-
High-Power Single-FrequencyFiber Lasers
by
Weihua Guan
Submitted in Partial Fulfillmentof the
Requirements for the DegreeDoctor of Philosophy
Supervised by
Professor John R. Marciante
The Institute of OpticsEdmund A. Hajim School of Engineering and
Applied Sciences
University of RochesterRochester,New York
2009
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ii
To my parents.
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Curriculum Vitae
The author received the B.E in Electrical Engineering from Xi’an
Jiaotong
University in 1999. He received the M.E in Electrical
Engineering from
Peking University in 2002. During the Master period, he did his
research
on L-band EDFA and Optical Add Drop Multiplexers (OADMs) in the
State
Key Laboratory for Local Optical Networks and Novel Optical
Communica-
tion Systems. In Sept. 2002, he started his PhD study at the
Institute of
Optics, University of Rochester. He received the Master’s degree
of Science in
Optics in 2005 from the Institute of Optics, University of
Rochester. He car-
ried out his doctoral research at the Laboratory for Laser
Energetics under
the direction of Prof. John R. Marciante.
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iv
Journal Publications
Weihua Guan and John R. Marciante, “1 W Single-Frequency Hybrid
Bril-louin/Ytterbium Fiber Laser,” Submitted to Optics Letters.
Weihua Guan and John R. Marciante, “Power scaling of
single-frequency hy-brid Brillouin/ytterbium fiber lasers,”
Submitted to IEEE Journal of QuantumElectronics.
Weihua Guan and John R. Marciante, “Complete elimination of
self-pulsationsin dual-clad ytterbium-doped fiber lasers at all
pumping levels,” Optics Let-ters, Vol. 34, No. 7, pp. 815-817,
March 15, 2009.
Weihua Guan and John R. Marciante, “Pump-Induced, Dual-Frequency
Switch-ing in a Short-Cavity, Ytterbium-Doped Fiber Laser,” Optics
Express, Vol. 15,No. 23, pp. 14979-14992, Nov. 12, 2007.
Weihua Guan and John R. Marciante, “Single-Polarization, Single
Frequency,2-cm Ytterbium-Doped Fiber Laser,” Electronics Letters,
Vol. 43, No. 10, pp.558-559, May 10, 2007.
Weihua Guan and John R. Marciante, “Dual-Frequency Operation in
a Short-Cavity Ytterbium-Doped Fiber Laser,” IEEE Photonics
Technology Letters,Vol. 19, No. 5, pp. 261-263, March 1, 2007.
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Presentations
Weihua Guan and John R. Marciante, “Suppression of
Self-Pulsations in DualClad, Ytterbium-Doped Fiber Lasers,”
Conference on Lasers and Electro-Optics(CLEO), San Jose, CA,USA,
May 2008.
Weihua Guan and John R. Marciante, “Single-Frequency Hybrid
Brillouin/Ytt-erbium Fiber Laser,” Frontiers in Optics, Rochester,
NY, October 2008.
Weihua Guan and John R. Marciante, “Elimination of
Self-Pulsations in Dual-Clad, Ytterbium-Doped Fiber Lasers,”
Frontiers in Optics, Rochester, NY, Oc-tober 2008.
Weihua Guan and John R. Marciante, “Dual Frequency Ytterbium
DopedFiber Laser,” IEEE Lasers and Electro-Optics Society (LEOS)
Annual Meet-ing, Montreal, Quebec, Canada, November 2006.
Weihua Guan and John R. Marciante, “Gain Apodization in
Highly-DopedFiber DFB Lasers,” Frontiers in Optics, Rochester,NY,
October 2006.
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Acknowledgments
When I finish the PhD study at the Institute of Optics, I want
to ac-
knowledge a lot of people that have been helpful to me, both in
academic and
non-academic aspects.
The first person I would like to acknowledge is my advisor,
Prof. John
R. Marciante, without whom, this thesis would not have been
possible. He
always gives me good advice on my research. He is very
supportive for ex-
perimental projects. His research and development experience in
fiber optics
makes him extremely helpful in research discussions. His
organization and
project management skills have been assets for the students. He
cares for
the students, making sure the students are on the right track.
He keeps the
students work under a happy environment.
I would like to acknowledge Prof. Govind P. Agrawal, from whom
I
learned a lot. He shared his intelligence and knowledge with the
students
in the courses and daily conversations. The talks with him had
been proven
to be very helpful and insightful. I learned a lot from his
supreme mathemat-
ical skills and physical insightfulness.
I am indebted to Prof. Duncan T. Moore. His vision in optical
engineer-
ing and system design broadens my knowledge in optics field.
During his fully
scheduled days, he met with students on weekends to make sure
the students
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vii
are on track. I acknowledge him for his support in the early
phase of my
graduate study.
I would like to thank Prof. Wayne H. Knox, who gave me helpful
sug-
gestions in the senior years of my PhD study. He shared his
excellence and
experience with interesting stories and quotes.
I would like to acknowledge my committee, Prof. Govind P.
Agrawal,
Prof. Thomas G. Brown and Prof. Roman Sobolewski, for their
guidance and
time.
I would like to acknowledge the faculty and staff of the
Institute of Op-
tics. The professors have been great in the courses and I felt
lucky to have the
opportunity to take their courses. I would thank Joan Christian,
Gina Kern,
Lissa Cotter, Besty Benedict, Gayle Thompson, Noelene Votens for
their help
in my study period.
I benefited a lot from the interactions with the scientists and
engineers
working in the Laboratory for Laser Energetics (LLE). I would
like to ac-
knowledge Prof. David Meyerhofer, Dr. Jonathan Zuegel, Dr.
Christophe
Dorrer, Dr. Seung-Whan Bahk, Dr. Jake Bromage. They gave me a
lot of help
especially in the sharing of equipment and scientific
discussions. I would like
to thank Kathie Freson, Jennifer Hamson, Jennifer Taylor, Lisa
Stanzel from
the illustrations group of LLE for their help in the preparation
of figures for
publications. I appreciate Giuseppe Raffaele-Addamo from
electronics shop of
LLE for his generous help on my high power laser diode driver
unit. I appre-
ciate Joseph Henderson in mechanical shop of LLE for his help on
mechanical
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viii
manufactures.
I would like to thank my groupmates Zhuo Jiang and Lei Sun.
The
discussions with them have been interesting and exciting.
I would like to thank the classmates from the Institute of
Optics, with
whom I feel I was studying with the most intelligent people. I
learned a lot
from them and I enjoyed the interactions with them. I wish them
successes
in the future.
I would like to acknowledge the support of the Frank J. Horon
fellowship
from the Laboratory for Laser Energetics, University of
Rochester. I would
like to acknowledge the supporting departments and agencies.
This thesis
work was supported by the U.S. Department of Energy Office of
Inertial Con-
finement Fusion under Cooperative Agreement No.
DE-FC52-92SF19460 and
DE-FC52-08NA28302, the University of Rochester, and the New York
State
Energy Research and Development Authority. The support of DOE
does not
constitute an endorsement by DOE of the views expressed in this
thesis.
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Abstract
Single frequency laser sources are desired in many applications.
Various
architectures for achieving high power single frequency fiber
laser outputs
have been investigated and demonstrated.
Axial gain apodization can affect the lasing threshold and
spectral modal
discrimination of DFB lasers. Modeling results show that if
properly tailored,
the lasing threshold can be reduced by 21% without sacrificing
modal dis-
crimination, while simultaneously increasing the differential
output power
between both ends of the laser.
A dual-frequency 2 cm silica fiber laser with a wavelength
spacing of
0.3 nm was demonstrated using a polarization maintaining (PM)
fiber Bragg
grating (FBG) reflector. The output power reached 43 mW with the
optical
signal to noise ratio (OSNR) greater than 60 dB. By thermally
tuning the
overlap between the spectra of PM FBG and SM FBG, a single
polarisation,
single frequency fibre laser was also demonstrated with an
output power of
35 mW. From the dual frequency fiber laser, dual frequency
switching was
achieved by tuning the pump power of the laser. The dual
frequency switching
was generated by the thermal effects of the absorbed pump in the
ytterbium
doped fiber.
Suppression and elimination of self pulsing in a watt level,
dual clad
ytterbium doped fiber laser was demonstrated. Self pulsations
are caused by
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the dynamic interaction between the photon population and the
population
inversion. The addition of a long section of passive fiber in
the laser cavity
makes the gain recovery faster than the self pulsation dynamics,
allowing
only stable continuous wave lasing.
A single frequency, hybrid Brillouin/ytterbium fiber laser was
demon-
strated in a 12 m ring cavity. The output power reached 40 mW
with an OSNR
greater than 50 dB. To scale up the output power, a dual clad
hybrid Bril-
louin/ytterbium fiber laser was studied. A numerical model
including third
order SBS was used to calculate the laser power performance.
Simulation
shows that 5 W single frequency laser output can be achieved
with a side
mode suppression ratio of greater than 80 dB. Experimentally, a
1 W single
frequency dual-clad fiber laser was demonstrated with an OSNR of
greater
than 55 dB.
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Table of Contents
List of Tables · · · · · · · · · · · · · · · · · · · · · · · · ·
· · · · · · xvi
List of Figures · · · · · · · · · · · · · · · · · · · · · · · ·
· · · · · · · xvii
Chapter 1
Introduction · · · · · · · · · · · · · · · · · · · · · · · · · ·
· · · · · · 1
1.1 High Power Fiber Lasers . . . . . . . . . . . . . . . . . .
. . . . . 1
1.1.1 Doping Ions and Laser Efficiency . . . . . . . . . . . . .
. 4
1.1.2 Double Cladding Fiber Structure . . . . . . . . . . . . .
. 9
1.1.3 Thermal Effects and Optical Damage . . . . . . . . . . .
10
1.1.4 Beam Quality . . . . . . . . . . . . . . . . . . . . . . .
. . 11
1.2 Single Frequency Fiber Lasers . . . . . . . . . . . . . . .
. . . . 11
1.2.1 DFB Fiber Lasers . . . . . . . . . . . . . . . . . . . . .
. . 12
1.2.2 Short Cavity DBR Fiber Lasers . . . . . . . . . . . . . .
. 13
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1.2.3 Ring Cavity Fiber Lasers with Embedded Filters . . . . .
