5 HfF+ Structure

66

5.1

Fits to HfF+ Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

5.2

Molecular Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

5.3

Comparison with Theory and Other Experimental Data . . . . . . . . . . . .

71

5.4

Isotope Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

6 Conclusion

76

Bibliography

80

Appendix
A Hollow Cathode Lamp Data86

B Velocity-Modulation Data98

C Frequency Comb Velocity-Modulation Data

104

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TablesTable
1.1

Summary of predictions of electron EDM values for various extensions to theStandard Model. [1]

1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3Comparison of recent and ongoing eEDM experiments (ongoing experimentvalues often represent order of magnitude estimates). The figure-of-merit for√evaluating the sensitivity of an eEDM experiment is Eef f τ N , where Eef f is
the effective electric field on the unpaired electron, τ is the coherence time,and N is the number of measurements that can be counted in some reasonable
experimental integration time. . . . . . . . . . . . . . . . . . . . . . . . . . .

4

3.1

Isotope averaged values of fit of line positions with 1σ errors. . . . . . . . . .

26

4.1

Distances between optical elements in enhancement cavity and between themode-matching optics and the enhancement cavity. Optical elements are labeled in Figure 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

x
5.1

Fitted constants for observed transitions in 180 HfF+ . Quoted uncertainties
are 95% except for the 3 Π1 − 1 Σ+ (3, 1) transition for which uncertainties arequoted at the level for which the manual fit would noticeably fail. Note thatthe values for the 1 Π1 − 1 Σ+ (0,0) transition were extracted from a fit to
the average line positions as the individual isotopes are not fully resolved as
discussed in Chapter Three. ◦ 3 Π1 − 1 Σ+ (3, 1) values assigned using a manual
fit due to challenges of determining the line centers of the weaker lines inthe dense spectrum. ∗ Value fixed to the fitted value from the 1 Π1 − 1 Σ+ (0, 1)transition. + Isotope averaged value with the error due to the uncertainty inthe electronic isotope shift.

5.2. . . . . . . . . . . . . . . . . . . . . . . . . . .

69

Derived constants for observed states in 180 HfF+ . Values for T0 are given
from the minima of the potential not the lowest vibrational level. The error
quoted is the statistical error and does not include any model dependent
systematics. The dominant source of the statistical error is the uncertainty in
αe . The model dependent error would arise from ωe χe for which we see good
agreement for the 1 Σ+ state with a non-model dependent value. . . . . . . .

5.3

71

Comparison of derived molecular constants for 180 HfF+ with the theoretical
calculations of Petrov et al. [2] and results from Barker et al. [3]. The
theoretical values of Be were computed from the equilibrium bond length. . .

5.4

73

Fitted constants for the 1 Π1 ←1 Σ+ (0, 1) band for four different isotopes of
HfF+ . The 176 HfF+ lines at 5.21% abundance were excluded from fitting due
to their lower intensity. Quoted uncertainties are 95%. . . . . . . . . . . . .

5.5

75

Fitted constants for the 3 Π1 ←1 Σ+ (2, 0) band for three different isotopes of
HfF+ . The 179 HfF+ lines at 13.63% abundance and the 176 HfF+ lines at 5.21%
abundance were not fit. Quoted uncertainties are 95%. ∗ Values fixed basedon expected rotational isotope shift.

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

75

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FiguresFigure1.1

An Electron EDM violates both time and parity invariance. The dipole moment of the electron must lie either parallel or anti-parallel to the electron’s
spin. Here the electron EDM is represented by the blue arrow labeled d andalso graphically by the charge separation on the sphere. The electron’s spin
is represented by the red arrow labeled s and graphically by the white arrow
representing a ring of current. Under a reversal of time, the sign of the spin
is reversed while the sign of the electron’s dipole moment stays the same.
Under a reversal of parity, the sign of the dipole moment is flipped while thespin stays the same. The amount of CP-violation and thus T-violation given
CPT invariance required to give a non-negligible electron EDM cannot be
accounted for with the Standard Model. . . . . . . . . . . . . . . . . . . . .

1.2

Energy levels of HfF+ relevant to the eEDM experiment.

2

Two different

schemes for spin readout, state selective photodissociation (a) and laser induced fluorescence (LIF) (b), are shown. States shown as dashed lines including the 1 Σ+ ground state may be utilized for state preparation. The exact
energies of these states will help to determine the best method for spin readout.

7

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2.1

Qualitative sketch of the hollow cathode lamp apparatus. The glow from thedischarge at the center of the hollow cathode (orange square) is imaged into
the grating monochromator. As the AC motor scans the grating, the light at
different wavelengths is imaged onto the photomultiplier tube and recordedby the oscilloscope. The cross-section of the cathode on the left shows the Hfsleeve press fit inside the copper cathode with the HfF4 powder placed inside.

2.2

9

Fluorescence from the hollow cathode lamp versus frequency. Locations ofstrong lines from the most likely atomic sources of fluorescence in this frequency range as well as several HfF bands and one HfF+ band at 13,000 cm−1
are also shown and are offset for clarity. The position of both the band origin
and the R-branch band head are indicated for the HfF and HfF+ bands foundusing other spectroscopic techniques. Note that a majority of the features
shown are not accounted for by the atomic lines published in the NIST stronglines tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

11

Regions of hollow cathode lamp spectra that contain known HfF+ (a) or HfF(b) bands. Panel (a) shows data from hollow cathode lamp containing HfF4
in the region of the 1 Π1 − 1 Σ+ (0,0) band of HfF+ , which has a band originof ∼ 13002.3 cm−1 (cyan line) and an R-branch band head at ∼ 13005 cm−1
(dashed cyan line), as well as other unidentified bands. The feature at ∼13,008 cm−1 is too blue to be the bandhead and lacks a sharp edge on theblue side. Panel (b) shows a region containing three HfF bands identified
using laser-induced fluorescence in a supersonic beam (cyan lines) as well as astrong argon line (red line) and many unidentified features. The dashed cyanlines indicate the approximate location of the R-branch band heads for the
HfF bands, which show reasonable agreement with the data. . . . . . . . . .

12

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3.1

Optical layout for velocity-modulation spectroscopy. The light from the diodelaser is spatially filtered, is split and balanced by polarization optics, andpasses through the discharge in both directions before being focused onto thebalanced photodetector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

16

Schematic of discharge electrode and endcap. (a) Side view of endcap withwater cooled electrode. The hollow stainless steel electrode is connected via a
glass-to-metal seal to the rest of the endcap with a wire spot welded into theinside to provide the electrical connection. (b) Top view of endcaps showing
the Brewster angle windows and ultratorr connections to the central quartz
or alumina tube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

18

Schematic of discharge electronics. A function generator drives two 2 kW
audio amplifiers at ∼10 kHz. Two homebuilt step-up transformers provide theimpedance matching for the discharge. For the diode laser VMS measurements
typical values were a discharge current of ∼100 mApk−pk and ∼22 Vrms acrossthe primary side of the transformer. . . . . . . . . . . . . . . . . . . . . . . .

3.4

20

Strength of hafnium line versus temperature. The 550 ◦ C temperature necessary to produce a reasonable vapor pressure of HfF4 is much lower thaninitially expected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5

21

Compilation of scans covering HfF+ band with isotope averaged fit (offset forclarity). Note that the high rotational temperature (800 K) results in a large
number of lines present. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

xiv
3.6

(a) Confirmation of an Ω=0 to Ω=1 transition. The red arrows point to thelocations of the P(1) and Q(0) predicted by the fit. The visibility of theR(0) and Q(1) lines confirms that the sensitivity is high enough to see theP(1) and Q(0) lines if they were present. (b) Region of HfF+ band in whichisotope shifts cancel. In this region of the spectra the rotational isotope shift
is canceled by the electronic and vibrational isotope shifts. The difference inenergy for an electronic transition for 178 HfF+ relative to 180 HfF+ is given by180∆E = (1− µµ178)Erot +(∆T0,iso +∆ων,iso ), where µ180 and µ178 are the respective

reduced masses, Erot is the rotational contribution to the transition energy,and (∆T0,iso + ∆ων,iso ) is the sum of the electronic and vibrational isotope
shift. From this we see that (∆T0,iso + ∆ων,iso ) ∼ −0.04 cm−1 , which matches
the structure seen in (a) where at the band origin Erot ∼ 0.
3.7

. . . . . . . . .

27

Summary of wavelength ranges and time constants of scans taken with thediode laser. All data was taken at a ∼ 100 mApk−pk discharge current witha voltage drop of ∼ 1 kV across the discharge and a sensitivity of 3x10−7
Hz−1/2 . No HfF+ bands other than the one spanning 12,950-13,000 cm−1
were identified in the wavelength range covered. The low current of these scanscompared to those in Chapter Four limited the size of the HfF+ absorption
signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

xv
4.1

Qualitative sketch of a frequency comb velocity-modulation spectroscopy system. Light from the frequency comb is coupled into a bow-tie ring cavity
containing an AC discharge tube. The AC discharge modulates the ions’Doppler shift and thus the ion absorption signal. The transmitted light isdispersed by a VIPA etalon and a grating to produce a 2-dimensional image
in which every comb tooth is resolved and detected. A fast demodulation
camera performs lock-in detection of each comb tooth at the discharge modulation frequency to recover ion-specific absorption signals across each comb
mode. (Figure courtesy B. Baxley(JILA).) . . . . . . . . . . . . . . . . . . .

4.2

32

Optical layout for frequency comb velocity-modulation spectroscopy. Light
from the frequency comb passes through λ/2 and λ/4 clean-up waveplates
and a optical isolator to achieve the appropriate polarization and prevent
back reflections to the comb. The two lenses are placed to match the modeof the collimated comb light from the fiber to the mode of the optical cavity
at the input couplers (M1 and M4). Depending on the polarization of thelight before the polarizing beam splitter (set by a LCR) light travels either
clockwise or counter-clockwise through the cavity. M2 and M3 are 99.99%
reflectivity mirrors with a 100 cm radius of curvature, and M1 and M4 are
98% reflective flat input couplers. A translation stage and long travel PZTon M4 coupled with a fast PZT on M2 allow for tuning of the cavity length
to match the free spectral range of the cavity to the repetition rate of the
comb. After traveling through the cavity, the outcoupled light passes throughthe AOM (used for measuring the DC intensity on the lock-in camera) andthen to the imaging system. A glass wedge is used to pick-off a small amountof the outcoupled light to measure the transmitted spectrum with a grating
spectrograph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

xvi
4.3

Full comb-spectrum and transmitted comb-spectrum measured with grating
spectrometer. The transmitted spectrum is less than the full comb bandwidth
due to the spectral filtering that results from the two Brewster angle windows
on the discharge tube. By using a grating to select different regions of thefull comb spectrum for producing the locking signal, the spectral region of thetransmitted spectrum can be changed. . . . . . . . . . . . . . . . . . . . . .

4.4

39

Schematic of imaging system. The VIPA provides high resolution, 1 GHz, but
low free spectral range, 100 GHz. The grating resolves the 100 GHz FSR of
the VIPA. The first f = 50 mm cylindrical lens focuses the light into the VIPAat the junction of the anti-reflective and high reflectivity coatings. After the
VIPA, the cylindrical telescope expands the beam to fill the grating. The
f=250 mm imaging lens then maps the differences in wavevector to differencesin position at the detector plane of the camera. The inset shows a picture of
the actual optics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5

41

Image of the transmitted comb light. The comb modes are spaced by therepetition rate of the comb (2.97 GHz) in the vertical direction and by the
FSR of the VIPA (94 GHz) in the horizontal direction. The size of a comb
mode is the same as the pixel size and in the horizontal direction the grating
resolution was chosen such that each FSR is spaced by one pixel to fit the
maximum number of comb modes on the camera. Because of the spacing andthe chirp of the spacing due to the VIPA, comb modes are sometimes split
across more than one pixel as can be seen in the zoomed in portion of the figure. 44

xvii
4.6

Sketch of selection of comb modes by spectrum generator code. In this simplified example, four comb modes fit within one free spectral range of the VIPAand the free spectral range is not an integer multiple of four leading to thechirp in the location in the VIPA direction. The red and blue circles represent
the comb mode locations. The number adjacent to that comb mode represents
the frequency offset of that mode scaled by the repetition rate. The red circles
represent the locations of the comb modes which would be concatenated by
the spectrum generator code. . . . . . . . . . . . . . . . . . . . . . . . . . .

4.7

48

Data Analysis Part 1. (a) The DC intensity is shown as the initial 2-dimensional
image, which is then fit to obtain the comb mode positions as shown in red
circles over part of the image. This fit calibrates the DC intensity as a function of the comb frequency, which is shown on the right. An additional 10
kHz amplitude modulation was applied to the transmitted light to measure
the DC intensity. In (b) the signal at 10 kHz (with only the discharge modulation) is shown for one image. The red circles on the two-dimensional image
are the positions of the comb modes determined from the global fit to the DC
intensity in (a). Note the bright and dark pixels, which represent the spectral
signature of nitrogen absorption. The difference between the clockwise andcounterclockwise signal extracted from the fit for one image is also shown withthe points spaced by the repetition rate of the comb. The running medianplotted with the data is subtracted from each image before they are interleaved. 50

xviii
4.8

Data Analysis Part 2. The resulting processed spectrum (blue) from all theimages (each processed similarly but with stepped cw laser frequency) is shownin (a). The predicted positions from molecular constants reported by Collet
et al. [4] for the (4,2) N+
2 band (red) are also shown in (a). The full recorded
spectrum was acquired in 20 images with 1500 simultaneous channels in lessthan one hour, with a sensitivity of 3 x 10−7 fractional absorption per detection channel. The zoomed-in panels (b) show three different portions of thespectrum with data points indicating the actual sampling period. . . . . . .

4.9

51

Repetition rate of the frequency comb during data acquisition. Each flat
plateau occurs during a measurement of the 1500 simultaneous channels. The
repetition rate is then stepped by scanning the cw laser such that each comb
mode with frequency fn = f0 +nfrep is scanned over 3 GHz to fully sample
the spectrum. As n ∼ 105 , stepping frep by 30 kHz steps the comb mode
frequency by 3 GHz. In the lower panel, the fluctuation of frep during a single
measurement is shown. The 200 Hz drift in frep corresponds to a maximal 300
kHz uncertainty in the comb modes not locked to the cw laser, which is well
below the 30 MHz resolution of the wavemeter. . . . . . . . . . . . . . . . .

54

4.10 Noise contributions as a function of the power in a comb mode. The HfF+
data was acquired with comb mode powers ranging from 1 nW to 20 nW.
The sensitivity of the system is given by the fractional noise divided by afactor of 30 due to the enhanced path length. The values for the camera noise
and technical noise are for switching rapidly, ∼ 3X s−1 , between clockwise
and counterclockwise directions of propagation as single direction measurements stopped averaging down at ∼ 1 Hz bandwidth. (Actual time to switch
directions is ∼ 1 ms, 300 ms is the time for data acquisition.) . . . . . . . . .

55

xix
4.11 Fluorescence from the discharge as a diagnostic tool. The light from the centerof the discharge tube passes through the mirror if it is below ∼ 700 nm and is
focused into a fiber connected to the grating spectrometer. (a) N2 signature
versus a He only discharge. The sharp spikes in the He only case are atomic Hetransitions. Note the suppression of the He fluorescence when large amountsof N2 are present. The clear N2 signature can also be used when checking
for leaks as it is quite distinctive albeit smaller even for significantly lower
partial N2 pressures. (b) Spectral signature the corresponds to the presenceof HfF+ is the broad background lifting off of the baseline. The additional
sharp features are mostlhy atomic transitions of fluorine. . . . . . . . . . . .

58

4.12 Schematic of the system. An 18 W Coherent Verdi pumps both the CW
Ti:Sapphire laser, used as a frequency reference, and the 3 GHz repetition
rate Ti:Sapphire comb. The CW laser is read by a wavemeter with an accuracy of 30 MHz. The comb is referenced to the CW laser via a beatnote
with a single comb tooth. Feedback to the comb’s repetition rate is achieved
through the pump power (high bandwidth and small range) and through the
comb cavity length (low bandwidth and large range). 95% of the comb lightand 5% of the CW light are combined in a fiber which is coupled to theenhancement cavity. Liquid crystal retarders and a polarizing beam splitterare used to select the direction of light propagation around the enhancement
cavity while maintaining all locks. The enhancement cavity is locked to the
comb. Cavity transmitted light is dispersed by the 2-dimensional imaging system, qualitatively sketched in Fig. 4.6, and recorded by the lock-in camera,
which demodulates each pixel at the discharge modulation frequency. The
CW light serves as a reference marker on the lock-in camera image. . . . . .

59

xx
4.13 Full spectral region scanned with HfF+ present. There are four HfF+ bands
that have been identified so far and these are highlighted in Figure 4.14. The
isolated sharp features are most likely due to atomic transitions. The increased
fractional signal at both edges is due to the camera noise starting to be thedominant noise source due to the decrease in comb power at the edges of the
comb bandwidth. This summary plot was generated by interpolating multiple
scans onto a 0.001 cm−1 grid and then averaging them together. . . . . . . .

61

4.14 Region of spectrum with HfF+ lines with smaller regions showing the characteristic velocity-modulation lineshape. Fits to the bands identified are discussed in Chapter Five. In the lower plots, the structure surrounding thestrongest lines is due to additional HfF+ molecular absorption. The fractional
sensitivity measured without HfF+ present is 3x10−7 . . . . . . . . . . . . . .

62

4.15 Improvement of signal-to-noise ratio as a function of splitting ratio. The
signal-to-noise ratio improves dramatically as the intensity for the two directions of propagation becomes more similar as long as common-mode technicalnoise dominates the noise. If the single direction measurements are dominatedby the shot-noise, the signal-to-noise ratio for coherent subtraction is the sameas single direction propagation. If the camera noise begins to dominate dueto the decrease in power as the splitting ratio approaches 50%, the signal-tonoise ratio will decrease. The challenge of achieving a splitting ratio of better
than 52%/48% also limits the gains in signal-to-noise ratio improvement. . .

64

xxi
4.16 Comparison of coherent subtraction and single direction measurements The
signal to noise for coherent subtraction (using a 52/48 beam splitter) surpasses that of single direction measurements by a factor of 10 in cases where
we have sufficient optical power to dominate over the camera readout noise.
Both measurements of a single N+
2 line were made with the cw Ti:Sapphirelaser. The dashed lines are a calculation of the expected lineshape. The
slight modification in the lineshape for the coherent subtraction is due toan additional differential phase shift of approximately π/20 between the twocounter-propagating beams. . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1

65(a) Data and fits to the four HfF+ bands identified with frequency comb
velocity-modulation spectroscopy. Linestrengths for the fit positions are scaledby isotopic abundance, by an approximate rotational temperature of ∼ 640
K, and by an overall transition strength to match the measured fractional
absorption. (b) Region of the recorded spectra containing the 1 Π1 − 1 Σ+ (0, 1)
band as well as at least three other HfF+ bands with the fit for the 1 Π1 −1

Σ+ (0, 1) band offset for clarity. (c) All five isotopes of HfF+ are present andresolved for transitions with ∆v 6= 0. The additional structure with HfF+
present (blue) consists of lines belonging to the other HfF+ bands while thered line (HfF+ absent) shows the level of sensitivity of the system in thisregion.
5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

Residuals from the fit to the 1 Π1 − 1 Σ+ (0, 1) band of 180 HfF+ . The fit spans∼100 cm−1 reaching J values as high as 70 due to the high rotational temperature. This fit provides high accuracy in the determination of the molecularconstants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

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6.1

Identified and predicted HfF+ rotational band locations from 11,000 to 15,000
cm−1 (909-668 nm) for up to v0 and v00 equal to 3. The relative intensity is
scaled by the predicted dipole moment, by thermal factors assuming an approximate vibrational and electronic temperature based on a rotational temperature of 640 K, and by the Franck-Condon factors. The relative intensities
for the four bands measured with frequency comb velocity-modulation spectroscopy agree with the experimental data and the uncertainty in the relative
intensity for bands with no experimental data is a factor of 3-4. The colorof the point indicates the lower state of the band, either 1 Σ+ (blue) or 3 ∆1(red), while the shape of the point indicates the upper state of the band. The
five identified HfF+ bands discussed in Chapter Five are shown in green withthe outline bolded. For those bands which arise from states for which noexperimental information about molecular constants exists, crude error bars
for the energy uncertainty have been added. The gray region represents the
spectral-range covered and relative sensitivity with frequency comb velocitymodulation spectroscopy. Even with the four bands of HfF+ accounted forin the spectral region covered with the frequency comb, we still have ∼100
lines unidentified in the 12,100 to 12,200 cm−1 corresponding on this scale
to a relative intensity of 2x10−3 . While we have been unable to assign theselines due to the overlapping bands, it is possible that they belong to a fifth
3−1
HfF+ band such as the 3 Σ+to the blue or
0 − ∆1 band predicted 1000 cm

the 3 Π2 − 3∆1 band shifted up from the predicted relative intensity currently
off the lower edge of what is plotted here. . . . . . . . . . . . . . . . . . . . .

