high-power fiber lasers
DESCRIPTION
High-power fiber lasersTRANSCRIPT
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
ii
To my parents.
iii
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.
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
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
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.
ix
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
x
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
xiii
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
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
xvii
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
xviii
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 L1
L. (b) The difference
in gain threshold between mode one and mode zero versus L1
L. . 44
3.7 The output power ratio from fiber ends versus L1
L. . . . . . . . . 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 L1
L= 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
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
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
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
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 =2π2n7p2
12
cλ2pρ0υaΓB
(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 (<=20 m) to make sure the lasers worked
in the single-frequency regime. In one of these experiments [60], the narrow
linewidth of 75 Hz was achieved with a pump laser linewidth of 7.5 KHz. The
SBS process was proven to be able to reduce the phase noise of the pump
source and produce ultra narrow-bandwidth single-frequency output.
1.3 Thesis Outline
The rest of this thesis is organized as follows.
In chapter 2, the laser models which will be used in the following chap-
ters are reviewed. The coupled mode equations are derived from perturba-
tion theory. The relaxation oscillation frequency of ytterbium doped fiber
laser is derived from space-independent quasi-three level rate equations. The
space-dependent model based on power-propagation equations is introduced
for fiber lasers including stimulated Brillouin scattering in active fibers.
In Chapter 3, the effects of gain apodization on the performance 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 fundamental matrix model
based on coupled wave equations is reviewed and applied to our case of fiber
lasers. Second, the physics of gain apodization in DFB lasers are explored
and compared to conventional DFB configurations. Third, the impact of gain
apodization on phase shifted DFB fiber lasers is investigated. Finally, issues
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 = ε0 ~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 ε, which can be described as a perturba-
tion in the polarization ~P . It can be written as
~D = ε0 ~E + ~P + ~Ppert = ε ~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 = µε∂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 = y[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 =
jωε0
4A+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
dN2
dt= Rp − φ(BeN2 −BaN1)− N2
τ
dφdt
= 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+Nτ
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)τ
dδφ(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 = − 1
t0±√
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 +N0
τ(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 +
fNt
N0
) (2.28)
In the case where p is complex, the relaxation oscillation angular fre-
quency ω can be extracted and written as
ω =
[x− 1
τcτ(1 +
fNt
N0
)
]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 ± gB
1
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 i
12
F i21 F i
22
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 (jβiBLi)
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 = k2
i − (∆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
P1
P2
=
∣∣∣∣∣A−(0)
A+(L)
∣∣∣∣∣2
= |F21|2 (3.7)
where P1
P2presents 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
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
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.
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 L1
Lis close to 0.7; the gain threshold is 17.9% less
compared to the uniform DFB fiber laser (L1
L= 1). For gain lengths L1
Lless
than unity, the longitudinal distribution of light extends into the unpumped
region, creating an effectively higher reflectivity. Since no gain is extracted
43
from this region, the effective grating strength is increased, thus creating a
lower gain threshold. For values of L1
Lthat 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 L1
L
of 0.7 for a normal DFB laser in figure 3.5, the optimum gain apodization
profile will be where L1
Lis 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,
44
Figure 3.6: (a) The lowest-mode gain threshold versus L1
L. (b) The difference
in gain threshold between mode one and mode zero versus L1
L.
45
Figure 3.7: The output power ratio from fiber ends versus L1
L.
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 L1
Lin figure 3.7. The power ratio from both ends of
the fiber changes monotonically with the apodization gain length L1
L. Higher
output power from the pumped end of the cavity can be obtained at the opti-
mum pumped length L1
Lfor 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.
46
3.5 Thermal and Splicing Phase Effects
It was shown in the former section that gain apodization can have a
beneficial impact on phase shifted DFB lasers. It has been previously shown
that DFB laser performance can be improved both by changing the location
of the phase shift and by varying κ along the laser axis [70,71]. To obtain the
highest single-frequency output from DFB fiber lasers, the gain apodization
length, the phase shift location, and the coupling coefficient profile must all
be optimized. While this presents a challenging numerical problem, genetic
algorithms have proven useful in optimizing laser and amplifier designs [72].
While the lasing threshold itself will determine the gain apodization pro-
file for a given DFB laser, this effect can be intentionally introduced. Two sep-
arate sections of photosensitive fiber, only one of which is doped with active
ions to provide gain, can be spliced together before a grating is written into
the fiber. As shown in the former section, with the π phase shift in the mid-
dle of the fiber laser, lower thresholds of up to 21% can be expected without
sacrificing the modal discriminations.
In this two-section DFB fiber laser, pump induced thermal effects in
active fibers can influence the performance of the gain apodized DFB fiber
laser. The temperature distribution along the ytterbium fiber can be calcu-
lated numerically. Assuming the pump power is deposited in the cylindrical
core region with radius s, the temperature distribution can be calculated as
47
follows [73]
∆T (r, z) =ηPv(z)s2
2bh− ηPv(z)s2
2kln(r/b) (r≥s) (3.8)
The temperature in the active fiber core can be approximated by the
temperature where r = s. In the above equation, s is the pump absorption
radius of 2.5 µm, b is the cladding radius of 62.5 µm, η is the conversion ef-
ficiency of the absorbed pump into heat of 5%, Pv is the power absorbed per
unit volume, h is the air’s convective parameter of 81.4 W/m2 /K, and k is the
thermal conductivity of 1.38 W/m/K. Suppose the pump power of 300 mW has
been absorbed in the length of 8 cm active fiber, the increased temperature
∆T is about 5 degrees. Using a coefficient of temperature expansion (CTE)
of 5× 10−7, the gain threshold change of the gain apodized DFB fiber laser is
calculated to be less than 10−4; therefore, the influence of the temperature on
gain thresholds can be ignored. Similarly, the output ratio from both ends of
the proposed gain-apodized, two-section DFB fiber-laser variation under dif-
ferent temperatures is less than 10−4 and can also be ignored. The working
wavelength of the phase-shifted DFB fiber laser will shift by 0.035 nm. Proper
thermal controller is required to eliminate the laser wavelength shift.
In building the two-section DFB fiber laser by splicing an active and a
passive section of fiber, there may be an uncontrolled phase shift at the inter-
face of the doped and undoped fiber. Using the fundamental matrix model, the
normalized 0th order gain threshold under different splicing induced phase
shifts is shown in figure 3.8. Figure 3.8 shows that, while the value of the
48
Figure 3.8: Gain thresholds of the proposed DFB two section fiber laser with
different splicing phase shifts.
Figure 3.9: The normalized gain thresholds, gain discrimination, and output-
power ratios of the gain-apodized DFB laser under different splicing phase
shifts, when L1
L= 0.65.
49
gain threshold changes as a phase defect is introduced, the nature of the phe-
nomenon does not because the minimum remains at L1
L= 0.65. Figure 3.9
shows the normalized values of gain thresholds, gain discriminations, and
output power ratios under different phase shifts when the gain apodization
L1
Lis set to 0.65. The gain threshold for a gain apodized DFB laser is less than
a DFB fiber laser with a uniform gain profile, provided the splicing-induced
phase shift is less than 0.6 π. In the case where the gain threshold is reduced
(splice phase less than 0.6 π), the reduction in modal discrimination is less
than 15%. Likewise, the output power ratio from both ends of the laser can
also be reduced, as L1
L= 0.65. In fact, for a splice phase between 0.3 π and 0.6
π, the output power is actually larger on the side of the fiber with no gain.
Figures 3.8 and 3.9 show that the beneficial impact of gain apodization
might be reduced by an abrupt phase change under splicing. However, this
problem can be avoided by post heating the splice and creating a tapered
diffusion, as is commonly done when splicing ytterbium doped fibers to con-
ventional undoped single mode fibers.
3.6 Chapter Summary
In summary, if the axial gain profile in phase shift DFB fiber lasers is
properly tailored, the lasing threshold can be reduced by 21% without sacri-
ficing modal discrimination, while simultaneously increasing the differential
output power between both ends of the laser.
50
Chapter 4
Linear Cavity Single Frequency and
Dual-Single Frequency Fiber Lasers
4.1 Dual Single Frequency Fiber Laser
Dual-frequency fiber lasers are attractive for applications in ranging,
communications, and interferometers [74–76], and have been the focus of in-
tensive research [77–81]. They have been demonstrated with a high-birefringence
fiber-Bragg grating (FBG) in a ring cavity [77], a multimode FBG in a linear
cavity [78], self-seeded multimode Fabry-Perot (FP) laser diodes [79], dual
FBG’s with a circulator in a ring cavity [80], multiple bandpass filters in a
ring cavity [81], and FBG’s with multiple phase shifts in linear or ring cav-
ities [83, 84]. Most of the demonstrated dual-wavelength lasers operate in a
multimode regime at each of the dual wavelengths [77–81].
In this section, we demonstrated a room-temperature, dual-single-frequency,
51
linear-cavity, silica fiber laser [45]. A polarization maintaining (PM) FBG and
a single-mode (SM) FBG are used to generate two single modes at two fre-
quencies with two orthogonal polarizations in a linear cavity. This configura-
tion can be realized using standard commercial components.
4.1.1 Enabling Dual-Frequency Lasers
In ytterbium-doped fiber lasers, the ytterbium can be treated as a spec-
tral homogenous broadening medium and thus permits only a single lasing
mode. In a linear cavity, however, a standing wave will be formed between
the two reflectors and thus spatial-hole burning (SHB) occurs. Additionally,
polarization-hole burning (PHB) is similar to SHB in the sense that differ-
ent polarizations will extract different gains from the active medium [85] and
thus affect lasers with birefringent components. Furthermore, gain satura-
tion enhances the dual-frequency lasing through the modal competition pro-
cess [83, 84]. Generally, the combined effects of SHB, PHB, gain saturation,
thermal effects, and nonlinearities determine the modal behaviors of the fiber
lasers.
4.1.2 Experimental Results
The demonstrated dual single-frequency fiber laser is shown in figure
4.1. A section of 1.5 cm highly ytterbium-doped silica glass fiber with an
absorption rate of 1700 dB/m at 976 nm is spliced between two FBG’s. A
wavelength division multiplexer is used to couple the pump light at 976 nm
52
Figure 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.
Figure 4.2: The measured transmission spectrum of the PM FBG using an
ASE source.
53
Figure 4.3: The optical spectrum of the laser with 43 mW output power.
Figure 4.4: The measured output spectrum of the fiber laser on the scanning
FP spectrometer. The output laser is set to 43 mW.
54
Figure 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.
into the laser cavity. The SM FBG has a center wavelength of 1029.3 nm and
a 3 dB bandwidth of 0.46 nm with a peak reflectivity of 99%.
Figure 4.2 shows the measured transmission spectrum of the PM FBG
when seeded with an unpolarized amplified spontaneous emission (ASE) source.
Because of the differential modal refractive index along the fast and slow
axis, the grating exhibits two peak-reflection wavelengths, one for each po-
larization. The wavelength spacing between the two reflection peaks is 0.31
nm. Both of the peak reflectivity wavelengths lie in the reflection band of the
SM FBG under ambient-temperature conditions. Each of the reflection bands
of the PM FBG has a 3 dB bandwidth of 0.06 nm and a 55% reflectivity for
the corresponding polarizations. As discussed in the former section, the gain
competition between polarizations at these two wavelengths determines the
spectral properties of the laser.