16
1.2.4 Brillouin Ring Fiber Lasers . . . . . . . . . . . . . . .
. . 18
1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 20
Chapter 2
Theoretical Models of Fiber Lasers · · · · · · · · · · · · · · ·
· · · 23
2.1 Coupled-Mode Theory in Periodic Structure . . . . . . . . .
. . . 23
2.2 Space-Independent Rate Equations . . . . . . . . . . . . . .
. . . 27
2.3 Space-Dependent Laser Model . . . . . . . . . . . . . . . .
. . . . 30
2.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . .
. . . . . 33
Chapter 3
Gain Apodized Single Frequency DFB Fiber Lasers · · · · · · · ·
34
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 34
3.2 Fundamental Matrix Model . . . . . . . . . . . . . . . . . .
. . . 35
3.3 Gain Apodization Physics . . . . . . . . . . . . . . . . . .
. . . . 38
3.4 Gain Apodization in Phase Shifted DFB Lasers . . . . . . . .
. . 43
3.5 Thermal and Splicing Phase Effects . . . . . . . . . . . . .
. . . 46
3.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . .
. . . . . 49
Chapter 4
Linear Cavity Single Frequency and Dual-Single Frequency
Fiber
Lasers · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
· · · · · · 50
4.1 Dual Single Frequency Fiber Laser . . . . . . . . . . . . .
. . . . 50
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4.1.1 Enabling Dual-Frequency Lasers . . . . . . . . . . . . . .
51
4.1.2 Experimental Results . . . . . . . . . . . . . . . . . . .
. . 51
4.1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . .
. . . 57
4.2 Single Polarization Single Frequency Fiber Laser . . . . . .
. . 57
4.2.1 Experimental Results . . . . . . . . . . . . . . . . . . .
. . 58
4.2.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . .
. . . 62
4.3 Pump Induced Dual Frequency Switching in Ytterbium Doped
Fiber Lasers . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 62
4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
. . . 62
4.3.2 Experimental Results . . . . . . . . . . . . . . . . . . .
. . 64
4.3.3 Modeling and Simulations . . . . . . . . . . . . . . . . .
. 66
4.3.4 Discussions and Conclusions . . . . . . . . . . . . . . .
. 76
4.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . .
. . . . . 78
Chapter 5
Elimination of Self Pulsing in Dual Clad Ytterbium Doped
Fiber
Lasers · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
· · · · · · 80
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 80
5.2 Experimental Demonstration . . . . . . . . . . . . . . . . .
. . . 82
5.3 Nonlinear Effects and Self Pulsing Dynamics . . . . . . . .
. . . 88
5.4 Discussions and Chapter Summary . . . . . . . . . . . . . .
. . . 89
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Chapter 6
Power Scaling of Single-Frequency Hybrid Ytterbium/Brillouin
Fiber Lasers · · · · · · · · · · · · · · · · · · · · · · · · · ·
· · · · · · 90
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 90
6.2 Numerical Model . . . . . . . . . . . . . . . . . . . . . .
. . . . . 92
6.3 Experimental Verification . . . . . . . . . . . . . . . . .
. . . . . 94
6.3.1 Full Injection Locking and Gain Saturation . . . . . . . .
99
6.3.2 Partial Injection Locking . . . . . . . . . . . . . . . .
. . . 104
6.4 Power Scaling of Single Frequency Hybrid
Brillouin/Ytterbium
Fiber Laser . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 107
6.4.1 1-W Single-Frequency Hybrid Brillouin/Ytterbium Fiber
Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 116
6.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . .
. . . . . 121
Chapter 7
Conclusion and Future Work · · · · · · · · · · · · · · · · · · ·
· · · 123
7.1 Thesis Conclusion . . . . . . . . . . . . . . . . . . . . .
. . . . . . 123
7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 126
Bibliography · · · · · · · · · · · · · · · · · · · · · · · · · ·
· · · · · 128
Appendix A: Mode Selection and Nonlinear Effects · · · · · · · ·
151
Appendix B: Scanning Fabry-Perot Spectrometer · · · · · · · · ·
156
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Appendix C: Measurement of Relative Intensity Noise · · · · · ·
160
Appendix D: Single Frequency Fiber Laser Linewidth · · · · · ·
162
D.1 Spontaneous-Emission-Limited Laser Linewidth . . . . . . . .
. 162
D.2 Laser Linewidth Enhancement Factor . . . . . . . . . . . . .
. . 167
D.3 Laser Linewidth Measurement . . . . . . . . . . . . . . . .
. . . 169
Appendix E: Numerical Methods · · · · · · · · · · · · · · · · ·
· · 174
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xvi
List of Tables
4.1 Parameters used for the laser pump simulation . . . . . . .
. . . 69
4.2 Parameters used for the thermal calculation . . . . . . . .
. . . 72
6.1 Additional physical parameters used for the simulation . . .
. . 101
6.2 Wave-dependent parameters for the simulation . . . . . . . .
. . 103
6.3 Additional physical parameters used for the simulation . . .
. . 109
6.4 Wave dependent parameters for the simulation . . . . . . . .
. . 110
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List of Figures
1.1 Energy levels of Y b3+. . . . . . . . . . . . . . . . . . .
. . . . . . . 6
1.2 Absorption (solid) and emission (dotted) cross sections for
a yt-
terbium doped germanosilicate host. . . . . . . . . . . . . . .
. . 7
1.3 Energy levels of Nd3+. . . . . . . . . . . . . . . . . . . .
. . . . . 7
1.4 Schematic drawing of a double-clad fiber. . . . . . . . . .
. . . . 9
2.1 Energy levels of a typical quasi-three level laser system. .
. . . 27
2.2 Schematic diagram of laser power amplification. . . . . . .
. . . 30
3.1 Schematic diagram of a periodic active waveguide. . . . . .
. . . 35
3.2 Schematic of (a) a gain-apodized DFB fiber laser, (b) a
uniform
DFB fiber laser, and (c) a uniform DFB fiber laser with end
re-
flector R2 = tanh2(κL2). . . . . . . . . . . . . . . . . . . . .
. . . . 38
3.3 Gain thresholds of the different DFB fiber-laser
configurations
shown in figure 3.2. The black triangular mode in the center
is
the zeroth order mode of the DFB laser (c). . . . . . . . . . .
. . 40
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3.4 Schematic of (a) the modal frequencies of a gain-apodized
DFB
fiber laser with L1=0.5 cm, L2=2.5 cm, and a reflection
spectrum
of a 3 cm fiber Bragg grating. (b) The modal frequencies of a
0.5
cm uniform gain DFB fiber laser and a reflection spectrum of
a
0.5 cm fiber Bragg grating. . . . . . . . . . . . . . . . . . .
. . . . 41
3.5 The gain thresholds of the lowest-order mode as a function
of a
gain-apodization profile. . . . . . . . . . . . . . . . . . . .
. . . . 42
3.6 (a) The lowest-mode gain threshold versus L1L
. (b) The difference
in gain threshold between mode one and mode zero versus L1L
. . 44
3.7 The output power ratio from fiber ends versus L1L
. . . . . . . . . 45
3.8 Gain thresholds of the proposed DFB two section fiber
laser
with different splicing phase shifts. . . . . . . . . . . . . .
. . . . 48
3.9 The normalized gain thresholds, gain discrimination, and
output-
power ratios of the gain-apodized DFB laser under different
splicing phase shifts, when L1L
= 0.65. . . . . . . . . . . . . . . . . 48
4.1 Configuration of the dual single-frequency fiber laser. PM
is
the power meter, PD is the photodetector, ESA is the
electrical
spectrum analyzer, OSA is the optical spectrum analyzer, and
FP is the Fabry-Perot spectrometer. . . . . . . . . . . . . . .
. . 52
4.2 The measured transmission spectrum of the PM FBG using
an
ASE source. . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 52
4.3 The optical spectrum of the laser with 43 mW output power. .
. 53
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xix
4.4 The measured output spectrum of the fiber laser on the
scan-
ning FP spectrometer. The output laser is set to 43 mW. . . . .
. 53
4.5 Measured RIN spectrum of each wavelength independently
(dot-
ted and thin solid lines) and both wavelengths
simultaneously
(thick solid lines) with the laser operating at 43 mW of
output
power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 54
4.6 The experimental setup of a single-polarization,
single-frequency,
silica fiber laser. . . . . . . . . . . . . . . . . . . . . . .
. . . . . 58
4.7 The measured spectra of SM FBG at 50 oC and PM FBG at 22
oC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 58
4.8 The measured optical signal-to-noise ratio of the
single-polarization,
single-frequency fiber laser at 35 mW output power. . . . . . .
. 59
4.9 The spectrum of the single-polarization, single-frequency
fiber
laser in a F-P scanning spectrometer at 35 mW output power. .
59
4.10 The relative intensity noise of the single-frequency laser
at 35
mW output power. . . . . . . . . . . . . . . . . . . . . . . . .
. . 59
4.11 Measured transmission spectrum of the PM and SM FBGs at
room temperature. . . . . . . . . . . . . . . . . . . . . . . .
. . . 63
4.12 Measured laser output power as a function of pump current.
. . 65
4.13 Measured laser power as a function of pump current. The
blue
curve represents the power at 1029.1 nm, the red curve
repre-
sents the power at 1029.4 nm. . . . . . . . . . . . . . . . . .
. . . 65
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xx
4.14 Calculated pump distribution along the 1.5 cm active fiber
at
different pump levels. . . . . . . . . . . . . . . . . . . . . .
. . . 70
4.15 Calculated thermal distribution along the fiber laser
cavity at
different pump levels. . . . . . . . . . . . . . . . . . . . . .
. . . . 71
4.16 The spectra of PM and SM gratings under different pump
levels.