79

Chapter 1
Introduction”At first sight, the search for an edm in a molecule would seem unprofitable.” – Sandars,
1967 [5]1.1Electron EDM as a probe of physics beyond the Standard Model
The existence of an electron electric dipole moment breaks both time invariance andparity invariance, as shown in Figure 1.1. The size of the electron EDM predicted by theStandard Model is tiny, while various extensions to the Standard Model predict values thatare more than 10 orders of magnitude larger [1]. (See Table 1.1 for a list of predictedvalues.) Because of this, electron EDM measurements do not have a Standard Model background that needs to be accounted for. This is in contrast to experiments such as the parity
non-conservation measurements using cesium, where there is a non-zero Standard Modelcomponent [6]. The zero background nature of the electron EDM in the Standard Model
combined with the current predictions for the extensions to the Standard Model mean thata reduction in the current limit (1.05x10−27 [9]) by an order of magnitude or two would put
serious constraints on those models.
Most electron EDM experiments using atoms and molecules are conducted in the sameway. (A summary of the state of electron EDM experiments is given in R. Stutz’s thesis [7].)An electron spin resonance experiment to determine the spin-flip frequency is performed
between two spin states with an electric and usually a magnetic field present. The sign of

2

d

s

P

-P

-++

T

-T

++
--

d s

++
-d s

Figure 1.1: An Electron EDM violates both time and parity invariance. The dipole momentof the electron must lie either parallel or anti-parallel to the electron’s spin. Here the electron
EDM is represented by the blue arrow labeled d and also graphically by the charge separationon the sphere. The electron’s spin is represented by the red arrow labeled s and graphicallyby the white arrow representing a ring of current. Under a reversal of time, the sign of thespin is reversed while the sign of the electron’s dipole moment stays the same. Under areversal of parity, the sign of the dipole moment is flipped while the spin stays the same.
The amount of CP-violation and thus T-violation given CPT invariance required to give a
non-negligible electron EDM cannot be accounted for with the Standard Model.

3
Model
Standard Model
Canonical CP Violation
Left-Right Symmetric
Higgs dynamics
Supersymmetry

delectron [e-cm]< 10−40
10−26
10−26 − 10−28> 10−27< 10−27

Table 1.1: Summary of predictions of electron EDM values for various extensions to theStandard Model. [1]the magnetic field or electric field is then flipped and the splitting between the two spinstates is measured again; if there is a permanent electron EDM, the spin-flip frequency isdifferent for the magnetic and electric fields configured as parallel versus antiparallel [8]. In√
this type of experiment the sensitivity figure-of-merit is Eef f τ N , where Eef f is the electricfield the electron experiences, τ is the spin resonance coherence time, and N is the numberof spin-flips that can be measured in a reasonable amount of time. Recent results from the
Hind’s group with YbF have lowered the current limit to 1.05x10−27 e-cm [9] from the limitof 1.6x10−27 e-cm on the electron EDM that was set by the Cummins group in 2002 using a
Tl beam [10].
1.2

The Advantage of Molecular Ions and the JILA eEDM experiment√
To improve upon the current limit, the figure-of-merit Eef f τ N must be increased.

In 1967, Sandars first proposed using molecules for EDM measurements due to the verylarge effective electric fields that could be achieved [5]. While molecules offer the prospect ofgreatly increased effective electric fields, the additional complexity of their rich structure also
poses additional experimental challenges. A number of experiments are currently underway
using neutral polar diatomic molecules [9, 11, 12, 13, 14], and very recently, the Hinds group
set a new limit on the electron EDM using YbF [9]. A table of recently completed andongoing eEDM experiments, Table 1.2, demonstrates different approaches to improving the4
Table 1.2: Comparison of recent and ongoing eEDM experiments (ongoing experiment values often represent order of magnitude estimates).
The figure-of-merit for evaluating the√sensitivity of an eEDM experiment is Eef f τ N , where Eef f is the effective electric field on
the unpaired electron, τ is the coherence time, and N is the number of measurements thatcan be counted in some reasonable experimental integration time.
location
Berkeley
Amherst
LBNL
Texas
Penn State
Yale
Imperial
Oklahoma
ACMEMichigan
JILA

ref
[10]
[15]
[16]
[17]
[11, 18, 19, 20, 21][8, 9][12, 22, 23][13, 24][14][25]

species
Tl
Cs
Cs
Cs
Cs
PbO
YbF
PbF
ThO
WC
HfF+ -ThF+

Elab [V/cm] Eef f [V/cm] τ [s]5

1.23x10
4x103
105
105
105
10
7x104
7x104
102
5

7

7x10
4.6x105107
107
107
2.5x1010
2.9x1010
2.9x1010
1011
5.4x1010
3-9x1010

N [1/s]−3

2x10
1.5x10−211
1
8x10−510−3
2 x 10−3
10−3
0.2-1

9

10
109
105
∼10

eEDM limit [e-cm]
1.6x10−27
5.5x10−26
1.9x10−22
10.5x10−28-

figure-of-merit for experiments using both atomic and molecular systems.
To achieve large, i.e. GV/cm, effective internal electric fields, the molecule must bepolarized. For some molecules, such as YbF, rotational states of opposite parity must be
mixed to fully polarize the molecule, leading to a requirement of kV/cm lab electric fields [8].
The lab electric field needs to fully polarize a molecule does not necessarily need to be thatlarge. Molecules with an electronic state containing closely spaced levels of opposite parity
due to the coupling of the electronic angular momentum to the rotation of the molecule,Ω-doubling, can be fully polarized in fields as small as a few volts per cm [26].
The presence of Ω-doubling offers another key advantage over atomic systems – aninternal chop. First realized by DeMille in their use of the 3 Σ+ state of PbO [11], the upperand lower manifold of the Ω-doublet have an identical Zeeman splitting as long as the gfactors are not significantly impacted by mixing with different higher lying states. However,the splitting due to the electron’s electric dipole moment interacting with the effective electricfield is equal in magnitude for both the upper and lower manifolds, but opposite in sign.
Molecular ions offer the additional promise of long coherence times since they are5easier to trap than neutral molecules. As long as the molecule has the desired ground stateor long-lived metastable state, a given molecule can be interrogated for longer than in abeam experiment, although at the cost of a reduction in number of spin-flips measured
simultaneously (N). (See Table 1.2.) For molecular ions, a state which is polarizable with
only a few volts per cm is necessary due to the additional challenges of working in a trapwith rotating fields that limit the electric fields that can be applied. Meyer et al. took theconstraints of working with molecular ions into account and proposed using molecular ionswith 3 ∆1 ground states or low-lying metastable states, specifically HfH+ , HfF+ and PtH+
[26]. Further consideration of 3 ∆ state molecules has also led to the proposal to use ThF+
[24]. Since HfF+ and ThF+ have been successfully created and trapped, the JILA eEDM
experiment has chosen to work with these two species [25, 7]. Many additional technologicalchallenges that come with doing precision measurements using molecules inside an ion trap
are discussed in Leanhardt et al. [25] as well as Stutz [7].
1.3

The Trouble with Molecular Ions: The Need for Spectroscopic Information
One of the challenges in using molecular ions for precision measurement is a lack ofavailable spectroscopic data. For the eEDM experiment, multiple states will be needed forstate preparation and spin readout. Figure 1.2 shows a schematic of some of the energy
levels of interest for HfF+ . Knowledge of the exact energies of these states will guide design
choices for other aspects of the eEDM experiment. For example, if the 3 Φ2 of HfF+ state is
too low in energy, the quantum efficiencies of photomultiplier tubes (PMTs) will make using
laser-induced fluorescence impractical.
When this work was begun, no experimental data existed for either HfF+ or ThF+ . Recent work using pulsed-field-ionization zero-kinetic-energy measurements (PFI-ZEKE) fromthe Heaven group has provided information about the lowest lying states of HfF+ [3] andThF+ . Theoretical calculations exist for the states of HfF+ , but due to the challenges of

6including so many electrons in the calculations, the uncertainties are thousands of wavenumbers [2, 27]. We chose to focus on HfF+ first rather than ThF+ due to the fact that thestates of ThF+ are even less certain.
The large theoretical uncertainties in the energy levels of HfF+ and ThF+ coupledwith the need for information about multiple electronic states necessitates a method ofspectroscopy that covers a broad spectral bandwidth. The requirement for sufficient signal
strength to cover a wide spectral range in a reasonable amount of time puts a lower limiton the number of ions needed. As a result, spectroscopy of HfF+ or ThF+ needs to occur in
an apparatus other than the experimental apparatus for the electron EDM measurement as
the latter was designed for precise control of a small population of ions [25, 7]. Furthermore,
many methods of producing molecular ions also produce their neutral counterparts often inhigh concentrations. Spectroscopic tools that discriminate between ionic and neutral species
reduce crowding of the spectrum and prevent strong neutral transitions from overwhelming
weak ionic signals. With molecular transitions, the ability to resolve individual rotationallines is key to state identification, and thus high resolution is needed.
1.4

Overview of Thesis
Chapter Two briefly covers our unsuccessful attempt to perform low-resolution spec-

troscopy without any ion-discrimination using a hollow cathode lamp of HfF+. Chapter
Three discusses velocity modulation spectroscopy and our first measurements of electronictransitions in HfF+ . Chapter Four details our development of a novel spectroscopic technique that provides high-resolution, broad spectral-bandwidth, ion-discrimination and highsensitivity. Chapter Five summarizes our current knowledge about the structure of HfF+
and its implications for the JILA eEDM experiment. Chapter Six concludes the thesis.

7pre-dissociative states
3

F
2

1

P
1
3
P
1
(a)

3

+
P
0
(b)

3
1

D
1

+
“science state”
S

not to scale
Figure 1.2: Energy levels of HfF+ relevant to the eEDM experiment. Two different schemes
for spin readout, state selective photodissociation (a) and laser induced fluorescence (LIF)(b), are shown. States shown as dashed lines including the 1 Σ+ ground state may be utilized
for state preparation. The exact energies of these states will help to determine the best
method for spin readout.

Chapter 2
Spectroscopy with a Hollow Cathode Lamp

Hollow cathode lamp (HCL) spectroscopy provides very broad spectral bandwidthwith a minimal amount of technological investment. After the need for a method of ion
spectroscopy outside of the electron EDM measurement apparatus was realized, these properties made HCL spectroscopy an appealing choice. An initial attempt at spectroscopy of
HfF+ was therefore conducted by resolving the emission from the hollow cathode lamp witha grating monochromator. Four thousand wavenumbers around 14,500 cm−1 were scannedwith a resolution of ∼ 0.6 cm−1 (0.04 nm) to search for transitions arising from HfF+ .
Although transitions were seen that corresponded to locations of Hf and possibly HfF transitions, the remaining unassigned features proved too complicated to extract the locationsof transitions of HfF+ . A restrospective comparison of the data with a now known HfF+
transition illustrates this difficulty. (See Figure 2.3 and associated discussion.)
A schematic of the system is shown in Figure 2.1. A hollow copper cathode was placedin a glass tube with a stainless steel anode on either side. In order to minimize contaminationof the spectra with copper lines and to avoid damage to the copper cathode, a hafnium sleeve
was press fit inside the copper cathode and then a small amount of HfF4 powder was added.
The discharge heats the HfF4 , creating a sufficient vapor pressure, which is then ripped apartto form HfF, HfF+ , as well as Hf and Hf+ . (With the hafnium sleeve absent, atomic copperand atomic copper ion transitions were both observed in the emission spectra.) Argon was
continually flowed through the discharge cell with a pressure of approximately 750 mtorr.

9

350 V
1A

grating
monochromator

H 2O
HfF4
AC
motor

Hf sleeve
copper cathode

PMT

Figure 2.1: Qualitative sketch of the hollow cathode lamp apparatus. The glow from thedischarge at the center of the hollow cathode (orange square) is imaged into the grating
monochromator. As the AC motor scans the grating, the light at different wavelengths isimaged onto the photomultiplier tube and recorded by the oscilloscope. The cross-section of
the cathode on the left shows the Hf sleeve press fit inside the copper cathode with the HfF4
powder placed inside.

10
Four 100 ohm ballast resistors each capable of handling 50 W were used to provide a total
ballast resistance of 100 ohms with the capacity to handle 200 W in order for the discharge
to run stably at 350 V with 1 A of current. The large amount of power dissipated required
water cooling of the copper cathode.
The fluorescence that passed through the exit slit of the monochromator was collectedwith a photomultiplier tube (Hamamatsu R3896). The slits on the monochromator wereclosed as far as possible to achieve the highest possible wavelength resolution, 0.04 nm. An
external motor was connected to the handle of the monochromator to scan it in a controlled
fashion and the PMT output was recorded with an oscilloscope. The NIST table of strong
atomic lines for argon, helium, and copper were used to calibrate the monochromator leading
to a frequency accuracy of ∼1.5 cm−1 (0.1 nm).
The complete frequency range scanned over is shown in Figure 2.2. In the spectral
region covered, numerous transitions from argon, singly charged argon, fluorine, hafnium,and zirconium (the most significant contaminant in the hafnium sleeve) listed in the NIST
strong lines tables are present, yet they account for only a fraction of the total features
observed. One of the challenges of doing spectroscopy this way is that there is no way todiscriminate between the signal from neutral molecules and atoms, such as HfF or Hf, andionic molecules and atoms, such as HfF+ or Hf+ . Furthermore, the low resolution, 0.04
nm, prevents resolution of individual rotational transitions for a given molecular electronictransition and makes discrimination between atomic lines and molecular bandheads difficult.
Figure 2.3 shows two separate regions of the recorded spectra that illustrate some of
the challenges in assignment of the measured lines. In Figure 2.3 (b), the locations of theband origins of two HfF bands (solid cyan line), obtained using laser induced fluorescence ofa beam of HfF at a later date, allow for the tentative identification of the peaks immediately
to the blue as the R-branch band head of those transitions (dashed cyan line). The highrotational temperatures expected in the discharge suggest that most prominent feature will
be due to the band head, in this case of the R-branch to the blue (dashed cyan line), and not11Intensity [arb. units}

1.5

1

0.5

0
12,50013,50014,50015,50016,500-1Frequency [cm ]data

HfF and HfF+ band origins

Transitions from NIST strong lines
table for Ar, Ar+, F, Hf, and Zr

HfF and HfF+ approximate
location of the R branch bandhead

Figure 2.2: Fluorescence from the hollow cathode lamp versus frequency. Locations of stronglines from the most likely atomic sources of fluorescence in this frequency range as well as
several HfF bands and one HfF+ band at 13,000 cm−1 are also shown and are offset forclarity. The position of both the band origin and the R-branch band head are indicated for
the HfF and HfF+ bands found using other spectroscopic techniques. Note that a majorityof the features shown are not accounted for by the atomic lines published in the NIST stronglines tables.

12

(b)
0.1

Intensity [arb. units]

Intensity [arb. units]

(a)

0.08
0.06
0.04
0.02
0

12,99013,00013,010-1Frequency [cm ]13,020

1
0.8
0.6
0.4
0.2
0
14,20014,30014,40014,500-1Frequency [cm ]Figure 2.3: Regions of hollow cathode lamp spectra that contain known HfF+ (a) or HfF (b)bands. Panel (a) shows data from hollow cathode lamp containing HfF4 in the region of the1Π1 − 1 Σ+ (0,0) band of HfF+ , which has a band origin of ∼ 13002.3 cm−1 (cyan line) andan R-branch band head at ∼ 13005 cm−1 (dashed cyan line), as well as other unidentified
bands. The feature at ∼ 13,008 cm−1 is too blue to be the bandhead and lacks a sharpedge on the blue side. Panel (b) shows a region containing three HfF bands identified using
laser-induced fluorescence in a supersonic beam (cyan lines) as well as a strong argon line
(red line) and many unidentified features. The dashed cyan lines indicate the approximate
location of the R-branch band heads for the HfF bands, which show reasonable agreementwith the data.

13
the Q-branch around the band origin (solid cyan line). Despite the identification of thesefeatures, in the 300 cm−1 spanned by the plot in Figure 2.3, there are still approximately
nineteen features unassigned.
Looking backwards from our velocity-modulation experiments (See Chapter 3), in Figure 2.3 (a), we can also examine the frequency range of the data for the 1 Π1 −1 Σ+ (0,0) band.
Although multiple features are present around 13,000 cm−1 , there is not a band present that
corresponds to the HfF+ band whose band origin is indicated by the solid cyan line andwhose R-branch band head is indicated by the dashed cyan line. The lack of a bandhead
near the HfF+ bandhead in the HCL data suggests that the sensitivity in these scans was too
low to measure electronic transitions of HfF+ , especially since the 1 Π1 −1 Σ+ (0,0) transitionis one of the strongest [2]. The gain and integration time per point were chosen such thatthe many unassigned features did not saturate the PMT and thus could have been increased.
Increasing the sensitivity would not have reduced the forest of unassigned features and so
only small windows of the spectral region covered would yield additional information.
Although the hollow cathode lamp provided broad spectral coverage quickly with someevidence for HfF production, the low resolution and lack of ion-specificity made this a poor
spectroscopic tool for identifying electronic transitions of HfF+ . These measurements highlighted the need for ion-specific, high resolution, and high sensitivity measurements across a
broad spectral range. Chapters three and four discuss our attempts to answer this need.

Chapter 3
Velocity-Modulation Spectroscopy with a Diode Laser

One of the great challenges of spectroscopy of ionic molecules is the presence ofabundant neutral species. The development of velocity-modulation spectroscopy (VMS) in
Richard Saykally’s group [28] overcame this problem by providing ion-specific spectroscopy
as well as high sensitivity. Over the past thirty years, this technique has been used andrefined by many groups to achieve high sensitivity for the study of a wide range of molecularions in the visible and infrared [29].
To achieve high sensitivity and ion specificity, velocity-modulation spectroscopy relieson the modulation of the ions’ drift velocity in an alternating current discharge. The modulation of the drift velocity of the ions relative to a laser passing through the sample ofions results in a modulated Doppler shift and thus in a modulated absorption. After lock-in
detection at the discharge frequency, the absorption signal has a dispersive-looking lineshapewith the zero-crossing at the transition frequency. The characteristic lineshape of velocity
modulation spectroscopy can be seen in Figure 3.6 (b).
The ionic and neutral species are both produced at twice the discharge frequency by thecreation of the discharge every half cycle. Asymmetries in the discharge configuration can
result in a non-negligible absorption signal from neutral molecular and atomic transitions at
the discharge frequency. Counter-propagating beams can be used to remove this commonmode absorption signal as well as other common-mode technical noise as the sign of thedemodulated ion absorption signal depends on the instantaneous velocity of the ions relative

15
to the direction of propagation of the laser.
3.1

Experiment Design
Velocity-modulation requires the detection of absorption in a laser that passes throughthe sample of ions modulated by an alternating-current discharge. Figure 3.1 shows the
optical layout of our system to achieve this. A SDL diode laser tunable from 769 - 792 nm is
split with a polarizing beam cube such that it passes through the discharge in both directions
with the beams crossing in the center of the discharge. Each beam is then focused onto abalanced photodetector (ThorLabs PDB130A). Polarization optics are used to balance thepower of the beams to approximately one part in a thousand. The difference signal between
the two beams is demodulated at the discharge frequency (10 kHz) with a SRS lock-in
detector and the analog output is recorded by the computer. The wavelength for each point
was recorded by a wavemeter that was occasionally calibrated against the rubidium D2 lineat ∼780 nm.
To further reduce the noise caused by background drifts and electrical and opticalpick-up from the discharge, we implemented a double demodulation scheme following Lan,
Tholl, and Farley [30]. A chopper wheel running at 10 Hz was added to the optical path.
The difference signal was then first demodulated on one lock-in at the discharge frequency(10 kHz) with a short enough time constant (1 ms) to be insensitive to the chopper wheel.
That demodulated signal was then fed to a second lock-in detector and demodulated at the
chopper frequency of 10 Hz with the time constant set to reach the desired sensitivity (0.3 10 s).
A key component of a velocity-modulation spectroscopy scheme is the discharge as it
both produces the sample of interest and provides the modulation of the ions’ drift velocity.
The discharge apparatus consists of a center tube made of either alumina or quartz to withstand the oven temperature and of two endcaps each housing an electrode. (See Figure 3.2.)
The center tube is housed within an oven for the production of a sufficient vapor pressure

16

Wavemeterf = 20 mm

25
f=

f=

50

Thorlabs
PDB130Ammmm

f = 20 mm

SDL

l
/2
ISO

BS

PBS
ChopperWheel

10 m
m pinhole
3 mm pinhole

Figure 3.1: Optical layout for velocity-modulation spectroscopy. The light from the diodelaser is spatially filtered, is split and balanced by polarization optics, and passes through thedischarge in both directions before being focused onto the balanced photodetector.

