GENERATION OF FEMTOSECOND PULSES AT NEW
WAVELENGTHS IN FIBER LASERS
A Dissertation
Presented to the Faculty of the Graduate School
of Cornell University
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
by
Yuxing Tang
August 2017
©c 2017 Yuxing Tang
ALL RIGHTS RESERVED
GENERATION OF FEMTOSECOND PULSES AT NEW WAVELENGTHS IN
FIBER LASERS
Yuxing Tang, Ph.D.
Cornell University 2017
The advancement of ultrafast laser sources has enabled both new discoveries
in fundamental research and the development of many practical applications, such
as nonlinear microscopy, biomedical imaging, micro-machining, laser surgery, etc.
Among the light sources, solid-state laser systems, such as Ti:sapphire lasers, have
long served as the workhorses to provide the femtosecond pulses with high peak
power and average power. As versatile as they are, the accompanying prices to pay
are the bulky size, high cost, requirement for water-cooling and high sensitivity to
alignment, which are all impediments for their operations outside the laboratory
environments.
Rare-earth doped fiber lasers come naturally with the advantages of being
alignment-free, compact and possessing large surface-to-volume ratio for efficient
heat dissipation at low costs. Single mode fibers provide nearly perfect gaussian
beams for applications. Starting from the year of 2006 when the all-normal disper-
sion fiber lasers were discovered, researchers have devoted great efforts to studying
nonlinear pulse evolution and ultrashort pulse generation in fiber lasers with high
energy and peak power. Multiple novel pulse evolutions have been thoroughly ex-
plored, including dissipative soliton, passive self-similar, amplifier similariton and
extended self-similar evolution. The performance of fiber lasers have also been
boosted up to the comparable level as that of the Ti:sapphire lasers.
With all these pulse evolutions and mechanisms studied and well understood,
there is still work to be done about fiber lasers, such as the exploration of femtosec-
ond pulse generations in the few-cycle regime and at other new wavelengths. The
study of few-cycle pulse generation directly from fiber oscillators is first explored
in an erbium laser with comb-like dispersion decreasing fiber, based on the idea of
the extended self-similar evolution.
Most of the fiber laser developments have been focused on using the gain
medium of erbium and ytterbium with emission peaks around 1030 nm and 1550
nm. Thulium (Tm) doped and thulium-holmium codoped fibers emit light cov-
ering 1.7 to 2.1 µm with a 100-nm wide gain bandwidth, which opens the door
for many other applications. The benefits of operating Tm fiber lasers with net
normal dispersion cavities are explored in this thesis. A Tm-doped fiber laser with
hybrid pulse evolution with elements of self-similar evolution is demonstrated, with
a 4-fold improvement in peak power over previous results. Similar thoughts can be
applied to erbium doped ZBLAN fiber lasers operating at 2.8 µm, at which wave-
length only soliton lasers have been realized. Simulations predict promising results
in cavities with dispersion management and some initial experimental results are
presented.
To further pursue the femtosecond pulse generation over 3 µm into the mid-
infrared region, a fiber-based laser system employing the soliton self-frequency shift
in fluoride fibers is investigated. Soliton pulses with 100-fs duration, nanojoule-
level energies are generated, with wavelength tunable from 2-4.3 µm. Compared
with previous reports, the pulse energies are two orders of magnitude higher and
the range of wavelengths is extended by 1000 nm.
Finally, future directions and the extension of the work in the thesis are dis-
cussed.
BIOGRAPHICAL SKETCH
Yuxing Tang was born in Xi’an, Shaanxi, China in 1990. Brought up in a
family of an engineer and a doctor, he is always curious about the world around
him and like to ask various questions starting from a young age. He got interested
in math and natural sciences in middle school. Following high school in 2008, he
decided to major in electronics engineering in Peking University. Thanks to the
comprehensive course settings of the program, he also got a chance to study all
the undergraduate physics courses in addition to learning how to design integrated
and digital circuits.
Intrigued by the optics course he took in the sophomore year, he wanted to
learn more about nonlinear optics and participated in a research group to work
on laser experiments, such as building up fiber lasers for medical applications. He
found it fascinating and enjoyed the process of obtaining the rainbow-like light from
the ”mysterious” processes he couldn’t understand at that time. Upon graduation
with a bachelor’s degree in 2012, he opted to pursue Ph.D. studies in nonlinear
optics and fiber lasers.
Luckily, he got enrolled in the Applied Physics Ph.D. program at Cornell Uni-
versity. After the first year’s exploration of the research directions in the depart-
ment, he decided to join the fiber laser group of Prof. Frank Wise in the summer
of 2013. Since then, he’s greatly enjoyed his time in the group researching on
nonlinear fiber optics and ultrashort pulse generations at new colors.
After completing his Ph.D., he will step on a new journey in San Francisco, CA
to work as a quantitative analyst and hope to explore his technical and modeling
skills in other areas as well.
iii
To my wife, Yi, and my parents for their constant support and unwavering love.
iv
ACKNOWLEDGEMENTS
First, I would like to give my sincere thanks to Prof. Frank Wise for all of
his help, support, and guidance during my time at Cornell. I truly appreciate the
opportunity of learning from someone as knowledgable and kind as Frank, and it
has been an honor to be a member of his team. I learnt from him not only the
knowledge of physics and ways of thinking, but also the philosophy of how to make
life-changing choices.
I also want to thank my committee members, Prof. Clifford Pollock and Prof.
Jeffrey Moses, for their support and guidance in various aspects of my academic
life.
I give special thanks to my wife Yi Xin for all of her help, encouragement,
support for all these years, and to my parents for always being there and supporting
me whenever I couldn’t get the lasers mode-locked. I am indebted to them for all
their care and understanding.
I also appreciate the help and friendship of my group members, including Will
Renninger, Hui Liu, Erin Lamb, Peng Li, Jun Yang, Logan Wright, Zhanwei Liu,
Zimu Zhu, Walter Fu, Zachary Ziegler, Andrei Isichenko and Pavel Sidorenko. I
additionally thank Logan for the insightful discussions with him and his help all
along.
I have also enjoyed the chance to work with some excellent collaborators and
am grateful to them too. In particular, I thank Andy Chong, Kriti Charan, Tianyu
Wang and Prof. Chris Xu for collaborating with me.
Finally, I want to thank the National Science Foundation for providing financial
support for my work presented in this thesis.
v
TABLE OF CONTENTS
Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 Introduction 1
1.1 Organization of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Pulse propagation in fiber . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Generalized nonlinear Schrödinger equation . . . . . . . . . 4
1.2.2 Simulating the fiber laser cavity . . . . . . . . . . . . . . . . 6
1.3 Pulse evolutions in fiber lasers . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 Soliton lasers . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.2 Dispersion-managed soliton lasers . . . . . . . . . . . . . . . 12
1.3.3 Dissipative soliton lasers . . . . . . . . . . . . . . . . . . . . 13
1.3.4 Passive self-similar lasers . . . . . . . . . . . . . . . . . . . . 14
1.3.5 Amplifier similariton lasers . . . . . . . . . . . . . . . . . . . 16
1.3.6 Extended self-similar lasers . . . . . . . . . . . . . . . . . . 18
1.4 Fiber laser components . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4.1 Spectral filters . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4.2 Saturable absorbers . . . . . . . . . . . . . . . . . . . . . . . 23
Bibliography 28
2 Self-similar pulse evolution in a fiber laser with comb-like disper-
sion decreasing fiber 31
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Er fiber laser with comb-like dispersion-decreasing fiber . . . . . . . 33
2.2.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3 Yb fiber laser with comb-like dispersion-decreasing fiber . . . . . . . 42
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Bibliography 45
3 High energy pulses from thulium fiber lasers 47
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Laser design and numerical simulations . . . . . . . . . . . . . . . . 49
3.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
vi
Bibliography 57
4 Generation of intense 100-fs solitons tunable from 2 to 4.3 µm in
fluoride fiber 59
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Description of fibers . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.5 Stability measurement . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Bibliography 75
5 High energy pulses from erbium ZBLAN fiber lasers 78
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2 Laser design and numerical simulations . . . . . . . . . . . . . . . . 80
5.2.1 Chalcogenide fiber . . . . . . . . . . . . . . . . . . . . . . . 80
5.2.2 Laser design and simulation setup . . . . . . . . . . . . . . . 82
5.2.3 Simulation results for dispersion-managed soliton lasers . . . 84
5.2.4 Simulation results for hybrid self-similar lasers . . . . . . . . 87
5.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3.1 Fiber cleaving . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3.2 Soliton laser . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Bibliography 94
6 Future Directions 96
6.1 Thulium fiber lasers with all normal-dispersion cavity . . . . . . . . 96
6.1.1 Modulation instability in Tm fiber laser . . . . . . . . . . . 96
6.1.2 Dissipative soliton Tm fiber laser . . . . . . . . . . . . . . . 98
6.1.3 Amplifier similariton Tm fiber laser . . . . . . . . . . . . . . 100
6.2 Divided-pulse Tm laser . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.3 Modelocked praseodymium fiber laser at 5 µm and its application . 106
Bibliography 109
A Appendix A: Fiber parameters 111
A.1 Mathematica code for calculating the fiber parameters . . . . . . . 111
vii
LIST OF TABLES
2.1 Values of Parameters Used in Simulation . . . . . . . . . . . . . . . 36
2.2 Parameters Used in Simulation for Yb DDF laser . . . . . . . . . . 42
3.1 Values of parameters used in Tm laser simulation . . . . . . . . . . 50
5.1 Values of parameters used in the Er ZBLAN laser simulations . . . 84
6.1 Values of fiber parameters used in the Tm ANDi laser . . . . . . . 99
6.2 Values of fiber parameters used in the Tm amplifier similariton laser101
viii
LIST OF FIGURES
1.1 Schematic of the all-normal dispersion fiber laser: BPF: bandpass
filter; PBS: polarizing beam splitter; ISO: isolator; QWP: quarter
waveplate; HWP: half waveplate; SMF: single mode fiber. . . . . . 7
1.2 The simulation results of the Yb fiber laser cavity: (a) output spec-
trum; (b) output chirped pulse profile; (c) transform limited pulse;
(d) pulse evolution in spectral and temporal domains. (position 1-
11: passive fiber; position 12-21: gain fiber; position 22-31: passive
fiber; position 32: SA; position 33: output port.) . . . . . . . . . . 9
1.3 Schematic of the dispersion managed soliton design. . . . . . . . . 12
1.4 The concept of the extended self-similar laser design. . . . . . . . 18
1.5 Transmission function of a Birefringent filter made of a 12-T (6 mm
thick) quartz plate near 1030 nm. . . . . . . . . . . . . . . . . . . 21
1.6 A Lyot filter, consisting of a sequence of birefringent crystals (BC)
and polarizers (P). Each birefringent plate has the thickness half of
the previous one. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.7 Transmission function of a Lyot filter made of a 6 T and 12 T
quartz plate near 1900 nm. The bandwidth is 35 nm and the free
spectral range is 150 nm. . . . . . . . . . . . . . . . . . . . . . . . 22
1.8 Schematic of a grating-based filter. . . . . . . . . . . . . . . . . . . 23
1.9 A 1-nm bandwidth grating filter at 1030 nm. (a) The transmission
profile of the filter; (b) The incident and diffracted angle of the
beam; The dependence of the introduced (c) 2nd order and (d) 3rd
order dispersion on distance between the collimator and the grating. 24
1.10 Illustration of saturable absorbers promoting pulse generation. . . 24
1.11 Illustration of NPE saturable absorber (PBS: polarizing beam split-
ter; QWP: quarter waveplate; HWP: half waveplate). . . . . . . . 26
2.1 Comparison of the GVD values of the ideal DDF (green) with the
comb-like DDF (blue) at 1550 nm. The lengths of the segments are
14, 6, 5, 20, 5 and 50 cm. The upper and lower red dashed lines
are for the ideal DDF at 1700 and 1400 nm, respectively. . . . . . . 35
2.2 (a) Schematic of the Er DDF laser: PBS: polarizing beam splitter;
QWP: quarter-wave plate; HWP: half-wave plate. (b)-(d) Simu-
lation results for an ideal (red) or a comb-like DDF (black), with
net cavity dispersion of -0.087 ps2. (b) Evolution of pulse dura-
tion (SA, saturable absorber; SF, spectral filter). (c) Evolution of
spectral bandwidth. (d) Evolution of misfit parameter. . . . . . . . 37
2.3 (a) Measured mode-locked spectrum of the Er DDF laser. (b) Fit-
ting of the cross correlation signal to a parabola. (c) Interferometric
autocorrelation of the dechirped pulse using two-photon absorption.
(d) Radio frequency spectrum of the mode-locked laser with reso-
lution bandwidth of 10 Hz and span range of 100 kHz. . . . . . . . 39
ix
2.4 (a) The pulse spectrum at the input and output end of the peak
power testing segment with coupled-in energy of 0.54 nJ. (b) Com-
parison of the spectral broadening in experiments (in black) and
simulations (Gaussian pulses; in red) at different input energy level. 40
2.5 (a) Pulse energy and dechirped pulse duration at various cavity dis-
persions. (b) Pulse chirp and deviation of dechirped pulse duration
from the transform limit at various cavity dispersions. . . . . . . . 41
2.6 (a) Schematic of the Yb DDF laser: PBS: polarizing beam splitter;
QWP: quarter-wave plate; HWP: half-wave plate. (b) Comparison
of the GVD values of the ideal DDF (green) with the comb-like
DDF (blue) at 1030 nm. (c)-(d) Comparison of the output spectra
and dechirped pulse (using comb DDF) with transform limit pulse. 43
3.1 Schematic of laser: BRP: birefringent plate; IF: interference filter;
PBS: polarizing beam splitter; ISO: isolator; WDM: wavelength-
division multiplexer. . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 (a) Simulated variation of pulse duration and spectral bandwidth
along the cavity (SA: saturable absorber, SF: spectral filter). The
pigtail segment of WDM is neglected for clarity; (b) pulse shape
evolution compared with a parabolic pulse; inset: comparison of
pulse shape after UHNA4 (black) with Gaussian pulse (blue) and
parabolic pulse (red) in log scale. . . . . . . . . . . . . . . . . . . . 51
3.3 (a, b) Comparison between experimental and simulation results
(squares: experimental data, triangles: simulation results); (c)
mode-locked spectra at various cavity GDD with the 16-nm fil-
ter (blue: 0.08 ps2, green: 0.17 ps2, red: 0.26 ps2, black: 0.37 ps2);
(d) mode-locked spectra with varying filter BW at cavity GDD ∼
0.28 ps2 (blue: 16 nm filter BW, black: 35 nm filter BW, red: 40
nm filter BW). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4 Experimental (a) output spectrum, (b) autocorrelation of dechirped
pulse (inset: autocorrelation of transform limited pulse), (c) radio-
frequency spectrum (span range: 200 kHz, RBW: 1 kHz) corre-
sponding to 7.6 nJ pulse energy. . . . . . . . . . . . . . . . . . . . 54
4.1 The parameters of the fluoride fibers used in the experiments. The
group velocity dispersion (GVD) of the fundamental mode and the
propagation loss for the (a) InF3 fiber; (b) ZrF4 fiber. The horizon-
tal blue dashed line indicates the position of the zero GVD value.
The vertical blue dashed line indicates the single mode cutoff wave-
length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Simulation results for the (a) spectral output, (b) spectral evolu-
tion, (c) temporal output and (d) temporal evolution in a 2-m long
InF3 fiber with 40-nJ and 70-fs pulses coupled into the fundamental
mode. RS: Raman soliton of interest at the longest wavelength. . . 66
x
4.3 Schematic of the experimental setup for SSFS in fluoride fibers:
LPF: long pass filter. LPF1 cuts on at 1.8 µm. LPF2 cuts on at
2.85 µm or 3.75 µm. . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4 (a) The spectra and (b) the interferometric autocorrelation of the
input light source at 1.9 µm. . . . . . . . . . . . . . . . . . . . . . 68
4.5 Measured spectra and the interferometric autocorrelations of the
most-redshifted solitons at (a,b) 3.2 µm; (c,d) 3.9 µm; (e,f) 4.3 µm
in the InF3 fiber. The red lines indicate the spectra after noise
reduction. A deconvolution factor of 1.54 was assumed to obtain
the indicated pulse durations. . . . . . . . . . . . . . . . . . . . . 69
4.6 Measured full spectra in a 2-m long InF3 fiber with shifted solitons
at 3.9 µm. PSD: power spectral density. . . . . . . . . . . . . . . . 70
4.7 Spectra of the most redshifted soliton in the 2-m long InF3 fiber at
different wavelengths in (a) experiments; (b) numerical simulations.
The dotted lines show the range which could not be covered by
available spectrum analyzers. The soliton and input-pulse energies
are indicated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.8 The stability measurement of the power and center wavelength vari-
ation of (a,b) the pump pulse; (c,d) the seed pulse; (e,f) the output
soliton at 3.6 µm. SD: standard deviation. . . . . . . . . . . . . . . 72
5.1 Group velocity dispersion of the chalcogenide fiber IRF-S-6.5. . . . 81
5.2 (a) Measurement of n2 of the chalcogenide fiber IRF-S-6.5 using
peak power test; (b) Extrapolation of n2 at longer wavelengths
using Miller’s formula. . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3 Schematic of the Er ZBLAN laser with dispersion compensation:
PBS: polarizing beam splitter; ISO: isolator; BPF: bandpass filter;
M: mirror; DM: dichroic mirror. . . . . . . . . . . . . . . . . . . . . 83
5.4 (a) Simulated variation of pulse duration and spectral bandwidth
with net cavity dispersion of -0.0084 ps2 (Page 22: spectral filter,
Page 23: output port); (b) comparison between the output pulse
and transform-limit pulse; (c) output spectrum with FWHM band-
width of 60 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.5 (a) Simulated variation of pulse duration and spectral bandwidth
with net cavity dispersion of 0.012 ps2 (Page 22: spectral filter,
Page 23: output port); (b) output chirped pulse with duration of
177 fs; (c) comparison between the de-chirped pulse and transform-
limit pulse; (d) output spectrum with FWHM bandwidth of 137
nm, corresponding to 106 fs. . . . . . . . . . . . . . . . . . . . . . . 86
xi
5.6 (a) Simulated pulse duration and spectral bandwidth with net
cavity dispersion of 0.1 ps2 (Page 22: spectral filter, Page 23:
output port); (b) comparison between the de-chirped pulse and
transform-limit pulse; (c) misfit parameter evolution compared
with a parabolic pulse; (d) the output spectrum with FWHM band-
width of 110 nm and pulse energy of 188 nJ. . . . . . . . . . . . . . 88
5.7 The end view, front view and back view of the bad angle cleave
(top row) and good angle cleaves (bottom row). . . . . . . . . . . . 90
5.8 Schematic of the Er ZBLAN soliton laser: PBS: polarizing beam
splitter; ISO: isolator; BPF: bandpass filter; M: mirror; DM:
dichroic mirror. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.9 Picture of the Er ZBLAN soliton laser.Line with red color indicates
the pump beam and the green one is the signal light. . . . . . . . . 91
5.10 Slope efficiency of the Er ZBLAN soliton laser. . . . . . . . . . . . 92
6.1 (a) Mode-locked spectrum from the Tm fiber laser in Figure 3.1;
(b) Zoomed-in spectrum in THz unit with multiple peaks indicated. 98
6.2 The schematic of the all normal-dispersion Tm laser design with
filter bandwidth of 50 nm. . . . . . . . . . . . . . . . . . . . . . . . 99
6.3 (a) Simulated output chirped pulse; (b) output spectrum with 150-
nm bandwidth and 280-nJ pulse energy; (c) spectral and temporal
evolution along the cavity; (d) comparison between the de-chirped
pulse (90 fs) and the transform-limit pulse (70 fs). . . . . . . . . . 100
6.4 (a) Evolution of the misfit parameter comparing the pulse profile
to a parabola; (b) output spectrum with 87-nm bandwidth; (c)
spectral and temporal evolution inside the cavity; (d) comparison
between the de-chirped pulse (97 fs) and the transform-limit pulse. 101
6.5 Schematic for the divided-pulse Tm fiber laser: BRP: birefringent
plate; PBS: polarizing beam splitter; ISO: isolator; F.R: Faraday
Rotator; M: mirror. . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.6 (a) Modelocked spectrum; (b) autocorrelation of the Tm soliton
laser in divided-pulse configuration. . . . . . . . . . . . . . . . . . . 104
6.7 Schematic for the divided-pulse Tm fiber laser with NPE for mode-
locking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.8 Mode-locked spectrum (a) and autocorrelation (b) of the Tm soli-
ton laser in divided-pulse configuration, without SESAM. . . . . . 105
6.9 Broadband emission in the 3∼5 µm wavelength region from a diode-
pumped Pr3+ doped chalcogenide fiber source. . . . . . . . . . . . 108
xii
CHAPTER 1
INTRODUCTION
The advancement of scientific research and industrial applications is accompa-
nied by the development of versatile laser sources. Among them, ultrafast laser
sources enable the researchers to dive into the regime of nonlinear optical sci-
ences, as well as various important applications, including biomedical imaging,
laser surgery, frequency metrology and material processing. It wouldn’t be possi-
ble to observe and study the most fundamental and interesting nonlinear optical
phenomena without these energetic femtosecond pulses. The continued and rapid
growth in applications will also require and benefit from the developments of ul-
trafast laser sources.