The red curves represent the reflection spectra of the SM
FBG,
the blue curves represent the reflection spectra of the PM FBG.
74
4.17 Calculated threshold gain discrimination between the fast
and
slow axes as a function of the pump current. . . . . . . . . . .
. 75
4.18 Measured laser power as a function of the PM FBG
tempera-
ture. The blue curve represents the power at 1029.1 nm, the
red curve represents the power at 1029.4 nm. . . . . . . . . . .
. 76
5.1 Schematic diagram of the ytterbium-doped fiber laser. D1 is
the
dichroic mirror, L1 and L2 are aspheric lenses, and FBG is
the
fiber Bragg grating. . . . . . . . . . . . . . . . . . . . . . .
. . . . 83
5.2 The output power as a function of the pump power for
fiber
lasers with four different cavity lengths. The active fiber
length
is 20 m in all four cases. . . . . . . . . . . . . . . . . . . .
. . . . 83
5.3 The self-pulsing dynamics of laser 1 when the pump power
is
3.2 W. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 83
5.4 The self-pulsing dynamics of laser 1 when the pump power
is
7.2 W. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 84
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xxi
5.5 The self pulsing characteristics of the fiber lasers with
four dif-
ferent cavity lengths. The active fiber length is 20 m in all
four
cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 84
6.1 The schematic of a general single-frequency hybrid
Brillouin/ytterbium
fiber laser. ISO is the isolator. YDF is the dual-clad
ytterbium-
doped fiber. . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 92
6.2 Schematic diagram of the hybrid Brillouin/ytterbium-doped
fiber
laser. WDM is the wavelength division multiplexer. YDF is
the
ytterbium doped fiber. SM fiber is the passive single-mode
fiber. 94
6.3 The laser output power as a function of 976 nm pump at
three
different Brillouin pump powers. . . . . . . . . . . . . . . . .
. . 95
6.4 The laser output spectrum on the optical spectrum
analyzer
with 370 mW of 976-nm pump and 9 mW of Brillouin pump.
The OSA resolution is 0.01 nm. . . . . . . . . . . . . . . . . .
. . 95
6.5 The laser output spectrum on the scanning FP
spectrometer
with 370 mW of 976-nm pump and 9 mW of Brillouin pump. . .
96
6.6 The power distributions of the optical waves in the active
and
passive fiber. Pp=370 mW, Pb=9 mW. . . . . . . . . . . . . . . .
. 102
6.7 Simulated and measured output power as a function of
Bril-
louin pump power when the pump power Pp is 370 mW. . . . . .
104
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xxii
6.8 The simulated OSNR versus the measured OSNR as a
function
of the Brillouin pump power with the 976 nm pump kept at 370
mW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 106
6.9 The single-frequency laser output power as a function of
the
pump power when the Brillouin pump power is 400 mW. The
first-order Stokes power is the output power from the
coupler,
and the second-order Stokes power is the power before the
isolator.111
6.10 The power distribution of the 915-nm and Brillouin pump
pow-
ers, the first-order Stokes wave, and the second-order
Stokes
wave. The 915-nm pump power is 10 W, and the Brillouin pump
power is 400 mW. The pump combiner has an insertion loss of
0.5 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 112
6.11 The required Brillouin pump power for full injection
locking as
a function of output power. . . . . . . . . . . . . . . . . . .
. . . . 113
6.12 The third-order Stokes power and the side-mode
suppression
ratio (SMSR) as a function of the laser output power when
the
Brillouin pump power is 400 mW. . . . . . . . . . . . . . . . .
. . 113
6.13 The pump power at which the second-order Stokes wave
reaches
threshold as a function of output coupler ratio. . . . . . . . .
. . 114
6.14 The laser output power and the required Brillouin pump
power
at the second-order Stokes wave thresholds with different
cou-
pler ratios. . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 115
-
xxiii
6.15 The side mode suppression ratio (SMSR) of lasers with
different
coupler ratios working at the second-order Stokes wave
threshold.115
6.16 Schematic diagram of the single frequency hybrid
Brillouin/ytterbium
fiber laser. ISO is the high power isolator. YDF is the
dual-clad
ytterbium-doped fiber. LD is laser diode. . . . . . . . . . . .
. . . 117
6.17 The output power versus the pump power. . . . . . . . . . .
. . . 117
6.18 The normalized OSA spectrum of the Brillouin seed and
laser
output when the output power is 1 W. The red curve is the
Bril-
louin seed, the blue curve is the laser output. The OSA
resolu-
tion is 0.02 nm. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 118
6.19 The laser output spectrum on the scanning F-P
spectrometer
when the output power is 1 W. . . . . . . . . . . . . . . . . .
. . 118
A.1 Schematic drawing of a helical core fiber [16]. . . . . . .
. . . . 151
A.2 An air-clad, ytterbium-doped large-mode-area fiber can
produce
high beam quality and single-mode, high-power laser outputs
(a). Ytterbium-doped rods form a triangularly-shaped large-
mode-area core (b) [20]. . . . . . . . . . . . . . . . . . . . .
. . . 153
A.3 SBS was suppressed by changing the doping ratio of
ytterbium,
germanium, and aluminum in active fiber [23]. . . . . . . . . .
154
B.4 Alignment of F-P spectrometer RC-110 using a laser beam. . .
. 156
B.5 Use a laser beam to align the mirrors of the F-P
interferometer. 157
-
xxiv
B.6 The ramp waveform without correction (left) and with
programmed
correction (right). . . . . . . . . . . . . . . . . . . . . . .
. . . . . 158
D.7 The phasor model for a single spontaneous emission for
the
laser field [61]. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 164
D.8 Schematic of delayed self-heterodyning measurement of
laser
linewidth [38]. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 170
D.9 Phasor of total optical field at the detector [61]. . . . .
. . . . . . 171
-
1
Chapter 1
Introduction
1.1 High Power Fiber Lasers
A new field of science and engineering emerged after the first
laser
demonstration in ruby crystal by Maiman in 1960 [1]. One year
later, Snizter
demonstrated the first fiber laser in Nd-doped fiber [2]. The
side pumping ge-
ometry that was used led to low laser efficiency, and the output
beam was spa-
tially multimode. In 1973, a longitudinally pumped Nd-doped
fiber laser was
reported by Burrus and Stone resulting in increased efficiency
and a single
spatial mode [3]. In 1987, the first erbium-doped fiber
amplifier (EDFA) was
demonstrated by D.N. Payne’s group [4]. The commericalization of
EDFA re-
duced the cost for long haul optical communications. The erbium
doped fiber
laser (EDFL) was demonstrated following the EDFA, but produced
limited
output power due to its low erbium doping density. For this
reason, Nd-doped
fiber lasers (NDFL) and Yb-doped fiber lasers (YDFL) are still
preferred today
-
2
for high power fiber laser applications.
The first YDFL was demonstrated by Etzel in 1962 [5]. There has
been
some debate about the choice between Nd and Yb as the optimum
dopant for
lasing. NDFLs have lower lasing thresholds due to the four-level
energy level
structure, and therefore attracted more interest in the early
days. Although
YDFLs work in the quasi-three level regime, Y b3+ has a lower
quantum defect
compared to Nd3+, and there are no ion-quenching effects in Y
b3+ doped laser
systems. The ion-quenching effect in Nd3+ doped fiber lasers can
lead to laser
efficiency degradation and self-pulsing of the laser output. For
these reasons,
Y b3+ is considered as a more appropriate gain medium than Nd3+
for high
power fiber lasers.
Fiber lasers have many advantages over solid-state glass lasers.
Fiber
lasers have compact volume, good thermal management, high beam
quality,
high laser efficiency and low noise floor. Laser efficiency over
80% can be
achieved in dual-clad fiber lasers. Because fiber lasers have
better thermal
management and a circular single mode waveguide, better beam
quality can
be achieved in fiber lasers. For example, due to thermal
lensing, a flash lamp-
pumped Nd:YAG laser can only offer limited output beam quality.
The inho-
mogeneous distribution of temperature along the cross section of
the glass rod
leads to the thermal lensing, which degrades the beam quality.
In fiber lasers,
the heat is easier to dissipate due to the increased
surface-to-volume ratio of
the fiber.
In addition, fiber lasers can be alignment free and therefore
are easier
-
3
to maintain. For the above reasons, fiber lasers are perferred
over solid-state
lasers in many applications.
To achieve high powers from fiber lasers, many obstacles have to
be over-
come. First, sufficient pump power has to be coupled into the
laser gain
medium. For single-mode active fiber, the single-mode laser
diode can only
provide up to Watt-level pump power; therefore the output power
of the fiber
laser system is limited to Watt level. To achieve higher pump
powers, the
dual-clad pumping technique has to be used so that high power
multimode
laser diodes can be utilized as pump sources. High-power
multimode laser
diodes have been developed to the point that hundreds of kWs can
be achieved
by combining laser diode arrays.
Fiber laser systems can be damaged by high optical powers. With
a dam-
age threshold of about 5 W/µm2, the fiber tends to be damaged
with optical
intensity beyond this value. In most high-power fiber laser
systems, the op-
tical damage tends to occur in the end facets and the splicing
points because
the interfaces have lower damage thresholds than the bulk fused
silica. The
extra heat generated by pump absorption in high-power fiber
lasers can nor-
mally be dissipated effectively without extra cooling units due
to the large
surace-to-volume ratio of the fiber.
Stimulated Brillouin scattering (SBS) and stimulated Raman
scattering
(SRS) play an important role in high power CW laser systems.
These non-
linear effects are induced by the interaction between the
optical wave and
the acoustic and optical phonons. They become important as the
intensity in
-
4
the fiber core increase. SBS is the main limitation of the
output power for
narrow-band signals. SRS can generate a Raman Stokes wave with a
13 THz
frequency down shift which will reduce the laser output power at
the signal
wavelength.
In the following sections, issues important for high power fiber
lasers
are reviewed. Spectrum, beam quality [65], and output power are
important
laser characteristics. For the interest of this thesis only
continuous wave (CW)
high-power fiber lasers have been covered. A full review on CW
and pulsed
high power fiber lasers can be found in the author’s master
essay [7].