55
The output power of the laser reaches 43 mW with 490 mW of pump
power, with lasing threshold at 10 mW of pump power. The optical signal-
to-noise ratio (OSNR) has been measured with an optical spectrum analyzer.
When the output power is 43 mW, the OSNR is measured to be greater than
60 dB as shown in figure 4.3. In the dual-frequency fiber laser shown in figure
4.1, the OSNR is limited by the residual ASE noise.
The single mode operation at each lasing wavelength is verified with a
FP spectrometer. Figure 4.4 shows the scanning spectrum of the laser modes
when the output power is 43 mW. The free spectral range (FSR) is 150 GHz.
With a finesse of 150, the FP spectrometer has a resolution of 1 GHz. Since
the fiber laser has a 2 cm cavity, corresponding to 5.1 GHz in modal frequency
spacing, the multiple modes caused by the fiber laser cavity can be well re-
solved by the FP spectrometer.
The relative intensity noise (RIN) has been measured by using an elec-
trical spectrum analyzer. The measurement is limited by the bandwidth of
the photodiode detector with the cutoff frequency of 1 GHz. The FP cavity is
used to filter out each wavelength by applying a bias voltage but not a scan-
ning signal. In this way, the RIN of each wavelength can be independently
measured. Figure 4.5 shows the RIN of each filtered lasing wavelength and
the RIN of the total laser output with both wavelengths on two scales with
the laser set to 43 mW output power. In the three cases, the RIN is limited by
the shot noise beyond 60 MHz. The noise peak at the frequency of 1.5 MHz
is caused by relaxation oscillations of the fiber laser. The additional noise be-
56
yond 20 MHz is due to the mode competition between the two frequencies and
does not appear in the case where both frequencies are measured together.
The polarization of the fiber-laser output has been measured with a
quarter wave plate and a polarizer. Each frequency shows a single polar-
ization with a polarization excitation ratio (PER) of >20 dB. The two polar-
izations at the dual frequencies are orthogonal to each other, as expected from
the PM FBG.
The dual-frequency operation of this laser is stable under perturbations
of pump power. In the working regime where the output power is close to
43 mW, the ratio of peak power at each wavelength changes with the pump
current as 0.02 dB/mA. Therefore, a 1% change in pump power leads to 5%
change in relative peak power. In practice, pump-power can be readily con-
trolled with commercial diode laser drivers to better than 0.01%, which leads
to less than a 0.05% relative peak-power variation between two fiber laser
wavelengths. This demonstrates that the laser has a highly stable dual-
frequency output.
Dual-frequency lasers can be customized by appropriately designing the
PM FBG. Different wavelength spacing can be achieved by controlling the
birefringence of the PM FBG, while changing the period of the grating will
make the laser work at different wavelengths. The output power of the dual-
frequency laser can be scaled up by optimization of the PM FBG reflectiv-
ity and imposing higher pump power. The relaxation-oscillation peak can be
reduced by using a negative-feedback circuit on the pump laser [86]. The
57
orthogonal output polarizations enable the fiber laser to work in a single-
polarization, single-frequency regime by using a polarization filtering compo-
nent.
4.1.3 Conclusions
In conclusion, a dual-frequency 2 cm silica fiber laser with a wavelength
spacing of 0.3 nm has been demonstrated using a PM FBG reflector. The
birefringence of the PM FBG was used to generate the two single mode lasing
frequencies of orthogonal polarizations. The single-mode operation in each
wavelength has been verified. The output power reaches 43 mW with the
OSNR of greater than 60 dB. The fiber laser shows stable dual frequency
output under pump variations.
4.2 Single Polarization Single Frequency Fiber Laser
Single frequency fiber lasers can be used for coherent communications,
holographic imaging, optical data storage [118], laser ranging [117], and laser
interferometers [89]. In 2004, a single-frequency, custom phosphate glass
fiber laser was demonstrated with 200 mW output power [38]. Using custom
photosensitive Er/Yb doped fiber, single polarization, single frequency output
was generated in both distributed feedback (DFB) and distributed Bragg re-
flector configurations [90]. In another experiment, a Fabry-Perot cavity com-
posed of Faraday-rotator mirrors is used to generate the single-polarization,
58
Figure 4.6: The experimental setup of a single-polarization, single-frequency,
silica fiber laser.
Figure 4.7: The measured spectra of SM FBG at 50 oC and PM FBG at 22 oC.
single-frequency fiber laser output [91]. In 1999, injection locking was used
to make a DFB fiber laser work in a single polarization [92]. In this section,
we demonstrate, for the first time, a single-frequency, single-polarization sil-
ica fiber laser with a polarization-maintaining fiber Bragg grating (PM FBG)
in a short linear cavity. The fiber laser works with a single frequency, single
polarization output under all pump levels, and is made from commercially
available fiber and components [44].
4.2.1 Experimental Results
The experimental setup of the single-polarization, single-frequency laser
is shown in figure 4.6. A wavelength division multiplexer (WDM) is used
59
Figure 4.8: The measured optical signal-to-noise ratio of the single-
polarization, single-frequency fiber laser at 35 mW output power.
Figure 4.9: The spectrum of the single-polarization, single-frequency fiber
laser in a F-P scanning spectrometer at 35 mW output power.
Figure 4.10: The relative intensity noise of the single-frequency laser at 35
mW output power.
60
to couple the 976 nm pump light into the laser gain medium. The 1.5 cm
active silica fiber is highly doped with ytterbium with an absorption rate of
1700 dB/m at 976 nm. It is spliced between two FBG’s. The single-mode
(SM) FBG has a center wavelength of 1029.5 nm and a peak reflectivity of
99%. The 3 dB bandwidth is 0.46 nm. The PM FBG has different modal
refractive index along the orthogonal fast and slow axes, which enable the
two reflection wavelengths with 0.3 nm spacing. Both of the reflection bands
have 3 dB bandwidths of 0.06 nm and show the reflectivity of 55% at the
corresponding wavelengths. The environment temperature is 22 oC and the
thermal controller is set to 50 oC. The measured transmission spectra of
the SM FBG at 50 oC and the PM FBG at 22 oC are shown in figure 4.7. An
amplified spontaneous emission source was used as the broadband seed in the
measurement. Overlapping of the spectrum of FBG’s by temperature tuning
is used to generate the single frequency, single polarization output. With the
thermal controller on the SM FBG, its reflection band is tuned to overlap the
wavelength peak of 1029.4 nm of the PM FBG. The other peak of the PM FBG
lies outside the reflection band of the SM FBG.
The laser output power has been measured from the PM FBG with a
power meter. The laser pump threshold is 10 mW. With a commercial laser
diode pump of 490 mW, the laser output power reaches 35 mW. Higher ef-
ficiency can be achieved by optimizing the reflectivity of the PM FBG. The
optical signal-to-noise ratio (OSNR) of the laser output has been measured
with an optical spectrum analyzer, to be greater than 65 dB as shown in fig-
61
ure 4.8. There is only one lasing mode in the whole spectrum of the ytterbium
gain band.
The single-mode behavior of the fiber laser has been verified with a scan-
ning Fabry-Perot (F-P) spectrometer. The FSR is 30 GHz and the finesse is
150, giving a resolution of about 200 MHz. Since the mode spacing of the
laser cavity is approximately 5 GHz, the lasing modes of the cavity can be
well resolved by the F-P cavity. Figure 4.9 shows the single-frequency laser
spectrum from the scanning F-P cavity. Although three F-P modes can be
supported in the 3 dB reflection band of the PM FBG, the curvature of the
PM FBG reflection spectrum provides sufficient longitudinal-mode discrimi-
nation to enable only a single mode to operate at each polarization. In the
measurement, no mode hopping is observed.
The relative intensity noise (RIN) of the fiber-laser output has been mea-
sured with an electrical spectrum analyzer. The measurement is limited by
the bandwidth of the photodiode detector with the cutoff frequency of 1 GHz.
Figure 4.10 shows the measurement result. The peak at 10 MHz is caused by
the relaxation oscillations of the fiber laser, in agreement with the measured
upper-state lifetime of 0.17 ms of this highly ytterbium-doped fiber. The RIN
beyond 80 MHz is shot-noise limited and less than -130 dB/Hz.
The state of polarization of the laser has been measured with a quarter
wave plate, a polarizer, and a power meter. The quarter wave plate is put be-
tween the laser output end and the polarizer to turn the lasing light into lin-
early polarized light, since the WDM is not a polarization-maintaining (PM)
62
WDM. The single polarization has been verified. The polarization extinction
ratio is measured to be greater than 20 dB under different pump levels. The
laser operates stably in single frequency and single polarization in terms of
power variation and the wavelength drift. Over a 2 h time period, the power
rms deviation is less than 0.9 % and the peak-to-valley wavelength drift is
less than 0.02 nm.
4.2.2 Conclusions
In summary, a single polarization, single frequency, ytterbium-doped sil-
ica fiber laser has been demonstrated in a 2 cm linear cavity. The output
power reaches 35 mW with an OSNR greater than 65 dB. A PM FBG is used
as the polarization dependent reflector to generate the single polarization out-
put with the polarization excitation ratio greater than 20 dB. The laser works
stably for two hours under laboratory conditions.
4.3 Pump Induced Dual Frequency Switching in
Ytterbium Doped Fiber Lasers
4.3.1 Introduction
Frequency-switchable fiber lasers [77, 93–96] have been proven to play
an important role in wavelength-routing wavelength-division-multiplexer (WDM)
networks. Although much effort has been focused on semiconductor lasers
[97], fiber lasers are considered to be desirable candidates as switchable sources
63
Figure 4.11: Measured transmission spectrum of the PM and SM FBGs at
room temperature.
for photonic networks. Multiple-frequency switchable fiber lasers have been
realized by designing different thresholds at various wavelengths [93] or by
changing the polarization of the various lasing modes [77, 94–96]. Such sys-
tems require overlapping multiple cavities or polarization controllers. In this
section, dual frequency switching is demonstrated in a single linear, fiber
laser cavity without any polarization controlling component. The laser fre-
quency switching is caused by the pump-induced thermal effects of the two
fiber Bragg gratings (FBG’s), and can therefore be controlled by current tun-
ing of the pump laser. This phenomenon can be used to design dual frequency
switchable fiber lasers by carefully aligning the spectra of the two gratings
along a short linear cavity [46].
64
4.3.2 Experimental Results
The experimental setup schematic is similar to figure 4.1. A 1.5 cm sec-
tion of highly ytterbium doped silica fiber was spliced between two FBG’s.
The pump absorption rate of the active fiber is 1700 dB/m. The polarization
maintaining (PM) FBG has two reflection peaks with 0.3 nm spacing due to
the different modal average index along the orthogonal fast and slow axes.