Solid state lasers, represented by mode-locked titanium sapphire (Ti:sapphire)
lasers, have long served as the main driving forces for ultrafast sciences and dom-
inated the laser markets. Perceived as a gift from the nature, Ti:sapphire crystal
has great material properties and a hugely broad gain bandwidth in the visible
range, which supports the generation of sub-10 fs pulses. With the discovery of
Kerr-lens mode-locking in 1991 [1] and two decades of engineering, Ti:sapphire
lasers have achieved a huge success and largely used as femtosecond light sources.
However, their widespread applications are still limited by a few factors, such as
the high cost, bulky size, cooling requirement and high sensitivity to the align-
ment. Laser technicians or staff with special training are required to operate the
Ti:sapphire lasers routinely and continuously. This, along with their bulky sizes,
prevent them from being used by doctors in the real clinical environments.
Fiber lasers possess the intrinsic properties of being robust, compact and cost
effective. They have grown into important high energy ultrashort pulsed sources
1
both in scientific research and applications. With the maturity of manufacturing
fiber components at low cost, it becomes quick and easy to assemble a fiber laser
using fibers out of the shelves and splicing them together. The large gain provided
by the highly-doped gain fibers enable fiber lasers to tolerate large losses and high
output coupling ratio. The light is guided inside the waveguide and has a perfect
Gaussian beam profile in single mode fibers. The prospect of all-fiber-format lasers
will totally eliminate the free space components and shrink the size of femtosecond
fiber lasers to that of a briefcase. It requires no or minimal water cooling due to
the fiber’s large surface-to-volume ratio for efficient hear dissipation.
Another main reason for the fast development of fiber lasers is the discovery
of various pulse evolutions in laser cavities. Since the discovery of the all normal
dispersion (ANDi) fiber laser [2], the performance has increased by orders of mag-
nitude from the previous soliton and dispersion-managed soliton lasers, starting
to compete with the performance of Ti:sapphire lasers. It remains a question to
see how short pulses fiber lasers can generate directly from oscillators, since these
broadband pulses can replace the role of those from Ti:sapphire lasers.
Despite the improvement in output pulse energy and durations, fiber lasers
still lag behind solid state lasers in delivering light at new colors. Fiber-based light
sources at new wavelengths are greatly needed for various types of applications
and their developments also open doors for new ideas. It’s interesting and excit-
ing to see how the idea of operating fiber lasers in the normal dispersion regime
works for other wavelengths and what the scaling rules of performance are. In
particular, people are becoming increasingly interested in developing light sources
into the mid-infrared region over 3 µm, which is beneficial to a full range of new
applications.
2
1.1 Organization of thesis
This thesis focuses on the study of generating few-cycle pulses directly from fiber
lasers as well as the routes to obtain femtosecond pulses from fiber lasers at new
wavelengths. The detailed introductions are organized as follows.
Chapter 1 introduces the basic concepts and modeling of pulse propagation in
optical fibers and fiber lasers. Multiple important pulse evolutions are summarized.
Finally, a brief description of the important components used in fiber lasers is
presented.
Chapter 2 presents the work on generation of sub-40 fs pulses in an erbium (Er)
fiber laser with a comb-like dispersion decreasing fiber. A hybrid pulse evolution
with elements of passive self-similar evolution is discovered and the pulse perfor-
mance shows a 4-fold improvement over previous results. Promising simulation
results of applying the same idea to a ytterbium (Yb) fiber laser are discussed.
Chapter 3 describes the study of a thulium (Tm) fiber laser at 2 µm operating
at net normal dispersion regime. Its behavior with varying cavity dispersion and
filter bandwidth is studied systematically. This is the first Tm fiber laser that
reaps the benefits of normal dispersion operation.
Chapter 4 demonstrates the routes to obtain 100-fs solitons, continuously
wavelength-tunable over 2-4.3 µm through the soliton self-frequency shift (SSFS)
in fluoride fibers. The pulse energies are 2 orders of magnitude higher than those
previously achieved by SSFS, around 3 µm, and the range of wavelengths is ex-
tended by 1000 nm. Peak power ranges from 20 to 75 kW are achieved across the
tuning range.
3
Chapter 5 describes the efforts to develop normal dispersion Er ZBLAN fiber
lasers at 2.8 µm. Promising simulation results show an order of magnitude im-
provement over those of the previous soliton lasers at this wavelength. Some initial
experimental results are presented.
Finally, future directions of these work will be discussed in Chapter 6.
1.2 Pulse propagation in fiber
The propagation of optical fields in fibers is governed by the generalized nonlinear
Schrödinger equation (GNLSE), which can be derived from the Maxwell’s equa-
tions. The derivations are presented in [3] in detail. Some main results related to
this thesis are listed below.
1.2.1 Generalized nonlinear Schrödinger equation
Normally, we start with the equation below:
1 ∂2E ∂2P ∂2∇2 L PNLE− = µ0 + µ0 , (1.1)
c2 ∂t2 ∂t2 ∂t2
where E(r, t) is the electric field, c is the speed of light in vacuum. PL and PNL
are the linear and nonlinear portions of the induced polarization and µ0 is the
magnetic permeability in free space.
Under the assumptions of slowly varying envelop approximation (SVEA) and
the optical fields being quasi-monochromatic (∆ω ≪ 1), the slowly varying part
ω0
4
A(z, t) of the full electric field E(r, t) can be factored out:
1
E(r, t) = x̂(F (x, y)A(z, t) exp [i(β0z − ω0t)] + c.c.), (1.2)
2
where F(x,y) is the spatial distribution of the field in the cross section of fibers and
c.c. stands for the complex conjugate. With a further assumption that the optical
field maintains the polarization along propagation, we can regard this as a scalar
model. In practice, it works quite well, but if needed, a vector code modeling two
polarization directions is used. After factoring out the rapidly varying part of the
field, we can arrive at the GNLSE:
∞
∂A α n∑− n+1βn ∂ A+ A i =
∂z 2 n! ∂tn
n=2
( )( )
∫
i ∂ t
iγ 1 + A(z, t) R(t′)|A(z, t− t′)|2dt′ . (1.3)
ω0 ∂t −∞
where R(t) is the normalized nonlinear response function, βn are the n-th order
of the fiber’s dispersion. Linear loss is introduced through the term α. Nonlinear
coefficient γ = n2(ω0)ω0 models the Kerr nonlinear effect in fibers. One point needs
cAeff
to be made here that γ scales inversely with the third power of wavelength, which
indicates the potential benefits of lasers at longer wavelengths.
The nonlinear response function R(t) incorporates both the electronic and the
nuclear contributions and takes the form:
R(t) = fRhR(t) + (1− fR)δ(t− te) (1.4)
where te represents the instantaneous electronic response time [3]. The proportion
5
of contribution from delayed Raman response is controlled by fR, which is 0.18 for
silica fibers. It’s hard to determine the exact form of the Raman response function
hR(t), which is approximated in simulations by a simple model:
τ 2 + τ 2
h 1 2R(t) = exp(−t/τ2)sin(t/τ1) (1.5)
τ 21τ2
The parameters τ1 and τ2 are two time constants unique to the material and
are chosen to match the actual Raman gain spectrum. For silica glasses, values of
τ1 = 12.2fs, τ2 = 32fs are used [4].
This equation can model 3rd order nonlinear effects, including self-phase mod-
ulation, stimulated Raman scattering, self-steepening, and shock formation. It is
capable of modeling pulses as short as a few optical cycles, which is sufficient for
the scope of the thesis.
Sometimes, when the pulse duration is long enough (> 100 fs), a simplified
version of GNLSE is used, given by the nonlinear Schrödinger equation (NLSE).
∂A −iβ2 ∂
2A
= + iγ|A|2A, (1.6)
∂z 2 ∂t2
This equation can be solved using the inverse scattering method and yields the
important soliton solution when the group velocity dispersion β2 is anomalous.
1.2.2 Simulating the fiber laser cavity
As the illustration of numerical simulation of a fiber laser cavity, we consider
the all-normal dispersion fiber laser design in Figure 1.1 working at 1 µm. Single
6
output
QWP QWP
ISO PBS
BPF HWP
SMF SMF
gain fiber
Figure 1.1: Schematic of the all-normal dispersion fiber laser: BPF: bandpass filter;
PBS: polarizing beam splitter; ISO: isolator; QWP: quarter waveplate; HWP: half
waveplate; SMF: single mode fiber.
mode fiber is Hi1060 and the gain fiber is Yb doped fiber. The laser is mode-locked
through nonlinear polarization evolution (NPE), implemented by wave plates and
polarizing beam splitter (PBS). PBS also serves as the output port for the cavity.
The isolator ensures unidirectional operation of the laser. The band-pass filter is
a critical element for stabilizing a positively chirped pulse circulating inside the
normal dispersion cavity.
Pulse propagation inside the cavity is modeled using the following NLSE with
gain term:
∂A(z, τ) β 22 ∂ A(z, τ)
+ i = iγ|A(z, τ)|2A(z, τ) + g(Epulse)A(z, τ) (1.7)
∂z 2 ∂τ 2
A(z, τ) is the electric field envelope, where z is the propagation distance and
τ models the local time in the frame moving at the group velocity of the pulse.
Group velocity dispersion (GVD) is modeled using β2, which is 23 fs
2/mm for
Hi1060 at 1 µm. γ is the Kerr nonlinear coefficient of the single mode fiber, with
the value of 0.0047 /(W·m). The gain term only appears in the gain segment and
7
the saturation of the gain fiber is modeled through g(Epulse) = g0/(1+Epulse/Esat),
where g0 is the small signal gain corresponding to 30 dB. Epulse is the total intra-
∫ T /2
cavity pulse energy in a round trip, which is R |A(z, t)|2dt. The modeling of
−TR/2
the pump power level is introduced via Esat.
Mode-locking is achieved using NPE, but it’s modeled in simulations as an ideal
saturable absorber with monotonically-increasing transmission curve of T (t) =
1 − l0/(1 + P (t)/Psat), where l0 is the unsaturable loss, with l0 = 1 for a perfect
SA [5]. Pt is the instantaneous pulse power and Psat is the saturation power of
the SA, which is normally 1∼4 kW in simulations. The linear loss and the output
coupling ratio of 70% are used. The simulation can be seeded with Gaussian white
noise or a pulse, which may speed up the convergence process.
Split-step Fourier method is used in this thesis [3], by modeling the disper-
sion part D̂ in frequency domain and the nonlinear part N̂ in time domain using
fourth-order Runge Kutta method. Propagation is conducted over a very small
longitudinal distance h each time and the accuracy of the method is limited on the
order of O(h2). The accuracy of the algorithm can be improved by employing the
symmetric split-step Fourier method, where the dispersion part is split into two
halves and modeled symmetrically before and after the nonlinear part. Then the
accuracy of O(h3) can be achieved.
The converged solution for the cavity with 2-m SMF, 1-m gain fiber and 1-m
SMF is shown in Figure 1.2. A 10-nm bandwidth filter is used and the saturation
energy for the gain segment is 5.7 nJ. Under these conditions, the laser gener-
ates chirped pulses with 8-nJ energy and 3.8-ps duration. The spectra has the
characteristic bat-man ears and steep edges [6]. With increasing pump power, or
increasing saturation energy of the gain fiber, higher pulse energy and broader
8
Figure 1.2: The simulation results of the Yb fiber laser cavity: (a) output spectrum;
(b) output chirped pulse profile; (c) transform limited pulse; (d) pulse evolution in
spectral and temporal domains. (position 1-11: passive fiber; position 12-21: gain
fiber; position 22-31: passive fiber; position 32: SA; position 33: output port.)
spectrum is produced.
1.3 Pulse evolutions in fiber lasers
Pulse generation in this thesis is mainly focused on mode-locking technique for
pulses in the picosecond and femtosecond range [7]. There are other pulse gener-
ation methods as well, such as gain switching, cavity dumping, Q-switching, but
these typically produce pulses with nanosecond durations.
Mode-locking, in frequency domain, can be understood as locking of the phases
of different longitudinal modes in the cavity. In time domain, it is a pulsed mode
9
operation of the laser, as opposed to the continuous-wave (CW) operation. The
separation between the pulses is the round trip time. Mode-locking can be real-
ized in two ways: active and passive mode-locking. Active mode-locking is em-
ployed by periodically modulating the cavity loss at a matched frequency of the
mode spacing. Usually it is done using acousto-optic, electro-optic modulator or
a Mach-Zehnder integrated-optic modulator. Picosecond pulses can be generated,
but further downscaling will be limited by the speed of the modulations or elec-
tronics. Passive mode-locking can generate pulses with much shorter durations
down to femtosecond range. It makes use of the pulse itself to introduce intensity-
dependent loss into the cavity using a saturable absorber. It’s an element which
has higher transmission for higher intensity. Some initial higher background noise
in the cavity, after many round trips, will win the competition with other fluctua-
tions and eventually become the circulating pulse in the cavity. This technique is
easy to implement and has superior performance over other active mode-locking
techniques.
Fiber lasers are great tools and stages for these physical processes to interact
with each other and study the nonlinear pulse evolutions when the lasers are mode-
locked. With stable mode-locking, the pulse characteristics and pulse evolutions
can be very different depending on the elements in the cavity. The rest of this
section is to go over the various pulse evolutions discovered in fiber lasers so far.
The theories and characteristics of these pulse evolutions not only help to identify
the specific regime the laser operates in, but also serve as a guidance to design
fiber lasers for different needs.
10
1.3.1 Soliton lasers
The story of fiber lasers starts with soliton lasers. When the laser is made up of
only anomalous dispersion fibers, it can be modeled using the NLSE:
∂A 2
= −iβ2 ∂ A + iγ|A|2A, (1.8)
∂z 2 ∂t2
This well-studied equation is integrable and has static solutions called solitons.
They have minimal breathing in spectral and temporal domain. These solitons
are the result of balance between anomalous GVD and third order nonlinearity
modeled by γ, and can be described as:
√ ( )t
A(t, z) = A sech exp iθz, (1.9)
τ
One important relation between the relevant parameters is the soliton area
theorem [8]:
2|β2|
Eτ = . (1.10)
γ
where τ is the pulse width and E is the pulse energy. For each combination of
the values β2 and γ, we have a family of soliton solutions under the soliton area
theorem.
As the solitons traverse in the cavity, they experience perturbations such as
gain and loss. Solitons adjust themselves by shedding excessive energy in the form
of dispersive waves and these can interfere with the original solitons to exhibit the
11
normal                                    anomalous 
 
Figure 1.3: Schematic of the dispersion managed soliton design.
characteristic Kelly sidebands [9]. These features on the output spectra are the
signature of the soliton mode-locking inside a laser cavity.
However, these Kelly sidebands set the limits on how high the soliton’s energy
can be. Usually, the pulse energy is limited to 100-pJ level in single mode fiber
lasers. The corresponding pulse duration is in the picosecond range. Higher pump
power results in the soliton fission and multi-pulsing.
1.3.2 Dispersion-managed soliton lasers
To further improve the pulse energy, a strategy called dispersion management was
developed by introducing a dispersion map in the cavity. Here we focus on the net
cavity dispersion near 0. One way is to introduce alternating segments of normal
and anomalous dispersion fibers [10]. Other methods such as fiber Bragg gratings,
telescope grating systems or prisms are also used.
These dispersion-managed (DM) solitons get stretched in the normal disper-
sion part and get compressed back in the anomalous dispersion fibers, having a
large breathing ratio in the time domain, as shown in Figure 1.3. This breathing
evolution in temporal domain can reduce the peak power, thus the nonlinear phase
accumulation to allow stable pulse energies to increase by an order of magnitude.
12
The net cavity dispersion can be slightly anomalous or normal. Solutions with
the shortest pulse duration exist on the slightly anomalous dispersion side, and
the higher energy happens on the net normal side. Up to now, some of the world-
record shortest pulse generations from fiber oscillators are still from DM solitons
[11, 12, 13].
A more thorough and theoretical summary of dispersion-managed solitons can
be found in [14].
1.3.3 Dissipative soliton lasers
Next, we are moving towards the direction of net cavity dispersion being large and
normal. A good summary of the pulse evolutions in the normal dispersion regime
can be found in [15].
In 2006, dissipative solitons were discovered by Andy Chong and coworkers
[2] in an all-normal dispersion fiber laser at 1 µm. They removed the dispersion
compensation elements in the cavity and managed to stabilize the highly chirped
high-energy solution by introducing a bandpass filter. The output spectra do
not share any similar features as previous solutions, but have steep edges and
batman ears. This is the result from the balance between normal dispersion, self-
focusing nonlinearity, dissipation by spectral filtering and saturable absorption.
The equation used to model this is the cubic-quintic Ginzburg-Landau equation
(CQGLE) [16, 17]:
( )
∂A 1 2
= gA+ − D ∂ Ai + (α + iγ)|A|2A+ δ|A|4A, (1.11)
∂z Ω 2 ∂t2
13
where g models the gain or loss, D is the 2nd order dispersion, and α and γ together
model the property of the SA. The gain filter is introduced by the Ω term and γ
is the Kerr nonlinear coefficient.