17of HfF4 . Several generations of endcap and electrode designs were tested before the final
configuration was settled upon. The discharge assembly needed removable endcaps so thatthe HfF4 could be place inside the central tube. (The initial oven could only accommodatea straight tube.) Concerns about needing to clean or replace the windows on the endcaps
also motivated a design in which the endcaps were easy to remove. Based on previous design
attempts, we knew the electrode surface needed to be far enough from the glass of the endcap
to prevent melting of the glass from localized heating due to the discharge. Water cooling of
the electrode was also needed to prevent excessive heating of the glass-to-metal seal and toprevent sputtering of the stainless steel. The endcaps also needed connections for the helium
buffer gas that needed to be continuously flowed through the system.
To meet these requirements, a hollow stainless steel electrode with a wire spot-weldedto the inside was connected via a glass-to-metal seal to a glass flange (Figure 3.2 (a).) The
glass portion of the electrode can then be clamped to the endcap to achieve a seal that holds
vacuum. The stainless electrode is one inch in diameter and the BK7 glass tube of the endcap
that surrounds it is 2 inches in diameter to provide a sufficient distance between electrodeand glass. The two inch diameter BK7 tube was then fused onto a two inch diameter tube
in a tee configuration; the matching diameter was chosen to ensure the robustness of theendcap. The two inch diameter tee is then tapered down to a half inch outer diameter tube
on either end so that it can be attached to the central discharge tube and to reduce theprobability of the window getting coated. At the window end of the endcap, a 83 ” thick pieceof BK7 is attached with a high temperature wax (Crystalbond). On each endcap there arealso two threaded connectors for the gas flow. On both endcaps, gas flows in on the connector
closest to the window to help keep it clean. On one endcap, the second gas connection serves
as the main inlet and on the other, it serves as the outlet. (See Figure 3.2 (b).)
A schematic of the discharge electronics is shown in Figure 3.3. A function generator
drives two 2 kW commercial audio amplifiers at ∼10 kHz. Two homebuilt step-up transformers with a turn ratio of 25:1 provide the impedance matching to drive the ∼ 2kΩ discharge

18

+V

(a)

H 2O
H2O

(b)

oven region

He

He

He

Figure 3.2: Schematic of discharge electrode and endcap. (a) Side view of endcap with watercooled electrode. The hollow stainless steel electrode is connected via a glass-to-metal seal
to the rest of the endcap with a wire spot welded into the inside to provide the electricalconnection. (b) Top view of endcaps showing the Brewster angle windows and ultratorr
connections to the central quartz or alumina tube.

19
given the 8 Ω output impedance of the audio amplifiers. (The impedance matching is imperfect due to changes in the discharge impedance from the initial design.) The transformers
on the secondary side are wired such that the voltage drop from each electrode to groundis one-half the voltage drop across the discharge. 100 Ω ballast resistors are used on the
discharge side as well as two sense resistors of 1 Ω that are used to monitor the discharge
current. A low value of 100 mApk−pk was chosen due to a concern that higher currents would
result in the HfF+ molecules being ripped into their atomic constituents. (Later measurements discussed in Chapter 4 used much higher currents ranging from 250 to 500 mApk−pk .)For these measurements the voltage across the primary side of the transformers was ∼ 22
Vrms leading to a voltage drop of ∼1.1 kV across the discharge.
3.2

Characterizing the System
Since no electronic transitions of HfF+ had been measured at the time of the design ofthe velocity-modulation system, a previously studied N+
2 band as well as a neutral hafnium
transition were used for testing. Production of N+
2 ions allowed for determination of thedischarge settings and for the phase of the lock-in detection. Multiple lines from the (7,4)
A2 Πu −X2 Σ+g band previously recorded by Collet et al [4] and Liu et al [31] were measured
as a first test of the system. The N+
2 lines enabled us to test everything except for the
production of HfF4 vapor and thus HfF+ with the oven.
In order to characterize the oven performance, we used a transition in neutral hafnium
at 784.8 nm. In order to be able to see a neutral transition, we rely on concentrationmodulation instead of the velocity-modulation of the ions. Since the discharge is responsible
for producing the HfF, HfF+ , Hf and Hf+ from the HfF4 vapor, the abundance of these
species and thus the absorption from them is modulated at twice the discharge frequency(2x10 kHz). Asymmetries in the discharge may even produce a concentration-modulation
signal at the discharge frequency, which is suppressed by using counter-propagating beams
as the concentration-modulation is common mode between the two beams. To deliberately

20

100 W
1W

1W
100 W
V

V

25:1 step -up
transformer
V

25:1 step -up
transformer
V

2 kW audio
amplifier~2 kW audio
amplifierfunction
generator

Figure 3.3: Schematic of discharge electronics. A function generator drives two 2 kW audio
amplifiers at ∼10 kHz. Two homebuilt step-up transformers provide the impedance matchingfor the discharge. For the diode laser VMS measurements typical values were a discharge
current of ∼100 mApk−pk and ∼22 Vrms across the primary side of the transformer.

H f peak signal [nW ]

21
300

200

100

0

400

500

600

700

Tube Furnace Temperature [C]Figure 3.4: Strength of hafnium line versus temperature. The 550 ◦ C temperature necessary
to produce a reasonable vapor pressure of HfF4 is much lower than initially expected.

measure the absorption from the concentration-modulation of hafnium at twice the dischargefrequency, we therefore must detect only one of the beams rather than the difference of thecounterpropagating beams. The strength of the hafnium line was then measured as a functionof oven temperature to determine the appropriate operating temperature of 550 ◦ C as shownin Figure 3.4. As one source quoted a sublimation temperature of HfF4 of 968 ◦ C [32] and we
could not obtain a vapor curve, we purchased an oven capable of reaching 1500 C anticipatinga need for high temperatures to produce a reasonable vapor pressure of HfF4 . The much
lower temperature of 550 ◦ C eventually allowed for the use of more compact home-built
ovens. These temperature measurements were later repeated with one of the HfF+ lines toconfirm that the optimal operating temperature for HfF+ measurements was the same as for
Hf.
The sensitivity of the velocity-modulation system was characterized both with andwithout the discharge running. With the discharge off, the fractional noise was 8x10−8 Hz −1/2 ;while with the discharge on, the fractional noise is 3x10−7 Hz −1/2 . This corresponds to approximately a factor of 5 above shot-noise with the discharge off and a factor of 20 above
shot-noise with the discharge on. The fractional noise with the discharge off is most likely

22
due to air currents. For most of the data shown here, the discharge tube was located 17
inches above the optics table in the high temperature oven resulting in air currents being
particularly problematic. Pick-up at the discharge frequency causes offsets on the lock-inthat fluctuate and thus the noise with the discharge running is greater. For all of the noise
measurements, the noise was greater than the noise-equivalent-power of the detector (15√pW/ Hz).
3.3

First Experimental Data of HfF+ Spectra
Scans with the diode laser were started after the system had been characterized withN+
2 and Hf starting with the wavelength region around 770 nm. HfF4 was loaded into thebottom of the central discharge tube and the oven was heated to 550 ◦ C. Helium buffer
gas was flowed through the discharge continuously such that the total pressure was 5 torr.
For these scans, the discharge current was ∼ 100 mApk−pk and the voltage drop across thedischarge was ∼ 1 kV. A series of lines with first-derivative lineshapes characteristic of ions
was seen near 13,000 cm−1 . The lines disappeared when the oven was turned off, whichmeant it was not due to one of the N+
2 bands located in this wavelength range. This seriesof lines was tentatively identified as belonging to a HfF+ band, which was confirmed by thefit.
Figure 3.5 shows the scans covering the HfF+ band. 60 cm−1 of bandwidth was covered
with rotational transitions with up to J=50 seen as the discharge plus the oven results ina relatively high temperature. The high rotational temperature provides many lines for thefit. Subsequent scans were acquired to see the band origin with higher sensitivity. Highsensitivity around the band origin was necessary to identify the low J lines, which have low
thermal population, so that the values of Ω could be assigned to the upper and lower states.
The electronic states of HfF+ can be approximated as Hund’s case (a) or in the case ofstrongly spin-orbit mixed states, Hund’s case (c) [26]. As described in Ref. [35], for Hund’scase (a), the interaction of the nuclear rotation (N) with the electronic motion, both spin

23(S) and orbital (L), is very weak while the electronic motion is coupled very strongly tothe internuclear axis. For Hund’s case (a), both the orbital angular momentum (L) and thespin (S) process around the internuclear axis with projection Λ and Σ respectively along theinternuclear axis with Ω = Λ + Σ. The total angular momentum, J, is then given by the
coupling of the electronic motion (L + S) to the nuclear rotation (N). Since J cannot havevalues smaller than Ω, for a given Ω, J = Ω, Ω + 1, Ω + 2,... As the rotational energy is
proportional to J(J+1) and transitions from one vibronic state to another are only allowed
for J = 0, +/ − 1, examination of rotational bands for the presence of transitions for low
values of J can thus lead to identification of the value of Ω for the upper and lower states[35].
Hund’s case (c) is very similar to Hund’s case (a) except that in this case the interaction
between L and S is stronger than the interaction with the internuclear axis [35]. In Hund’scase (c), Λ and Σ are not well-defined quantum numbers but rather L and S couple to form
the resultant vector Ja , which has a projection Ω along the internuclear axis. Ja couples tothe nuclear rotation to form the total angular momentum, J as in case (a). Again as in case(a), J cannot be smaller than Ω, leading to J = Ω, Ω + 1, Ω + 2,... and giving us the same
tool to identify Ω for the upper and lower states from a rotational band [35]. Despite thestrong spin-orbit mixing for some of the states of HfF+ and ThF+ such as the 1 Π1 and 3 Π1[64], we still denote all of the states using the molecular term symbol, 2S+1 ΛΩ , as if Λ is agood quantum number.
As hafnium has five isotopes with relative abundances of 35.10%(180), 13.63%(179),
27.30%(178), 18.6%(177), and 5.21%(176) [33], we expected to see some isotope structure in
the measured spectrum. Initially, when this data was acquired, we did not anticipate seeing
the odd isotopes due to the hyperfine structure washing out the lines. Later measurements
discussed in Chapters 4 and 5 demonstrated that the hyperfine splittings are small enoughthat we see all five isotopes.
The presence of multiple isotopes allows us to determine the change in vibrational

24

-6

3

x 10

data
2.5

P
Q
R

2

Fractional Absorption

1.5
1
0.5
0
-0.5
-1
-1.5
-2
12,95012,96012,97012,98012,99013,00013,010-1Frequency [cm ]Figure 3.5: Compilation of scans covering HfF+ band with isotope averaged fit (offset forclarity). Note that the high rotational temperature (800 K) results in a large number of lines
present.

25
quantum number with some assumption of reasonable vibrational energies. In this case,∆v = v 0 − v 00 = 0 as there is no sign of vibrational isotope splitting for ∆v > 0, where v 0 andv 0 are the vibrational quantum numbers of the upper and lower states respectively . Because∆v = 0, we did an isotope averaged fit rather than fitting to lines from only one isotope of
hafnium, i.e. 180 HfF+ , as only in portions of the spectra was there partially resolved isotope
structure. For part of the spectrum, the electronic, vibrational and rotational isotope shifts
cancel or nearly cancel resulting in a single lineshape. (See Figure 3.6 (b).) Attempts to fit
the lineshapes and to extract the positions of each individual isotope were unsuccessful.
Lines belonging to the P (J00 -J0 =1), Q (J00 -J0 =0) and R (J00 -J0 =-1) branches were identified. All three branches were then fit to the following:
E(J 00 , J 0 ) = ν0 +(B 0 −P∗(p+2q))J 0 (J 0 +1)+D0 J 02 (J 0 +1)2 −B 00 J 00 (J 00 +1)−D00 J 002 (J 00 +1)22
(3.1)

where P = −1 for ∆J = 0 and P = 1 for ∆J = +/−1, ν0 is the energy difference between thev = 0 levels of the electronic states, B0 and B00 are the rotational constants of the upper andlower states respectively, (p+2q) is the Ω-doubling of the upper state (this parameterization
is discussed in Ref. [34]), and D0 and D00 are the respective anharmonic distortion termsof the upper and lower state. Higher order Ω-doubling terms were not included as D0 was
already consistent with zero. The values obtained by the fit are given in Table 3.1 and thelocations of the P, Q, and R branches from the fit are shown graphically in Figure 3.5 alongwith a compilation of the data. The values for B00 and B0 are consistent with the species being
HfF+ as opposed to a species with the mass of N+
2 . The rotational constant in wavenumbers
is given byB=h
8π 2 cµre2(3.2)

where h is Planck’s constant, µ is the reduced mass, c is the speed of light in cm/s and re is
the equilibrium bond length [35]. Petrov et al. [2] had calculated values of re ranging from1.78 to 1.87 Å for different electronic states of HfF+ , leading to values of B ranging from

26
Table 3.1: Isotope averaged values of fit of line positions with 1σ errors.
parameter
B00
B0
D00
D0
(p+2q)
ν0value [cm−1 ]0.30489(10)0.28117(10)1(1) x 10−7
1(1) x 10−7
3.55(2) x 10−4
13002.229(3)0.281 to 0.310 cm−1 . A more detailed comparison with theory follows in Chapter 5.
The presence or absence of low J lines allows for the determination of Ω for the upperand lower state. If the R(0) line is present, Ω00 = 0 and Ω0 ≤ 1. If the P(1) line is missing,then Ω0 > 0. Additional confirmation comes from looking at the Q(0) and Q(1) line locations.
If Q(0) is missing, Ω00 > 0 or Ω0 > 0 and if Q(1) is present, Ω00 ≤ 1 and Ω0 ≤ 1. The region of
the spectrum containing the low J lines is shown in Figure 3.6. With R(0) and Q(1) present
and P(1) and Q(0) missing, Ω00 = 0 and Ω0 = 1.
From comparison with theory [2], it would initially appear that the Ω00 = 0 and Ω0 = 1
transitions is the 3 Π1 −1 Σ+ (0,0) transition since the predicted value for T0 is only 300 cm−1
away from the theory prediction. However, the sign of the Ω-doubling corresponds not to
the 3 Π1 − 1 Σ+ transition, but to the 1 Π1 − 1 Σ+ transition [64]. The identification of morestates of HfF+ as discussed in Chapter 4 led to the confirmation that this is actually the1Π1 − 1 Σ+ (0,0) band. The structure of HfF+ as identified so far is discussed in Chapter 5.
3.4

Extending Velocity-Modulation Spectroscopy
The fact that the 1 Π1 − 1 Σ+ (0,0) is 2100 cm−1 from the predicted value [2] highlights

one of the limitations of doing VMS with a cw laser when there is no prior spectroscopic
information. Even with cw lasers that are relatively easy to scan, it takes a long time tocover several thousand wavenumbers of bandwidth. With the diode laser, we sampled small

27

(a)

-6

1.5

x 10

R(24) + Q(3)Fractional Absorption

1
Q(5)
0.5

R(23)

Q(4)
Q(2)

R(0)

Q(1)

0
-0.5
-1

Q(0) missing

P(1) missing-1.5
13,001.5

13,002

13,002.5

13,003-1Frequency [cm ]

(b)
data
3

P

Q

R

x 10-6

Fractional Absorption

2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
12,96012,962

12,964

12,966

12,968

12,970-1Frequency [cm ]Figure 3.6: (a) Confirmation of an Ω=0 to Ω=1 transition. The red arrows point to thelocations of the P(1) and Q(0) predicted by the fit. The visibility of the R(0) and Q(1)lines confirms that the sensitivity is high enough to see the P(1) and Q(0) lines if they were
present. (b) Region of HfF+ band in which isotope shifts cancel. In this region of the spectra
the rotational isotope shift is canceled by the electronic and vibrational isotope shifts. The
difference in energy for an electronic transition for 178 HfF+ relative to 180 HfF+ is given by
180)Erot + (∆T0,iso + ∆ων,iso ), where µ180 and µ178 are the respective reduced∆E = (1 − µµ178
masses, Erot is the rotational contribution to the transition energy, and (∆T0,iso + ∆ων,iso )is the sum of the electronic and vibrational isotope shift. From this we see that (∆T0,iso +∆ων,iso ) ∼ −0.04 cm−1 , which matches the structure seen in (a) where at the band origin
Erot ∼ 0.

28
regions of the accessible bandwidth as is shown in Figure 3.7, but did not fully cover the
wavelength range of the laser. The sampling approach was made to try to leverage the highrotational temperature, which results in each band covering ∼100 cm−1 with lines spaced
by less than 0.5 cm−1 . By sampling the spectrum in non-contiguous 1 cm−1 sections, we in
effect gain information about a much wider frequency range as we would see multiple lines
if there is a band in the vicinity. However, once potential lines are seen, the whole band stillneeds to be scanned out. Instead of pushing to cover a wider energy range with cw lasers, we
switched to frequency comb velocity-modulation spectroscopy which is discussed in Chapter
Four.

0
600

1

2

3

4

5

6

7

8

9

10

650

700

750
-1

850

900
-1

frequency [cm ] - 12,000 cm

800

950

1000

1050

Figure 3.7: Summary of wavelength ranges and time constants of scans taken with the diode laser. All data was taken at a ∼100 mApk−pk discharge current with a voltage drop of ∼ 1 kV across the discharge and a sensitivity of 3x10−7 Hz−1/2 . No HfF+
bands other than the one spanning 12,950-13,000 cm−1 were identified in the wavelength range covered. The low current of thesescans compared to those in Chapter Four limited the size of the HfF+ absorption signal.

time constant [sec]1129

Chapter 4
Development of Frequency Comb Velocity-Modulation Spectroscopy1

The application of optical frequency combs to a variety of spectroscopic techniques
has resulted in ultrahigh sensitivity systems with simultaneous high resolution, absolute frequency accuracy, and spectral bandwidth, all with dramatic reduction in data acquisition
times [37, 38, 39, 40, 41, 42, 43]. Compared to broadband incoherent light sources or tunable cw lasers, frequency combs offer both broad spectral bandwidth and spectrally narrowlines with well-defined frequencies given by fn = f0 + n ∗ frep set by only two parameters, the carrier-envelope-offset frequency, f0 , and the repetition rate, frep [44]. Applicationsof direct frequency comb spectroscopy have included breath analysis [45], tomography ofsupersonically-cooled molecular jets [46], demonstration of high-sensitivity molecular fingerprinting with ammonia [42] and detection of impurities in semi-conductor specialty gases[47]. The development and use of these spectroscopic techniques has been focused on neutral
atomic and molecular species as opposed to ionic species. Prior to the research discussed below, no techniques have been capable of ion-specific high-sensitivity modulation spectroscopy
on every parallel detection channel over a broad spectral range [36]. The integration of velocity modulation spectroscopy [29, 28] and cavity-enhanced direct frequency comb spectroscopy[39, 43] provides a way to achieve very broad spectral bandwidth ion-sensitive spectroscopy
without sacrificing high-resolution or high-sensitivity.
1

Portions of this chapter have been previously published in Sinclair et al. [36].