Both of the reflection bands have 3 dB bandwidths of 0.06 nm and exhibit a
peak reflectivity of 55% at the corresponding wavelengths. The single mode
(SM) FBG has a center wavelength of 1029.3 nm and a peak reflectivity of
99%. The transmission spectra of the PM FBG and SM FBG are measured
with an unpolarized amplified spontaneous emission (ASE) source and are
shown in figure 4.11. The spectra of the SM and PM FBGs overlap at room
temperature. A WDM is used to couple the 976 nm pump light into the active
fiber.
The laser output characteristics were measured from the transmission
end of the PM FBG with a power meter, a Fabry-Perot (FP) scanning spec-
trometer, and an optical spectrum analyzer (OSA). In the measurement, the
FP spectrometer had a 1 mm cavity length, corresponding to a free spectral
range of 150 GHz. With the finesse of 150, the FP cavity had a frequency res-
olution of 1 GHz. Since the 2 cm laser cavity dictates a 5 GHz mode spacing,
the FP spectrometer can resolve all longitudinal modes of the laser resonator.
The pump current-output power characteristics of the dual-frequency
65
Figure 4.12: Measured laser output power as a function of pump current.
Figure 4.13: Measured laser power as a function of pump current. The blue
curve represents the power at 1029.1 nm, the red curve represents the power
at 1029.4 nm.
66
laser are shown in figure 4.12. The available output power was measured
to be 16.5 mW. The maximum pump power in the experiment was 270 mW
at 850 mA. The rollover of the curve is due to the thermal effects, as will be
explained in the next section. The single longitudinal mode property of each
frequency was verified in previous reports [44,45].
Dual frequency switching can be achieved by tuning the power of the
pump laser. The peak power at each lasing wavelength as a function of the
pump current is displayed in figure 4.13. This figure shows that the pump
current can be selected to generate either single frequency or dual frequency
output. The laser emits equal powers at two lasing peaks when the pump
current is 250 mA, 430 mA, or 640 mA. The output power ratio of the two
lasing wavelengths differs at other pump currents. In the single frequency
working regime, at pump currents of 100 mA, 310 mA, 490 mA, and 850 mA,
the OSNR of the laser is greater than 50 dB. Additionally, the power rms
variation is <0.9% and the peak-to-valley wavelength deviation is <0.02 nm
over a 2 h period [44]. In the dual frequency working regime, the relative
peak power variation between two lasing peaks is less than 0.05% for a typical
diode pump controller [45].
4.3.3 Modeling and Simulations
The switching phenomenon demonstrated in this laser is achieved via
pump induced thermal effects. The absorption of pump light in the active fiber
leads to asymmetric heating in the laser cavity, which causes the reflection
67
spectra of the FBGs to shift with respect to each other [43]. The round trip
net gain at each of the two lasing peaks will thus be altered in the laser
cavity. The lasing thresholds of the two lasing peaks are therefore changed,
leading to differential laser gain and output. To model this phenomenon, the
axial distribution of absorbed pump power is calculated along the active fiber,
from which the temperature is calculated along the entire fiber laser cavity.
After the temperature distribution is calculated, the spectra of the two FBGs
are calculated using the temperature profile in the FBGs with the transfer
matrix method [63]. The gain thresholds at the two lasing wavelengths are
thus derived at different pump levels and phenomenologically describe the
dual frequency switching in this laser cavity.
To obtain the absorbed pump power along the fiber, a simulation is car-
ried out using an iteration method [98]. Similar to erbium doped fiber lasers,
ytterbium doped fiber lasers can be modeled as a quasi two level system [99].
In this case, the distribution of pump power, lasing signal power, and ASE
power can be described in the steady-state (i.e., cw) regime by [98]:
N2(z)N =
Pp(z)σapΓpλp+Γs∫σaPT (z,λ)λ dλ
Pp(z)(σap+σep)Γpλp+hcAτ
+Γs∫
[σe(λ)+σa(λ)]PT (z,λ)λ dλ
dPp(z)dz = −Γp [σapN − (σap + σep)N2 (z)]Pp (z)− αp (z)Pp (z)
±dP±(z,λ)dz = Γs [σe (λ) + σa (λ)]N2 (z)− σa (λ)NP± (z, λ)
+ Γsσe (λ)N2 (z)P0 (λ)− αs (z, λ)P± (z, λ)
(4.1)
where Pp(z) is the pump power; P±(z, λ) is the signal, PT (z, λ) is P+(z, λ) +
P−(z, λ), and ASE power per unit wavelength traveling in the forward or
backward direction; A is the fiber core cross section; Γp/s is the overlap fac-
68
tor of the pump/signal with the doping ions; αp/s is the scattering coefficient
at the pump/signal wavelength; N is the total population density and N2 is
the upper-state population density; σe(λ) and σa(λ) are the emission and ab-
sorption cross sections, respectively, whose values are taken from previous
work [100]; and τ is the upper state spontaneous lifetime, measured as 0.17
ms for this highly ytterbium doped fiber. The spontaneously emitted power
P0(λ) per unit wavelength is defined as
P0 (λ) = 2hc2/λ3 (4.2)
Equation 4.1 are solved numerically with a spectral bandwidth of 0.05
nm using the finite difference method. This bandwidth provides sufficient
resolution to model the observed laser behavior. The boundary conditions at
the two FBG’s are
P+ (0, λ1,2) = R0 (λ1,2)P− (0, λ1,2)
P− (L, λ1,2) = RL (λ1,2)P+ (L, λ1,2)
(4.3)
where λ1 and λ2 are the peak reflection wavelengths of the PM FBG, corre-
sponding to the lasing peaks. Since the FBG bandwidths are very small, the
reflection spectra can be approximated as
R0 (λ) =
RPM, λ = λ1, λ2
0, otherwise
RL (λ) =
RSM, λ = λ1, λ2
0, otherwise
(4.4)
69
Table 4.1: Parameters used for the laser pump simulation
Parameter Value
λP 976 nm
λS 1029.1 nm and 1029.4 nm
τ 0.17 ms
σap 2.47 ×10−24m2 [100]
σep 2.48 ×10−24m2 [100]
A 30 µm2 [100]
αP 0.001 m−1
αS 0.005 m−1
ΓP 0.85
ΓS 0.82 [98]
L 1.5 cm
RPM 0.55
RSM 0.99
70
Figure 4.14: Calculated pump distribution along the 1.5 cm active fiber at
different pump levels.
where RPM is the PM FBG peak reflectivity and RSM is the SM FBG
peak reflectivity. When equation 4.1 are solved using equations 4.2, 4.3 and
4.4 and the parameters from table 4.1, the total signal power converges at
an error of less than 10−5. The results are displayed in figure 4.14, which
shows the pump power attenuation along the fiber due to the absorption by
the ytterbium ions.
The temperature distribution in the laser is described with the following
heat-conduction equation [73]:
ρcv∂T (r, z, t)
∂t− k∇2T (r, z, t) = ηpv (r, z) (4.5)
where pv (r, z) is the pump power absorbed per unit volume, η is the conversion
efficiency of the absorbed pump into heat, and k is the thermal conductivity.
Due to the relative sizes of the core and cladding, pv can be simplified by
71
Figure 4.15: Calculated thermal distribution along the fiber laser cavity at
different pump levels.
assuming the pump power is uniformly absorbed across the core as [73]
pv (z) =
1πa2
dPp(z)dz
, in the core
0, in the cladding(4.6)
The boundary condition at the fiber-air interface is [73]
−k∇T (r) |r=b = h [T (b)− T0] (4.7)
where h is the heat transfer coefficient, b is the cladding radius of the active
fiber, and T0 is the ambient environmental temperature. Assuming azimuthal
symmetry, equations 4.5, 4.6 and 4.7 are solved in both the active and passive
sections of the fiber with the parameters from table 4.2 using the finite dif-
ference method [132]. The resultant steady state temperature distribution
along the fiber is shown in figure 4.15. Due to the single end pump geometry,
the heat generation is higher on the PM FBG end of the laser; therefore, the
temperature rise and thermal gradient in the PM FBG are larger than those
of the SM FBG, becoming very pronounced at higher pump powers.
72
Table 4.2: Parameters used for the thermal calculation
Parameter Value
η 0.05
h 81.4 W/m2 /K [73]
k 1.38 W/m/K [73]
a 3 µm
T0 22 oC
b 62.5 µm
The thermal distribution in the PM and SM FBGs will chirp the two
gratings and shift their central wavelengths. The PM FBG has a peak reflec-
tivity of 55% and a grating length of 5 mm, giving a coupling coefficient of κPM
= 1.6 cm−1. The SM FBG has a peak reflectivity of 99% and a grating length
of 4 mm, yielding a coupling coefficient of κSM = 7.1 cm−1. The coefficient of
thermal expansion of the active silica fiber is 5 ×10−7 /oC [101]. Starting with
these parameters, the FBG spectra with different thermal distributions are
calculated with the transfer matrix method [63]. In this method, the chirped
gratings are segmented into many sections, each having a uniform period of
Λ = Λ0(1 + CTE∆T ), where Λ0 is the grating period at room temperature, CTE
is the coefficient of thermal expansion, and ∆T is the grating temperature
rise. Each section is described with a fundamental transfer matrix. The ma-
trix elements are derived from standard coupled mode theory for a uniform
73
grating and are given by the equation of 3.5 and 3.6 in Chapter 2. The total
transfer matrix describing the chirped grating can be written as
[F ] =N
Πi=1
[F ]i (4.8)
and the reflection spectrum of the grating is [63]
R =∣∣∣∣F21
F11
∣∣∣∣2 (4.9)
The reflection spectrum of each FBG is calculated using the equations
of 3.6, 4.8 and 4.9. the calculated temperature distributions shown in 4.15.
Fig. 4.16 shows the detailed spectra of two FBGs under different pump cur-
rents. From the figure, the SM FBG is only minimally affected by the pump
induced heating, while the PM FBG is substantially thermally chirped. The
lasing wavelength shift is less than 0.01 nm as the pump current is increased.
Assuming that the gain curve in the measured wavelength region is uniform,
the gain threshold is determined by the reflectivities of the two FBG’s and can
be described by the simple equation
GthR1 (λ)R2 (λ) = 1 (4.10)
The laser wavelengths are therefore determined by the product of the
reflection spectra of the two FBGs. The result of generating this product from
the spectra is shown in figure 4.16. As the pump power is increased, the
induced temperature rise causes the spectrum of the PM FBG to shift with
respect to that of the SM FBG. As the peaks of the PM FBG shift, their overlap
with the structured spectrum of the SM FBG also changes, creating threshold
74
Figure 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.
75
Figure 4.17: Calculated threshold gain discrimination between the fast and
slow axes as a function of the pump current.
conditions for each peak that vary with pump current. The gain thresholds
at the two peaks (along the fast and slow axes) are calculated according to
equation 4.10. Figure 4.17 shows the threshold gain difference between the
two spectra peaks along the two polarization axes. When the gain difference
is near zero, the two modes have equivalent threshold gain and the two lasing
modes generate equal power. When the gain difference is not zero, differential
output power is produced. For example, when the pump current is 160 mA,
having the differential gain threshold shown in figure 4.10, the two wave-
lengths operate with output powers having 10 dB mode discrimination, as
can be seen in figure 4.13. Furthermore, the calculated multiple switching
currents at 250 mA, 430 mA, and 640 mA are in excellent agreement with the
switching currents measured in figure 4.13. As the pump current is increased,
the spectrum of the PM grating shows a pronounced shift to the longer wave-
length, while the SM grating shows only a minor shift. The overlap of the PM
76
Figure 4.18: Measured laser power as a function of the PM FBG temperature.
