ABSTRACT
Title of dissertation: OPTICAL NANOFIBER FABRICATION AND
ANALYSIS TOWARDS COUPLING ATOMS
TO SUPERCONDUCTING QUBITS
Jonathan Hoffman, Doctor of Philosophy, 2014
Dissertation directed by: Professor Luis Orozco
Department of Physics
We describe advancements towards coupling superconducting qubits to neutral
atoms. To produce a measurably large coupling, the atoms will need to be on the
order of a few micrometers away from the qubit. A consequence of combining super-
conducting qubits and atoms is addressing their operational constraints, such as the
deleterious light effects on superconducting systems and the magnetic field sensitiv-
ity of superconducting qubits. Our group proposes the use optical-nanofiber-based
optical dipole traps to confine atoms near the superconductor. Optical nanofibers
(ONFs) have high-intensity evanescent waves that require less power than equivalent
standard dipole traps.
This thesis focuses on the fabrication and analysis of the behavior of ONFs.
First we present the construction of the pulling apparatus. We outline the necessary
steps for a typical pull, detailing the cleaning and alignment process. Then we
examine the quality of the fibers by measuring their transmission and comparing
our results to other reported measurements, demonstrating a two-order of magnitude
decrease in loss.
Next we present the modal evolution in ONFs using simulations and spec-
trogram analysis. We identify crucial elements to improve the transmission and
demonstrate understanding of the modal dynamics during the pull.
Then we study higher-order modes (HOMs) with ONFs using the first excited
TE01, TM01, and HE21 modes. We demonstrate transmissions greater than 97% for
780 nm light when we launch the first excited LP11 family of modes through fibers
with a 350 nm waist. This setup enables us to launch these three modes with high
purity at the output, where less than 1% of the light is coupled to the fundamental
mode.
We then focus on the identification of modes on the ONF waist. First we
use Rayleigh scattering to identify the modal content of an ONF. Bulk optics can
convert the modes in the ONF, and we observe the controllable conversion of su-
perpositions of modes. Finally, we use an evanescently-coupled tapered optical fiber
probe that allows for the identification of the fundamental mode beating with HOMs
and compare the results to simulations.
OPTICAL NANOFIBER FABRICATION AND ANALYSIS
TOWARDS COUPLING ATOMS TO SUPERCONDUCTING
QUBITS
by
Jonathan Hoffman
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2014
Advisory Committee:
Professor Luis A. Orozco, Chair/Advisor
Professor Steven L. Rolston, Co-Advisor
Professor Mario Dagenais, Dean’s Rep
Professor Frederick C. Wellstood
Professor Christopher J. Lobb
Dr. Fredrik Fatemi
Professor William D. Phillips
c© Copyright by
Jonathan Hoffman
2014
Dedication
To my parents and their endless quest to ensure my happiness and well-being.
ii
Acknowledgments
This thesis is the culmination of six years of work and I could not have done
it without the support of my friends, family, and coworkers. I would like to start by
thanking my family. My parents always allowed me the freedom to pursue whatever
interests I had. They were only concerned with my happiness and encouraged me
to search for a major that piqued my curiosity. My sister is always there for me,
even if it is to inform me my opinion is wrong, often before I state it.
Being a part of the Atoms on SQUIDs team has provided me the opportunity
to work with wonderful people.
First, I must thank my advisor Professor Luis Orozco. His tireless dedication
to and enthusiasm for science and education are the traits I admire most in him.
Over the last five years he has always provided an open door for questions and a
willingness to help. I am fortunate to have an advisor I can rely on to send me
thesis edits within a few hours regardless of the time I sent them, be it midday or
midnight. I must also thank Luis for years of spoiling me and all of his students
with excellent dinners whether they were home-cooked meals or at a restaurant.
I owe a great deal to my co-advisor, Professor Steve Rolston. Your excellent
physical intuition has provided me with an opportunity to learn your approach to
physics. Whenever we have a result that contradicts your intuition we often joke
that we must have made a mistake and more often that not, we have. I truly
appreciate his openness to discussion, be it science or general life advice.
I thank Professor Frederick Wellstood for his dedication to the project and
iii
taking the time to understand all results, even if they are outside his field. If he is
satisfied with my reasoning I can be confident that it is sound.
I have to thank Professor Christopher Lobb for introducing me to the Atom’s
on SQUIDs project. His insight is only surpassed by his good humor. More so than
anything else, as an insomniac, I am envious of his ability to sleep during meetings.
Yet somehow, he always has a better idea of whats going on than I do.
I am incredibly grateful for my collaboration with Dr. Fredrik Fatemi. His
knowledge and intuition for optics has placed me in an unbelievably lucky position
and I am appreciative for everything he has taught me. Thank you for working with
me, even though your ten-year-old son solves a Rubik’s cube 28 years faster than I
can.
Jeffrey Grover and I have worked together on this project nearly from its
inception. I have probably spent more time with him than any other human being
over the last 5 years. I am very lucky to have you as a coworker and friend. Your
superb knowledge of science and current experiments is invaluable and you never
give yourself enough credit for your abilities. I will always appreciate your ability to
keep everything in perspective and I was lucky to have a lab mate that understood
what the truly important things are in life.
Most of the work in this thesis was done in collaboration with Sylvain Ravets.
I have met few people with his drive and no one else with his ability to get things
done. It is impressive. The long hours pulling fibers, in the unreasonably hot clean
room (at least cleanish room), were undercut by your humor and enthusiasm for
problem solving. I think few people quite understand how far a few feet can feel,
iv
especially after the hundredth or so failed attempt at carrying a small piece of glass
across those few feet. Thank you for all the help and the friendship that we share,
it’s some sort of the best.
Pablo Solano’s calm demeanor, genuine interest, and physical intuition make
him a pleasure to work with. I can say with confidence that I’m leaving the project
in better hands. My back thanks you for reminding me that we are physicists and
not weightlifters when I had the idea to lift the 100 kg granite slab.
Peter Kordell was an undergraduate that did not deserve the slight of being
called an undergraduate. His dedication to working through the poorly documented
FIMMPROP led to great results producing dispersion curves and beautiful mode
evolution simulations.
It was serendipitous that I fell into my housing situation my first year of
graduate school. I should probably come clean and admit that I spent two minutes
looking for housing before my first year: approximately one minute emailing Andres
to tell him of my interest in rooming together and a promise that I would help look
for housing, and the other minute emailing Matt to express my interest in the place
he found. Sorry Andres, I never looked for housing, I was just efficient at emailing.
The timing of these emails was fortuitous, since I ended up living with some of
my closest friends in graduate school. Andres, Matt, and later Anjor and I not only
worked through many assignments together, but with two almost proposals and two
almost adoptions I think that almost makes us family. Their friendship along with
David, Rangga, Eric , Evan, and Rufus really helped create a tight-knit community
that I will always cherish. Rufus’ relentless drive is something I aspire to and his
v
continual outwitting of me pushes me to be more creative and strategically minded.
The person I owe the greatest deal of gratitude to is Kristen Voigt. No, I
didn’t forget you, I just wanted to build a bit of tension while you read this. There
is no way for me to adequately express the extent of my appreciation and love for
you. I don’t know how I would have gotten through the last few years without your
support, affection, and organization. I’m pretty hopeless at organization. Thank
you, you are the most wonderful person I will ever know.
As per Eric’s request I acknowledge spongebob square pants. Not his help just
his existence as a comedic entity for children.
If you actually read my thesis, thank you for (potentially) bringing this docu-
ment past the average readership of publications at four reads.
This work was supported by the NSF through the PFC at the JQI.
vi
Table of Contents
List of Figures x
List of Abbreviations xiii
1 Introduction 1
1.1 Quantum computing background . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Physical examples . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 DiVincenzo criteria . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Atoms coupled to superconducting qubits . . . . . . . . . . . . . . . . 4
1.2.1 The interaction and its realization . . . . . . . . . . . . . . . . 6
1.2.2 Proof of principle . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.3 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Optical nanofibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1 Intensity profile and trapping potential . . . . . . . . . . . . . 12
1.4 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Optical nanofiber fabrication 18
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 The fiber-pulling apparatus . . . . . . . . . . . . . . . . . . . 20
2.2.2 Transmission monitoring setup . . . . . . . . . . . . . . . . . . 27
2.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.1 Motor control . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.2 Measurement of the flame width . . . . . . . . . . . . . . . . . 34
2.4 The pulling process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.1 Cleaning procedure . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.2 Alignment procedure . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5.1 Microscopy validation . . . . . . . . . . . . . . . . . . . . . . 40
2.5.2 Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5.3 Spectrogram analysis . . . . . . . . . . . . . . . . . . . . . . . 50
2.6 Power Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
vii
2.7 Dust studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 Optical nanofiber spectrogram analysis 59
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Motivation and construction overview . . . . . . . . . . . . . . . . . . 60
3.3 Modal evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3.1 Modes in a cylindrical waveguide . . . . . . . . . . . . . . . . 61
3.4 Adiabaticity in fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4.1 Adiabaticity criterion . . . . . . . . . . . . . . . . . . . . . . . 68
3.4.2 Transmission of a tapered fiber section . . . . . . . . . . . . . 71
3.4.3 Genetic algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4.4 Fully adiabatic fiber . . . . . . . . . . . . . . . . . . . . . . . 74
3.4.5 Utilizing non-adiabaticity . . . . . . . . . . . . . . . . . . . . 78
3.5 Analysis of transmission signal . . . . . . . . . . . . . . . . . . . . . . 83
3.5.1 Single mode section . . . . . . . . . . . . . . . . . . . . . . . . 85
3.5.2 From single mode to multimode . . . . . . . . . . . . . . . . . 85
3.5.3 Single mode again . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.6 Spectrograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.6.1 Modeling the pull . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.6.2 Identifying the modes . . . . . . . . . . . . . . . . . . . . . . . 93
3.7 Applications of the quality of the pull . . . . . . . . . . . . . . . . . . 98
3.7.1 Internal parameters: algorithm input parameters . . . . . . . 99
3.8 Understanding the losses . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.8.1 Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.8.2 Coupling to higher-order modes . . . . . . . . . . . . . . . . . 105
3.8.3 Rayleigh scattering . . . . . . . . . . . . . . . . . . . . . . . . 105
3.8.4 Systematic effects . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4 Efficient guidance of higher-order modes 108
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.2 Mode propagation in an optical fiber . . . . . . . . . . . . . . . . . . 110
4.2.1 First family of excited modes . . . . . . . . . . . . . . . . . . 110
4.2.1.1 Adiabaticity condition . . . . . . . . . . . . . . . . . 112
4.2.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 113
4.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.2.3.1 Varying the fiber type . . . . . . . . . . . . . . . . . 116
4.2.3.2 Transmission measurements . . . . . . . . . . . . . . 117
4.2.3.3 Spectrograms . . . . . . . . . . . . . . . . . . . . . . 119
4.2.3.4 Imaging the fiber output . . . . . . . . . . . . . . . . 120
4.2.3.5 Varying the angle . . . . . . . . . . . . . . . . . . . . 121
4.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
viii
5 Higher-order mode identification and control 126
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2 IDIOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.3 Rayleigh scattering measurements . . . . . . . . . . . . . . . . . . . . 127
5.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.3.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 135
5.3.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 139
5.3.3.1 Mode identification . . . . . . . . . . . . . . . . . . . 139
5.3.3.2 Mode control . . . . . . . . . . . . . . . . . . . . . . 143
5.3.4 Surface scattering and mode cutoffs . . . . . . . . . . . . . . . 150
5.3.5 Optical fiber probe . . . . . . . . . . . . . . . . . . . . . . . . 159
5.3.6 Photodiode motor jog . . . . . . . . . . . . . . . . . . . . . . 164
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6 Conclusions and Outlook 171
6.1 Hybrid System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.2 ONF studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
A Fiber Modes 177
A.1 Cylindrical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 177
A.2 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
A.3 Cylindrically symmetric dielectric waveguide . . . . . . . . . . . . . . 179
A.4 Two layer step index fiber . . . . . . . . . . . . . . . . . . . . . . . . 182
A.4.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 184
A.4.2 The propagation constant and fiber modes . . . . . . . . . . . 186
A.4.3 Solving for the propagation constants . . . . . . . . . . . . . . 187
A.4.4 Quasilinear polarization . . . . . . . . . . . . . . . . . . . . . 192
A.5 Power Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
A.5.1 HElm and EHlm mode normalizations . . . . . . . . . . . . . 196
Bibliography 201
ix
List of Figures
1.1 Lumped-element LC resonator . . . . . . . . . . . . . . . . . . . . . . 8
1.2 HFSS simulation of the lumped-element LC resonator magnetic field . 9
1.3 Aluminum on sapphire lumped element resonator . . . . . . . . . . . 10
1.4 Fiber trap geometry, intensity profile, and trapping potential . . . . . 13
2.1 A schematic of the fiber pulling setup and photograph. . . . . . . . . 22
2.2 An image of the flame nozzle. . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Transmission monitoring setup for the fundamental mode. . . . . . . 27
2.4 Physical model of taper linking process. . . . . . . . . . . . . . . . . . 32
2.5 Measurement of the effective flame size . . . . . . . . . . . . . . . . . 36
2.6 Fiber alignment image of left and right fiber clamps. . . . . . . . . . 39
2.7 Measured vs simulated profile for a three angle microfiber. . . . . . . 42
2.8 SEM image of an ONF. . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.9 Normalized transmission during the pull for the fundamental mode. . 45
2.10 Loss in dB/mm vs fiber radius comparing previous groups results
with our ONFs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.11 Spectrogram of transmission. . . . . . . . . . . . . . . . . . . . . . . . 50
2.12 Transmission and spectrogram comparison for an uncleaned fiber. . . 53
2.13 Core- to cladding-guidance transition simulated in FIMMPROP . . . 56
3.1 Schematic of the stretched fiber and calculated intensity profile. . . . 63
3.2 Dispersion relations for various modes inside the fiber. . . . . . . . . 67
3.3 Adiabatic criterion for a fiber taper presented as taper angle vs radius. 70
3.4 FIMMPROP simulation of transmission in a section of fiber taper
(from 25.5 to 23.5 µm). . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.5 Optimal adiabatic tapers calculated with the genetic algorithm for
99.90% and 99.99% transmission. . . . . . . . . . . . . . . . . . . . . 76
3.6 Optimal taper profile using the genetic algorithm for 99.90% and
99.99% transmission. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.7 Simulation of fiber profile for 99.99% transmission with optimized
length given by the genetic algorithm. . . . . . . . . . . . . . . . . . . 79
3.8 Mode evolution for a 2 mrad linear fiber down to 250 nm radius. . . . 82
x
3.9 Transmission through a 2 mrad fiber as a function of time during the
manufacturing process and corresponding waist radius evolution. . . . 84
3.10 Detailed view of the transmission of the 2 mrad pull shown in Fig. 3.9. 86
3.11 Schematic showing modal evolution in the transition region. . . . . . 88
3.12 Simulation of fiber stretch for fiber sample 1.a . . . . . . . . . . . . . 91
3.13 Transmission spectrogram of a 2 mrad pull. . . . . . . . . . . . . . . 92
3.14 Flow chart of spectrogram overlap process . . . . . . . . . . . . . . . 94
3.15 Simulated taper profile from step N and βi(r(z)) . . . . . . . . . . . 95
3.16 Simulation of the differences between the fundamental mode and the
first four excited modes of family one. . . . . . . . . . . . . . . . . . . 96
3.17 Identification of the modes beating for the 2 mrad tapered fiber spec-
trogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.18 Normalized transmission through the fiber as a function of time dur-
ing the manufacturing process for a two-angle pull from 2 to 0.75
mrad taper. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.19 Spectrogram of the transmission data and mode identification overlap
for the two-angle pull. . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.20 Study of the asymmetry of a two-angle pulled fiber. . . . . . . . . . . 103
4.1 Effective indices of refraction vs V number with corresponding HOMs.112
4.2 Experimental setup for HOM efficient guidance . . . . . . . . . . . . 113
4.3 HOM transmission as a function of initial fiber radius and spectrogram116
4.4 HOM core-cladding transition . . . . . . . . . . . . . . . . . . . . . . 120
4.5 HOM transmission as a function of taper angle and FIMMPROP
simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.1 Beat length vs ONF radius . . . . . . . . . . . . . . . . . . . . . . . . 130
5.2 Inverse beat length vs ONF radius . . . . . . . . . . . . . . . . . . . 131
5.3 Inverse beat frequency simulation for rw = 390 nm with Ω = 1 mrad
taper. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.4 HOM transverse mode profiles . . . . . . . . . . . . . . . . . . . . . 133
5.5 HOM interference mode profile . . . . . . . . . . . . . . . . . . . . . 134
5.6 HOM identification experimental setup . . . . . . . . . . . . . . . . . 135
5.7 Rayleigh Scattering mode identification imaging system . . . . . . . . 137
5.8 HOM 50x montage transverse polarization . . . . . . . . . . . . . . . 140
5.9 Rayleigh Scattering depicting HOMs leaking transitioning from core
to cladding modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.10 Scattered power along ONF waist . . . . . . . . . . . . . . . . . . . . 142
5.11 Scattered power vs position and FFT for two input polarizations. . . 145
5.12 Conversion of HE21 and TM01 to HE21 and TE01 as a function of
half-wave plate angle . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.13 Conversion of HE21 and TM01 to HE21 and TE01 longitudinal and
transverse components as a function of half-wave plate angle . . . . . 147
5.14 Contrast in the relative frequency band for HOM mode conversion . 148
5.15 Pure mode conversion as a function of half-wave plate angle . . . . . 149
xi
5.16 HE21 cutoff observable in Rayleigh scattering intensity . . . . . . . . 151
5.17 Rayleigh scattering vs position and Spectrogram for Ω = 0.5 mrad
and rw = 300 nm ONF . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.18 HOM spectrogram overlap and radius extraction. . . . . . . . . . . . 157
5.19 HOM spectrogram overlap and radius extraction with TM01 mode. . 158
5.20 Image of fiber probe setup . . . . . . . . . . . . . . . . . . . . . . . . 160
5.21 Power collected through fiber probe as a function of position along
ONF waist. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.22 FFT of power collected using the fiber probe. . . . . . . . . . . . . . 162
5.23 Probe power collected on a PD while jogging the motors. . . . . . . . 165
5.24 Spectrogram of data from Fig. 5.23. . . . . . . . . . . . . . . . . . . . 166
5.25 FFT of power collected from ONF waist from 5.23 and table of ex-
tracted beat frequencies. . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.26 Spectrogram with overlap. . . . . . . . . . . . . . . . . . . . . . . . . 168
5.27 Measured vs Simulated radius and deviation from simulation. . . . . 169
6.1 Hybrid quantum system setup . . . . . . . . . . . . . . . . . . . . . . 173
6.2 Γ1D calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
A.1 Cylindrically symmetric waveguide, with constant layers of index of
refraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
A.2 Two layer cylindrically symmetric waveguide . . . . . . . . . . . . . . 182
A.3 Determining the HE1m and EH1m propagation constants . . . . . . . 188
A.4 Determining the TE0m and TM0m propagation constants . . . . . . . 189
A.5 Effective indices of refraction vs V number . . . . . . . . . . . . . . . 190
A.6 Region of closest approach for neff . . . . . . . . . . . . . . . . . . . 192
A.7 HE11 Intensity with quasilinear polarization. . . . . . . . . . . . . . . 193
A.8 Ratio of power in an annulus surrounding an ONF. . . . . . . . . . . 194
xii
List of Abbreviations
Ω the fiber taper angle
APD avalanche photodiode
FC fiber channel / ferrule connector
FWHM full width at half maximum
HOM Higher-order mode, specifically TE01, TM01, and HE21
HWHM half width at half maximum (3 dB point)
HWP half-wave plate
JC Jaynes-Cummings
JQI Joint Quantum Institute
Lw length of the fiber waist
MOT magneto-optical trap
NISP Nanoscale Imaging Spectroscopy and Properties Laboratory
NIST National Institutes of Standards and Technology
NSF National Science Foundation
ONF Optical nanofiber
PBS Polarizing beam splitter
PC Polarization controller
PM polarization maintaining
PZT piezo-electric transducer
QED quantum electrodynamics
QWP quarter-wave plate
Rb Rubidium
rw radius of the optical nanofiber waist
SEM scanning electron microscope
SS Stainless steel
UHV ultra-high vacuum
xiii
Chapter 1: Introduction
1.1 Quantum computing background
The field of quantum computation arguably began with Feynman’s paper [1] on
quantum simulation. While the Deutsch-Jozsa algorithm [2] was the first example of
a quantum algorithm that outperformed classical computation, it was Peter Shor’s
work [3] concerning prime factorization that garnered widespread interest. This
work gave quantum computation national security implications since the widely
used RSA encryption, a public key encryption named after Rivest, Shamir, and
Adleman, relies on the difficulty involved in factoring large numbers into primes.
With the advent of quantum error correction codes [4–7] the feasibility of quantum
computation, while still daunting, significantly improved.
Although Shor’s [3] and Grover’s [8] algorithms give credence to the potential
application of a quantum processor the concept of quantum simulation [1, 9, 10] is
of fundamental interest in physics. Full control of a qubit, understanding of all its
interactions, manipulating and measuring it is a worthy study on its own, and opens
insights into our understanding of physical phenomena. Analog and Digital quan-
tum simulators can be used to study classically intractable problems [11]. A fully
functioning quantum computer, one that could run Shor’s or Grover’s algorithm,
1
might not be necessary for these types of problems. In principle, a quantum simu-
lator could study specific Hamiltonians, modeling various physical systems such as
quantum magnetism [12–15] and high-temperature superconductivity [16–18].
1.1.1 Physical examples
There are a wide variety of physical incarnations of qubits, each with its own
benefits and limitations. Some qubits exhibit better isolation from their environ-
ment, having longer coherence times, while others have fast gate times, or more
straightforward means of scalability. Examples include atoms [19–21], ions [22–24],
electrons on helium [25], superconducting qubits [26,27], NV centers [28] and Phos-
phorus donors in Silicon-28 [29].
Many of these systems have seen great advances in the last decade. For in-
stance, the ground state hyperfine splitting in a Rb atomic qubit was demonstrated
to have a 2 second coherence time in 2004 [19]. Addressing a singular 13C nucleus
near an NV center [28] has shown coherence times on the order of seconds. Recently,
the longest observed coherence time was measured in Ionized Phosphorus Donors
in Silicon-28. Exhibiting a room temperature coherence time of 39 minutes, and a
coherence time in excess of 180 minutes at cryogenic temperatures [29].
Qubits based on Josephson junctions have seen remarkable advances in the
last decades with the development of the charge [30], phase [31–33], and flux qubits
[34–36]. Later the development of circuit Quantum Electrodynamics (cQED) [37],
Transmons [38], Fluxonium [39], and Transmons in 3D cavities [40], have seen the
2
coherence times improve by about five orders of magnitude [27].
Despite major advancements in all of these systems it is not clear that any
single qubit currently satisfies the DiVincenzo criteria [41, 42] : a set of conditions
describing the necessary qualities this qubit must satisfy in order to construct a
universal quantum computer.
1.1.2 DiVincenzo criteria
Briefly, a quantum computer according to DiVincenzo [41,42] will need:
1. Scalability and well defined Hilbert Space: A quantum computer requires full
knowledge of the energy level structure of a given quantum system. Having
a well-defined two level system, a qubit, allows the system to be controllably
addressed and manipulated.
2. Initialization: It is necessary to be able to place a qubit in a predefined initial
state. This allows for calculations to begin with known conditions.
3. Long coherence time relative to gate operation time: The time scale of in-
formation loss to the environment, decoherence, must be long relative to the
time it takes to perform an operation. In this way, the quantum computer
can perform many operations before control of the qubit is lost. Furthermore,
decoherence can lead to errors in calculations over time.
4. Universal set of quantum gates: A quantum computer must be able to perform
a controllable sequence of unitary operations that are functionally complete, a
3
set of operations that can be used reach any desired state in the computational
Hilbert space.
5. Qubit specific strong measurement: Finally, it is necessary to measure final
state of each qubit with high fidelity.
DiVincenzo also developed two requirements for quantum communication:
1. Stationary to flying qubits: This conversion allows for the concept of compu-
tation nodes for computation and the ability to convert and send the result
elsewhere.
2. Transmission of flying qubits: Conversion of a stationary to flying qubit is not
quite stringent enough a condition, the flying qubit must be capable of being
transported successfully to another site.
1.2 Atoms coupled to superconducting qubits
The idea behind a hybrid quantum processor is to exploit the best aspects of
different types of qubits. Our group proposes a scheme to couple the ground state
hyperfine transition of neutral 87Rb atoms to a superconducting (SC) qubit. Atomic
systems exhibit excellent isolation from the environment [19], while condensed mat-
ter systems benefit from technological developments in microfabrication, leading to
a scalable qubit architecture [43].
For the ease of discussion here we will use a flux qubit, but the ultimate
SC qubit may be a transmon, fluxonium, or other qubit and is not important for
4
the conceptual understanding of the system. A flux qubit consists of one or more
Josephson junctions in a micrometer-sized superconducting loop [35, 44, 45]. The
supercurrent can exist in a superposition of clockwise and counterclockwise flow.
Many designs involve three or more junctions with one junction smaller than the
others and chosen so that its Josephson inductance dominates the geometrical loop
inductance. We can use the magnetic field from this SQUID loop to drive the atomic
transition. The flux qubit can possess a relatively long coherence time, on the order
of 10 µs [46] at the flux degeneracy point in a four-junction SQUID [47], and a
tunability range of over a GHz [48].
The atomic qubit in our approach consists of two magnetic sublevels on one
of the hyperfine ground state manifolds of 87Rb, between the |5S1/2;F = 1〉 and
|5S1/2;F = 2〉 states. These states are separated by about 6.83 GHz and one of the
Zeeman pairs forms a clock transition that is insensitive to magnetic field to first
order.
We note that there are similar proposals that focus on creating a memory
by coupling ions [43, 49, 50], neutral atoms [51, 52], or molecules [53, 54] to a SC
stripline resonator [55]. Nitrogen-vacancy (NV) centers have taken a prominent role
in these designs because of the ease with which these natural ion traps integrate
with SC systems [56]. Recent results demonstrate strong coupling of NV centers to
a stripline resonator [25,57]. It is worth noting that most other proposals involving
neutral atoms and superconducting circuits focus on magnetic traps to confine the
atoms [52,58–60] rather than optical traps.
5
1.2.1 The interaction and its realization
A flux qubit in its excited state can induce magnetic dipole transitions in the
ground state hyperfine manifold of 87Rb. The coupling occurs via the interaction
Hamiltonian, −
∑
iµi ·B, where µi is the magnetic dipole moment of the ith in-
dividual atom, and B is the magnetic field from the SC qubit. To calculate the
strength of the coupling we must first determine the strength of the magnetic field.
Assuming that only geometrical inductance is present the average magnetic
field associated with a single microwave photon is:
B =
√
µ0~ω
2Veff
, (1.1)
where ω is the frequency of the photon, µ0 the permeability of free space, and Veff
the effective mode volume. We take the effective mode volume for a square SQUID
loop with 10 µm × 10 µm sides and width of 5 µm to be about 10 × 10−16 m3,
assuming the field is confined within 5 micrometers above and below the loop. For a
photon at 6.8 GHz this gives a magnetic field of about 10 nT, which for the typical
moment of µ = 1.4× 1010 Hz/T gives a coupling strength of roughly 100 Hz for one
atom.
The field associated with a single quantum flux, Φ0 = h/2e = 2 ∗ 10−15Tm2,
in a loop of this size corresponds to 0.2 Gauss, a much higher value than the single
photon. The coupling of one magnetic flux to the atomic magnetic moment would be
lowered by some geometric factor determined by the specific shape of the magnetic
field obtaining a single flux coupling larger than 100 Hz. Further understanding of
6
the field distribution and the inductance of the SQUID will help narrow the range
of this number.
1.2.2 Proof of principle
As a proof of principle, our short-term goal is to first couple atoms to a high-Q
lumped-element superconducting LC resonator, tuned to the 87Rb hyperfine split-
ting [61]. The SC resonator affords a few advantages over a SC qubit. First, the
fabrication of the SC resonator is simpler. There are fewer production steps neces-
sary and resonators less sensitive and less prone to destruction than a qubit. Second,
the length of the inductor can be made relatively long (about 1 mm). This length is
beneficial because it increases the possible number of trapped atoms and therefore
the coupling strength. Here, the interaction results from a (Cavity QED) Jaynes-
Cummings Hamiltonian and the coupling strength of the system scales with the
square root of the number of atoms. Having many atoms trapped in a small volume
is difficult so if the length of the inductance line, which carries the magnetic field,
can be made long, we can potentially increase the number of atoms and therefore
the coupling strength.
7
13
2
380 μm
330
 μm
35 μm
Figure 1.1: Lumped-element LC resonator. 1: microwave transmission line. 2:
interdigitated capacitor. 3: meandering inductor.
Figure 1.2.2 shows our 2010 design for a thin-film lumped-element LC super-
conducting resonator. This niobium resonator consists of a meandering inductor and
an interdigitated capacitor coupled to a transmission line. At T=12 mK and on res-
onance at 6.863 GHz, the transmission through the microwave line decreases by 1.5
dB, and the loaded quality factor is 40,000. We simulate the electromagnetic fields
of the SC resonator with the software package, High Frequency Structure Simulator
(HFSS), to determine the position of maximal magnetic field and uniformity.
8
1.000e-3
8.750e-4
7.500e-4
6.250e-4
5.000e-4
3.750e-4
2.500e-4
1.250e-4
0.000e+0
H Field [A/M]
35 µm
Ground Plane
Figure 1.2: Results from a HFSS simulation of the magnetic field produced by the
lumped-element LC resonator.
The coupling strength of an atom to the resonator depends on the specific
geometry but may not reach the single photon value that we estimate for the SQUID.
It would be lower by an order of magnitude, for example, if the resonator inductor
is ten times longer than the SQUID. Figure 2, shows that the magnetic field outside
of the SC for the early design. The magnetic field lines encircle the edges of the
meandering inductor. We estimate the effective mode volume of this LC resonator
to be around 1×10−13 m3, which yields a coupling on the order of 10 Hz per atom.
For this estimate, we assume the atoms to be about 10 µm above the SC surface,
independent of the device being a SQUID or lumped resonator [62].
The strongest interaction will occur if the resonator is on resonance with the
hyperfine splitting of 87Rb. Fabricating a resonator with a resonance that overlaps
9
with on the Rb hyperfine splitting is challenging [61], making an in situ tuning
technique desirable. In Ref. [61] we employ an Al pin as a frequency tuner by
placing it above the inductor and using a piezo-electrically driven stage to change
the inductance of the resonator. Using this method, we demonstrate tuning of
the resonator within 2 kHz (well within the bandwidth of the resonator) of the
hyperfine splitting in 87Rb, with only minor degradation of the Q due to the tuning
mechanism [61].
~0.9 mm
Figure 1.3: Aluminum on Sapphire lumped element resonator fabricated by J.
Hertzberg [64].
10
Fig. 1.2.2 shows the 2013 resonator design that J. B. Hertzberg developed [64].
This design increases the inductor line length from a few hundred micrometers to
nearly a millimeter. At 13 mK this resonator has a resonance of 6.140651 GHz with
a high power Q of over 257,000. Eventually one could even imagine using the LC
resonator as an impedance matcher between the atoms and a SC qubit.
1.2.3 Constraints
The resonator operates inside a Triton 200 Cryofree Dilution Refrigerator from
Oxford Instruments at 12 mK. The 200 µW cooling power of the mixing chamber
and degradation of the quality factor of SC resonators in the presence of light or a
strong magnetic field present an unusual set of constraints for trapping atoms. In
particular, these design constraints make the magneto-optical trap (MOT) and a
conventional optical dipole trap difficult to implement [63]. Also, studies of light
interaction with SC circuits shows tight light constraints leading to quasiparticle
formation and degradation of Q [64–67].
These constraints necessitate a unique atomic trap. With this in mind we
wish to use a trap that requires no magnetic field and minimizes the dissipated or
scattered optical power from the atomic trap. One way to reduce the power is to
use an optical dipole trap that has a small mode field area. To realize this we seek
to employ an evanescent wave-based dipole trap outside an optical nanofiber whose
diameter is smaller than the wavelength of input light [68–71], see Fig. 1.4.
11
1.3 Optical nanofibers
ONFs can be tailored to have tightly bound evanescent fields, see Sec. 1.3.1 and
Appendix A. The tight confinement of the mode field allows us to achieve the same
field strength for significantly less input power when compared to a conventional
optical dipole trap formed by focusing a gaussian beam. Furthermore, unlike a
gaussian beam, which will expand more rapidly the tighter the focus, we do not face
the same design limitations with ONFs: as long as there is waveguide the light will
propagate without expending radially. Also, incredibly low losses are achievable
with ONFs, see Chap 2, making them an ideal candidate for the hybrid system.
Finally, ONFs offer a simple geometry that is well-suited for coupling atoms to SC
qubits.
1.3.1 Intensity profile and trapping potential
Figure 1.4 (a) Shows the ONF trapping schematic [72]. We couple linearly
polarized red- and blue-detuned light from the D2 transition in 87Rb into an ONF.
The red-detuned light is orthogonally polarized to the blue-detuned light, to mini-
mize light shifts on the atomic levels [69,71,73]. Red-detuned light is also launched
through the fiber output to create a standing wave on the fiber waist, forming a 1D
lattice on either side of the fiber.
12
Figure 1.4: (a) The trapping schematic for a two-color ONF optical dipole trap. (b)
An inset showing the red- and blue-detuned light surrounding the ONF waist. (c)
The intensity profile for horizontally polarized 1064 nm light launched into an 180
nm radius waist ONF. Where the color scale indicates increasing intensity from blue
to red. (d) The intensity profile for vertically polarized 730 nm nm light propagating
on an 180 nm radius waist ONF. (e) The trapping potential taken along a slice of the
abscissa from the fiber surface with 7.5 mW at 730 nm and 4.5 mW in a standing
wave configuration for 1064 nm light.
13
The inset in Fig. 1.4 (b) shows the ONF waist with the evanescent wave formed
by the trapping fields. Here we can see the two lattices that form around the ONF
waist on either side. Notice that the red- and blue-detuned light have different decay
lengths from the ONF surface, see Appendix A.
Figure 1.4 (c) shows the intensity profile for a 1064 nm horizontally polarized
red-detuned input beam on a 180 nm radius ONF. The black circle represents the
cladding-air boundary of the ONF. Similarly, Fig. 1.4 (d) displays the intensity
profile for a 730 nm vertically polarized blue detuned input beam for a 180 nm
radius ONF. The intensity profiles inside and outside the fiber are [70,74]
|Ein|
2 = gin
[
J20 (hr) + uJ
2
1 (hr) + fJ
2
2 (hr)
+
(
uJ21 (hr)− fpJ0(hr)J2(hr)
)
cos [2 (φ− φ0)]
]
(1.2)
|Eout|
2 = gout
[
K20(qr) + wK
2
1(qr) + fK
2
2(qr)
−
(
wK21(qr)− fpK0(qr)K2(qr)
)
cos [2 (φ− φ0)]
]
, (1.3)
where Jn and Kn are Bessel functions of the first and second kinds of order n
respectively, r is the radius from the center of the fiber, and g is a normalization
constant. The terms h, q, f, fp and w, given explicitly in [70], are functions that
depend on the fiber radius and the propagation constant β of the input field.
The intensity outside the fiber falls off with a characteristic decay length pro-
portional to q =
√
β2 − k2 (see Fig. 1.4), where k is the free space wave vector.
In Fig. 1.4 (c) and (d) we observe the difference in decay lengths: the 1064 nm
intensity has longer decay length than the 730 nm intensity. Furthermore, there
is a discontinuity across the cladding-air boundary. This occurs because the ra-
14
dial and longitudinal components of the electric field for the fundamental mode are
discontinuous, see Appendix A.
Light shifts due to the optical power can be problematic for this type of trap,
and the shifts include contributions from the longitudinal component of the electric
field [75]. This longitudinal component, which is inherent to the fundamental mode
of a fiber, leads to vector light shifts [71, 73, 75, 76]. In our scheme the standing
wave formed by the counter-propagating red-detuned beam cancel the longitudinal
component of the electric field, making the blue-detuned light the only source of
vector light shifts. In fact, at the trapping site along the horizontal axis of the red-
detuned beam, the longitudinal component from the blue-detuned light is zero in
this configuration. We also note that Refs. [71,73] employ a trap which significantly
reduces light shifts in a Cs based ONF trap by finding wavelengths that minimize
or cancel the effect, the so called magic wavelengths [77,78].
Fig. 1.4 (e) shows the trapping potential realized by sending two color light,
red- and blue-detuned from the D2 transition of Rb, through the fiber, with 7.5 mW
at 730 nm and 4.5 mW at 1064 nm light in a standing wave configuration . Here,
we take a slice of the potential along the abscissa from the fiber surface. The red
curve corresponds to the attractive red-detuned light, the blue curve correspond to
the repulsive blue-detuned light, the green curve is the van der Waals potential, and
the black curve is the total potential. We see this forms a trap that is more than
one mK deep at a few hundred nanometers from the fiber surface.
As a result of the close proximity of the atoms to the fiber surface means
we must include the van der Waals interaction in the trapping potential. We do
15
not, however, use the exact van der Waals interaction between an atom and a
nanowire [79] but instead treat the system as if the atoms were located next to an
infinite dielectric, where the C3 coefficient is 8.46×10−49J· m3 [80, 81]. Given the
distances of atoms from the fiber surface, it appears that we may be in a cross-over
region between the van der Waals and Casimir-Polder regimes [105].
Changing the radius of the fiber, the wavelengths of the red- and blue-detuned
beams, and the relative powers of the two-color trap allows us to change the trapping
distance of the atoms from the fiber surface. This distance plays a critical role in
atom-light interactions and is explains why ONFs are of interest in atomic physics.
The optical depth (OD) per atoms can reach 10% in certain configurations and these
type of traps allow for studies of long-range interactions [82].
1.4 Thesis overview
This thesis focuses on the fabrication of ONFs for coupling atoms to supercon-
ducting qubits. In Chapter 2 we present the ONF fabrication process with particular
attention to obtaining high transmission and adiabatic tapering for reducing light
losses. Here we will discuss the entire fabrication process and algorithm and report
on measurements comparing results to simulation. We demonstrate that following
this process yields the lowest loss ONFs reported. In Chapter 3 we discuss modal
analyses of ONFs using simulations and spectrogram analysis. Using a genetic algo-
rithm we produce tighter bounds on ONF taper geometries that yield desired trans-
mission. Furthermore, using simulations we can identify the entire modal evolution
16
during a fiber taper using a spectrogram analysis. We next present studies of HOM
guidance in Chapter 4, in which we demonstrate the highest reported transmission
of HOMs through ONFs. Chapter 5 discusses modal identification techniques for
HOMs on the ONF waist using Rayleigh scattering and evanescent coupling with
tapered fiber probes. Finally, Chapter 6 includes a brief conclusion and outlook
followed by an Appendix discussing the mode structure in an ONF.
17
Chapter 2: Optical nanofiber fabrication
2.1 Introduction
Optical nanofibers have seen widespread use in science and engineering appli-
cations during the last thirty years [83, 84]. The tight confinement of the evanes-
cent field around the optical nanofiber [70], unique light geometries provided by
the fiber modes [72, 85, 86], low loss, and promise of improved atom-light interac-
tion [69,71,87–89] have led to increased interest in the physics community. Optical
micro- or nanofibers are used for sensing and detection [90,91], and coupling light to
resonators [91–96], NV centers [97], or photonic crystals [98–100]. Optical nanofiber
fabricated systems can be connected to an existing fiber network to provide appli-
cations in quantum information science [101].
The development of atom traps around optical nanofibers affords new avenues
of research [69–71] including hybrid qubit systems. These qubits can benefit from
high transmission nanofibers through a reduction in unwanted stray light fields pro-
duced from non adiabatic mode excitation and reduced laser power requirements.
Here we present the tools and procedures necessary to create ultrahigh transmission
nanofibers.
Some previous work focuses on the effect of the post-pull environment concern-
18
ing the humidity and air purity [102]. Here we focus on the critical pre-pull steps
necessary to achieve an ultrahigh transmission, before handling the known environ-
mental effects. Following the protocols described below we have produced fibers
with 99.95 % transmission when launching the fundamental mode. We have also
launched higher-order modes [103, 104] through the fabrication process below and
achieve transmissions of greater than 97% for the first family of excited modes, see
Chap. 4. This level of transmission requires a thorough optimization of the pulling
algorithm and fiber cleaning procedure. In this chapter we describe our technique
and the apparatus that produces fibers that we will use in a series of experiments
towards building a hybrid quantum system [105,106].
Our pulling technique is based on an existing methodolgy [84]. It requires
two pulling motors and a stationary oxyhydrogen heat source. This flame brushing
method allows us to reliably produce optical nanofibers with controllable taper ge-
ometries and a uniform waist. With our setup the waist can vary in length from 1
to 100 mm, and we can achieve radii as small as 150 nanometers [107–111]. Rather
than sweep the flame back and forth over the fiber, we keep the flame stationary;
this action reduces the creation of small air currents, which could lead to nonuni-
formities on the fiber waist and is equivalent to transforming to the rest frame of
the flame. This approach is applicable to other pulling techniques as well, so there
would be no need to scan a heat source.
Other common techniques for optical nanofiber production make use of micro-
furnaces, fusion splicers, chemical etchants, and CO2 lasers [112–117]. Chemical
etching generally produces lower transmission than other heat and pull methods and
19
offers less control over the shape of the taper and the length of the waist. A CO2
laser produces high-transmission optical nanofibers but the final diameter is limited
by the power and focus of the laser. Here we present, to the best of our knowledge,
result on ONF fabrication; with the highest reported transmission, corresponding
to a loss of 2.6 × 10−5 dB/mm for the fundamental mode [84,118], with controllable
taper geometries and long fiber waists. These fibers were specifically developed to
be suitable for meeting constraints of use in cryogenic environments [105].
2.2 Experimental setup
This sections describes the details of the experimental setup. Our work follows
the originally Mainz and currently Vienna group [111].
2.2.1 The fiber-pulling apparatus
The fiber-puller apparatus (see Fig. 2.1 and Table 2.1) consists of a heat source
that brings the glass to a temperature greater than its softening point (1585◦ C for
fused silica [119]) and two motors that pull the fiber from both ends. We use
two computer-controlled motors, Newport XML 210 (fiber motors), mounted to a
precision-ground granite slab with dimensions 12” × 48” × 4”, flat to 3.81 µm on
average. The granite slab serves two purposes: it’s weight suppresses vibrations and
it provides a flat surface. The weight of the granite slab, exceeding 100 kg, damps
the recoil from the fiber motors as they change direction at the end of every pull
step. Without a flat surface the motors will not work to specification, leading to
20
distortion of the ONF during the pull: the pitch or yaw of the motor can vary the
distance between the fiber and the flame, changing the effective size of the flame
and pulling the fiber in unintended directions (negating any pre-pull alignment), see
Fig. 2.1(b). The motors are mounted to the granite by L-bracket adapters designed
to not deform the motors from the the granite surface. The granite is mounted on
an optical breadboard at three points so that surface imperfections of the optical
table do not distort the granite slab.
21
8 5 7 
6 
3 
9 10 11 12 
13 
14 15a 
15b 
1 2 4 
16 
15c
17
13 2
45
67
8
15
14
16
17
a)
b)
Figure 2.1: (a) A schematic of the fiber pulling setup (view from above). (b) An
image of the pulling apparatus. 1) Fiber motors. 2) Granite slab. 3) Optical
breadboard. 4) Adapter plates. 5) L brackets. 6) XYZ fiber alignment flexure
stages. 7) Fiber holders. 8) Adjustment screws. 9) Gas flow meters. 10) Filters.
11) Valves. 12) Pipes. 13) Filter. 14) Nozzle. 15) (a) Illumination system, (b)
Optical microscope, and (c) CCD. 16) Flame positioning stepper motor. 17) 2 MP
USB microscope positioned orthogonally to the fiber. The entire apparatus is inside
a cleanroom rated to ISO Class 100.
22
The fiber motors have 210 mm of motion with a minimum step size of 0.01
µm and an on-axis accuracy of 3 ± 1.5 µm (1 in Fig. 2.1). The resolution of the
motors is much smaller than any other relevant length scale in our system, although
the accuracy is not less than the fiber radius. This is sufficient for the motors
to be suitable for pulling ONFs. The fiber motors are controlled with a Newport
XPS controller, which allows us to implement trajectories with a relatively constant
acceleration, resulting in jerkless motion during each step of the pull.
23
Item Part Description
1 Newport XML 210 Computer-controlled high precision motor
2 Granite slab 12” × 48” × 4”, Flat to 3.81 µm
3 Newport VH3660W 3’ x 5’ workstation
4 Adapter Plate Adapts metric XML 210 to 466A
5 L brackets Adapts XML 210 to granite
6 Newport 466A Compact XYZ fiber alignment flexure stages
7 Newport 466A-710 Double arm bare fiber holder double V-grooves
8 Newport DS-4F High precision adjuster
8 Newport AJS100-0.5 High precision small knob adjustment screw
9 Omega FMA 5400/5500 Gas flow meters
10 Swagelok SS-4F-7 Particulate filter, 7 micron pore size
11 Swagelok SS-4P4T Valve to close the flow of gas
12 Swagelok SS-FM4SL4SL4-12 Stainless steel flexible tubing
13 GLFPF3000VMM4 “Mini Gaskleen filter” from Pall
14 Custom SS nozzle 29, 200 µm holes in a 1x2 mm2 array
15 Optical microscope Microscope objective, CCD, and illumination system
15a Illumintation System Kohler illumination system
15b Mitutoyo M Plan APO 10X Microscope objective, 0.28 NA
15c Flea2G CCD camera 2448 x 2048 pixels, 3.45 x 3.45 µm2 pixels
16 Thorlabs DRV014 50 mm Trapezoidal Stepper Motor Drive
17 USB microscope 200x, 2 MP USB microscope
18 Platinum wire Platinum catalyst to ignite flame
19 Clean room ISO class 100 cleanroom
Table 2.1: List of equipment parts for the pulling apparatus.
24
We attach Newport 466A flexure stages (6 in Fig. 2.1) with a Newport 466A-
710 fiber clamp (7 in Fig. 2.1) to the XML 210 (1 in Fig. 2.1). We position the
v-grooves of the fiber clamp on each stage at the minimum separation allowed by
the parameters of a given pull, which is typically 3 cm. Separating the fiber clamps
at the minimum distance minimizes the fiber sag during the pul, which can result in
the ONF breaking during the pulling process. The v-grooves of the fiber clamps on
the left and right fiber motors must be aligned within a few micrometers to achieve a
high transmission. We align the v-grooves using Newport DS-4F (8 in Fig. 2.1) and
AJS100-0.5 (8 in Fig. 2.1) micrometers, attached to the flexure stages to allow for
three axis translation and use an in situ optical microscope (15(a)-(c) in Fig. 2.1).
The optical microscope includes a Mitutoyo M Plan APO 10X infinity-corrected
objective and a Point Grey Flea2G CCD camera (15(a)-(c) in Fig. 2.1). The Flea2G
has 2448 ×2048 pixels, each having an area of 3.45 µm ×3.45 µm pixels. With this
long working distance microscope objective each pixel corresponds to 0.345 µm
×0.345 µm in the image. We illuminate the microscope with a Ko¨hler illumination
system composed of a thermal light source, two condenser lenses, and two apertures.
We use an oxyhydrogen flame to heat the fibers, in a stoichiometric mixture
of hydrogen and oxygen to ensure that water vapor is the only byproduct. Stainless
steel gas lines introduce the hydrogen (red) and oxygen (green) to two Omega FMA
5400/5500 flow meters (9 in Fig. 2.1). The flow rates are set to 30 mL/min and 60
mL/min for oxygen and hydrogen respectively. Directly after the flow meters is a
coarse particle filter (10 in Fig. 2.1), followed by a valve for safety (11 in Fig. 2.1).
The gases mix in a tee after a flexible stainless steel tube (12 in Fig. 2.1). The gas
25
mixture is then filtered with a high quality 3 nm filter (13 in Fig. 2.1). Finally the
hydrogen-oxygen mixture exits through a custom-made nozzle with 29 holes of 200
µm diameter in a 1×2 mm2 area (14 in Fig. 2.1, see Fig. 2.2). The nozzle serves as
a flame arrestor, while still allowing for the gas flow to be in the laminar regime.
1 mm
2 mm
Figure 2.2: An image of the flame nozzle. The holes are arranged in a grid that is
1 mm × 2 mm.
We ignite the flame using a resistively-heated platinum wire as a catalyst.
This process is clean and results in less deposition of particulate on the fiber. The
nozzle is clamped to a Thorlabs DRV014 motor (16 in Fig. 2.1), the flame motor,
that translates the flame in front of the fiber for the duration of the pulling process.
26
This flame motor introduces and removes the heat source to the fiber. During the
pull we fix the distance between the nozzle and the front edge of the fiber to about
0.5 mm. We have found experimentally that a distance of 0.4-0.6 mm provides the
proper heat distribution from our flame and does not result in the fiber breaking
during the pulling process.
The entire pulling apparatus is inside a nominally specified ISO Class 100
cleanroom. If any fiber buffer remains or dust lands on the fiber at any time the
transmission will degrade (see Sec. 2.7).
2.2.2 Transmission monitoring setup
 