1.1.1 Doping Ions and Laser Efficiency
Ytterbium and neodymium are appropriate doping candidates for
high-
power fiber lasers due to their energy level structures. They
have slightly
different energy transition mechanisms and can both work in the
1060 nm
region. To get the strongest absorption, neodymium needs to be
pumped at
808 nm and ytterbium needs be pumped at 976 nm. When operating
at 1060
nm, neodymium behaves as a four level system while ytterbium
behaves as a
quasi-three level system. Therefore, neodymium systems show
lower thresh-
olds than those fiber lasers built with ytterbium. However,
ytterbium is free
from the self-quenching effect while neodymium is not.
Therefore, a higher
ion concentration can be reached in ytterbium fiber lasers for
larger pump
power absorption. Additionally, ytterbium has lower quantum
defect com-
pared to neodymium. For these reasons, ytterbium is more
attractive than
-
5
neodymium as a doping element for high power fiber lasers.
There are some effects that affect rare-earth doped fiber
lasers. The ion
concentration quenching effect reduces the quantum efficiency
(the percent-
age of input photons (pump photons) which contribute to the
stimulated pho-
ton emission) of an ion doped system as the concentration of
ions is increased.
This ion quenching effect happens in Nd3+ doped fiber laser
systems but does
not exist in Y b3+ doped systems. The primary physical process
behind the
ion quenching effect in Nd3+ is cross-relaxation. In this
process, one excited
ion transfers part of its energy to a neighboring ion in the
ground state, af-
ter which both ions are left in an intermediate state. Since the
energy gap
between the intermediate state and the ground state is small,
both ions non-
radiatively decay to the ground state. In the process, one
photon is lost, reduc-
ing the stimulated emission. There is another physical process
that induces
inefficiency in Er3+ systems due to the energy levels of the
doping ions. This
process is cooperative upconversion, where two excited ions at
the metastable
level interact with each other. One of the excited ions
transfers its energy to
the neighboring excited ion, after which the first ion falls to
the ground state
while the second ion is excited into a higher energy state. From
the higher
energy state, the second ion relaxes into the metastable level
again through
multiphonon emission, generating heat. In the cooperative
upconversion, one
photon is lost, reducing the stimulated emission, and
transferred into heat.
Y b3+ is free from both of the cross-relaxation and the
cooperative upconver-
sion processes. Therefore, ytterbium can be doped in host silica
with high
-
6
Figure 1.1: Energy levels of Y b3+.
concentration and produce high laser efficiency.
To get a better understanding of fiber lasers, the spectroscopic
proper-
ties of the doping ions need to be investigated. In fiber
lasers, the most widely
used gain media are glass fibers doped with rare-earth ions due
to their high
solubility. Y b3+ exhibits a narrow absorption peak at 976 nm.
It also has a
broad emission bandwidth at longer wavelengths. The
spectroscopic proper-
ties do not change significantly in different glass hosts. The
relatively long
meta-stable lifetime in ytterbium ions enables high quantum
efficiency in
fiber lasers. Figure 1.1 and figure 1.2 show the energy levels
and corre-
sponding emission and absorption cross sections of Y b3+ ions
[8, 41]. Figure
1.3 shows the energy levels of Nd3+ ions [10].
The energy level diagrams explain why Y b3+ can be used in high
effi-
ciency, high output-power laser systems. The ground energy
manifold 2F7/2
and the excited energy manifold 2F5/2 constitute the two
manifolds of Y b3+
energy level diagram. The excited manifold splits into three
sublevels while
-
7
Figure 1.2: Absorption (solid) and emission (dotted) cross
sections for a ytter-
bium doped germanosilicate host.
Figure 1.3: Energy levels of Nd3+.
-
8
the ground manifold splits into four sublevels. This is due to
the Stark ef-
fect, where the atomic spectral lines split under an applied
electrical field.
The energy diagram of Y b3+ in figure 1.1 shows no intermediate
state be-
tween the ground and excited energy manifolds so that the
cross-relaxation
process does not happen. Additionally, the large energy gap
between the two
manifolds leads to little possibility of multi-phonon emission
from the excited
manifold, and there is no excited-state absorption for Y b3+
ions. For these two
reasons, there is no cooperative upconversion in ytterbium doped
fiber lasers.
Without the cross relaxation and cooperative upconversion
processes, little
concentration quenching occurs in ytterbium laser systems.
Additionally, the absorption and emission peaks can be matched
with
the energy level diagram in figure 1.1. As shown in the
spectroscopic diagram
in figure 1.2, peak A in the emission and absorption spectra
corresponds to
the energy transfer between the level e and level a. Peak B
matches the
absorption from level a to f and g. Peak C corresponds to the
transitions
from level b, which can produce re-absorption and lead to higher
thresholds
in the Y b3+ laser systems working around 1000 nm. The emission
spectrum
peak D corresponds to the energy transitions from level e to the
levels of b,
c and d. The emission spectrum at E corresponds to the
transition from the
level f, generating weak emissions around 900 nm wavelength. The
broad
absorption spectrum of the Y b3+ ions enable the easy
configuration of the
pump wavelength. Depending on the requirement of the laser
system, the
laser signal wavelength can be configured in the range from 970
nm to 1200
-
9
Figure 1.4: Schematic drawing of a double-clad fiber.
nm due to the wide emission spectrum of ytterbium.
1.1.2 Double Cladding Fiber Structure
Double-cladding pumping technology has been developed to go
beyond
the power limitations of single-mode laser pump diodes. For
these single-
spatial-mode laser pump diodes, the powers are normally limited
to below
1 W. However, with the development of the spatial multimode pump
diodes,
the pump power of a single emitter can reach 10 W. With arrays
and pump
combiners, kilowatt level pump power can be achieved, but with
spatially
multimode beam quality. Cladding pumping was developed to
transfer the
multimode pump power into the small core fiber laser systems.
When the
laser output power is scaled up, the core size limits the laser
power to a certain
amount due to the optical damage, thermal effects and nonlinear
effects in the
fiber medium.
Figure 1.4 shows the schematic drawing of a double clad fiber
[11]. The
two cladding structure of the fiber makes it different from
regular fibers. In
a dual clad fiber, the inner cladding confines highly multimode
pump light,
-
10
while the core confines the signal light to a single spatial
mode. The inner
cladding is designed with high numerical aperture (NA) to couple
more pump
light into the laser medium. The inner cladding is normally
designed to be
non-circular to enable more pump light reflections and therefore
more ab-
sorbed pump light by the active ions in the core. Different
inner cladding
shapes lead to different pump absorption efficiencies.
Additionally, an offset
core leads to a higher pump absorption efficiency. In [12], four
times higher
pump absorption efficiency was achieved by using an offset core
and rectangu-
lar inner cladding fiber compared with a symmetrical core and
circular shape
inner cladding active fiber.
1.1.3 Thermal Effects and Optical Damage
Thermal effects can be significant in high-power fiber lasers.
Fortu-
nately, the fiber geometry provides a large surface-to-volume
ratio and the
heat can be easily dissipated. Additionally, less than 15% pump
energy is con-
verted into heat due to the high quantum efficiency of the Y b3+
gain medium.
In some circumstances, there are some thermal effects for the
fiber coatings,
which can be minimized by proper heat sinking.
High optical power in laser systems can damage the fiber.
Optical dam-
age thresholds vary in different active fibers. In [13], an
optical intensity of
6.5 W/µm2 has been achieved in the fiber laser without optical
damage. As-
suming the CW damage threshold for a fiber is about 5 W/µm2, a
minimum
core area of 200 µm2 is required for a fiber laser with 1 kW
output power.
-
11
This core area normally produces multimode beam in the laser
output. For-
tunately, various mode selection techniques enable single mode
output beam
from a multimode core fiber.
1.1.4 Beam Quality
High beam quality output beams are required in most high-power
laser
systems. Therefore, the fiber laser has to work in the single
mode regime.
However, due to the optical damage and unwanted nonlinear
effects in high-
power fiber lasers, large core sizes are required. Mode
selection techniques
were developed to solve this problem [14]. A more detailed
review in the
progress of mode selection techniques and nonlinear effects in
fiber lasers can
be found in the appendix.
1.2 Single Frequency Fiber Lasers
A single-frequency fiber lasers operate in a single longitudinal
mode.
They are desired in sensing, ranging, high resolution
spectroscopy and inter-
ferometry. Additionally, a stable single-frequency laser source
is needed for
OMEGA laser at the Laboratory for Laser Energetics, University
of Rochester.
The above applications motivate the research in high-power
single-frequency
fiber lasers. This field has evolved slowly compared to that of
high-power
multi-longitudinal-mode fiber lasers, as described in the above
section. Sin-
gle frequency output can be generated from distributed feedback
(DFB) fiber
-
12
lasers, short cavity distributed Bragg reflector (DBR) fiber
lasers, ring cavity
fiber lasers with embedded narrow-bandwidth filters, Brillouin
fiber lasers,
injection locked fiber lasers. In all of these schemes, higher
output powers are
always desired from single-frequency fiber lasers.
1.2.1 DFB Fiber Lasers
DFB fiber lasers offer single longitudinal mode output by
resonantly cou-
pling the forward and backward lasing waves along the active
gratings. Gen-
erally, DFB fiber lasers work on the modal frequencies where
Bragg condition
satisfies in the active fiber grating. The Bragg condition can
be written as [25]
sin θi − sin θr = mλ/(nΛ) (1.1)
where θi and θr are the incident angle and diffraction angle of
the light, Λ is
the grating period, λ is the wavelength of the optical wave in
vacuum, m is
the Bragg diffraction order. While the Bragg condition leads to
many possible
modes from the distributed feedback structure, uniform DFB fiber
lasers tend
to work in two symmetric lasing modes of +1 order and -1 order
with the same
thresholds. This leads to mode hopping between the two modes
because the
lasing occurs at either of the two modes with equal
probability.