The blue curve represents the power at 1029.1 nm, the red curve represents
the power at 1029.4 nm.
FBG spectrum with the side lobes and the center reflection band of the SM
FBG spectrum are crucial for having multiple switch points in this dual fre-
quency switching phenomenon. This can be verified by observing the shifting
of the PM FBG peaks across the side lobes and nulls of the SM FBG in figure
4.16. Additionally, the walk off of the two spectra generates the total output
power rollover observed experimentally in figure 4.12. If the temperature dif-
ference between the two FBG’s becomes large enough, the output power will
be zero.
4.3.4 Discussions and Conclusions
Given the grating detuning mechanism, the dual-frequency switching
can be induced thermally without changing the pump current. Holding the
pump power constant at 270 mW, the temperature of the PM FBG is varied
77
between 25oC and 35oC. Under these conditions, the laser output spectra re-
produces the switching behavior as shown in Fig. 4.18. While both the pump
power and the temperature varied in the pump induced switching experi-
ment, the temperature variation alone generates the dual frequency switch-
ing in this experiment; therefore, the thermal chirp of the gratings is not the
main contributor of the dual frequency switching. The dual frequency switch-
ing DBR fiber laser can be designed according to the application. The side
lobes of the SM FBG can have high reflectivities as the product of the grat-
ing coupling coefficient κ and the grating length L becomes large. To achieve
the dual frequency switching from the fiber laser with a temperature rise, the
left reflection peak of the narrowband PM FBG should be overlapped with
the left side lobe of the SM FBG, the other reflection peak of the PM FBG
should be overlapped with the right edge of the SM FBG center reflection
band. Alternatively, to achieve the dual frequency switching with a temper-
ature reduction, the right reflection peak of the narrowband PM FBG should
be overlapped with the right side lobe of the SM FBG, the other PM FBG
reflection peak should be overlapped with the left edge of the SM FBG cen-
ter reflection band. Note that ambient temperature changes will affect both
gratings similarly and no switching will occur, unless one of the gratings is
temperature controlled. In either case, thermally induced spectral overlap
variation will generate dual frequency switching from the carefully designed
DBR fiber laser.
It is also important to note that the complexities due to the spectral side
78
lobes of the FBG are not required if only a single switching point is desired.
Overlapping one peak of the PM FBG with the SM FBG will generate a single
frequency output. By thermally tuning the PM FBG to make the other peak
overlap with the SM FBG, the former peak walks out of the SM FBG reflection
band, and a single switching point can be realized to achieve dual-frequency
switching.
In conclusion, using a short linear cavity composed of a section of highly
ytterbium doped fiber surrounded by two FBG’s, dual frequency switching is
achieved by tuning the pump power of the laser. The dual frequency switch-
ing is generated by the thermal effects of the absorbed pump in the ytterbium
doped fiber. At each frequency, the laser shows single longitudinal mode be-
havior. In each single mode regime, the OSNR of the laser is greater than
50 dB. The dual frequency, switchable, fiber laser can be designed for various
applications by careful selection of the two gratings.
4.4 Chapter Summary
In this chapter, linear cavity single frequency fiber lasers have been
studied and demonstrated. Firstly, a dual-frequency 2 cm silica fiber laser
with a wavelength spacing of 0.3 nm has been demonstrated using a polar-
ization maintaining (PM) fiber Bragg grating (FBG) reflector. The output
power reaches 43 mW with the optical signal to noise ratio of greater than
60 dB. Secondly, by thermally tuning the overalap between the spectra of
79
PM FBG and SM FBG, a single polarisation, single frequency fibre laser has
been demonstrated with an output power of 35 mW. Thirdly, dual frequency
switching is achieved by tuning the pump power of the laser in the dual single
frequency fiber laser. The dual frequency switching is studied and explained
as being generated by the thermal effects of the absorbed pump in the active
fiber.
80
Chapter 5
Elimination of Self Pulsing in Dual
Clad Ytterbium Doped Fiber Lasers
5.1 Introduction
High-power, high-beam-quality, stable continuous-wave (cw) fiber lasers
are desired in sensing, ranging, telecommunications, and spectroscopy [24,
102]. Although high output powers have been achieved in many high-power
fiber laser systems [24], self pulsing often occurs in cw fiber lasers under
specific pumping and cavity conditions [103]. Generally, self pulsing in fiber
lasers can be classified as sustained self pulsing (SSP) and self mode locking
(SML). SSP is the periodic emission of optical pulses at a repetition rate cor-
responding to the relaxation oscillation frequency of the inversion and photon
populations. SML is the periodic emission of optical pulses with a rate cor-
responding to the cavity round trip time [103]. Both of the regimes can be
81
described by the interaction of the photon population and the population in-
version [103,104].
Although the self-pulsations typically occur at the lower end of the pump
power range, the pulses caused by these instabilities carry sufficient opti-
cal energy to cause catastrophic damage to the fiber laser, particularly when
they are allowed to occur for extended periods of time. For this reason, there
have been intensive investigations on self-pulsation suppression in cw fiber
lasers. Electronic feedback has been used on the pump laser to shift the
gain and phase to minimize relaxation oscillations [105]. Auxiliary pump-
ing near the lasing wavelength sustains the population inversion in the gain
medium, thereby preventing rapid gain depletion and minimizing the relax-
ation oscillations [106]. The fast saturable gain of a semiconductor optical
amplifier included within the fiber-laser cavity prevents large signal buildup
in the fiber laser and suppresses the self-pulsing behavior [107]. The nar-
row passband of a λ/4 -shifted fiber Bragg grating (FBG) structure in a ring
cavity limits the number of longitudinal cavity modes and suppresses self-
pulsations [108]. Using double-ended pumping or a short active fiber can re-
duce or suppress self pulsations by providing more uniform pumping and re-
moving saturable absorption as a initialization mechanism [109, 110]. Using
a ring cavity with an isolator was shown to suppress self pulsations by elim-
inating backscattered stimulated Brillouin scattering (SBS) as a pulse initia-
tion method [111]. Self-pulsations in fiber lasers are completely described by
the interaction of the photon and inversion populations, but the initiation of
82
self-pulsations can be enhanced by additional mechanisms, such as saturable
absorption of under-pumped gain and nonlinear reflections from SBS. Due to
the various initialization mechanisms, significantly changing the cavity con-
figuration can result in a laser whose instabilities occur at a different range
of pumping levels than the original laser. This is particularly important in
the context of the previous work aimed at eliminating self-pulsations in fiber
lasers, all of which have demonstrated their reduction or suppression at only
a single or limited range of pumping levels. In this chapter, we demonstrate a
new technique to suppress self-pulsations in fiber lasers by addressing their
root cause: the dynamic interaction of the laser field and the gain. By increas-
ing the round-trip time in the laser cavity with a long section of passive fiber,
the relaxation oscillation dynamics are changed and self-pulsations are effec-
tively suppressed. Most importantly, we demonstrate that with sufficiently
long fiber, the self-pulsations can be completely eliminated at all pump power
levels [112].
5.2 Experimental Demonstration
The experimental setup is shown in figure 5.1. The 25 W pump light
at 915 nm is delivered by the pump coupling fiber F1, which has a 200-µm
diameter and 0.22 numerical aperture (N.A.). L1 and L2 are aspheric lenses
with focal lengths of 27 mm and 13.5 mm, respectively. The overall pump
coupling efficiency is 75%. The laser gain medium F2 is a 20 m, dual clad,
83
Figure 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.
Figure 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.
Figure 5.3: The self-pulsing dynamics of laser 1 when the pump power is 3.2
W.
84
Figure 5.4: The self-pulsing dynamics of laser 1 when the pump power is 7.2
W.
Figure 5.5: The self pulsing characteristics of the fiber lasers with four differ-
ent cavity lengths. The active fiber length is 20 m in all four cases.
85
ytterbium doped, single mode fiber with an absorption rate of 0.5 dB/m at
915 nm. This ytterbium-doped fiber has a 130 µm cladding diameter with an
NA of 0.46. The fiber has a core diameter of 5 µm with an NA of 0.12. One
end of the fiber is spliced into an FBG having a 3 dB bandwidth of 0.36 nm
and > 99% reflectivity at a center wavelength of 1080 nm. The other end of
the active fiber is cleaved perpendicularly, providing a 4% reflection at the
fiber air interface. The dichroic mirror D1 is inserted between L1 and L2 to
couple the laser output signal into a 2 GHz bandwidth optical detector and
a 600 MHz bandwidth oscilloscope to measure laser dynamics. Three addi-
tional configurations are characterized in this experiment. In these alternate
configurations, three long sections of passive fiber F3 (329 m, 1329 m, and
2329 m) are spliced into the laser cavity between the active fiber F2 and the
FBG. The four lasers are designated as laser 1 (20-m cavity), laser 2 (349-m
cavity), laser 3 (1349-m cavity), and laser 4 (2349-m cavity).
The lasing properties of the four configurations have been characterized.
The output power characteristics are shown in figure 5.2. The four lasers
have lasing thresholds that increase slightly with cavity length (0.68 W, 0.69
W, 0.72 W and 0.75 W, respectively). At maximum pump power, the output
power drops from 1.55 W to 1.40 W. Both of these characteristics are due
to the scattering loss of the passive fiber sections, which has an attenuation
rate of 1.2 dB/km at the lasing wavelength. The low efficiency of the laser is
caused by unoptimized laser mirror reflectivities and difficulties splicing the
single-mode FBG to the nonround dual-clad active fiber.
86
Both SSP and SML have been observed in laser 1. A cw optical output
is observed with low pump powers since the rate of gain damping dynamics,
which is proportional to the signal intensity, is slow compared to the pumping
rate of the laser system. Therefore cw operation is preferred. As the injected
pump power is increased beyond 2.0 W, quasi-periodic optical pulses, induced
by undamped relaxation oscillations, are observed in the SSP regime. Figure
5.3 shows an example of such pulsations when the pump power is 3.2 W. The
pulse period is around 20 µs, which agrees with the calculated relaxation os-
cillation frequency of the laser. As the pump power is tuned higher to 6.6 W,
SML pulsing at a rate corresponding to a cavity-round-trip time is observed.
This regime occurs because the gain medium is pumped hard enough to re-
cover the population inversion in a single-cavity-round-trip time. Figure 5.4
shows an example of such pulsations when the pump power is 7.2 W. The mea-
sured pulse period of 290 ns corresponds to the round-trip time of the laser
cavity. As the pump power is further increased beyond 7.5 W, the laser once
again operates in the cw regime because the gain is replenished more rapidly
than the time it takes for the pulse to complete a round-trip through the laser
cavity.