31
4.1

Overview
High resolution spectroscopy of molecular ions is typically performed using velocity-

modulation spectroscopy (VMS), first demonstrated in Saykally’s group [29, 28]. Velocitymodulation spectroscopy achieves high sensitivity via lock-in detection of the absorption
signal from a laser which passes through a sample of ions with a modulated Doppler shift
due to an alternating current discharge [28]. Although recent experiments have pushed
to higher sensitivity [48, 49] as well as higher accuracy [50, 51], these systems still relyon slow scans of continuous wave (cw) lasers. In addition, broad spectral bandwidth has
been achieved via emission spectroscopy [52, 53], but without the methods of enhancing
sensitivity of the current experiment. Velocity-modulation spectroscopy of HfF+ including
some of the limitations was discussed in Chapter Three. Cavity-enhanced direct frequency
comb spectroscopy has demonstrated broad simultaneous bandwidth with high resolution
and high sensitivity [39, 43], and thus is an ideal candidate to integrate large instantaneous
bandwidth with high-resolution VMS by enabling lock-in detection simultaneously on every
comb tooth.
To integrate these two powerful spectroscopic techniques, several key features areneeded, as sketched in Figure 4.1. An alternating-current discharge tube for the production and modulation of molecular ions must be placed within an optical enhancement cavity,with the cavity length actively stabilized such that the frequency comb light can be coupled
into the cavity. The cavity needs a ring geometry to provide a single direction of propagationof the light relative to the instantaneous motion of the ions, which enables lock-in detection
at the discharge frequency (10 kHz). For frequency accuracy that is not limited by theimaging system resolution, the imaging system must be able to resolve each individual comb
mode. Finally, the detector used must be able to demodulate the intensity of each comb
mode at the discharge frequency of 10 kHz to recover the absorption signal. The design
considerations as well as the acquisition of a thousand wavenumbers of the spectra of HfF+

32

Figure 4.1: Qualitative sketch of a frequency comb velocity-modulation spectroscopy system. Light from the frequency comb is coupled into a bow-tie ring cavity containing an
AC discharge tube. The AC discharge modulates the ions’ Doppler shift and thus the ion
absorption signal. The transmitted light is dispersed by a VIPA etalon and a grating to
produce a 2-dimensional image in which every comb tooth is resolved and detected. A fastdemodulation camera performs lock-in detection of each comb tooth at the discharge modulation frequency to recover ion-specific absorption signals across each comb mode. (Figure
courtesy B. Baxley(JILA).)33
are both discussed in this chapter. The analysis of the HfF+ bands measured is discussed in
Chapter Five.
4.2

Experiment Design

4.2.1

Cavity Design

To increase the effective absorption path length, a ring cavity was placed around thedischarge tube. The ring geometry provides a well defined direction of propagation of thelaser relative to the instantaneous velocity of the ions and allows for the use of counterpropagating beams for additional noise suppression. The existence of the discharge tube
inside the cavity places limitations on the achievable finesse. Additionally, the presence ofthe discharge tube windows causes spectral filtering of the comb bandwidth coupled into thecavity due to group velocity dispersion.
For the cavity, a bow-tie configuration was chosen as shown in Figure 4.2. On eitherside of the approximately 1 meter discharge tube are 100 cm radius of curvature mirrors.
On the short arm of the bow-tie are the flat input and output couplers with 98% reflectivity.
The higher the finesse, the more sensitive the system but issues of intra-cavity losses andgroup velocity dispersion become more problematic. The reflectivity of the input couplers
was chosen as a compromise between matching the losses due to the Brewster angle windows
to maximize the transmitted intensity and maximizing the finesse. The ratio of cavityenhanced signal to single-pass absorption signal for a ring cavity is given by F /π, where F
is the cavity finesse. The finesse of the cavity is given by
√
π grt
F =
1 − grt(4.1)√
where grt = R Tcav is the probability a photon makes one round trip (round trip gain), R is
the reflectivity of the input couplers, and Tcav is the transmissivity of the Brewster windows

34[54]. The ratio of transmitted intensity to incident intensity is given by
Itrans(1 − R)2 Tcav√=
Iinc(1 − R Tcav )2(4.2)[54]. For the current system Tcav ∼ 0.97 due to imperfect alignment of the Brewster angle
windows on the discharge tube and thus F ∼ 88 and Itrans /Iinc ∼ 32%. Itrans /Iinc has
been measured to be ∼ 30% with the single-frequency reference laser (thus avoiding issuesof spectral filtering for the measurement).
The free spectral range (FSR) of the cavity, ignoring intra-cavity dispersion, is given
by FSR ≈ c/L, where c is the speed of light and L is the length of the cavity. We wanted tomatch every comb mode, spaced by 2.97 GHz, to every 25th cavity resonance, which resulted
in a cavity FSR of ∼119 MHz. By making the bow-tie as compact as possible, we achieved
the required optical path length of 254 cm. Fine tuning of the cavity length to match the
comb’s repetition rate is achieved via mounting one input coupler on a translation stage.
Additional, finer control is possible via a long-travel (80 µm) piezoelectric transducer (PZT)
on the translation stage. The long-travel PZT has enough range to keep the cavity lengthand thus the FSR matched to the repetition rate of the comb as the repetition rate is scanned
to fully sample the spectrum.
Once the required optical pathlength and desired cavity configuration were known,
we wanted to confirm that we had designed a stable resonator and to calculate the radiusof curvature and beam diameter of the cavity mode at different locations of the cavity.
Knowledge of the radius of curvature and beam waist at the input coupler is necessary formatching the incident light to the cavity mode. We also wanted to know the size of the beamat the focus inside the discharge tube as well as at the two curved mirrors to get a sense of
the size of the beam through the discharge region.
The so-called ABCD matrices and the q-parameter of the beam can be used to calculate
the waist and radius of curvature of the beam as a function of position. The beam waist,

35
ω(z), at a given position, z, is given by
λ1−Im(q) ==2πω (z)
B

r
1−(

A+D 2
)
2

(4.3)

where λ is the wavelength of the light, and the radius of curvature at a given position, z, isgiven by
Re(q) =

1
D−A
=
R(z)
2B

(4.4)

[54]. For any point in the cavity, we can also define a quantity m = (A+D)/2, where if −1 ≤m ≤ 1, the resonator is stable. In order to find the q-parameter, the ABCD matrix canbe calculated
 n with byusing the propagation through a medium with index of refractiond 1 0
1 n 
, and the reflection from a spherical mirror with curvature R, ,length d, 



2
0 1
1
R
where R is positive if the center of curvature lies in the positive direction of the incident ray

propagation. For example, the ABCD matrix for the focus between the two curved mirrors
is given by:








1
A B  1 2 Ldisch  1


=


 
0
1
0
C D





t
1
nf s  

1(L1 − Ldisch ) − t  1
2

1
0


1 L3 1 L2  1 0 1








−2
0 1
0 1
1
0
R





1




0 1 L2




0 1



−2
R


1
(L1 − Ldisch ) − t 1
2

1



0



t
1
nf s  

1




0

1
L
2 disch 1

where Ldisch = 101 cm is the length of the discharge tube; t=0.1 cm is the thickness of
the window; nf s = 1.5 is the index of refraction of fused silica at 800 nm; L1=110 cm isthe distance between the two curved mirrors (M2 and M3 in Figure 4.2); R=100 cm is theradius of curvature of the two curved mirrors; L2 = 67 cm is the distance between the curved
mirror and the input coupler; and L3 = 7 cm is the distance between the two input couplers.
From this we get that the beam diameter at the waist at the center of the discharge tube
is ∼250 µm, at the curved mirrors it is ∼615 µm, and at the waist between the two input

36
Table 4.1: Distances between optical elements in enhancement cavity and between the modematching optics and the enhancement cavity. Optical elements are labeled in Figure 4.2.
Optical elements

Distance [cm]M2 and M3
M1 and M2
M3 and M4
M1 and M4
f=70 mm lens and f=50 mm lensf=50 mm lens and PBS
PBS and M1
PBS and M4

110
67
67
8
17
47
44
44

couplers it is ∼500 µm. Also, the value of m = −0.92, so the resonator is stable.
To couple light into one mode of the cavity, both the beam waist and radius of curvatureof the incident beam must be matched to the waist and radius of curvature of the cavity
mode at the input coupler. ABCD matrices were used to calculate the propagation of thebeam from the collimated output of the fiber to the input coupler. Matlab’s ability to solve
systems of nonlinear equations was utilized to determine the appropriate distances for twolenses to be placed before the cavity by matching the real and imaginary parts of the qparameter of the cavity to the incident light at the input coupler. Both lenses needed to
go before the beamsplitter so that the beam would be well matched to the cavity for bothclockwise and counterclockwise propagation. (See Figure 4.2.) Two lenses were needed dueto the space constraints and the very gentle focus to match the two parameters.
One of the additional complications of this system is the presence of astigmatism. With
the bow-tie cavity, the angle of incidence for the curved mirrors is 15◦ and thus the effective
radius of curvature in the tangential plane (plane of incidence) is given by Re = Rcos(θ),
while in the sagittal plane (perpendicular to the plane of incidence) it is given by Re =R/cos(θ). For θ = 15◦ and R=100 cm, Re = 104 cm in the sagittal plane and Re = 97
cm in the tangential plane. Additionally, the optical path through the Brewster window is

37

M3

M2

M1

comb +
cw ref

M4
FC

FC
LCR

AO
M

FC

IS
O

LCR
imagingsystem

l
/4
l
/2

PBS
LCR

f=70 mm

LCR
f=50 mmgrating
monochromator
FC = fiber collimator
ISO = Faraday isolator
AOM = acousto-optic modulator
LCR = liquid crystal retarderbreadboard
M1, M2, M3, M4 = cavity mirrors
Polarization optics for controlling clockwise vscounterclockwise propagation not shown

Figure 4.2: Optical layout for frequency comb velocity-modulation spectroscopy. Light fromthe frequency comb passes through λ/2 and λ/4 clean-up waveplates and a optical isolator
to achieve the appropriate polarization and prevent back reflections to the comb. The twolenses are placed to match the mode of the collimated comb light from the fiber to the modeof the optical cavity at the input couplers (M1 and M4). Depending on the polarization of
the light before the polarizing beam splitter (set by a LCR) light travels either clockwise orcounter-clockwise through the cavity. M2 and M3 are 99.99% reflectivity mirrors with a 100
cm radius of curvature, and M1 and M4 are 98% reflective flat input couplers. A translation
stage and long travel PZT on M4 coupled with a fast PZT on M2 allow for tuning of thecavity length to match the free spectral range of the cavity to the repetition rate of the comb.
After traveling through the cavity, the outcoupled light passes through the AOM (used formeasuring the DC intensity on the lock-in camera) and then to the imaging system. A glass
wedge is used to pick-off a small amount of the outcoupled light to measure the transmitted
spectrum with a grating spectrograph.

38
different in the two planes, but this is a smaller effect than the difference in effective radiusof curvature. Despite this difference, the incident light is mode-matched to the cavity using
spherical lenses and as a consequence there will always be some coupling into higher-ordermodes. With a single-frequency laser, i.e. no spectral filtering, 85-90% of the light is coupled
into the lowest order mode (TEM00 ).
In order to determine the spectral bandwidth of the frequency comb that can be coupled
into the cavity simultaneously, we need to reconsider the free spectral range of the cavity
taking intra-cavity dispersion into account. The free spectral range of the cavity is given by
F SR(ω) =c∂φ
L + c ∂ω(4.5)

∂φ
is the intrawhere c is the speed of light, L is the optical path length of the cavity and ∂ω

cavity dispersion [55, 56]. The variation of the FSR as a function of frequency is then given
by∂F SR(ω)∂ 2φ= −F SR(ω)2 2
∂ω
∂ω

(4.6)

2

∂ φ
where ∂ω
2 is the group delay dispersion (GDD). The variation in the cavity FSR is non-negligible primarily due to the presence of the discharge tube windows inside the cavity. The
discharge tube windows are 1 mm thick fused silica, which has a group velocity dispersion(GVD) of 36 fs2 /mm at 800 nm. The fused silica windows replaced the BK7 windows
described in Chapter Three to reduce the window thickness from 3 mm to 1 mm. Using the
GVD for fused silica at 800 nm, an initial FSR of 120 MHz, and a 1 MHz cavity linewidth,
the change in frequency for a cavity resonance to be a cavity linewidth away from the barecavity resonance occurs across a ∼ 6 THz spectral width. As is shown in Figure 4.3, ∼ 6.5
THz (15 nm at 830 nm) of the full comb spectrum is transmitted through the cavity. (Byusing a grating to select the portion of the full comb bandwidth for locking the frequency
comb to the cavity, different 6.5 THz portions of the full comb bandwidth can be coupled
into the cavity.) As is discussed in the next section, the imaging system has a spectral
bandwidth of ∼ 4.4 THz (10 nm at 830 nm) so this is sufficient.

39

1transmitted spectrum
full comb bandwidth
Intensity[arb. units]

0.8

0.6

0.4

0.2

0
760

780

800

820
Wavelength [nm]

840

860

Figure 4.3: Full comb-spectrum and transmitted comb-spectrum measured with grating
spectrometer. The transmitted spectrum is less than the full comb bandwidth due to thespectral filtering that results from the two Brewster angle windows on the discharge tube.
By using a grating to select different regions of the full comb spectrum for producing thelocking signal, the spectral region of the transmitted spectrum can be changed.

40
4.2.2

Imaging System

The cavity-transmitted light is coupled via fiber to a two-dimensional lock-in imaging
system designed for massively parallel detection. A 1 GHz resolution crossed-dispersion
system consisting of a titled VIPA etalon [57, 58] and a grating is used to resolve eachindividual comb mode [40, 45]. A Heliotis C2 “smart pixel” lock-in camera [59, 60], whichuses a combination of CCD and CMOS technology, is used to demodulate and read out
each pixel at the discharge modulation frequency. The lock-in camera uses a microlens arrayto achieve close to a hundred percent filling factor even though the active area is only ten
percent of the total area. The use of the microlens array also makes the camera sensitive
to horizontal tilt and thus careful alignment of the camera is required. A schematic of theimaging system is shown in Figure 4.4.
A VIPA is a high resolution element that consists of a plate with an entrance stripe withan anti-reflection coating, a highly reflective coating above the entrance stripe (R≥99.9%),
and an exit surface that is partially reflective (R∼99.6%). Unlike a usual etalon, all incident
wavelengths are transmitted. Similar to an etalon, the VIPA generates repeating modeorders with the free spectral range set by the plate thickness (94 GHz for this experiment).
A grating is then used to resolve overlapping mode orders of the VIPA, and thus only requires
a resolution at the level of the VIPA FSR. A detailed discussion of the design considerations
for a VIPA etalon and grating crossed-dispersion system is found in Chapter 6 of M. Thorpe’s
thesis [56]. In this system, the first VIPA used was designed for 100 MHz resolution and thus
was coated for F ∼ 1000. Due to the technical challenges of constructing a high resolution
VIPA, the actual resolution was only on the order of 1 GHz, but alignment through it often
resulted in aberrations and in a low efficiency of ∼ 1%. With a second VIPA designed for 1
GHz resolution (F ∼ 100), alignment was much easier and 10% of the light is transmitted
through the VIPA.
The imaging system was designed with the goal of placing a comb mode onto a single

41

Top View
grating

f = 250 mm

cy.l lensf = 75 mm
cyl. lensf = 30 mm

lock-in camera

VIPAcyl. lensf = 50 mm

fiber
collimator

Partial Side View
cyl. lensf = 50 mm

cyl. lens cyl. lens
VIPA f = 30 mm f = 75 mm grating

Figure 4.4: Schematic of imaging system. The VIPA provides high resolution, 1 GHz, but
low free spectral range, 100 GHz. The grating resolves the 100 GHz FSR of the VIPA.
The first f = 50 mm cylindrical lens focuses the light into the VIPA at the junction of
the anti-reflective and high reflectivity coatings. After the VIPA, the cylindrical telescope
expands the beam to fill the grating. The f=250 mm imaging lens then maps the differencesin wavevector to differences in position at the detector plane of the camera. The inset showsa picture of the actual optics.

42
pixel, which has dimensions of 86.3 x 50 µm, and of spacing the comb modes in the grating(horizontal) direction by every other pixel to maximize the total number of comb modes
imaged while retaining the ability to individually resolve them. In the VIPA (vertical)
direction, prior experience led us to choose an imaging lens with f ∼ 250 mm so that one
FSR nearly fills the vertical direction of the camera. In the grating (horizontal) direction,
the beam diameter after the VIPA is 2 mm, leading to a bi = 5 mm beam diameter after thehorizontal telescope. The beam diameter at the focus on the lock-in camera is then given byfimaging
250mm
bcamera = 4λ≈ 51 µm. When measured using a conventional CCD= 4∗800nm
π
bi
π
5mm

camera with 13 x 13 µm pixels, the horizontal width was found to be ∼ 65 µm. In the VIPA
direction, the width was found to be ∼ 50 µm.
The maximum resolution of a grating is given by ∆λ = λ/N , where λ is the wavelengthof the light and N number of grating lines illuminated. At 800 nm, given the 1800 lines/mm
pitch of the grating and a 5 mm diameter beam, the maximum resolution is ∼ 42 GHz,
which is sufficiently below the VIPA free spectral range of 94 GHz. To place the comb
modes separated by one FSR on every other pixel, necessitates a 172.6 µm spacing formodes spaced by 0.22 nm at a center wavelength of 800 nm. The grating equation gives theangle of the first order off of the grating as follows:sin(β) = Gλ − sin(α)(4.7)

where G is the grating pitch of 1800 lines/mm, λ is the wavelength of the light, and α is theincident angle. From this we get that the change in angle as a function of wavelength is
∆β ≈

G
∆λ
cos(β)(4.8). Using ray-tracing matrices, the position from the center axis is given by
xf = fimaging G∆λ/cos(β)(4.9)

where fimaging is the focal length of the imaging lens, di is the distance from the grating to
the imaging lens, ∆λ is the change in wavelength, and β is the angle of the first order off of

43
the grating. Given fimaging = 250 mm, G=1800 lines/mm, and ∆λ = 0.22 nm, an angle ofβ = 55◦ is needed to produce xf = 172.6 µm. From the grating equation, for λ = 800 nm,
this requires an angle of incidence of 38◦ . Using the CCD camera, the comb modes had ahorizontal spacing of ∼ 170 µm and a vertical spacing of ∼ 200 µm.
After the characterization with the conventional CCD camera, the comb light was thenimaged onto the lock-in camera by applying a well-calibrated 10 kHz amplitude modulation
to the light using the AOM shown in Figure 4.2. The resulting image of the incident comb
light is shown in Figure 4.5. As can be seen in Figure 4.5, the chirp in the VIPA spacing
results in some comb modes spanning more than a single pixel. The identification of thecenter of each comb mode and resulting summing scheme are discussed in the following
section on spectrum generation.
A calibration of the lock-in camera was performed by measuring the shot-noise withboth a LED and the cw Ti:sapph laser. In both cases, the light was modulated at 10
kHz as the lock-in camera does not measure the DC intensity. Since the readout noise isnon-negligible, the noise in counts is the sum in quadrature of the readout and shot-noise
contributions,rnoise =

2
σreadout+

1NC(4.10)

where σreadout is the readout noise, C is the calibration factor in photons per count and N
is the number of counts measured in a single frame. From this, C, the calibration factor, is
3000 photons per count.
Additionally, the maximum number of cycles of demodulation that the camera canperform at 10 kHz is limited to 150. Above this, the camera reads a lower value of the power
than it should. Issues of camera saturation arising from 10 kHz electrical pickup necessitatedrunning at between 10 and 50 demodulation cycles per frame so this was not a limitation on
the acquisition of data. At 10 cycles of demodulation at 10 kHz, saturation occurs at 100
percent modulation depth with ∼ 800 pW on a pixel.