For practical applications, DFB fiber lasers are often designed
with one π
phase shift in the middle of active gratings [26]. The π phase
shift enables the
lasers work in zeroth order with the lowest lasing threshold
among the multi-
ple longitudinal modes. In these cases, a high intensity region
is formed in the
-
13
phase shift region, which limits the achievable output power
from the laser.
Another limitation for the output power is the absorbed pump
power in the
short section of active fiber grating. For this reason DFB fiber
lasers normally
have a low efficiency of a few percent and low output powers in
the milliwatt
regime. Recently a dual-clad active fiber DFB fiber laser was
demonstrated
with an output power of 160 mW using injected multimode pump
power of 12
W [27]. To improve the efficiency of DFB fiber lasers, various
design models
and techniques have been proposed. Phase shift location,
coupling strength,
and active fiber length have been optimized to achieve high
output powers
from fiber lasers [28–31].
Multiple wavelength DFB fiber lasers have been demonstrated by
super-
posing multiple Bragg gratings with different central reflection
wavelengths
along a single active fiber [32]. The same goal can be achieved
by simply cas-
cading DFB fiber lasers with different Bragg wavelengths [33].
The dynamic
behavior of highly nonlinear fiber DFB lasers has been analyzed
theoreti-
cally [34].
1.2.2 Short Cavity DBR Fiber Lasers
Single-frequency output can be generated from short cavity DBR
fiber
lasers. A normal DBR laser composes of one section of active
fiber and two
Bragg gratings as laser mirrors. The active fiber has to be
short enough to
enable a single longitudinal mode operation of the laser. For
laser mirrors
formed by two fiber Bragg gratings of 0.01 nm bandwidth, the
laser cavity is
-
14
normally limited to less than 10 cm to achieve single frequency
operation.
Single-frequency DBR fiber lasers in the low output regime have
been
demonstrated with Nd-doped and Er-doped silica fibers [36, 37].
A 200-mW
single frequency DBR fiber laser has been demonstrated with a
highly doped
phosphate glass fiber in 2004 [38]. The absorbed pump power
along the single
mode active fiber limited the output power. To scale up the
output power to
a higher level, dual-clad pumping technique was used to
demonstrate a watt-
level single-frequency fiber laser in 2005 [39]. Further
research shows that
spatial hole burning (SHB) tends to make DBR fiber lasers work
in a multi-
longitudinal mode regime and thus limits the length of the laser
cavity of
single-frequency fiber lasers. For this reason, in one
experiment, a twist-mode
technique was used in a 20-cm long DBR laser cavity. Two short
sections of
polarization-maintaining fiber were spliced to the active fiber
to rotate the
polarization of the modes. The laser cavity length was
effectively doubled by
using this method [40]. The standing wave in the linear cavity
was broken
down by utilizing a fiber-based quarter-wave-plate in both
travelling wave
directions. SHB was eliminated by changing the lasing light from
linearly
polarized to circularly polarized. Single-frequency output power
up to 1.9
W was generated with this scheme with an external coupled 10-cm
grating
cavity and a side-pumping architecture.
While in most experiments SHB limited the available active fiber
length
for single-frequency operation, in some experiments it has been
utilized for
achieving the single-frequency operation in fiber lasers. It has
been reported
-
15
that under certain circumstances the SHB in an unpumped section
of a standing-
wave cavity can stabilize the single-frequency laser output
instead of disturb-
ing it [41]. The former effect overrides the latter under
certain conditions.
The active fiber must have a large pump absorption cross section
so that the
pump power can be absorbed in a short section of fiber, leaving
the rest of the
fiber unpumped. The large pump cross sections will generate a
reduction of
the SHB effect in the pumped section due to the short pumped
fiber. There-
fore the stabilizing effect in the unpumped region will surpass
the destabiliz-
ing SHB effect in the pumped region. Stable single-frequency
output without
mode hopping was achieved by utilizing the unpumped section of
the SHB in
the linear fiber laser cavity [41].
Power scaling of single-frequency DBR fiber lasers can be
achieved with
active photonic crystal fiber (PCF). The larger mode-area of the
low-NA PCF
is critical for the power scaling while maintaining
single-spatial-mode beam
quality. By using the active fiber with PCF cladding and highly
doped large
area core in a DBR fiber laser configuration, high output power
was achieved
in 2006 [42]. The single frequency fiber laser output was 2.3 W
with the 3.8
cm active phosphate glass fiber with a photonic crystal cladding
and a large
core mode area of 430 µm2. The beam quality was the
single-spatial-mode
beam quality of M2 = 1.2.
Thermal effects influence the performance of single-frequency
DBR fiber
lasers. Thermal fluctuations in the active fiber lead to mode
hopping and
intensity noise. Mode hopping can be suppressed with the aid of
temperature
-
16
controllers [40].
1.2.3 Ring Cavity Fiber Lasers with Embedded Filters
A single-frequency laser wave can be generated with a ring
cavity fiber
laser having embedded narrow-band filters. With an isolator in
the cavity,
the laser wave travels unidirectionally and can therefore
eliminate the SHB
induced by the spatially dependent gain saturation of the
standing waves.
With an inserted narrow-bandwidth filter, a single longitudinal
mode can be
selected. In a 1990 experiment [47], a single frequency fiber
laser was demon-
strated by using a tunable band-pass filter in the laser cavity.
The active fiber
length was 15-m erbium-doped fiber. The laser can be tuned by
2.8 nm by
tuning the 1-nm-bandwidth band-pass filter. However, the output
power was
only 2 mW due to the single-mode pump laser diode of 78 mW. A
much higher
power single-frequency fiber ring laser was demonstrated in 2005
[40]. The
gain medium was 11-cm long highly Er/Yb doped phosphate-glass
fiber. The
output power was 700 mW without any mode hopping using a side
pumping
scheme. The single-longitudinal-mode output was selected by
using a sub-
cavity formed by two FBGs with 5 cm spacing. The mode hopping
was elimi-
nated by the sub-cavity. In another experiment in 1999, single
frequency laser
output was generated by using a ring resonator filter [49]. The
mode hopping
was also suppressed by the inserted ring resonator. In another
experiment,
a narrow-band filter was generated by SHB effect in a ring laser
cavity by
forming a standing wave in the unpumped active fiber [50].
Single-frequency
-
17
laser output was achieved in this ring cavity with an output
power of 1.4 mW.
The laser linewidth was measured to be 7.5 KHz.
Multiple-single frequency fiber lasers are of interests in some
applica-
tions. These lasers operate with multiple wavelengths and each
of the wave-
length works in the single frequency regime. In one experiment
in 2004, a
Lyot-Sagnac filter was used as an embedded band-pass filter for
generating
multiple-single frequency output from a ring cavity [51].
While isolators are used in most ring cavity fiber lasers, there
were
some single frequency ring fiber lasers where the waves travel
in both di-
rections. In one experiment, the SHB was eliminated by inducing
differential
losses for clockwise and counter-clockwise traveling waves. The
homogeneous
broadening of the Nd doped fiber made the laser work in the
single frequency
regime [52].
Wide tunability is desired in many laser applications. While
short DBR
fiber lasers are simple schemes for generating single frequency
output, it
is very hard to achieve wide tunability from them. Ring cavity
fiber lasers
are free from this limitation. In one experiment, a 45-nm tuning
range was
achieved from a fiber laser by utilizing a compound ring cavity
[53]. Two cou-
plers were connected to form a compound ring. The compound fiber
ring was
embedded into the main ring cavity. Additionally, a tunable
band-pass filter
was used in the main cavity to achieve the wide tunability, with
an output
power of 20 mW. In another paper, the combination of a tunable
bandpass fil-
ter and a fiber Fabry-Perot filter enabled a 42 nm tunable
single frequency Er-
-
18
doped fiber laser. The linewidth of the laser was measured to be
6 KHz [54].
1.2.4 Brillouin Ring Fiber Lasers
Brillouin ring fiber lasers can generate single-frequency laser
output.
SBS provides a spectral filtering effect that selects out a
single longitudinal
mode as laser output. The Brillouin gain bandwidth is close to
20 MHz in a
normal silica fiber and only allows one longitudinal mode to
exist for a ring
cavity shorter than 16 m.
When a beam of light is injected into a section of fiber used as
the Bril-
louin gain medium, the pump light is scattered by the refractive
index grating
associated with a traveling acoustic wave. The acoustic wave is
traveling for-
ward, and the scattered light is down shifted to the Stokes
frequency. The
interference between the pump wave and the Stokes wave induces a
density
and pressure variation along the fiber by the electrostriction
effect, which
forms a traveling index grating and drives the acoustic wave.
Electrostriction
is the effect that materials tend to be compressed under the
presence of an
electric field. It is the coupling mechanism for generating the
acoustic wave
in the Brillouin gain medium. To be more specific, for a
molecule under the
electrical field of E, the force acting on the molecule can be
written as [55]
F =1
2α∇(E2) (1.2)
where α is the molecule polarizability. When the pump light is
intense enough,
the acoustic wave and the Stokes wave reinforce each other in
the scattering
-
19
process. Therefore both of the waves grow to large
amplitudes.
The Brillouin gain coefficient is used to describe the strength
of the SBS
process. The gain spectrum of the SBS process is related to the
acoustic damp-
ing time (phonon lifetime) of the fiber material. Due to this
reason, the SBS
gain spectrum is as narrow as 20 MHz. To be more specific, the
SBS gain can
be written as [56]
g(Ω) = g0(ΓB/2)
2
(Ω− ΩB)2 + (ΓB/2)2(1.3)
where ΓB is the damping rate of the acoustic waves. It can be
written as
ΓB = 1/TB where TB is the acoustic lifetime of about 10 ns. ΩB
is the Stokes
frequency shift of about 15 GHz at 1 µm. The peak gain
coefficient g0 can be
written as [56]
g0 =2π2n7p212cλ2pρ0υaΓB
(1.4)
where n is the refractive index, p12 is the longitudinal
elasto-optic coefficient
related to electrostriction effect, ρ0 is the material density,
λp is the pump
wavelength, υa is the acoustic velocity in the fiber.