Without the filter, self-consistent solutions do not exist in a highly chirped fiber
cavity. By cutting the pulses in both the temporal and spectral domains, the filter
forces the pulses to go back to the starting state and circulate for the next round
trip. The chirped pulses are far from the transform limited duration of the spectra
and can accumulate large nonlinear phase shift with high energies. These output
pulses can be dechirped to close to the transform limited duration by grating pairs.
A good example of the DS soliton laser is shown in Figure 1.2 above. Thus,
the resulting peak power can be orders of magnitude higher than the ordinary
soliton lasers. Currently, single mode fiber lasers with best performances have
been achieved using this evolution with 100-fs duration and 30-nJ pulse energy
[18, 6]. These numbers are starting to compete with those from the Ti:sapphire
lasers on the market.
Further studies continue to reveal the properties of the DS solutions, such as
the DS-version of the soliton area theorem [5], giant chirped oscillator [19], Raman
dissipative solitons [20], etc.
1.3.4 Passive self-similar lasers
Even with the normal-dispersion operation of fiber lasers, the excessive nonlinear
phase accumulation imposes the ultimate limit on the highest achievable pulse en-
ergies. Under the interplay of dispersion and nonlinearity in fibers, pulses undergo
the process called wave breaking [21], where they develop side lobes on the spectra
14
and oscillations on the pulse wings. They can not be de-chirped close to transform
limited durations any more. Then the idea of the wave-breaking-free pulses was
put forward [22], which share parabolic intensity profiles and linear chirps in the
time domain.
These are the wave-breaking-free solutions of the NLSE with normal dispersion
in the high intensity regime. Along further propagation, pulses maintain their
parabolic shapes and propagate self-similarly without any distortion and wave-
breaking. This form-invariant propagation is the origin and the meaning of the
word self-similar and sometimes these pulses are called similaritons.
These pulses share the temporal profile in the following analytical form:
( ( )2)
A2(0, t) = A20 1−
t
, t ≤ τ (1.12)
τ
After conducting the Fourier transform of the parabolic profile in the time
domain, we can find it’s also a parabola in the frequency domain. It then took a
decade until the first self-similar laser was developed by Ilday et al. [23]. The main
component of the laser is the dispersion map, formed by the normal dispersion
fiber and a dispersive delay line (DDL), without any extra introduced spectral
filters. The pulse evolution is defined by the dispersion map with minimal spectral
evolution. The pulse duration increases in the normal dispersion fiber segment
and gets compressed in the DDL. The temporal breathing ratio can be as high as
10∼50. The output chirped pulse has a parabolic shape before the DDL and can
be de-chirped close to the transform limited duration using a grating pair outside
the cavity.
In contrast with the self-similar evolution in the gain fiber, which admits a local
15
nonlinear attractor, this pulse shaping process is based on the whole cavity. The
pulses have minimal spectral evolution and the DDL is primarily responsible for
returning the chirped pulses to their original pulse durations in the cavity. These
are the asymptotic solutions of the NLSE with only nonlinearity and normal GVD,
without any gain term. With the features of the pulses propagating in the time
domain and to differentiate it from the amplifier similariton lasers, we call them
passive self-similar lasers. The maximum pulse energy achievable using dissipative
soliton lasers and passive self-similar lasers are comparable.
1.3.5 Amplifier similariton lasers
The asymptotic solution of the NLSE with normal dispersion and constant gain
term without bandwidth limit [24]:
∂A iβ 22 ∂ A g
= − + iγ|A|2A+ A, (1.13)
∂z 2 ∂t2 2
also has the property of propagating self-similarly in the gain fiber. Distinct
from the above passive self-similar solutions, these are the local nonlinear attractors
in the gain fiber, called amplifier similaritons. Any input arbitrary pulses for
the gain fiber eventually converge to the parabola-shaped solutions and scale self-
similarly along propagation. The features of the asymptotic solution depends solely
on the input pulse energy E and the gain fiber parameters with the form:
√
( ( )2)
− tA(z, t) = A iφ(z,t)0(z) 1 e , t ≤ τ (1.14)
τ(z)
The pulse phase φ(z, t) and pulse chirp δω(t) have the form of:
16
( ) ( )
1 2 − 1φ(z, t) = φ0 + 3γ A0(z) g t2, (1.15)2g 6β2
− ( )∂φ(z, t) 1
δω(t) = = g t, (1.16)
∂t 3β2
The pulse amplitude A0(z) and duration τ(z) scale as exp(gz/3), where g is the
constant gain coefficient. The exponential growth of the bandwidth will eventually
be limited by the gain bandwidth.
The stabilization of the amplifier similariton solution inside a laser cavity was
realized around the same time by 3 groups of people. Oktem et al. [25] make use
of the anomalous dispersion fiber to transform the pulses after the gain fiber into
solitons, which are also nonlinear attractors, and prepare them as the seed input
for the gain fiber. Aguergaray et al. [26] rely on a km-long Raman gain fiber to
provide enough propagation length for the input pulses to reach the asymptotic
solution. Renninger et al. [27] conceive the idea of employing a narrow filter to
stabilize the amplifier similariton in an all-normal-dispersion fiber laser without
dispersion compensation.
The attractive feature of the amplifier similariton is that it’s a local nonlinear
attractor, which allows the manipulation of other parts of the cavity. For example,
the net cavity dispersion can be changed by introducing dispersion compensation
outside the gain fiber, without disturbing the nonlinear attractor in the gain fiber,
as long as the proper seed pulse is provided [28]. In contrast, passive self-similar
solutions are defined by the average cavity parameters in a narrow parameter space.
Amplifier similaritons have a large breathing ratio in both temporal and frequency
domains, which is different from DS solitons and passive self-similar lasers.
17
Figure 1.4: The concept of the extended self-similar laser design.
Some of the nice high-energy short pulse generations are based on this evolution,
where 40-fs pulses from Yb fiber laser and 70-fs pulses from Er fiber laser are
obtained [29, 30]. Generating broader bandwidth using amplifier similaritons are
limited by the gain bandwidth [31]. The chirp of the pulses is disrupted and the
pulse quality is degraded. Other methods need to be developed to pursue shorter
pulse generation beyond the gain bandwidth limits.
1.3.6 Extended self-similar lasers
The conventional way to make shortest pulses from fiber lasers is to operate them
with net cavity dispersion near 0, but this method is ultimately limited by the
bandwidth of the gain medium. For example, pulses from Yb fiber lasers are
limited to durations of 30 fs due to the 40-nm gain bandwidth. And this evolution
only generates pulse energies of a fraction of a nanojoule. It requires new wisdom
to generate mode-locked spectrum beyond the gain bandwidth limit in fiber lasers.
Chong et al. came up with the idea employing amplifier similariton as the
local nonlinear attractor to make the significant spectral broadening happen after
the gain segment in a highly nonlinear fiber [32]. As long as the pulses can reach
the self-similar solutions in the gain fiber, the nonlinear pulse propagation in the
passive fiber can be stabilized. These are all possible with the help of the narrow
18
spectral filter. The linearly chirped parabolic pulses in the amplifier will maintain
the parabolic shape and linear chirp in the passive highly nonlinear fiber. This is
the reason why this evolution is called extended self-similar. This method demon-
strated 1-nJ pulses with 20-fs duration after dechirping at 1 µm, which are the
shortest pulses ever generated directly from a Yb fiber oscillator.
1.4 Fiber laser components
This thesis is mainly focused on building up fiber lasers and mode-locking them
to obtain ultrashort pulses. A summary of the important components that appear
in a fiber laser is necessary. This part is to introduce different kinds of individual
components, such as spectral filters and saturable absorbers (SA) in the laser
cavity.
1.4.1 Spectral filters
As we have seen in the descriptions of dissipative soliton and amplifier similariton
lasers, spectral bandpass filters are the critical and core elements for pulse shaping.
They are employed to cut the intra-cavity chirped pulses in spectral and temporal
domain, to enable a self-consistent solution in the cavity. In fiber lasers, we mainly
use birefringent filters and grating filters, due to their easy implementation and
flexibility.
19
Birefringent filters
Birefringent filters work based on the wavelength-dependent phase shift difference
between the ordinary (o-wave) and extraordinary waves (e-wave) in the birefringent
plate [33]. When a linearly polarized light is launched into the birefringent plate, it
splits into these 2 components. The phase delay difference between them, combined
with a second polarizer placed after the birefringent plate will induce a sinusoidal
transmission curve with respect to the wavelengths.
When the incident angle of the input light with respect to the principal axis of
the material is set to 45 degree, the distribution of the intensity between o-wave
and e-wave is the same. Then the transmission function through the birefringent
plate is:
(
2 π(ne −
)
no)d
T = cos , (1.17)
λ
where ne and no are the refractive indices of the material for e-wave and o-wave.
And d is the thickness of the birefringent plate. If quartz is used as the birefringent
material, the bandwidths near ∼ 10nm can be created in the near infrared color
range. An 8-nm bandwidth filter using a 12-T quartz plate has the transmission
curve shown in Figure 1.5. Attractive properties of this filter are that it can achieve
a modulation depth of 100% and the filter peak positions can be tuned simply by
rotating the plates.
20
1.0
0.8
0.6
0.4
0.2
0.0
960 980 1000 1020 1040 1060 1080
Wavelength HnmL
Figure 1.5: Transmission function of a Birefringent filter made of a 12-T (6 mm
thick) quartz plate near 1030 nm.
Lyot filters
Lyot filters, invented by Bernard Lyot, also make use of the material birefringence
to produce a narrow bandpass filter based on the wavelength-dependent power
transmission. However, unlike birefringent filters introduced above, Lyot filters
are made up of a sequence of birefringent plates with each plate being half the
thickness of the previous one, as shown in Figure 1.6. The light can be considered
as containing ordinary and extraordinary polarization components, which respec-
tively experience a different phase delay. The relative phase delay depends on
the wavelength, thus making the transmission of optical power at a subsequent
polarizer wavelength-dependent.
The key feature of the multiple-plate design is to obtain a sharper/narrower fil-
ter with a larger free spectral range. This is particularly useful when it’s employed
in a gain medium with broad gain bandwidth and to avoid the gain competition
between multiple lasing peaks. One illustration of the Lyot filter is shown in Figure
1.7 to produce a 35-nm bandwidth filter at 1900 nm. With more plates introduced
21
Transmission
P BC P BC P BC P 
Figure 1.6: A Lyot filter, consisting of a sequence of birefringent crystals (BC) and
polarizers (P). Each birefringent plate has the thickness half of the previous one.
1.0
0.8
0.6
0.4
0.2
0.0
1800 1850 1900 1950 2000 2050 2100
Wavelength HnmL
Figure 1.7: Transmission function of a Lyot filter made of a 6 T and 12 T quartz
plate near 1900 nm. The bandwidth is 35 nm and the free spectral range is 150
nm.
into the sequence, the filter peaks become sharper.
Grating filters
Grating filter is another option to provide a singly-peaked filter with a nice Gaus-
sian transmission profile due to the Gaussian mode in the single mode fiber. Typ-
ically, it’s formed by the combination of a fiber-pigtailed collimator and a grating.
The spectra of the incident beam is dispersed and spread out by the grating spa-
tially and the acceptance collimator collects only a small portion of the wide beam,
22
Transmission
Figure 1.8: Schematic of a grating-based filter.
as shown in Figure 1.8. The bandwidth of the grating filters is usually in the range
of a few nanometers and is mainly dependent on the line density of the grating. To
gain access to a broader filter bandwidth, one should either employ a grating with
lower line density, such as 50 lines/mm, or replace the grating with a much less
dispersive element, e.g. a prism. Other factors, such as the beam size, mode field
diameter, distance of the collimator to the grating, focal length of the collimator
all have influence on the filter transmission profile.
There’s one point to be made that grating filters can introduce a large amount
of dispersion, which influences pulse evolutions under certain circumstances. Nor-
mally, group delay dispersion with a negative sign is introduced and the amount of
dispersion increases with the number of lines of the grating used. In Figure 1.9, the
1-nm bandwidth filter at 1030 nm is formed using a 1000 lines/mm grating. The
introduced dispersion is ∼ -0.1 ps2, which needs to be taken into consideration.
1.4.2 Saturable absorbers
Saturable absorbers (SA) are elements that provide decreasing loss as the intensity
of the incident light increases. This nonlinear feature helps to select and promote
23
a b
c d
Figure 1.9: A 1-nm bandwidth grating filter at 1030 nm. (a) The transmission
profile of the filter; (b) The incident and diffracted angle of the beam; The de-
pendence of the introduced (c) 2nd order and (d) 3rd order dispersion on distance
between the collimator and the grating.
Figure 1.10: Illustration of saturable absorbers promoting pulse generation.
some slightly higher intensity fluctuations initially out of the noise background and
gradually build them up into mode-locked pulse trains.
There are two categories of saturable absorbers used in lasers, the artificial
saturable absorbers and the material-based saturable absorbers. The former ones
include Kerr-lens mode-locking [34], nonlinear polarization evolution (NPE) [35,
36], nonlinear optical loop mirror (NOLM). The latter ones include semiconductor
24
saturable absorber mirrors (SESAM) [37], quantum dots [38], carbon nanotubes
[39], and multiple novel materials. Two types of SAs, SESAM and NPE, are
discussed below.
Semiconductor saturable absorber mirrors
SESAMs for passive mode-locking were invented by Prof. Ursula Keller for solid
state lasers. They are the optical devices based on the semiconductor materials
and can control the magnitude and phase of the optical absorption and reflection
at different wavelengths. The optical characteristics of the modulation depth, sat-
uration fluence, and non-saturable loss are key parameters for laser mode-locking.
In case of slow response time from SESAM relative to the pulse durations, the
dynamics of SESAM can be complex and needs to be modeled through:
∂l 2
= − l −∆R − |A| l, (1.18)
∂t τres Esat
where l is the time-dependent saturable loss, ∆R is the saturable loss or modulation
depth, τres is the response time, and Esat is the saturation fluence of the SESAM.
SESAMs have the advantage of being environmentally stable for self-starting
mode-locked lasers. However, they are prone to damage at high power operations,
and they typical have low modulation depth of ∼ 10%. These impede their use in
the high power fiber lasers.
25
output
QWP HWP QWP
PBS
fiber
Figure 1.11: Illustration of NPE saturable absorber (PBS: polarizing beam splitter;
QWP: quarter waveplate; HWP: half waveplate).
Nonlinear polarization evolution
Nonlinear polarization evolution (NPE) [40] is the artificial SA of choice for lasers
in this thesis. It is highly tunable with ultrafast response time and deep modulation
depth. It works by converting the nonlinear polarization rotation of light in fibers
to amplitude modulation via polarization-dependent loss.
As shown in Figure 1.11, NPE is implemented by 1 polarizer beam spitter
(PBS) and 3 waveplates, including two quarter waveplate (QWP) and one half
waveplate (HWP). The linearly polarized light after the PBS is set to elliptically
polarized light by the quarter waveplate, then launched into the fiber. Self-phase
modulation and cross-phase modulation of the two orthogonally polarization ellipse
produce intensity-dependent phase shifts. After the fiber, a half waveplate and a
quarter waveplate set the polarization back to linear and try to match the axis
to the second polarizer. Intensity-dependent loss is induced through the intensity-
dependent rotation of the polarization plane. The PBS usually serves as the output
port as well.
The transmission curve was derived in [41]. If the angles between the principle
26
axis of the polarizer with QWP, QWP and HWP are α1, α2 and β1, the normalized
output intensity after the second polarizer is:
1
I = (1−sin(2α1)sin(2α2)+cos(2α1)cos(2α2)cos(2(α1+α2−2β1+∆φ))) (1.19)
2
It depends on the intensity through the term ∆φ, which is the rotation angle
of the polarization in the Kerr medium (fiber). By changing the angles of the
waveplates to certain positions, the transmission of the NPE will act as a SA. The
drawback of the NPE is the lack of environmental stability. The retardation of
the waveplates depends on the ambient temperature and the polarization rotation
is changed if the fiber segment is touched or moved. Under these circumstances,
the operating state of SA will be changed and the laser may lose its mode-locking
state.
27
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30
CHAPTER 2
SELF-SIMILAR PULSE EVOLUTION IN A FIBER LASER WITH
COMB-LIKE DISPERSION DECREASING FIBER1
Here we demonstrate an erbium fiber laser with self-similar pulse evolution
inside a comb-like dispersion-decreasing fiber. We show numerically and experi-
mentally that the comb-like dispersion-decreasing fiber works as well as an ideal
one, and offers major practical advantages. The existence of a nonlinear attractor
is verified by the invariant pulse chirp over a wide range of net cavity dispersion
in experiments. The laser generates 1.3-nJ pulses with parabolic shape and lin-
ear chirp, which can be dechirped to 37 fs. Comb-like dispersion-decreasing fiber
should enable the generation of high energy few-cycle pulses directly from a fiber
oscillator.
2.1 Introduction
Lasers with few-cycle pulse durations are attractive for applications in frequency
combs, high-harmonic generation, and ultrafast spectroscopy, among others. The
few-cycle regime in the near infrared region is currently dominated by solid-state
lasers, such as Ti:sapphire, Cr:forsterite and Cr:YAG [2], due to their broad gain
bandwidths. Fiber lasers have started to compete with solid state lasers in terms of
pulse energy and average power, but they still lag behind in obtaining high energy
few-cycle pulses directly from an oscillator. Ultimately, mode-locked fiber lasers
that generate octave-spanning spectra and few-cycle pulses will be very attractive
for applications.
1Much of the work presented in this chapter was published in Optics Letters [1].
31
Ultrashort pulses are obtained routinely from fiber lasers with cavity dispersion
close to zero [3], but the pulse energy is limited. Sub-30 fs pulses were obtained
from Yb fiber lasers with pulse energies up to 0.7 nJ [4, 5], which corresponds
to nonlinear phase shifts up to π. To obtain high-energy ultrashort pulses, pulse
evolutions that can tolerate large nonlinear phase are desirable. Self-similar pulse
evolution [6, 7, 8] has allowed the generation of 10-nJ and 42-fs pulses at 1030 nm
[9]. However, the gain bandwidth limitation has to be circumvented to reach the
few-cycle pulse regime.
Chong et al. [10] made use of the nonlinear attractor in the gain fiber and
extended self-similar evolution in a highly-nonlinear passive fiber to obtain 20-
fs pulses. However, the pulse accumulated a complex phase, and multiphoton
intrapulse interference phase scan was required to dechirp the pulse. Dispersion-
decreasing fiber (DDF) with normal dispersion is formally equivalent to a gain fiber
and it supports self-similar pulse evolution, but without any bandwidth limitation
[11]. Liu and coworkers [12] exploited extended self-similar evolution in a DDF
made of tapered photonic crystal fiber to obtain 6-cycle pulses at 1 µm, dechirped
with a standard grating pair. However, the fabrication of DDF is difficult [13]. Re-
cently, aspects of self-similar pulse evolutions have been demonstrated in passive
comb-like DDF by Kibler et al. [14] and Méchin et al. [15]. It was demonstrated
[14] that an ideal DDF with a hyperbolic dispersion profile can be approximated by
alternating segments of two ordinary fibers with different dispersion values (Fig-
ure 2.1). This so-called comb-like fiber offers a simple, engineerable, and practical
alternative to a real DDF [16].