MotorStage MotorStageFlame
780 nminput
PD 1
PD 2
Figure 2.3: The experimental setup to monitor the transmission when launching the
fundamental mode during ONF fabrication.
Figure 2.3 shows the transmission monitoring setup. Using a 780 nm Vortex
laser, we launch light into a fiber and split the light with a 50/50 in-fiber beam
27
splitter. One output of the beam splitter goes to a Thorlabs DET36A photodetector
and records the laser power. The other output is connected to a FC connectorized
fiber that we fusion splice to Fibercore SM800 fiber. We then place the SM800 fiber
in the fiber puller and record the power of light through the fiber at the output of the
fiber puller using another DET36A. We record data for the duration of the pull on a
DPO7054 Tektroniks oscilloscope in high resolution mode set to collect 107 samples.
We normalize the signal through the fiber puller to the laser drift throughout the
pull.
2.3 Algorithm
We pull our fibers using a flame brushing technique [107–109, 111]. A section
of fiber, less than a millimeter in length, is brought to its softening point using a
clean oxyhydrogen flame and then pulled by two high-precision motors.
To choose the parameters for a pull we developed an algorithm1, based on the
work of the originally Mainz and currently Vienna group [111], that calculates the
trajectories of the motors needed to produce a fiber with the desired final radius,
length of uniform waist, and taper geometry. The tapers are formed by a series
of small fiber sections that are well approximated by lines, allowing us to form a
linear taper with a given angle down to a radius of 6 µm. This connects to an
exponential section that smoothly reduces and connects to a uniform section with a
1The program is available at the Digital Repository of the University of Maryland (DRUM) at
http://hdl.handle.net/1903/15069.
28
submicron radius, typically 250 nm. The slope of the linear taper section generally
varies between 0.3 and 5 mrad. The algorithm divides the pull into steps defined by
their pulling velocity and the traveling length of the flame. It recursively calculates
the parameters, starting from the desired final radius, rw, until reaching the initial
radius, r0.
Full details on the algorithm can be found in Ref. [120]. The key points are:
• The fiber volume is conserved during the pulling process so that the radius rn
of the fiber waist at step n, is related to the radius rn−1 of the waist at step
n− 1, according to the equation
rn = rn−1 exp
(
−t0,nvf,n
2L0
)
, (2.1)
where t0,n is the time during which the flame fully sweeps through a point on
the fiber, vf,n is the velocity at which the fiber motors move apart, and L0 is
the length of the heating region.
• We assume that the fiber is pulled symmetrically with respect to the center
of the flame and that the distribution of the pulling velocities inside the flame
is linear. Using this assumption, we derive a differential equation that we can
solve to find an expression for t0,n:
t0,n =
L0
vf,n
ln
(
2vb,n + vf,n
2vb,n − vf,n
)
. (2.2)
With these results the ratio rn/rn−1 is fixed by the geometric constraints we impose
on our fiber. We fix the velocity of the burner vb,n = vb and use Eqns. 2.1 and 2.2
29
to calculate vf,n and t0,n. Typically we fix vb at 2 mm / s while vf can vary from 10
µm /s to more than 100 µm/s.
We want each subsection in the taper region to smoothly connect to neigh-
boring sections. We achieve this by properly linking the taper in each consecutive
step. The model requires that vb,n > vf,n/2 to ensure the flame can reach previously
thinned points. Fig. 2.4 presents a physical picture of the taper linking process.
Where Fig. 2.4(a) and (b) shows the evolution of the fiber taper and the flame in
step n and n+ 1 respectively.
Here we can identify three heating classes, which we distinguish by color: fiber
sections that begin in the flame(light blue), fiber sections that are fully swept by
the flame (green), and fiber sections that end in the flame (dark blue). The fiber
sections that begin in the flame form the taper in step n. This occurs because each
marker on the fiber is swept by the flame for a different period of time and therefore
thinned differently. The ONF waist in step n forms when the flame fully sweeps
through that section of fiber. Here each part of the glass sees the same integrated
heat distribution of the flame and therefore thins uniformly. Finally, the region of
fiber in which the flame ends forms the other side of taper in step n.
The dashed black line in Fig. 2.4 separates the evolution of the fiber and flame
in step n from step n + 1. We note that the z-axis flips and zeroes between steps.
Specifically, the outer edge of the flame in step n becomes the zero of the z-axis in
step n+ 1.
We link the tapers by tracking the evolution of the first section of glass that
the flame fully sweeps in step n. In Fig. 2.4 we represent this marker with a thick
30
red line. This marker represents the taper-waist link in step n. By tracking the
evolution of this section of glass from step n to step n + 1 and bringing the outer
edge of the flame in contact with this point in step n+ 1 we are able to link tapers.
31
t = 0
t
z
t= t0,n
t = tend,n - t0,n
0 L0 zend,nLn
Taper: Starts in flame
Taper: Ends in flame
Uniformly swept waist
Taper-waist link
t = tend,n
tn+1 = tend,n+1 
tn+1 = tend,n = 0
0L0zend,n+1 Ln+1
tn+1 = t0,n+1 
tn+1 = tend,n+1 - t0,n+1
t
z
a)
b)
Figure 2.4: The taper linking process. a) The evolution of the flame and taper in
step n. b) The flame evolution in step n + 1. We note that the the horizontal axis
flips between steps such that the outer edge of the flame in step n becomes the zero
of the z-axis in step n + 1. There are three heating classes in a given step: fiber
sections that begin in the flame(light blue), fiber sections that are fully swept by the
flame (green), and fiber sections that end in the flame (dark blue). These regions
form the taper, waist, and taper, in step n respectively.
32
Using conservation of volume we relate the length of the waist Lw,n−1 in step
n− 1, to the length of the waist Lw,n in step n:
r2n−1 Lw,n−1 = r
2
n Lw,n + r
2
n−1
(
vb,n −
vf,n
2
)
t0,n. (2.3)
This relates Lw,n to the distance Ln, the distance the flame sweeps in step n,
and also relates it to Lw,n, vf,n, and t0,n by
(Ln − vbt0,n) (vb,n − vf,n/2) = vb,nLw,n. (2.4)
Once we experimentally establish L0 and vb (see Sec. 2.3.2) and fix r0 for a
given fiber we specify the desired rw, Lw and Ω for the given pull and the algorithm
calculates the necessary parameters to produce a fiber with desired geometry.
2.3.1 Motor control
The model produces a velocity profile that is a square wave in time. Experi-
mentally, we approximate the square wave in three parts:
1. A ramp up of the pulling motors to vb,n ± vf,n/2
2. A constant pull velocity equal to vb,n ± vf,n/2
3. A ramp down of both fiber motors to zero velocity.
Here vb,n is the velocity of the flame in step n and vf,n is the velocity that the
fiber motors move apart. The addition of vb,n arises from the transformation to the
rest frame of the flame. Typically, vb,n (2 mm/s) is an order of magnitude greater
33
than vf,n (0.01 mm/2 - 0.1 mm/s). When transforming to the rest frame of the
flame, both motors move in the direction the flame would have swept in that step.
The motor whose pull velocity is in the same direction as the flame motion will lead
while the other motor will lag. We have verified this sequence using the position
encoders on the motors. This allows us to record the trajectory of the motors. We
have separately verified this by looking at the output of a Michelson interferometer
with one arm spanning the two motorized stages.
2.3.2 Measurement of the flame width
One fundamental experimental parameter of the algorithm is the effective size
L0 of the flame. This corresponds to the effective length of the fiber inside the flame
that melts and thins during the pulling process. The softening point for the fused
silica used by Fibercore for the SM800 fiber occurs at 1585◦ C. The best way to
estimate L0 is to pull fibers measure the resulting geometry and compare to the
algorithm, since our flame is not visible by eye.
For reproducible conditions, we need to working distance between the fiber
and the nozzle. Ideally the fiber is always in the same part of the flame and always
sees the same distribution of temperature. We set the distance from the end of the
nozzle to the center of the fiber to 400 ± 50 µm before each pull and verify the
distance with a microscope.
We measure L0 by fixing the flame and letting both motors move apart at
a constant velocity. Conservation of volume leads to an exponential profile with a
34
waist of length L0, and the radius profile is given by :
rw = r0 exp
(
−
thvf
2L0
)
, (2.5)
where th is the heating time and r0 the unmodified radius of the fiber. We use our
imaging system (15 in Fig. 2.1)to measure the radius of the waist of the fiber for
different values of vf th, and fit ln (r0/rw) to extract L0. The measurement consists
of fixing the pulling velocity at 0.05 mm/s, varying the heating time from 2 to 32
s, and then measuring the final radius of the waist. We limit ourselves to times less
than 40 seconds to stay within the 2 µm resolution of our imaging system.
Plotting ln (r0/rw) as a function of vf th we obtain a fit (see Fig. 2.5) with a
reduced χ2 of 1.07. Here we extract the slope of the fit which corresponds to the
effective width of the flame resulting in L0 = 0.753 ± 0.014 mm. This parameter
should be checked from time to time as the pulling apparatus is used since it can
vary by a small amount.
35
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Ln
(r w
/r 0
)
1.41.21.00.80.60.40.20.0
vf.th (mm)
Figure 2.5: ln(rw/r0) as a function of vf th. The blue markers are the experimental
points, and the red line is the fit. From the slope of the fit we extract L0 =
0.753± 0.014 mm.
Measuring the length of the waist or fitting the profiles of the taper to an
exponential are less accurate methods than the above procedure because Eq. 2.5
assumes a uniform hot zone, L0. In this measurement we keep the flame fixed,
which means that our hot zone is not uniform. During the actual pulling procedure
we sweep, which creates an effective uniform hot zone. Here, the section of fiber
located at the central point of the flame is thinned the most, as a result it is more
accurate to measure the profile of the fiber after tapering and find the smallest
radius to extract the value of L0.
36
2.4 The pulling process
The setup for an ultrahigh transmission pull involves a series of cleaning and
alignment steps. We outline this procedure in this section.
2.4.1 Cleaning procedure
Obtaining high transmission through an ONFr requires careful attention to
the pre-pull cleanliness of the fiber. If any particulate remains from the fiber buffer
or if dust arrives on the fiber before being introduced to the flame, the particulate
will burn and greatly diminish the final transmission, see Sec. 2.7. Furthermore, we
found that evaporate from solvents can decrease transmission.
Our cleaning procedure starts by mechanically removing the protective plastic
buffer to expose the glass of the fiber to the flame. Then we use isopropyl alcohol
on lens tissue to remove larger particulate. A few wipes of acetone2 are then applied
with class 10a cleanroom wipes from Ted Pella, in order to dissolve smaller remnants
of the buffer. A final cleaning with methanol using class 10a cleanroom wipes
removes evaporate left from the previous solvents. After, we carefully lay the fiber
into the grooves of the fiber clamps on the pulling apparatus and image the entire
2We used acetone for the data shown here; however, we generally would not recommend its use
because it can prolong the cleaning process. SM800 fibers have a buffer made of dual acrylate,
which dissolves in acetone. Acetone is fine for chemical removal of the buffer when heated or
paired with other chemicals, but when cleaning with a wipe, the acetone can spread small buffer
particulate along the stripped portion of fiber, which can burn when introduced to the flame.
37
length of cleaned fiber using the optical microscope. If there is any visible dust,
particulate, or evaporate, within the 2 µm resolution of the optical microscope, we
repeat the cleaning procedure over. If the fiber is clean, we proceed to align it.
2.4.2 Alignment procedure
We start the alignment procedure by tensioning the fiber. We tension the fiber
by moving the fiber motors apart in 200 µm increments until the fiber slides through
the fiber clamps, which typically takes 800 µm of total displacement. This allows
the fiber to reach a uniform tension. However, early measurements showed the
fiber to be overtensioned: introducing the fiber to the flame will yield an immediate
thinning, even if the motors are stationary. To prevent this, we then untension the
fiber in 20 µm increments until the fiber buckles. We observe the buckling process
(the fiber bending inwards and then straightening under the inward force from the
motors) with a 2 MP USB digital microscope mounted orthogonally to the flame
above the center of the fiber, see 17 in Fig. 2.1.
The buckling process results in no loss in transmission and no thinning visible
through the optical microscope upon introducing the fiber to the flame, at least
within the microscope’s resolution of ±2 µm.
38
a) b)
250 µm 250 µm
Figure 2.6: Images of fiber sections (a) next to the left and (b) right fiber clamps.
The two images are separated by 3 cm and show excellent lateral alignment.
Once the fiber is properly tensioned, we align the fiber such that the sections
of fiber directly next to the left fiber clamp and right fiber clamp are equidistant
from the optical microscope and at the same height. We translate each section of
fiber in front of the optical microscope using the fiber motors, see Fig. 2.1, and align
the height and focus of each fiber using the micrometers attached to the flexure
stage until both images overlap. If we see a sag in the fiber caused by the buckling
we carefully retention the fiber in 5 µm steps until the fiber is straight as in Fig. 2.6.
The microscope objective has a 3 µm depth of field, so by matching the diameter
of the lensed light from the cladding and core we ensure that the v-grooves of the
fiber clamps on each motor are equidistant from the camera, and therefore the
nozzle of the flame. This alignment is on the order of micrometers over a length of
39
centimeters. Once the images overlap, see Fig. 2.6, the fiber is ready to be pulled.
2.5 Results
Here we discuss the results obtained from following the procedures outlined
above. First we validate the pulling algorithm and simulation of the taper geom-
etry using microscopy. Then we present details on the transmissions achieved by
following cleaning and alignment procedures. Finally, we detail methods to aid in
understanding the entire modal evolution during the fiber pull as a final check on
the quality of the ONFs we produce.
2.5.1 Microscopy validation
We validate the accuracy of our simulation of the expected fiber profile using
both an in situ optical microscope and a scanning electron microscope (SEM) to
examine fibers after pulling. Figure 2.7 (a) shows the measured (blue markers) and
simulated profiles (red lines) of a fiber taper imaged optically. The taper profile is
designed to have three angles, 5, 2, and 3 mrad, on sections that taper down to radii
of 50, 35 and 25 µm, respectively. An exponential profile smoothly links the radius
of 25 µm to the the final radius of 15 µm. The final radius is chosen to be well
above the resolution of our optical microscope to allow it to be precisely measured.
The length of the uniform waist is chosen to be 5 mm long.
The data in Fig. 2.7(a) is a compilation of optical microscope images taken of
the entirety of the tapered fiber. We use an edge finding technique to measure the
40
profile of the fiber at different cuts. The error in the measured radius is dominated
by a systematic error of ±2.5 µm due to the finite resolution of the imaging system.
We first use an image of the unmodified fiber, which has a known diameter of 125.1
µm, to determine the pixel to micron conversion. The number of pixels measured
for an unmodified fiber has an error of a few pixels as a result of the resolution of
the optical microscope. We then binarize the gray levels of the pixels and choose
a threshold such that the diameter of the unmodified fiber matches the pixel count
from the previous measurement. The edge finding technique itself has an error of
about 0.5 pixels for a flat length of fiber resulting from the binarization process.
Figure 2.7 (b) displays the relative difference between the measured image radius
and the simulated radius normalized to the expected radius. The largest deviation
is slightly larger than 2%, while the RMS value is 0.0187. This verifies the accuracy
of our algorithm and pulling apparatus for larger radius tapers.
We use a SEM to measure the ONF profile below a micrometer to verify that
our ONFs truly achieve the desired diameter. Figure 2.8, shows a SEM image of a
ONF, coated with graphite, with an expected diameter of 500 nm, and a measured
diameter of 536 ± 12 nm. The error is systematic, coming from the scaling factor
associated with the SEM calibration. We attribute this small disagreement to ther-
mal air current forces that push the fiber away from the nozzle at the end of the pull
when the fiber is thin. We could compensate for this in the algorithm by adjusting
the effective hot zone as the fiber tapers, but we have not found it necessary to do
so.
41
20
30
40
50
60
70
Ra
diu
s (μ
m
)
0 1 2 3 4 5 6−3
−2
-1
0
1
2
Fiber axis (cm)
Re
lati
ve 
Dif
fer
en
ce 
(x 1
0-2 ) 
a)
b)
Figure 2.7: Profile of a multiple-angled, linearly tapered fiber. (a) The blue dots
are measurements taken using the optical microscope and the red curve represents
the intended profile shape from the fiber tapering simulation. The pull was for a
final radius of 15 µm. The taper profile was designed to have three angles, 5, 2,
3 mrad, that taper down to radii of 50, 35 and 25 µm, respectively. The error in
each measurement is dominated by a systematic error of ±2.5 µm. (b) The relative
difference between the expected profile and the measured profile with an RMS value
of 0.0187. 42
Figure 2.8: A SEM image taken in the Nanoscale Imaging Spectroscopy and Prop-
erties Laboratory (NISP) lab at UMD. We measure a radius of 536 ± 12 nm, the
intended diameter of the waist is 500 nm. In order to image the fiber we coat it
with a layer of graphite.
2.5.2 Transmission
Figure 2.9(a) shows the transmission as a function of time during the pull for
an ONF with a 2 mrad angle taper to a radius of 6 µm and exponential profile to
reach a final waist radius of 250 nm, with a fiber waist length of 5 mm. It achieves a
43
transmission of 99.95 ± 0.02 %, corresponding to a loss of 2.6 × 10−5 dB/mm when
taken over the entire stretch. The error listed in the transmission is the standard
deviation. We see from Fig. 2.9 (b) and (c) a histogram of the data ranges we use to
find the mean for the average value at the beginning of the pull and the average value
at the end of the pull. We obtain a standard deviation of 1.0 ×10−4 for the data
in Fig. 2.9 b) and 1.0 ×10−4 for the data in c). This leads to a standard deviation
of the final transmission of 2.3 ×10−4. There are possible systematic errors such as
drifts in the amplifier gain, but we expect them to be of the same order or smaller
than the quoted standard deviation.
44
−5 −4 −3 −2 −1 0 1 2 3 4 50
12
34
56
7
Cou
nts
 (x 
105
)
Transmission - Running Average(x 10-4)
0
4
8
12
16
Cou
nts
 (x 
103
)
−5 −4 −3 −2 −1 0 1 2 3 4 5
Transmission - Running Average(x 10-4)
0 50 100 150 200 250 300 350 400 450 5000.9800
0.9850
0.9900
0.9950
1.0050
1.0000
0 50 100 150 200 250 300 350 400 450 5000.980
0.9850
0.990
0.9950
1.0000
1.0050
Tra
nsm
issi
on 
Nor
ma
lize
d
Time (sec)
a)
b)
c)
Figure 2.9: a) The normalized transmission as a function of time during the pull
through an ONF with an angle of 2 mrad to a radius of 6 µm and exponential profile
to a final waist radius of 250 nm. The length of the waist is 5 mm. The fiber has
a final transmission of 99.95 ±0.02 % or equivalently a loss of 2.6 × 10−5 dB/mm.
b) Histogram of data taken from the beginning of the pull before higher order mode
excitation. c) Histogram of transmission data taken after the pull ended.
45
The final transmission is determined by taking the mean of the data after the
pull ends, delineated by the red line in Fig. 2.9, and dividing by the value of the
normalized signal at the beginning of the pull, which we must determine. We take
a cumulative average of the transmission from the beginning of the pull until just
before any higher order modes are excited, see Sec. 2.5.3. Using this we see that
the transmission steadies at 99.95 %. We find this to be a fair method because
there is no detectable loss over this range and no beating between modes in the
signal since we have yet to excite any higher order modes, and by checking the
cumulative average we show that the transmission listed is steady. We also note
that this provides a lower bound on the transmission through the fiber.
46
10-4
10-5
10-3
10-2
10-1
100
101
Figure 2.10: Propagation loss as a function of ONF radius as compiled in Ref. [84]
edited to include our results for optical ONF loss in dB/mm. The smaller, solid
gray and black squares, circles, and triangles, represent previous results for differ-
ent pulling techniques: self-modulated taper-drawing (SMTD), the flame-brushing
technique (FBT), and the modified flame-brushing technique (MFBT). The solid
red circle represents our result for loss when launching the fundamental mode. The
open red circle scales the fundamental mode result to the effective radius to compare
results with equivalent V numbers at 1550 nm. The solid blue square represents our
loss when launching the higher-order modes. Similarly the open blue square scales
the result to the effective radius.
47
Much previous work on high transmission ONFs was focussed on using telecom
light at 1550 nm. In Fig. 2.10 we plot the propagation loss as a function of ONF
radius for different pulling techniques as compiled from Ref. [84] and references
therein. We extend the axes to overlay our results. Figure 2.10 shows that the
lowest loss for previous work on the fundamental mode is on the order of 10−3
dB/mm at 1550 nm, with final radii of between 440-600 nm. Our result for the
fundamental mode has a loss of 2.6 × 10−5 dB/mm when the loss is taken over
the entire 84 mm stretch. If the loss is only attributed to the 5 mm waist this
corresponds to 4.34× 10−4 dB/mm. These results mark an improvement of two
orders of magnitude over previous work [84,121,122]. For higher-order modes pulls,
our loss of 5 ×10−4 dB/mm, when taken over the entire stretch, has less than the
previous results for fundamental mode launches [84, 121, 122], see Chap 4 for more
details. This loss exceeds the expected loss in an untapered fiber at 780 nm by more
than an order of magnitude which is given by 3.3 dB/km.
Since the V number is proportional to the fiber radius divided by the input
wavelength we compare our results at a wavelength of 780 nm to the results in
Fig. 2.10 at 1550 nm by scaling our final radius by a factor of 2. The solid red
circle and blue square in Fig. 2.10 represent the actual radius of the pull while the
open red circle and blue square are designed to scale our results to equivalent V
numbers for inputs at 1550 nm and represent an effective radius. This means our
effective final radius for the fundamental mode is 500 nm and for the higher modes
640 nm. This ultra-high transmission is reproducible to better than 1% over time
with the same fiber, when following the cleaning and alignment procedure outlined
48
in Sec. 2.4.
Using a numerical Maxwell’s equations solver, FIMMPROP [123], we simulate
the expected transmission through a fiber with the same profile as in our pulls. We
find the expected transmission to be 99.97% [103], through a one−sided taper profile
matching the 2 mrad pull depicted in Fig. 2.9, which is consistent with our experi-
mental result that measures the transmission through the entire ONF. Furthermore,
when launching the next family of modes through the fiber the FIMMPROP sim-
ulations were well-matched to the achieved transmissions [104]. This suggests that
we are not limited by imperfections in the pulling apparatus.
49
2.5.3 Spectrogram analysis
0.15
0.1
0.05
0 60 65 70 75 80
No
rm
ali
ze
d B
ea
t F
req
ue
nc
y
Fiber Length (mm)
Figure 2.11: Spectrogram of the transmission data from Fig. 2.9. The curves corre-
spond to higher-order mode excitations of the same symmetry as the fundamental
mode: EH11, HE12, and HE13.
We analyze the quality of the ONF using a spectrogram, a short-time Fourier
transform of the transmission data, also sometimes referred to as the Gabor Trans-
form. The spectrogram allows us to extract the entire modal evolution in the ONF
during the pull. Each curve in the spectrogram corresponds to the evolution of the
spatial beat frequency between the fundamental mode and excited modes propa-
50
gating in the fiber, while the contrast corresponds to the energy transferred from
the fundamental mode to the higher-order mode. We use simulations to identify all
modes that are excited during the pull. A detailed description and full theoretical
background, can be found in Ref. [103] and in Chap. 3.
Figure 2.11 is a spectrogram of the transmission data from Fig. 2.9. We see
that for a successful 2 mrad pull with SM800 fiber we expect to observe a few
higher order mode excitations. If modes are excited that are asymmetric to the
fundamental mode, we know the cylindrical symmetry of the fiber was broken during
the pulling process [103], which can aid in identifying and fixing the error in the
pulling apparatus.
It is worth noting that the modal excitation remains in the family of the
same symmetry as the fundamental mode. To the best of our knowledge, this is
the first report of modal excitation that remained purely in the symmetric family
of modes. Previous work has seen asymmetric excitations to the TE01, TM01, and
HE21 modes [112,124]. In Sec. 2.7, we demonstrate that such coupling to asymmetric
modes can occur for an uncleaned fiber.
2.6 Power Measurements
Once the pull is complete the fiber is transferred to a HV chamber beneath
a HEPA filter. Without keeping the fiber in a clean environment the transmission
will degrade [102] as dust accumulates on the surface of the ONF waist and taper,
which will cause the fiber to break under high optical powers in vacuum due to local
51
heating [125].
Ion pumps produce small leakage currents that can charge the ONF [126].
To prevent the leakage current from impinging on the ONF, we avoided having
direct line of sight between the ion pump and the ONF by placing the ion pump
on an elbow. Between the elbow and the ion pump we placed a grounding mesh to
prevent the electric field from penetrating past the mesh. With this arrangement a
250 nm radius ONF has withstood the application of more than 400 ± 12 mW from
a Ti:Sapphire laser at 760 nm in HV conditions. To the best of our knowledge, this
is the highest optical power reported through an ONF.
2.7 Dust studies
We find that the presence of particulate on the optical fiber even before the
pull begins can compromise the quality of the ONF: it will degrade the transmis-
sion, excite higher order modes, change the modal evolution, and scatter light. If
any particulate accumulates on the fiber before the pull, the maximum possible
transmission for a given taper geometry will not be achieved. Using FIMMPROP,
as described in Sec. 2.5.2, we could estimate the ideal transmission for a given ge-
ometry, and if the observed transmission deviates, it could possibly be attributed
to a lack of proper cleaning. If the ONF environment is not clean or has a high
humidity the transmission will decrease after a pull is finished [102]. Furthermore,
if dust accumulates on the ONF surface, the fiber will not withstand high optical
powers when the fiber is transferred to vacuum.
52
10-310-20.75
0.80
0.85
0.90
0.95
1.00
Radius (mm)
No
rm
ali
ze
d T
ran
sm
iss
ion
Fiber Length (mm)N
orm
ali
ze
d B
ea
t F
req
ue
nc
y
60 65 70 75 800.00
0.05
0.10
a) b)
TM01, TE01,
     HE21
Figure 2.12: (a) Transmission signal for an uncleaned 2 mrad tapered fiber. The
transmission of this fiber is 80.50 %. (b) Spectrogram of the transmission data. We
can distinguish the excitation of many higher order modes. Of special interest are
the curves identified, the asymmetric TE01, TM01, and HE21 modes.
We find that if the fiber is not properly cleaned before pulling, the final trans-
mission can vary by a few percent. Figure 2.12 displays the extreme case of me-
chanically stripping the buffer and not cleaning the fiber at all before pulling. Here,
the transmission is only 80.5% for a 2 mrad taper down to rw = 250 nm, leading to
more than a 19% loss in transmission when compared to a properly cleaned fiber.
The spectrogram in Fig. 2.12 (b) shows excitation to an asymmetric mode: TE01,
TM01, and HE21, identified by arrows, that were not present when the fiber was
properly cleaned. It is further interesting that there is more energy transferred to
these asymmetric modes than any other modes.
Repeating what we stated above, before every pull, we follow the cleaning
53
procedure described in Sec. 2.4.1. After imaging the fiber, we decide whether or
not we should start the pull or restart the cleaning process. We restart if anything
is obstructing the light, which we shine perpendicular to the fiber axis, traveling
through the fiber reaching the CCD. When there is particulate attached on top of
or below the fiber, we use a wipe with methanol and remove it. If there is nothing
observable within the resolution of the optical microscope we proceed with the pull.
When we do not follow these criteria the reproducibility in the transmission will vary
by a few percent from one pull to the next. When we apply this cleaning method,
the variability between runs is better than 1 %.
Particulate can come in various forms: remnants of plastic buffer, residue
left after the solvent evaporates, or the small dust particulate floating in the air.
We believe the most common source to be the buffer. Since we use a mechanical
fiber stripper to remove the buffer, micro or macroscopic pieces of buffer remain
on the fiber after stripping. We apply wipes to remove the buffer remnants. This
removal process can be imperfect because mechanical strippers are not designed to
make contact with the actual glass of the fiber. Buffer remnants are a particularly
insidious form of particulate because it is a plastic that is generally designed with a
higher index of refraction than the cladding to help remove cladding modes. If the
buffer remains it may burn into the fiber during the pulling process. This higher
index irregularity can lead coupling to higher-order modes.
During the pull we find signatures if the fiber was not properly cleaned. These
include a large decrease in transmission and the excitation of higher-order modes. If
there is initially loss or beating in the transmission this is a sign that the fiber was
54
not properly cleaned; this is displayed in Fig. 2.12(a). The fiber starts with a single
mode in the core and therefore there should be no beating between modes and
negligible losses in the initial pulling process. Eventually the fundamental mode
becomes a cladding mode, as the tapering process reduces the effective index of
refraction below the index of refraction of the cladding.
55
Figure 2.13: Multimedia View showing the transition of the intensity in the fun-
damental mode from a core to a cladding mode. The intensity of each frame is
renormalized to aid visualization. The modes are obtained using the numerical
solver FIMMPROP.
56
As the fiber tapers during a pull and the effective index of refraction of the
fundamental mode approaches the index of refraction of the cladding, the mode
begins to leak from the core. This is when the fundamental mode can couple to
higher-order modes. For the SM800 fiber used in this study, the transition occurs
at a radius of 19.4 µm. Fig. 2.13 show a simulation of the fundamental mode as the
fiber tapers. Here, we see that as the radius reduces, the effective index of refraction
approaches the index of refraction of the cladding and we can see the mode starts
to leak from the core. If the beating between higher order modes occurs before this
point, this is evidence that the fiber was damaged. We typically found that an ONF
with a 2 mrad geometry, if handled properly, only couples to three higher0-order
modes: EH11, HE12, and HE13 [103]. In contrast we can identify more than twenty
excited modes as a result of the buffer remnants in Fig. 2.12(b).
We believe that chemically removing the buffer could be beneficial to the fiber
transmission. In principle chemical removal can lead to less mechanical damage to
the fiber and complete removal of the buffer. While this is not a critical issue, since
our transmission is in good agreement with simulations from FIMMPROP, but it
could improve reproducibility and ease the cleaning process. Preliminary tests using
acetic acid to remove the buffer were promising.
2.8 Conclusion
We provide the necessary procedures to clean, prepare, and pull an ultrahigh
transmission ONF in a reproducible way. The work is validated through microscopy,
57
and we present the transmission results of a standard 2 mrad pull yielding a trans-
mission of 99.95 ± 0.02% or loss of 2.6 × 10−6 dB/mm, an improvement of two
orders of magnitude for the fundamental mode over previous work. When launch-
ing higher-order modes we have losses of 5 ×10−4 dB/mm. The transmission results
are in excellent agreement with transmission simulations, suggesting that the lim-
iting factor in transmission comes from a lack of pre-pull cleanliness. The cleaning
protocol greatly improves the reproducibility for ultrahigh transmission fibers and
produces the first recorded tapers without asymmetric modal excitation. The pre-
pull cleanliness is critical to achieving ultrahigh transmission ONFs. These fibers
can achieve efficient guidance with short, controllable taper lengths and are usable
for various atomic physics applications. During the manuscript writing process we
became aware of similar independent work [127].
58
Chapter 3: Optical nanofiber spectrogram analysis
3.1 Introduction
Understanding mode coupling in an optical waveguide [128] is important for
good control of transmission and proper design of optical connections between fibers.
This is especially true for tapered optical fibers with sub-wavelength waists, where
light propagates in a mode that exhibits a large evanescent component propagat-
ing outside the waveguide. ONFs are useful for probing nonlinear physics, atomic
physics, and other sensing applications [69, 70, 121, 129]. As the light propagates
through the taper, it successively encounters regimes where the fiber is single mode,
multimode and then single mode again. Careful design of the tapers can lead to
ultra-low loss fibers [130]. Adiabatic tapering criteria give an upper limit on how
steep a taper can be, but are too vague for optimization of transmission. Here we
are interested in giving quantitative bounds and constraints on the taper geometry.
Using a spectrogram analysis of the transmission signal through the fiber [124],
we are able to identify the modes excited during the tapering process and extract
the coupling to each of these modes. From this analysis, we show the importance
of the geometry control and the fiber cleanliness to reach transmissions as high as
99.95% in commercial fibers at 780 nm. After reaching the cutoff radius, the excited
59
modes couple to radiative modes [131] and diffract outside of the fiber.
The analysis we describe here provides a full model the electromagnetic field
evolution in an ONF. This is crucial for modeling the coupling between light on the
ONF and atoms trapped around the fiber [68, 96]. When atoms are trapped in the
evanescent field around ONF waist, we will need to know the coupling coefficients
between the modes of the field and the atoms to understand the trap depth. This
chapter details the modal evolution in the fiber and provides perspectives on the
design of even more adiabatic fibers or other control protocols [132], making them
usable in extreme conditions [105].
This chapter presents our diagnostics, and characterization on the fabricated
ONFS. We first discuss the experimental motivation in Sec. 3.2. Section 3.3 presents
the dynamics of the modal evolution in ONFs. We then study in Sec. 3.4 adiabaticity
in tapered fibers. Section 3.5 analyzes the transmission signal in more detail. We
introduce the spectrogram to analyze the transmission [124] in Sec. 3.6. In Sec. 3.7
we identify crucial elements to improve the transmission. Section 3.8 looks into the
other losses present in the fiber. Finally, in Sec. 3.9 is the conclusion of the paper.
3.2 Motivation and construction overview
Controlling neutral atoms with optical dipole traps is a promising approach to
implementating of a variety of applications [133]. These traps are based on produc-
ing a force from off-resonant interaction between light and atoms in the presence of
an intensity gradient: detuning the light below an atomic resonance attracts atoms
60
to be attracted to the most intense region. This is the same effect used to pro-
duce optical tweezers [134, 135]. Applying light above resonance pushes the atoms
to intensity minima, which requires more complicated geometries to implement a
trap [136, 139–141]. One drawback of optical tweezers obtained by tightly focusing
a laser beam comes from diffraction, which limits the trapping volume extension in
the axial direction. One solution to this limitation is the use of ONFs [68, 69, 71].
ONFs allow confinement and guidance of trapped atoms over a few centimeters
in the axial direction and appear to be well-suited for integration with other de-
vices [88, 96,142,143].
3.3 Modal evolution
3.3.1 Modes in a cylindrical waveguide
The theoretical description of electromagnetic modes in a cylindrical waveg-
uide using Maxwell equations can be found in several references e.g. [74, 131] and
is described in detail in Appendix A. The modal fields vary as exp [i(βlmz − ωt)]
where βlm is the propagation constant of the mode of order (l,m). Analysis re-
veals the propagation of light inside a two-layered step index fiber depends on the
V -parameter of the fiber,
V =
2pi
λ
a
√
n21 − n
2
2, (3.1)
where a is the core radius, n1 is the core index of refraction, n2 is index of refraction of
the surrounding medium, and λ is the wavelength in free space. The relation between
βlm and the V -parameter is equivalent to the dispersion relation of mode (l,m).
61
For our tapers, we can approximate the fiber as a two-layer step index cylindrical
waveguide in two regions: At the beginning of the taper, the light is confined to
the core and guided through the core-to-cladding interface. In the waist, what was
initially the core in the center of the fiber is now negligible (acore ≈ 10 nm  λ).
The light is then guided through the cladding-to-air interface.
62
z0
z(L) w(L)
L
Core-claddingguidance Taper Waist: Cladding-airguidance
L
0
0.19
15
60
0
1
Normalized position
Fiber radius (µm)
No
rma
lize
d p
ow
er
(a)
(b)
ΩR
Figure 3.1: (a) Schematic of the stretched fiber. At a given time, the fiber is
composed of two tapers and an uniform waist of radius r and length w. The total
stretch is equal to L. (b) Calculated intensity profile versus radius for a fiber equal
to 60 µm, 15 µm and 190 nm. Note that the position axes are not quantitative,
and have been scaled to make the plots visible. The profiles are normalized to their
maximum power.
As we continuously decrease the fiber radius during the pull, the fundamental
mode leaks from the core to the cladding. In that region, the presence of the core,
63
the cladding, and the air influence the mode (see Fig. 3.1). A proper treatment has
to take into account all of those interfaces. Accordingly, we model our fibers using a
three-layered structure, and we calculate the dispersion relations for a series of modes
using a commercial fully vectorial finite difference mode solver FIMMWAVE [123].
Figure 3.2 shows a plot of neff = β/k0 as a function of the radius of the SM800 fiber
described in Sec. 3.2.
We pull a SM800 fiber from Fibercore that has a numerical aperture of 0.12 and
a cutoff wavelength of 794 nm. Using the Sellmeier coefficients provided by Fiber-
core, we determine the core (ncore = 1.45861) and the cladding (nclad = 1.45367)
indices of refraction. The pull is divided into approximately 100 steps, such that
the taper is composed of a series of sections small enough to be considered linear,
see Chap. 2. Our tapers are generally composed of a section with a constant few
mrad taper angle that reduces the fiber to a radius of 6 µm, and then connects
to an exponential section that gently reaches submicrometer radii (on the order of
250 nm). The central waist is uniform and its length can be between 5 mm and
10 cm. A pull generally lasts for a few hundreds of seconds.
We are interested in modes that are initially launched into the untapped core
and guided by the core-to-cladding interface. Core modes have most of their energy
contained in the core, and their effective indices of refraction satisfy nclad < neff <
ncore. Figure 3.2 shows that the HE11 mode effective index is initially greater than
nclad = 1.45367 (green curve indicated by an arrow). In typical optical fiber, the
step index of refraction between the core and cladding is small. This small change
in the index of refraction leads to an approximation to the fiber modes known as
64
Linear Polarized (LP) modes. Some higher-order modes from the LP11 family, a
set of degenerate modes that are composed of the HE21, TM01, and TE01 modes,
may be accepted in the core, close to the cutoff condition. We note that the fiber
cutoff wavelength is 792 nm > 780.24 nm, so strictly speaking, our fibers are not
in the single mode regime. Experimentally, we filter higher-order modes that have
been launched or excited with a 1.27 cm diameter mandrel, effectively placing us
into the single mode regime.
When the fiber radius decreases, nHE11eff approaches nclad. Around the point
where nHE11eff = nclad (R = 19.43 µm in Fig. 3.2) the core becomes too small to
support the fundamental mode. The mode progressively leaks into the cladding
to be guided by the cladding-to-air interface. The characteristic length-scale of the
waveguide is R λ, and many modes can be guided by the cladding to air interface
(nair < neff < ncore), together with the fundamental mode. As long as R  λ,
the air has little influence on the effective index of many of the accepted modes
(neff ≈ nclad for all the modes shown in Fig. 3.2). However, the effective indices of
the modes are so close to each other that the modes interact and exchange energy
easily. For that reason, this is the critical region of the taper, where the adiabaticity
condition is the most stringent. By symmetry, for a fully cylindrical fiber intermodal
energy transfer will only happen between modes of the same family (one color in
Fig. 3.2). Energy transfers between modes from different families are a consequence
of the azimuthal asymmetries in the fiber.
Further decreasing R, we observe that the modes effective indices approach
nair = 1. The dispersion curves separate, and adiabaticity can again be easily
65
achieved. When the index of refraction of a mode reaches nair, the mode is not
guided by the fiber anymore and radiates into the air. This radius, specific to each
mode, is called its cutoff. The highly-excited modes leave the fiber first, and the
number of modes allowed in the waveguide decreases progressively (see Fig. 3.2(c)).
Under 0.3 µm, the only mode that can propagate is the HE11 mode, whose index
asymptotically approaches 1. The fiber is once again single-mode.
66
10 30
50
0
nclad
ncore
1.0
0 1
2
0123456
1.0
1.2
1.4
R (µm)l order
n eff
123456
0 0.2 0.4 0.6 0.8 11.0
1.2
1.4
R (µm)
n eff
n eff
(b)
(a)
(c) HE11
HE11
Figure 3.2: Dispersion relations for various modes TE0m, TM0m, HElm and EHlm
(l = 1 to 5) as a function of radius calculated for a three-layer model using FIMM-
PROP. Here, nair = nvacuum = 1. We show the first few modes of each family. (a)-(b)
Three dimensional representation of the dispersions for different mode families. (c)
Plot showing smaller values of R.
67
3.4 Adiabaticity in fibers
Achieving high transmission in ONFs requires precise control of the fiber taper
geometry. This is particularly critical in the part of the taper where the mode
escapes from the core and leaks into the cladding before coupling back to the core
[107,131]. High transmission through ONFs is indicative of the quality of the taper
section as well as the cleanliness of the fiber during and after the pull [102,130].
3.4.1 Adiabaticity criterion
The taper geometry determines the mode conversion in a taper. If a taper
is too short (taper angle too steep), the mode evolution is non-adiabatic, and we
observe a drop in transmission. On the other hand, as the taper is lengthened,
the mode conversion is more adiabatic. In the limit of a very shallow angle, the
transmission can reach 100%, since all the energy remains in the fundamental mode
throughout the evolution.
Following these ideas, we can derive an adiabaticity criterion [131, 137, 138]
that relates the characteristic taper length zt, to the characteristic beating length
between two modes zb. We define: zt as the length associated with the tapering
angle Ω at radius R, defined by:
zt =
R
tan(Ω)
, (3.2)
zb as the beat length between two modes (the spatial frequency of the beating):
zb =
2pi
β1 − β2
=
λ
neff,1 − neff,2
. (3.3)
68
Here β1 is the fundamental mode propagation constant at radius R and β2 is the
propagation constant at radius R of the first excited mode with the same symmetry
as the fundamental mode (EH11). Equation (3.3) relates the beat length to the
inverse of the vertical distance between two curves in Fig. 3.2. Mode conversion in a
taper is adiabatic when the fiber is long enough: zt  zb [131]. We note that if the
two modes have nearly the same neff , zb is large, making the adiabaticity condition
more difficult to satisfy. The choice of EH11 gives the most stringent condition on
the fiber length, as it produces the shortest beat length between the fundamental
mode and any mode with symmetry l = 1. Nevertheless, this condition remains too
vague when one wants to optimize the taper geometry for a given transmission.
Using the dispersion relations from FIMMPROP, we can solve the equation
zt = zb, to find when the taper angles for which the beat length equals the taper
length. The blue curve in Fig. 3.3 separates the plane into two regions: in order to
be adiabatic, taper angles need to be much smaller than the ones indicated on the
curve. Above the curve, the angles correspond to non-adiabatic tapers.
69
0 10 20 30 40 50 60
10
100
Fiber radius (μm)
Ta
per
 an
gle
 (m
rad
)
50
Non-adiabatic
Adiabatic
Figure 3.3: Upper boundary for the taper angle Ω as a function of the radius of
the fiber set by zb = zt. Note the logarithmic scale for the vertical axis. The core to
cladding diameter ratio for this fiber of 2.535/62.55 is fixed for the entirety of the
pull. ncore = 1.45861 and nclad = 1.45367.
Figure 3.3 gives an upper limit on the taper angle at a specific radius using
the condition zb = zt from Eqs. 3.2, 3.3. It does not provide any quantitative
information on the intermodal energy transfers for a given taper: calculations in
Sec. 3.4.4 show that the angles in Fig. 3.3 lead to large energy transfers. We are
interested in producing fibers with high transmissions, greater than 99.90%, and we
need to find the optimal geometry necessary to reach a specific transmission.
70
3.4.2 Transmission of a tapered fiber section
We perform numerical simulations with FIMMPROP to explore the parameter
space and find the optimal adiabatic profile for a given transmission. For these
numerical studies we assume fiber tapers from a 62.55 µm radius down to a 250 nm
radius and use the indices of refraction for our SM800 fiber (see Sec. 3.2). We divide
the taper into 32 discrete series of linear sections. At the end of each section we
project the output field into the first family of modes (the 15 first modes of family 1)
to obtain the transmitted amplitude and phase information in terms of the excited
modes. The S-matrix, relating input and and output, contains all the mode phases
and amplitudes necessary to relate the input and output fields of each section.
71
0 0.2 0.4 0.6 0.810.0
5.0
0.0
(1-
No
rm
aliz
ed 
HE
11
 po
we
r)1
0-4  
0.0
5.0
10.0
0 0.2 0.4 0.6 0.8
−π
−π/2
0
π/2
π
Ω (mrad)
HE
11 
to E
H 11
pha
se 
diff
ere
nce
(b)
(a)
(No
rm
aliz
ed 
EH
11
 po
we
r)1
0-4  
Figure 3.4: Transmission of one section (tapering from 25.5 to 23.5 µm) as a function
of angle when the input is the fundamental mode. (a) Amplitude of the fundamental
HE11 (continuous blue) and the first higher-order mode EH11 (dashed green) (b)
Phase difference between the fundamental and the first higher order mode.
Figure 3.4 shows the modal evolution in a tapered section when the input is
in the fundamental mode. When Ω is small (or the L is large), the modal evolution
is adiabatic and the transmission approaches unity for small taper angles, as seen
72
in the plot for the normalized power in the HE11 mode in Fig. 3.4(a). When Ω
increases, some energy couples to higher-order modes, and the fundamental mode
transmission decreases. For the small angles considered here, Fig. 3.4(a) shows
energy transfer to one mode only (EH11 mode dashed green curve). Energy transfer
to to other modes (HE12 mode and higher) is negligible within the resolution of
the plot. The oscillations in the transmission are due to modal dispersion in the
fiber, which leads to spatial beating: two modes see different indices of refraction
and accumulate a phase difference as they propagate through the fiber (see Sec. 4.2
and Chap. 5). The phase accumulation increases and can become large for small
angles (or increased fiber length). In the particular situation of Fig. 3.4(b) where
only two modes beat together, the EH11 power reaches a local maximum for phase
differences of 2pin, where n is an integer, and local minimum for phase differences
of npi where n is an odd integer.
The situation is more complex when more than two modes are excited. Con-
sequently, there exist some situations where large intermode energy transfers during
the propagation still results in good fundamental mode transmission. Thanks to
mode spatial interferences, most of the energy can couple back to the fundamental
mode during the propagation. In this case, one relies on interference to reduce the
loss of transmission when the taper is non-adiabatic.
73
3.4.3 Genetic algorithm
To obtain the total transmission T , after calculating the projection on the
fundamental mode of the full S-matrix, we find the product of the S-matrices for
all the sections. Ideally we want to to find the shortest tapered fiber that gives
a target transmission. For this task, we use the genetic algorithm function from
MATLAB to find an optimal solution. This approach is expected to be efficient
for large problems and allows the use of information from previous runs to improve
the computing time. This can be contrasted with naive unweighted MonteCarlo
methods or other optimization techniques that use deterministic approaches.
We typically use a population size of 500, a crossover probability of 0.7, a
mutation probability of 0.025 and a number of generations of 500 as parameters for
the algorithm. The genetic algorithm can probe a large parameter space: for each
section, we have calculated 1500 S-matrices, and use angles in the range between
10 µrad and 1.57 rad. We run the algorithm more than 1000 times with different
sets of parameters to try to find the global minimum.
3.4.4 Fully adiabatic fiber
We define total transmissions greater than T = 0.9990 as a fully adiabatic
fiber. In this section, we discuss simulation results on fibers with limited intermode
energy transfers during the pull. This means that the power contained in the fun-
damental mode never deviates much from T at any point in the taper. In this case,
the interference between higher order modes plays a minimal role in the final trans-
74
mission. We note that we benefit from the robustness with respect to variation in
parameters that is associated with an adiabatic process. We obtain the most strict
condition on the angles that can be used to reach a specific transmission. We run
the algorithm with the added condition that the transmission of each small taper
section must be greater than the target in the total transmission T . That way, we
make sure that the fundamental mode transmission is greater than T at 32 points
in the taper. Between those points the fundamental mode power can oscillate, but
remains constrained around T , limiting the intermode energy transfers in the taper.
75
0 10 20 30 40 50 60
1.0
10
Fiber radius (μm)
Ta
pe
r a
ng
le 
(m
rad
)
 