Extensive research effort has been put into single-frequency
Brillouin
fiber lasers [57–60]. In these experiments, the pump frequency
had to be
resonant with the fiber ring cavity to achieve pump intensity
enhancement
sufficient to generate SBS in a short (20 m) length of fiber. A
tunable coupler
or a piezo-electric controller was used to adjust the
accumulated phase in
the cavity to be an integer multiple of 2π. Alternatively, a
tunable laser can
be used as the Brillouin pump source in these lasers. The
cavities of these
-
20
lasers had to be reasonablely short (
-
21
surrounding the engineering of gain apodization into DFB fiber
lasers are
discussed.
In chapter 4, single-frequency fiber lasers based on short
linear cavities
are demonstrated. First, we demonstrate a room-temperature, dual
single-
frequency, linear-cavity, silica fiber laser. A
polarization-maintaining (PM)
fiber Bragg grating (FBG) and a single-mode (SM) FBG are used to
gen-
erate two single frequencies with two orthogonal polarizations
in a linear
cavity. Second, we demonstrate a single frequency, single
polarization sil-
ica fiber laser by adjusting the spectral overlap between the PM
FBG and
the SM FBG using a thermal controller. The fiber laser provides
a single-
frequency, single-polarization output under all pump levels.
Third, dual-
frequency switching is demonstrated in a linear fiber laser
cavity without any
polarization-controlling component. The laser frequency
switching is caused
by pump-induced heating of the two FBGs, and can therefore be
controlled
by current tuning the pump laser. This phenomenon can be used to
design
dual-frequency switchable fiber lasers by carefully aligning the
spectra of the
two FBGs.
In chapter 5, we demonstrate a new technique to suppress self
pulsa-
tions in fiber lasers by addressing their root cause: the
dynamic interaction
of the laser field and the gain. By increasing the round trip
time in the laser
cavity with a long section of passive fiber, the relatively fast
pumping rate
forbids the population dynamics and the self pulsations are
effectively sup-
pressed. Most importantly, we demonstrate that with sufficiently
long fiber,
-
22
the self pulsations can be completely eliminated at all pump
power levels.
In chapter 6, a single-frequency, hybrid Brillouin/ytterbium
fiber laser
is demonstrated in a 12-m ring cavity. The output power reaches
40 mW with
an optical signal-to-noise ratio (OSNR) greater than 50 dB. The
laser works
stably without mode hopping under ambient environmental
conditions. As
the Brillouin pump is increased, the laser evolves from partial
injection lock-
ing to full injection locking at the Stokes wavelength. A
coupled-wave model
is used to describe the partial injection locking. When the
laser is fully in-
jection locked, the output power decreases as the Brillouin pump
is increased
due to the gain saturation induced by Brillouin pump
amplification in the yt-
terbium doped fiber. A space-dependent model including
second-order SBS
is included to describe this gain saturation. Excellent
agreement is achieved
between the simulation and the measurement results. To scale up
the output
power, a dual-clad hybrid Brillouin/ytterbium fiber laser is
proposed. Nu-
merical model including third-order SBS is included to calculate
the laser
performance. Simulation shows that 5-W single-frequency laser
output can
be achieved from the dual-clad hybrid Brillouin/ytterbium fiber
laser with
a side-mode-suppression ratio greater than 80 dB.
Experimentally, a 1 W
single-frequency fiber laser is demonstrated with an OSNR of
greater than
55 dB using this dual-clad hybrid Brillouin/ytterbium laser
configuration.
In chapter 7, the primary conclusions of the thesis are
presented along
with directions for future research on high-power
single-frequency fiber lasers.
-
23
Chapter 2
Theoretical Models of Fiber Lasers
2.1 Coupled-Mode Theory in Periodic Structure
Coupled-mode theory is widely used in describing periodic
wavegudes.
This section presents the derivation of coupled-mode equations
in DFB struc-
tures using perturbation theory, following Yariv and Pollock’s
procedures [61,
62]. Assuming that the periodic structure has a cross secion of
single mode
fiber, the electrical field of the eigenmodes of the structure
satisfy the wave
equation of
∇2 ~E = µ∂2 ~D
∂t2(2.1)
where µ is time-invariant.
The electrical flux in a dielectric medium can be written in
terms of
polarization ~P :
~D = �0 ~E + ~P (2.2)
Therefore, the dielectric medium changes the electrical flux by
a polarization
-
24
value ~P . Additionally, the periodic index structure leads to
periodic deviation
from the average dielectric constant �, which can be described
as a perturba-
tion in the polarization ~P . It can be written as
~D = �0 ~E + ~P + ~Ppert = � ~E + ~Ppert (2.3)
By putting equation 2.3 into the wave equation 2.1, the new wave
equation
with the perturbation polarization as the driving term is
[62]
∇2 ~E = µ�∂2 ~E
∂t2+ µ
∂2 ~Ppert∂t2
(2.4)
Standard perturbation theory technique can be used to solve
equation 2.4 [61,
62]. The eigenmodes of the unperturbed fiber can be solved by
setting the
driving term ~Ppert to zero. The eigenmodes of the waveguide
form a complete
set. Therefore, a solution of the perturbed fiber waveguide can
be written in
terms of a superposition of the eigenmodes. Assuming the
polarization direc-
tion of the electrical field in fiber waveguide does not change
during propa-
gation and is aligned with y axis, the perbutation term ~Ppert
should have the
same polarization direction. The electrical field in the
perturbed single mode
fiber can be written as
~E = ŷ[12A+(z)ε(x, y)e−j(βz−ωt) + 1
2A−(z)ε(x, y)ej(βz+ωt) + c.c.] (2.5)
where ε(x, y) is the spatial amplitude distribution of the
eigenmode, A± are
the amplitudes of the forward and backward travelling waves, β
is the propa-
gation constant. Putting the general solution 2.5 into the
perturbation equa-
-
25
tion 2.4, the new perturbation equation can be written in the
scalar form as
12(∂
2A+
∂z2− 2jβ ∂A+
∂z)ε(x, y)e−j(βz−ωt) + 1
2(∂
2A−
∂z2+ 2jβ ∂A
−
∂z)ε(x, y)ej(βz+ωt) + c.c. = µ ∂
2
∂t2Ppert
(2.6)
where many terms have been eliminated because the eigenmode
satisfies the
unperturbed equation. In the small perturbation cases, the
envelopes changes
slowly with z, therefore, the second derivative terms can be
neglected. The
new equation can be written as
−jβ ∂A+∂zε(x, y)e−j(βz−ωt) + jβ ∂A
−
∂zε(x, y)ej(βz+ωt) + c.c. = µ ∂
2
∂t2Ppert (2.7)
Multiplying both sides of the equation with ε∗(x, y) and
integrating over
the x, y plane yields
∂A−
∂zej(βz+ωt) − ∂A
+
∂ze−j(βz−ωt) + c.c. =
−j2ω
∂2
∂t2
∫∫xyPpert(x, y)ε
∗(x, y)dx dy (2.8)
due to the eigenmode relation that∫∫xy ε
∗(x, y)ε(x, y)dx dy = 1. To make the
forward and backward waves have maximum coupling efficiency, the
right
hand driving term of equation 2.8 should have the same spatial
phase and
temporal frequencies as the left hand terms. In this case, the
perturbation
can be written in the form
Ppert(z, t) = ε0∆n2(z)
[A+
2ε(x, y)e−j(βz−ωt) +
A−
2ε(x, y)ej(βz+ωt) + c.c.
](2.9)
Substituting equation 2.9 into equation 2.8, the coupling
equation between
the forward and backward waves can be written as
∂A−
∂z−∂A
+
∂ze−2jβz =
jωε04
A+e−2jβz∫∫ ∞−∞
∆n2(z)ε∗εdxdy +jωε0
4A−
∫∫ ∞−∞
∆n2(z)ε∗εdxdy
(2.10)
-
26
In a uniform section of periodic structure, the refractive index
can be written
as
∆n2(z) = ∆n01
2
[ej(
2πΛz−φ) + e−j(
2πΛz−φ)
](2.11)
where ∆n0 is the amplitude of the index modulation, Λ is the
period of the
DFB structure, φ is the phase of the periodic structure at z =
0. Due to spa-
tial phase matching considerations, the coupling between the
forward wave
A+ and the backward wave A− requires that the index modulation
∆n2(z)
contains the periodic terms with spatial frequencies close to 2β
and −2β. If
we denote ∆β = β− πΛ
, and extract the matching terms in equation 2.10, then
the coupled equations can be written as
∂A−
∂z= jωε0
8A+e−j(2∆βz−φ)
∫∫∞−∞∆n0ε
∗(x, y)ε(x, y)dxdy
∂A+
∂z= jωε0
8A−ej(2∆βz−φ)
∫∫∞−∞∆n0ε
∗(x, y)ε(x, y)dxdy
(2.12)
If we denote the coupling coefficient as
κ =jωε0
8
∫∫ ∞−∞
∆n0ε∗(x, y)ε(x, y)dxdy (2.13)
then the coupled amplitude equations can be written as
∂A−
∂z= κA+e−j(2∆βz−φ)
∂A+
∂z= κA−ej(2∆βz−φ)
(2.14)
If the gain coefficient of the uniform fiber waveguide is g,
then the coupled
equations with gain are [63]
∂A−
∂z= κA+e−j(2∆βz−φ) − gA−
∂A+
∂z= κA−ej(2∆βz−φ) + gA+
(2.15)
Equation 2.15 are the widely used coupled mode equations for DFB
fiber
lasers.
-
27
Figure 2.1: Energy levels of a typical quasi-three level laser
system.
2.2 Space-Independent Rate Equations
Ytterbium doped fiber lasers are quasi-three level systems.