The physics underscored here imply that when the pumping rate is suf-
ficiently fast compared to the relaxation oscillation dynamics, the gain will
always be replenished before a pulse can build up in the cavity. The dynamics
in the SSP regime are dependent on the cavity length such that the relax-
ation oscillation frequency becomes smaller with increasing cavity length, as
87
governed by conventional laser theory. The dynamics in the SML regime are
directly dependent on the cavity length since the laser mode locks to the cav-
ity round-trip time. Therefore, by sufficiently increasing the cavity length, all
self-pulsation dynamics can be made slow compared to the pumping rate and
all self-pulsations will be eliminated.
The modulation depth of the pulsations, defined as the ratio of the peak
to valley value of the modulation to the peak value, indicates the competition
between self-pulsing and cw working regimes. Figure 5.5 shows the modula-
tion depth as a function of pump power for the four laser cavities. A modula-
tion depth of zero means the laser output is strictly cw, with no self pulsations
at any time. The measurements were taken up to a maximum pump level of
17 W. but all four of the lasers had a modulation depth of zero beyond 8 W.
As predicted by fiber-laser rate equations [124], figure 5.5 shows that the
modulation depth decreases as the fiber-laser cavity length is increased, in-
dicating a stronger tendency toward cw operation. However, the pump range
where self-pulsations occur also decreases drastically with increasing cavity
length. Laser 2 has an instability range that is less than 19% of that of laser
1, while laser 3 has an instability range that is less than 7% of that of laser
1. For laser 4, the instability range reduces to zero and no self-pulsations
occur over the entire pump range, from lasing threshold up to 23 times above
threshold.
88
5.3 Nonlinear Effects and Self Pulsing Dynamics
For fiber lasers having long cavity lengths such as in laser 4, nonlin-
ear effects such as stimulated Raman scattering (SRS), SBS, and the Kerr
effect, can have a significant impact on the laser operation, particularly at
high power levels. In the experiments described above, no SRS spectra above
the noise floor were observed, but SRS can be induced at higher pump lev-
els. For example, a laser with a 1-km cavity length has an SRS thresh-
old of about 5 W. SRS can be mitigated with appropriate filters, such as
wavelength-division multiplexers, in-line short-pass filters, or hole-assisted
single-polarization fibers [114]. Large-mode-area fiber can also be used in
fiber laser systems to suppress SRS in long fiber lasers. For example, higher-
order-mode (HOM) fiber with a mode-field diameter of 86 µm [19] can increase
the nonlinear threshold by a factor of 200 compared to normal single-mode
fiber, which mitigates SRS in the kilometer-long laser.
SBS and the Kerr effect can destabilize high-power fiber lasers by in-
ducing self pulsations. For a non-resonant fiber of 2349 m, the SBS threshold
is 0.7 W for a signal of the bandwidth of our laser [56]. In the laser, the SBS
threshold is even lower owing to resonance. Therefore, our laser operates
well above the SBS threshold. The mode spacing of the 2 m laser was 50 kHz,
implying millions of modes within the 0.36 nm FBG bandwidth. Any SBS-
induced Stokes-shifted modes cannot be distinguished from the Fabry-Perot
cavity modes. The threshold for Kerr-induced nonlinearity is 0.2 W [109]. The
89
laser reported here is operating well in excess of this value and is clearly in
the regime where such nonlinearities can play a significant role in initiating
self-pulsation behavior. However, no such instabilities are observed, since the
passive fiber changes the underlying physics of self-pulsations regardless of
their initiation mechanism.
5.4 Discussions and Chapter Summary
Using long lengths of passive fiber to suppress self-pulsing has many
advantages over other methods. No active components or electronics are re-
quired, resulting in reduced system complexity. This method does not require
free-space alignment and can be easily integrated into existing laser systems;
2.5 km of fiber can be implemented on a spool that is 12 cm across and less
than 2 cm high. This technique can be applied to fiber-laser systems working
at 1 µm (Yb) and 1.5 µm (Er), where low-loss passive fiber is available. The
theoretical loss limit of chalcogenide fiber [115] will also make it possible for
this method to be eventually applied to 2 µm fiber lasers (e.g., Tm).
In summary, suppression and elimination of self-pulsing in a watt-level,
dual-clad, ytterbium-doped fiber laser have been demonstrated. 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. This scheme provides a simple and practical method for eliminating
self-pulsations in fiber lasers at all pumping levels.
90
Chapter 6
Power Scaling of Single-Frequency
Hybrid Ytterbium/Brillouin Fiber
Lasers
6.1 Introduction
Single-frequency fiber lasers have potential applications in sensing [116],
ranging [117], data storage [118], communications [119], and interferome-
try [89]. Watt-level single frequency output has been achieved from distributed-
feedback fiber lasers [27], short linear-cavity fiber lasers [23], and ring-cavity
fiber lasers with narrow-bandwidth filters [40]. Pump coupling and thermal
effects limit the output powers of these lasers. Another approach for gener-
ating high-power single-frequency sources is using master-oscillator/power-
amplifier (MOPA) schemes with a single-frequency seed [24, 66]. However,
the amplified spontaneous emssion (ASE) noise degrades the optical signal-
91
to-noise ratio (OSNR) of these single frequency sources. Futhermore, single-
frequency performance is limited by multiple-order stimulated Brillouin scat-
tering (SBS) in fiber amplifiers due to the narrow bandwidth of the seed [66].
From the system complexity point of view, a MOPA architecture is more com-
plicated than a single oscillator.
Stimulated Brillouin scattering has a narrow gain bandwidth and can
therefore be used as a narrowband filter for generating single-frequency laser
output. Hybrid Brillouin/rare-earth fiber lasers have been demonstrated by
using the conventional gain in a rare-earth doped fiber and the nonlinear
Brillouin gain in a section of passive fiber [120, 122, 123]. Tens of milliwatts
of single-frequency output has been achieved, and is limited by the single-
spatial-mode pump power. Higher single-frequency output powers from hy-
brid Brillouin/rare-earth fiber lasers have not been studied yet. In this chap-
ter, a numerical model with multiple-order stimulated Brillouin scattering
(SBS) is used to study the power scalability of hybrid ring fiber lasers to high
powers. In section 6.2, the general model is presented describing the light
and population in the Yb-doped fiber, including multiple-order SBS. In sec-
tion 6.3, a hybrid laser is built and compared to simulation results to validate
the model. In section 6.4, the model is applied to a dual-clad ring cavity fiber
laser to investigate the power scability. In section 6.5, a 1 Watt single fre-
quency fiber laser is demonstrated.
92
Figure 6.1: The schematic of a general single-frequency hybrid Bril-
louin/ytterbium fiber laser. ISO is the isolator. YDF is the dual-clad
ytterbium-doped fiber.
6.2 Numerical Model
The general architecture of a single-frequency hybrid Brillouin/ytterbium
fiber laser is shown in figure 6.1. The active fiber can be core pumped or dual-
clad ytterbium doped fiber depending on spatial mode properties of the pump
laser. The ring cavity length must be less than 16 m to allow single frequency
output using the 20 MHz Brillouin gain bandwidth in the fiber medium. The
pump coupler can be a wavelength division multiplexer (WDM) or a dual-clad
pump combiner. The pump laser wavelength can be at 915 nm or 976 nm. The
Brillouin pump must be a single-longitudinal-mode laser source.
The numerical model describes the interaction between the pump wave,
the Brillouin pump wave and multiple orders of Stokes waves. Specifically,
in the active fiber section, the general coupled wave equation can be written
93
as [66]
dP±idz
= ±[(σei + σai )n2 − σai ]N0ΓiP±i ∓ αiP±i ± gBΨP±i (P∓i−1 − P∓i+1)± gSB(P∓i−1 − P±i )
(6.1)
Where Pi is the power of the ith optical wave including the pump wave and
Brillouin seed, and + and − denotes the propagation direction. σei and σai
are the emission and absorption cross sections of the ytterbium ions at the
ith optical wavelength. N0 is the doping density and Γi is the overlap factor
between the ith optical wave and the doping ions. αi is the attenuation factor
at the ith wavelength, Ψ is the inverse of the effective mode area, and gB
is the net Brillouin gain coefficient. gSB = gBΨPN is the gain coefficient for
the spontaneous process, where PN = hν∆ν and ∆ν is the linewidth of the
Brillouin pump beam in Hz. n2 is the population inversion in the active fiber
section and can be written as [66]
n2 =
∑iσai Γi(P
+i + P−i )(Ahc/λi)
−1
1τ2
+∑i
(σei + σai )Γi(P+i + P−i )(Ahc/λi)−1
(6.2)
where A is the effective mode area, τ2 is the upper state lifetime, and λi is
the wavelength of the ith optical wave in the ytterbium fiber. In applying
this model, the effect of SBS on the pump wave is neglected due to its large
spectral bandwidth. The Pi−1 term is not valid for the Brillouin seed. The
performance of single-frequency hybrid Brillouin/ytterbium fiber lasers can
be modeled by solving Equations 6.1 and 6.2 with appropriate boundary con-
ditions for each wave in the cavity. The finite difference method is used to
discretize the power along the fiber [132], and the relaxation method is used
94
Figure 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.
to find the steady state solution by iterating many round trips in the ring cav-
ity, where the result of the last round trip is used for the initiation value for
the next round trip calculation.
6.3 Experimental Verification
A single-frequency hybrid Brillouin/ytterbium fiber laser was built to
validate the numerical model presented in Section 6.2. The experimental
setup of the hybrid fiber laser is shown in Fig. 6.2. A wavelength division mul-
tiplexer (WDM) is used to couple the 976 nm pump power into the core of the
ytterbium-doped fiber. The active fiber is 40 cm long and highly ytterbium-
doped with a pump absorption rate of 12 dB/cm. A 50-kHz-bandwidth laser
source (Koheras) at 1053 nm is seeded through a 80/20 coupler into the laser
cavity. When the Brillouin pump is injected into the cavity, it is first ampli-
fied by the ytterbium-doped fiber. The passive single mode fiber is about 10
95
Figure 6.3: The laser output power as a function of 976 nm pump at three
different Brillouin pump powers.
Figure 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.
96
Figure 6.5: The laser output spectrum on the scanning FP spectrometer with
370 mW of 976-nm pump and 9 mW of Brillouin pump.
m in length and functions as the Brillouin gain medium. The SBS stokes
wave is generated in the single mode fiber and circulates in the ring cavity.
The isolator with an insertion loss of 1.5 dB at 1053 nm is used to prevent
the laser from injection locking at the Brillouin pump wavelength. The 80/20
coupler couples the Brillouin pump into the cavity and the laser light out of it
through the 80% port, while only 20% of the laser light remains in the cavity.
Although many coupling configurations were tested, this configuration was
experimentally found to yield the highest output power by balancing the re-
quired Brillouin pump power against the cavity loss, which is compensated
by the ytterbium-doped fiber amplifier.
The laser power is measured using another coupler, and back calculated
to obtain the laser output power as shown in Fig. 6.2. Fig. 6.3 displays the
output power as a function of the pump power for three different Brillouin
pump powers. When the Brillouin pump is 9 mW, the laser threshold is 125
mW, and the laser output power reaches 40 mW at a pump power of 370
97
mW. The high threshold is due to the low coupling ratio of the coupler (20%).