 
0.5
5.0
Figure 3.5: Simulation of optimal adiabatic tapers calculated with the genetic al-
gorithm for T = 99.90% (blue crosses and continuous line) and T = 99.99% (red
dots and and dashed line), where the intermode energy transfers are limited. Each
marker corresponds to the optimum angle for a section. The lines are guides for the
eye.
Figure 3.5 shows results from the genetic algorithm for optimized adiabatic
fiber tapers using target transmissions of 99.90% and 99.99%. Here we plot the
taper angle as a function of the fiber radius. We observe similar behavior in Fig. 3.3:
large taper angles are allowed for large fiber radii, there is a minimum around the
transition region at 20 µm, before the angle increases again at smaller radii. For
T = 0.9999, the optimal taper in Fig. 3.5 (red dashed curve) shows angles as low
as 0.4 mrad, 30 times smaller than the zb = zt criteria. The results in Fig. 3.5 give
76
precise bounds on adiabaticity, with minimum power transmitted to higher-order
modes. This last point ensures that this algorithm is insensitive to phase effects: the
final transmission is not a consequence of constructive interference between several
modes and will be independent of perturbation to the fiber geometry.
0 1 2 3 4 5
10
20
30
40
50
60
z (cm)
Fi
be
r r
ad
ius
 (μ
m
)
0
Figure 3.6: Simulation of optimal taper profiles for T = 99.90% (continuous blue
line) and T = 99.99% (dashed red line). The profiles are only based on the dots
from Fig. 3.5 and not on the continuous lines. Note that the horizontal axis scale is
in centimeters whereas the vertical axis scale is in microns.
Figure 3.6 shows the corresponding optimized taper profiles corresponding to
T = 0.9990 (blue continuous line) and T = 0.9999 (red dashed line). Strikingly,
for T = 0.9999 the optimized adiabatic taper is only 4.5 cm long, on the order of
typical non-adiabatic tapers lengths produced with a heat-and-pull method [144].
For example the 2 mrad taper presented in Sec. 3.5 is ≈ 6 cm long and still presents
77
non-adiabaticities. Note however that in Fig. 3.6, Ω varies continuously as a function
of z, and can be large at the beginning of the pull. Experimentally, we show below
(see Sec.3.7) that abrupt variations of Ω during the pull can induce detrimental
asymmetries in the taper. With our apparatus, we have precise control of the
taper geometry for linear and exponential profiles [144], see Sec. 2.3. Using only a
single linear taper section in the same way as above would require a linear taper
angle Ω ≈ 0.5 mrad, and a substantially increased length. One could chose to use
smaller clad-fibers [103] or to chemically pre-etch fibers, in principle allowing shorter
adiabatic tapers.
3.4.5 Utilizing non-adiabaticity
Limiting intermodal energy transfers in a taper to arbitrarily small values is
possible, but can be impractical due to large taper lengths. An alternative approach
consists of allowing large energy transfers, yet reaching high transmissions by careful
design and phase control in the fiber. As we discuss in Sec. 4.2, different modes
interfere together as they propagate in the taper. Taking advantage of this spatial
beating, we can design fibers with particular phase combinations that allow high
transmission, despite the presence of non-adiabaticities. In this section, we run the
genetic algorithm with only a condition on the final transmission (T ≥ 0.9999):
intermodal energy transfers in each section is no longer limited. Using this non-
adiabaticity, it is possible to produce short high-transmission tapers.
78
0.93
0.94
0.95
0.96
0.97
0.98
0.99
HE
11 m
ode
 no
rma
lize
d p
ow
er
0 1 2 3 40
10
20
30
40
50
60
z axis (mm)
Fib
er r
adi
us 
(μm
)
0 20 40 601
10
100
Fiber radius (μm)
Tap
er a
ngl
e (m
rad
)
 