Figure 2.1
shows the energy level diagram of a typical quasi-three level
laser [64]. The
lower laser level 1 is a sublevel of the ground level. The
sublevels are assumed
to be in thermal equilibrium. When the pumping rate and
population inver-
sion are uniform along the fiber axis, the ytterbium laser can
be described
with a space-independent model. Assuming that the population of
the ground
level and the upper level are N1 and N2, the rate equations for
the population
and photons are [64]
N1 +N2 = Nt
dN2dt
= Rp − φ(BeN2 −BaN1)− N2τdφdt
= Vaφ(BeN2 −BaN1)− φτc
(2.16)
where Rp is the pumping rate, φ is the photon number, Nt is the
total popu-
lation density, τ is the metastable level lifetime, Va is the
volume of the gain
-
28
medium, τc is the photon lifetime, Be and Ba can be written
as
Be =σecnV
Ba =σacnV
(2.17)
where n is the refractive index of the active fiber, V is the
modal volume in
the laser cavity, σe and σa are the emission and absorption
cross sections of
ytterbium doped fiber, c is the light velocity in vacuum.
Although continuous wave lasers are predominantly studied in
this the-
sis, there are many cases where self pulsing occurs in CW fiber
lasers. Relax-
ation oscillation is the most important physical mechanism that
leads to self
pulsations.
Starting from the space-independent laser rate equation 2.16, an
analyt-
ical form of the self-pulsing condition can be derived. If we
use the notation
that f = σaσe
, N = N2 − fN1, then equation 2.16 can be written as [64]
dNdt
= Rp(1 + f)− (σe+σa)cnV φN −fNt+N
τ
dφdt
= VaσecnV
Nφ− φτc
(2.18)
For any pulsing behavior starting from small perturbations, the
popula-
tion inversion and photon number can be written as
N(t) = N0 + δN(t)
φ(t) = φ0 + δφ(t)
(2.19)
where δN � N0, δφ� φ0. Substituting equation 2.19 into equation
2.18, after
the very small product δNδφ is ignored, the equation takes the
linear form of
dδN(t)dt
= − (σa+σe)cnV
(φ0δN(t) +N0δφ(t))− δN(t)τdδφ(t)dt
= VaσecnV
φ0δN(t)
(2.20)
-
29
Differentiating the photon population equation and substituting
in the
inversion population yields a single equation for δφ
d2δφ
dt2+ (φ0
c
nV(σa + σe) +
1
τ)dδφ
dt+σe(σa + σe)c
2Van2V 2
N0φ0δφ = 0 (2.21)
This equation has the solution of the form
δφ = δφ0 exp(pt) (2.22)
After substitution into the equation of 2.21, a simple equation
of p can be
written as [64]
p2 +2
t0p+ ω2 = 0 (2.23)
where2t0
= φ0cnV
(σa + σe) +1τ
ω2 = σe(σa+σe)c2VaN0φ0
n2V 2
(2.24)
p has the solution of
p = − 1t0±√
1
t20− ω2 (2.25)
If p is real, i.e, equation 2.21 has two solutions of
exponential decays, there
will be no pulsing for the laser. The following condition must
hold
1
t0> ω (2.26)
To write the condition in a more explicit form, equations 2.24
are used
with the note that in quasi-three level fiber lasers φ0 can be
written as [64]
φ0 =nV
N0(σe + σa)c
fNt +N0τ
(x− 1) (2.27)
-
30
Figure 2.2: Schematic diagram of laser power amplification.
where x = RpRcp
is the pumping rate. Therefore, the condition for a
quasi-three
level fiber laser to be free from self-pulsations is
τcτ>
4(x− 1)x2
(1 +fNtN0
) (2.28)
In the case where p is complex, the relaxation oscillation
angular fre-
quency ω can be extracted and written as
ω =
[x− 1τcτ
(1 +fNtN0
)
]1/2(2.29)
Equation 2.28, 2.29 govern the relaxation oscillation dynamics
in ytterbium-
doped fiber lasers, which behave as quasi-three level
systems.
2.3 Space-Dependent Laser Model
A model can be applied to fiber lasers that describes the
spatial depen-
dence of the pump power and population inversion. To derive the
space-
dependent model, a section of active gain medium dz is
investigated. Fig-
ure 2.2 shows the schematic diagram of the laser power
amplification along
a section of gain medium [65]. If we consider a laser signal
wavefront with
-
31
power P (z, t) travelling along the +z direction in the
population inverted gain
medium of length dz, the equation of signal amplification can be
derived as
follows. If the energy density in the dz section is ρ(z, t), due
to the energy
conservation law, the rate of stored energy is the injected
energy flux minus
the output energy flux, plus the stimulated emitted energy flux.
The relation
can be written as [64]
∂
∂t[ρ(z, t)dz] = P (z, t)− P (z + dz, t) + Γ(σeN2 − σaN1)P (z,
t)dz (2.30)
where σe and σa are the stimulated emission cross-section and
the stimulated
absorption cross-section, N2 and N1 are the populations of the
upper level and
the lower level, Γ is the overlap factor between the active ions
and the signal
mode. Considering that P (z, t) = υgρ(z, t) where υg is the
signal group velocity,
the z dependent power amplification equation can be written
as
∂P (z, t)
∂t+ υg
∂P (z, t)
∂z= υgΓ(σeN2 − σaN1)P (z, t) (2.31)
Incorporating scattering loss and spontaneous emission, the
laser power
equations can be written as
1
υg
∂P (z, t)
∂t+∂P (z, t)
∂z= Γ(σeN2 − σaN1)P (z, t)− αP (z, t) + 2σeN2hνδν (2.32)
where the term of 2σeN2hνδν represents spontaneous emission at
the signal
frequency ν in two orthogonal polarizations, δν is the signal
bandwidth, α is
the scattering loss coefficient of the laser medium.
If the laser signal has a narrow bandwidth and generates SBS
waves,
the above equation must be modified to correctly describe the
power propa-
-
32
gation along the active fiber. In these cases, the spontaneous
scattering is
the mechanism that leads to the multiple order Stokes waves,
therefore, the
spontaneous emission is normally negligible compared to the
sponetaneous
scattering. If the laser operates in the continuous wave regime,
the laser
signal and multiple order SBS power propagation equations can be
written
as [66]
dP±idz
= ±[σeiN2 − σaiN1]ΓiP±i ∓ αiP±i ± gB1
AeffP±i (P
∓i−1 − P∓i+1)∓ gSB(P∓i−1 − P±i )
(2.33)
where ± and ∓ stand for the wave propagation directions, i
stands for the ith
optical wave, Aeff is the effective mode area of gain medium, gB
is the SBS
gain coefficient, and gSB is the spontaneous scattering gain
coefficient which
can be written as
gSB = gB1
Aeffhν∆νi (2.34)
where ∆νi is the optical bandwidth of the ith optical wave.
The population inversion in equations 2.32 and 2.33 can be
written as
n2 =
∑iσai Γi(P
+i + P
−i )(Ahc/λi)
−1
1τ2
+∑i
(σei + σai )Γi(P
+i + P
−i )(Ahc/λi)
−1 (2.35)
where n2 = N2/(N1 +N2) and τ2 is the metastable level
lifetime.
Equations 2.32 and 2.33 can be solved with finite difference
method to-
gether with the equation 2.35 to obtain the longitudinal power
profiles of the
waves and population inversion in the laser cavity.
-
33
2.4 Chapter Summary
In this chapter, various models for fiber lasers have been
reviewed and
derived. First, coupled mode equations in distributed feedback
fiber lasers
were derived with perturbation theory. Second, a
space-independent rate
equation model for quasi-three level fiber lasers was reviewed.
The relax-
ation oscillation frequency was derived from the rate equations.
Finally, a
space-dependent laser model was reviewed for fiber lasers,
including stimu-
lated Brillouin scattering.
-
34
Chapter 3
Gain Apodized Single Frequency DFB
Fiber Lasers
3.1 Introduction
DFB fiber lasers show the advantage of high stability with
relative struc-
ture among the various ways of generating single frequency fiber
laser sources
[67–69]. In this chapter, the effects of axial gain apodization
on the perfor-
mance of DFB fiber lasers are investigated for the first time.
In particular,
the impact of gain apodization on threshold behavior is explored
along with
its effect on output power and mode discrimination. First, the
physics of gain
apodization in DFB lasers are explored and compared to
conventional config-
urations. Secondly, the impact of gain apodization on phase
shifted DFB fiber
lasers is investigated. Finally, issues surrounding the
engineering of gain
apodization into DFB fiber lasers are discussed. The
investigation shows that
-
35
Figure 3.1: Schematic diagram of a periodic active
waveguide.
if properly tailored, ideally the lasing threshold can be
reduced by 21% with-
out sacrificing modal discrimination, while simultaneously
increasing the dif-
ferential output power between both ends of the laser [35].
3.2 Fundamental Matrix Model
Although DFB lasers are widely used for single-mode operation,
their
mode spectrum is more complicated. In a uniform index-coupled
DFB fiber
laser without phase shift or end mirrors, DFB lasers can operate
in one of two
degenerate longitudinal modes, symmetrically located along the
Bragg fre-
quency of the grating. Nominally, only a single mode runs due to
fabrication
imperfections that cause slight asymmetry.
The coupled-mode theory can be used to analyze the threshold
behav-
ior in simple DFB lasers. Figure 3.1 illustrates the schematic
of the coupling
between forward and backward waves in a DFB structure. To derive
the fun-
damental matrix model, the coupled mode equations 2.15 are
rewritten
∂A−
∂z= κA+e−j(2∆βz−φ) − gA−
∂A+
∂z= κA−ej(2∆βz−φ) + gA+
(3.1)
-
36
To solve the above equations, the following notations are
used:
EA(z) = A+ exp(−jβz)
EB(z) = A− exp(+jβz)
(3.2)
where β is the propagation constant of the optical wave in the
laser medium.