32
2.2 Er fiber laser with comb-like dispersion-decreasing
fiber
Ultrashort pulse generation in erbium (Er) fiber lasers has followed a similar de-
velopment path. Sub-45 fs pulses at 1550 nm were obtained based on dispersion
management [17, 18, 19] with pulse energies below 0.5 nJ. Sub-50 fs pulses were
generated, but with a long pedestal expanding over 800 fs, which indicates un-
controlled nonlinear phase [20]. Self-similar pulse evolution has been employed to
generate 3.5-nJ and 70-fs pulses [21]. A single-cycle light source was realized by
coherently combining two continuua driven by the same fiber laser [22]. Despite
the significant progress people have made at this important wavelength, the gen-
eration of few-cycle pulses with nanojoule-level energy directly from an oscillator
remains a challenge.
Here we demonstrate an Er fiber laser with self-similar evolution in a comb-
like DDF in the cavity. Numerical simulations show that the pulse evolution and
performance with the comb-like DDF are very similar to those that would be
obtained with a real DDF. The nonlinear attractor is studied by varying the cavity
dispersion over a large range, and confirmed by the invariant chirp on the output
pulses. The experimental results agree well with numerical simulations. The laser
generates 1.3-nJ pulses with a parabolic shape and linear chirp. These can be
dechirped to 37 fs duration, which corresponds to 7 optical cycles. The pulse
energy is a 4-fold improvement over previous Er fiber lasers with comparable pulse
duration. Factors that limit the laser performance are discussed.
33
2.2.1 Modeling
Pulse propagation inside an ideal DDF is governed by a nonlinear Schrödinger
equation with the group velocity dispersion (GVD) varying with the propagation
distance z according to Eq. 2.1 [11]. The dispersion decreasing rate is given by Γ,
and β20 denotes the GVD at z = 0.
β20
β2(z) = (2.1)
1 + Γz
The full-width-half-maximum bandwidth of the asymptotic self-similar solution
can be calculated using the method of steepest descent [23]:
√ 2 ( )1/3λc EinγΓ
∆λ = 2 (1 + Γz)1/3 (2.2)
2πc 2β220
where λc is the center wavelength of the pulse, c is the speed of light, Ein is the input
pulse energy to the DDF, and γ is the fiber nonlinearity coefficient. Evaluation of
the spectral bandwidth at the end of the DDF yields:
( )1/3
∝ EinΓ∆λ (2.3)
Aeffβ20β2
where β2 is the GVD at the end of the DDF and Aeff is the effective mode area,
assumed to be constant along the fiber. A DDF optimized to obtain the broadest
spectra should have small core area, low dispersion on both fiber ends, and a large-
dispersion decreasing rate. In addition, the loss in the cavity should be kept small
to maximize the input energy.
We use Eq. 2.3 as a figure of merit to choose the fibers to form the comb-like
34
DDF. It is desirable for one fiber to have dispersion as close to zero as possi-
ble, which led us to choose PWG1-XP (Nufern). The choice of the second fiber
is a trade-off between minimizing β20 and maximizing Γ. Among commercially-
available fibers, UHNA7 (Nufern) offers the largest bandwidth. The total length
of DDF is chosen to be 1 m, to avoid the overdriving of nonlinear polarization
evolution (NPE). We take the minimum segment length to be 5 cm, considering
practical issues such as splicing. To find the optimum segment lengths, a simplex
algorithm [24] was used to fit the comb-like DDF to an ideal hyperbolic DDF with
the same initial and final dispersion, and constant fiber nonlinearity. We started
with 2 segments in optimization and stopped at 6 segments, where the results are
similar to an ideal DDF. The fitting result is shown in Figure 2.1. Use of the
soliton number N2 ∝ 1 to include the differences in nonlinearity of the fibers
Aeffβ2
produces similar results for the optimum segment lengths. The dispersion profiles
of an ideal DDF at 1400 nm and 1700 nm are shown to illustrate the dispersion
variation across the expected spectral range. The large and negative third-order
dispersion (TOD) of UNHA7 helps compensate the TOD of the cavity.
60
40
20
0
0 20 40 60 80 100
Fiber length (cm)
Figure 2.1: Comparison of the GVD values of the ideal DDF (green) with the
comb-like DDF (blue) at 1550 nm. The lengths of the segments are 14, 6, 5, 20, 5
and 50 cm. The upper and lower red dashed lines are for the ideal DDF at 1700
and 1400 nm, respectively.
35
2
GVD (fs /mm)
Numerical simulations are conducted assuming the laser setup of Figure 2.2 (a)
and using appropriate terms of the generalized nonlinear Schrödinger equation for
each segment, including stimulated Raman scattering and self-steepening. Values
of the fiber parameters are listed in Table 2.1. A relatively-short (1 m) gain fiber
is chosen for two reasons: we do not want the pulse to be stretched too much
before entering the DDF, and we do not want the pulse to reach the asymptotic
self-similar form in the gain medium. Instead, we rely on the nonlinear attractor
in the DDF to stabilize the pulse evolution. The gain segment is modeled with
30-dB small signal gain, a 40-nm Gaussian gain spectrum centered at 1550 nm,
and a saturation energy of 10 nJ. Normal-dispersion fibers (NDF) are inserted
to reduce the repetition rate to 10∼20 MHz. Other fiber segments are 0.5 m of
SMF28e and 0.5 m of OFS980 as the pigtails of the collimators and wavelength-
division multiplexers, respectively. A 5-nm bandwidth Gaussian filter with peak
transmission of 50% is assumed. It introduces dispersion of -0.11 ps2 due to the
combination of the grating and collimator. The output coupling ratio is 70%,
while the linear cavity loss is 60% to reflect the collimator pair coupling efficiency.
A lumped saturable absorber is modeled with modulation depth of 100% and
saturation power of 2 kW.
Table 2.1: Values of Parameters Used in Simulation
Fiber Aeff GVD TOD
Name (µm2) (fs2/mm) (fs3/mm)
UHNA7 7.6 31 -153
PWG1-XP 18 7.8 1.1
Er110 29 15 50
OFS980 44 4.5 109
NDF 48 3.8 83
SMF28e 85 -23 86
Simulations converge over a large range of parameters, and typical solutions are
shown in Figure 2.2. The comb-like DDF functions similarly to the ideal one, as
36
a Output
600/mm HWP QWP HWP
SMF28
Isolator
PBS
DDF
QWP NDF
SMF28
Er
diode diode
SM NDF Er DDF SM SA SF SM NDF Er DDF SM SA SF
SM NDF Er DDF SM SA SF
Figure 2.2: (a) Schematic of the Er DDF laser: PBS: polarizing beam splitter;
QWP: quarter-wave plate; HWP: half-wave plate. (b)-(d) Simulation results for
an ideal (red) or a comb-like DDF (black), with net cavity dispersion of -0.087 ps2.
(b) Evolution of pulse duration (SA, saturable absorber; SF, spectral filter). (c)
Evolution of spectral bandwidth. (d) Evolution of misfit parameter.
they match well in both spectral and temporal pulse evolution. Pulse propagation
is mainly defined by the 5-nm filter and the DDF segment. The spectral breathing
ratio can be as large as 25, with bandwidth around 120 nm after the DDF seg-
ment. The pulse duration increases monotonically in the DDF, until the pulse is
compressed by the anomalous-dispersion pigtail and cut by the narrow filter. To
37
get further insight into the pulse-shaping process, we employ the misfit parameter
M, which is the root-mean-square deviation of the pulse intensity profile from a
parabolic pulse of the same energy. Due to the presence of intermediate-asymptotic
wings around the parabolic pulse core [25], we calculate the M parameter based on
the central portion of the pulse, which contains 95% of the pulse energy (Figure 2.2
(d)). The pulse is primarily attracted to the self-similar solution in the DDF, with
a small contribution from the gain fiber. The M parameter decreases sharply in
the initial stage of the DDF and remains close to zero. The parabolic shape is
maintained upon further propagation inside the SMF until the spectrum is cut by
the filter.
2.2.2 Experiments
We built a laser as in Figure 2.2(a) with a comb-like DDF by simply splicing
six segments of fiber together. Total loss of 0.25 dB is introduced by the comb-
like structure. NPE is implemented with the wave plates and a polarizing beam
splitter, which also serves as the output port. A half-wave plate in front of the
grating is used to optimize the transmission efficiency. Typically, mode-locking
with multi-pulsing is initiated at a pump power of 1 W by rotating the waveplates,
and a single-pulsing state is obtained by decreasing the pump power to 500 mW.
Single-pulsing is confirmed with an autocorrelator with 180-ps scan range and a
fast detector with 30-ps rise time.
A representative set of experimental results is shown in Figure 2.3. The cross-
correlation of the output pulse and the dechirped pulse fits a parabola well, which
is evidence of the self-similar pulse shaping. Output pulses with 1.3-nJ energy
and spectral bandwidth of 240 nm at the -20-dB level are obtained, in good agree-
38
Figure 2.3: (a) Measured mode-locked spectrum of the Er DDF laser. (b) Fitting
of the cross correlation signal to a parabola. (c) Interferometric autocorrelation of
the dechirped pulse using two-photon absorption. (d) Radio frequency spectrum
of the mode-locked laser with resolution bandwidth of 10 Hz and span range of
100 kHz.
ment with the simulations. The pulses can be dechirped with a prism pair to 37
fs, within 10% of the transform limit of 34 fs. The pulse energy is a 4-fold im-
provement compared with Er fiber lasers that have reached this pulse duration.
The radio-frequency spectrum exhibits a contrast ratio over 85 dB, which confirms
stable mode-locking. The pulse quality was verified by launching the dechirped
pulses into a segment of PWG1-XP fiber. The observed spectral broadening in
Figure 2.4 agrees reasonably with results of numerical simulations and confirms
the approximately 30 kW peak power.
To demonstrate the nonlinear attractor in the DDF, the net cavity dispersion
was varied by inserting normal-dispersion fiber into the laser cavity. Mode-locking
is observed with net dispersion ranging from -0.1 ps2 to 0.3 ps2, at which point other
39
Figure 2.4: (a) The pulse spectrum at the input and output end of the peak power
testing segment with coupled-in energy of 0.54 nJ. (b) Comparison of the spectral
broadening in experiments (in black) and simulations (Gaussian pulses; in red) at
different input energy level.
states that resemble dissipative solitons start to appear. The pulse performance is
summarized in Figure 2.5. Over a large dispersion range, the pulse chirp is always
much smaller than, and relatively independent of the cavity dispersion. This is a
strong indication of the existence of a nonlinear attractor [26] in the DDF segment.
The chirp of the asymptotic solution is determined only by the parameters of the
DDF. The deviation of the dechirped pulse duration from the transform limit is
always less than 10%, as expected for a parabolic pulse with linear chirp.
2.2.3 Discussion
The performance achieved in this proof-of-concept demonstration of the piecewise
DDF inside a laser is significantly limited by losses in the cavity. The implementa-
tion of the filter by a diffraction grating and collimator is very lossy. In simulations,
if these losses are reduced from 7 dB to 1 dB, 25-fs pulses with peak power over
200 kW are generated. The use of a fiber Bragg grating filter in an amplifier sim-
ilariton laser to minimize the loss was recently reported [27], and this approach
could be applied to a laser with a DDF. Excessive nonlinear phase accumulation
40
Figure 2.5: (a) Pulse energy and dechirped pulse duration at various cavity disper-
sions. (b) Pulse chirp and deviation of dechirped pulse duration from the transform
limit at various cavity dispersions.
will then limit the performance.
The versatility of the comb-like DDF lies in the fact that we can tailor its
properties freely using different combinations of fibers. Limitations of the gain
spectrum are obviously eliminated, and it should be possible to mitigate other
factors that limit self-similar evolution [28, 29, 30]. Third-order dispersion can be
compensated in the fiber segments of the DDF (as is the case in the work here)
and stimulated Raman scattering can be managed by using short overall lengths.
41
2.3 Yb fiber laser with comb-like dispersion-decreasing
fiber
This flexible comb-like DDF will be valuable for fiber lasers at other wavelengths
as well. For example, it will help with Tm fiber lasers at 2 µm, where it is difficult
to design a similariton laser due to the lack of normal-dispersion gain fibers. With
this home-made DDF, the formation of a nonlinear attractor inside the passive
segment is possible.
Here I want to illustrate the application of comb-like DDF in a Yb fiber laser,
although the implementation will be complicated a bit by the need to use mi-
crostructure fibers to obtain different dispersions. Considering the intrinsic mate-
rial dispersions in fibers at 1 µm are ∼20 fs2/mm, it’s hard to obtain ordinary
fibers with GVD close to 0 to form the DDF structure. Special microstructure
fibers from NKT, such as NL-1050-ZERO-2, need to be considered as the low-
dispersion fiber to form the comb structure. Employing the Eq. 2.3 as the figure
of merit to choose the high-dispersion fiber, UHNA3 stands out as a choice. The
parameters of the fibers are shown in Table 2.2.
Table 2.2: Parameters Used in Simulation for Yb DDF laser
Fiber Aeff GVD TOD
Name (µm2) (fs2/mm) (fs3/mm)
UHNA3 4.5 60 -102
ZERO-2 3.8 2 0
Yb501 31 23 40
Hi1060 31 23 40
OFS980 14 27.3 57.5
If we assume a 6-segment structure for the DDF, with a total length of 1 m
and dispersion deceasing rate of Γ = 0.29/cm, the fitted lengths are: 2.3 cm, 1 cm,
42
a b
60
40
20
0
0 20 4 60 80 100
Fiber length (cm)
Figure 2.6: (a) Schematic of the Yb DDF laser: PBS: polarizing beam splitter;
QWP: quarter-wave plate; HWP: half-wave plate. (b) Comparison of the GVD
values of the ideal DDF (green) with the comb-like DDF (blue) at 1030 nm. (c)-
(d) Comparison of the output spectra and dechirped pulse (using comb DDF) with
transform limit pulse.
1.3 cm, 36 cm, 1 cm, 58 cm for an ideal DDF with β 220 = 60 fs /mm. The loss at
the filter is set to 0 assuming the employment of a fiber Bragg grating.
In simulations, pulses with 4 nJ energy and transform-limit duration of 5.6
fs are obtained. The pulse evolution in terms of the Misfit parameter and the
spectral bandwidth are very similar to the case where the DDF is ideal. The
chirp is linearized by the nonlinear attractor inside the DDF and the pulses can be
dechirped to 9.4 fs, close to the TL duration. In this cavity, the nonlinear phase
is ∼ 80 π. Further increase of the pulse energy is limited by the nonlinear phase
accumulation. But this laser has provided a route to the generation of few-cycle
pulses directly from a fiber laser at 1 µm.
43
2
GVD (fs /mm)
2.4 Conclusion
In conclusion, we have demonstrated a fiber laser with self-similar evolution in a
comb-like dispersion-decreasing fiber, which works as well as an ideal dispersion-
decreasing fiber but is much more practical to realize experimentally. The existence
of the nonlinear attractor inside the DDF is experimentally confirmed. The laser
generates 1.3-nJ pulses with 7-cycle duration at 1550 nm. This constitutes a 4-
fold increase in pulse energy compared to previous reports of this pulse duration.
Numerical results show great potential to obtain shorter pulses with much higher
energy by minimizing cavity losses.
Portions of this work were supported by the National Science Foundation
(ECCS-1306035).
44
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46
CHAPTER 3
HIGH ENERGY PULSES FROM THULIUM FIBER LASERS1
3.1 Introduction
Ultrafast laser sources that emit with a wavelength around 2 µm are attracting
increasing attention due to their various applications, which include medical treat-
ment, spectroscopy, environmental monitoring, and remote sensing. Among them,
fiber lasers that employ thulium-doped or thulium-holmium co-doped fibers stand
out for covering the broad range from 1.7 µm to 2.1 µm, with great potential to
produce sub-100 femtosecond pulses. The cross-relaxation process in highly-doped
thulium (Tm) fiber pumped by 793-nm diodes can be exploited to obtain quantum
efficiency near 2. Continuous-wave power as high as 1 kW has been obtained from
a Tm fiber amplifier [2].
There has been significant progress in the development of femtosecond fiber
lasers at ∼ 2 µm. Sub-500 fs pulses were realized in the first soliton Tm fiber
laser back in 1995 [3]. Pulse durations as short as ∼ 50 fs have been achieved in
dispersion-managed cavities, with pulse energy < 1 nJ [4, 5]. A cavity containing
a grating and telescope combination for the normal-dispersion segment generated
5.4-nJ and 216-fs pulses [6]. The pulse energy is the highest previously reported
for a femtosecond Tm fiber laser, but the modified grating system for dispersion
compensation compromises the practical benefits of fiber. A Ho-doped fiber laser
operating with net cavity dispersion near zero generated 160-fs pulses at ∼ 2.1 µm
[7].
1Much of the work presented in this chapter was published in Optics Letters [1].
47
Recent work has demonstrated the performance benefits of operating fiber
lasers at large normal dispersion, and in particular the highest pulse energies have
been obtained in the dissipative-soliton (DS) regime [8]. The DS and self-similar
(SS) pulse evolutions [9, 10, 11] allow the best combinations of high energy and
short pulse duration. Reports of normal-dispersion 2-µm fiber lasers are limited,
and the performance lags behind that achieved at shorter wavelengths. A steep-
sided spectrum with the characteristic cat-ear features, which is a signature of DS
evolution, has been observed in several reports [12, 13, 14]. However, in these
lasers the pulse energy is still well below that expected. An obvious challenge
to designing normal-dispersion lasers is the anomalous dispersion of passive and
standard Tm-doped silica fibers. Questions have also been raised about possi-
ble limitations that arise from gain-assisted modulation instability in lasers with
anomalous-dispersion gain media [15, 16]. Thus, there is substantial motivation to
investigate normal-dispersion Tm and Ho fiber lasers.
We report a study of a mode-locked Tm fiber laser with varying normal disper-
sion. It is difficult to reach the high-energy dissipative-soliton regime due to the
anomalous dispersion of most fibers at 2 µm. With large normal dispersion, the
laser exhibits elements of self-similar pulse evolution, and is the first Tm fiber laser
to achieve the performance benefits of normal-dispersion operation. The behavior
and performance with varying net cavity group delay dispersion (GDD) and filter
bandwidth are presented. Experimental results agree well with trends in numerical
simulations. The laser generates 7.6-nJ pulses, which can be dechirped to 130 fs
duration. The resulting peak power is 4 times higher than that of previous Tm
fiber lasers. This is a major improvement over the performance of previous Tm
fiber lasers. Factors that limit the performance, and routes to further performance
increases, will be discussed.
48
3.2 Laser design and numerical simulations
Tm-doped gain fibers typically exhibit large anomalous dispersion, which differs
from Yb-doped and some Er-doped gain fibers. Numerical simulations were per-
formed to assess the feasibility of reaching the DS regime in a laser with large
anomalous dispersion in the gain medium. As a practical matter, the use of some
anomalous-dispersion passive fibers (as component pigtails) is also unavoidable.
The simulations were based on the ring cavity of Figure 3.1 and the parameters
of the fibers are listed in Table 3.1. The fiber pigtails of the WDM and two col-
limators are all SMF28e, with a total length of 2 m. The gain fiber (TH512 from
CorActive) has a core diameter of 9 µm and NA of 0.16, and is modeled with 30
dB small-signal gain, 100-nm Gaussian gain bandwidth at the center wavelength
of 1920 nm, and saturation energy of 5 nJ. Due to the anomalous dispersion of
the gain fiber and SMF28e, the laser will have a dispersion map and we need to
consider its role in pulse propagation.