 
(a)
(c)
(b)
1.00
0204060 50 30 10Fiber radius (μm)
Figure 3.7: Simulation of fiber profile for 99.99% transmission with optimized length
given by the genetic algorithm. (a) Taper angles and fiber radius, squares are from
the simulation and the continuous curve is a guide for the eye. (b) Fiber radius and
fiber length with a final length of 3.7 mm using the results in (a). (c) Fundamental
mode transmission as a function of fiber radius for the optimized adiabatic fiber
(red dashed line) and the optimized non-adiabatic fiber (green continuous line).
In this case we use the genetic algorithm to search for the profile with the
shortest fiber length that has a 99.99% total transmission in the fundamental mode.
Figure 3.7(a) shows that the taper angles allowed here are much larger than the
ones presented above in the adiabatic case (Fig. 3.5). At large fiber radii, the taper
79
angle reaches ≈ 100 mrad. Closer to the transition region, the minimal taper angle
can still be as large as 2 mrad. From the the taper angles used here, we know that
the fundamental mode is not propagating adiabatically in this taper. Figure 3.7(b)
shows the corresponding profile. Figure 3.7(c) shows the simulated transmission
for this fiber as a function of position for a total transmission of a 99.99% (green
continuous curve). This fiber would have a 3.7 mm length, a factor of 12 shorter
than in the adiabatic case (red dashed curve) calculated using FIMMPROP. This
greatly reduces the length requirements for high-transmission fibers, which could be
useful for our application.
In the optimized adiabatic case (red dashed curve Fig. 3.7(b)), we confirm that
the power contained in the fundamental mode is close to 99.99% throughout the
taper. Higher-order modes excitations are negligible, and the evolution is adiabatic.
However, for the non-adiabatic simulation (green curve Fig. 3.7(b)), we observe
large energy transfers to higher-order modes. Around R = 23 µm, more than 7%
of the energy has been transferred to higher-order modes. However, using this
particular geometry, the resulting phase combinations lead to high-transmission in
the fundamental mode.
The fact that non-adiabaticity can lead to high-transmission with shorter ta-
pers is potentially useful for taper design. However, exploiting non-adiabaticity
requires paying close attention to the geometry because there is sensitivity to the
phases of the modes: deviations from the calculated profile might lead to situa-
tions where mode interference causes large losses, with less energy ending in the
fundamental mode than initially expected. One would need to reproduce the calcu-
80
lated geometry as accurately as possible. As discussed above, producing the taper
in Fig. 3.7 with a continuously varying angle is not the best option for us, due
to the presence of large angles and possible experimental asymmetries. Moreover,
this particular taper length (3.7 mm) is likely to be too small in comparison to the
heating-zone size (0.75 mm in our experiment) to accurately produce such a profile.
Our typical profiles start with a linear section (Ω of a few mrad) from 62.5 µm down
to 6 µm radius, followed by an exponential section down to 250 nm radius. We calcu-
late with FIMMPROP the HE11 mode evolution through such a taper (Ω =2 mrad)
and show that it benefits from non-adiabatic effects, leading to high-transmission.
81
0102030405060
0.996
0.997
0.998
0.999
1.000
Fiber radius
HE
11
 no
rm
aliz
ed 
pow
er
0
0.001
0.002
0.003
0.004
0.005
HE11
EH11 HE12
EH12
HE13
Higher-order modes power
Figure 3.8: Mode evolution for a 2 mrad linear fiber down to 250 nm radius. During
the propagation through the taper, some energy is transferred from the fundamental
HE11 (blue thin continuous line) to 4 higher-order modes EH11 (green long dashed
line), EH12 (light blue dotted line), HE12 (red dashed dot line), and EH13 (purple
thick continuous line). The final transmission through one taper is 99.97% on the
HE11 fundamental mode.
We start by using FIMMPROP to examine geometries we can produce with
good accuracy using our fiber puller. Figure 3.8 shows the transmission of the
first few modes of family l = 1 through a 2 mrad taper. We create a taper with
FIMMPROP that reproduces the experimental profile, which has been validated
with microscopy measurements [144], Sec. 2.5.1. Initially, all the power is contained
in the fundamental mode. Around R = 23 µm, ≈ 0.4% of the energy is transferred
to higher-order modes because of non-adiabaticities (up to HE13, the fifth mode
82
of family l = 1). This illustrates that non-negligible higher-order mode excitations
can be observed below the zb = zt limit (the taper angle Ω = 2 mrad is at least a
factor of five below the zb = zt limit everywhere in the taper). Those modes beat
together, and by the end of the taper, 99.97% of the energy is transmitted through
the fundamental mode. For different taper angles, we observe that our typical
tapers benefit from non-adiabaticity (see Sec. 3.5). Although there is still room for
optimization, the simplicity of the linear geometry makes it the ideal candidate for
our application.
3.5 Analysis of transmission signal
To test the results from FIMMPROP and the pulling algorithm we evaluate
the quality of a pull by monitoring the transmission of a few mW from a 780.24
nm laser through a fiber during the pulling process following the setup described
in Sec. 2.2.2. Figure 3.9(a) shows a plot of transmission as a function of time
for a successful 2 mrad pull. The transmission and incident power fiber outputs
are connected to two Thorlabs DET10A photodetectors that deliver a signal to a
Stanford Research Systems SR570 low-noise differential preamplifier. A Tektronix
DPO7054 digital oscilloscope set on high resolution mode and sample rate of 10-
20 ksample/s records the data. The fiber is thinned during the pull, and as its
radius decreases, we observe different notable features in the transmission signal.
Figure 3.9(b) shows the relation between time and radius for the particular pull
of Fig. 3.9(a) calculated using the algorithm for fiber pulling that was validated in
83
Sec. 2.5.1 with a deviation from the experimental measurements lower than 8% at
all diameters.
0 100 200 300 4000.980
0.985
0.990
0.995
1.00
No
rm
aliz
ed 
Tra
nsm
iss
ion
0 100 200 300 4000
10
20
30
40
50
60
Time (s)
Wa
ist 
Ra
diu
s (µ
m)
(a)
(b)
Figure 3.9: (a) Measure transmission through a 2 mrad fiber as a function of time
during the manufacturing process. (b) Evolution of the waist radius during the pull,
calculated from the algorithm described in [144]. The final radius is 250 nm.
84
3.5.1 Single mode section
We note that even before we pull the fiber it is not single mode (V ≈ 2.45) at
the light wavelength we use to measure the transmission. However, during the first
100 seconds (down to 25 µm radius), we observe a constant transmission. Thus a 2-
mrad taper is completely adiabatic in this region (see Fig. 3.3) and we can conclude
that the fundamental mode is confined to the core and does not couple to any other
modes.
3.5.2 From single mode to multimode
As the fiber radius decreases, the fundamental mode effective index approaches
the cladding index of refraction (see Fig. 3.2). The fiber core becomes too small
to support the fundamental mode, which progressively leaks into the cladding to
become guided by the cladding-to-air interface. The point where the fundamental
mode leaks into the cladding is nHE11eff = nclad, at R = 19.43 µm. At that point, the
waveguide is so large in comparison to the wavelength of the light that the fiber is
multimode (V ≈ 170). The dispersion relation curves of all the modes are close to
each other (see Fig. 3.2(a)), and the modes can easily interact. Figure 3.5 shows
that the tapering angle has to be smaller than 0.3 mrad in order to be adiabatic in
that region. The transmission signal shows mode beating (see Fig. 3.10).
85
350 360 370 380 390 400 4100.982
0.984
0.986
0.988
0.99
0.992
0.994
0.996
0.998
1
Time (s)
No
rm
aliz
ed 
Tra
nsm
iss
ion
26 24 22 20 18 16 14 12 10Radius (µm)
Figure 3.10: Detailed view of of the transmission of the 2 mrad pull shown in
Fig. 3.9, when the radius of the waist is near 20 µm. We see small oscillations
in the transmission signal, due to the beating between the fundamental mode and
higher-order modes excited at a radius of 20 microns. The top vertical scale is the
fiber radius at the waist.
Energy transfers to higher-order modes occur during the transition from core to
cladding because of this non-adiabaticity. For a cylindrically symmetric fiber, such
a transfer of energy is only possible from the fundamental mode to other modes of
order l = 1 (by symmetry). Once they have been excited in the fiber, those modes
coexist and propagate together with different propagation constants, given by the
dispersion relation curves (green curves in Fig. 3.2). The optical path length inside
86
the fiber is:
(L)n =
∫
fiber
nneff (z)dz. (3.4)
According to Eqn. (3.4) different modes will accumulate different phase differences.
The modulation observed in the transmission signal around radius of 20 µm is a
signature of the presence of higher-order modes beating together.
87
(a)
(b)
Figure 3.11: Schematic showing modal evolution in the transition region. All the
power is initially contained in the fundamental mode (blue profile). When the
core of the fiber becomes too small compared to the wavelength, the light escapes
into the cladding (green arrows) and some higher-order modes can be excited (red
profile). The radius of the waist is equal to 20 µm, so that the excited modes do
not experience any cutoff as they propagate through the waveguide. (a) The length
of the fiber is an integer number of beating lengths. (b) Length of the fiber not an
integer of beating lengths. The mode profiles were calculated with FIMMPROP.
The beating signal is related to the relative phases when light couples back
into the fiber core as the R increases in the second taper. When R reaches 19.43 µm
88
in the second taper, energy couples back into the core. Although the two tapers
are identical, the presence of beating between modes breaks the symmetry (see
Fig. 3.11). Depending on the fiber length, the phase accumulation between the
modes leads to a different field distribution entering the core at 19.43 µm. The
fraction of energy that can couple back into the core depends on the field distribution
at this point. If the modes travel through an integer number of beat lengths, the
field distribution returns to its initial input. The reciprocity theorem implies that
all the energy couples back into the core. If the modes experience a non-integer
number of beat lengths, the field distribution is different from what it was initially
and only a fraction of energy can couple back into the core: the rest of the energy
couples to cladding modes. The cladding light is not detected since we filter the
higher-order modes placing a mandrel wrap in front of the detector. At the fiber
output, we only observe on the detector the light that coupled back into the fiber
core.
3.5.3 Single mode again
As we continue to thin the fiber, the modes’ effective indices approach the
air index of refraction. When R (equivalently, the V -parameter) becomes small
enough, the excited modes cut off and couple to radiation modes in air. On the
other hand, the fundamental mode’s effective index asymptotically approaches nair
without reaching a cutoff. A small enough fiber can consequently be single mode
again after all the higher-order modes cutoff. For the SM800 fiber, the single-mode
89
cutoff occurs at 300 nm radius as we can calculate in Appendix A. After this cutoff,
we should not see beating and the transmission should be steady again.
We measure a transmission of 99.950(23)% where the dispersion of the distri-
bution is 5.8 × 10−3 and the dispersion on the mean is 1.2 × 10−5. Possible sys-
tematic effects related to the fiber cleanliness and the detectors and amplifiers long
term stability prevent us from giving a better bound than 0.023% to the measured
uncertainty in the transmission, but T is close to unity, both in the measurement
and in the simulation. Note that the simulation Sec. 3.4.5 looks at the propagation
through a single taper. In the present case, light goes through two tapers, support-
ing one reason why the measured transmission of 99.95% is slightly smaller than the
simulated one of 99.97%.
3.6 Spectrograms
Much additional information about the mode evolution during a pull can be
obtained from a spectrogram plot. A spectrogram is a local, windowed Fourier
transforms of the transmission signal as a function of time. We use the spectrogram
function in MATLAB with a window of 8192 points and an overlap of 7000 points.
Figure 3.13 shows an example of a spectrogram found from the transmission
data in Fig. 3.9(a)). The modulation in the transmission does not have a single
frequency. The frequency is chirped for various reasons. First, the stretch of the
fiber is not a linear function of time.
90
0 10 20 30 40 50 60 70 80 90 10020
30
40
50
60
70
80
90
Step
Str
etc
h (
mm
)
Figure 3.12: Simulation of the stretch as a function of step for fiber sample 1.a.
Fiber sample 1.a has the following properties: Ω = 2 mrad, Lw = 5 mm, and rw =
250 nm.
Its form depends on the chosen pulling parameters, and can be calculated
using our algorithm, see Fig. 3.12. Second, the propagation constants of the modes
are not only radius-dependent but the way they evolve also depends on the mode.
The difference between two curves varies as a function of R, which means that the
phase does not accumulate at a constant rate.
stretchvsstep.eps
91
Fiber Length (mm)
No
rm
aliz
ed 
Fre
que
ncy
40 50 60 70 800
0.05
0.1
30 20 10 4 2 1 0.5 0.3
Radius (µm)
Figure 3.13: Transmission spectrogram of a 2 mrad pull (see the time evolution
in Fig. 3.9) as a function of the stretch L of the fiber, showing the chirp of the
beating frequency and the abrupt end of the beating. The top vertical scale shows
the waist radius calculated from the algorithm. The colormap corresponds to the
power spectral density (PSD).
Each curve in the spectrogram signals the interaction between two modes
beating at a given frequency. They all appear when the fiber enters the multimode
regime (R ≈ 19.43 µm). The presence of these curves indicates non-adiabaticities in
the pull. The curves terminate before the end of the pull, at a point that corresponds
to the cutoff of one of the two beating modes. We now have the task to identify
which modes are excited, how they are excited, and if there is a way to suppress their
excitation. Given the specificity of the phase accumulation for a couple of modes,
92
it is possible to label the modes excited during the pull and use the spectrogram as
a diagnostic to evaluate the adiabaticity and symmetry of the fibers.
3.6.1 Modeling the pull
The phase accumulation between two modes is a function of their optical path
length, which depends on the geometry of the fiber at a time t (see Eq. (3.4) above).
Let the stretch at a time t be L, then the phase accumulation between two modes
is:
Φi,j(L) =
∫ L
0
[βi(r(z))− βj(r(z))] dz. (3.5)
We can then define the with spatial frequency Ki,j [124] due to modes i and
j.
Ki,j(L) =
1
2pi
dΦi,j
dL
. (3.6)
Typically we will be interested in the case where mode i is the fundamental
mode and we will need to simulate K1,j for higher-order modes.
3.6.2 Identifying the modes
Fig. 3.14 outlines the procedure we use to identify the modes in producing a
spectrogram.
93
Figure 3.14: Flow chart of the spectrogram outline. Ellipses represented simulations
from FIMMPROP. Rectangles are taken from the fiber pulling simulation. Rounded
rectangles implies a combination of the two simulations.
We start with the simulated profile at the end of a step from the pulling al-
gorithm described in [144] and in Sec. 2.3, and Fig. 3.15. We use the dispersion
relations obtained with FIMMPROP to calculate the differences, ∆βi,j, in propaga-
tion constants for mode i and mode j.
94
0 10 20 30 40 50 600
20
40
60
z (mm)
r (μ
m)
0 10 20 30 40 50 601.4475
nclad
ncore
z (mm)
n eff
(a)
(b)
Figure 3.15: a) Simulated taper profile from pulling algorithm. b) Plot of neff from
FIMMPROP as a function of the profile for multiple modes.
By integrating numerically ∆βi,j(z) at each step, we obtain Φi,j(L). A nu-
merical differentiation of Φi,j with respect to L gives us the evolution of the spatial
frequency as a function of step (see Fig. 3.16). From our simulation of the pull, we
know the stretch as a function of time, and we can plot the evolution of the spatial
frequency as a function of time.
95
0 20 40 60 800.00
0.01
0.02
z (mm)
0 20 40 60 80 970
10
20
30
Steps
Φ 1,
j (ra
d)
80 970.0
0.5
1.0
Steps
K 1,j
 (μm
-1 )
n 1,1
-n 1,
j
70
(a)
(b)
(c)
Figure 3.16: Simulation of the differences between the fundamental mode and the
first four excited modes of family one for a 2mrad taper profile to rw of 250 nm.
(a) ∆β1,j as a function of length along the fiber axis (difference between the indices
of refraction at step 75). (b) Phase accumulation Φ1,j as a function of step. (c)
Spatial frequency K1,j of the beating as a function of step. The lines (long dashed
red, continuous blue, short dash black and long-short dash green) join the calculated
points.
We calculate the spatial frequency for a thousand pairs of modes with different
96
radial symmetry (l =1 to 6) and azimuthal order (m = 1 to 20), and we map them
on a measured spectrogram, see Fig. 3.17. Using the overlap, we can then identify
and label the curves observed on the spectrogram by looking for their overlap with
the experimental curves. Figure 3.17 shows excellent matching without any scale
adjustments.
Fiber Length (mm)
No
rm
aliz
ed 
Fre
que
ncy
40 50 60 70 800
0.05
0.1 HE12EH12HE13
30 20 10 4 2 1 0.5 0.3
Radius (µm)
Figure 3.17: Identification of the modes responsible for the beating in the 2 mrad
tapered fiber spectrogram shown in Fig. 3.13. Only modes from family one are
beating.
We can identify in the spectrogram (2-mrad tapered fiber) the signature of
beating between four modes. All those modes have the symmetry l = 1 (see Fig. 3.2)
as expected for a cylindrically symmetric fiber. We observe higher-order mode exci-
tation up to the fifth mode of family l = 1, which is consistent with the simulations
97
(Sec. 3.4.5). The total HE11 transmission is 0.9995, meaning that 0.05% of the
energy has been transferred to the other modes. If we assume that all the losses
come from the transfer of power to other modes, this power would radiate into the
air when those modes reach cut off. The contribution of other losses like Rayleigh
scattering are expected to be much smaller. The power spectral density (PSD),
which defines the colormap in a spectrogram, gives a representation of how the re-
maining power is distributed between the higher-order modes as a function of time.
By plotting the PSD at different times, we evaluate the power contained in each
branch contributing to the beating. Below R = 4 µm, those contributions are al-
most constant, and the higher-order mode relative power is distributed as follow:
5.5 ± 0.5% in HE12, 9 ± 0.5% in EH12 and 85.5 ± 0.5% in HE13. Note that we
only resolve the beating between the fundamental mode and one excited mode. The
beating between excited modes exists, but this second order effect is too weak to be
visible in the spectrogram.
3.7 Applications of the quality of the pull
We can use the spectrogram analysis to design and diagnose its quality while
pulling a fiber. The number of modes excited and which modes are excited give us
information about the adiabaticity, asymmetries and the quality of the fiber after
the pull.
98
3.7.1 Internal parameters: algorithm input parameters
The beating amplitude and higher-order modes excitations seen in Fig. 3.9
and Fig. 3.13 show that the angle of tapering near the critical region at 19.43 µm,
is non-adiabatic. A shallower taper angle around that region could lead to a more
adiabatic transition. Following this idea, we study a fiber with a 2 mrad angle until
a radius of 20 µm, and then decrease the angle to 0.75 mrad. After R = 6 µm, the
pull is exponential down to R = 250 nm.
99
0 200 400 600 8000.94
0.95
0.96
0.97
0.98
0.99
1.0
No
rm
aliz
ed 
Tra
nsm
iss
ion
0 200 400 600 800 10000
10
20
30
40
50
60
Time (s)
Wa
ist 
Ra
diu
s (µ
m)
(a)
(b)
Figure 3.18: (a) Measure of normalized transmission through the fiber as a function
of time during the manufacturing process. (b) Evolution of the radius of the waist
during the pull. Based on the algorithm, we initially taper the fiber with a 2 mrad
angle until a radius of 20 µm; the angle changes to 0.75 mrad until the radius of the
fiber is equal to 6 µm, where the radius exponentially decreases down to 250 nm.
We see that the transmission at the end of the pull is only 97.850% from
100
Fig. 3.18. This corresponds to a transfer of energy to the higher-order modes larger
than 3%, a factor of sixty worse than in the linear 2 mrad pull (Fig. 3.9). The beating
amplitude is much larger than in the 2 mrad case. This is surprising since this pull
is designed to be more adiabatic, and simulations with FIMMPROP confirm that
we still expect a transmission T ≥ 99.90%.
Fiber Length (mm)
No
rm
aliz
ed 
Fre
que
ncy
90 100 110 120 1300
0.02
0.04
0.06
0.08
0.1
0.12
TE01TM01
HE21
EH11HE12
EH12HE13
4 3 2 1 0.5 0.3
Radius (µm)
Figure 3.19: Spectrogram of the transmission data shown in Fig. 3.18. The solid
black curves are the one given by the simulation. The modes are labeled on the
figure. The total transmission in the fundamental is 0.97850. For R ≤ 2 µm we
calculate from the PSD that the remaining energy is distributed between seven
higher-order modes as follow: TE01 (0.08%), TM01 (0.05%), EH11 (0.35%), EH12
(0.05%), HE12 (98.4%), HE13 (0.2%) and HE21 (0.87%).
The spectrogram in Fig. 3.19 shows coupling to the TE01, TM01, and HE21
101
modes, which do not belong to the family of the fundamental mode. The largest
transfer is still to the same family, with a different distribution. Coupling to other
families should not be observed for a fiber with cylindrical symmetry. This suggests
that our multiple angle tapers introduce some asymmetries in the fiber. We imaged
the fiber using an optical microscope near the angle change regions (see Fig. 3.20)
to further investigate the decrease of transmission.
102
0 20 40 60 800
50
100
150
z (mm)
Ed
ge 
(pix
)
0 20 40 60 80−0.02
0
0.02
0.04
0.06
z (mm)
Re
lati
ve 
dis
tan
ce
(a)
(b)
(c)
2 mrad
0.75 mrad
80 µm
10 mm
Figure 3.20: Study of the asymmetry of a pulled fiber. The fiber has a 10 µm radius
with an angle change from 2 mrad to 0.75 mrad at 20 µm. (a) 100 images taken
with an optical microscope stacked and horizontally compressed to enhance any
asymmetries. (b) Profile of the bottom edge (blue curve) and top edge (red curve)
of the fiber. The abrupt change in angle at 20 µm introduces an asymmetry at this
radius. (c) Relative difference between the two edges (normalized by the diameter
of the fiber) as a function of z.
Figure 3.20 shows that the bottom angle of the fiber exceeds the top angle.
Although the measured diameters are as expected, superimposed plots of the top and
bottom edges show that there are imperfections around the transition. We observe
103
a peak at the transition radius (R ≈ 20 µm) in the distance between the edges. We
believe that the excitation of higher-order modes at this radius is a consequence of
this asymmetry. We do not observe the same imperfection around the transition
region for a 2 mrad flat fiber. The abrupt change in angle exacerbates imperfections
in the pulling process by introducing some asymmetries. This results further support
that single-angle linear tapers are good candidates for our application. Further
increasing adiabaticity would require to decrease Ω, leading to large taper lengths.
Because of geometrical and handling constraints, we find it ideal to work with 2 mrad
tapers. To work with steep and multiple angles might require a smaller flame or a
more symmetric heating.
3.8 Understanding the losses
3.8.1 Losses
Understanding the losses in ONFs is important for our future applications
[105], see Chap. 1 and Chap. 6, which require knowledge of such photon loss. We
identify two main loss mechanisms that contribute to the final losses: coupling
to higher-order modes through non-adiabaticities and scattering around the ONF
waist [131]. Systematic effects like the presence of impurities on the fiber surface,
or asymmetries in the pull, enhance the losses through those mechanisms.
104
3.8.2 Coupling to higher-order modes
We observe beating in the transmission doing the fiber pulls due to the effect
of non-adiabaticities in the taper. The presence f the beats is a signature of energy
transfer to higher-order modes. As we reach the single mode regime, the higher-order
modes cut off. They can not be guided by the fiber anymore, and they diffract out as
radiative modes. In a plane transverse to the fiber, one can observe a characteristic
diffraction pattern further supporting that this effect is the most important for the
pulls considered in this study.
3.8.3 Rayleigh scattering
Rayleigh scattering is present in any real glass due to local defects and in-
homogeneities, leading to scattering of light and attenuation of the transmitted
signal [131]. The attenuation coefficient due to Rayleigh scattering for a fused silica
optical fiber is small at a wavelength of 780.24 nm. However by imaging a fiber that
is carrying 1 mW of optical power, it is possible to directly observe the scattered
light. Experimentally, it is particularly visible on the fiber waist, but the resulting
attenuation is on the order of 3 dB/km. This is small enough that we could ne-
glect it when discussing profile optimization. Further study of Rayleigh scattering
is presented in Chap. 5.
105
3.8.4 Systematic effects
The transmission varies drastically with the surface condition of an ONF.
When the fiber is initially dirty, the spectrogram analysis during a pull shows the
excitation of more modes corresponding to more losses, see Sec. 2.7. We attribute
this to the presence of impurities on the surface of the fiber at the beginning or
during the pull. Particulate on the fiber waist leads to losses through coupling to
higher-order modes or scattering. The cleanliness of the fiber is critical before and
during the pull. Such imperfections are avoidable by properly cleaning the fiber and
imaging the fiber prior to a pull as explained in [144], see Sec. 2.4. All the pulls
presented in this work were done after applying the cleaning procedure described
in [144], see Sec. 2.4.
3.9 Conclusion
Reaching high transmissions is important for many ONF applications. In this
chapter, we described an algorithm for calculating the optimum taper profile and
length for a given transmission, or equivalently the optimum transmission for a
given taper length. This approach gives more precise bounds for achieving adi-
abaticity and yields design for a suitable taper geometry. We found that in our
experiments, the transition from the single-mode regime to the multimode regime
is non-adiabatic, inducing excitations of higher-order modes during the tapering.
Having a good control of the taper geometry is crucial for limiting losses and the
propagation of different modes during the pull leads to a characteristic beating pat-
106
tern in the transmission. Plotting the spectrogram of the transmission signal and
using a model of the fiber pulling process, we were able to identify the modes ex-
cited during the pull. This gives information for the analysis of the quality of a
fiber and the understanding of loss factors. This technique can be applied to the
manufacturing of even more adiabatic fibers.
107
Chapter 4: Efficient guidance of higher-order modes
4.1 Introduction
ONFs with a waist diameter smaller than the wavelength of the guided light
are currently used for non-linear optics, atomic physics, sensing, and fiber cou-
pling [69, 70, 121]. The intense evanescent field outside of an ONF is particularly
interesting for atom trapping and strong atom-photon coupling. Tapering standard
optical fiber down to submicron diameters has been a successful fabrication tech-
nique for a variety of applications [88, 107, 130, 142, 143], in which transmissions
of the fundamental mode of the waveguide reaching more than 99 % have been
achieved [122, 130, 144], see Chap 2. Until now, the efficient guidance of higher-
order modes has not been observed, due to the ease with which they couple to other
modes, leading to large losses. This restricts most work with ONFs to the single
mode regime, where the diameter is small enough to only support the fundamental
mode HE11 [69, 96].
This chapter reports measurements with ONFs using the first excited TE01,
TM01, and HE21 modes, which have azimuthal, radial, and hybrid polarization
states, respectively. Prior work on HOM propagation has shown ≈30% transmission
of this LP11 family of modes [145], with a mode purity at the fiber output of ≈70%,
108
which gives a total transmission of only 20 %. Here, by carefully controlling the taper
geometry, and by choosing a commercially available 50 micron reduced-cladding-
diameter fiber we demonstrate transmissions greater than 97% for 780 nm light in the
first excited LP11 family of modes through fibers with a 350 nm waist. Furthermore,
we present a setup that enables us to efficiently launch these three modes with
exceedingly high purity at the output, where less than 1% of the light is coupled
to the fundamental mode [146]. We follow the work of Ref. [111] to fabricate the
ONFs and use a series of diagnostics during the pull to monitor the quality of the
fiber described in Chap. 2 and Chap. 3. We record the total transmission of the
light and image the mode exiting the fiber for the duration of the pull. Analysis
of the transmission as a function of time for different types of fibers allows us to
estimate which modes are excited during the pull as well as their relative energies
of excitation through the use of spectrograms [103], see Chap. 3.
These results open the way to efficiently use HOMs in ONFs. Unlike the HE11
mode, HOMs experience a cutoff at a finite radius. This allows improved control
of the evanescent field extent at large radii, enabling stronger fibers and improved
handling characteristics. Our work enables the usage of HOMs for atomic physics
applications. In particular, the spatial interference between several of those modes
can create unique evanescent field distributions on the waist, providing an easy and
self-consistent way to break the symmetry along the propagation axis, suppressing
the need to create a standing wave. This is particularly relevant to atomic physics
applications for the realization of a one color, blue-detuned and state insensitive
trapping potential for atoms [85].
109
4.2 Mode propagation in an optical fiber
We describe the modes in a cylindrical waveguide using Maxwell’s equations
in Chap 3 and Appendix A.
The propagation of light inside a two-layered step-index fiber, consisting of a
core of radius a and refractive index ncore surrounded by a cladding of radius R and
refractive index nclad, depends on V , given by Eqn. 3.1.
V plays an important role in our tapers, since the radius, a in Eqn. 3.1, varies
enough that the interfaces seen by the modes change as they propagate through the
taper. At the beginning of the taper, the light is confined in the core, and guided
by the core-to-cladding interface with Vcore as in Eqn. 3.1, and nclad < neff < ncore.
At the end of the taper, the core is negligible (acore ≈10 nm  λ) and the light
is guided by the cladding-to-air interface with nair < neff < nclad. Between these
two regimes, the light leaves the core, and the relevant radius in Eq. 1 becomes
R, which is much larger than a. Due to this radius increase, and the large index
difference between nclad and nair, V >> 1. The fiber becomes highly multimode, as
the number of bound modes is proportional to V 2/2 [131]. Maintaining adiabaticity
through this transition is critical.
4.2.1 First family of excited modes
Figure 4.1 (a) shows the effective index of refraction vs V for several low-order
modes in an ONF. When V < 2.405, the fiber supports only the HE11 mode. For
a typical ONF, ncore ≈ 1.5 and nclad = 1. The TE01 and TM01 modes are allowed
110
for V > 2.405, and the HE21 mode is allowed for V > 2.8. As long as V remains
lower than 3.8, only the four modes mentioned above are allowed. In the weakly
guided regime1, the modes of interest are known as the LP11 family. Prior work has
emphasized propagation of the HE11 mode, where V ≤ 2.405 [69, 96, 143]. We are
interested in selectively exciting and guiding the LP11 family through an ONF, in
a regime where 2.405 ≤ V ≤ 3.8. When expanded to free space, these modes have
the intensity and polarization profiles shown in Fig. 4.1 (b).
1We initially launch into the actual LP11 family, because at the entrance, the fiber is weakly
guided. The taper takes us into the strong guiding regime where the LP11 splits into TE01, TM01,
and HE21. We will refer to the set of modes (TE01, TM01, and HE21) during the entirety of
the pull as the LP11 family. This simplifies discussions when referring to the full set of modes,
especially in reference to excitations to modes or families with the same symmetry, i.e. LP12 for
TE02, TM02, and HE22.
111
1.5
1.0
1.4
1.3
1.2
1.1
V
n eff
TE01 TM01 HE21
HE11
HE12TE01 TE02
HE21
TM01 TM02
HE22
(a)
(b)
0 2 4 6 8
Figure 4.1: (a) neff indices of several low-order modes in an ONF with nclad = 1.5,
surrounded by vacuum (nair = 1). Below V ≈ 3.8, two families of modes exist. In
this work, we are emphasizing the LP11 family (circled). By symmetry, modes in
this family can interfere with the TM02, TE02, HE22 family, which may be excited
through non-adiabatic processes. (b) Intensity and polarization profiles of the LP11
family of modes considered in this work.
4.2.1.1 Adiabaticity condition
Using the adiabatic criterion described in Sec. 3.4, we calculate a minimum
Ω of a few milliradian for the limiting case of zt = zb for the HOMs. This implies
that a taper requires sub-milliradian Ω to achieve adiabaticity in the region where
light leaves the core of the fiber and becomes a cladding mode. We note that this
is a more stringent condition than for the fundamental mode as shown in Fig. 3.3.
Therefore, choosing an optical fiber with a small Ri and thus a reduced Vclad is
highly advantageous to maintaining adiabaticity [145]. Additionally, such a fiber
112
reduces the overall drawing time, length requirements of the pulling apparatus, and
the overall length of the taper.
4.2.2 Experimental setup
PD
MotorStage MotorStageFlame
π-plate
780 nminput
CCD
(a)
zWaistΩ
R
ancorenclad
nair
r(b)
Figure 4.2: (a) Simplified diagram of the experimental setup. A Gaussian beam
passes through a pi-phase plate and is coupled into the fiber to be drawn. On the out-
put of the fiber, a photodetector (PD) and camera (CCD) monitor the transmission.
Typical beam images are shown. (b) Schematic of an ONF.
Figure 4.2 shows a diagram of our experimental setup. Here, we review the
mode preparation, the pulling process, and the detection and analysis. We efficiently
excite the TM01, TE01, and HE21 modes using a fiber-based Cylindrical Vector
Beam (CVB) generation method [146, 147]. The fiber is drawn using a heat-and-
pull method [111,116,144], described in Chap 2. We measure the transmission while
pulling, recording simultaneously the output of the photodiode to a digital storage
oscilloscope and beam profiles on a Charge-Couple Device (CCD) camera.
113
A New Focus Vortex laser delivers a TEM00 Gaussian beam at λ=780.24 nm.
We spatially filter this beam using a polarization-maintaining optical fiber, and
collimate with an asphere to a 1/e2 diameter of 630 microns. The Gaussian beam
passes through a phase plate that imparts a pi phase shift on half of the beam,
producing a two-lobed beam that approximates a TEM01 free-space optical mode.
Fig. 4.2 shows profiles of the beam before and after the phase plate. We couple the
beam into the fiber using a matched asphere. The coupling coefficients to the fiber
modes are determined by the beam symmetry. The inversion of the polarization
over half the incident beam allows us to selectively excite with high efficiency TM01,
TE01, and HE21, which have cylindrically symmetric intensity profiles [146]. The
fundamentalHE11 mode is only excited through aberrations in the beam. To achieve
efficient coupling into the LP11 family, we wrap the fiber with two or three windings
around a 4-mm diameter mandrel that attenuates any modes higher than the LP11
family. Fig. 4.2(a) shows typical input (black and white) and output (color) modes.
We follow the same ONF pulling process described in Chap. 2 and Chap. 3.
To observe all possible mode cutoffs, the fibers are typically drawn to R ≈280 nm,
at which point only the fundamental HE11 mode propagates. We vary Ω from 0.4
mrad to 4 mrad, resulting in pull times lasting between 100 to 1000 seconds. The
output side of the fiber is held straight with no mode filtering. We follow the mode
evolution for the duration of the pull by monitoring the transmission of a few mW of
laser power through the fiber. The transmitted beam is monitored both by a CCD
and by a photodetector (PD). The beamsplitter shown in the figure is tilted to as
small an angle as possible to eliminate polarization-dependent reflections. Another
114
PD, not shown in Fig. 4.2 (a), measures the input laser power during the pull to
normalize the transmission signal. We record PD signals with a Tektronix DPO7054
oscilloscope with 16-bit resolution and a sample rate of 1-10 ksamples/s.
4.2.3 Results
We first analyze the improvements gained by choosing a fiber with a reduced
cladding radius. Ref. [145] looked at the improvement obtained by reducing the
initial R from 125 µm to 80 µm. Moving from 80 µm to 50 µm fibers, we improve the
transmission from 10 % to 51 % for a 2-mrad taper. A spectrogram analysis depicts
fewer and weaker excitations of higher-order modes. Second, we demonstrate an
improvement in adiabaticity through the control of the taper geometry. Our ability
to vary Ω allows for marked improvement in guidance efficiency through the ONF,
increasing from 16 % at 4 mrad up to 97.8 % at 0.4 mrad.
115
4.2.3.1 Varying the fiber type
110250
0.2
0.4
0.6
0.8
1
Norm
alized
 trans
missio
n
110400
0.2
0.4
0.6
0.8
1
Radius (μm)
Norm
alized
 trans
missio
n
Radius (μm)
Norm
alized
 Frequ
ency
110250
0.01
0.02
0.03
0.04
0.05
Norm
alized
 Frequ
ency
110400
0.01
0.02
0.03
0.04
0.05
Radius (μm)
Radius (μm)
(a)
(b) A B C D
52%
10%
A B C
(c)
R0 λcutoff NA Vcore,R0 RLP11c Vclad,Rc
(µm) (nm) (µm)
Thorlabs SM980G80 40 920 0.18 2.9 34 288
Fibercore SM1500 25 1396 0.3 4.3 13 119
Figure 4.3: Evolution while tapering of the transmission through fibers with a half
angle of 2 mrad as a function of the radius of the waist. (a) Fiber with an initial
diameter of 80 µm. (b) Fiber with an initial diameter of 50 µm. The spectrograms
associated with those transmission curves give a clear picture of the power transfers
during the pull. Note the logarithmic scales on the horizontal axis. (c) (1.0 MB)
Movie of the evolution of the beam transmitted through the fiber measured on the
CCD during a 2 mrad pull of the 50 µm fiber. We record one frame every second,
and display them at a 7 frames per second speed. Sections A, B, C, and D are
described in the text. The properties of the fibers used are summarized in the table.
116
4.2.3.2 Transmission measurements
Figure 4.3 shows typical transmissions obtained when exciting the fiber with
a combination of LP11 modes. We plot the normalized transmission during a single
pull as a function of the fiber waist radius. We identify four distinct regimes in
Fig. 4.3. The modes are initially confined to the core (regime A in Fig. 4.3). Adi-
abaticity can easily be achieved, and the transmission is steady. Because this fiber
initially also supports the LP02 and LP21 modes and because our launch can weakly
excite these modes, we observe a slight drop in transmitted power near R = 20 µm,
when these modes become cladding modes. Regime B occurs after the light has es-
caped from the core to the cladding. Because the cladding is typically much larger
than the core (R/a > 10), and because nclad − nair  ncore − nclad, V increases
by over two orders of magnitude (Vclad ≈ 200). If the core-cladding transition is
not adiabatic, modes of similar symmetry are excited. In particular, we observe
transfers of energy to the LP12 family which contains the modes TE02, TM02, and
HE22. The interaction between those modes results in mode beating inside the
fiber, and an oscillation in the amount of output light. The oscillations continue
to R ≈ 0.7 µm (regime C), where for a typical silica fiber, Vclad ≈ 6. From this
cutoff location, it is clear that much of the beating behavior is due to the LP12
modes which have been excited through non-adiabatic transitions. The pull extends
through the TM01, TE01, HE21 cutoffs (regime D). After reaching the cutoff radius
near R = 290 nm, very little light reaches the photodetector - typically less than
1-2% - indicating low population of the fundamental mode. Removing the phase
117
plate to restore the HE11 mode shows that losses were negligible in our tapers for
this mode, meaning that we reach a mode purity of 98-99%. Mode purity is of
fundamental importance for ONF applications, and in particular for atomic physics
where polarization control and stability are required.
We have compared the transmissions obtained using fibers of diameter 80 µm
and 50 µm, and observed the beneficial effects of smaller clad fibers for Ω = 2 mrad.
This taper angle is chosen because it is non-adiabatic and highlights the effect of fiber
diameter on adiabaticity. The improvement in adiabaticity is clear in Fig. 4.3, with a
transmission of 52% obtained for the 50-µm fiber, compared to the 10% transmission
of the 80-µm fiber. The table in Fig. 4.3 compares the properties of the 80-µm and
50-µm fibers. Note that the SM1500 fiber initially supports the LP01, LP11 as well
as the next families of excited modes LP02 and LP21 (Vcore,R0 = 4.3). These modes
are substantially filtered, though not completely, by winding the fiber around a 4
mm-diameter rod. It is worth noting the substantial difference in adiabaticity for
the HOMs compared to the fundamental mode. Fig. 2.9 shows a transmission of
99.95% for the fundamental mode with the same 2 mrad taper angle.
Using the numerical aperture (NA) and the cutoff wavelength provided by
Thorlabs and Fibercore, we are able to derive using Eq. 3.1 the initial V value
Vcore,R0 , the radius at which the LP11 family escapes from the core to the cladding
RLP11c and the corresponding V -number at that radius Vclad,Rc for a wavelength of
780 nm. From those numbers, we see that reducing the radius of the cladding allows
the LP11 family to leak from the core to the cladding at a smaller radius in the fiber.
This results in a significantly reduced Vclad when the modes escape from the core:
118
By reducing the cladding diameter from 80 µm to 50 µm, the number of available
modes decreases by more than an order of magnitude.
4.2.3.3 Spectrograms
In non-adiabatic propagation, the LP11 modes couple to higher-order modes
of the same symmetry, belonging to families LP1m (m ≥ 2). Because they propa-
gate with different propagation constants during the pull, they accumulate a phase
difference leading to interference in the amount of light recoupled into the core.
Since the photodetector only measures core light, this interference leads to oscilla-
tions in the transmission (Fig. 4.3). Plotting the spectrogram of the transmission
signal [103, 124], we directly observe the contribution of various pairs of modes in
the beating. Fig. 4.3 shows the spectrograms for the 80-µm and 50-µm diameter
fiber pulls. Each line in the spectrogram is specific to the beating between a LP01
or LP11 mode and another mode of similar symmetry excited during the pull. The
curve ends when one of the modes reaches its cutoff: the energy is then lost via
coupling to radiative modes.
The number of lines observed in a spectrogram is directly related to the excita-
tion of higher-order modes through non-adiabatic processes from a single launched
mode: the more lines present in the spectrogram, the less adiabatic the pull is.
Moreover, the colormaps in Fig. 4.3 are normalized in such a way that the intensity
of each red line gives the strength of the energy transferred. It is clear from Fig. 4.3
that more intense lines are present in the 80-µm fiber than in the 50-µm, further
119
supporting our observation of more stringent adiabaticity requirements for fibers
with a large cladding radius. Improvements could be achieved by pre-etching the
fiber to a smaller diameter so that the initial core clad ratio R/a is further reduced.
4.2.3.4 Imaging the fiber output
We monitor the transmitted beam with a CCD, and obtain Movie 1 showing
both the core and cladding light as a function of time throughout the tapering
process. Using a microscope objective, we first image the end of the fiber to observe
the core-guided light (Movie 1 in Fig. 4.3). The movie shows oscillations that result
from mode competition, which modulates the amount of light that exits the fiber in
the core.
350 400 450 500 550 600 6500
0.5
1
Time (s)
Nor
ma
lize
d p
owe
r (a) (b)
(b)
(a)
Figure 4.4: Amount of light (normalized) exiting the fiber from the core (blue
curve) and from the cladding (red curve). The signals are out of phase, confirming
the transfer of energy between modes during the tapering. We observed the two
simultaneously by using the two reflections from a thick beamsplitter. (a)-(b) (1.1
MB) Movie 2 shows the evolution of the beam transmitted through an ONF during
a portion of a pull, where the power is high enough to observe the cladding light.
120
Movie 2 in Fig. 4.4 (a) also depicts the transfer of energy from the core to the
cladding. We image both the core and the cladding light simultaneously by using
strong and weak reflections from a thick beamsplitter plate with an AR-coated front
surface. Fig. 4.4 shows the normalized fractions of energy exiting the fiber from the
cladding and from the core. The two signals are out of phase. We note that the
sum of the energy contained in the core and the energy contained in the cladding
does not add up to the total energy input in the fiber. Outside the taper region,
cladding light becomes highly scrambled and lost through the fiber buffer, resulting
in both the speckle observed in Movie 2 and the reduced total transmitted power.
Residual cladding light is spatially filtered from hitting the photodetector, so that
the observed oscillations Fig. 4.3 are due only to core-guided light.
4.2.3.5 Varying the angle
The measurements in this section use the reduced-cladding Fibercore SM1500.
For this fiber, the LP11 modes transition to cladding modes near R = 13 µm. By
using a reduced-diameter fiber, we observed a drastic improvement of the trans-
mission of the LP11 modes. To further improve the adiabaticity, it is necessary to
look into more details of the tapering process itself. We have lowered Ω to improve
the transmission over what is observed in the previous section. Fig. 4.5 shows the
results of draws using Ω = 4, 2, 1, 0.75 and 0.4 mrad. Although each plot shows
the same qualitative behavior as described for the 2-mrad pull, the strength of the
features depends on Ω.
121
0.40.60.81
0
0.2
0.4
0.6
0.8
1.0
R (μm)
Nor
ma
lize
d tr
ans
mis
sion
(a)
(b)
Ω (mrad)
Tra
nsm
issi
on
0.30.50.70.9
IncreasingAdiabaticity20
40
60
80
100
0.40.751234
97.8%
84.3%
51.5%
91.2%
16.6%
Figure 4.5: (a) Transmission through a SM1500 ONF for Ω = 4, 2, 1, 0.75, 0.4 mrad
depicted in green, blue, purple, yellow, and red respectively. The transmission plots
have the fundamental mode subtracted out (typically about 1 %) to accurately
describe the total transmission of the LP11 modes. The horizontal axis for the 4
mrad pull was renormalized to take into account extra tension in the fiber due to
the rapidity of the pull. (b) Simulated (red circles) and experimental (blue lines)
final transmissions through the fiber as a function of angle. Decreasing Ω enables
us to improve the transmission of the LP11 family up to 97.8 %.
The free-space mode at the fiber input has a spatial polarization that is an
equal superposition of HE21 and TM01 (or TE01). However, mode conversion occurs
where the fiber is wound around the mode-filtering mandrel so that the distribution
122
entering the ONF is unknown. Within the ONF waist, which is held fixed and
straight, mode conversion is unlikely to occur so that the desired mode profile can
be achieved after the pull [146]. By R = 0.45 µm, only the TM01, TE01, and HE21
modes are confined, with a small contribution in the fundamental HE11. The HE12
mode achieves cutoff at Vclad = 2.8 (R ≈ 330 nm), earlier than the TM01 and TE01
modes, which reach cutoff at V = 2.4 (R ≈ 290 nm). At R ≈ 330 nm, the power
is reduced by the HE21 content, which is determined by the initial superposition of
states entering the ONF.
The transmitted power in the LP11 mode family is 16.6 % for Ω = 4 mrad.
The transmitted power drops sharply near R = 13 µm, and undergoes strong oscil-
lations between 10-80 %. For Ω = 2 mrad, the oscillations below R = 13 µm are
reduced, with the transmission fluctuating between 30-90 %, and the transmitted
power improves to 51.5%. Further decreasing the angle to Ω = 1, 0.75 and 0.4
mrad, improves the transmission to 84.3 %, 91.2 % and 97.8 % respectively, where
uncertainty is dominated by systematic effects that should be less than 1 %. Fig. 4.5
also shows excellent agreement between the experimental results and those obtained
using commercial waveguide propagation software [123]. For those calculations, the
propagation was modeled using the targeted fiber geometry. In our transmission
measurements, the scaling between time and radius is made using a separate algo-
rithm that models the dynamics of the pull based on conservation of volume [144],
see Chap. 2. For Ω = 0.4 to 2 mrad, the mode cutoff positions we obtain using the
scaling from the algorithm correspond directly to what is expected theoretically. We
also observe excellent agreement between the experimental transmission measure-
123
ments and the calculations, confirming the accuracy of our pulling procedure. We
were not limited by systematic effects that might include accumulation of contami-
nants for longer pulls, and asymmetric profiles for faster ones. For the fastest pull
(4 mrad), we observed a discrepancy due to systematic effects, and we must apply
a different scaling to match the theoretical cutoffs.
For Ω = 0.4 mrad, the most shallow angle studied, the amplitude of the os-
cillations, which is directly related to the energy transfer to undesired modes, is
reduced to a few percent. The observed transmissions are due to losses into and out
of the waist. Because the fiber is symmetric, the normalized transmitted power is
the square of that in the waist. For Ω = 0.4 mrad, this leads to 98.9 % power in
the waist. We believe that such a fiber is usable for various applications involving
HOMs. We note that reaching adiabaticity for HOMs requires fibers that are sub-
stantially longer than for the fundamental mode. For the HE11 mode, transmissions
greater than 99% can easily be achieved for Ω up to 5 mrad [103], see Chap. 3. We
also observed that when we remove the pi-phase plate from the launch, the transmis-
sion in the fundamental mode is essentially equal to the transmission before pulling,
as adiabaticity is strongly satisfied for this mode. During the tapering process, the
higher-order modes escape from the core earlier than the fundamental HE11 mode.
When the HE11 mode finally transitions to a cladding mode, R has decreased, so
that V and the number of available modes to couple to is smaller. The reduction
in R also leads to an increase in the difference between adjacent propagation con-
stants, allowing less mode interaction and a steeper Ω. Mode conversion also occurs
throughout the fiber and not just at the core-cladding transition point, of course,
124
but this region has the most stringent adiabatic criterion.
4.2.4 Conclusion
We have demonstrated propagation of higher-order modes in ONFs using the
TE01, TM01, and HE21 modes. By tapering the fiber with angles near 0.4 mrad and
using a commercial, off-the-shelf fiber with 50 µm diameter, we have achieved trans-
mission efficiency of 97.8% with excellent mode purity, a factor of four higher than
previous work, and more than one order of magnitude improvement on mode purity.
Critical to this work was a spectrogram analysis of the modes present during the
pulling. Our experimental results agree with simulations of the propagation through
the taper. High transmissions of LP11 modes with high purity is a promising tool
for atomic physics, expanding the possible intensity and polarization configurations
of evanescent fields surrounding the ONF.
125
Chapter 5: Higher-order mode identification and control
5.1 Introduction
We demonstrate efficient guidance of higher-order modes through an ONF in
Chap. 4 . The identification and control of the modes propagating on the ONF
waist is crucial for excellent atom trapping and other applications of ONFs [85] .
This chapter shows our successful control of the propagation modes where we can
launch a pure mode or a superposition of the LP11 family.
5.2 IDIOM
The first attempt at modal identification was to use Interferometric Decompo-
sition Into Optical Modes (IDIOM) [148], a method developed at the Naval Research
Laboratories (NRL). This technique is used to decompose the output of a multimode
fiber into its constituent modes and identify the relative occupancy of each mode by
beating the output of the fiber with a reference gaussian beam. Here we wanted to
use IDIOM to show that a pure mode input resulted as a pure mode output. The
premise being if a mode superposition is preserved from input to output, we know
the mode composition on the ONF waist.
126
In the first tests we pulled past the cutoff of the LP11 family, hoping to see the
spectral weight of each mode drop off as the radius moved below its cutoff radius.
But the decomposition showed components of the HE21 mode even after cutoff. The
implication is that there is mode conversion somewhere along the fiber taper. In
further tests, even when we launch a pure mode, the fiber leads to mode conversion.
Whether this is an issue inherent to ONFs or a technical issue requires more
study. There are technical issues with the fiber clamps and fiber bends that could
potentially lead to mode conversion. In addition our experimental setup has limited
space since all the optics must fit on a breadboard that can sit on the fiber pulling
motors, see Fig. 2.1. For these reasons, we decided it necessary to pursue other
means to identify the modes on the ONF waist.
5.3 Rayleigh scattering measurements
Since the ONF appears to behave as a bulk optic, converting a pure mode input
to a superposition of modes output, it is necessary to probe the ONF on the fiber
waist itself to gain information about the mode distribution. The Rayleigh scattering
in fibers arises from induced dipole radiation dependent on the polarization of the
mode [149, 150], we can use the radiated light from the ONF waist to ascertain
information about the local modal distribution.
127
5.3.1 Theory
We work with Fibercore SM1500 fiber, which is multi-mode with input light
at 795 nm initially admitting not only the fundamental mode but the next family
of modes known as the LP11 family. These modes are labelled the TE01, TM01, and
HE21 modes. An unfortunate consequence of working with SM1500 fiber is that
the V number exceeds 3.8. From Fig. 4.2.1 we see that the fiber also admits the
next family of modes. Therefore, a clean launch of the LP11 family requires careful
alignment.
Upon launching a mode into a fiber, if we could spatially resolve the transverse
profile of the Rayleigh scattering, we could identify the modes [151]. However, given
that we typically pull to waist radius below 400 nm, we do not have the optical
resolution. It is also possible that since the surface scattering dominates it could
obscure the bulk scattering in the image 5.3.4.
The Rayleigh scattered power is proportional to the local intensity. Therefore,
by observing the modulation in the Rayleigh scattered power along the fiber as a
function of position, resulting from the beating between modes, we can identify the
modes present in the fiber. In Chap. 2 we observed the phase accumulation from
this beating. Here we can directly measure and determine the sets of beat lengths
as a function of position. Using the data from Fig. 4.2.1 and reparametrizing the
horizontal axis in terms of radius, we can obtain the propagation constants as a
function of radius. Then using Eqn. 3.3, we plot beat lengths for the modes of
interest as a function of fiber radius in Fig. 5.1. We see that for an ONF with a
128
waist radius of 390 nm we can expect the beat length to vary from a few microns for
any HOM beating with the fundamental mode, to tens of microns for HOMs beating
together. Note, that due to the orthogonality of the polarizations at all points in
space the TE01 and TM01 will never interfere.
To extract the beat lengths we perform the following procedure:
1. Image the entire length of an ONF over the tapered region and waist.
2. Align and concatenate the images.
3. Sum over the transverse profile creating data representing power scattered vs
position along the fiber axis.
4. Take a spectrogram of the power scatter vs position data.
129
100 0.50 0.75 1.00 1.25 1.50
Be
at 
len
gth
 (µ
m)
 