3 GHz spacing

44

100 GHz spacing

Figure 4.5: Image of the transmitted comb light. The comb modes are spaced by therepetition rate of the comb (2.97 GHz) in the vertical direction and by the FSR of the VIPA
(94 GHz) in the horizontal direction. The size of a comb mode is the same as the pixel
size and in the horizontal direction the grating resolution was chosen such that each FSR isspaced by one pixel to fit the maximum number of comb modes on the camera. Because of
the spacing and the chirp of the spacing due to the VIPA, comb modes are sometimes split
across more than one pixel as can be seen in the zoomed in portion of the figure.

45
4.2.3

Generating a Spectrum

Once a series of images each with comb modes spaced by 2.97 GHz but stepped by 100200 MHz are acquired, the absorption signal for each comb mode must be extracted from each
image and then that information is interleaved to produce a conventional looking velocitymodulation spectrum. The two stages of generating a spectrum are identifying the locationof each comb mode on an image and putting the absorption signal from that location into
the spectrum. In order to construct a spectrum from the data, the LabView data acquisition
program records for each data image acquired: the signal at the discharge frequency (10kHz) for both clockwise and counterclockwise propagation, a “chopper” image produced byapplying a well-calibrated 10 kHz amplitude modulation to the transmitted light for bothclockwise and counterclockwise propagation, the wavelength of the cw reference laser fromthe 30 MHz accuracy wavemeter, and the repetition rate of the comb from a frequency
counter. For the data and “chopper” images, the data acquisition program averages overthe number of frames and sets of frames taken by the camera, but not the number of
demodulation cycles. The “chopper” images allow for the identification of the location ofeach comb mode as well as the determination of the DC intensity for each comb mode andthus the fractional absorption.
To identify the locations of each comb mode on an image, a global fit to the grid formedby the pixels is used. A global fit is necessary instead of simply counting by a fixed number ofpixels from a given starting location due to the chirp in the comb mode spacing. (See Figure4.5.) The fit starts with a small region and the fit region is expanded to eventually include
the whole image for higher polynomial orders. For the full image, in the VIPA direction, thefit to the comb mode positions is quartic and in the grating direction, the fit to the comb
mode positions is quadratic. For each region, the fitting program maximizes the sum of theweighted intensity of each predicted mode averaged over the number of predicted modes inthat region. The weighting function consists of a quadratic arch centered on the predicted46
comb mode location to minimize issues of comb modes split across two pixels. A portion of
the global fit is shown in Figure 4.7 (a).
Once the location of each comb mode is known for the first image of the scan, all of
the information acquired by the data acquisition program discussed above, in addition to
the following, is utilized by a data analysis program to generate the spectrum. In order tocompute the appropriate DC intensity, the modulation depth for the AOM that applies the
10 kHz modulation needs to be determined. Since the data acquisition program does notaverage over the number of demodulation cycles, the number of demodulation cycles2 forboth the data images and the “chopper” images must be input. The output from the global
fit program is passed to the data analysis program as a 3-dimensional array of positions
on the grid formed by the pixels where first dimension corresponds to an integer VIPA row
index, the second dimension corresponds to the grating column index and the third dimension
corresponds to the pixel location in either the VIPA or grating direction. As more than one
FSR of the VIPA fits onto the camera, the pixel from which we start counting a single FSR
can be chosen to avoid portions of the camera that have lower incident power or a poorer
fit. In order to determine the absolute frequency of a given comb mode, the position of the
cw reference laser on the first image and the value of the beatnote frequency between the cw
reference laser and the frequency comb are input. From previous measurements, we know
the sensitivity of the camera (∼3000 photons/count) and the FSR of the VIPA (94 GHz).
The analysis program first extracts the appropriate number of comb spots from theglobal fit output and condense that 3-dimensional array output into a 2-dimensional arrayof size Lx2, where L is given by the number of comb modes per FSR times the number ofcolumns in the grating direction that contain a full FSR. For each grating column, the code
starts counting from the first spot that is above the user-input starting pixel. Because thecode draws a flat line from which to start counting, it is important to keep track of whether
2

For our specific Heliotis lock-in camera, the actual number of demodulation cycles is two less than thenominal value entered into the control program.

47
the first comb mode location above the start pixel for a given column is above or below the
first comb mode location for the previous column. The nuances of keeping track of the comb
modes are illustrated in Figure 4.6. Once this operation is completed, the index of the arrayof grating (x) and VIPA (y) locations corresponds to the relative frequency of each comb
mode.
The analysis code also calculates the distance between each neighboring comb mode in
the grating (x) and VIPA (y) directions for the first image. For the last comb mode locationin each FSR it uses the distance between the previous mode and that mode. Rather thanrun the global fit code for each individual image, the locations for each subsequent image areinterpolated using the distance between comb modes in the first image. The interpolation
maps the frequency step taken by the cw laser to a step in pixel position via the following
for the mth comb mode:

xf inal,m =∆ym∆f p 2
2 cos(arctan(∗ ∆xm + ∆ym)) + xinitial,m
frep
∆xm

(4.11)

yf inal,m =∆f p 2∆ym
2 sin(arctan(∗ ∆xm + ∆ym)) + yinitial,m
frep
∆xm

(4.12)

where ∆f = fcw,f inal − fcw,initial , fcw is the frequency of the cw reference laser, frep is therepetition rate of the comb, ∆xm = xinitial,m+1 − xinitial,m , ∆ym = yinitial,m+1 − yinitial,m , andm is the index of the comb mode.
For the first image it is also necessary to determine the absolute frequency of one comb
mode in addition to the relative frequencies of all the modes determined earlier. To do this,the comb mode at index n that is located closest to the cw laser spot is found. The frequency
difference between that comb mode and the cw laser is just the beatnote frequency, fbeatnote .
Frequencies can then be assigned for every data point. The frequency of the mode withindex m, for image i, is given byfi,m = (m − n) ∗ frep,i + fcw,i + fbeatnote(4.13)

Free Spectral
Range

VIPA Direction

48

50
35 40 45 4930
6
15 20 25
39 44 48
34
29
5 10
38 43
14 19 24
4 9 13 18 23 28 33
37 42 47
32
3 8
27
17 22
36 41 4631
2 7 122645
16 21
30 35 40
1 6 1125
15 20
34 39 445 10
19 24 29
Grating Direction

Figure 4.6: Sketch of selection of comb modes by spectrum generator code. In this simplified
example, four comb modes fit within one free spectral range of the VIPA and the free
spectral range is not an integer multiple of four leading to the chirp in the location in the
VIPA direction. The red and blue circles represent the comb mode locations. The number
adjacent to that comb mode represents the frequency offset of that mode scaled by therepetition rate. The red circles represent the locations of the comb modes which would beconcatenated by the spectrum generator code.

49
where fcw,i is the frequency of the cw laser for that image and frep,i is the repetition rate of
the comb for that image.
The intensity for a given comb mode is determined by the sum over one to four pixels
depending on the center of the mode with respect to the center of the pixel. If the center ofthe comb mode is in the left half of the pixel, the intensity is summed over that pixel andone pixel to the left in the grating direction. Likewise, if the center is in the right half ofthe pixel, the intensity is summed over that pixel and one pixel to the right in the grating
direction. If the center is in the bottom 40% of the pixel, the intensity is summed over thatpixel and one pixel below in the VIPA direction, and if the center is in the upper 40% of the
pixel, the intensity is summed over that pixel and one pixel above in the VIPA direction.
If the center is in the middle 20% of the pixel, just that pixel is summed over in the VIPA
direction. The intensity at 10 kHz and the DC intensity are then given by the resulting
double sum. After summation over the appropriate number of pixels, the intensity at 10
kHz is normalized by the DC intensity.
After normalizing the intensity, a running median is calculated for each image to remove
noise that is common across neighboring comb teeth. The width of the range over which to
calculate the median is determined via a user input and is typically 6 comb modes. Notethat this is for each image, so that those 6 comb modes would span ≈ 18 GHz, which ismuch wider than any features we are trying to retain. The raw data and running medianare shown for a test of our system with a N+
2 band in Figure 4.7. Note that the highpoints in Figure 4.7 (b) correspond to N+
2 absorption and are not removed by the mediansubtraction. After median subtraction, the spectra with ∼ 3 GHz spacing for each image
are interleaved together. At this point, additional smoothing across neighboring frequencies
can also be applied, which now reflects smoothing across points from different images. The
final product as well as several intermediate steps of the spectrum generation are shown in
Figures 4.7 and 4.8.

50

DC image
DC intensity [nW]

(a)

20

10

0
11910

fractional absorption

Signal at 10kHz

(b)

11930

11950

11970

11990

12010

12030

12050

wavenumber [cm-1 ]

-4

x 10
2

data
running median

0
-2
11910

11930

11950

11970

11990

12010

12030

12050

-1

wavenumber [cm ]Figure 4.7: Data Analysis Part 1. (a) The DC intensity is shown as the initial 2-dimensional
image, which is then fit to obtain the comb mode positions as shown in red circles overpart of the image. This fit calibrates the DC intensity as a function of the comb frequency,
which is shown on the right. An additional 10 kHz amplitude modulation was applied tothe transmitted light to measure the DC intensity. In (b) the signal at 10 kHz (with only
the discharge modulation) is shown for one image. The red circles on the two-dimensional
image are the positions of the comb modes determined from the global fit to the DC intensityin (a). Note the bright and dark pixels, which represent the spectral signature of nitrogen
absorption. The difference between the clockwise and counterclockwise signal extracted fromthe fit for one image is also shown with the points spaced by the repetition rate of the comb.
The running median plotted with the data is subtracted from each image before they areinterleaved.

51

-4

fractional absorption(a)Processed Spectrum

x 10
2
1
0
-1

data
predicted N2+ lines

-2
11910

11930

11950

11970

11990

12010

12030

12050

wavenumber [cm-1 ]

(b)
-4

2

-4

x 10

2

-4

x 10

2

1

1

1

0

0

0

-1

-1

-1

-2
11923

-2
11998

11924

11925
-1

wavenumber [cm ]

11999

12000
-1

wavenumber [cm ]

x 10

-2
12011

12012

12013
-1

wavenumber [cm ]Figure 4.8: Data Analysis Part 2. The resulting processed spectrum (blue) from all theimages (each processed similarly but with stepped cw laser frequency) is shown in (a). The
predicted positions from molecular constants reported by Collet et al. [4] for the (4,2) N+
2band (red) are also shown in (a). The full recorded spectrum was acquired in 20 images with
1500 simultaneous channels in less than one hour, with a sensitivity of 3 x 10−7 fractional
absorption per detection channel. The zoomed-in panels (b) show three different portions of
the spectrum with data points indicating the actual sampling period.

52
4.2.4

Control of the Frequency Comb

A cw Ti:sapphire laser serves as a frequency reference for stabilizing the comb andprovides a marker on the two-dimensional spectral image. The cw laser is locked to its
internal reference cavity, and feedback to the comb’s repetition rate locks a comb mode to
the cw laser. The comb mode referenced to the cw laser is controlled by acting on boththe comb laser cavity length (slow feedback via a PZT) and the pump power (fast feedback
via an acousto-optic modulator) [61, 62]. The comb is not fully fixed because only one
degree of freedom is locked, because the feedback mostly drives the repetition rate, the offset
frequency is essentially left free running. The enhancement cavity is then locked to the comb
via a dither lock (at 50 kHz) using the spectrally filtered reflection from the cavity. The
enhancement cavity length is controlled by both a fast PZT mounted directly to one of thecavity mirrors following Briles et al [63] to achieve high bandwidth and a slow, long travel
PZT mounted inside a translation stage to keep the cavity length in range of the fast PZT.
We found it critical to feed forward a 10 kHz compensation voltage onto the lock of theenhancement cavity to compensate for the 10 kHz electrical pickup from the discharge, aneffect seen in other cavity-based velocity-modulation experiments [51] as well. A detailed
schematic of the optical layout is shown in Figure 4.12.
In Figure 4.9, the repetition rate of the comb is shown over the acquisition of multiple
images to form a fully sampled spectrum. Since a single comb mode with frequency, fn =f0 + n ∗ frep , is locked to the cw laser, the repetition rate change gives a measure of the driftof the carrier-offset-frequency, f0 , which is left free-running. From the noise in frep during asingle measurement (lower panel Figure 4.9), we can determine the amount of uncertainty
in the frequency of a comb mode that is m modes away from the comb mode that is locked
to the cw laser. With 1500 comb modes in a single image and the cw reference always on
the image as a marker, the maximal uncertainty is ∼ 300 kHz for a comb mode that is 1500
comb modes away from the fixed reference given the 200 Hz level fluctuations in frep , which

53
is well below the 30 MHz resolution of the wavemeter used to determine the frequency ofthe cw laser.
To achieve this level of quietness of f0 , control of the air currents around the pumpbeam and control of the cooling water flow through the comb laser were both necessary. A
box was placed around the pump beam to reduce air currents. Appropriate cooling waterflow further reduced the fluctuations in the repetition rate from 1-2 kHz to 100-200 Hz overthe course of a measurement. A flow gauge and valve were installed on the chiller for the
comb. The flow was lowered until the noise on frep when a comb mode was locked to thecw laser was no longer reduced. Reducing the flow too much causes inadequate cooling andlong-term thermal drifts. Minimizing the drift in the offset frequency is necessary not only
for accurate wavelength metrology, but also to keep the transmitted spectrum the same overthe course of a measurement, especially over the timescale of switching directions.
4.3

Noise and the Sensitivity of the System
With the discharge running, 10 kHz electrical pickup, while minimized as much as

possible by placement and isolation of the discharge electronics, remains a problem. A
10 kHz compensation voltage is fed forward onto the lock of the enhancement cavity tocompensate for this. Despite this, residual 10 kHz noise from the discharge is present on
the cavity transmitted light due to incomplete cancelation by the feed-forward mechanism
resulting in baseline offsets. These offsets arise from frequency-to-amplitude-noise conversion
in the enhancement cavity. Since the offsets have common values across neighboring comb
modes, they are removed by subtracting the median value across a number of neighboring
comb modes. This method of rejecting common mode noise, inherent in this comb-based
spectroscopy, is not available for conventional cw systems.
The technical noise in our current system, due to incomplete common-mode noise
rejection, lies above the readout noise on the camera, which in turn lies above the shot-noise
limit. A summary of the relative contributions of the technical, camera, and shot-noise for54

2.970 GHz - frep [kHz]

475
470
465
460
455
450
445
440
0

100

200

300

400

500

600

700

800

measurement number
2.970 GHz - frep [kHz]

469.2
469
468.8
468.6
468.4

50

55

60

65

70

75

80

85

measurement number
Figure 4.9: Repetition rate of the frequency comb during data acquisition. Each flat plateau
occurs during a measurement of the 1500 simultaneous channels. The repetition rate is thenstepped by scanning the cw laser such that each comb mode with frequency fn = f0 +nfrep
is scanned over 3 GHz to fully sample the spectrum. As n ∼ 105 , stepping frep by 30 kHz
steps the comb mode frequency by 3 GHz. In the lower panel, the fluctuation of frep during
a single measurement is shown. The 200 Hz drift in frep corresponds to a maximal 300 kHz
uncertainty in the comb modes not locked to the cw laser, which is well below the 30 MHz
resolution of the wavemeter.

55

-3

Fractional Noise [1/Hz-1/2 ]10

shot-noise
camera noisetechnical noise

-4

10

-5

10

-6

10

-1

10

0

10

1

10

2

10

Power in a Comb Mode [nW]Figure 4.10: Noise contributions as a function of the power in a comb mode. The HfF+
data was acquired with comb mode powers ranging from 1 nW to 20 nW. The sensitivityof the system is given by the fractional noise divided by a factor of 30 due to the enhanced
path length. The values for the camera noise and technical noise are for switching rapidly,∼ 3X s−1 , between clockwise and counterclockwise directions of propagation as single direction measurements stopped averaging down at ∼ 1 Hz bandwidth. (Actual time to switch
directions is ∼ 1 ms, 300 ms is the time for data acquisition.)

56
varying comb mode powers is shown in Figure 4.10. Note that the sensitivity of the system isgiven by the fractional noise divided by F /π ∼ 30 as discussed earlier. Initial measurementsof the noise with the light traveling in a single direction of propagation through the cavity
did not average down for bandwidths below 1 Hz. Similar problems were seen in the camera
noise as measured across different pixels with no light incident on the camera. By rapidly
switching between clockwise and counterclockwise beam propagation through the ring cavity
via polarization control and subtracting the two images, we are able to reduce the impact ofdrifts in the gain of each individual pixel on long time scales, remove other common modenoise, and further increase the rejection of neutral molecular signals while retaining the ion
absorption signals.
Our achieved sensitivity is 3 x 10−7 single-pass fractional absorption for each measurement channel, with 1500 simultaneous channels covering 150 cm−1 , corresponding to
4 x 10−8 Hz−1/2 (spectral element)−1/2 absorption sensitivity. One full spectrum, consisting of 30 interleaved images takes 30 minutes to acquire. For other high-accuracy cw-laser
velocity-modulation experiments using cavity-enhancement, the sensitivity is ≈ 10−5 [51],which is much lower than our current sensitivity due to the additional common-mode noisesuppression discussed earlier. We note that for a cw laser system to achieve the same levelof sensitivity across the spectral bandwidth we cover in a single acquisition, it would need asensitivity of 4 x 10−8 Hz−1/2 and the ability to scan across 150 cm−1 continuously.
4.4

A Thousand of Wavenumbers of HfF+
As a diagnostic tool, the fluorescence from the plasma in the discharge tube was imaged

behind one of the high reflectivity cavity mirrors into a fiber collimator leading to a grating
spectrograph (OceanOptics). The spectral region of fluorescence imaged was limited towavelengths below the coating curve of the cavity mirror. By looking at the changes influorescence, we could tell when the oven was running low on HfF4 . Additionally, this wasa more sensitive test of small leaks than the pressure gauge on the system due to the easily

57
identified N2 signature. The spectral signature of a Helium-only discharge, a discharge with∼ 10% N2 present, and a discharge with the oven hot with HfF4 present can be seen in
Figure 4.11.
For the following spectra, a 2.5 meter long bow-tie enhancement-cavity with a finesseof 100 surrounds a 1 meter long discharge tube, which is driven at 10 kHz with ≈ 400
mApk−pk . The discharge tube contains HfF4 , which is heated to 800 K, and helium buffer
gas is flowed through the discharge tube such that the total pressure is 5 torr. A Gigaoptics
3 GHz repetition rate Ti:saphhire comb is coupled into the 120 MHz free-spectral-range
enhancement-cavity traveling either clockwise or counterclockwise around the bow-tie cavity,with every 25th cavity resonance matched to a comb mode. Spectral filtering of the cavity
due to group velocity dispersion (GVD) limits the transmitted bandwidth to 300 cm−1 . The
windows on the discharge tube are at Brewster’s angle to minimize intra-cavity loss and arethin to reduce GVD. A schematic of the system is shown in Figure 4.12.
A summary of the entire spectral region covered with multiple scans is shown in Figure
4.13. Multiple overlapping scans, taken across the whole region, were interpolated onto
a 0.001 cm−1 grid and then averaged together. For each scan, the fluorescence from thedischarge tube was monitored to ensure that HfF+ was present. Additional scans with theoven off were taken in the regions in which molecular bands were seen as an initial check thatthey arose from HfF+ . From Figure 4.13, two obvious bands can be seen as well as a number
of isolated sharp features that are most likely atomic transitions. In the region with the two
obvious bands, we have identified four different HfF+ bands as shown in Figure 4.14 with ∼150 as yet unidentified presumably HfF+ lines in the same region as the four identified bands.
The fits to all four bands are discussed in detail in Chapter Five. The lower panels in Figure
4.14 show the characteristic velocity-modulation lineshape of the HfF+ lines. The structurearound the strong lines is not the noise floor but additional HfF+ lines. The fractional noisefor the region with the HfF+ bands was 4x10−8 Hz−1/2 (spectral element)−1/2 as can be seenin Figure 5.1 in Chapter Five.