To shift the cavity GDD to the normal regime, ultra-high NA fibers (Nufern)
are employed [17, 18]. In these fibers the strong waveguide dispersion exceeds
the material dispersion. The mismatch of core sizes with other fibers can cause
self-lasing within the cavity due to reflections at the splice points. Among the ultra-
high NA fibers, UHNA4 has the lowest splice loss (0.4 dB) with SMF28e, and is
employed after the gain segment to avoid reflections that can disrupt mode-locking.
UHNA7 is designed for relatively low loss (∼ 20 dB/km) and has a splice loss of
0.7 dB with SMF28e, so it is used in front of the gain fiber for GDD control. The
combination of a long UHNA7 segment and a short UHNA4 segment (Figure 3.1)
yields net normal cavity GDD, with minimal loss and reflections. The splice loss
is negligible between other fibers and modeled using the same values between fiber
49
segments in simulations. The output coupling ratio and linear cavity loss are
both assumed to be 70%. A lumped saturable absorber with 100% modulation
depth and saturation power of 1 kW is assumed. A variety of filters were tried in
simulations and a 16-nm Gaussian filter is used to achieve the broadest spectra for
initial demonstrations.
BRP 
Lyot  
filter 
Output 
ISO 
SMF28e SMF28e 
PBS 
IF 
UHNA7 UHNA4 
Waveplates 
SMF28e Tm fiber 
SMF28e 
WDM 
pump 
 
Figure 3.1: Schematic of laser: BRP: birefringent plate; IF: interference filter; PBS:
polarizing beam splitter; ISO: isolator; WDM: wavelength-division multiplexer.
Table 3.1: Values of parameters used in Tm laser simulation
Fiber Aeff GVD TOD Length
(µm2) (fs2/mm) (fs3/mm) (m)
UHNA7 11 90 -33 4.2
UHNA4 23 93 154 1.2
Tm 78 -73 96 1
SMF28e 98 -71 104 2
Simulations produce stable pulses for cavity GDD up to ∼ 10 ps2 with the 16-
nm filter. An example of a typical converged solution is shown in Figure 3.2 with
GDD ∼ 0.3 ps2. Pulse propagation is strongly affected by the filter, with negligi-
ble contribution from gain filtering. The pulse duration and spectral bandwidth
50
breathe by a factor of 5 as the pulse traverses the cavity. The pulse broadens in the
normal-dispersion sections and compresses in the anomalous-dispersion sections,
and is positively-chirped and far from the transform limit throughout the cavity.
These features are qualitatively similar to the evolution in a passive self-similar
laser [9]. However, the strong spectral breathing contrasts with the evolution in
a passive self-similar laser; on the other hand, it is quite similar to that of lasers
with self-similar evolution in the gain segment [10, 11]. The misfit parameter M
is the root-mean-square deviation of the pulse shape from a parabolic pulse with
the same energy. We see that the pulse reshapes dramatically in the first segment
of UHNA fiber, and approximates a parabola well (M ∼ 0.02) after the second
UHNA segment until it is filtered. The maximum stable pulse energy is 15 nJ.
The cavity dispersion was varied by changing the length of UHNA7. Even with
very large dispersion, the dispersion map evidently hinders the formation of DSs.
The hybrid self-similar pulse evolution is a consequence of the joint action of the
strong dispersion map and the filter.
SMF UH7 Tm UH4 SMF SA SF SMF UH7 Tm UH4 SMF SA SF
Figure 3.2: (a) Simulated variation of pulse duration and spectral bandwidth along
the cavity (SA: saturable absorber, SF: spectral filter). The pigtail segment of
WDM is neglected for clarity; (b) pulse shape evolution compared with a parabolic
pulse; inset: comparison of pulse shape after UHNA4 (black) with Gaussian pulse
(blue) and parabolic pulse (red) in log scale.
51
3.3 Experiments
We constructed a laser as in Figure 3.1. Pump light at 1569 nm is provided by a
home-built erbium fiber laser and coupled into the cavity through a wavelength-
division multiplexer (WDM) with 2-by-2 ports. The pigtails on the WDM and
collimators amount to a total of 2 m of SMF28e. An interference filter with 16-nm
bandwidth and peak transmission of 90% was used initially. The wave plates and
PBS implement nonlinear polarization evolution (NPE), which also provides the
output port. The isolator ensures unidirectional operation of the laser. The output
power is measured after a long-pass filter that cuts on at 1650 nm, to eliminate
residual pump light. The lengths of UHNA fibers and the filter bandwidth were
varied to study the behavior of the laser. The resulting pulse repetition rate was
in the range of 20-30 MHz. Operation with a single pulse in the cavity was verified
with an autocorrelator with 150-ps scan range and a detector with 30-ps risetime.
The pulse width was obtained from the interferometric autocorrelation based on
2-photon photoconductivity of a silicon detector, and a deconvolution factor of
1.45 was assumed.
The influence of cavity GDD was studied by varying the length of UHNA7.
Initial lengths of 2.2 m of UHNA7 and 1.2 m of UHNA4 yield GDD ∼ 0.1 ps2. The
dispersion was gradually increased by adding UHNA7 before the gain fiber, without
changing other fiber segments. This laser can be mode-locked for dispersion as
large as 0.37 ps2. Mode-locking was usually initiated at high pump power through
fine adjustment of the wave plates. Typically, the laser started in a multi-pulsing
state, and a single-pulsing state was obtained by reducing the pump power.
The output pulse parameters are summarized in Figure 3.3. The pulse chirp,
dechirped and transform-limited pulse duration, and the pulse energy all increase
52
with cavity GDD, while the spectrum narrows. The experimental data and simu-
lation results exhibit similar trends. The output pulse chirp is less than the cavity
GDD, as is the case for self-similar lasers [11]. In contrast, for DS lasers without a
dispersion map, the pulse chirp is comparable to or larger than the cavity GDD.
Figure 3.3: (a, b) Comparison between experimental and simulation results
(squares: experimental data, triangles: simulation results); (c) mode-locked spec-
tra at various cavity GDD with the 16-nm filter (blue: 0.08 ps2, green: 0.17 ps2,
red: 0.26 ps2, black: 0.37 ps2); (d) mode-locked spectra with varying filter BW at
cavity GDD ∼ 0.28 ps2 (blue: 16 nm filter BW, black: 35 nm filter BW, red: 40
nm filter BW).
The length of UHNA4, where most of the NPE occurs, was varied to optimize
the performance. We used 4.2 m of UHNA7 and kept all other components fixed.
Best performance was obtained with 1.1 m of UHNA4. The laser emitted 7.6-nJ
and 3-ps chirped pulses with spectral bandwidth of 80 nm at the -10 dB level,
53
which could be dechirped to 130 fs (within 10% of the transform limit) via a
grating pair (Figure 3.4). The minor discrepancy from the transform-limited pulse
is probably due to nonlinear phase in the pulse and higher-order dispersion of the
grating pair. After dechirping with gratings with overall efficiency of 65% the
peak power is ∼ 40 kW, which is 4 times the peak power of previous Tm fiber
lasers. By minor adjustments of the cavity the pulse energy could be increased to
8.3 nJ, with 150-fs pulse duration. Finally, the radio-frequency spectrum confirms
stable mode-locking. The dynamic range of the measurement is limited by the
detector and spectrum analyzer, but for frequency offsets in the range of 100 kHz
(covering relaxation oscillations), any sidebands are more than 70 dB below the
first harmonic.
Figure 3.4: Experimental (a) output spectrum, (b) autocorrelation of dechirped
pulse (inset: autocorrelation of transform limited pulse), (c) radio-frequency spec-
trum (span range: 200 kHz, RBW: 1 kHz) corresponding to 7.6 nJ pulse energy.
54
3.4 Conclusion
Although the numerical simulations indicate that stable pulses should be obtained
at energies as high as 15 nJ, we observed experimentally that the pulse energy
was limited by multi-pulsing. A low multi-pulsing threshold is often a symptom
of too-narrow filter bandwidth in normal-dispersion lasers, so we investigated the
influence of filter bandwidth on the performance. A single birefringent filter was
first tried for larger bandwidth. However, mode-locking was hard to achieve, and
we suspect that this can be attributed to the multiple transmission bands of the
filter within the Tm gain spectrum. A Lyot filter (indicated in Figure 3.1) was used
to suppress the secondary transmission peaks. With available birefringent plates,
this combination can provide a filter bandwidth of 35 nm or 40 nm and free spectral
range of 140 or 160 nm respectively, which ensures a single-peaked pass-band within
the gain spectrum. The laser was mode-locked with very similar pulse energy and
bandwidth using several different filters (Figure 3.3d), with the performance still
limited by multi-pulsing. Information about the pulse evolution in the cavity can
be inferred from the pulse extracted from the unused port of the WDM. For the 35-
nm and 40-nm Lyot filters, the spectral bandwidth after passing through the WDM
is 23 nm and 28 nm, respectively. Thus, we tentatively conclude that the WDM
contributes to an overall filter in the cavity, and this limits the performance. In
simulations, pulse energies larger than 20 nJ are reached with a filter bandwidth of
30 nm. The pulse energy in the current cavity is limited by accumulated nonlinear
phase of ∼ 10 π before wave-breaking occurs in the simulations. Gain fibers with
lower anomalous dispersion, or even normal dispersion, are desirable to ultimately
increase the pulse energy by reaching the DS regime. DS-like spectra have been
produced by a laser with normal-dispersion gain fiber [14].
55
In conclusion, we have demonstrated a Tm fiber laser operating at large nor-
mal dispersion with hybrid self-similar evolution. Despite technical limitations in
realizing suitable intracavity filters, the design reported here achieves 7.6-nJ and
130-fs pulses. These are so far the shortest pulses with highest energy in this dis-
persion regime, and yield a 4-fold improvement in peak power over previous Tm
fiber lasers.
Portions of this work were supported by the National Science Foundation
(ECCS-1306035). I want to thank Erin Lamb and Logan Wright for helpful dis-
cussions.
56
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58
CHAPTER 4
GENERATION OF INTENSE 100-FS SOLITONS TUNABLE FROM
2 TO 4.3 µM IN FLUORIDE FIBER1
There is great interest in sources of coherent radiation in the mid-wave infrared
(3-5 µm), and instruments based on fiber can offer major practical advantages.
This range, and much broader, can be covered easily by supercontinuum genera-
tion in soft glass fibers, but with low power spectral density. For applications that
require intense ultrashort pulses, fiber sources are quite limited. In this letter, we
report a fiber-based system that generates 100-fs pulses with 5 nJ energy, contin-
uously wavelength-tunable over 2-4.3 µm through the soliton self-frequency shift
(SSFS) in fluoride fibers. The pulse energies are two orders of magnitude higher
than those previously achieved by SSFS around 3 µm, and the range of wavelengths
is extended by 1000 nm. Peak power ranges from 20 to 75 kW are achieved across
the tuning range. Numerical simulations are in good agreement with the experi-
mental results, and indicate the potential for few-cycle soliton generation out to
5.6 µm. Fiber-integrated sources of femtosecond pulses tunable across this region
should be valuable for mid-infrared applications.
4.1 Introduction
The mid-infrared (MIR) spectral region is of great importance to both fundamental
research and applications. Mid-infrared spectroscopy [2] offers unique capabilities
to detect and identify various molecules in different environments, including real-
time breath analysis [3], trace gas detection, and early cancer diagnostics [4]. The
3-5 µm region is a transparent window in the atmosphere, and is suitable for
1Much of the work presented in this chapter was published in Optica [1].
59
LIDAR and remote sensing. These technologies can be greatly improved through
the development of intense coherent MIR sources, which will also facilitate studies
in strong-field physics, high-harmonic generation and attosecond science [5].
New scientific discoveries and practical applications in the MIR region will re-
quire versatile light sources with high spectral density and continuous tunability
over a wide spectral range. Currently, MIR light sources are mainly based on solid-
state laser systems. Quantum cascade lasers [6] emit from 3 to 25 µm. They are
usually operated continuous-wave, and reach watt-level powers [7]. Demonstra-
tions of short pulse operation are rare [8]. Kerr-lens mode-locked Cr2+:ZnS lasers
cover a broad range around 2.4 µm, with impressive few-cycle pulses [9]. Dif-
ference frequency generation (DFG) using near-infrared lasers can provide pulsed
sources over a broad MIR range [10]. Microjoule-level pulses, tunable over 3-20
µm, were obtained from a solid-state optical parametric oscillator (OPO) pumped
by a Ti:sapphire amplifier system [11]. The output from the DFG process can
be boosted to higher pulse energy by optical parametric chirped pulse amplifica-
tion [12]. However, all these solid-state systems require bulky and complex optical
systems.
Fiber-based light sources compete with solid-state systems due to their com-
pactness, low cost, and maintenance-free operation. MIR supercontinuum span-
ning more than three octaves (1.4-14.3 µm) was generated in a chalcogenide
fiber [13]. The power spectral density is naturally low, and many applica-
tions would benefit from intense short pulses. Mode-locked erbium (Er) ZBLAN
(ZrF4BaF2LaF3AlF3NaF) fiber lasers have recently been demonstrated at 2.8 µm
but are not tunable [14, 15]. Fiber-based light sources employed DFG to gener-
ate femtosecond pulses with 10-pJ energy tunable over the MIR [16, 17]. MIR
60
frequency combs serve as increasingly-important tools for massively parallel spec-
troscopy with unprecedented speed, precision, sensitivity and spectral coverage
[18]. They have been realized by degenerate OPO systems with fiber-based seed
lasers and bulk crystals [19, 20], which motivates the development of fiber-based
high-energy MIR femtosecond pulses.
The soliton self-frequency shift (SSFS) in optical fibers provides high-quality
femtosecond pulses with wide spectral tunability [21, 22]. Recently, SSFS has been
explored to reach the MIR region [23]. Standard silica fibers are limited to about
2.5 µm, while germanium-doped silica fibers reach almost 3 µm. To obtain soliton
shifting further into the MIR, soft-glass fibers with low phonon energies are desir-
able. Koptev et al. achieved spectral coverage up to 2.65 µm in microstructured
tellurite fibers with 100-fs soliton pulses [24]. However, sub-50 pJ pulse energies
were obtained, limited by the high nonlinearity of the fiber. Impressive results with
soliton wavelength up to 3.42 µm were realized in a chalcogenide fiber pumped at
2.8 µm [25]. The picojoule-level energy was limited by the high nonlinearity of the
chalcogenide fiber and the low pump pulse energy.
Fluoride glass fibers have drawn attention due to their wide MIR transparency
window, relatively high strength, and low background loss [26]. Compared to
tellurite and chalcogenide fibers, they have much smaller nonlinear indices and
shorter zero-dispersion wavelengths (ZDWs). Based on these properties, 100-fs
solitons with roughly 100 times the energy (10-nJ level) or peak power of solitons
in tellurite or chalcogenide fibers are expected. Approximately 0.5-nJ pulses of
unspecified duration at 2.5 µm were obtained in an early experiment on SSFS in
fluoride fiber [27]. Recently, Salem et al. demonstrated supercontinuum spanning
1.8 octaves in this kind of fiber, pumped by a thulium (Tm) fiber amplifier which
61
was in turn seeded by a Raman-shifted erbium fiber laser [28]. In that work, a
dispersion-flattened fiber was pumped with 10-nJ and 100-fs pulses at 2.1 µm,
which is near the ZDW.
Here we demonstrate a fiber-based system that generates 100-fs, nanojoule-
energy soliton pulses via SSFS in fluoride fibers. Relative to experiments on con-
tinuum generation [28], we enhance Raman soliton formation by pumping the flu-
oride fiber in a region with strong anomalous dispersion, using shorter pulses with
about 15 times higher peak power. The generated Raman solitons are continuously
tunable from 2 to 4.3 µm. The pulse energy is a 2-order-of-magnitude improve-
ment over previous Raman solitons at wavelengths beyond 2.5 µm. Numerical
simulations of the soliton propagation are in good agreement with the experimen-
tal results, and indicate the potential for few-cycle soliton generation above 5 µm.
The work reported here demonstrates that SSFS can serve as an important source
of coherent ultrashort pulses with broad tunability for applications in the 3-5 µm
range.
4.2 Description of fibers
We employed two kinds of fluoride fibers for the study of SSFS (Thorlabs models
P3-32F-FC-2 and P3-23Z-FC-1). The indium fluoride (InF3) fiber has a core diam-
eter of 9 µm and numerical aperture of 0.26. The single-mode cutoff wavelength is
3.2 µm, and the propagation loss is specified as lower than 0.5 dB/m over 2.0-4.5
µm. The ZDW for the InF3 fiber is ∼ 1.71 µm. The zirconium fluoride (ZrF4)
fiber has a core diameter of 9 µm and NA of 0.19. The single-mode cutoff wave-
length is 2.3 µm, and the propagation loss is specified as lower than 0.2 dB/m over
62
2.0-3.7 µm. The zero-dispersion wavelength for the ZrF4 fiber is ∼ 1.62 µm. The
parameters of the fluoride fibers used in the experiments are plotted in Figure 4.1.
The InF3 fibers have benefits over the ZrF4 fibers in their capability to transmit
wavelengths beyond 5.4 µm [29]. The fibers are protected in patch cables, with
both ends terminated by angled ferrule connectors to reduce reflections.
InF ZrF
3 4
Figure 4.1: The parameters of the fluoride fibers used in the experiments. The
group velocity dispersion (GVD) of the fundamental mode and the propagation
loss for the (a) InF3 fiber; (b) ZrF4 fiber. The horizontal blue dashed line indicates
the position of the zero GVD value. The vertical blue dashed line indicates the
single mode cutoff wavelength.
To achieve a tunable light source with a spectrally-separated shifted soliton,
pumping deep in the anomalous dispersion regime of fiber with low nonlinearity
is desirable. In this way, a smaller number of fundamental solitons is excited, and
energy conversion to the most-redshifted soliton is maximized [22]. Our experi-
ments are based on pumping at 1.9 µm, which is the longest wavelength we can
reach with stable pulses.
4.3 Numerical simulations
Numerical simulations were performed to assess the feasibility of reaching the MIR
region through SSFS in these fibers.
63
Considering the single-mode cutoff wavelength of the fibers and our focus on
the 3-5 µm region, simulations were conducted assuming only fundamental-mode
propagation. We employed the generalized nonlinear Schrödinger equation, in-
cluding dispersion terms up to the 8th order, stimulated Raman scattering, self-
steepening, and propagation loss increasing exponentially with wavelength beyond
4 µm [30]. Due to the large variation of the effective area of the mode over this
wide wavelength range, the frequency-dependence of the effective area is accounted
for by introducing a first-order correction to the timescale τshock associated with
self-steepening and optical shock formation at the pump wavelength [31].