The 40-cm length of highly doped ytterbium fiber is long compared to the
pump absorption length but is necessary to shift the gain peak, making the
free running laser work at 1051 nm, which is sufficiently close enough to the
Stokes wavelength of 1053.05 nm for easy injection locking. Fig. 6.3 shows
that as the Brillouin pump power is increased from 9 mW to 18 mW, the laser
threshold increases and thus output power decreases. This is due to the gain
saturation in the ytterbium-doped fiber by the Brillouin pump, which leads to
less available gain for amplification of the Stokes wave and thus lower output
power.
The optical signal to noise ratio (OSNR) was measured with an optical
spectrum analyzer (OSA) at 40 mW output power. The laser output spectrum,
shown in Fig. 6.4, indicates an OSNR greater than 50 dB. Additionally, the
Brillouin pump wavelength is 1053.05 nm, while the fiber laser wavelength
is 1053.1 nm. The frequency difference between the Brillouin pump and the
laser output matches the expected Stokes shift of the SBS at the working
wavelength.
Single frequency output is verified with a scanning Fabry-Perot (FP)
spectrometer. The FP cavity length is 15 cm, corresponding to a free spectral
range (FSR) of 1 GHz. The FP cavity has a finesse of 160, giving a resolution
of 6.25 MHz. Since the laser cavity length is 12 m, corresponding to 16 MHz
mode spacing, the resonant modes of the ring cavity can be resolved by the FP
spectrometer. The measurement result shown in Fig. 6.5 clearly shows single
98
frequency operation. Although multiple cavity modes can in principle operate
within the about 20 MHz SBS gain profile, the SBS lineshape provides sig-
nificant longitudinal mode discrimination such that only a single mode will
operate in the ring cavity. No mode hopping is observed in the measurement
at any pump level.
The hybrid laser shows high stability. From the oscilloscope, the laser
output is stably continuous wave for over 2 hours without any self pulsations.
The high stability can be explained by the saturation of the ytterbium-doped
fiber amplifier due to Brillouin pump preamplication in the fiber laser [125].
When the Brillouin pump is sufficiently large, there is no free gain in the satu-
rated ytterbium-doped fiber amplifier, and thus no extra population inversion
is built up.
When there is no Brillouin pump injected into the laser cavity, the free
running laser works with multiple longitudinal modes at 1051 nm. In this
sense, the Brillouin/ytterbium fiber laser can be treated as a ytterbium fiber-
laser injection locked by the Stokes wave generated in the ring cavity [128],
where the isolator prevents the laser from injection locking to the Brillouin
pump. For a conventional ring cavity Brillouin laser, the pump-coupling ratio
depends on the cavity loss and the round-trip phase shift in the cavity, which
must be an integer multiple of 2π to achieve the intensity enhancement (maxi-
mum intensity). For this reason, piezo controllers or tunable couplers are nor-
mally required to maintain resonance between the cavity and the Brillouin
pump wavelength [57–59, 126]. This added complication yields a Brillouin
99
pump-intensity enhancement determined by the finesse of the resonator [57],
which can be typically on the order of 10 to 1000. In the hybrid-laser con-
figuration, an isolator in the cavity prevents the pump wave from traveling
multiple round-trips and, thus, eliminates the need for cavity or frequency
control. This results in a loss of Brillouin pump intensity, which is compen-
sated for by inserting a pumped active fiber in the cavity, i.e., a hybrid laser.
In the demonstrated configuration shown in Fig. 6.2, the short-length-fiber
cavity makes the laser less influenced by mechanical and thermal fluctua-
tions and, therefore, entirely free from mode hopping [129]. Furthermore, the
demonstrated hybrid fiber laser generates higher output power than the in-
jected Brillouin pump, as shown in Fig. 6.3, making the laser advantageous
to previously demonstrated hybrid lasers [120,121].
6.3.1 Full Injection Locking and Gain Saturation
Applying Equations 6.1 and 6.2 to the active fiber section, the interaction
between the 976 nm pump, the Brillouin pump and multiple order Stokes
waves can be written as
dP−Pdz
= −[(σeP + σaP )n2 − σaP ]N0ΓPP−P + αPP
−P
dP+0
dz= [(σe0 + σa0)n2 − σa0 ]N0Γ0P
+0 − α0P
+0 − gBΨP+
0 P−1 − gSBP+
0
dP−1dz
= −[(σe1 + σa1)n2 − σa1 ]N0Γ1P−1 + α1P
−1 − gBΨP+
0 P−1 − gSBP+
0 + gBΨP−1 P+2 + gSBP
−1
dP+2
dz= [(σe2 + σa2)n2 − σa2 ]N0Γ2P
+2 − α2P
+2 + gBΨP−1 P
+2 + gSBP
−1
(6.3)
100
Where PP is the 976 nm pump wave, P0 is the Brillouin pump wave, P1 is the
first order Stokes wave. P2 is the second order stokes wave. The spontaneous
emission has been neglected in the equation because spontaneous Brillouin
scattering dominates the noise [66]. The population inversion n2 is identical
to the term in Equation 6.2 where the inversion is induced by the four waves
described by Equation 6.3. σei and σai are the emission and absorption cross
sections of the ith wave, whose values are taken from reference [100]. The
upper state lifetime is measured as 0.17 ms for the highly doped fiber [44].
The polarized Brillouin gain coefficient is 4.6 × 10−11 m/W. In single-mode
fibers, the Brillouin gain is reduced by a factor of 0.667, yielding gB=3 × 10−11
m/W [130,131].
In the passive fiber section, the same set of equations are applied with
n2 and the stimulated absorption and emission cross sections set to zero. The
boundary conditions for the ring cavity fiber laser are
PP (L1) = Pp
P0(0) = ρPb
P1(0) = βρP1(L1 + L2)
P2(0) = 0
(6.4)
Where Pp is the 976 nm pump power and Pb is the Brillouin pump power.
L1 and L2 are the lengths of the ytterbium-doped and passive fiber respec-
tively. The coupling ratio of the coupler ρ is 20%. The insertion loss of the
isolator β is 0.6. Relating to Figure 6.2, the point z=0 is where the Brillouin
pump is coupled into the cavity. The positive z axis is towards the SM fiber
101
Table 6.1: Additional physical parameters used for the simulation
L1 40 cm
L2 10 m
∆ν 50 KHz
A 30 ×10−12 m2
N0 1.25 ×1026
h 6.63 ×10−34
end. The isolator typically has an insertion loss of 20-30 dB, such that the
power at z=0 is small but non-zero. Including this would make the calcula-
tion significantly more complex without significantly improving the accuracy
of the result. Equations 6.2, 6.3 and 6.4 are solved with the finite difference
method [132], and the relaxation and shooting methods [133]. The loss co-
efficient of the fiber is 3 × 10−5m−1. The additional parameters used in the
simulation are shown in tables 6.1 and 6.2.
When the 976 nm pump power is 370 mW and the Brillouin pump power
is 9 mW, the power distributions of the optical waves inside the laser cavity
are shown in Fig. 6.6. The 976 nm pump PP is quickly absorbed in the active
fiber due to the high ytterbium doping density in the active fiber. Experimen-
tally, the active fiber length of 40 cm was chosen such that the free running
laser peaks (1051 nm) were sufficiently close to the Brillouin Stokes wave-
length (1053.05 nm) to easily injection lock the laser. The Brillouin pump
102
Figure 6.6: The power distributions of the optical waves in the active and
passive fiber. Pp=370 mW, Pb=9 mW.
103
Table 6.2: Wave-dependent parameters for the simulation
Subscript λ Γ
P 976 nm 0.8
0 1053 nm 0.9
1 1053.05 nm 0.9
2 1053.1 nm 0.9
is amplified by the ytterbium-doped fiber before entering the passive fiber.
The 9 mW Brillouin pump is effectively turned into a 50 mW seed for SBS
generation in the passive fiber. This feature makes this particular hybrid
configuration power scalable without scaling the narrow-line Brillouin pump
power.
Fig. 6.6 shows that the first-order Stokes wave is generated from the
scattering of the Brillouin pump in the passive fiber and amplified by the
active fiber. Note that the Brillouin seed is not fully converted into the first
Stokes wave. A longer fiber length could be used for full conversion, but would
lead to multiple-longitudinal-mode operation. Due to the intracavity isola-
tor, the second-order Stokes wave is 100 dB lower than the first-order Stokes
wave. The absence of second-order Stokes wave motivates the power scaling
of this hybrid configuration.
When the laser is fully injection locked to the Brillouin Stokes wave,
Fig. 6.3 shows that the measured output power decreases as the Brillouin
104
Figure 6.7: Simulated and measured output power as a function of Brillouin
pump power when the pump power Pp is 370 mW.
pump power is increased. This can be explained by the gain saturation in-
duced by the Brillouin pump amplification in the ytterbium doped fiber. Fig-
ure 6.7 shows the simulated laser output power versus the seeded Brillouin
pump as the 976 nm pump is kept at 370 mW. The simulation result agrees
with the measured output power reduction, also shown in the figure, which
is due to the gain saturation generated by Brillouin pump in the active fiber.
When the Brillouin pump power is increased for the fiber laser, the gain in
the active fiber is more deeply saturated and therefore the first order Stokes
wave power decreases.
6.3.2 Partial Injection Locking
When the Brillouin pump power is relatively low, the hybrid Brillouin/ytt-
erbium fiber laser will not be fully injection locked to the Stokes wave. The
measured OSNR versus the Brillouin pump power is shown in Fig. 6.8 with
105
976 nm pump power of 370 mW. When the Brillouin pump is less than 4 mW,
the laser is not injection locked, and the laser operates with the free-running
cavity modes. When the Brillouin pump is between 4 mW and 9 mW, the laser
is partially injection locked to the Brillouin Stokes wave. When the Brillouin
pump is greater than 9 mW, the laser is fully injection locked at the Stokes
wavelength of the pump. To model the partial injection locking of the hybrid
Brillouin/ytterbium fiber laser, a revised numerical model based on Equations
6.1 and 6.2 is applied to the fiber laser. In the active fiber section, the inter-
action between the 976 nm pump wave, the first order Brillouin Stokes wave
and the free running laser wave can be written as
dP−Pdz
= −[(σeP + σaP )n2 − σaP ]N0ΓPP−P + αPP
−P
dP+0
dz= [(σe0 + σa0)n2 − σa0 ]N0Γ0P
+0 − α0P
+0 − gBΨP+
0 P−1 − gSBP+
0
dP−1dz
= −[(σe1 + σa1)n2 − σa1 ]N0Γ1P−1 + α1P
−1 − gBΨP+
0 P−1 − gSBP+
0
dP−Fdz
= −[(σeF + σaF )n2 − σaF ]N0ΓFP−F + αFP
−F
(6.5)
where PF is the free running laser signal wave. There is no SBS term for
PF because of its large bandwidth. Due to the isolator in the ring cavity, the
second-order SBS is negligible as verified in section 3.1. The free-running
laser wavelength is 1051 nm. In the passive fiber section, the same set of
equations is applied with n2 and the stimulated absorption and emission cross
sections set to zero.