58

Intensity [arb units]

Grating
Spectrometer

60,000N2 present
He only
40,00020,000

0
450

500

550

600

650

700

750

800

Intensity [arb units]

Wavelength [nm]4000
He + HfF4
He only

3000
2000
1000
0

450

500

550

600

650

700

750

800

Wavelength [nm]Figure 4.11: Fluorescence from the discharge as a diagnostic tool. The light from the centerof the discharge tube passes through the mirror if it is below ∼ 700 nm and is focused intoa fiber connected to the grating spectrometer. (a) N2 signature versus a He only discharge.
The sharp spikes in the He only case are atomic He transitions. Note the suppression of the
He fluorescence when large amounts of N2 are present. The clear N2 signature can also be
used when checking for leaks as it is quite distinctive albeit smaller even for significantly
lower partial N2 pressures. (b) Spectral signature the corresponds to the presence of HfF+ isthe broad background lifting off of the baseline. The additional sharp features are mostlhy
atomic transitions of fluorine.

59

Laser beams

Cyl Lens

10 kHzFiber coupler
FI Faraday Isolator
Liquid crystal
variable retarder
Grating

VIPA

Lock-in Camera

Electronics

ComputerFI

Scan out

MBR 110cw Ti:Sapph
30 MHzWavemeterFI

3 GHz
Ti:Sapph
Comb

15 MHz
F->VAOM

18W Verdi

95/5
B/S

50 kHzLoop
FilterFigure 4.12: Schematic of the system. An 18 W Coherent Verdi pumps both the CW
Ti:Sapphire laser, used as a frequency reference, and the 3 GHz repetition rate Ti:Sapphirecomb. The CW laser is read by a wavemeter with an accuracy of 30 MHz. The comb isreferenced to the CW laser via a beatnote with a single comb tooth. Feedback to the comb’s
repetition rate is achieved through the pump power (high bandwidth and small range) andthrough the comb cavity length (low bandwidth and large range). 95% of the comb lightand 5% of the CW light are combined in a fiber which is coupled to the enhancement cavity.
Liquid crystal retarders and a polarizing beam splitter are used to select the direction of lightpropagation around the enhancement cavity while maintaining all locks. The enhancement
cavity is locked to the comb. Cavity transmitted light is dispersed by the 2-dimensional
imaging system, qualitatively sketched in Fig. 4.6, and recorded by the lock-in camera,
which demodulates each pixel at the discharge modulation frequency. The CW light serves
as a reference marker on the lock-in camera image.

60
Although the scanned region covers from 11,700 cm−1 to 12,700 cm−1 no additional
HfF+ bands were seen outside the 12,100 to 12,300 cm−1 region. An additional unidentified
band was seen around 12,400 cm−1 , but it was present with HfF+ absent (oven shut off). At
the edges of the scan range, the lower comb power resulted in higher fractional readout noise
and a less stable lock which caused the fractional noise to increase as can be seen in Figure
4.13. To increase the spectral bandwidth covered, the frequency comb would need to bebroadened and the intensity of each comb mode on the camera would need to be increased.
Different options for doing this are discussed briefly in Chapter Six.
4.5

Coherent Subtraction
An additional way to increase signal-to-noise is to use coherent subtraction; i.e., using

a balanced Sagnac-type interferometer as shown in the blue inset to Figure 4.16. With aring cavity, it is possible to have counter-propagating beams through the sample and tocancel common-mode noise at the output of the dark port of the resulting interferometer.
The signal-to-noise using coherent subtraction increases as the splitting ratio approaches
50% between the two beams until camera readout noise or shot-noise dominates. Figure
4.15 shows the improvement in signal-to-noise ratio as a function of splitting ratio when thecommon mode technical noise dominates the noise. We have tested coherent subtraction versus a single direction of propagation using a photodiode and single-channel lock-in detector
with our CW laser and demonstrate a factor of 10 gain in the signal-to-noise (Figure 4.16).
Coherent subtraction, unlike counter-propagating beams in a multipass cell, is sensitive tophase as well as amplitude of the light field; thus, the resulting lineshape is dependent on
both absorption and dispersion and can vary if there are additional differential phase shifts
between the two directions. These phase shifts in effect change the relative sensitivity toabsorption and dispersion. We have made a simple model (dashed lines in Figure 4.16) that
reproduces the lineshape modification. In our current experiment, the power per comb mode
is too low for us to take advantage of this coherent subtraction approach, but this problem

613

x 10-5

fractional signal

2
1
0
-1
-2
-3
11,70011,90012,10012,30012,50012,700-1

frequency [cm ]Figure 4.13: Full spectral region scanned with HfF+ present. There are four HfF+ bands
that have been identified so far and these are highlighted in Figure 4.14. The isolated sharp
features are most likely due to atomic transitions. The increased fractional signal at bothedges is due to the camera noise starting to be the dominant noise source due to the decreasein comb power at the edges of the comb bandwidth. This summary plot was generated byinterpolating multiple scans onto a 0.001 cm−1 grid and then averaging them together.

62

3

1

1 +
P
(0,1)
1 <-- Sx 10-5
1

1 +
P
(1,2)
1 <-- S

3
3

1

+

P
(2,0)
1 <-- S+P
(3,1)
1 <-- S2

fractional signal

1

1
0
-1
-2

-3
12,00012,05012,10012,15012,20012,25012,30012,350-1

frequency [cm ]-55fractional signal

fractional signal

1.5 x 10
1
0.5
0
-0.5
-1
-1.5
12,16012,161

frequency [cm-1]

12,162x 10-60-512,27012,271

12,272-1

frequency [cm ]Figure 4.14: Region of spectrum with HfF+ lines with smaller regions showing the characteristic velocity-modulation lineshape. Fits to the bands identified are discussed in Chapter
Five. In the lower plots, the structure surrounding the strongest lines is due to additional
HfF+ molecular absorption. The fractional sensitivity measured without HfF+ present is
3x10−7 .

63
could be overcome in the future by using either a more sensitive camera or a comb withmore power.
The lineshape simulations used to model the data in the coherent subtraction scheme
were done by applying a cosine modulation in time to the center frequency for both the absorption (assumed Gaussian lineshape) and dispersion terms (obtained via Kramers-Kronig)
and then taking the cosine transformation and selecting the appropriate modulation harmonic. This was done at a variety of laser frequencies to generate the lineshape. The
amplitude and linewidth of the single direction simulation was scaled to the measured value
and this value was fed into the coherent subtraction simulation. The constant phase offset(most likely due to the beamsplitter) added to the coherent subtraction simulation was chosen to fit the measured lineshape (about π/20 radians). The beamsplitter intensity splitting
ratio between transmission and reflection was 52%/48%.

Signal to Noise Improvement

64

50
50

40
30

40
30
20

20

10
0.45

0.475

0.5

10
0
0 0.1 0.2 0.3 0.4 0.5
Reflectance of Beam Splitter

Figure 4.15: Improvement of signal-to-noise ratio as a function of splitting ratio. The signalto-noise ratio improves dramatically as the intensity for the two directions of propagation
becomes more similar as long as common-mode technical noise dominates the noise. If thesingle direction measurements are dominated by the shot-noise, the signal-to-noise ratio forcoherent subtraction is the same as single direction propagation. If the camera noise begins
to dominate due to the decrease in power as the splitting ratio approaches 50%, the signal-tonoise ratio will decrease. The challenge of achieving a splitting ratio of better than 52%/48%
also limits the gains in signal-to-noise ratio improvement.

65

x10

SinglePass (x10)
Coherent
Subtraction

Figure 4.16: Comparison of coherent subtraction and single direction measurements The
signal to noise for coherent subtraction (using a 52/48 beam splitter) surpasses that of single
direction measurements by a factor of 10 in cases where we have sufficient optical power todominate over the camera readout noise. Both measurements of a single N+
2 line were madewith the cw Ti:Sapphire laser. The dashed lines are a calculation of the expected lineshape.
The slight modification in the lineshape for the coherent subtraction is due to an additional
differential phase shift of approximately π/20 between the two counter-propagating beams.

Chapter 5
HfF+ Structure1

Currently, five different bands of HfF+ have been identified; one with single-frequency
velocity-modulation spectroscopy, discussed in Chapter Three, and four with the frequency
comb velocity-modulation system described in Chapter Four. The full spectrum acquiredwith the frequency comb velocity-modulation system can be seen in Figure 4.13, with Figure
5.1 showing the four bands we have identified so far; the fifth band acquired with singlefrequency velocity-modulation spectroscopy can be seen in Figure 3.5 and Figure 3.6. Furthermore, we have a collection of as yet unassigned lines in the midst of the other bands
in the frequency comb velocity-modulation data, indicating the presence of at least one
more band at around 12200 cm−1 . As illustrated in Figure 5.1, the spectrum from about
12100-12300 cm−1 is extremely congested due to the presence of many bands, each with fiveisotopes, and the high temperature of our oven, which results in observed J00 values up to
about 70. The power of frequency comb velocity-modulation spectroscopy is demonstrated
by our ability to identify the overlapping 1 Π1 − 1 Σ+ (0, 1) and 3 Π1 − 1 Σ+ (3, 1) bands despite
the difference in linestrengths and an offset in band origin of only ∼1 cm−1 . This chapter
will discuss the fits to each of the bands and the molecular constants that can be extracted
from them.
1

Much of the discussion on the structure of HfF+ will be discussed in a forthcoming publication from

K.C. Cossel et al.

67
-5

x 10

(a)
fractional absorption

3
2
1
0
-1
-2
12,05012,10012,15012,20012,25012,300-1Frequency [cm ]
+

(b)

1

+

+

+

data HfF present

3+data HfF absent
Line Positions from Fit

2

176

+

HfF

1

177

HfF+

0

178
179

HfF+

12,14012,1601

12,18012,200Frequency [cm-1]

12,24012,260-5

x 10-5

0.5

0
-0.5
-1
12,133

12,22012,134Frequency [cm-1]

12,1352

x 10

fractional absorption

-2

+

HfF

-1

fractional absorption

fractional absorption

1

1

fit for P
(2,0)
1 - S
3
1 +
fit for P
(3,1)
1 - S

Data and Fit to P
(0,1) band
1 - S

x 10-5(c)
fractional absorption

3

1 +fit for 1P
(0,1)
1 - S
1
1 +
fit for P
(1,2)
1 - S

data HfF present

1
0
-1

-2
12,178

12,179Frequency [cm-1]

12,180

2

180

+

HfF

x 10-510-1-2
12,212

12,213Frequency [cm-1]

Figure 5.1: (a) Data and fits to the four HfF+ bands identified with frequency comb velocitymodulation spectroscopy. Linestrengths for the fit positions are scaled by isotopic abundance,by an approximate rotational temperature of ∼ 640 K, and by an overall transition strength
to match the measured fractional absorption. (b) Region of the recorded spectra containing
the 1 Π1 − 1 Σ+ (0, 1) band as well as at least three other HfF+ bands with the fit for the1
Π1 −1 Σ+ (0, 1) band offset for clarity. (c) All five isotopes of HfF+ are present and resolved
for transitions with ∆v 6= 0. The additional structure with HfF+ present (blue) consists oflines belonging to the other HfF+ bands while the red line (HfF+ absent) shows the level ofsensitivity of the system in this region.

12,214

68
5.1

Fits to HfF+ Bands
We fit each band with the energy expression,

E(J 00 , J 0 ) =ν0 + (B 0 −PP∗ (p + 2q))J 0 (J 0 + 1) + (D0 − ∗ (pD + 2qD ))J 02 (J 0 + 1)2 (5.1)
2
2− B 00 J 00 (J 00 + 1) − D00 J 002 (J 00 + 1)2(5.2)

where B0 and B00 are the rotational constants for a given vibrational level of the upper andlower state, D0 and D00 are the centrifugal distortion terms for a given vibrational level of theupper and lower state, (p + 2q) is the Ω-doubling of the excited state [34] with (pD + 2qD )the next higher-order of the Ω-doubling, and P is a parity term with P = −1 for ∆J = 0
and P = 1 for ∆J = +/ − 1. A summary of the fitted constants for each band is given in
Table 5.1 and the molecular constants that can be derived for the states observed are given
in Table 5.2. For the bands for which the centrifugal distortion term is consistent with zerodue to a sufficiently large uncertainty, the corresponding higher-order term of the Ω-doubling
was not included in the fit.
All five isotopes of Hf were resolved for transitions with ∆v 6= 0 allowing for the
extraction of molecular constants for a particular isotope. The rich isotope structure also
yielded the change in vibrational quantum number for those bands. For the 1 Π1−!1 Σ+ (0, 0)
band, we were not able to fully resolve the isotope structure as discussed in Chapter Three.
Therefore, we fit to estimated line centers and scaled the fitted constants by the reduced
masses and isotopic abundances to obtain the final constants in Table 5.1. For this band wewere able to establish that ∆Ω = 1 since we observed an R(0) line but no P(1) line, as canbe seen in Figure 3.6. For the other four bands, the congestion of the spectrum prevented asimilar confirmation of the values of Ω for the upper and lower states.

69
1

1 +

Pi1 <- S (0,1)

0.015

-1

Residual (cm )

0.01
0.005
0
-0.005R
Q
P-0.01-0.015
-100

-80

-60

-40

-20

0

20

-1

Offset from band origin (cm )Figure 5.2: Residuals from the fit to the 1 Π1 − 1 Σ+ (0, 1) band of 180 HfF+ . The fit spans∼100 cm−1 reaching J values as high as 70 due to the high rotational temperature. This fit
provides high accuracy in the determination of the molecular constants.

Table 5.1: Fitted constants for observed transitions in 180 HfF+ . Quoted uncertainties are
95% except for the 3 Π1 − 1 Σ+ (3, 1) transition for which uncertainties are quoted at the level
for which the manual fit would noticeably fail. Note that the values for the 1 Π1 − 1 Σ+ (0,0)
transition were extracted from a fit to the average line positions as the individual isotopes
are not fully resolved as discussed in Chapter Three. ◦ 3 Π1 − 1 Σ+ (3, 1) values assigned using
a manual fit due to challenges of determining the line centers of the weaker lines in the dense
spectrum. ∗ Value fixed to the fitted value from the 1 Π1 − 1 Σ+ (0, 1) transition. + Isotope
averaged value with the error due to the uncertainty in the electronic isotope shift.
1Π1 ←1 Σ+ (0, 1) 1 Π1 ←1 Σ+ (1, 2)
ν0 [cm ]12217.369(2)12136.012(3)00−1B [cm ]
0.30335(3)0.30180(5)B0 [cm−1 ]0.28115(3)0.27973(5)D00 [10−7 cm−1]
1.88(8)1.80(14)
D0 [10−7 cm−1 ]1.81(8)1.74(14)−4−1(p+2q) [10 cm ]3.69(2)2.68(4)(pD +2qD ) [10−9 cm−1 ] 9.7(7)7.4(14)−1

Π1 ←1 Σ+ (2, 0) 3 Π1 ←1 Σ+ (3, 1)◦
12304.400(3)12216.87(1)0.30481(4)0.30335∗
0.28096(4)0.27958(20)1.78(12)1.8(2)
1.78(11)
1.8(2)-3.82(1)-3.8(2)-0.3(10)3

1

Π1 ←1 Σ+ (0, 0)13002.229(6)+0.30474(20)0.28104(20)
1(2)
1(2)
3.55(4)-

70
5.2

Molecular Constants
The molecular constants that can be derived from the fit values listed in Table 5.1 aregiven in Table 5.2. These values enable the prediction of other transitions from the samestates as well as comparison with theory.
The electronic rotational constant is given by [35]1
Be = Bν + αe (v + ),
2

(5.3)

where Bν is the rotational constant for a given vibrational level and αe is the vibrationalcorrection and thus by measuring Bν for multiple vibrational levels, we can extract αe and
Be . Likewise, the electronic centrifugal distortion term is given by1
De = Dν + βe (ν + ).
2

(5.4)

The energy of an electronically excited state as measured from the minima of the potential
with a zero reference for the electronic ground state is given by1
1
1
1
T0 = ν0 − ωe0 (v 0 + ) + ωe χ0e (v 0 + )2 + ωe00 (v 00 + ) − ωe χe00 (v 00 + )2
2
2
2
2

(5.5)

where ν0 is the band origin for a given transition, ωe is the vibrational constant, and ωe χeis the anharmonicity term. Initially, it would seem that the (0,0), (0,1), and (1,2) bands of1

Π1 −1 Σ+ do not provide enough information to extract ωe and ωe χe for both 1 Π1 and 1 Σ+states as well as T0 for the 1 Π1 state. However, assuming a Morse potential,ωe χe =

αe2 ωe2 αe ωe
+
+ Be
36Be3
3Be

(5.6)

[35]. Using this assumption, we obtain the molecular constants given in Table 5.2. The
errors quoted represent the propagation of the statistical 95% uncertainty levels from the fit
assuming a Morse potential and thus do not reflect any model dependent errors.
The vibrational constant, ωe , can also be calculated from the relationship of the rotational constant to the centrifugal distortion,
s
ωe =

4Be3
De

(5.7)

71
180

+

Table 5.2: Derived constants for observed states in HfF . Values for T0 are given from theminima of the potential not the lowest vibrational level. The error quoted is the statisticalerror and does not include any model dependent systematics. The dominant source of thestatistical error is the uncertainty in αe . The model dependent error would arise from ωe χe
for which we see good agreement for the 1 Σ+ state with a non-model dependent value.
1Σ+T0 [cm−1 ]0−1
Be [cm ]0.30568(8)−1ωe [cm ]791.1(1)ωe χe [cm−1 ]3.10(16)−3−1
αe [10 cm ] 1.55(6)

1

3
Π1
Π1
13046.3(3) 10933(3)0.28186(4) 0.28441(50)702.6(7)713(1)
2.70(15)
2.61(26)
1.42(6)1.38(20)[35]. From this definition of ωe we obtain 780(17) cm−1 compared to 791.1(1) cm−1 asderived above for the 1 Σ+ state, 719(22) cm−1 compared to 713(1) cm−1 for the 3 Π1 state,
and 703(16) cm−1 compared to 702.6(7) cm−1 for the 1 Π1 state. However, the uncertainty
in the centrifugal distortion from the fits causes the uncertainty in the vibrational constant
to be much greater than when it is calculated using the definition of ωe χe given in Equation(5.6). The good agreement of our value of ωe χe for the 1 Σ+ state using the Morse potential
assumption with that of the Heaven group [3] as shown in Table 5.3 supports the Morsepotential assumption, which allows us to achieve much more accurate values of the derived
molecular constants.
5.3

Comparison with Theory and Other Experimental Data
A comparison of some of the derived molecular constants with theoretical calculations

from Petrov et al. [2] is presented in Table 5.3. Initially, the predicted value of 12,686
cm−1 for the energy of the 3 Π1 state led to the identification of the band at ∼ 13, 000
cm−1 measured using single-frequency velocity-modulation spectroscopy as the 3 Π1 −1 Σ+(0,0) band. However, if the band at ∼ 13, 000 cm−1 was the 3 Π1 −1 Σ+ (0,0) band, theΩ-doubling was expected to be negative as opposed to the positive value we measured [64].