 HE21 TE01
HE21 TM01
HE11 TE01
HE11 TM01
HE11 HE21
101
102
103
104
Fiber radius (µm)
Figure 5.1: The beat length between various modes as a function of fiber radius for
a fiber with indices of refraction nCore = 1.48693; nClad = 1.45424, and nAir = 1.0
simulated using FIMMPROP.
The spectrogram will show all the spatial frequencies that are changing as a
function of position in the fiber. These are the inverse spatial angular beat lengths
given by 1/ beat length. Figure 5.2 shows the inverse beat lengths as a function
of radius, displaying the value we actually expect to extract from the spectrogram.
We can then overlap the observed beat lengths with the difference in propagation
130
constants and identify the modes that are in the fiber as a function of position.
0.50 0.75 1.00 1.25 1.500.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Fiber radius (µm)
Inv
ers
e B
ea
t L
en
gth
 x1
05  
( 1
/m
)
 HE21 TE01
HE21 TM01
HE11 TE01
HE11 TM01
HE11 HE21
Figure 5.2: The inverse beat length between the modes in Fig. 5.1 as a function of
fiber radius.
Using the above method we can identify the modes on the waist.
131
0 5 10 15 200
2
4
6
8
Fiber axis (mm)
Fr
eq
. x
 1
04
 (1
/m
)
HE21 TE01
HE21 TM01
Figure 5.3: Inverse beat frequency simulation for rw = 390 nm with Ω = 1 mrad
taper and Lw = 10 mm.
We can use the simulation of the ONF profile, see Chaps. 2 and 3, and the
inverse beat frequencies to simulate the inverse beat frequency as a function of
position for a given ONF. Figure 5.3 shows such a simulation for the HE21 mode
beating with the TE01 (red) and TM01 (blue) modes for a fiber with a 1 mrad taper
to a waist radius of 390 nm and 10 mm length waist.
To demonstrate control we need to launch and manipulate the manifold of
132
HOMs in a controllable and efficient manner. Following Ref. [146] we use a vector
beam generator described in Sec. 5.3.2 and bulk optics to rapidly manipulate the
input mode launched into the ONF and therefore the mode propagating on the ONF
waist.
-400 -200 0 200 400
-400
-200
0
200
400
x (nm)
y (
nm
)
-400 -200 0 200 400
x (nm)
-400 -200 0 200 400
x (nm)
Figure 5.4: The vector profile inside a 450 nm radius fiber for the transverse compo-
nents of the TE01, TM01 and HE21 modes respectively. The black circle represents
the cladding air interface at 450 nm. The indices of refraction are chosen to be 1.45
and 1 for the cladding and air respectively.
Figure 5.4 displays the vector profile for the transverse components of a 450
nm radius fiber for the HOMs. By placing two HWPs with a relative angle of 45
degrees between their fast axes in front of a TM01 mode will output a pure TE01
mode in the fiber, which can be seen in Fig. 5.4.
Similarly, when launching a superposition of the HE21 and TM01, which pro-
duces a two lobed mode with polarization perpendicular to the null, we can com-
pletely convert to a superposition of HE21 and TE01 using a HWP with a relative
133
angle of 45 degrees. Fig. 5.5 shows the transverse profiles of the HE21 and TM01
interfering together and the HE21 and TE01 modes interfering respectively.
-400 -200 0 200 400
-400
-200
0
200
400
x (nm)
y (
nm
)
-400 -200 0 200 400
-400
-200
0
200
400
x (nm)
y (
nm
)
a) b)
Figure 5.5: The vector profile inside the fiber for the transverse component HOM
interfering. a) HE21 and TM01 modes interfering b) HE21 and TE01 modes inter-
fering. The black circle represents the cladding of the fiber with a radius of 450
nm.
These calculations confirm what we had stated earlier in this chapter that by
launching a known mode and using bulk optics we should be able to rapidly convert
the mode on the waist to any desired superposition of states.
134
5.3.2 Experimental setup
PD
Motor
Stage
Motor
StageFlame
π-plate
780 nm
  input
CC
D
Imaging system
Vector Beam 
Generator
λ/2 
Figure 5.6: The experimental setup to measure the Rayleigh scattering from an
ONF.
The experimental setup, as depicted in Fig. 5.6, has three components: the
ONF fiber launch and vector beam generation setup, the ONF Rayleigh scattering
imaging setup, and the ONF output imaging setup. The HOM generation differs
from Sec. 4.2.2. Here we generate a desired mode instead of an unknown superpo-
sition. First we launch light from a 795 nm laser into a polarization maintaining
fiber. This spatially filters the mode and sets the polarization. The output of the
polarization maintaining fiber travels through a phase plate. The phase plate im-
parts a pi phase shift onto half of the beam. This phase shift creates a two lobed
135
mode that when projected onto the fiber basis will excite a superposition of either
HE21 and TE01 modes or HE21 and TM01 modes. The next step is to launch this
superposition into the vector beam generator.
The vector beam generator consists of HI1060 fiber that we thread through
a Thorlabs In-Line Fiber Polarization Controller (PC). The HI1060 fiber has a V
number below 3.8 and admits only the HE11, TE01, TM01, and HE21 modes. The
PC then allows us to control the output mode of the HI1060 fiber. We can then
produce a pure mode, or any superposition of modes from the vector beam generator.
The output of the vector beam generator is then carefully coupled into the ONF.
136
1)
2)
3)
4)
5)
2) 3)
4)
1)
a)
b)
Figure 5.7: (a) The imaging system used for Rayleigh scattering measurements. (b)
The optics for the imaging system as detailed in Tab. 5.1
137
Item Description
1 Mitutoyo NIR 10X objective
2 Achromatic Doublet, f=200 mm
3 795 nm Bandpass filter
4 Mounted Calcite Beam Displacer
5 Luca Andor EMCCD
Table 5.1: List of equipment parts for the Rayleigh scattering imaging system.
Once a pull is completed we use the fiber motors to step along and image the
Rayleigh scattering from the ONF using the imaging system depicted in Fig. 5.7
(see Tab. 5.1). The Rayleigh scattering imaging system is composed of a Mitutoyo
M Plan Apo NIR optimized infinity corrected objective, either 10x or 50x with NAs
of 0.26 and 0.65 respectively. This allows us to choose a shorter or longer depths
of field, or use the minimum spatial resolution necessary depending on the topic of
study. The objective collects and collimates the Rayleigh scattering from the ONF.
The light then passes through a 200 mm achromat doublet (part number here).
Then we place a 795 nm bandpass filter to prevent any room light from adding
unwanted background to the images. Finally we use a calcite polarizer as a walk
off to separate the longitudinal and transverse polarization of the radiated light.
The light is then imaged on an Andor Luca EMCCD. The calcite walk-off limits the
spatial resolution of the imaging system to 3 µm. Since the transverse resolution is
worse than a few microns we cannot observe HOMs beating with the fundamental,
138
as shown in Fig. 5.1.
The ONF output is collimated with an asphere and then imaged on a Thorlabs
Beam Profiler. We control the focus of the Rayleigh scattering image system with
micrometers. While we control the positioning of the ONF relative to the imaging
system along the fiber axis using the two high precision XML 210 fiber motors. This
allows us to precisely move the fiber, image the entire fiber length, and calibrate the
Rayleigh scattering imaging system.
To control the mode [146] we use bulk optics: by placing combinations of
HWPs and QWPs we can efficiently convert modes.
5.3.3 Experimental results
5.3.3.1 Mode identification
Using the experimental setup described in Sec. 5.3.2 we launch a HOM into
an ONF and step the fiber motors in set increments to take a series of images of the
Rayleigh scattering. We then align and concatenate the images together. Figures 5.8
displays approximately 300 concatenated images taken with a Mitutoyo 50X M Plan
Apo NIR optimized infinity corrected objective. Each individual image captures a
200 µm field of view which we then crop to 150 µm. We use the XML 210 motors to
move in controlled 150 µm steps and use features from different images to calibrate
the length scales. Figure 5.8 displays the transverse polarization of the Rayleigh
scattering. We adjust the gain, exposure time and averaging so that we can image
the ONF without saturating the pixels.
139
Figure 5.8: HOM 50x montage transverse polarization with the power scale in
arbitrary units.
Tracing Fig. 5.8 from the left edge to the right we observe that the mode
starts as a core mode. As described in Sec. 3.5, the mode is initially guided by the
core-cladding interface. The imaging system has the resolution to actually observe
the Rayleigh scattering focusing along the fiber axis as the core mode follows the
tapering core. The null in the center of the scattering is an expected consequence of
HOM Rayleigh scattering. Superpositions of HOMs can have a two-lobed intensity
pattern that when scattering and imaged on a plane can have a null in the center.
As the fiber continues to taper and reduces below a radius of 13 µm we see that
the mode can no longer be supported by the core and leaks into the cladding.
Here we observe a beautiful oscillation pattern indicative of modes interfering. The
interference pattern increases in spatial frequency as the fiber tapers and traces
out the diameter of the cladding as the mode focuses down to the ONF waist in
140
which the power scattered increases as surface scattering begins to dominate the
bulk scattering.
~ 0.75 mm
~ 26 µm
Figure 5.9: Rayleigh Scattering depicting HOMs transitioning from core to cladding
modes.
Fig. 5.9, captures in a single image, a HOM superposition leaking from core
to cladding guidance in a fiber taper using SM1500 fiber. Using the Mitutoyo 10X
objective the field of view of the image is 750 µm. We see that the mode starts as a
core mode and leaks into the cladding. The transition occurs at a radius of 13 µm
in Fig. 5.9, which is consistent with the transmission data in Sec. 4.2.3.1.
We can spatially resolve transverse modal information from the Rayleigh scat-
tering when the mode is a core mode. Once the mode escapes into the cladding the
imaging system can resolve the transverse profile until a diameter of about 2-3 µm.
141
If we were able to spatially resolve the transverse Rayleigh scattering on the ONF
waist we could follow the work done in [151]. Unfortunately, we do not have the spa-
tial resolution. Therefore, as described in Sec. 5.3.1 we average out the transverse
scattering information by summing over the columns. When we sum the columns we
are able to observe variations in scattered power along the fiber axis as in Fig. 5.10.
Figure 5.1 displays the expected beat lengths as a function of radius and since we
expect the signal to be chirped we take a spectrogram. A typical spectrogram of
the data is displayed in Fig. 5.10.
0.2
0.4
0.6
0.8
1.0
Sca
ttere
d P
owe
r (A
.U.)
Position along fiber axis (mm)
Bea
t Le
ngth
 x 1
04 (1
/m)
0 2 4 6 8 10 12 14 16 18 20
0.5
1.0
1.5
2.0
2.5
3.0
3.5
a)
b)
Figure 5.10: a) Scattered power along the ONF waist after summing over the
columns. b) A spectrogram of aligned concatenated images displaying the HE21
mode beating with the TM01 mode.
Fig. 5.10 (a) shows the power scattered as a function of position near the ONF
142
waist for a 1 mrad pull with rw = 390 nm. The change in oscillation frequency in
scattered power is quantified in the spectrogram in Fig. 5.10 (b). We see that the
inverse beat frequency first increases, then decreases implying a long beat length.
The inverse beat frequency reaches a steady value along the fiber waist, and then
reproduces the symmetric pattern on the other side of the taper.
Using Figs. 5.2 and 5.3 we can identify that the only inverse beat frequency
that matches near 390 nm is the value for the HE21 mode beating with the TM01.
We can also see that rw for this fiber is slightly smaller than predicted at about
360 nm. Furthermore, the general shape of the mode profile is qualitatively the
same as the shape of the blue curve in Fig. 5.3. This distinct shape arises from
the crossing of the neff curves for the HE21 mode and TM01 at around 430 nm,
which is identifiable in Fig. 4.2.1. Fig. 5.2 comes from the sum of the transverse and
longitudinally polarized scattered light. If we were to plot the scattered power for
the transverse and longitudinal components and take each respective spectrogram
we would see the same profile in either image. Therefore, it is easy to see that this
signal must arise from the TM01 mode beating with the H21 since the TE01 mode
should have no longitudinal component 1.
5.3.3.2 Mode control
To demonstrate mode control on the ONF waist we launch an input mode that
projects onto a superposition of the HE21 and TM01 modes, see Fig. 5.5. Rotating
1We are never actually able to completely eliminate the longitudinal component of the scatter-
ing.
143
a HWP converts the superposition into a sum of HE21 and TE01. We image the
longitudinally and transversally polarized Rayleigh scattering across the waist and
then take the Fourier transform of the power scattered vs position for various HWP
angles. Figure 5.11 (a) displays the power scattered as a function of position along
the ONF waist when launching a superposition of the HE21 and TM01 modes. We
note that the offset between the blue curve, the launching of a superposition of
HE21 and TM01, and the red curve,the launching of a superposition of the HE21
and TE01, is added to ease the visualization of the profiles. We changed the mode
by rotating a HWP at the input of the ONF launch by 44 degrees.
144
0.5 1 1.5 2 2.5 3 3.51
2
3
4
5x 10
4
Fiber axis (mm)
Sc
atte
red
 Po
we
r (a
.u.)
2 4 6 8 10
x 104
0
0.5
1
1.5
2
2.5x 10
6
Freq. (1/m)
Ab
s. F
FT
 (a.
u.)
Freq. x104 (1/m)
a) b)
Figure 5.11: (a) Scattered power vs position along the fiber axis with HWP angle
offset by 44 degrees. This converts the mode from a HE21 and TM01 superposition
(blue) to a HE21 and TE01 superposition (red). The offset is added for visualization.
(b) Absolute value of the Fourier transform for the Rayleigh scattering. We see the
HE21 and TM01 peak at about 2.5 × 104 1/m in the blue curve (and not in the
red curve) and we observe the HE21 and TE01 peak at about 8 × 104 1/m in
the red curve (and not in the blue curve) indicating we have fully converted the
superposition of modes.
Figure 5.11 (b) shows Absolute value of the Fourier transform for the Rayleigh
scattering. We see the HE21 and TM01 peak at about 2.5 × 104 1/m in the blue
curve (and not in the red curve) and we observe the HE21 and TE01 peak at about
8 × 104 1/m in the red curve (and not in the blue curve) indicating we have fully
converted the superposition of modes.
Figure 5.12 shows a more detailed description of the above process for the
145
longitudinally and transversely scattered light. We now collect the scatter over
the ONF waist for 2 degree rotations of HWP. Then we take the absolute value of
the Fourier transform squared and plot each of these curves as a vertical slice in
Fig. 5.12.
Half−wave plate angle (degrees)
Inve
rse b
eat l
engt
h x 1
04  (1
/m)
0 20 40 60 80
2
4
6
8
10 Half−wave plate angle (degrees)
Inve
rse b
eat l
engt
h x 1
04  (1
/m)
0 20 40 60 80
2
4
6
8
10
a) b)
0.5
1.0
1.5
2.0x 10
4
1
2
3
4
5
x 104HE21 - TM01
HE21 - TE01
HE21 - TM01
Figure 5.12: Conversion of HE21 and TM01 to HE21 and TE01 as a function of
half-wave plate angle a) Longitudinal component only. b) Transverse component
only.
We see that at a HWP angle of zero we start with the HE21 and TM01 modes
beating together since the signal occurs in the longitudinal plot. As the HWP angle
rotates we see that the the magnitude of the Fourier transform drops for the HE21
and TM01 while it grows for the the HE21 and TE01. At a 40 degree rotation we
see that a superposition of HE21 and TE01 occupies the ONF waist while there is
no TM01 within the contrast of the Fourier transform. This conversion occurs at
40 degrees rather than 45 because the a HWP angle of 0 does not correspond to
a maximum in the absolute value of the Fourier transform in the HE21 and TM01
146
frequency band. Rotating the HWP another 45 degrees to 85 degrees brings in a
maximum in the HE21 and TM01 frequency band and a minimum in the HE21 and
TE01 frequency band, completely as expected.
Half−wave plate angle (degrees)
Inv
ers
e b
ea
t le
ng
th 
x 1
04  
(1/
m)
0 20 40 60 80
2
4
6
8
10
1
2
3
4
5
6
x 104
HE21 - TE01
HE21 - TM01
Figure 5.13: Conversion of HE21 and TM01 to HE21 and TE01 with longitudinal
and transverse components summed.
Figure 5.13 is the sum of the longitudinal and transverse components in
Fig. 5.12. It is clearer here that the background in the other frequencies is more
uniform that in the individual longitudinal and transverse components.
147
0 25 50 75 1000.0
0.5
1.0
1.5
2.0
2.5
3.0
Half−wave plate angle (degrees)
FF
T 
(a
.u.
) 
Figure 5.14: Conversion of HE21 and TM01 to HE21 and TE01 specifically in the
proper frequency bands.
Fig. 5.14 displays the absolute value of the Fourier transform from Fig. 5.13
in the relevant frequencies bands for the inverse beat. The red circles represent the
presence of the HE21 and TM01 modes, while the blue circles depict the HE21 and
TE01 modes. We see that the two signals are 90 degrees out of phase (45 degrees in
HWP angle) as expected.
148
Half−wave plate angle (degrees)
Inv
ers
e b
ea
t le
ng
th 
x 1
04
 (1
/m
)
0 20 40 60 80
2
4
6
8
10 10 30 50 70 90
HE21 and TE01
HE21 and TM01
Figure 5.15: Pure mode conversion as a function of half-wave plate angle
The last step necessary to demonstrate modal control on the ONF waist is to
send in a pure mode. If there is a pure mode, there should be no frequency spectrum
in the spectrogram or Fourier transform. Since a pure mode input undergoes mode
conversion through the ONF we cannot simply launch a pure mode into the fiber.
Instead, we image the waist and look at the Fourier transforms in real time and
tweak the input until there is no peak in the frequency spectrum. We place QWP
and HWP in front of the fiber launch and adjust the PC to change the fiber launch
into the ONF. Fig. 5.15 demonstrates the result of this arbitrary input. At a HWP
angle of zero degrees there is a small signal from the HE21 and TM01 modes. As
149
the HWP is rotated the TM01 mode comes in and out as the TE01 mode is excited.
Finally at 90 degrees we see that there is no beat signal at all. Implying a pure
mode on the waist.
This technique does not uniquely identify the mode on the ONF waist nor does
it express the determine the distribution of modes, without some further calibrations.
It seems within reason, that if we could spatially resolve and identify the mode at
a radius of a few microns the mode conversion would be minimal to the waist.
Furthermore, if we were able to both identify the mode at a few microns on input
and output of the waist it is likely that the mode on the ONF waist is identical.
5.3.4 Surface scattering and mode cutoffs
When tapering a fiber below 325 nm, we pass the HE21 cutoff. Fig. 5.16
shows the longitudinal, transverse, and total Rayleigh scattering from the ONF as
a function of position with a 1 mrad angle tapered to a 300 nm radius waist that is
1 cm long. Here we observe a number of phenomena.
150
10 20 30 40 50
2
4
6
8
10
Fiber axis position (mm)
To
tal 
po
we
r (a
.u.
)
2
3
4
5
6
Tra
n. 
po
we
r (a
.u.
)
2
3
4
5
Lo
ng
. p
ow
er 
(a.
u.)
Lw
HE21 cutoff
HE21 loss
Lw
HE21 loss
a)
b)
c)
Figure 5.16: Launching a superposition of HOMs into a fiber with Ω = 1 mrad, rw
= 300 nm, and Lw = 1 cm. The three panels correspond to scattering collected with
(a) longitudinal polarization, (b) transverse polarization, and (c) total scatter as a
function of length respectively. The long dashed gray line denotes the HE21 cutoff
and from that point we observe only the propagation of the TE01 mode. The short
dashed red lines show the power loss from the HE21 mode ejecting from the ONF.
Finally, the continuous black lines designate the ONF waist.
151
First, from a position of 4 mm to 14 mm on the fiber axis we see that the total
power is steady. That this region, a taper section, has a steady scattering rate, is
indicative of bulk scattering and is consistent with Lambert-Beer law. The fact that
there is a steady increase in the scattering rate as the fiber continues to taper, from
14 mm to 29 mm is consistent with surface scattering effects. The intensity of the
mode at the fiber surface increases as the radius decreases and surface imperfections
begin to dominate over bulk scattering.
The next interesting behavior occurs at 29 mm on the abscissa, denoted by
the dashed gray line in Fig. 5.16. This is the point that the HE21 mode cuts off
at a radius of 325 nm. We see an immediate drop in the Rayleigh scattering, most
pronounced in the longitudinal component in panel (a) of Fig. 5.16.
In each panel of Fig. 5.16 we see a flat region between 30.5 mm and 40.5 mm
in the scattering. This region, denoted Lw corresponds to the ONF waist. We see
that it is 10 mm long, consistent with the length specified by the pulling algorithm
for the ONF.
The difference in height between the red dashed lines in the total scattered
power plot represents the power loss from the HE21 mode ejection as observed in
the decrease in scattered light. We choose two symmetric regions on opposite sides
of the waist, such that we are looking at similar radii, to make the comparison.
Interestingly, the scattering stays fairly consistent along the the rest of the
ONF . The data is consistent with having the HE21 and TE01 modes excited. Since
the HE21 mode is ejected from the waveguide, and we see that scattering still exists
in the transverse direction, we know there are other modes left in the fiber. The
152
only mode that can propagate with no z-component in the mode profile is the TE01
mode at that radius. Therefore, only the TE01 mode is left in the waveguide from
the point of the HE21 cutoff. Note there is still scattering along the ONF waist
in the longitudinal component. This could be caused by a small TM01 or HE11
background that is not observable in the spectrogram. Ref. [151] notes that the
longitudinal scattering was never fully eliminated even in unmodified fibers.
153
40
Fiber axis position (mm)
Inve
rse 
bea
t len
gth 
(x 1
04 )
38 40 42 44 46 48 50 52 54
5
10
1
2
3
1
2
3
Tran
. po
wer
 sca
t. (a
.u.)
Lon
g. p
owe
r sc
at. (
a.u.
) a)
b)
c)
HE21 - TE01
Figure 5.17: (a) Longitudinally polarized scattered power vs position. (b) Transver-
sally polarized scattered power vs position. (c) Spectrogram of power scattered. The
red dashed line at 53 mm represents the HE21 mode cutoff.
Using the spectrogram we can extract the radius of the fiber as a function
of position. Fig. 5.17 shows the longitudinal and transverse power scatter and the
spectrogram of the total power scattered for an ONF with a taper geometry of Ω
= 0.5 mrad and rw = 300 nm. We see at 53.6 m that the HE21 mode cuts off as
154
represented by the red dashed line in Fig. 5.17. This is observable in not only the
longitudinal and transverse power scatter but also in the spectrogram in which the
signal disappears. Similar to Fig. 5.16 we see that the magnitude of the longitudinal
scatter decreases significantly in Fig. 5.17 implying only the TE01 mode propagating
in the waveguide from that point. From the inverse beat length in spectrogram at
the cutoff we know exactly what modes are beating and identify them as the HE21
and TE01 modes.
Here we extract the radial profile as a function of position along the fiber axis
in various ways:
1. Use the simulated profile for the fiber in regions where the beating in the
spectrogram is clean and overlap the expected beat lengths for that profile
with the spectrogram.
2. Manually extract the beat length from the spectrogram and from there invert
the beat lengths to obtain the radius as a function of position.
3. Suppose an exponential taper with an unknown coefficient and vary the coef-
ficient until the overlap with the spectrogram is optimized.
The taper profile occasionally deviates from the simulation enough to make
1 an imperfect choice. It helps to identify the mode but not necessarily extract
the radius as a function of position. Commonly the fiber is thinned and stretched
farther than our simulation predicts. So that using the fiber pulling simulation leads
to decent overlaps on one side of the taper but not both. The issue then becomes
155
in defining the zero The simulated profile seems to fit either side of the taper well
locally when varying the origin of the taper. When overlapping the simulation with
the experimental data, there is an error in where we define our zero. That is we
do not know exactly the radius of the fiber from our measurements. We know how
far we have moved and the pixel conversion, but not the radius of the fiber as a
function of position. Therefore, we need to redefine the zero of the simulated radius
vs position to overlap with the scattered power.
2 is both time consuming and difficult to automate for pulls that do not have
clean spectrograms. The speckle in the scattering leads to frequency components
that obfuscate the inverse beat frequencies we wish to extract. This broadband
signal in the Fourier transform makes it difficult to automate the process of inverse
beat length extraction.
We have found that 3 is the easiest to implement in practice. Fig. 5.18 demon-
strates the overlap, in white, of the spectrogram. Here we use four separate expo-
nential sections and vary the coefficients in each until the overlap appears to agree.
156
Inve
rse b
eat l
engt
h (x 
104 1
/m)
2
4
6
8
10
0.0
0.5
1.0
1.5
Fibe
r rad
ius (
µm)
38 40 42 44 46 48 50 52 54Fiber axis position (mm)
38 40 42 44 46 48 50 52 54Fiber axis position (mm)
Region I
Region 2 Region 4
Region 3
Cutoff
a)
b)
Figure 5.18: (a) A spectrogram of the data from Fig.5.17 with a white line repre-
senting the expected beat length for an extracted radius profile. (b)The extracted
radius as a function of position along the fiber axis from stitching together four
exponential geometries. Each region with a different coefficient is marked with red
dashed lines.
Fig. 5.18 (b) shows the corresponding extracted radius as a function of position.
With the extracted radius information, it could be possible to perform further studies
of the power scattered as a function of radius for various modes.
157
Fre
qu
en
cy 
x 1
04  
(1/
m)
1
2
3
2 4 6 8 10 12 14 16 180.00
0.02
0.04
0.06
0.08
0.10
Fiber axis (mm)
Re
lat
ive
 di
ffe
ren
ce
a)
b)
Figure 5.19: (a) Spectrogram overlapped with fiber simulation (solid white line)
and four exponential taper sections (dashed black line). (b) The deviation in the
measured overlap of extract radius to the simulation normalized to the simulated
radius. The red line marks the RMS value.
Fig. 5.19 displays a similar means of extracting the fiber radius, this time for a
fiber intended to have a 390 nm radius waist. Therefore, the HOMs do not cutoff in
the spectrogram and we can clearly see the section that corresponds to the waist. We
employ a similar technique as done before: start with the simulated fiber profile, fit
the profile with an exponential, vary the coefficient in multiple exponential sections
158
until the overlap matches nicely. Despite, the 104 (1/m) difference on the fiber waist
between the measured and simulated fiber this is only equal to a 30 nm difference
in radius. Therefore on the fiber waist, with a HWHM (3 dB point) of about 1000
(1/m) this leads to an error in diameter of ± 5 nm.
Fig. 5.19 (b) displays the deviation in the extracted radius from the simula-
tion. The red line denotes the RMS value of 0.069.
It is worth noting that this is an in situ method of measuring the radius of
the ONF. We do not require a high NA imaging system and can extract the radius
of the fiber waist down to about 640 nm diameter. If we were able to resolve the
beating of the fundamental mode with HOMs, see Sec. 5.3.5, we could measure the
radius of fibers down to 560 nm diameter. This is all for 780 nm light. If we were
to scale to the equivalent V number using 405 nm input light we should be able to
use this technique to measure the radii of ONFs down to a waist diameter of 290
nm without requiring high NA optics.
5.3.5 Optical fiber probe
The Rayleigh scattering provides an efficient means of measuring the modes
on the ONF waist but the speckle leads to noise in the spectrogram observable in
Fig. 5.12. These frequencies in the Fourier transform obscure the signal we wish to
extract. Another means of detecting the mode information is to directly access the
evanescent field. The Rayleigh scattering occurs from a radiating dipole polarized in
the direction of the mode field. If we directly measure the variation in the evanescent
159
field intensity along the fiber axis the signal could be improved. To achieve this we
use a tapered fiber probe.
Here, we use evanescent coupling from the ONF waist under study to the
field of the fiber probe. For example, we produce a tapered fiber with a 3 µm radius
waist and directly bring the probe fiber into contact with the ONF waist from below.
Some of the light propagating along the ONF waist will couple into the fundamental
mode of the probe fiber. We then image the output of the fiber probe with a 5X
Mitutoyo objective and 795 nm bandpass filter on an Andor Luca EMCCD.
2
3
4
5
6
7
1
Figure 5.20: The fiber probe setup. 1) Direction of the input HOM beam. 2) ONF.
3) Aluminum fiber holder. 4) Probe fiber. 5) DRV014 stepper motors for positioning
probe fiber. 6) XML 210 motors for positioning the ONF. 7) Output beam from the
probe fiber to EMCCD.
Fig. 5.20 shows the experimental setup for the fiber probe measurement. A
3 µm radius waist fiber probe (4 in Fig. 5.20) is epoxied onto an aluminum fiber
holder (3 in Fig. 5.20). Two DRV014 Thorlabs stepper motors (fiber probe motors)
160
bring the fiber probe into contact with the ONF of interest(5 in Fig. 5.20). The fiber
probe motors allow us to bring the fiber probe into contact from below the ONF and
position the waist of the fiber probe in contact with the ONF (2 in Fig. 5.20). We
raise the fiber probe motors in 50 µm steps until we observe light output from the
fiber probe on the EMCCD. Then using the XML 210 pulling motors (6 in Fig. 5.20)
we can step the ONF along the fiber probe, typically in 0.5 µm steps, and measure
the collected power as a function of fiber position.
0 0.050 0.100 0.150 0.200 0.2500
50
100
150
200
Co
lle
cte
d 
po
we
r (
a.
u.
)
Fiber position (mm)
Figure 5.21: Power collected through the fiber probe as a function of position along
ONF waist. The data is collected in 500 nm steps.
161
Fig. 5.21 shows the power collected as a function of fiber position in which
we collected data in steps of 500 nm through the probe fiber. The dimensions of
the ONF are rw = 400 nm, Ω = 1 mrad, and Lw = 10 mm. Here we intentionally
launched the fundamental mode to test the resolution of the technique. The signal
shows multiple modes beating together with a few wavelengths on the order of 4 µm
and a longer wavelength on the order of 100 µm.
0 1 2 3 4 510
2
103
Ab
s F
FT
 (a
.u
.)
Inverse beat frequency (x 105 1/m)
HE11-HE21
HE11-TE01
HE21-TE01
HE21-TM01
HE11-TM01
Figure 5.22: FFT of optical power collected using the fiber probe from Fig. 5.21.
Here we can identify five frequencies detailed in Tab. 5.2. We note the y-axis is a
log scale.
162
Figure 5.22 show the absolute value of the Fourier transform of the data from
Fig. 5.21. Here we observe five peaks corresponding to the HE21 − TM01, HE21 −
TE01, HE11 − TE01, HE11 − TM01, and HE21 −HE11 modes beating together.
Mode Sim. freq.(1/m) Exp. freq. (1/m)
HE11 −HE21 2.80×105 2.84×105 ±0.06×105
HE11 − TM01 2.66×105 2.68×105 ±0.06×105
HE11 − TE01 2.09×105 2.20×105 ±0.07×105
HE21 − TE01 0.70×105 0.64×105 ±0.10×105
HE21 − TM01 0.13×105 0.12×105 ±0.08×105
Table 5.2: Table comparing simulated inverse beat frequencies for a 380 nm ONF
with extracted. The error bars correspond to the HWHM (3 dB point) relative to
each peak.
Table 5.2 compares the inverse beat frequencies simulated in Fig. 5.2 to the
frequencies extracted from Fig. 5.22. The values for the simulated frequency were
chosen for an ONF with a 380 nm radius waist using the largest highest frequency
peak in Fig. 5.22 to identify the radius and mode. The error bars correspond to the
HWHM from the maximum of each peak. We see that the frequencies are in excellent
agreement for a 380 nm fiber and note that this method has a finer resolution than
the optics of the Rayleigh scattering method. Here we can both observe and identify
the fundamental mode interfering with HOMs which was beyond the resolution of
the Rayleigh scattering method.
163
5.3.6 Photodiode motor jog
We employ a more efficient means of data collection using a transimpedance
amplifier set to a gain of 109 V/A to amplify the power collected on a photodiode
(PD) from the fiber probe. This method allows us to observe the mode profile on
an oscilloscope in real time as we sweep the motors. Using the jog mode of the
motors, we are able to set a fixed velocity for both fiber pulling motors, and we
sweep the ONF across the fiber probe. We typically sweep with velocities from 1
to 50 micrometers per second. The data is collected on a DPO7054 for hundreds to
thousands of seconds, with a one sample collected per ms.
164
0 5 10 15 20 250.6
0.62
0.64
0.66
0.68
0.7
Fiber axis (mm)
Powe
r (a.u.)
20 20.5 21 21.5 22 22.5 23 23.5 24 24.50.62
0.64
0.66
0.68
0.7
Fiber axis (mm)
Powe
r (a.u.)
20 20.05 20.1 20.15 20.2 20.25 20.3 20.350.62
0.64
0.66
0.68
0.7
Fiber axis (mm)
Powe
r (a.u.)
a)
b)
c)
Figure 5.23: Probe power collected using a PD while jogging the pulling motors at
20 µm/s. (a) Power collected while sweeping from the input taper across the ONF
waist and output taper. (b) A zoom in along 20 mm to 25 mm. (c) A magnification
of 20 mm to 20.35 mm.
Fig. 5.23 displays the probe power collected using the above method while
jogging the motors at 20 µm/s. Fig. 5.23 (a) shows a sweep from the input taper
(0-12 mm), where we can see many modes interacting. The next two panels are
magnifications show the beating from a 3 µm scale to tens of micrometers.
165
Fiber axis (mm)
Fr
e
qu
e
n
c
y 
(x1
05
 