106
Figure 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.
The boundary conditions for equations 6.5 are
PP (L1) = Pp
P0(0) = ρPb
P1(0) = βρP1(L1 + L2)
PF (0) = βρPF (L1 + L2);
(6.6)
Where β and ρ have the same values as the previous section. These boundary
conditions explicitly show the competition between the Stokes wave P1 and
the free-running wave PF . The parameters used in the simulation are the
same as the parameters shown in tables 6.1 and 6.2.
When the 976 nm pump is kept at a constant level, the injection lock-
ing quality of the laser, defined by the OSNR, varies as the Brillouin pump
is changed. The simulated OSNR is plotted with the measured OSNR in
Fig. 6.8. When the free running laser power is greater than or equal to
107
the Brillouin Stokes power, the OSNR is defined as 0 dB. As one can see in
Fig. 6.8, excellent agreement is achieved between the simulation and mea-
surement results. When the Brillouin pump is below the threshold value of 9
mW, the fiber laser is not fully injection locked. Beyond this threshold value,
the laser operates in a single frequency, fully injection locked to the Brillouin
Stokes wave.
6.4 Power Scaling of Single Frequency Hybrid
Brillouin/Ytterbium Fiber Laser
The numerical model presented in the previous section describes single-
frequency hybrid fiber lasers with high accuracy and can therefore be used to
explore power scaling. Higher output power can be expected from a single-
frequency Brillouin/ytterbium fiber laser when the pump power is further in-
creased. The model of equation 6.1 can be used to predict the performance
of high-power single-frequency Brillouin/ytterbium fiber lasers, but the laser
configuration must change. In high-power fiber lasers, the pump power is cou-
pled into the dual-clad active fiber by using pump combiners and multimode
pump laser diodes. This means that the active fiber must be much longer to
absorb the pump power. In order to maintain the single-frequency operation,
the total fiber length should be less than 16 m. Therefore, the passive fiber
must likely be moved from the system. As such, the active fiber provides both
the material gain and the nonlinear medium that produces the SBS Stokes
108
wave. In typical high-power fiber amplifiers, SBS degrades the operation of
the amplifier [66]. In the demonstrated configuration, it is used as an advan-
tage to produce single-frequency operation at high power levels.
For the numerical study, the active fiber length for the laser shown in
figure 6.1 is 15 m and the passive fiber length is zero. The coupler has a
coupling ratio of 70/30 where 70% of the laser light is kept in the ring cavity.
The coupled wave equations 6.1 in the active fiber can be written as
dP−Pdz
= −[(σeP + σaP )n2 − σaP ]N0ΓPP−P + αPP
−P
dP+0
dz= +[(σe0 + σa0)n2 − σa0 ]N0Γ0P
+0 − α0P
+0 − gBΨP+
0 P−1 − gSBP+
0
dP−1dz
= −[(σe1 + σa1)n2 − σa1 ]N0Γ1P−1 + α1P
−1 − gBΨP+
0 P−1 − gSBP+
0 + gBΨP−1 P+2 + gSBP
−1
dP+2
dz= +[(σe2 + σa2)n2 − σa2 ]N0Γ2P
+2 − α2P
+2 + gBΨP−1 P
+2 + gSBP
−1 − gBΨP+
2 P−3 − gSBP+
2
dP−3dz
= −[(σe3 + σa3)n2 − σa3 ]N0Γ3P−3 + α3P
−3 − gBΨP+
2 P−3 − gSBP+
2 + gBΨP−3 P+4 + gSBP
−3
dP+4
dz= +[(σe4 + σa4)n2 − σa4 ]N0Γ4P
+4 − α4P
+4 + gBΨP−3 P
+4 + gSBP
−3
(6.7)
Where PP is the 915 nm pump, P0 is the Brillouin pump, Pi is the ith order
Stokes wave. The fourth order Stokes is included to make the third order
Stokes wave more accurate. The loss coefficient of the fiber is 3 × 10−5m−1.
The other parameters used for the simulation are shown in table 6.3 and
table 6.4.
109
Table 6.3: Additional physical parameters used for the simulation
L1 15 m
∆ν 1MHz
A 30× 10−12m2
N0 1.25× 1025
h 6.63× 10−34
The boundary conditions are
PP (L1) = Pp
P0(0) = ρPb
P1(0) = βρP1(L1)
P2(0) = 0
P3(0) = P3(L1)/(βρ)
P4(0) = 0
(6.8)
Where L1 is the active fiber length. Pp is the 915 nm pump and Pb is the
Brillouin pump. The insertion loss of the isolator β is 0.6. The coupling ratio
of the coupler ρ is 70%.
Equations 6.7, 6.8, and 6.2 are solved with finite difference and shoot-
ing methods. The details of the numerical calculation can be found in the
appendix. With the Brillouin pump power set at 400 mW, the first-and second-
order Stokes powers are shown in Fig. 6.9. The first-order Stokes output
power reaches 1 W before the second-order Stokes power starts to degrade
110
Table 6.4: Wave dependent parameters for the simulation
Subscript λ Γ
P 915 nm 0.01
0 1080 nm 0.9
1 1080.05 nm 0.9
2 1080.1 nm 0.9
3 1080.15 nm 0.9
4 1080.2 nm 0.9
the laser efficiency. Due to the isolator, the second-order SBS does not res-
onate as a laser mode, which enables high-power single-frequency output.
Fig. 6.10 shows the power distribution of the first-and second-order Stokes
waves with the 915 nm pump and Brillouin pump in the active fiber on linear
and logarithmic scales. The output power is 1.2 W and Brillouin pump power
is 400 mW. From the figure, one can see that 80% of the 915 nm pump power
is absorbed by the active fiber. The Brillouin pump propagates from the left-
hand side and is amplified in the active fiber and coupled into the first-order
Stokes wave. The first-order Stokes wave propagates from right-hand side
first attenuating before being amplified. This attenuation is due to cascaded
SBS where part of the first-order Stokes wave is transferred to the second-
order Stokes wave, which grows rapidly towards the right end of the active
111
Figure 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.
fiber. The intracavity power of the first-order Stokes wave is nearly 4 W at
the left-hand side, where 30% of the light is coupled out as laser output.
The minimum Brillouin pump power required for full injection locking
is also calculated and shown in figure 6.11. The required Brillouin pump
power for a 2 W single-frequency output power is 400 mW. Below this Bril-
louin pump threshold, the laser is only partially injection locked with rela-
tively low OSNR.
Due to the directional properties of SBS, the single-frequency perfor-
mance of the laser is ultimately limited by the third-order Stokes wave, which
propagates in the same direction as the first-order Stokes and is not blocked
by the isolator. The laser output side-mode-suppression ratio (SMSR) is deter-
112
Figure 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.
113
Figure 6.11: The required Brillouin pump power for full injection locking as a
function of output power.
Figure 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.
114
Figure 6.13: The pump power at which the second-order Stokes wave reaches
threshold as a function of output coupler ratio.
mined by the power ratio between the first-order Stokes wave and the third-
order Stokes wave. As the laser output power varies between zero and 2 W,
the simulated SMSRs are shown in figure 6.12. When the output power is 1.8
W, the third order Stokes power becomes pronouned and the SMSR degrades
to 50 dB.
The output coupler plays an important role in the single frequency hy-
brid Brillouin/ytterbium fiber laser. The coupler ratio changes the ratio be-
tween the intracavity power and the output power and also affects the amount
of Brillouin pump that is injected into the laser cavity. Fig. 6.13 shows the
second-order Stokes threshold pump powers versus the coupler ratio. The
threshold pump power decreases as the coupler ratio is increased due to laser
intracavity first-order Stokes power. Fig. 6.14 shows the first-order Stokes
output power and the required Brillouin pump power with different coupling
ratios when the pump powers are set at the second-order Stokes threshold
115
Figure 6.14: The laser output power and the required Brillouin pump power
at the second-order Stokes wave thresholds with different coupler ratios.
Figure 6.15: The side mode suppression ratio (SMSR) of lasers with different
coupler ratios working at the second-order Stokes wave threshold.
116
pump powers. From this figure one can see that as the coupler ratio is in-
creased from 0.1 to 0.8, lower available output power at the second-order
Stokes threshold. However, as a higher output power is achieved from the
laser with a lower coupler ratio, a higher Brillouin pump seed is required to
fully injection lock the Stokes output. The SMSR of the laser as a function
of the coupling ratio is shown in Fig. 6.15. The output SMSR remains above
80 dB regardless of coupler ratios since the laser is operating at the second
order SBS threshold. From Fig. 6.13, Fig. 6.14, and Fig. 6.15 one can see that
up to 5 W single frequency output power can be achieved in the hybrid Bril-
louin/ytterbium fiber laser at high efficiency with a side-mode-suppression
ratio greater than 80 dB.
SBS generates the first-order Stokes output but also limits the output
power with the onset of second-order Stokes. Therefore, higher output power
can be achieved with large mode area fiber. Increasing the mode area reduces
the intensity at a fixed power level. Since the second-order Stokes generation
is an intensity dependent phenomenon, its threshold will scale with the mode
area. Comparing to the results for a 6-µm core, 20-µm core ytterbium-doped
fiber can be used to scale up the output power of the single-frequency laser by
a factor of 10.
6.4.1 1-W Single-Frequency Hybrid Brillouin/Ytterbium Fiber Laser
Single-frequency Brillouin/rare-earth fiber lasers have been demonstrated
with output power up to tens of milliwatts [120,122,123], which is limited by
117
Figure 6.16: Schematic diagram of the single frequency hybrid Bril-
louin/ytterbium fiber laser. ISO is the high power isolator. YDF is the dual-
clad ytterbium-doped fiber. LD is laser diode.
Figure 6.17: The output power versus the pump power.
the relatively low pump power available from single-spatial mode pump laser
diodes. In this section, a watt-level single-frequency hybrid Brillouin/ytterbiu-
m fiber laser is demonstrated using multimode pump laser diodes and dual-
clad ytterbium-doped fiber.
The experimental setup is shown in figure 6.16. A fused tapered pump
combiner (OFS) is used to couple the pump light from two 915-nm laser diodes
into the inner clad of the dual-clad ytterbium-doped fiber. The maximum
118
Figure 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 Brillouin seed, the
blue curve is the laser output. The OSA resolution is 0.02 nm.
Figure 6.19: The laser output spectrum on the scanning F-P spectrometer
when the output power is 1 W.
119
achievable pump power is 14 W after the pump combiner. The active fiber
is 12 m long and ytterbium-doped with a pump absorption rate of 0.5 dB/m.
The total cavity length is 15 m. The 2 mW seed source is a 1-MHz bandwidth
distributed-feedback semiconductor laser source at 1080 nm. This is ampli-
fied to 300 mW by a dual-clad ytterbium-doped fiber amplifier then coupled
through a 70/30 coupler into the hybrid laser cavity. The active fiber func-
tions as the Brillouin gain medium. The SBS Stokes wave is generated in the
passive and active fiber and circulates in the ring cavity. An isolator with an
insertion loss of 1.5 dB at 1080 nm is used to prevent the laser from injection
locking to the Brillouin pump. The 70/30 coupler couples the Brillouin pump
into the cavity and the laser light out of it through the 30% port, while 70%
of the laser light remains in the cavity.