72
The additional data from the frequency comb velocity-modulation spectroscopy resolved
this confusion with the identification of the ∆v = 1 band at 12,217 cm−1 with positiveΩ-doubling and a higher energy upper state and the ∆v = 2 band at 12,304 cm−1 withnegative Ω-doubling and a lower energy upper state. The predicted energies of the 1 Π1
and 3 Π1 states are ∼1800 cm−1 above the predicted values, but the fine structure splitting
between the two agrees with the predicted values to within the uncertainty of the derived
energy from the fit. Preliminary theoretical calculations with a modified core potential shift
the manifold of Π states so that their positions more closely match the measured values[64]. (This will be discussed further in a forthcoming publication from K.C. Cossel et al.)Additional unpublished work from E.R. Meyer has shown similar discrepancies in the stateenergies, but good agreement in Be and αe [65].
In addition to theoretical comparisons, in Table 5.3 molecular constants of the v = 0
level of the 1 Σ+ state can be compared to the experimental results from Barker et al. using
pulsed-field-ionization zero-kinetic-energy measurements (PFI-ZEKE) to study the low-lying
states of HfF+ [3]. We find that values of Be , ωe , and ωe χe agree within the errors. The
agreement in ωe χe supports our assumption of a Morse potential for calculations of ωe , T0and ωe χe . Due to directly measuring a large number of rotational lines for each HfF+ band,our values of Be are considerably more precise. In addition to the 1 Π1 −1 Σ+ (0,0) and (0,1)
bands, we measure the energy of the v = 1 level of the 1 Σ+ state relative to the v = 0 level
to be 784.841(6) cm−1 , which is again consistent with the Heaven group’s value of 785.8(10)cm−1 [3].
5.4

Isotope Shifts
For the stronger transitions, we can fit not only the 180 HfF+ bands, but also thosecorresponding to other isotopes of Hf. Tables 5.4 and 5.5 contain the fit values for multiple
isotopes for the 1 Π1 ←1 Σ+ (0, 1) band and the 3 Π1 ←1 Σ+ (2, 0) band, respectively. For the
00
00
00
0
0
0
rotational constants for the 1 Π1 ←1 Σ+ (0, 1) band, B179
, B178, B177, B179,B178, and B177
all

73

Table 5.3: Comparison of derived molecular constants for 180 HfF+ with the theoretical calculations of Petrov et al. [2] and results from Barker et al. [3]. The theoretical values of Be
were computed from the equilibrium bond length.
this work
1

Σ+
3
Π1
1
Π1
1

Σ+
3
Π1
1
Π1

theory [2] other exp [3]T0 [cm−1 ]00010933(3)12686–
13046.3(3) 14784–−1
ωe [cm ]791.1(1)751
791.2(10)713(1)
687–
702.6(7)679–

this work

theoryBe [cm−1 ]0.30368(8) 0.3092
0.2844(5) 0.2835
0.28186(4) 0.2805
ωe χe [cm−1 ]3.10(16)–
2.61(26)–
2.70(15)–other exp
0.304(5)––2.95(20)––

74
scale with the ratio of reduced masses as expected to within the error. The uncertainty inthe values of D make comparison with the expected isotope shift impossible.
The ability to resolve all five isotopes of Hf for transitions with ∆v 6= 0 aids in band
identification as the vibrational isotope shift for ∆v = v 0 − v 00 = 2 versus ∆v = 1 is obviously
different. For those bands with sufficient signal strength and uncrowded spectral region, i.e.
1

Π1 −1 Σ+ (0, 1) and 3 Π1 −1 Σ+ (2, 0), we can fit to multiple isotopes and use the expectedvibrational isotope scaling to extract the electronic isotope shift. While only the 180 HfF+
band was fit for the 3 Π1 −1 Σ+ (3, 1) and 1 Π1 −1 Σ+ (2, 1) bands, the vibrational isotope shiftseen is consistent with a ∆v = 2 and ∆ν = −1 transition respectively.
From the differences in band origin, we can extract information about the electronic
isotope shift of a transition by assuming that the vibrational isotope shift scales like theratio of the square root of the reduced masses, which is an extremely good approximation.
For a given isotope, the electronic energy
T0,i = ν0,i − ∆ων,i

(5.8)

1
1
1
1
0
00
∆ων,i = ωe,i(v 00 + ) + ωe χ00e,i (v 00 + )2
(v 0 + ) − ωe χ0e,i (v 0 + )2 − ωe,i
2
2
2
2

(5.9)

where

and thus the electronic isotope shift relative to 180 HfF+ is given byrµ180∆T0 = T0,180 − T0,i = ∆ν0 − (1 −)∆ων,180 .
µi(5.10)For 178 HfF+ , from the 1 Π1 −1 Σ+ (0, 1) band, ∆T0 = −0.073(3) cm−1 . From observing
the cancelation of the rotational isotope shift with the combination of the electronic andvibrational isotope shifts for the 1 Π1 −1 Σ+ (0, 0) band as shown in Figure 3.6, the total
electronic and vibrational isotope shift of 178 HfF+ relative to 180 HfF+ was -0.04 cm−1 . Using
the values for the vibrational constant and anharmonicity term, this results in an electronic
isotope shift of ∆T0 = -0.064(7) cm−1 of 178 HfF+ relative to 180 HfF+ , which is consistent withthe shift found for the 1 Π1 −1 Σ+ (0, 1) band. For the 3 Π1 −1 Σ+ (2, 0) band, the electronic
isotope shift between 180 HfF+ and 178 HfF+ is ∆T0 = -0.048(4) cm−1 .

75Table 5.4: Fitted constants for the 1 Π1 ←1 Σ+ (0, 1) band for four different isotopes of HfF+ .
The 176 HfF+ lines at 5.21% abundance were excluded from fitting due to their lower intensity.
Quoted uncertainties are 95%.
180

179
178
HfF+
HfF+
HfF+
abundance [%]
35.10
13.63
27.30
−1
ν0 [cm ]12217.369(2) 12217.198(4) 12216.997(2)B00 [cm−1 ]0.30335(3)0.30350(6)0.30369(5)0−1B [cm ]
0.28115(3)0.28129(7)0.28146(4)00−7D [10 cm−1]
1.81(8)1.70(19)
1.87(9)D0 [10−7 cm−1 ]1.81(8)1.64(18)
1.80(9)−4−12(p+2q) [10 cm ]3.69(2)
3.72(5)3.68(2)−9−12(pD +2qD ) [10 cm ] 9.7(7)10.4(25)
9.2(8)

177

HfF+
18.6
12216.818(4)0.30387(7)0.28163(7)1.90(17)
1.83(16)
3.72(4)10.4(14)

Table 5.5: Fitted constants for the 3 Π1 ←1 Σ+ (2, 0) band for three different isotopes of HfF+ .
The 179 HfF+ lines at 13.63% abundance and the 176 HfF+ lines at 5.21% abundance were notfit. Quoted uncertainties are 95%. ∗ Values fixed based on expected rotational isotope shift.
180

178
177
HfF+
HfF+
HfF+
abundance [%]
35.10
27.30
18.6
−1
ν0 [cm ]12304.400(3) 12305.183(2) 12305.581(3)00−1B [cm ]
0.30481(5)0.30519(4)0.30531∗B0 [cm−1 ]0.28096(4)0.28131(4)0.28142∗
D00 [10−7 cm−1]
1.78(12)
1.89(12)1.80(8)0−7−1
D [10 cm ]1.78(11)
1.88(11)1.80(8)−4−12(p+2q) [10 cm ]-3.82(1)-3.82(1)-3.84(2)−9−12(pD +2qD ) [10 cm ] –––Chapter 6
Conclusion

In this work, different spectroscopic techniques have been both unsuccessfully (hollow
cathode lamp spectroscopy) and successfully (single-frequency and frequency-comb velocitymodulation spectroscopy) used to identify bands of HfF+ . The first spectroscopic information for HfF+ came from our measurement of the 1 Π1 −1 Σ+ (0,0) band measured withsingle-frequency velocity-modulation spectroscopy with a sensitivity of 3x10−7 Hz−1 as discussed in Chapter Three. Single-frequency velocity-modulation spectroscopy also enabled
the characterization of the discharge and oven system. The development of a new powerfulspectroscopic tool [36], frequency comb velocity-modulation spectroscopy (Chapter Four),allowed us to cover a thousand wavenumbers of spectral bandwidth and to identify an additional four HfF+ bands. The achieved sensitivity for frequency-comb velocity-modulation
spectroscopy was 4x10−8 Hz−1/2 (spectral element)−1/2 with 1500 simultaneous detection
channels spanning 150 cm−1 of bandwidth. For a 30 minute acquisition using 30 interleaved
images to densely sample the whole spectrum, this corresponded to a 3x10−7 single-passfractional absorption sensitivity for each of 45,000 measurement channels. Identification ofbands arising from multiple vibrational levels of the same electronic states of HfF+ enables
the prediction of additional 3 Π1 −1 Σ+ and 1 Π1 −1 Σ+ rotational bands with only a few
wavenumbers of uncertainty. The spectroscopic information from all five HfF+ rotational
bands was presented in Chapter Five. Molecular constants for the 1 Σ+ , 3 Π1 , and 1 Π1 states
were extracted, which can be compared with theory as well as other experimental results for

77
the 1 Σ+ and which may lead to improvements in the theoretical predictions for electronic
states that couple well to the 3 ∆1 metastable state. The resolution of all five isotopes ofhafnium allows for examination of the isotope shifts and provides a confirmation that the
bands do in fact arise from HfF+ .
As discussed in Chapter One, the science state of interest of HfF+ for the JILA eEDM
experiment is the metastable 3 ∆1 state, not the 1 Σ+ ground state [25], and thus additional
survey spectroscopy of HfF+ is needed as all the bands currently identified arise from the 1 Σ+ground state. Figure 6.1 summarizes the five bands identified so far as well as the spectral
region currently covered with frequency comb velocity-modulation spectroscopy. The relative
intensity covered with the single-frequency velocity-modulation measurements is not shown,
but the relatively low discharge current of 100mAppk limited the strength of the absorption
signal. Additionally, Figure 6.1 shows the potential location and relative intensity of otherbands from 11,000 to 15,000 cm−1 with crude error bars representing the large theoreticaluncertainties for band locations. Theory values were taken both from Petrov et al.[2] aswell as recent preliminary work from Petrov et al.[64] and E.R. Meyer[65]. Work is on-goingboth to use single-frequency velocity-modulation spectroscopy with a cw Ti:sapph laser tosample portions of the spectral region from 11,000 to 15,000 cm−1 likely to contain states ofinterest and to broaden the ti:sapph comb to cover 11,000 to 15,000 cm−1 with the required
power per comb mode. Since comb sources enable efficient non-linear optical generation,
which allows for spectral broadening and access to spectral regions ranging from the UV to
the mid-IR [66, 67, 47], modifications to the imaging system, such as an InGaAs camera, astreak camera similar to Thorpe et al. [39], or an optical frequency up-conversion system,
would be all that would be required to push this technique even further to those wavelengths,
thus opening the door to even more applications.
The impetus for the development of frequency comb velocity-modulation spectroscopy
was the lack of experimental data on the electronic structure of molecular ions such as
HfF+ and ThF+ , which are ideal for a highly sensitive search for the electron electric dipole

78
moment [26]. Coupled with the large theoretical uncertainties (thousands of wavenumbers)[27], the lack of experimental data necessitated broad survey spectroscopy to understandand assign the molecular energy level structure. However, this technique has a wide rangeof potential applications because broad bandwidth, precision spectroscopy of molecular ionsis of interest in precision measurement, astrochemistry, and physical chemistry. Beyond
molecules of interest to eEDM experiments, precision spectroscopy of molecular ions such as+H+
3 [48] and CH5 [68] can provide a test of ab initio theory and yield new insights into few-

body quantum dynamics, respectively. In astrochemistry, numerous ions and radicals havebeen discovered in the interstellar medium and many more species have been hypothesized[69, 70]. It is believed that the source of the unidentified diffuse interstellar bands, currently
the longest standing question in astronomical spectroscopy, is molecular (possibly ionic), but
accurate identification will require matching the spectra of candidate species to astronomical
observations over wide and often times disparate spectral windows [71, 72]. Additionally,
understanding the chemistry of molecular clouds in circumstellar gases has been enabled by
the identification of molecular ions in the laboratory [73]. Efficient survey spectroscopy ofmolecular ions is thus of interest beyond the JILA electron EDM experiment.

79+

Location of HfF Bands

Relative Intensity

100

10-1

10-2

10-3

10-4
11,00012,00013,00014,00015,000-1Frequency [cm ]lower state
1 +S= blue3D
1 = redupper state
3
+
P
0
3
P
1
1
P
1
3
+
S
0
3
F
2

Figure 6.1: Identified and predicted HfF+ rotational band locations from 11,000 to 15,000
cm−1 (909-668 nm) for up to v0 and v00 equal to 3. The relative intensity is scaled by
the predicted dipole moment, by thermal factors assuming an approximate vibrational andelectronic temperature based on a rotational temperature of 640 K, and by the FranckCondon factors. The relative intensities for the four bands measured with frequency comb
velocity-modulation spectroscopy agree with the experimental data and the uncertainty inthe relative intensity for bands with no experimental data is a factor of 3-4. The color of
the point indicates the lower state of the band, either 1 Σ+ (blue) or 3 ∆1 (red), while theshape of the point indicates the upper state of the band. The five identified HfF+ bands
discussed in Chapter Five are shown in green with the outline bolded. For those bands whicharise from states for which no experimental information about molecular constants exists,
crude error bars for the energy uncertainty have been added. The gray region represents
the spectral-range covered and relative sensitivity with frequency comb velocity-modulation
spectroscopy. Even with the four bands of HfF+ accounted for in the spectral region covered
with the frequency comb, we still have ∼100 lines unidentified in the 12,100 to 12,200 cm−1
corresponding on this scale to a relative intensity of 2x10−3 . While we have been unable
to assign these lines due to the overlapping bands, it is possible that they belong to a fifth
3−1
HfF+ band such as the 3 Σ+to the blue or the 3 Π2 − 3∆1
0 − ∆1 band predicted 1000 cm
band shifted up from the predicted relative intensity currently off the lower edge of what isplotted here.

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Appendix A
Hollow Cathode Lamp Data

All data acquired with the hollow cathode lamp as discussed in Chapter Two areplotted in 250 cm−1 segments. The first image is a summary of the spectral region covered.

87

Intensity [arb. units}

1.5

1

0.5

0
12,50013,50014,50015,50016,500-1Frequency [cm ]data

HfF and HfF+ band origins

Transitions from NIST strong lines
table for Ar, Ar+, F, Hf, and Zr

HfF and HfF+ approximate
location of the R branch bandheadIntensity [arb. units}

Intensity [arb. units}012,750

0.5

1

1.5

0
12,500

0.5

1

1.5

-1

12,650-1

12,900+12,95012,700HfF and HfF+ approximate
location of the R branch bandhead

HfF and HfF band origins

Frequency [cm ]12,850Frequency [cm ]12,600Transitions from NIST strong lines+
table for Ar, Ar , F, Hf, and Zr

data

12,80012,55013,00012,75088

Intensity [arb. units}

Intensity [arb. units}0
13,250

0.5

1

1.5

0
13,000

0.5

1

1.5

-1

13,150-1

13,400+13,45013,200HfF and HfF+ approximate
location of the R branch bandhead

HfF and HfF band origins

Frequency [cm ]13,350Frequency [cm ]13,100Transitions from NIST strong lines+
table for Ar, Ar , F, Hf, and Zr

data

13,30013,05013,50013,25089Intensity [arb. units}

Intensity [arb. units}0
13,250

0.5

1

1.5

0
13,000

0.5

1

1.5

-1

13,150-1

13,400+13,45013,200HfF and HfF+ approximate
location of the R branch bandhead

HfF and HfF band origins

Frequency [cm ]13,350Frequency [cm ]13,100Transitions from NIST strong lines+
table for Ar, Ar , F, Hf, and Zr

data

13,30013,05013,50013,25090Intensity [arb. units}

Intensity [arb. units}

013,750

0.5

1

1.5

0
13,500

0.5

1

1.5

-1

13,650-1

13,900+13,95013,700HfF and HfF+ approximate
location of the R branch bandhead

HfF and HfF band origins

Frequency [cm ]13,850Frequency [cm ]13,600Transitions from NIST strong lines+
table for Ar, Ar , F, Hf, and Zr

data

13,80013,55014,00013,75091

Intensity [arb. units}

Intensity [arb. units}0
14,250

0.5

1

1.5

0
14,000

0.5

1

1.5

-1

14,150-1

14,400+14,45014,200HfF and HfF+ approximate
location of the R branch bandhead

HfF and HfF band origins

Frequency [cm ]14,350Frequency [cm ]14,100Transitions from NIST strong lines+
table for Ar, Ar , F, Hf, and Zr

data

14,30014,05014,50014,25092

Intensity [arb. units}

Intensity [arb. units}

0
14,750

0.5

1

1.5

0
14,500

0.5

1

1.5

-1

14,650-1

14,900+14,95014,700HfF and HfF+ approximate
location of the R branch bandhead

HfF and HfF band origins

Frequency [cm ]14,850Frequency [cm ]14,600Transitions from NIST strong lines+
table for Ar, Ar , F, Hf, and Zr

data

14,80014,55015,00014,75093Intensity [arb. units}

Intensity [arb. units}0
15,250

0.5

1

1.5

0
15,000

0.5

1

1.5

-1

15,150-1

15,400+15,45015,200HfF and HfF+ approximate
location of the R branch bandhead

HfF and HfF band origins

Frequency [cm ]15,350Frequency [cm ]15,100Transitions from NIST strong lines+
table for Ar, Ar , F, Hf, and Zr

data

15,30015,05015,50015,25094Intensity [arb. units}

Intensity [arb. units}0
15,750

0.5

1

1.5

0
15,500

0.5

1

1.5

-1

15,650-1

15,900+15,95015,700HfF and HfF+ approximate
location of the R branch bandhead

HfF and HfF band origins

Frequency [cm ]15,850Frequency [cm ]15,600Transitions from NIST strong lines+
table for Ar, Ar , F, Hf, and Zr

data

15,80015,55016,00015,75095

Intensity [arb. units}

Intensity [arb. units}0
16,250

0.5

1

1.5

0
16,000

0.5

1

1.5

-1

16,150-1

16,400+16,45016,200HfF and HfF+ approximate
location of the R branch bandhead

HfF and HfF band origins

Frequency [cm ]16,350Frequency [cm ]16,100Transitions from NIST strong lines+
table for Ar, Ar , F, Hf, and Zr

data

16,30016,05016,500

16,25096

Intensity [arb. units}

0
16,500

0.5

1

1.5

-1

16,650+16,700HfF and HfF+ approximate
location of the R branch bandhead

HfF and HfF band origins

Frequency [cm ]16,600Transitions from NIST strong lines+
table for Ar, Ar , F, Hf, and Zr

data

16,55016,75097

Appendix B
Velocity-Modulation Data

Single-frequency velocity-modulation spectroscopy data for the 1 Π1 −1 Σ+ band as
discussed in Chapter Three are presented along with the line locations from the fit plottedin 5 cm−1 segments. The first image is a summary of the spectral region covered. Additional
higher signal-to-noise scans were taken over select regions to determine the value of Ω0 andΩ00 through the presence or absence of the low J lines.

99

-6

3

x 10

data
2.5

P
Q
R

2

Fractional Absorption

1.5
1
0.5
0
-0.5
-1
-1.5
-2
12,95012,96012,97012,98012,990-1Frequency [cm ]13,00013,010Fractional Absorption

Fractional Absorption

Fractional Absorption

-6

x 10

-2
12,960

-1

0

1

2

3

-6

x 10

-2
12,955

-1

0

1

2

3

-6

x 10

-2
12,950

-1

0

1

2

3

12,961

12,956

12,951

12,953-1

12,958-1

12,963Frequency [cm ]12,962Frequency [cm ]12,957Frequency [cm-1 ]12,952

12,964

12,95912,954

12,965

12,96012,955

P
Q
R

data

P
Q
R

data

P
Q
R

data

100

Fractional Absorption

Fractional Absorption

Fractional Absorption

-6

x 10

-2
12,975

-1

0

1

2

3

-6

x 10

-2
12,970

-1

0

1

2

3

-6

x 10

-2
12,965

-1

0

1

2

3

12,976

12,971

12,966

12,968-1

12,973-1

12,978Frequency [cm ]12,977Frequency [cm ]12,972Frequency [cm-1 ]12,967

12,97912,974

12,96912,98012,975

12,970

P
Q
R

data

P
Q
R

data

P
Q
R

data

101

Fractional Absorption

Fractional Absorption

Fractional Absorption

-6

x 10

-2
12,990

-1

0

1

2

3

-6

x 10

-2
12,985

-1

0

1

2

3

-6

x 10

-2
12,980

-1

0

1

2

3

12,991

12,986

12,981

12,983-1

12,988-1

12,993Frequency [cm ]12,992Frequency [cm ]12,987Frequency [cm-1 ]12,982

12,994

12,98912,984

12,995

12,99012,985

P
Q
R

data

P
Q
R

data

P
Q
R

data

102

Fractional Absorption

Fractional Absorption

Fractional Absorption

-6

x 10

-2
13,005

-1

0

1

2

3

-6

x 10

-2
13,000

-1

0

1

2

3

-6

x 10

-2
12,995

-1

0

1

2

3

13,006

13,001

12,996

12,998-1

13,003-1

13,008Frequency [cm ]13,007Frequency [cm ]13,002Frequency [cm-1 ]12,997

13,00913,004

12,99913,01013,00513,000

P
Q
R

data

P
Q
R

data

P
Q
R

data

103

Appendix C
Frequency Comb Velocity-Modulation Data

On the following pages, all of the data acquired with frequency comb velocity-modulation
spectroscopy with HfF+ present are plotted in 5 cm−1 segments. For the four HfF+ rotational
bands identified, the line positions from the fit are shown offset for clarity. The first figure
shows a summary of the spectral region covered and a summary of the region containing thefour identified bands. There are many unassigned weaker lines but at the time this thesis
was completed it was not clear whether these indicate the presence of yet another band, orwhether instead these lines are aliases from nearby high-intensity lines.