1/
m
)
0 10 20 30
1
2
3
4
5
6
Figure 5.24: Spectrogram of data from Fig. 5.23 taken with the
Fig. 5.24 shows a spectrogram of the data from Fig. 5.23. We use a bin of
1024 points and overlap of 1019 points which is equivalent to 20.48 mm bins with a
20.38 mm overlap. It is clear that this 1 mrad taper was lossy since we can observe
many modes cutting off on the input taper. On the ONF waist we observe 6 peaks:
5 corresponding to known modes and 1 occurring from a nonlinearity. At the taper
output we observe a cleaner signal since the higher-order mode excitations were
cutoff before the waist.
166
1 2 3 4
10
0
10
2
10
4
Freq. (x105 1/m)
A
bs
. F
FT
 (a
.u
.)
Mode Measured freq. (1/m)
HE11 −HE21 2.82×105± 0.04× 105
HE11 − TM01 2.65×105± 0.025× 105
HE11 − TE01 2.11×105± 0.02× 105
HE21 − TE01 7.0×104± 0.125× 104
HE21 − TM01 1.50×104± 0.125× 104
Figure 5.25: FFT of power collected from ONF waist from 5.23 and table of
extracted beat frequencies.
167
Fig. 5.25 displays the absolute value of the FFT of the power collected cor-
responding to the ONF waist from Fig. 5.23. The error bars are extracted from
the HWHM from the peak. We observe the peaks for each of the allowed modes
of propagation, all in excellent agreement with simulated frequencies for a 380 nm
radius fiber, see Table. 5.2.
We follow the same method described earlier for extracting the radius. This
time we vary two exponentials for either side of the taper and overlap for each of
the five modes, see Fig. 5.26.
Fr
eq
ue
nc
y 
(x
10
5  
1/
m
)
Fiber axis (mm)
10 20 30
0.5
1
1.5
2
2.5
3
Figure 5.26: Spectrogram of data from Fig. 5.23 with radius extraction overlap
(white curves).
168
We compare measured radius to the expected radius form simulation in Fig. 5.27.
Here we see that the agreement is slightly better with an RMS value of 0.06. The
waist radius of 380 nm is closer to the expected radius of 400 nm.
0 10 20 30 400
2
4
6
Fiber axis (mm)
Ra
diu
s (m
icro
ns)
0 10 20 30 400
0.02
0.04
0.06
0.08
0.1
Fiber axis (mm)
Ra
diu
s d
evi
atio
n
a)
b)
Figure 5.27: a) Measured radius (blue) and simulated radius (red) vs position. b)
Deviation of the simulated radius with the red line representing the RMS value of
0.06.
169
Using the HWHM frequency range for the HE21 mode beating with TM01
mode we arrive at an error of ±6 nm. Once again it is worth reiterating that these
beat frequency measurements allow for nondestructive precise measurements of the
fiber radius.
5.4 Conclusion
We demonstrated the ability to identify and control the mode on the ONF
waist in this chapter. We use two different techniques: Rayleigh scattering and
evanescent coupling with a fiber probe. We note that the ONF appears to act as
a mode converter when light propagates through the waveguide. Whether this is
a fundamental property of the ONF or simply a technical constraint imposed by
working with fiber clamps and in tight confines inside the clean room is still up for
debate. Further study is required using IDIOM on microfibers to better understand
this phenomena. Rayleigh scattering measurements inherently have more noise that
the fiber probe technique, but provide a faster real time analysis on the mode distri-
bution. The fiber probe provides an improved resolution compared to the Rayleigh
scattering technique and is capable of distinguishing between the fundamental mode
beating with any of the HOMs. It worth noting that these techniques are accurate
means of measure the fiber radius.
170
Chapter 6: Conclusions and Outlook
In the preceding five chapters, we have described the fabrication and analysis of
ONFs usable for a hybrid quantum system composed of a superconducting resonator
and cold trapped 87Rb atoms.
In Chapter 2 we outlined the fabrication process providing details on the clean-
ing, alignment, and full pulling process. We also provided details on the derivation
of the pulling algorithm. Using microscopy we have verified that our pulling geom-
etry is in excellent agreement with simulation. We have also report the lowest loss
ONFs with 2.6 ×10−5 dB/mm marking an improvement of 2 orders of magnitude
when compared to previous work. When following the pulling and transfer proce-
dures outlined we report the highest transmission reported through an ONF with
400 mW of optical power transferred through.
Chapter 3 details an analysis of the modal evolution in an ONF taper. We were
able to identify the higher-order modes the fundamental mode couples to during the
tapering process Using a genetic algorithm we are able to provide better bounds on
ONF geometries to produce desired transmission given specified length constraints.
In Chapter 4 we demonstrate the highest reported transmissions through ONFs
when intentionally launching HOMs. Here, we are able to demonstrate comparable
171
losses when launching HOMs to losses reported through ONFs when launching the
fundamental mode. Finally in Chapter 5 we provide a detailed analysis identifying
the mode structure on the ONF waist using both Rayleigh scattering and evanescent
coupling.
Below we describe the outlook and next steps.
6.1 Hybrid System
Fig. 6.1 shows our groups current vision of the setup for the hybrid quantum
system. The Rayleigh scattering from an ONF, while approximately parts in a
million, will still form quasiparticles in the SC. Therefore we want to minimize the
cross section of scattered light from the ONF by using the SC resonator geometry to
our advantage. With this in mind, we polish the Sapphire substrate to the inductor
edge. This allows us to place the ONF in the plane of the chip, parallel to the 100
nm-400 nm layer of deposited Al rather than above the entire chip. This reduces
the light scattered cross section by at least a factor of 20.
172
Figure 6.1: The setup of the hybrid quantum system. The superconducting lumped-
element LC resonator was fabricated from a thin-film of Al deposited on a sapphire
chip. The sapphire chip was polished by Nesco Lettsome at Neocera so that the
edge of the sapphire chip was at the edge of the inductor. The ONF is in the plane
of the SC resonator and is 10 µm from the edge of the inductor. Atoms are trapped
in standing wave maxima approximately every 500 nm.
The ONF trap requires that atoms are cooled for efficient loading. Therefore
the atoms will need to be cooled and trapped elsewhere in the setup. One approach
that the group has pursued is using a pyramid-type MOT [153, 154] in a chamber
anchored to the 3.5 K stage of the dilution refrigerator which has Watts of cooling
power. Here we can place the necessary magnetic coils away from the supercon-
173
ducting chip. However, this means that we will need an optical conveyor belt to
transport the atoms from the MOT along the ONF waist to the chip. We need
to determine the maximum distance we can transport the atoms while minimizing
atomic loss. This process will require great care, for example using adiabatic proto-
cols [155] or dynamic controls [132]. The atomic transport process is an important
design consideration. Fortunately we can study this process atroom temperature.
To introduce atoms into the dilution refrigerator one approach the group is
currently developing is a 2D MOT atomic beam source [156]. The 2D MOT would
attach to one of the ports on the vacuum jacket of the dilution refrigerator and will
be used to load the pyramid MOT. A room temperature 2D MOT, could in theory,
be directly attached to the dilution refrigerator as a separate module.
6.2 ONF studies
There are many other interesting ONF issues that we did not explore in this
thesis. For example it would be interesting to optimize an ONF to maximize the
spontaneous emission back into the fiber, a term known as Γ1D. Fig. 6.2 shows a pa-
rameter search calculating Γ1D as a function of ONF radius and atomic displacement
from the ONF surface.
174
Pred
æ
15 mW
à
30 mW
ì
60 mW
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ
æ
à
à
à
à
à
à
à
à
à
à
à
à
à
à
à
à
à
à
à
à
à
à
à
à
à
à
ì
ì
ì
ì
ì
ì
ì
ì
ì
ì
ì
ì
ì
ì
ì
ì
ì
ì
ì
ì
ì
ì
ì
ì
ì
100 150 200 250 300
0
50
100
150
200
250
300
350
fiber radius HnmL
a
t
o
m
-
f
i
b
e
r
d
i
s
t
a
n
c
e
H
n
m
L
Γ
1D
Γ
0.05
0.10
0.15
0.20
0.25
Figure 6.2: Γ1D calculated in the parameter space of fiber radius and distance from
the fiber surface. This space is then searched for a 300 µK trap depth for 15 mW
(circle), 30 mW (square), and 60 mW (diamond) in the red-detuned trapping beam.
Using the information in Fig. 6.2 we could pull a fiber that maximizes Γ1D for
atoms located at realistic trapping sites. With an ONF maximized for Γ1D it will
be possible to pursue self-organization of atoms into crystalline structures [82].
175
This work opens new avenues for the studies of HOMS in ONFS using the
interference of HOMs we can create unique trapping geometries. A possible next
steps would be to design and fabricate an ONF system capable of loading and
trapping atoms using the HOMs that such an ONF supports. This would reduce the
level of optical power in the atom trap and reduce the Rayleigh scattering from the
ONF, two important parameters in the pursuit of coupling atoms to superconducting
devices.
176
Appendix A: Fiber Modes
Here we derive the bound modes for a step index fiber. We generally follow
the work of Refs. [74,131,157,158].
A.1 Cylindrical coordinates
∇ ·A =
1
r
∂r [r Ar] +
1
r
∂φAφ + ∂zAz (A.1)
∇×A =
(
1
r
∂φAz − ∂zAφ
)
rˆ + (∂zAr − ∂rAz) φˆ+
1
r
(∂r [r Aφ]− ∂φAr) zˆ (A.2)
∇2 = ∂2r +
1
r
∂r +
1
r2
∂2φ + ∂
2
z (A.3)
177
A.2 Maxwell’s Equations
Maxwell’s equations in a material with dielectric constant  and magnetic
permeability µ are given by
∇ · E =
ρ