The equations 3.1 can be solved analytically as [63]
EA(z) = [c1 exp(Γ1z) + c2 exp(Γ2z)] exp[(g − jβ)z]
EB(z) = {exp(−j(2∆β′z − φ)]/κ}[c1Γ1 exp(Γ1z) + c2Γ2 exp(Γ2z)]
exp[−(g − jβ)z](3.3)
where c1 and c2 are some constants and ∆β′ and Γ1,2 are written
as
∆β′ = ∆β + jg
Γ1 = j∆β − γ
Γ2 = j∆β + γ
γ2 = k2 − (∆β′)2
(3.4)
For some gain media that are not uniformly periodic, they can be
seg-
mented into many different sections, each of which is uniform.
For the ith
uniform section, according to the notations in figure 3.1, the
electrical fields
are related through a fundamental matrix [63] EA (zi+1)EB
(zi+1)
= F
i11 F
i12
F i21 Fi22
EA (zi)EB (zi)
(3.5)
-
37
where the matrix elements are written as
F i11 = [cosh (γiLi) + j∆β′iLi sinh (γiLi)/(γiLi)] exp (jβ
iBLi)
F i12 = −κiLi sinh (γiLi) exp [−j (βiBLi + φi)]/(γiLi)
F i21 = −κiLi sinh (γiLi) exp [j (βiBLi + φi)]/(γiLi)
F i22 = [cosh (γiLi)− j∆β′iLi sinh (γiLi)/(γiLi)] exp [−j
(βiBLi)]
(3.6)
where ∆β′i = ∆βi + jgi, γ2i = k2i − (∆β′i)2, βiB = π/Λi, Λi is
the period of the
ith section and Li is the length of the ith section. The matrix
form provides
a convenient and powerful tool for studying DFB laser behaviors.
Many key
parameters including the gain thresholds of all longitudinal
modes, output
power ratio from both ends of a DFB laser can be calculated with
the fun-
damental matrix model. With the above fundamental matrix
formalism, the
active gratings can be split into N sections, where the total
matrix will be
Ft = FNFN−1...F2F1. For a nonuniform DFB fiber laser, the
coupling coeffi-
cient κ and gain coefficient g can change with the position z.
For DFB fiber
lasers without a phase shift, the phase terms in equation 3.6
can be writ-
ten as φi = φi−1 + 2βiBLi−1 where i = 1, 2, 3, ...N . For
phase-shifted DFB fiber
lasers, the phase terms in equation 3.6 is φi = φi−1 + 2βiBLi−1
+ ∆φi where
i = 1, 2, 3, ...N . Adding the boundary conditions A+(0) = A−(L)
= 0, the gain-
threshold condition can be obtained from the relation Ft11 = 0.
Nominally,
this relation will produce a mode spectrum with different modes
appearing at
different frequencies ∆β.
For high-power operation, it is desirable not only to have a low
threshold,
but also to have most of the light coming out of only one side
of the cavity. By
-
38
Figure 3.2: Schematic of (a) a gain-apodized DFB fiber laser,
(b) a uniform
DFB fiber laser, and (c) a uniform DFB fiber laser with end
reflector R2 =
tanh2(κL2).
using the total matrix Ft, the output-power ratio from both ends
of the fiber
can be written as
P1P2
=
∣∣∣∣∣A−(0)A+(L)∣∣∣∣∣2
= |F21|2 (3.7)
where P1P2
presents the ratio of the power coupling out at z = 0 compared
to
z = L.
3.3 Gain Apodization Physics
To understand the physics introduced by gain apodization, we
apply
the formalism in the former section to three cases. In all
cases, the grating
strength κ and period Λ are kept constant and no phase shift
will be included.
The peak reflectivity of the grating is determined by R =
tanh2(κL) and, to
not lose generality, typical values for κ and L are chosen. In
all the following
sections, the coupling coefficient of the fiber grating is κ = 1
cm−1. The grating
-
39
length is 3 cm in most cases. Since the length under which the
gain will drop
from its maximum value to zero is very small, the gain
apodization along the
z axis will be approximated by a step function. The
gain-apodized DFB fiber
laser is schematically shown in figure 3.2(a), where the L1
section is highly
doped with uniform gain coefficient g, and L2 has no gain. This
case will be
compared to two other cases. The first, a DFB fiber laser of
length L1 and
uniform gain but no unpumped section, is shown in figure 3.2(b).
The sec-
ond case, shown in figure 3.2(c), is the same laser as shown in
figure 3.2(b),
but with a reflector at the end of the cavity where the grating
would be in
the apodized case. The reflectivity value is chosen to be the
peak reflectivity
of the unpumped fiber grating of case figure 3.2(a), namely, R2
= tanh2(κL2).
This value was chosen to directly compare to the apodized case
figure 3.2 (a).
The gain thresholds for these cases, where L1=2.5 cm and L2=0.5
cm
are shown in figure 3.3. The horizontal axis is the normalized
frequency
∆βL (L = L1 + L2), while the vertical axis is the normalized
gain thresh-
old gthL1. The gain is normalized with L1 since the value of gL1
relates to the
absorbed pump power at threshold. The mode spectra of the three
different
lasers is nearly identical, since the lasing cavities are of
nearly equal length.
When compared to the short DFB laser, the gain-apodized DFB
lasers show
nearly a 30% reduction in lasing threshold due to its passive
grating section.
The DFB with the reflector similarly shows a reduction in lasing
threshold for
its first-order mode. However, the threshold reduction applies
significantly to
all modes since the reflector is spectrally uniform. For the
gain-apodized DFB
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40
Figure 3.3: Gain thresholds of the different DFB fiber-laser
configurations
shown in figure 3.2. The black triangular mode in the center is
the zeroth
order mode of the DFB laser (c).
laser, whose passive section has spectral dependence, the
additional reflector
also aids in modal discrimination with higher-order modes.
It is also important to note that although the passive grating
system in-
troduces system asymmetry, the 0th order mode cannot reach
threshold since
the phase of the transition between the two sections is
maintained. Never-
theless, figure 3.3 demonstrates the advantage of a reduced
lasing threshold
without the penalty of decreased spectral purity.
Figure 3.4 shows the gain threshold for DFB lasers plotted with
the
Bragg grating reflection spectrum to understand the interplay of
active ver-
sus grating length. To exaggerate the physics, the active
portion of the gain-
apodized DFB fiber laser is chosen to be L1=0.5 cm, with the
passive portion
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41
Figure 3.4: Schematic of (a) the modal frequencies of a
gain-apodized DFB
fiber laser with L1=0.5 cm, L2=2.5 cm, and a reflection spectrum
of a 3 cm
fiber Bragg grating. (b) The modal frequencies of a 0.5 cm
uniform gain DFB
fiber laser and a reflection spectrum of a 0.5 cm fiber Bragg
grating.
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42
Figure 3.5: The gain thresholds of the lowest-order mode as a
function of a
gain-apodization profile.
longer, L2=2.5 cm. The mode spectrum of this laser and the
corresponding re-
flectivity of a 3 cm FBG are shown in figure 3.4 (a). For
comparison, figure 3.4
(b) shows the mode spectrum of a conventional 0.5 cm long DFB
laser along
with the reflectivity spectrum of a 0.5 cm FBG. It is clear from
these figures
that the mode spectrum of the gain-apodized laser is determined
by the entire
grating rather than by only the active portion.
Figure 3.5 shows the lowest modal-gain threshold versus
different gain
length L1 for the gain-apodized DFB laser. From this figure, it
is clear that
the minimum threshold for L1L
is close to 0.7; the gain threshold is 17.9% less
compared to the uniform DFB fiber laser (L1L
= 1). For gain lengths L1L
less
than unity, the longitudinal distribution of light extends into
the unpumped
region, creating an effectively higher reflectivity. Since no
gain is extracted
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43
from this region, the effective grating strength is increased,
thus creating a
lower gain threshold. For values of L1L
that are too small (less than 0.7 in this
case), the grating-length product becomes too small to produce
sufficient re-
flection, effectively increasing the laser threshold via reduced
feedback. Fig-
ure 3.5 demonstrates that gain apodization can decrease the
laser threshold
if properly tailored.
3.4 Gain Apodization in Phase Shifted DFB Lasers
It is convenient to avoid mode degeneracy by introducing a phase
shift
in the middle of the grating. As is well known, the π phase
shift will enable
a narrowband filter in the grating forbidden band, thereby
allowing the 0th
order mode to have a low lasing threshold [26]. Considering the
influence of
this geometry, it is instructive to understand the role of gain
apodization on
phase shifted DFB fiber lasers.
Figures 3.6 (a) and 3.6 (b) show the lowest mode gain threshold
and the
mode discrimination of the uniform gain, phase shifted DFB fiber
lasers. As
before, the total cavity length L is 3 cm and the coupling
coefficient is 1 cm−1.
The results show that the apodization with the lowest gain
threshold also has
nearly the largest mode discrimination. Slightly different to
the optimum L1L
of 0.7 for a normal DFB laser in figure 3.5, the optimum gain
apodization
profile will be where L1L
is close to 0.6. From figure 3.6(a), the gain threshold
can be reduced 21.2% compared to the normal phase-shifted DFB
fiber laser,
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44
Figure 3.6: (a) The lowest-mode gain threshold versus L1L
. (b) The difference
in gain threshold between mode one and mode zero versus L1L
.
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45
Figure 3.7: The output power ratio from fiber ends versus
L1L
.
with nearly the same modal discrimination, as shown in figure
3.6 (b).
Since the gain apodization has introduced system asymmetry, the
output-
power ratio from both ends of the laser will also be modified.
To investigate
these characteristics, the output-power ratio of equation 3.7 is
plotted against
the apodized gain length L1L
in figure 3.7. The power ratio from both ends of
the fiber changes monotonically with the apodization gain length
L1L
. Higher
output power from the pumped end of the cavity can be obtained
at the opti-
mum pumped length L1L
for the minimum threshold shown in figure 3.6 (a);
the power ratio can be increased by 12.4%. This asymmetry,
combined with
the 21.2% threshold reduction, can lead to a substantial
increase in output
power due solely to gain apodization.
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46
3.5 Thermal and Splicing Phase Effects
It was shown in the former section that gain apodizati