105

Summary of Frequency Comb Velocity-Modulation Data3

x 10-5

fractional signal

2
1
0
-1
-2
-3
11,70011,90012,10012,30012,50012,70012,25012,300-1

frequency [cm ]
-5

x 10

fractional absorption

3
2
1
0
-1
-2
12,05012,10012,15012,200-1Frequency [cm ]+data HfF present

1 +fit for 1P
(0,1)
1 - S
1
1 +
fit for P
(1,2)
1 - S

3

1

+

fit for P
(2,0)
1 - S
3
1 +
fit for P
(3,1)
1 - S

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
11725

−1

0

1

2

3

−5

x 10

−2
11720

−1

0

1

2

3

−2
11715

−1

0

1

2

3

−5

x 10

11726

11721

11716

11727

11722

11717

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

11728

11723

11718

11729

11724

11719

11730

11725

11720

106

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
11740

−1

0

1

2

3

−5

x 10

−2
11735

−1

0

1

2

3

−2
11730

−1

0

1

2

3

−5

x 10

11741

11736

11731

11742

11737

11732

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

11743

11738

11733

11744

11739

11734

11745

11740

11735

107

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
11755

−1

0

1

2

3

−5

x 10

−2
11750

−1

0

1

2

3

−2
11745

−1

0

1

2

3

−5

x 10

11756

11751

11746

11757

11752

11747

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

11758

11753

11748

11759

11754

11749

11760

11755

11750

108

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
11770

−1

0

1

2

3

−5

x 10

−2
11765

−1

0

1

2

3

−2
11760

−1

0

1

2

3

−5

x 10

11771

11766

11761

11772

11767

11762

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

11773

11768

11763

11774

11769

11764

11775

11770

11765

109

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
11785

−1

0

1

2

3

−5

x 10

−2
11780

−1

0

1

2

3

−2
11775

−1

0

1

2

3

−5

x 10

11786

11781

11776

11787

11782

11777

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

11788

11783

11778

11789

11784

11779

11790

11785

11780

110

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
11800

−1

0

1

2

3

−5

x 10

−2
11795

−1

0

1

2

3

−2
11790

−1

0

1

2

3

−5

x 10

11801

11796

11791

11802

11797

11792

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

11803

11798

11793

11804

11799

11794

11805

11800

11795

111

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
11815

−1

0

1

2

3

−5

x 10

−2
11810

−1

0

1

2

3

−2
11805

−1

0

1

2

3

−5

x 10

11816

11811

11806

11817

11812

11807

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

11818

11813

11808

11819

11814

11809

11820

11815

11810

112

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
11830

−1

0

1

2

3

−5

x 10

−2
11825

−1

0

1

2

3

−2
11820

−1

0

1

2

3

−5

x 10

11831

11826

11821

11832

11827

11822

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

11833

11828

11823

11834

11829

11824

11835

11830

11825

113

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
11845

−1

0

1

2

3

−5

x 10

−2
11840

−1

0

1

2

3

−2
11835

−1

0

1

2

3

−5

x 10

11846

11841

11836

11847

11842

11837

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

11848

11843

11838

11849

11844

11839

11850

11845

11840

114

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
11860

−1

0

1

2

3

−5

x 10

−2
11855

−1

0

1

2

3

−2
11850

−1

0

1

2

3

−5

x 10

11861

11856

11851

11862

11857

11852

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

11863

11858

11853

11864

11859

11854

11865

11860

11855

115

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
11875

−1

0

1

2

3

−5

x 10

−2
11870

−1

0

1

2

3

−2
11865

−1

0

1

2

3

−5

x 10

11876

11871

11866

11877

11872

11867

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

11878

11873

11868

11879

11874

11869

11880

11875

11870

116

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
11890

−1

0

1

2

3

−5

x 10

−2
11885

−1

0

1

2

3

−2
11880

−1

0

1

2

3

−5

x 10

11891

11886

11881

11892

11887

11882

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

11893

11888

11883

11894

11889

11884

11895

11890

11885

117

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
11905

−1

0

1

2

3

−5

x 10

−2
11900

−1

0

1

2

3

−2
11895

−1

0

1

2

3

−5

x 10

11906

11901

11896

11907

11902

11897

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

11908

11903

11898

11909

11904

11899

11910

11905

11900

118

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
11920

−1

0

1

2

3

−5

x 10

−2
11915

−1

0

1

2

3

−2
11910

−1

0

1

2

3

−5

x 10

11921

11916

11911

11922

11917

11912

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

11923

11918

11913

11924

11919

11914

11925

11920

11915

119

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
11935

−1

0

1

2

3

−5

x 10

−2
11930

−1

0

1

2

3

−2
11925

−1

0

1

2

3

−5

x 10

11936

11931

11926

11937

11932

11927

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

11938

11933

11928

11939

11934

11929

11940

11935

11930

120

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
11950

−1

0

1

2

3

−5

x 10

−2
11945

−1

0

1

2

3

−2
11940

−1

0

1

2

3

−5

x 10

11951

11946

11941

11952

11947

11942

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

11953

11948

11943

11954

11949

11944

11955

11950

11945

121

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
11965

−1

0

1

2

3

−5

x 10

−2
11960

−1

0

1

2

3

−2
11955

−1

0

1

2

3

−5

x 10

11966

11961

11956

11967

11962

11957

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

11968

11963

11958

11969

11964

11959

11970

11965

11960

122

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
11980

−1

0

1

2

3

−5

x 10

−2
11975

−1

0

1

2

3

−2
11970

−1

0

1

2

3

−5

x 10

11981

11976

11971

11982

11977

11972

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

11983

11978

11973

11984

11979

11974

11985

11980

11975

123

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
11995

−1

0

1

2

3

−5

x 10

−2
11990

−1

0

1

2

3

−2
11985

−1

0

1

2

3

−5

x 10

11996

11991

11986

11997

11992

11987

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

11998

11993

11988

11999

11994

11989

12000

11995

11990

124

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12010

−1

0

1

2

3

−5

x 10

−2
12005

−1

0

1

2

3

−2
12000

−1

0

1

2

3

−5

x 10

12011

12006

12001

12012

12007

12002

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12013

12008

12003

12014

12009

12004

12015

12010

12005

125

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12025

−1

0

1

2

3

−5

x 10

−2
12020

−1

0

1

2

3

−2
12015

−1

0

1

2

3

−5

x 10

12026

12021

12016

12027

12022

12017

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12028

12023

12018

12029

12024

12019

12030

12025

12020

126

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12040

−1

0

1

2

3

−5

x 10

−2
12035

−1

0

1

2

3

−2
12030

−1

0

1

2

3

−5

x 10

12041

12036

12031

12042

12037

12032

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12043

12038

12033

12044

12039

12034

12045

12040

12035

127

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12055

−1

0

1

2

3

−5

x 10

−2
12050

−1

0

1

2

3

−2
12045

−1

0

1

2

3

−5

x 10

12056

12051

12046

12057

12052

12047

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12058

12053

12048

12059

12054

12049

12060

12055

12050

128

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12070

−1

0

1

2

3

−5

x 10

−2
12065

−1

0

1

2

3

−2
12060

−1

0

1

2

3

−5

x 10

12071

12066

12061

12072

12067

12062

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12073

12068

12063

12074

12069

12064

12075

12070

12065

129

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12085

−1

0

1

2

3

−5

x 10

−2
12080

−1

0

1

2

3

−2
12075

−1

0

1

2

3

−5

x 10

12086

12081

12076

12087

12082

12077

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12088

12083

12078

12089

12084

12079

12090

12085

12080

130

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12100

−1

0

1

2

3

−5

x 10

−2
12095

−1

0

1

2

3

−2
12090

−1

0

1

2

3

−5

x 10

12101

12096

12091

12102

12097

12092

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12103

12098

12093

12104

12099

12094

12105

12100

12095

131

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12115

−1

0

1

2

3

−5

x 10

−2
12110

−1

0

1

2

3

−2
12105

−1

0

1

2

3

−5

x 10

12116

12111

12106

12117

12112

12107

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12118

12113

12108

12119

12114

12109

12120

12115

12110

132

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12130

−1

0

1

2

3

−5

x 10

−2
12125

−1

0

1

2

3

−2
12120

−1

0

1

2

3

−5

x 10

12131

12126

12121

12132

12127

12122

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12133

12128

12123

12134

12129

12124

12135

12130

12125

133

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12145

−1

0

1

2

3

−5

x 10

−2
12140

−1

0

1

2

3

−2
12135

−1

0

1

2

3

−5

x 10

12146

12141

12136

12147

12142

12137

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12148

12143

12138

12149

12144

12139

12150

12145

12140

134

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12160

−1

0

1

2

3

−5

x 10

−2
12155

−1

0

1

2

3

−2
12150

−1

0

1

2

3

−5

x 10

12161

12156

12151

12162

12157

12152

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12163

12158

12153

12164

12159

12154

12165

12160

12155

135

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12175

−1

0

1

2

3

−5

x 10

−2
12170

−1

0

1

2

3

−2
12165

−1

0

1

2

3

−5

x 10

12176

12171

12166

12177

12172

12167

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12178

12173

12168

12179

12174

12169

12180

12175

12170

136

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12190

−1

0

1

2

3

−5

x 10

−2
12185

−1

0

1

2

3

−2
12180

−1

0

1

2

3

−5

x 10

12191

12186

12181

12192

12187

12182

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12193

12188

12183

12194

12189

12184

12195

12190

12185

137

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12205

−1

0

1

2

3

−5

x 10

−2
12200

−1

0

1

2

3

−2
12195

−1

0

1

2

3

−5

x 10

12206

12201

12196

12207

12202

12197

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12208

12203

12198

12209

12204

12199

12210

12205

12200

138

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12220

−1

0

1

2

3

−5

x 10

−2
12215

−1

0

1

2

3

−2
12210

−1

0

1

2

3

−5

x 10

12221

12216

12211

12222

12217

12212

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12223

12218

12213

12224

12219

12214

12225

12220

12215

139

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12235

−1

0

1

2

3

−5

x 10

−2
12230

−1

0

1

2

3

−2
12225

−1

0

1

2

3

−5

x 10

12236

12231

12226

12237

12232

12227

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12238

12233

12228

12239

12234

12229

12240

12235

12230

140

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12250

−1

0

1

2

3

−5

x 10

−2
12245

−1

0

1

2

3

−2
12240

−1

0

1

2

3

−5

x 10

12251

12246

12241

12252

12247

12242

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12253

12248

12243

12254

12249

12244

12255

12250

12245

141

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12265

−1

0

1

2

3

−5

x 10

−2
12260

−1

0

1

2

3

−2
12255

−1

0

1

2

3

−5

x 10

12266

12261

12256

12267

12262

12257

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12268

12263

12258

12269

12264

12259

12270

12265

12260

142

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12280

−1

0

1

2

3

−5

x 10

−2
12275

−1

0

1

2

3

−2
12270

−1

0

1

2

3

−5

x 10

12281

12276

12271

12282

12277

12272

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12283

12278

12273

12284

12279

12274

12285

12280

12275

143

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12295

−1

0

1

2

3

−5

x 10

−2
12290

−1

0

1

2

3

−2
12285

−1

0

1

2

3

−5

x 10

12296

12291

12286

12297

12292

12287

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12298

12293

12288

12299

12294

12289

12300

12295

12290

144

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12310

−1

0

1

2

3

−5

x 10

−2
12305

−1

0

1

2

3

−2
12300

−1

0

1

2

3

−5

x 10

12311

12306

12301

12312

12307

12302

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12313

12308

12303

12314

12309

12304

12315

12310

12305

145

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12325

−1

0

1

2

3

−5

x 10

−2
12320

−1

0

1

2

3

−2
12315

−1

0

1

2

3

−5

x 10

12326

12321

12316

12327

12322

12317

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12328

12323

12318

12329

12324

12319

12330

12325

12320

146

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12340

−1

0

1

2

3

−5

x 10

−2
12335

−1

0

1

2

3

−2
12330

−1

0

1

2

3

−5

x 10

12341

12336

12331

12342

12337

12332

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12343

12338

12333

12344

12339

12334

12345

12340

12335

147

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12355

−1

0

1

2

3

−5

x 10

−2
12350

−1

0

1

2

3

−2
12345

−1

0

1

2

3

−5

x 10

12356

12351

12346

12357

12352

12347

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12358

12353

12348

12359

12354

12349

12360

12355

12350

148

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12370

−1

0

1

2

3

−5

x 10

−2
12365

−1

0

1

2

3

−2
12360

−1

0

1

2

3

−5

x 10

12371

12366

12361

12372

12367

12362

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12373

12368

12363

12374

12369

12364

12375

12370

12365

149

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12385

−1

0

1

2

3

−5

x 10

−2
12380

−1

0

1

2

3

−2
12375

−1

0

1

2

3

−5

x 10

12386

12381

12376

12387

12382

12377

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12388

12383

12378

12389

12384

12379

12390

12385

12380

150

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12400

−1

0

1

2

3

−5

x 10

−2
12395

−1

0

1

2

3

−2
12390

−1

0

1

2

3

−5

x 10

12401

12396

12391

12402

12397

12392

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12403

12398

12393

12404

12399

12394

12405

12400

12395

151

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12415

−1

0

1

2

3

−5

x 10

−2
12410

−1

0

1

2

3

−2
12405

−1

0

1

2

3

−5

x 10

12416

12411

12406

12417

12412

12407

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12418

12413

12408

12419

12414

12409

12420

12415

12410

152

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12430

−1

0

1

2

3

−5

x 10

−2
12425

−1

0

1

2

3

−2
12420

−1

0

1

2

3

−5

x 10

12431

12426

12421

12432

12427

12422

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12433

12428

12423

12434

12429

12424

12435

12430

12425

153

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12445

−1

0

1

2

3

−5

x 10

−2
12440

−1

0

1

2

3

−2
12435

−1

0

1

2

3

−5

x 10

12446

12441

12436

12447

12442

12437

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12448

12443

12438

12449

12444

12439

12450

12445

12440

154

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12460

−1

0

1

2

3

−5

x 10

−2
12455

−1

0

1

2

3

−2
12450

−1

0

1

2

3

−5

x 10

12461

12456

12451

12462

12457

12452

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12463

12458

12453

12464

12459

12454

12465

12460

12455

155

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12475

−1

0

1

2

3

−5

x 10

−2
12470

−1

0

1

2

3

−2
12465

−1

0

1

2

3

−5

x 10

12476

12471

12466

12477

12472

12467

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12478

12473

12468

12479

12474

12469

12480

12475

12470

156

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12490

−1

0

1

2

3

−5

x 10

−2
12485

−1

0

1

2

3

−2
12480

−1

0

1

2

3

−5

x 10

12491

12486

12481

12492

12487

12482

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12493

12488

12483

12494

12489

12484

12495

12490

12485

157

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12505

−1

0

1

2

3

−5

x 10

−2
12500

−1

0

1

2

3

−2
12495

−1

0

1

2

3

−5

x 10

12506

12501

12496

12507

12502

12497

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12508

12503

12498

12509

12504

12499

12510

12505

12500

158

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12520

−1

0

1

2

3

−5

x 10

−2
12515

−1

0

1

2

3

−2
12510

−1

0

1

2

3

−5

x 10

12521

12516

12511

12522

12517

12512

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12523

12518

12513

12524

12519

12514

12525

12520

12515

159

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12535

−1

0

1

2

3

−5

x 10

−2
12530

−1

0

1

2

3

−2
12525

−1

0

1

2

3

−5

x 10

12536

12531

12526

12537

12532

12527

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12538

12533

12528

12539

12534

12529

12540

12535

12530

160

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12550

−1

0

1

2

3

−5

x 10

−2
12545

−1

0

1

2

3

−2
12540

−1

0

1

2

3

−5

x 10

12551

12546

12541

12552

12547

12542

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12553

12548

12543

12554

12549

12544

12555

12550

12545

161

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12565

−1

0

1

2

3

−5

x 10

−2
12560

−1

0

1

2

3

−2
12555

−1

0

1

2

3

−5

x 10

12566

12561

12556

12567

12562

12557

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12568

12563

12558

12569

12564

12559

12570

12565

12560

162

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12580

−1

0

1

2

3

−5

x 10

−2
12575

−1

0

1

2

3

−2
12570

−1

0

1

2

3

−5

x 10

12581

12576

12571

12582

12577

12572

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12583

12578

12573

12584

12579

12574

12585

12580

12575

163

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12595

−1

0

1

2

3

−5

x 10

−2
12590

−1

0

1

2

3

−2
12585

−1

0

1

2

3

−5

x 10

12596

12591

12586

12597

12592

12587

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12598

12593

12588

12599

12594

12589

12600

12595

12590

164

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12610

−1

0

1

2

3

−5

x 10

−2
12605

−1

0

1

2

3

−2
12600

−1

0

1

2

3

−5

x 10

12611

12606

12601

12612

12607

12602

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12613

12608

12603

12614

12609

12604

12615

12610

12605

165

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12625

−1

0

1

2

3

−5

x 10

−2
12620

−1

0

1

2

3

−2
12615

−1

0

1

2

3

−5

x 10

12626

12621

12616

12627

12622

12617

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12628

12623

12618

12629

12624

12619

12630

12625

12620

166

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12640

−1

0

1

2

3

−5

x 10

−2
12635

−1

0

1

2

3

−2
12630

−1

0

1

2

3

−5

x 10

12641

12636

12631

12642

12637

12632

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12643

12638

12633

12644

12639

12634

12645

12640

12635

167

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12655

−1

0

1

2

3

−5

x 10

−2
12650

−1

0

1

2

3

−2
12645

−1

0

1

2

3

−5

x 10

12656

12651

12646

12657

12652

12647

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12658

12653

12648

12659

12654

12649

12660

12655

12650

168

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12670

−1

0

1

2

3

−5

x 10

−2
12665

−1

0

1

2

3

−2
12660

−1

0

1

2

3

−5

x 10

12671

12666

12661

12672

12667

12662

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12673

12668

12663

12674

12669

12664

12675

12670

12665

169

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12685

−1

0

1

2

3

−5

x 10

−2
12680

−1

0

1

2

3

−2
12675

−1

0

1

2

3

−5

x 10

12686

12681

12676

12687

12682

12677

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12688

12683

12678

12689

12684

12679

12690

12685

12680

170

Fractional Absorption

Fractional Absorption

Fractional Absorption

−5

x 10

−2
12700

−1

0

1

2

3

−5

x 10

−2
12695

−1

0

1

2

3

−2
12690

−1

0

1

2

3

−5

x 10

12701

12696

12691

12702

12697

12692

Frequency [cm ]−1Frequency [cm ]−1Frequency [cm−1]

12703

12698

12693

12704

12699

12694

12705

12700

12695

171