(A.4)
∇ ·B = 0 (A.5)
∇× E = −µ∂tH (A.6)
∇×H = ∂tE (A.7)
Using the curl equations we arrive at six separate equations
 ∂tEr =
1
r
∂φHz − ∂zHφ (A.8)
 ∂tEφ = ∂zHr − ∂rHz (A.9)
 ∂tEz =
1
r
(∂r [r Hφ]− ∂φHr) (A.10)
−µ ∂tHr =
1
r
∂φEz − ∂zEφ (A.11)
−µ ∂tHφ = ∂zEr − ∂rEz (A.12)
−µ ∂tHz =
1
r
(∂r [r Eφ]− ∂φEr) (A.13)
(A.14)
178
A.3 Cylindrically symmetric dielectric waveguide
n
1
n
2
n
3
n
4
n
1
n
2
n
3
n
4
r
1 r
2
r
3
r
4
n
5
n
r
Figure A.1: Cylindrically symmetric waveguide, with constant layers of index of
refraction.
For the moment, we envision a waveguide with radial layers of azimuthally
symmetric dielectric functions and free space magnetic permeability, see Fig A.1.
This symmetry implies solutions of the form,
E = E0 (r, φ) exp [i (ω t− β z)]
H = H0 (r, φ) exp [i (ω t− β z)]
179
inside each radial layer. Using the above it is clear that
∂tE = iωE
∂tH = iωH
∂zE = −iβE
∂zH = −iβH
Applying the above to equations A.8 we find
iωEr =
1
r
∂φHz + iβHφ
iωEφ = −iβHr − ∂rHz
iωEz =
1
r
(∂r [r Hφ]− ∂φHr)
−iωµHr =
1
r
∂φEz + iβEφ
−iωµHφ = −iβEr − ∂rEz
−iωµHz =
1
r
(∂r [r Eφ]− ∂φEr)
which we can now use to solve for the transverse components of the field.
iωEr =
1
r
∂φHz + iβHφ
=
1
r
∂φHz + iβ
1
−iωµ
(−iβEr − ∂rEz)
=
1
r
∂φHz +
β
ωµ
∂rEz +
iβ2Er
ωµ
Er =
−i
ω − β
2
µω
(
1
r
∂φHz +
β
ωµ
∂rEz
)
Er =
−iβ
µω2 − β2
[
−
µω
βr
∂φHz + ∂rEz
]
.
180
Similarly, we arrive at
Er =
−iβ
µω2 − β2
[
µω
βr
∂φHz + ∂rEz
]
(A.15)
Eφ =
−iβ
µω2 − β2
[
−
µω
β
∂rHz +
1
r
∂φEz
]
(A.16)
Hr =
−iβ
µω2 − β2
[
−
ω
βr
∂φEz + ∂rHz
]
(A.17)
Hφ =
−iβ
µω2 − β2
[
ω
β
∂rEz +
1
r
∂φHz
]
. (A.18)
Solving for the complete modal equations is reduced to finding the fields in the
longitudinal direction. We take µ = µ0 and k = µ0ω, using the wave equation we
find that
[
∂2r +
1
r
∂r +
1
r2
∂φ +
(
k2 − β2
)
]