8 ( )
∂A α n+1 n∑
+ A− i ∂ A ∂βn = iγ 1 + iτshock
∂z 2 n! ∂T n ∂T
n≥2
( )
∫ ∞
′ × | ′ ′A(z, T ) R(T ) A(z, T − T )|2)dT + iΓR(z, T ) (4.1)
−∞
The equation is solved with the standard split-step technique, with a 4th order
Runge-Kutta integration for the nonlinear step. A temporal resolution of 0.44 fs
and spectral resolution of 8.6 GHz are used. The frequency range extends over
1200 THz. A time window of 120 ps and step size of 1 µm in propagation are
employed. The convergence of the simulation results is confirmed by comparing
with those obtained using higher resolution.
The βn terms up to the 8th order are the dispersion values for the fundamental
mode in the fluoride fibers, calculated based on the Sellmeier equations with their
refractive indices and numerical aperture, measured by the manufacturer. The
linear propagation loss α is set to be 0.2 dB/m for the wavelengths shorter than
the pump wavelength and increases exponentially into the deep MIR region. The
frequency dependence of the loss is modeled as:
64
f−f0
α(f) = a× e− b + c (4.2)
where f is the frequency in the unit of PHz and f0 is the frequency at pump
wavelength. The coefficients for the InF3 fiber at pump wavelength of 1.9 µm are
a = 1.13× 10−10 dB/m, b=0.00434 PHz, c=0.0997 dB/m. The coefficients for the
ZrF4 fiber at pump wavelength of 1.9 µm are a = 1.12× 10−11 dB/m, b=0.00328
PHz, c=0.0423 dB/m.
The nonlinear coefficient γ is evaluated at the pump frequency ω0 and the
generally accepted value of nonlinear refractive index is n2 = 2.1 × 10−20 m2/W
[26], which is used in the simulations. Due to the large variation of the effective
area of the mode over this wide wavelength range, the frequency-dependence of
the effective area is accounted for by introducing a first-order correction to the
timescale τshock at the pump wavelength. The variation of the nonlinear refractive
index is negligible.
[ ]
1 − 1 dAeff (ω)τshock = (4.3)
ω0 Aeff(ω) dω ω0
The response function R(t) includes both instantaneous electronic and delayed
Raman contributions. The Raman response is modeled as a simple damped oscil-
lation in the temporal domain with two time constants τ1=16.67 fs and τ2=20.32 fs
[32]. The fraction of Raman contribution in the nonlinearity is taken to be fR=0.20
[33]. The effects of spontaneous Raman noise are introduced through ΓR(z, T ).
Numerical results of launching 40-nJ and 70-fs sech2-shaped pulses at 1.9 µm
into the fundamental mode of a 2-m long InF3 fiber are shown in Figure 4.2. The
soliton number N for the input pulse is ∼ 11. The spectral and temporal evolution
65
1.0
(a)
0. RS
0.0
(b)
200 10-
10-
10-100
10-
0 10
-
1 2 4
avelength (µm)
1.0
(c)
0.5 RS
0.0
(d)
200 101
100
100
10-

10-0
0 10 20
Time (ps)
Figure 4.2: Simulation results for the (a) spectral output, (b) spectral evolution,
(c) temporal output and (d) temporal evolution in a 2-m long InF3 fiber with
40-nJ and 70-fs pulses coupled into the fundamental mode. RS: Raman soliton of
interest at the longest wavelength.
of the pulses along the fiber exhibit the main features expected for fiber pumped in
the anomalous dispersion regime with femtosecond pulses. Higher-order temporal
solitons are initially formed, which then undergo fission under the perturbation
of the higher-order dispersion and Raman scattering to break up into a series of
constituent fundamental solitons. These experience frequency shifts, shed energy
66
Distance (cm) Intensity (A.U.) Distance (cm) Intensit ( )
Energy density (nJ/nm) Power (kW)
into dispersive waves in the normal dispersion regime, and eventually separate
spectrally and temporally. The most-shifted soliton (labeled RS in Figure 4.2) is
at 4.3 µm and has 80-fs duration and 5-nJ energy.
4.4 Experiments
The experimental setup is shown in Figure 4.3. An Er fiber chirped pulse ampli-
fication system delivers 550-fs pulses with 1.3 µJ energy at 1.55 µm and 1.5 MHz
repetition rate. These are shifted to longer wavelengths by SSFS in a polarization-
maintaining rod-type photonic crystal fiber (PCF) [34] with a mode-field diameter
of 55 µm. The output pulses are collimated by a lens with a focal length of 50
mm, and a long-pass filter (LPF) that cuts on at 1.8 µm is used to select the most-
redshifted soliton centered at 1.9 µm, which has 120-nJ pulse energy and 70-fs
duration (Figure 4.4). The fringes on the spectrum are due to interference with
residual light left from a second soliton centered at 1.7 µm. The pulse duration is
measured using an interferometric autocorrelator based on two-photon absorption
in a silicon photodetector. A deconvolution factor of 1.54 is assumed.
A half-wave plate is used to suppress residual unshifted light and other shifted
solitons due to the polarization dependence of the blocking range of the LPF.
These pulses are coupled into the fluoride fiber using an aspheric lens and colli-
mated at the output using a protected silver reflective collimator with a 7 mm focal
length. They then pass through a LPF which cuts on at 2.85 µm or 3.75 µm for
selecting the most-redshifted soliton. Several Fourier-transform optical spectrum
analyzers (OSAs), each covering a different wavelength range, are required to mea-
sure the output spectra. The pulse duration is measured using an interferometric
67
autocorrelator based on two-photon absorption in an InGaAs photodetector.
LPF1 LPF2
erbium fiber PCF rod fluoride
diagnostics
amplifier fiber
lens lens lens
Figure 4.3: Schematic of the experimental setup for SSFS in fluoride fibers: LPF:
long pass filter. LPF1 cuts on at 1.8 µm. LPF2 cuts on at 2.85 µm or 3.75 µm.
Figure 4.4: (a) The spectra and (b) the interferometric autocorrelation of the input
light source at 1.9 µm.
Experiments were conducted with a 2-m InF3 fiber. A 6-mm focal length lens
was used to couple light into the fiber. By varying the incident pulse energy from 3
to 120 nJ using neutral density filters, we are able to continuously tune the center
wavelength of the output pulses from 2 to 4.3 µm. The fiber is specified to be
single-mode for wavelengths beyond 3.2 µm. The excitation of higher-order modes
at the pump wavelength does not prevent the generation of Raman solitons in
the fundamental mode. Any shifted energy in the higher-order modes experiences
high loss upon passing the cutoff wavelength. This ensures a single-mode output
beam after the LPF (for wavelengths beyond 3.2 µm). Soliton pulses with few-
nanojoule energies and ∼ 100-fs durations are obtained at each wavelength. The
representative examples shown in Figure 4.5 correspond to peak powers between
45 and 75 kW. The LPF introduces some chirp on the pulses, which leads to a 10%
deviation from the transform limit. Further shifting to longer wavelengths should
be possible with higher pump energy. However, at higher pulse energy we observe
68
in fluoride fibers
Figure 4.5: Measured spectra and the interferometric autocorrelations of the most-
redshifted solitons at (a,b) 3.2 µm; (c,d) 3.9 µm; (e,f) 4.3 µm in the InF3 fiber.
The red lines indicate the spectra after noise reduction. A deconvolution factor of
1.54 was assumed to obtain the indicated pulse durations.
damage to the end face connector.
It is worth mentioning that even with the excitation conditions chosen to en-
hance Raman-soliton formation, broad supercontinuua are generated, as expected
from the numerical results above. When the most-shifted soliton is near 4 µm,
a continuum that spans more than two octaves is produced, with the short-
wavelength edge ∼ 0.8 µm. The complete output spectra were recorded using
three optical spectrum analyzers (ANDO AQ6315E: 0.35-1.7 µm, noise floor of -70
dBm/nm; Thorlabs OSA203: 1.0-2.5 µm, noise floor of -70 dBm/nm; and Thor-
69
labs OSA206: 3.3-8.0 µm, noise floor of -45 dBm/nm). The measured full spectra
of the supercontinuum for a 2-m long InF3 fiber with shifted solitons at 3.9 µm
is shown in Figure 4.6. The spectra data measured by AQ6315E and OSA203 are
stitched together at 1.67 µm. The gap from 2.5 µm to 3.3 µm could not be covered
by available spectrum analyzers, since the instrument that covers 1-5.6 µm was not
available when we did the experiments in the 2.5-3.3 µm range.
Figure 4.6: Measured full spectra in a 2-m long InF3 fiber with shifted solitons at
3.9 µm. PSD: power spectral density.
We estimate that about 35% of the incident light is coupled into the fundamen-
tal mode. Over the range of wavelengths generated, 10-20% of the energy coupled
into the fundamental mode of the fiber is converted to the most-shifted soliton.
The overall practical energy conversion efficiency is therefore 3.5-7%.
The experimental results agree well with the numerical simulations that assume
the measured coupling efficiency, for both the soliton wavelengths and energies.
The comparison for the most redshifted solitons is shown in Figure 4.7. The
nonlinear coefficient of the fiber exhibits strong wavelength dependence because the
70
mode area of the fiber increases with wavelength. Incorporating this dependence
into the simulation is crucial to accurate modeling of the long-wavelength edges of
the spectra.
Experiment Simulation
(a) (b) Input: 3.5 nJ
80 nJ
4.8 nJ 60 nJ
5 nJ 40 nJ
4 nJ 31 nJ
3.6 nJ 20 nJ
2.7 nJ 10 nJ
1.9 nJ 5 nJ
0.8 nJ 1 nJ
2 3 4 5 2 3 4 5 6 7
Wavelength (µm) Wavelength (µm)
Figure 4.7: Spectra of the most redshifted soliton in the 2-m long InF3 fiber at
different wavelengths in (a) experiments; (b) numerical simulations. The dotted
lines show the range which could not be covered by available spectrum analyzers.
The soliton and input-pulse energies are indicated.
For wavelengths below 3.2 µm, the output is not specified to be single-mode.
For the InF3 fiber, the V number is 3.6 at 2 µm and there are 3 guided modes. The
resulting M2 ∼ 2 might be adequate for many applications. To obtain a strictly
single-mode beam below 3.2 µm, which is the cutoff wavelength of the InF3 fiber,
experiments were conducted using a 1-m long ZrF4 fiber. The ZrF4 fiber provides
similar pulse performance as the InF3 fiber for wavelengths around 3 µm, but it
extends the wavelength region with single-mode output down to 2.3 µm. Numerical
simulations of soliton self-frequency shift (SSFS) in a 1-m long ZrF4 fiber with the
same input pulse parameters in the paper exhibit similar results, with the soliton
shifting out to 3.4 µm. Experimentally, output solitons up to 3.4 µm are obtained
with estimated pulse duration of 70 fs, as predicted by the simulation. The use
of the two fluoride fibers allows the generation of ∼ 100-fs pulses with spectral
71
Intensity (A.U.)
coverage over 2-4.3 µm and diffraction-limited output beam for wavelengths above
2.3 µm.
4.5 Stability measurement
We performed a long-term (taken to be one hour) stability measurement of the
pump pulses, the seed pulses and the output soliton pulses in terms of the center
wavelengths and pulse powers (Figure 4.8).
400
(a) Z[\]^ _`00653 (b) Me	 


300 ab^ 0.00012 S	 002
acd[^ fg__
Se	 

00
300
200
200
100 100
0 0
0.0063 0.0066
power (arb ) wavelength (nm)
600
(c)  . (d) "#$%& '(')*+
0.00032
 ,/& 5*(''(
 1 ! 36 1216   , #&
300
200
200
100
0 0
0.003 0.005 0.006
Seed power (arb ) Seed wavelength (nm)
500
(e) 789:; <=>?@A@ HIJKL NO53.1
; <=<> A
BC D E PQ RTUUL NO
8; @ <@
BFG E PVW TXYTIL
300 300
200 200
100 100
0 0
0.2 0.3 0.5 3500 3600
Soliton power (arb ) Soliton wavelength (nm)
Figure 4.8: The stability measurement of the power and center wavelength varia-
tion of (a,b) the pump pulse; (c,d) the seed pulse; (e,f) the output soliton at 3.6
µm. SD: standard deviation.
72
For the pump pulses near 1550 nm, the coefficient of variation (standard de-
viation/mean) of the power is 1.8 %. The coefficient of variation of the pump
wavelength is 0.002 %. Comparing the seed and output solitons, the coefficients
of variation of the power are on the same levels (8 % vs. 9 %). The coefficients
of variation of the wavelengths for the output soliton (1%) is 3 times larger than
that of the seed pulses (0.31%), sharing the same order of magnitude.
We dont observe much amplification of the instability during the SSFS process
in the fluoride fibers. We attribute this to the short pulse duration (70 fs) we used
for pumping, since for short pulse excitation, the SSFS process is known to remain
coherent even during the propagation in the anomalous dispersion regime [22].
Based on these measurements, this light source based on SSFS has the potential
of being useful provided stable and short input pulses are available.
4.6 Discussion
It should be possible to achieve significantly better performance than the results
reported here, by optimizing the conditions of the SSFS. For example, simulations
show that launching 40 nJ in the fundamental mode of a 5-m long InF3 fiber
yields solitons that reach 5 µm. This would correspond to 120 nJ incident on
the fiber. In the experiments described above, only 40 nJ out of the total 120 nJ
pulse energy is effectively utilized. With better coupling design and the use of bare
fibers to avoid damage, it should be possible to couple 80 nJ in the fiber. With
that energy coupled into a 2-m long InF3 fiber, simulations predict shifting out to
5.6 µm (Figure 4.7). Shifting to even longer wavelengths will be limited by the
large anomalous dispersion and high loss of the InF3 fiber.
73
While the source described here is an important step towards a practical, high-
performance instrument, several improvements should be considered. Our ap-
proach uses cascaded SSFS stages, so it is reasonable to expect high noise due to
nonlinear noise amplification and the coupling of spectral, temporal and intensity
fluctuations in the SSFS process. Our measurements show that the long-term sta-
bility of the source is similar to that of the 1.9 µm source. This implies that a stable
MIR source can be obtained provided the pump source is stable. The short input
pulses used here ensure that noise has a minimal influence on the soliton fission
and spectral-shifting processes. Further optimization of the SSFS process for high-
power pulses, for shifts to greater wavelengths, and for better noise performance
should be possible. As is well known, the soliton frequency shift is inversely pro-
portional to the fourth power of the pulse duration [35]. A source of even shorter
pulses at 2 µm will improve the performance in all respects. More studies will be
needed to optimize the design for specific parameters, but there is considerable
room for improvement from our current results: in the same InF3 fiber used here,
50-fs, ∼ 100 nJ solitons at 5 µm are consistent with the soliton area theorem.
4.7 Conclusion
In summary, 100-fs pulses with nanojoule energies, wavelength-tunable from 2-4.3
µm, are generated by the SSFS in fluoride fibers. With peak powers around 50 kW,
these are the most-intense mid-infrared pulses generated by a fiber source to date.
A fiber-based, broadly-tunable source of short pulses should facilitate numerous
applications in the important 3-5 µm region.
74
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77
CHAPTER 5
HIGH ENERGY PULSES FROM ERBIUM ZBLAN FIBER LASERS
5.1 Introduction
There are strong interests in developing ultrafast fiber lasers working at increas-
ingly longer wavelengths in the mid-infrared region above 2 µm. The reasons lie
not only in the practical advantages of fiber-format laser sources, but also in the
fact that ultrafast laser sources with new color capabilities open up doors for nu-
merous applications. The molecular fingerprint region covering 2-13 µm enables
the study of molecular structures and the dynamics in chemical reactions, as well
as the precision molecular spectroscopy using frequency combs. Applications re-
quiring pulses with high peak power, such as strong field physics, high-harmonic
generation, also benefit from the development of ultrafast fiber lasers at these
wavelengths.
This range can be partially covered by quantum cascade lasers (QCL) with
mostly continuous-wave operations, but only rare demonstrations for picosecond
pulse generations [1, 2]. The gain recovery time of QCL is on the order of a
few picoseconds, while the cavity time is ∼ 40-60 ps. This fast gain dynamics
is the main obstacle for stable passive mode-locking and impedes the formation
of ultrashort pulses. Other solid-state light sources based on different parametric
processes provide outstanding performances at these new colors, however, they are
bulky and require complex optical systems [3].
Fiber lasers that employ erbium (Er) doped ZBLAN (ZrF4BaF2LaF3AlF3NaF)
fibers stand out for covering the range from 2.7 µm to 2.9 µm, which supports
78
generation of sub-100 fs pulses. Continuous-wave power as high as 30 W [4] and
slope efficiency up to 50% [5] have been obtained by researchers from Laval Uni-
versity and Macquarie University. Recently there has been a surge in studies and
demonstrations of femtosecond fiber lasers centered around 2.8 µm. Sub-500 fs
pulses with peak power of 6.4 kW were realized in a mode-locked Er ZBLAN soli-
ton fiber laser [6] by nonlinear polarization evolution (NPE). Almost around the
same time, a soliton laser with similar design delivering 207-fs pulses with 3.5 kW
was obtained by the group in Laval University [7]. With further optimization of the
cavity in terms of the output coupling ratio and the length of the gain fiber, they
were able to obtain 270-fs pulses with 23-kW peak power [8]. To avoid the influence
of the strong water absorbtion lines near 2.8 µm, soliton lasers with holmium as
the gain medium working at 2.9 µm were also demonstrated [9]. These initial re-
ports have already shown great results and people are continuing to explore better
performance for these fiber lasers at new colors.
It has become well-known wisdom that operating lasers at large normal dis-
persion allows the best combination of high energies and short pulse durations
employing pulse evolutions like dissipative soliton or self-similar pulse evolutions
[10, 11]. Up to now, demonstrated mode-locked fiber lasers at 2.8 µm are all soliton
lasers, with only anomalous dispersion gain fibers in the cavity. It is interesting
and attractive to see how the performance scales if we introduce dispersion man-
agement elements and shift the net cavity dispersion to the normal regime. New
pulse evolutions may as well be discovered along the road.
Here I investigate the pulse evolution and output performance of mode-locked
Er ZBLAN fiber lasers with net normal dispersion. As is similar to the case of
Tm doped fiber lasers [12], fiber-format dispersion compensation is used due to
79
its flexibility to tune and practical advantages. The behavior and performance
with varying net cavity group delay dispersions (GDD) and filter bandwidths are
presented. Some initial experimental results will be presented.
5.2 Laser design and numerical simulations
5.2.1 Chalcogenide fiber
Single mode Er-doped ZBLAN fibers have anomalous dispersion. To shift the
cavity dispersion to the normal regime, chalcogenide fibers are used as normal
dispersion fibers at 2.8 µm for dispersion compensation.
Chalcogenide glasses have low phonon energies, transmit light in the mid-
infrared region from 2-10 µm. They normally exhibit 100 ∼ 1000 times higher
nonlinear refractive indices compared with silica glasses. Chalcogenide fibers drawn
from these glasses serve as great waveguides at this wavelength range. Single mode
chalcogenide fiber used in this experiment is made from As2S3 glasses with a trans-
mission range of 1.5-6 µm and single-mode cutoff wavelength at 2.46 µm. This
fiber is IRF-S-6.5 from IRflex with a core/cladding size of 6.5 µm/125 µm and NA
of 0.28. The dispersion profile kindly provided by the fiber supplier is shown in
Figure 5.1.