Ez(r, φ)
Hz(r, φ)



 = 0. (A.19)
Separation of variables yields solutions of the form




Ez(r, φ)
Hz(r, φ)



 =




RE(r)
RH(r)



 Exp (±lφ) (A.20)
where l is a nonnegative number. This leads to the differential equation
[
∂2r +
1
r
∂r +
(
k2 − β2 −
l2
r2
)]




RE(r)
RH(r)



 . (A.21)
The above is Bessel’s differential equation, which has solutions of the form,
R(r) =



c1Jl(hr) + c2Yl(hr), k2 − β2 > 0
c1Il(qr) + c2Kl(qr), k2 − β2 < 0
(A.22)
where we define h =
√
k2 − β2 and q =
√
β2 − k2.
Now we apply these types of solutions to a two layer step index fiber.
181
A.4 Two layer step index fiber
n
1
n
2
n
1
r
1
n
2
n
r
r
1
Figure A.2: Cylindrically symmetric waveguide, with a core of index of refraction
n1 out to radius of r1 and an infinite cladding with index of refraction n2, where
n1 > n2.
The fiber waveguide in the two layer model is composed of a core with con-
stant dielectric 1 out to a radius r = a, and an infinite cladding from r = a to
∞ with dielectric constant 2. These boundary conditions reduce the longitudinal
182
components to
Ez(r, φ) = AJl(hr)Exp [i (ωt− βz ± lφ)] , r < a
Hz(r, φ) = BJl(hr)Exp [i (ωt− βz ± lφ)] , r < a
Ez(r, φ) = CKl(qr)Exp [i (ωt− βz ± lφ)] , r > a
Hz(r, φ) = DKl(qr)Exp [i (ωt− βz ± lφ)] , r > a.
(A.23)
As a result of the azimuthal symmetry, there exist two degenerate solutions corre-
sponding to ±lφ. These solutions correspond to a left or right circulating solution
as the mode advances along the fiber axis. We have the freedom to choose either
set as our solution; however, for the moment we will keep the solutions as general
as possible.
Applying equation A.23 to equation A.15 we find for r < a
Er,± =
−iβ
h2
[
±
iµ0ωl
βr
BJl (hr) + AhJ
′
l (hr)
]
Exp [i (ωt− βz ± lφ)] (A.24)
Eφ,± =
−iβ
h2
[
±
il
r
AJl (hr)−
µ0ωh
β
BJ ′l (hr)
]
Exp [i (ωt− βz ± lφ)] (A.25)
Ez,± = AJl(hr)Exp [i (ωt− βz ± lφ)] (A.26)
Hr,± =
−iβ
h2
[
∓
i1ωl
βr
AJl (hr) +BhJ
′
l (hr)
]
Exp [i (ωt− βz ± lφ)] (A.27)
Hφ,± =
−iβ
h2
[
1ω
β
AhJ ′l (hr)±
il
r
BJl (hr)
]
Exp [i (ωt− βz ± lφ)] (A.28)
Hz,± = BJl(hr)Exp [i (ωt− βz ± lφ)] (A.29)
(A.30)
and for r > a
183
Er,± =
iβ
q2
[
±
iµ0ωl
βr
DKl (qr) + CqK
′
l (qr)
]
Exp [i (ωt− βz ± lφ)] (A.31)
Eφ,± =
iβ
q2
[
±
il
r
CKl (qr)−
µ0ωq
β
DK ′l (qr)
]
Exp [i (ωt− βz ± lφ)] (A.32)
Ez,± = CKl(hr)Exp [i (ωt− βz ± lφ)]Exp [i (ωt− βz ± lφ)] (A.33)
Hr,± =
iβ
q2
[
∓
i2ωl
βr
CKl (qr) +DqK
′
l (qr)
]
Exp [i (ωt− βz ± lφ)] (A.34)
Hφ,± =
iβ
q2
[
2ω
β
CqK ′l (qr)±
il
r
DKl (qr)
]
Exp [i (ωt− βz ± lφ)] (A.35)
Hz,± = DKl(qr)Exp [i (ωt− βz ± lφ)] (A.36)
(A.37)
where, J ′l (hr) =
∂Jl(hr)
∂(hr) , K
′
l (qr) =
∂Kl(qr)
∂(qr) .
A.4.1 Boundary Conditions
The parallel E and H fields must be continuous across the boundary at r=a.
This implies that Ez, Hz, Eφ and Hφ must be continuous across the boundary at
r = a. This results in
AJl (ha)− CKl (qa) = 0 (A.38)
BJl (ha)−DKl (qa) = 0 (A.39)
±
βl
h2a
AJl (ha) +
iµ0ω
h
BJ ′l (ha)±
βl
q2a
CKl (qa) +
iµ0ω
q
DK ′l (qa) = 0 (A.40)
−i1ω
h
AJ ′l (ha)±
βl
h2a
BJl (ha)−
i2ω
q
CK ′l (qa)±
lβ
q2a
DKl (qa) = 0 (A.41)
which can be recast in the form,
184











Jl (ha) 0 −Kl (qa) 0
0 Jl (ha) 0 −Kl (qa)
± βlh2aJl (ha)
iµ0ω
h J
′
l (ha) ±
βl
q2aKl (qa)
iµ0ω
q DK
′
l (qa)
−i1ω
h J
′
l (ha) ±
βl
h2aJl (ha) −
i2ω
q K
′
l (qa) ±
lβ
q2aDKl (qa)
























A
B
C
D












= 0. (A.42)
The determinant of the above equation must vanish in order for there to be a
nontrivial solution for A,B,C,D . This results in an eigenvalue problem for β,
[
1
ha
J ′l (ha)
Jl (ha)
+
1
qa
K ′l (qa)
Kl (qa)
] [
n21
ha
J ′l (ha)
Jl (ha)
+
n22
qa
K ′l (qa)
Kl (qa)
]
= l2
[(
1
ha
)2
+
(
1
qa
)2
]2
β2
k20
(A.43)
Now that we know a nontrivial solution exist we use the boundary conditions
to solve for the constants,
C
A
=
Jl (ha)
Kl (qa)
(A.44)
B
A
= ±
ilβ
µ0ω
[(
1
ha
)2
+
(
1
qa
)2
][
J ′l (ha)
haJl (ha)
+
K ′l (qa)
qaKl (qa)
]−1
(A.45)
D
A
=
BC
A2
=
B
A
Jl (ha)
Kl (qa)
−
l2β2
k20
[(
1
ha
)2
+
(
1
qa
)2
]2
. (A.46)
Once we obtain A, the power normalization coefficient (see Sec. A.5), we have a
complete set of solutions for the Electromagnetic fields for a given fiber with radius,
a, and indices of refraction, n1 and n2.
185
A.4.2 The propagation constant and fiber modes
We rewrite Eqn. A.43 as
0 =n21
[
J ′l (ha)
haJl (ha)
]2
+
(
(
n21 + n
2
2
) K ′l (qa)
qaKl (qa)
)[
J ′l (ha)
haJl (ha)
]
+

n22
K ′l (qa)
qaKl (qa)
−
l2β2
k20
[(
1
ha
)2
+
(
1
qa
)2
]2

 .
Solving for
[
J ′l (ha)
haJl(ha)
]
, we obtain
[
J ′l (ha)
haJl (ha)
]
= −
(n21 + n
2
2)
2n21
[
K ′l (qa)
qaKl (qa)
]
±R (A.47)
R =
√
√
√
√(n
2
1 − n
2
2)
2
(2n21)
2
[
K ′l (qa)
qaKl (qa)
]2
+
l2β2
k20
[(
1
ha
)2
+
(
1
qa
)2
]2
. (A.48)
Recalling the recursion relations,
J ′l (x) = Jl−1 (x)−
l
x
Jl (x)
K ′l (x) = −
1
2
[Kl−1 (x) +Kl+1 (x)] ,
we find
Jl−1 (ha)
haJl (ha)
=
(n21 + n
2
2)
4n21
[
Kl−1 (qa) +Kl+1 (qa)
qaKl (qa)
]
+
l
(ha)2
±R (A.49)
R =
√
√
√
√(n
2
1 − n
2
2)
2
(4n21)
2
[
Kl−1 (qa) +Kl+1 (qa)
qaKl (qa)
]2
+
l2β2
n21k
2
0
[(
1
ha
)2
+
(
1
qa
)2
]2
.
(A.50)
The (±)R solutions correspond to the EH and HE modes, respectively. These
are hybrid modes in which all components of the field Er, Eφ, Ez, Hr, Hφ, and Hz
exist. The distinction between EH and HE depends on whether the electric (EH)
or magnetic (HE) field dominates at a chosen reference point.
186
Two special cases of solutions arise when l = 0,
J1 (ha)
ha J0 (ha)
=



−n
2
2
n21
K1(qa)
qaK0(qa)
, (+)TM
− K1(qa)qaK0(qa) , (−)TE
(A.51)
We see that when l = 0 for the (+) solution both A and C vanish yielding a TM
mode. The (−) solution leads to B and D vanishing, resulting in a TE mode.
A.4.3 Solving for the propagation constants
To find the propagation constant for the HE, EH, TE or TM modes we must
solve their respective transcendental equations, A.49 or A.51. In Fig. A.3 we plot
the left hand side (blue) and right hand side (red) of each equation with a fixed V
number and l value and look for their crossings. Where the V number is given by
V = a k
√
n21 − n
2
2. It is worth noting that we can recast qa as a function of ha and
V number.
187
0 2 4 6 8 10
-1.0
-0.5
0.0
0.5
1.0
ha
EH1m rhs vs lhs
0 2 4 6 8 10
-1.0
-0.5
0.0
0.5
1.0
ha
HE1m rhs vs lhs
<
HE11 HE12
EH11
Figure A.3: We plot the left hand side (blue) and right hand side (red) of Eqn. A.49.
There exist two solutions circled with dashed black line for (−R) with l = 1 and the
normalized frequency V=6.5. These solutions correspond to the HE11 and HE12
modes. We see that for any V value, there will always be a crossing for the HE11
mode and therefore it is guided mode for and radius fiber. Unlike the HE modes,
the EH modes have only one solution to Eqn. A.49 for (+R) with l = 1 and the
normalized frequency V=6.5. This solution corresponds to the EH11 mode.
The multiple crossings for a given l value yield families of the same symmetry.
Modes are labeled by HElm where l corresponds to the radial symmetry or oscil-
lations of the mode and m the azimuthal symmetry. The intersections in the plot
yield higher m numbers for the HE1m modes which correspondingly have higher
propagation constants. This procedure can be repeated to find EHlm, TE0m, and
TM0m, see Figs. A.3 and A.4. Figure A.3 shows that for any V number there always
exists a crossing for the HE11 mode. This means that no matter how small the
waveguide radius, the HE11 mode is always guided. All other modes have cutoffs
188
and will radiate their energy into free space, or couple back to other modes in the
fiber, as the radius decreases.
0 2 4 6 8 10
-1.0
-0.5
0.0
0.5
1.0
ha
TE0m rhs vs lhs
<
0 2 4 6 8 10
-1.0
-0.5
0.0
0.5
1.0
ha
TM0m rhs vs lhs
<
TM01 TM02 TE01 TE02
Figure A.4: We plot the left hand side (blue) and right hand side (red) of
Eqn. A.51 with l=0 and V=6.5. From the crossings of the curves we find that
the TE01, TE02, TM01, and TM02 modes can propagate in a fiber with V = 6.5.
These plots are all for a V=6.5. If we vary V, which is effectively changing the
radius of the fiber, we can arrive at dispersion relations. The dispersion relations
describe the effective index of refraction, the propagation constant normalized to
the free space wave vector, each mode experiences for fibers with various core radii.
This effective index, neff is bounded by the indices of refraction of the core and the
infinite cladding. In the limit that V goes to infinity, neff of all modes approaches
ncore, and in the limit as V goes to zero, all modes cut off and radiate, except the
fundamental mode which asymptotically approaches ncladding.
189
0 2 4 6 8 101.0
1.1
1.2
1.3
1.4
V number
n e
ff
Figure A.5: The effective indices of refraction vs V number. Where ncore = 1.45
and ncladding = 1
Figure A.5 shows that as the V number of a fiber increases, or, for fixed indices
of refraction and k0, as the radius of the core of the fiber increases, the fiber becomes
highly multimode. A fiber is single mode, in the core, when the V number is less
than 2.405. Therefore, the first family of excited modes is cutoff and can no longer
propagate.
When working with a single mode fiber designed for propagation of 780 nm
light, such as SM800 from Fibercore, there is a finite core and finite cladding, unlike
in the two layer model. These fibers, typically have a core radius of 2-5 µm and
cladding radius of 62.5 µm. As a fiber is tapered the radius decreases, which naively
would imply that modes get cutoff. But, in a single mode fiber there exists only
one mode, the fundamental, in the core. There are no other modes to cutoff and
no modes to couple to as the eigenbasis changes. As the fiber tapers, this would
190
seem to imply that no other modes could get excited; however, modes can exists in
the cladding. This implies to two regimes, core modes and cladding modes. Core
modes have indices of refraction that satisfy ncore > neff > ncladding. While cladding
modes satisfy ncladding > neff > nair. A real ONF is not adequately represented by
the two layer model. Light initially guided by the core-cladding interface, leaks
into the cladding as neff drops below the index of refraction of the cladding and is
then guided by the cladding-air interface. Although the fundamental mode always
exists in this fiber, there is a discontinuity in the two layer model. If one plots
the dispersion relation for the fundamental mode for core-cladding and cladding-air
guidance as a function of radius we find that these two curves approach each other
at radius of 20µm, see Fig. A.6.
These dispersion relations are necessary in understanding the modal evolution
in a fiber during a taper.
191
16 18 20 22 241.45350
1.45355
1.45360
1.45365
1.45370
1.45375
1.45380
Cladding radius (µm)
n eff
Figure A.6: A magnification of the region of closest approach for the dispersion
relation of the HE11 mode for core-cladding (red) and cladding-air (blue) guidance.
These curves do not connect, exhibiting a limitation of the two layer model.
A.4.4 Quasilinear polarization
When we launch linearly polarized light into a fiber it will excite both the
±lφ solutions. We can represent the mode in what we call a quasilinear basis, as
represented below.
Elin =
1
√
2
[E+ ± E−] (A.52)
Hlin =
1
√
2
[H+ ±H−] . (A.53)
It is useful to think of this as analogous to representing linearly polarized light as a
superposition of left and right circularly polarized.
192
-200
200
0
-200
200
0
-200
200
0
-200
200
0
-20
0
200
0
-20
0
200
0
-20
0
200
0
-20
0
200
0
x (nm)
x (nm) x (nm)
x (nm)
y (n
m)
y (n
m)
y (n
m)
y (n
m)0.0
0.4
0.8
0.0
0.4
0.8
0.0
0.4
0.8
0.0
0.5
1.0
a) b)
c) d)
  
|Ex(r,ⱷ)|2
|Ex(a,ⱷ)|2
  
|Ey(r,ⱷ)|2
|Ey(a,ⱷ)|2
  
|Ez(r,ⱷ)|2
|Ez(a,ⱷ)|2
Int (a.u.)
Figure A.7: The HE11 mode for a 180 nm radius fiber with 780 nm light input
with indices of refraction n1 = 1.45367 and n2 = 1.0. (a) The Intensity profile. (b)
|Ex|2 normalized to the field at the fiber surface. (c) |Ey|2 normalized to the field
at the fiber surface. (d) |Ez|2 normalized to the field at the fiber surface.
Fig. A.7 (a) shows the Intensity profile of the fundamental mode with a radius
of 180 nm and for light with a 780 nm wavelength. We observe a discontinuity
across the cladding-air boundary with about 60% of the power propagating outside
the fiber tightly confined near the fiber surface, see Fig. A.8. Fig. A.7 (b-d) shows
|Ei(r, φ)|2/|Ei(a)|2 for i = x, y, z polarizations. There is a sizable longitudinal com-
193
ponent for the fundamental mode of the fiber. This longitudinal component can lead
to vector light shifts [75]. There are trapping regimes that minimize the longitudinal
component and therefore its deleterious effects [71]. Alternatively, there are means
of cleverly using the longitudinal component to ones advantage and selectively ad-
dressing arrays of trapped atoms on either side of the ONF [76], since the sign of
Ez is different on opposite sides of the fiber.
0 200 400 600 800 10000.0
0.2
0.4
0.6
0.8
1.0
Po
we
r r
ati
o
Thickness of annulus from fiber surface(nm)
Figure A.8: Power ratio as a function of the annulus thickness from the ONF
surface. Red: normalized (to power propagating outside the fiber) power contained
in an annulus of a given thicknesses from the fiber surface. Blue: same but with the
normalization to the total power propagating thorough the fiber. The dashed lines
are the asymptotic values.
194
Fig. A.8 displays the ratio of integrated power propagating through an ONF
with a 180 nm radius as a function of distance from the fiber surface. The red curve
is normalized to the power outside the fiber and the blue curve is normalized to the
total power. We see the red curve asymptotically approaches one while the blue
curve asymptotically approaches 0.61, the power outside the fiber.
A.5 Power Normalization
We have not yet determined all the coefficients to fully describe the modes of
the fiber. The missing coefficient, A, is derived by normalizing the modes to the
input power propagating through the fiber. In other words, the energy flux traveling
in the z direction is conserved:
P =< S >z,t (A.54)
< S >z,t =
1
2
∫
(E×H∗) · z r dr dφ (A.55)
=
A2
2
∫
ErH
∗
φ − EφH
∗
r r dr dφ. (A.56)
Specifically, the time averaged Poynting vector is a constant given by the input
power. We can solve analytically for the exact normalization. We note that the
integral is independent of φ and that we can break the integral into two components:
< S >z,t = A
2pi
∫ a
0
E(in)r H
∗(in)
φ − E
(in)
φ H
∗in
r r dr dφ
+ A2pi
∫ ∞
a
E(out)r H
∗out
φ − E
(out)
φ H
∗out
r r dr dφ
= A2pi (Din +Dout) (A.57)
195
Where Din and Dout represent the power inside and outside the fiber carried
by the guided mode. Once we solve for Din and Dout we can use Eqn. A.57 to solve
for A.
A.5.1 HElm and EHlm mode normalizations
Here we outline how to solve analytically for Din and Dout for the HElm and
EHlm modes.
Dout =pi
∫ ∞
a
(
ErH
∗
φ − EφH
∗
r
)
rdr
Dout =pi
∫ ∞
a
(
iβ
q2
[
iµ0ωl
βr
DKl (qr) + CqK
′
l (qr)
]
−iβ
q2
[
2ω
β
CqK ′l (qr) +
il
r
DKl (qr)
]
−
iβ
q2
[
il
r
CKl (qr)
µ0ωq
β
DK ′l (qr)
]
−iβ
q2
[
i2ωl
βr
CKl (qr)−DqK
′
l (qr)
])
rdr
Dout =pi
β2
q2
C2
∫ ∞
a
([
iµ0ωl
β
B
Kl (qr)
qr
+K ′l (qr)
] [
2ω
β
K ′l (qr) + ilB
Kl (qr)
qr
]
−
[
il
Kl (qr)
qr
−
µ0ω
β
BK ′l (qr)
] [
i2ωl
β
Kl (qr)
qr
−BK ′l (qr)
])
rdr
Defining s as
s =
µ0ω
ilβ
B =
[(
1
ha
)2
+
(
1
qa
)2
][
J ′l (ha)
haJl (ha)
+
K ′l (qa)
qaKl (qa)
]−1
we can rewrite Dout in the form
Dout =pi
β2
q2
C2
∫ ∞
a
([
−l2s
Kl (qr)
qr
+K ′l (qr)
]
ilβ
µ0ω
(−il)
[
2µ0ω2
l2β2
K ′l (qr)− s
Kl (qr)
qr
]
−il
[
Kl (qr)
qr
− sK ′l (qr)
]
ilβ
µ0ω
[
2µ0ω2
β2
Kl (qr)
qr
− sK ′l (qr)
])
rdr
Dout =pi
β2
q2
l2β
µ0ω
C2
∫ ∞
a
([
−l2s
Kl (qr)
qr
+K ′l (qr)
] [
2µ0ω2
l2β2
K ′l (qr)− s
Kl (qr)
qr
]
+
[
Kl (qr)
qr
− sK ′l (qr)
] [
2µ0ω2
β2
Kl (qr)
qr
− sK ′l (qr)
])
rdr.
196
Then, we can define N2 as
N2 =
√
2µ0
ω2
β2
=
√
n22k2
β2
=
n2k
β
which is the ratio of the free-space wavevector and that of the guided mode.
We also define
Z = pi
β2
q2
l2β
µ0ω
C2
which yields the following
Dout = Z
∫ ∞
a
([
−l2s
Kl (qr)
qr
+K ′l (qr)
] [
N22
l2
K ′l (qr)− s
Kl (qr)
qr
]
+
[
Kl (qr)
qr
− sK ′l (qr)
] [
N22
Kl (qr)
qr
− sK ′l (qr)
])
rdr
Dout = Z
∫ ∞
a




−sN22
Kl(qr)
qr K
′
l (qr) + l
2s2
(
Kl(qr)
qr
)2
+ N
2
2
l2 (K
′
l (qr))
2 − sKl(qr)qr K
′
l (qr)
+N22
(
Kl(qr)
qr
)2
− sKl(qr)qr K
′
l (qr)− sN
2
2
Kl(qr)
qr K
′
l (qr) + s
2 (K ′l (qr))
2



 rdr
Dout = Z
∫ ∞
a
(
−2s
Kl (qr)
qr
K ′l (qr)
[
1 +N22
]
+
(
Kl (qr)
qr
)2 [
N22 + l
2s2
]
+ (K ′l (qr))
2
[
N22
l2
+ s2
])
rdr
Dout = Z
∫ ∞
a
(
−2s
Kl (qr)
qr
K ′l (qr)
[
1 +N22
]
+
[
l2
(
Kl (qr)
qr
)2
+ (K ′l (qr))
2
][
N22
l2
+ s2
])
rdr.
Using the Bessel function relation
Kl (qr) =
qr
2l
(Kl+1 (qr)−Kl−1 (qr))
K ′l (qr) = −
1
2
(Kl+1 (qr) +Kl−1 (qr))
197
we arrive at
Dout = Z
∫ ∞
a
( s
2l
[
K2l+1 (qr)−K
2
l−1 (qr)
] [
1 +N22
]
+
1
4
[
[Kl+1 (qr)−Kl−1 (qr)]
2 + [Kl+1 (qr) +Kl−1 (qr)]
2]
[
N22
l2
+ s2
])
rdr
Dout = Z
∫ ∞
a
( s
2l
[
K2l+1 (qr)−K
2
l−1 (qr)
] [
1 +N22
]
+
1
2
[
K2l+1 (qr) +K
2
l−1 (qr)
]
[
N22
l2
+ s2
])
rdr
Dout =
Z
2l2
∫ ∞
a
([
sl
[
1 +N22
]
+N22 + l
2s2
]
K2l+1 (qr)
+
[
−sl
[
1 +N22
]
+N2 + l2s2
]
K2l−1 (qr)
)
rdr
Dout =
Z
2l2
∫ ∞
a
(
(1 + sl)
(
N22 + sl
)
K2l+1 (qr) + (1− sl)
(
N22 − sl
)
K2l−1 (qr)
)
rdr.
Now we perform a change of variable r → rq = qr,
Dout =
Z
2l2q2
[
(1 + sl)
(
N22 + sl
)
∫ ∞
qa
K2l+1 (rq) rqdrq + (1− sl)
(
N22 − sl
)
∫ ∞
qa
K2l−1 (rq) rqdrq
]
and applying the relation
∫ ∞
qa
K2l (rq) rqdrq = −
1
2
(qa)2
[
K2l (qa)−Kl−1 (qa)Kl+1 (qa)
]
we arrive at
Dout =
−Za2
4l2
[
(1 + sl)
(
N22 + sl
) [
K2l+1 (qa)−Kl (qa)Kl+2 (qa)
]
+ (1− sl)
(
N22 − sl
) [
K2l−1 (qa)−Kl (qa)Kl−2 (qa)
]]
Dout =
−pia2β
4µ0ω
β2
q2
(
Jl (ha)
Kl (qa)
)2 [
(1 + sl)
(
N22 + sl
) [
K2l+1 (qa)−Kl (qa)Kl+2 (qa)
]
+ (1− sl)
(
N22 − sl
) [
K2l−1 (qa)−Kl (qa)Kl−2 (qa)
]]
.
198
For the fundamental mode of the fiber, l = 1, this yields,
Dout =
−pia2β
4µ0ω
β2
q2
(
J1 (ha)
K1 (qa)
)2 [
(1 + s)
(
N22 + s
) [
K22 (qa)−K1 (qa)K3 (qa)
]
+ (1− s)
(
N22 − s
) [
K20 (qa)−K
2
1 (qa)
]]
We can similarly solve for Din, which results in
Din =
pia2β
4µ0ω
β2
h2
[
(1 + sl)
(
N21 + sl
) [
J2l+1 (ha)− Jl (ha) Jl+2 (ha)
]
+ (1− sl)
(
N21 − sl
) [
J2l−1 (ha)− Jl (ha) Jl−2 (ha)
]]
. (A.58)
Setting l = 1 we arrive at
Din =
pia2β
4µ0ω
β2
h2
[
(1 + s)
(
N21 + s
) [
J22 (ha)− J1 (ha) J3 (ha)
]
+ (1− s)
(
N21 − s
) [
J20 (ha) + J1 (ha)
2]] . (A.59)
Finally following the same procedure we can derive the power inside and out-
side the fiber for the TE01 and TM01 modes
DTE−OUT =
−pia2
4
µ0ω
β
q2
(
J1 (qa)
K1 (ha)
)2 [
K22 (qa)−K1 (qa)K3 (qa) +K
2
0 (qa)−K
2
1 (qa)
]
DTE−IN =
pia2
4
µ0ωβ
h2
[
J22 (ha)− J1 (ha) J3 (ha) + J
2
0 (ha) + J
2
1 (ha)
]
DTM−OUT =
−pia2
4
2ωβ
q2
(
J1 (ha)
K1 (qa)
)2 [
K22 (qa)−K1 (qa)K3 (qa) +K
2
0 (qa)−K
2
1 (qa)
]
199
DTM−IN =
pia2
4
1ωβ
h2
[
J22 (ha)− J1 (ha) J3 (ha) + J
2
0 (ha) + J
2
1 (ha)
]
.
We have implemented a Mathematica notebook in the lab that allows us to
calculate all of these parameters. We also obtain similar results when using a com-
mercial software, FIMMPROP [123].
200
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