Characterization of the nonlinear refractive index n2 in this As2S3 fiber was
conducted using peak power test and calibrated by the simulation of the spectral
broadening in the same fiber. Pulses with 250-fs durations at 1.92 µm were free-
space coupled into the 4.4-m long chalcogenide fiber, with both fiber ends angle
80
Figure 5.1: Group velocity dispersion of the chalcogenide fiber IRF-S-6.5.
Figure 5.2: (a) Measurement of n2 of the chalcogenide fiber IRF-S-6.5 using peak
power test; (b) Extrapolation of n2 at longer wavelengths using Miller’s formula.
cleaved. Output spectra from the other end of the fiber at different power level
were recorded, and their root-mean-squared bandwidths are calculated. In the
meanwhile, simulations were conducted to model the same experimental condi-
tions, assuming different levels of n2 compared to that of silica fibers. From the
comparison of spectral broadening results in Figure 5.2(a) assuming multiple levels
of n2, it can be inferred that n2 of the As2S3 fiber at 1920 nm is ∼ 90 times that
of silica fibers, with value of 207 ×10−20m2/W .
With the information of the material dispersion, values of n2 at other wave-
lengths can be extrapolated using Miller’s formula [13], as shown in Figure 5.2(b).
81
Considering that the band gap energy of As2S3 corresponds to the wavelength of
560 nm and n2 sharply decreases at multi-photon absorbtion wavelengths, the true
value of n2 in As2S3 fiber is expected to be less than 90 times that of silica fibers
at 2.8 µm. Further assessment can be done once a mode-locked fiber laser at 2.8
µm is available.
Another critical parameter of the chalcogenide fibers is the Raman coefficient.
Since the Raman peak gain scales with the third order nonlinear coefficient, a
roughly 100-times stronger Raman effect would be expected based on the high
nonlinearity of the material. The Raman coefficient in the simulations is calibrated
using the existing experimental data in [14]. The Raman gain coefficient at 1550
nm is verified to be 4.3 ×10−12 m/W. The Raman response function of the As2S3
fiber in simulations is modeled using the single Lorentzian Raman response model:
R(t) = fRhR(t) + (1− fR)δ(t− te) (5.1)
τ 2 + τ 2
h (t) = 1 2R exp(−t/τ2)sin(t/τ2 1) (5.2)τ1τ2
where values of τ1 = 15.5fs, τ2 = 230.5fs are used [15]. The contribution of
Raman part modeled by fR is 0.1.
5.2.2 Laser design and simulation setup
The schematic of the laser design with dispersion compensation is shown in Fig-
ure 5.3. Cavity is mainly formed by two segments of fibers: an Er-doped double-
cladding ZBLAN fiber, and a piece of chalcogeinde fiber (IRF-S-6.5). The Er-doped
82
output
pump
ISO
mirror
DM
PBS
BPF
waveplates
lens
lens
Er ZBLAN fiber chalcogenide fiber
Figure 5.3: Schematic of the Er ZBLAN laser with dispersion compensation: PBS:
polarizing beam splitter; ISO: isolator; BPF: bandpass filter; M: mirror; DM:
dichroic mirror.
ZBLAN fiber from Le Verre Fluoré has a core/cladding diameter of 15.4/260 µm
and NA of 0.12. It provides multimode pump guiding in the cladding with NA >
0.46 and ensures single mode operation above 2.5 µm. The pump absorption in the
cladding is 2.5∼3 dB/m at 976 nm. Splicing between the ZBLAN gain fiber and
chalcogenide fiber is non-trivial, so free-space coupling is used in between. Pump
light at 976 nm is provided by a multi-mode diode and free space coupled into the
gain fiber using a lens after passing through a dichroic mirror. Light coming out of
the chalcogenide fiber is collimated by another lens. The wave plates and polariz-
ing beam splitter (PBS) implement NPE for mode-locking, and PBS serves as the
output port. The isolator ensures unidirectional operation of the laser. A bandpass
filter is introduced for stabilization of the large normal dispersion operation.
Numerical simulations were conducted based on the laser design in Figure 5.3
for different fiber lengths and filter bandwidths. The parameters for fibers are
listed in the Table 5.1. The gain fiber is modeled with 30 dB small-signal gain,
100-nm Gaussian gain bandwidth centered at 2.8 µm. The free-space coupling
83
ratio of the signal light into the chalcogenide fiber and the gain fiber are 60% and
65% respectively. The output coupling ratio and cavity loss are both assumed
to be 70%. A lumped saturable absorber with 100% modulation depth and a
saturation power of 4 kW is used. The transmission curve of the SA is ideal and
monotonically increasing.
Table 5.1: Values of parameters used in the Er ZBLAN laser simulations
Fiber Aeff GVD TOD n2
(µm2) (fs2/mm) (fs3/mm) (10−20m2/W )
Er ZBLAN 224 -86 0 2.3
As2S3 fiber 45 208 -335 230
5.2.3 Simulation results for dispersion-managed soliton
lasers
First, we simulate the cavity with 1.2-m long chalcogenide fiber and 3-m long Er
ZBLAN fiber, resulting a net cavity dispersion of -0.0084 ps2. The gain saturation
energy is set to 0.45 nJ, which controls the pump power level. No filter is used in
this cavity.
A typical converged solution is shown in Figure 5.4. The spectral bandwidth
variation along the cavity is minimal, while the pulse duration varies significantly
between the normal and anomalous dispersion fibers. The pulses are up-chirped
and down-chirped at the end of the chalcogenide fiber and the gain fiber respec-
tively. Output pulses are slightly down-chirped with 220-fs duration close to the
transform limited. The pulse energy is limited to 0.5 nJ with accumulated nonlin-
ear phase of 0.5 π. Further increase in pump power leads to pulse breaking. The
above features are consistent with those of a dispersion-managed soliton laser.
84
Figure 5.4: (a) Simulated variation of pulse duration and spectral bandwidth with
net cavity dispersion of -0.0084 ps2 (Page 22: spectral filter, Page 23: output
port); (b) comparison between the output pulse and transform-limit pulse; (c)
output spectrum with FWHM bandwidth of 60 nm.
Introducing more chalcogenide fiber can shift the cavity dispersion from slightly
anomalous to the normal side. With 1.3-m long As2S3 fiber, the cavity dispersion is
now 0.012 ps2 and has converged solutions shown in Figure 5.5. The pulse evolution
changes drastically from the previous one. Pulse shaping is defined by the strong
dispersion map and the 100-nm wide gain filter together. Pulses get stretched in the
As2S3 fiber and compressed in the Er ZBLAN gain fiber with a breathing ratio over
10. Pulses are always up-chirped throughout the whole cavity. However, in contrast
with the nearly static spectral evolution in the previous design, significant spectral
broadening happens in the As2S3 fiber, and then the bandwidth shrinks down to
∼ 100-nm bandwidth, with a breathing ratio of ∼ 3. The spectral shrinking in
the gain segment is due to the intrinsic gain narrowing effect. The output pulses
can be de-chirped to 120 fs, within 15% of the transform limited duration. The
maximum pulse energy is limited to 3.6 nJ with an accumulated nonlinear phase
∼ 3 π.
85
Figure 5.5: (a) Simulated variation of pulse duration and spectral bandwidth with
net cavity dispersion of 0.012 ps2 (Page 22: spectral filter, Page 23: output port);
(b) output chirped pulse with duration of 177 fs; (c) comparison between the
de-chirped pulse and transform-limit pulse; (d) output spectrum with FWHM
bandwidth of 137 nm, corresponding to 106 fs.
This is a new kind of pulse evolution, different from all the previous evolutions.
It is the result of the interplay of the strong dispersion map, the pulse shaping in
the highly nonlinear passive fiber and the filter effect in the gain fiber. Pulses are
first shaped towards the asymptotic solutions of the NLSE with nonlinearity and
normal GVD, and become parabolic and get compressed in the gain fiber. A large
breathing ratio makes it different from the static dissipative solitons [16]. The gain
filter contributes to returning the pulse spectra back to the original state after a
round trip, which is different from the passive self-similar evolution [17]. Amplifier
similariton cannot exist in such a cavity with anomalous dispersion gain fiber. But
one thing to keep in mind is that the shape of the gain spectra is Gaussian in the
simulation, which is hardly the case in real fibers. These approximations, to some
extent, can help us understand the potential pulse evolutions.
86
5.2.4 Simulation results for hybrid self-similar lasers
We further move on to study the lasers with larger normal cavity dispersion and
introduce filters to stabilize the highly chirped pulses. Simulations converge for
cavity GDD up to 1 ps2.
Here we show an example of a typical converged solution with net cavity dis-
persion ∼ 0.1 ps2 in Figure 5.6. The fiber segment consists of a 1.2-m long As2S3
fiber and a 1.6-m long Er ZBLAN gain fiber. A 30-nm Gaussian filter with peak
transmission of 70% is used to achieve the broadest spectra for initial demonstra-
tions. The gain saturation energy of 240 nJ is employed. The maximum output
pulse energy can reach 188 nJ with transform limited duration of 120 fs. Chirped
pulses can be de-chirped within 15% of the transform limited duration, indicating
the linear chirp. Nonlinear phase accumulation is 35 π at this energy level.
It is shown in Figure 5.6(a) that the pulse evolution is strongly defined by the
dispersion map and the filter effects, resulting from both the gain bandwidth and
the introduced Gaussian filter. The breathing ratio is around 10 in both spectral
and temporal domains as the pulse traverses the cavity, which is distinct from
dissipative soliton or self-similar lasers. The pulses broaden significantly in the
normal-dispersion parts and compress in the anomalous-dispersion sections, and
are positively-chirped and far from the transform limited in the cavity.
These features are similar to those of the Tm laser at 2 µm [12]. Both of
the lasers consist of the highly nonlinear fibers with normal dispersion, where the
pulse is passively shaped towards a parabolic shape, as shown in Figure 5.6(c).
On the other hand, there also remains important differences between them. The
misfit parameter, which is the root-mean-square deviation of the pulse shape from
87
Figure 5.6: (a) Simulated pulse duration and spectral bandwidth with net cavity
dispersion of 0.1 ps2 (Page 22: spectral filter, Page 23: output port); (b) compar-
ison between the de-chirped pulse and transform-limit pulse; (c) misfit parameter
evolution compared with a parabolic pulse; (d) the output spectrum with FWHM
bandwidth of 110 nm and pulse energy of 188 nJ.
a parabolic pulse, drops close to 0.08 in the chalcogenide fiber, and then deviates
quickly from parabola when the pulse enters the anomalous gain fiber. While in
the Tm laser, the pulses can maintain the parabolic shape in the gain fiber due to
its minimal contributions to the pulse shaping. In the Tm laser, the spectra gets
back to the original point mainly due to the filter introduced. However, in this
laser, the spectra shrinking is the result of the joint actions of the gain narrowing
and the filter.
Compared with the pulse evolution in the cavity with GDD of 0.012 ps2 in
Figure 5.5, this laser shares some common features. The main difference of design
lies in the introduction of the extra narrow filter, which can accommodate much
higher intra-cavity energy. Further increase in the length of the chalcogenide fiber
pushes the cavity dispersion to a even larger value. Similar to the concept of
giant chirp oscillator [18], the pulse energy is scaled almost linearly with the cavity
88
dispersion. But the advantages of this laser is that the chirped pulses can always
be dechirped within 15% of the transform limited duration of 120 fs, which is
ultimately determined by the Er gain bandwidth at 2.8 µm. This is not true for
giant chirp oscillators.
5.3 Experiments
As a starting point to test the laser components and gain some understanding of
the fiber parameters, a soliton Er ZBLAN laser similar to that in [7] was built.
Then the chalcogenide fiber will be introduced in the cavity to change the cavity
dispersion to the normal regime. Since the light is free-space coupled into the
fibers in the cavity, the first step is to angle cleave the gain fiber to eliminate back
reflections.
5.3.1 Fiber cleaving
Both ends of the gain fiber are angle cleaved at 8◦ to avoid the parasitic lasing,
which can be detrimental to mode-locking. Some examples of bad and good angle
cleaves are shown in Figure 5.7. Due to different material properties, a fiber cleaver
with the capability of tuning the cleaving parameters is needed for Er ZBLAN
fibers. Vytran cleaver (LDC-400) is employed for this task in this thesis. With too
much tension applied along the fiber, the fiber end surface shows tension striations,
which cause damage when the fiber end is pumped with high power. Tension needs
to be set much lower compared to that used with silica fiber and the scribe delay
should be set to a high value to allow time for the scribe to propagate. The tension
89
Figure 5.7: The end view, front view and back view of the bad angle cleave (top
row) and good angle cleaves (bottom row).
used for Er ZBLAN fiber is 270 g and scribe delay is set to 1000 ms. Under these
settings, angle cleaves in the bottom row of Figure 5.7 can be obtained consistently,
and exhibit high threshold for self-lasing issues.
5.3.2 Soliton laser
We constructed the laser following the schematic in Figure 5.8 and the picture of
the setup is in Figure 5.9. Pump light at 976 nm is first collimated by a B-coated
lens (Thorlabs AC127-019-B-ML) and then focused into the Er ZBLAN fiber using
lens with focal length of 15 mm (Thorlabs LB5766). This CaF2 lens is uncoated to
ensure the high transmission (95%) at both the pump wavelength and the signal
wavelength (2.8 µm). The output power is measured after a long-pass filter that
cuts on at 1650 nm, to eliminate unabsorbed pump light. Spectral measurements
90
DM
976 nm
Er ZBLAN fiber MM pump
output
M
M
PBS
M waveplates
Figure 5.8: Schematic of the Er ZBLAN soliton laser: PBS: polarizing beam split-
ter; ISO: isolator; BPF: bandpass filter; M: mirror; DM: dichroic mirror.
Pump
Output
Laser light
Figure 5.9: Picture of the Er ZBLAN soliton laser.Line with red color indicates
the pump beam and the green one is the signal light.
were made using a Fourier-transform optical spectrum analyzer (OSA205), and
temporal measurements were made using a mercury cadmium telluride (MCT)
detector and electronic amplifier with a bandwidth of 1 GHz.
Slope efficiency of 17% is achieved with CW power of 400 mW obtained under
3-W incident pump power, shown in Figure 5.10. To operate the laser with CW
power level of 10∼30 W, end cap protection is then required, which is done by
splicing a short piece of AlF3 fiber to both gain fiber ends.
91
Figure 5.10: Slope efficiency of the Er ZBLAN soliton laser.
Currently the laser is prone to Q-switching due to large losses in the cavity,
including the coupling loss and the loss at the isolator. Due to the mismatch of
beam sizes at the entrance port of the isolator, a 50% loss is introduced. Further
optimization of the efficiency is desirable to minimize the losses and reach mode-
locking states of this laser.
5.4 Conclusion
In conclusion, the performance and behavior of the Er-doped ZBLAN fiber lasers
are studied at net normal dispersion regime in simulations, including both the near-
zero dispersion case and large normal dispersion case. A new kind of pulse evolution
distinct from the previous evolutions is presented. It shares some common features
of the passive self-similar lasers, but not the same. This pulse evolution can push
the laser to operate at very large normal dispersion regime, like a GCO laser,
but without loss of the capability to dechirp the output pulses within 15% of the
transform limited duration. With promising simulation results of 188-nJ pulse
energy and 140-fs de-chirped durations shown in this chapter, this hybrid self-
92
similar evolution awaits to be studied with further experimental evidence.
I want to thank Simon Duval and Vincent Fortin from Laval University for
their helpful guidance on fiber cleaving. I also want to thank Prof. Liejia Qian
from Shanghai Jiao Tong University for providing the dichroic mirror used in the
experiments.
93
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[2] C. Y. Wang, L. Kuznetsova, V. M. Gkortsas, L. Diehl, F. X. Kärtner, M. a.
Belkin, A. Belyanin, X. Li, D. Ham, H. Schneider, P. Grant, C. Y. Song, S.
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95
CHAPTER 6
FUTURE DIRECTIONS
6.1 Thulium fiber lasers with all normal-dispersion cavity
6.1.1 Modulation instability in Tm fiber laser
The operation of fiber lasers in the normal dispersion regime is well-known to
produce high-energy short pulses with dissipative soliton or self-similar evolutions.
This is one of the motivations to develop high energy fiber lasers at 2 µm, where the
benefits of normal-dispersion operation hasn’t been fully exploited. In the previous
chapter, a Tm fiber laser operating at large normal dispersion with hybrid self-
similar evolution has been demonstrated, with a 4-fold improvement in peak power.
But these performances are still not even close to the full potential of generating
femtosecond pulses in Tm fiber lasers.
As a follow-up study of the Tm fiber laser in Chapter 3 to further improve the
performance, efforts have been devoted to understand the role of the anomalous
dispersion in both the Tm gain fiber and the passive fibers. It’s already shown
that the modulation instability (MI) in optical fiber amplifiers and fiber lasers
with anomalous dispersion leads to continuous wave radiation breakup [1]. This is
a detrimental effect limiting the performance with the growth rate of the perturba-
tions in the anomalous amplifiers being superexpressive. The anomalous dispersion
significantly lowers the threshold for pulse break-up compared to those where the
gain dispersion is normal.
Even when the gain dispersion is normal, but with anomalous-dispersion seg-
96
ment after the gain, the performance of the Tm laser is still greatly limited. In
simulations, it shows a clear trend that with shorter and shorter passive fiber
(SMF28e) after the Tm gain fiber, the output pulse energy can be increased sig-
nificantly. Pulses with 100 nJ and transform limited duration of 100 fs can be
obtained solely by removing the passive fiber after the gain.
Some initial experiments were done to test this finding by cutting down the
SMF28e after the gain from 40 cm to 30 cm long. However, similar results were
obtained compared to the previous ones. Further cutting down was limited by
the practical issues, such as the minimum fiber pigtail length required by the
collimator. The nonlinear length in SMF28e for the pulses right after the gain
fiber (4-ps, 12-nJ) is ∼ 20 cm, which is at the same level where the fiber was
cut down to. So this initial test couldn’t help to prove the simulation results. In
the future, a customized collimator with normal dispersion fiber pigtail, such as
UNHA4 fiber, can be implemented to eliminate the anomalous passive fiber and
test this concern.
Another signature of MI happening inside the cavity is based on the fringes
sitting on the mode-locked spectrum in Figure 6.1. It is studied in [2] that the
positions of the adjacent fringes on the spectrum contains information of the net
cavity dispersion.
These fringes are different from the Kelly sidebands [3] we would observe in
a typical soliton laser. They have different mechanism and these sidebands can
cause long pedestals in the time domain. The relationship between the positions
of the sidebands and the net cavity dispersion can be modeled through Eq. 6.1
and Eq. 6.2, where fc is the center frequency in THz and fk is the k-th sideband
frequency from the center, and their distance is defined as ∆fk.
97
Figure 6.1: (a) Mode-locked spectrum from the Tm fiber laser in Figure 3.1; (b)
Zoomed-in spectrum in THz unit with multiple peaks indicated.
∆fk = fk − fc (6.1)
2π2β2L(∆f
2 2
k+1 −∆fk ) = 2π (6.2)
From the relationship above, the net cavity dispersion of the laser in Figure 3.1
is calculated to be GDD = β2L = 0.18ps
2. Compared with the dispersion value
(0.34 ps2) calculated using the simulation parameters, this number is only half of
that. But it’s close to the pulse chirp of 0.22 ps2 inferred from the de-chirping
process. So this method may provide some useful information to understand the
pulse evolution, since the comparable pulse chirp with the cavity GDD is a feature
of self-similar evolution.
6.1.2 Dissipative soliton Tm fiber laser
It’s desirable to set up the Tm laser with all normal-dispersion fibers in the cavity.
Due to the intrinsic anomalous material dispersion of silica fiber at 2 µm, the
98
output
QWP QWP
ISO PBS
UHNA BPF HWP UHNA
Fiber 50 nm Fiber
3 m 1 m
Tm gain fiber
30 cm
Figure 6.2: The schematic of the all normal-dispersion Tm laser design with filter
bandwidth of 50 nm.
way to obtain normal dispersion fibers is either through the small core design
of the fiber or doping the ions in different glass materials, such as the silicate
glass. Experimentally, a 30-cm long Tm gain fiber with normal dispersion was
demonstrated with a gain per unit length of 2 dB/cm [4]. Simulations of cavities
employing this 30-cm long gain fiber were conducted to assess the advantages. The
laser design is shown in Figure 6.2 with 3-m and 1-m long passive fiber (UHNA4)
on two sides of the gain fiber. The parameters used in the simulation are listed in
Table 6.1. A 50-nm bandwidth bandpass filter is used in the cavity.
Table 6.1: Values of fiber parameters used in the Tm ANDi laser
Fiber Aeff GVD TOD Length
(µm2) (fs2/mm) (fs3/mm) (m)
UHNA4 23 93 154 4
Tm 21 17 0 0.3
An example of a typical converged solution is shown in Figure 6.3 with GDD ∼
0.28 ps2. The features of the characteristic steep edges and bat-man ear spectra in-
dicate the operation regime of the laser to be dissipative soliton [5]. The maximum
pulse energy can reach 280 nJ with output spectra to support 70-fs pulses. The
pulses can be de-chirped using only second order dispersion compensation to 90
99
Figure 6.3: (a) Simulated output chirped pulse; (b) output spectrum with 150-nm
bandwidth and 280-nJ pulse energy; (c) spectral and temporal evolution along the
cavity; (d) comparison between the de-chirped pulse (90 fs) and the transform-limit
pulse (70 fs).
fs. These results are way beyond the performance of the Tm lasers demonstrated
with anomalous gain fiber and passive fibers inside the cavity.
6.1.3 Amplifier similariton Tm fiber laser
Another benefit of having normal dispersion Tm gain fiber is to enable the design
the amplifier similariton laser near 2 µm. The wisdom here is to employ lower
doped and longer gain fiber, and the cavity is stabilized by a narrow filter [6].
The laser design is similar to that in Figure 6.2, but with a 2-m long gain fiber
and 30-cm, 20-cm long passive fiber (UHNA3) on two sides of the gain fiber. The
parameters are listed in the Table 6.2. A 20-nm bandwidth bandpass filter is used
100
Figure 6.4: (a) Evolution of the misfit parameter comparing the pulse profile to a
parabola; (b) output spectrum with 87-nm bandwidth; (c) spectral and temporal
evolution inside the cavity; (d) comparison between the de-chirped pulse (97 fs)
and the transform-limit pulse.
in the cavity.
Table 6.2: Values of fiber parameters used in the Tm amplifier similariton laser
Fiber Aeff GVD TOD Length
(µm2) (fs2/mm) (fs3/mm) (m)
UHNA3 39 30.5 154 0.5
Tm 21 17 0 2
The converged solution with GDD ∼ 0.05 ps2 is shown in Figure 6.4. Fig-
ure 6.4(a) describes the evolution of the misfit parameter M , which is the root-
mean-square deviation of the pulse shape from a parabolic pulse with the same
energy. With M being close to 0, the pulse shape is more like a parabola. We can
see that the pulses start with Gaussian shape (M ∼ 0.14) propagate in the first
segment of passive fiber without too much changes in shapes, then get attracted to
101
the asymptotic solution in the gain fiber and obtain the parabolic shape. In Fig-
ure 6.4(c), the pulse duration and spectral bandwidth breathe by a factor of 4 and 8
respectively as the pulse traverses the cavity. The pulses can be de-chirped almost
as same as the transform limited pulses, due to the linear chirp. The amount of
pulse chirp inferred from the de-chirping process is ∼ 0.038 ps2, lower than the net
cavity dispersion, which is also a feature of amplifier similariton evolution. Further
improvement in the laser design is needed to optimize the laser performance. This
laser can also serve as the basis for obtaining high-energy short pulse generation
using the idea of extended self-similar evolution.
6.2 Divided-pulse Tm laser
It’s been demonstrated by Erin Lamb and coworkers in [7] that the soliton energy
can be increased by an order of magnitude employing the idea of divided pulse
amplification in a fiber laser. Other interesting pulse evolutions, especially those
in the normal dispersion regime, have also been demonstrated in her thesis. It
would be very interesting to see how this idea can be applied in Tm fiber lasers.
Theoretically, this idea can be applied to lasers at any wavelengths, as long as
the pulse dividing and combining can be done in the right way, which motives my
initial work towards this direction.
The working principle of this laser is to divide the input linearly-polarized
pulses using the birefringent crystals into multiple copies of subpulses with equal
energies, but perpendicular polarizations. After being amplified in the gain fiber,
these pulses are recombined together to form a single pulse with high intensity.
In this way, the soliton area theorem can be broken through and nonlinear phase
102
Output 
BRF 
lens QWP PBS HWP 15 T PBS 
F.R. Tm   dividers/ ISO 
SESAM M 
combiners 
M HWP M 
Figure 6.5: Schematic for the divided-pulse Tm fiber laser: BRP: birefringent plate;
PBS: polarizing beam splitter; ISO: isolator; F.R: Faraday Rotator; M: mirror.
accumulation can be reduced by 16 times, provided 4 crystals are used.
The cavity design in Figure 6.5 for the divided-pulse Tm fiber laser is similar
to that in [7]. The divider/combiner part in the cavity is formed with the same set
of Y V O4 crystals. After the pulses pass through the Faraday rotator twice, the
polarizations of the pulses are rotated 90 degree and they will be recombined on
their way back.
As a first step, the cavity is built up without the divider/combiner inside the
cavity. With no dispersion compensation, this laser has a cavity dispersion of
-0.4 ps2 and should mode-lock as a soliton laser. The birefringent plate with
thickness of 15 T serves as a 30 nm bandwidth filter to facilitate the mode-locking
process, since the Tm laser has a broad bandwidth and have multiple lasing peaks.
After aligning the laser without any self-lasing issue, the mode-locking can self
start once the pump power is turned up. The mode-locked spectrum is shown in
Figure 6.6(a). Distinctions from a regular soliton behavior are reduced spectrum
bandwidth, suppressed large-width soliton sidebands in Figure 6.6(b).
An important feature of the autocorrelation trace is that it always contains
103
Figure 6.6: (a) Modelocked spectrum; (b) autocorrelation of the Tm soliton laser
in divided-pulse configuration.
a remarkable pedestal which cannot be eliminated. Efforts have been made to
slowly decrease the pump power with different NPE settings, but the background
is consistently happening. From a paper on the impact of gain fiber dispersion on
the stability of soliton bound states [8, 9], this pedestal is the result of averaging
several irregular bunching solitons. This happens mainly due to the combination of
the slow response from SESAM and the anomalous dispersion from the gain fiber.
The slow response component of SESAM (∼ 10 ps) provides the attraction force
for the soliton bunching. But if a fast SA were used in the same cavity, bunching
is not observed for all pump power level. So in the case of anomalous Tm gain
fiber, NPE is the ideal option as SA to avoid this bunching effect.
To assess the idea of adding NPE segment into the cavity as a fast saturable
absorber, the laser is redesigned with an extra NPE-arm formed by a 7-m long
SMF28e fiber. The mode-locking is achieved by the combination of the SESAM
and NPE in this case. It is very clear from Figure 6.8 that the spectra is now a
typical soliton spectrum with sharp peaks at the position of the Kelly sidebands.
And the autocorrelation trace indicates there’re just discrete pulses without any
bunching effect. Single-pulsing state is hard to obtain with decreasing pump power.
104
Output 
BPF 
lens PBS HWP 16nm 
M 
F.R. Tm ISO PBS 
SESAM 
M SMF 
QWP 
Figure 6.7: Schematic for the divided-pulse Tm fiber laser with NPE for mode-
locking.
Figure 6.8: Mode-locked spectrum (a) and autocorrelation (b) of the Tm soliton
laser in divided-pulse configuration, without SESAM.
The continuous-wave lasing threshold is always reached first before the level for
the single pulse is reached. More efforts are needed to optimize the cavity to solve
this issue.
If the Tm gain fiber with normal dispersion is available, the NPE segment
is not necessary. The discrete pulsing states are still achievable even with slow
SESAMs. Further pursing the divided-pulse Tm laser is stopped at this stage, and
more work needs to be done to redesign the cavity dispersion and obtain the single
pulsing state. In this sense, the normal dispersion gain fiber is relevant, since it’s
105
the solution to avoid soliton bunching, provided fast SESAMs are not available.
6.3 Modelocked praseodymium fiber laser at 5 µm and its
application
Mid-infrared region has intrigued lots of interest in optical communities to develop
new kinds of ultrafast light sources for various applications. These can be focused
on generating femtosecond or picosecond pulses with new colors at different energy
levels. One of the important applications of mid-infrared lasers is supercontinuum
generation for molecular detection and identification. This leads us to think about
a more important question: what kind of fiber lasers are the most helpful for an
all-fiber version mid-infrared supercontinuum light source?
As is well known, to generate a flat and broad supercontinuum, it’s optimal
to launch the input pulses near the fiber’s zero dispersion wavelength (ZDW) at
the anomalous dispersion side, making use of the soliton fission and Raman effect
[10]. Among the mid-infrared materials with low loss over 3-12 µm region are
chalcogenide, silicon, silicon nitride, which all share the ZDW in the range of 4 ∼
5 µm. Aside from the advantages of fiber lasers in terms of compactness, minimal
maintenance, no requirement for water cooling, it will have huge impact if a pulsed
fiber laser operating at 5 µm is available. This is the key to realize an all-fiber
mid-IR supercontinuum light source.
People have already been thinking along this direction to build fiber lasers
at increasingly longer wavelengths, such as 2 µm [11, 12, 13] and 2.8 µm [14, 15].
Based on the study of pulse evolutions in modelocked 1 µm fiber lasers and the fact
106
that the nonlinear effect scales inversely with the third power of the wavelength,
it is advantageous and natural to develop fiber lasers with much higher energies at
longer wavelengths.
With fiber lasers developed at a few mid-infrared wavelength points, the most
direct way to obtain other colors is through soliton self-frequency shift (SSFS) [16].
Recently, the mW-level average power of the shifted soliton has been improved up
to watt level [17], covering 2.8 ∼ 3.6 µm. These results have shown some promising
aspects of an all-fiber version source, but still need more wavelength shifting to
reach broader region. A mode-locked fiber laser emitting femtosecond pulses near
5 µm would be a great replacement of the first stage of SSFS process, reducing the
intensity fluctuation and noises.
One of the appropriate host materials for the mid-infrared fiber lasers is chalco-
genide glass. Doping chalcogenide fibers with Pr3+ ions can have emission peaks
centered at 5 µm, when pumped at 2.04 µm [18, 19]. The gain bandwidth at 5 µm
is around 600 nm wide in Figure 6.9 [19], which supports pulse duration of sub-50
fs.
Although there have been a few theoretical and computational studies of Pr3+
doped chalcogenide fiber emission at 5 µm [15], no mid-infrared fiber lasers have
been demonstrated over 4 µm wavelength for two main reasons. First, the fabri-
cation of these fibers can be challenging [20, 21] and a fast growth in developing
fiber lasers at 5 µm wouldn’t be easy until all the components are well established.
Another reason is that both the upper and lower laser manifolds possess long life-
times compared to that of the ground state [21] and resonant pumping by quantum
cascade laser may be needed. Despite the difficulties currently we have in working
with Pr-doped fibers, the idea of developing such a source at 5 µm is still valuable
107
Figure 6.9: Broadband emission in the 3∼5 µm wavelength region from a diode-
pumped Pr3+ doped chalcogenide fiber source.
and should be kept in mind, in case the components are available in the future.
If we come back to the discussion about the light sources we want, intense
ultrashort pulses at 2.5-6 µm can be used for multi-photon MIR spectroscopy. This
is in line with the work on developing high-energy fiber lasers at 2.8 µm and making
use of the SSFS to get access to other color regions. Supercontinnum generation
at watt-level average power covering up to 13 µm in an all-fiber format would be
ideal for molecular detection, breath analysis and environmental monitoring, etc.
108
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M. Bernier, M. Piché, and R. Vallée, Opt. Lett. 41, 5294 (2016).
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110
APPENDIX A
APPENDIX A: FIBER PARAMETERS
A.1 Mathematica code for calculating the fiber parameters
This thesis involves simulations of single-mode fiber lasers using relevant fiber
parameters, such as dispersion and effective area. It’s helpful to show how to
implement these calculations using Mathematica. This section provides the code
used to calculate the parameters for single mode fibers.
Usually we obtain the fiber parameters such as the NA and core sizes from
manufacturers. These can help us calculate the waveguide dispersion. Most of the
fibers used in this thesis are silica fibers and we can calculate the material disper-
sion using the refractive index information for silica glass. These two dispersion
functions combined can yield the total dispersion at different wavelengths.
An illustration of using the code to calculate the dispersion of the fiber Hi1060
at 1 µm is shown below. The fiber Hi1060 has NA = 0.14 and core diameter of
5.3 µm, and it’s often used in the Yb fiber lasers. In the code, all the wavelength
terms are in the unit of µm and the effective area is in the unit of µm2.
cutoff:=0.97
NA:=0.14
c:=0.3
delta = 0.5 ∗ NA∧2/1.45∧2
0.00466112
111
a = cutoff ∗ 2.405/(2 ∗ π ∗ NA)
2.65203
V [λ ] = 2 ∗ π ∗ a ∗ NA/λ;
V [1.03]
2.2649
MFR[λ ] = a ∗ (0.65 + 1.619V [λ]∧(−3/2) + 2.879V [λ]∧(−6));
MFR[1.03]
3.04004
Aeff[λ ] = π ∗MFR[λ]∧2;
Plot[Aeff[λ], {λ, 0.8, 2.0},AxesLabel → { ∧"Wavelength (µm)", "Aeff (µm 2)"},
PlotStyle → {Blue,Thick},BaseStyle → {FontWeight → "Bold",FontSize → 16}]
Aeff HΜm^2L
200
150
100
Wavelength HΜmL
1.0 1.2 1.4 1.6 1.8 2.0
Aeff[1.03]
29.0341
√
n[λ ] = (1 + ((6.996166300 ∗ 10∧ − 1) ∗ λ∧2/(λ∧2− (4.67914826 ∗ 10∧ − 3)))+
112
(((4.07942600 ∗ 10∧ − 1) ∗ λ∧2)/(λ∧2− (1.35120631 ∗ 10∧ − 2)))+
(((8.97479400 ∗ 10∧ − 1) ∗ λ∧2)/(λ∧2− (9.79340025 ∗ 10∧1))));
n[1.03]
1.45124
Ka[V ] = (1 + Sqrt[2])/(1 + (4 + V ∧(4))∧(1/4));
b[V ] = 1− (Ka[V ])∧2;
aa[λ ] = D[V ∗ b[V ], {V, 2}]/.V ->2 ∗ π ∗ a ∗ NA/λ;
WvgdDisp[λ ] = −(delta ∗ n[λ] ∗ 1000/(λ ∗ .3)) ∗ (V [λ] ∗ aa[λ]);
WvgdGVD[λ ] = −(λ∧2) ∗WvgdDisp[λ]/(2 ∗ π ∗ .3);
WvgdTOD[λ ] = −(λ∧2) ∗D[WvgdGVD[λ], {λ, 1}]/(2 ∗ π ∗ .3);
WvgdFOD[λ ] = −(λ∧2) ∗D[WvgdTOD[λ], {λ, 1}]/(2 ∗ π ∗ .3);
Plot[WvgdDisp[λ], {λ, 0.8, 1.5},AxesLabel → {"Wavelength (µm)",
"Waveguide dispersion (ps/nm/km)"},PlotStyle → {Blue,Thick},
BaseStyle → {FontWeight → "Bold",FontSize → 16}]
Waveguide dispersion HpsnmkmL
Wavelength HΜmL
0.9 1.0 1.1 1.2 1.3 1.4 1.5
-5
-10
-15
113
Plot[WvgdGVD[λ], {λ, 0.8, 1.5},AxesLabel → {"Wavelength (µm)",
"Waveguide GVD (fs∧2/mm)"},PlotStyle → {Blue,Thick},
BaseStyle → {FontWeight → "Bold",FontSize → 16}]
Waveguide GVD Hfs^2mmL
20
15
10
5
Wavelength HΜmL
0.9 1.0 1.1 1.2 1.3 1.4 1.5
MatDisp[λ ] = −(λ) ∗D[n[λ], {λ, 2}] ∗ 1000/(.3);
MatGVD[λ ] = (λ∧3) ∗D[n[λ], {λ, 2}] ∗ 1000/(2 ∗ π ∗ .3∧2);
MatTOD[λ ] = −(λ∧2) ∗D[MatGVD[λ], {λ, 1}]/(2 ∗ π ∗ .3);
MatFOD[λ ] = −(λ∧2) ∗D[MatTOD[λ], {λ, 1}]/(2 ∗ π ∗ .3);
Plot[MatGVD[λ], {λ, 0.8, 1.5},AxesLabel → {"Wavelength (µm)",
"Material GVD (fs∧2/mm)"},PlotStyle → {Blue,Thick},
BaseStyle → {FontWeight → "Bold",FontSize → 16}]
Material GVD Hfs^2mmL
30
20
10
Wavelength HΜmL
0.9 1.0 1.1 1.2 1.3 1.4 1.5
-10
-20
Plot[{MatGVD[λ] +WvgdGVD[λ]}, {λ, 0.8, 1.5},AxesLabel →
114
{"Wavelength (µm)", "GVD (fs∧2/mm)"},PlotStyle → {Blue,Thick},
BaseStyle → {FontWeight → "Bold",FontSize → 16}]
GVD Hfs^2mmL
35
30
25
20
15
10
5
Wavelength HΜmL
0.9 1.0 1.1 1.2 1.3 1.4 1.5
WvgdGVD[1.03]
MatGVD[1.03]
4.44843
18.9889
beta[λ ] = MatGVD[λ] +WvgdGVD[λ];
MatGVD[λ] +WvgdGVD[λ]/.λ → 1.03
23.4373
MatTOD[λ] +WvgdTOD[λ]/.λ → 1.03
25.6112
MatFOD[λ] +WvgdFOD[λ]/.λ → 1.03
−9.58417
Plot[{MatTOD[λ] +WvgdTOD[λ]}, {λ, 0.8, 1.5},AxesLabel →
115
{"Wavelength (µm)", "TOD (fs∧3/mm)"},PlotStyle → {Blue,Thick},
BaseStyle → {FontWeight → "Bold",FontSize → 16}]
TOD Hfs^3mmL
90
80
70
60
50
40
Wavelength HΜmL
0.9 1.0 1.1 1.2 1.3 1.4 1.5
116