In this configuration, the required Brillouin seed power for a 1 W single-
frequency output power is 300 mW, in excellent agreement with figure 6.11.
When Brillouin seed is below this value, the laser is partially injection locked
to the Stokes wavelength and operates in the multiple frequency regime.
The single-frequency laser output is collected using another coupler with the
coupling efficiency of 4% and back calculated to achieve the output power.
Fig. 6.17 shows the output power as a function of the pump power when the
Brillouin seed is 300 mW. The laser pump threshold is 1.2 W, and the output
power reaches 1 W at a pump power of 10 W.
The Brillouin seed wavelength is 1080.20 nm, while the laser output
power is 1080.26 nm. The frequency difference between the Brillouin seed
120
and the laser output matches the Brillouin Stokes shift at the operating wave-
length.
The optical signal-to-noise ratio (OSNR) was measured with an optical
spectrum anlayzer (OSA) when the laser output power was 1 W. The laser
spectrum and the Brillouin seed spectrum were normalized to the same peak
power to compare the OSNRs. The noise of the Brillouin seed comes from
the noise of the semiconductor laser and the amplified spontaneous emission
(ASE) in the fiber amplifier. As shown in Fig. 6.18, the laser OSNR is greater
than 55 dB, while the OSNR of the 300 mW Brillouin seed has an OSNR of
about 35 dB. This demonstrates that the OSNR of the hybrid single frequency
fiber laser can be 20 dB higher than the seed source due to the intensity
and phase noise reduction of the SBS process [60, 126]. Therefore, a low-
noise high-power fiber laser can be achieved in this single frequency hybrid
Brillouin/ytterbium laser architecture.
The single-frequency output is verified with a scanning Fabry-Perot (FP)
spectrometer. The 15 cm long cavity length corresponds to a free spectral
range (FSR) of 1 GHz. With a finesse of 150, this FP spectrometer gives a res-
olution of about 6.7 MHz. Compared to the cavity mode spacing of 14 MHz for
the 15-m laser ring cavity, the cavity modes of the fiber laser can be resolved.
The single frequency operation is shown in Fig. 6.19. Although two cavity
modes can in principle coexist in the 20 MHz SBS gain bandwidth, the SBS
gain profile provides sufficient discrimination to make the laser operate in the
single frequency regime. Due to the relatively short length of the laser cavity,
121
no mode hopping is observed at all pumping levels in the measurement.
To achieve higher single-frequency output, higher pump power must be
coupled into the cavity. The NA mismatch between the fused tapered pump
combiner and the dual clad fiber generates heat and can burn the fiber in-
terface. This can be eliminated by proper engineering of the pump combiner
and dual-clad fiber. Additionally, the damage threshold of the isolator limits
the maximum power in the laser cavity. Currently commerical products are
limited to 30 W, but all-fiber isolators will likely remove this barrier in the
near future [127].
6.5 Chapter Summary
In summary, a coupled-wave rate-equation model, including multiple
order stimulated Brillouin scattering (SBS), was used for studying power
scaling of single-frequency hybrid Brillouin/ytterbium fiber lasers. A single-
frequency, Brillouin/ytterbium fiber laser with laser output of 40 mW and an
optical signal-to-noise (OSNR) greater than 50 dB was demonstrated to val-
idate the model. The numerical simulations agreed with the measurements
in both fully and partially injection-locked regimes. To scale up the laser out-
put power, a dual-clad single-frequency hybrid Brillouin/ytterbium fiber laser
was proposed. In this new configuration, the active Yb-doped fiber provided
the nonlinear SBS gain as well as the gain resulting from the excited Yb
ions. Numerical modeling including three Stokes orders showed that over 5
122
W of single-frequency laser output can be achieved with a side-mode suppres-
sion ratio (SMSR) greater than 80 dB. Beyond this power, multi-order SBS
affects the laser efficiency and SMSR. Experimentally, a high-power single-
frequency hybrid Brillouin/Ytterbium fiber laser was demonstrated with an
output power of 1 W. The laser worked in the single-frequency regime with an
optical signal-to-noise ratio (OSNR) greater than 55 dB.
123
Chapter 7
Conclusion and Future Work
7.1 Thesis Conclusion
High power fiber lasers have a lot of potential applications in many fields.
This thesis studied different approaches for achieving high power single fre-
quency outputs.
Axial gain apodization on the lasing threshold and spectral modal dis-
crimination of DFB lasers were studied. It was shown that if properly tai-
lored, the lasing threshold can be reduced by 21% without sacrificing modal
discrimination, while simultaneously increasing the differential output power
between both ends of the laser.
A dual single-frequency fiber laser was demonstrated using a polariza-
tion maintaining (PM) fiber Bragg grating (FBG) reflector. The birefringence
of the PM FBG was used to generate the two single mode (SM) lasing frequen-
124
cies of orthogonal polarizations. The SM operation in each wavelength was
verified. The output power reached 43 mW with the optical signal to noise ra-
tio of greater than 60 dB. The fiber laser showed stable dual frequency output
under pump variations. Additionally, a single polarization, single frequency,
ytterbium doped silica fibre laser was demonstrated by thermally tuning the
overalap between the spectra of PM FBG and SM FBG. The output power
reached 35 mW with an optical signal to noise ratio greater than 65 dB. The
laser output was in a single polarization and the polarisation extinction ra-
tio was greater than 20 dB. The laser worked stably for 2 h under laboratory
conditions. 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. At each frequency, the laser showed single longitudinal mode be-
havior. In each single mode regime, the optical signal to noise ratio of the
laser was greater than 50 dB. The dual frequency, switchable, fiber laser can
be designed for various applications by the careful selection of the two grat-
ings.
Suppression and elimination of self pulsing in a watt level, dual clad, yt-
terbium doped fiber laser was demonstrated. Self pulsations are caused by the
dynamic interaction between the photon population and the population inver-
sion. 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. This scheme provides a simple and practical method
125
for eliminating self pulsations in fiber lasers at all pumping levels.
Power scaling of single frequency hybrid ytterbium/Brillouin fiber laser
was investigated. A coupled-wave rate-equation model, including multiple or-
der stimulated Brillouin scattering (SBS), was used for studying the power
scaling of single frequency hybrid Brillouin/ytterbium fiber lasers. A single-
frequency, Brillouin/ytterbium fiber laser with laser output of 40 mW and an
optical signal-to-noise (OSNR) of greater than 50 dB was demonstrated to
validate the model. The numerical model simulation agreed with the mea-
surements in both fully injection regime and partially injection regime. To
scale up the laser output power, a dual-clad single-frequency hybrid Bril-
louin/ytterbium fiber laser was proposed and demonstrated. First order SBS
is generated in the active fiber and amplified for laser output. Numerical
modeling including third order Stokes wave showed that over 6 W of single-
frequency laser output can be achieved with a side-mode suppression ratio
(SMSR) of greater than 80 dB. Beyond this power, multi-order SBS affects the
laser efficiency and SMSR. Experimentally, a high power single frequency hy-
brid Brillouin/Ytterbium fiber laser was demonstrated with an output power
of 1 W. The laser worked in the single frequency regime with an optical signal
to noise ratio (OSNR) of greater than 55 dB.
126
7.2 Future Work
For the future work in this field, both theoretical and experimental work
can be further investigated.
Although axial gain apodization in DFB fiber lasers has been investi-
gated in Chapter 3, the transverse spatial apodization for a multiple spatial
mode DFB fiber lasers have not been studied yet. The beam quality and mode
decomposition in these DFB fiber lasers can be investigated for future study.
Single frequency fiber lasers have narrow bandwidth due to its sin-
gle longitudinal mode property. Although intensive work has been done on
linewidth enhancement factor of semiconductor lasers [141, 146], little work
has been done on single frequency fiber lasers. Compared to single mode semi-
conductor lasers, where the laser linewidth is broadened due to the refractive
index modulation generated by the carrier density fluctuation, fiber lasers
have a much smaller linewidth enhancement factor [38]. Further work on the
wavelength dependent linewidth enhancement factor of fiber lasers would be
of scientific value.
Further power scaling of the single frequency hybrid Brillouin/ytterbium
fiber lasers can be done in the future. In this thesis, a 1 Watt single frequency
fiber laser has been demonstrated, but a higher power can be expected. With
the progress of high power isolators, hybrid Brillouin/ytterbium fiber laser
can generate much higher single frequency output. Further work on this
would be beneficial.
127
As shown in Chapter 5, self pulsations in fiber lasers can be eliminated
by inserting a section of passive fiber. In some single frequency fiber lasers,
self pulsations have been observed [108]. Convenient ways of eliminating
self pulsations in single frequency fiber lasers are highly desired. Further
research in this field is expected.
128
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151
Appendix A:Mode Selection and Nonlinear Effects
High beam quality is required for many high power fiber laser systems.
To make lasers work in high power regime with fundamental beam quality
without nonliear effects and optical damage, mode selection techniques have
been developed.
Coiled fiber can be used to strip off the high-order modes to maintain
good laser-beam quality in multimode fiber [15]. Normally a low NA is de-
signed for the core to decrease the number of fiber modes and enhance the
discrimination between the modes. This coiling technique shows the advan-
tage of easy implementation in high power fiber laser systems.
The coiling technique can be integrated with the fiber by manufacturing
a helical-core fiber. In this kind of fiber there are differential losses for the
higher order modes and the fundamental mode. Single mode output can be
Figure A.1: Schematic drawing of a helical core fiber [16].s
152
achieved from the large multimode core without physically coiling the fiber.
Figure A.1 shows the shematic drawing of a dual-clad helical-core fiber where
P is the helical pitch and Q is the core offset.
By analyzing the differential losses of fiber modes [16], helical core fibers
have been demonstrated to be able to operate under conditions where nor-
mal coiled fibers physically cannot [16]. A recent experiment demonstrated a
single-mode output of 60 W at 1043 nm under 92 W pump power from a he-
lical core of 30-µm-diameter [17]. In the dual clad ytterbium doped fiber, the
helical core is offset by 100 µm with an 8.5 mm helical pitch and 0.087 NA.
The inner cladding has a diameter of 275 µm and 0.49 NA. A slope efficiency
of 87% was achieved in the laser. Higher output power can be expected with
single mode beam quality under increased launched pump power.
A single mode output can be achieved directly from a multimode fiber
when the refractive-index profile is carefully designed. This kind of fibers
are normally called large-mode-area fibers. They are typically designed to
have a large mode area and a low NA. In 1998 [18], by using large-mode-area
fiber, pulses with 0.5 mJ energy were produced with single mode beam quality
from a Q-switched fiber laser with 21 µm core diamter. The fiber cross section
consisted of a low-NA core region and an outer-ring raised refractive index
region. In this kind of fiber, the low-NA core reduced the number of guided
modes, while the outer-ring of raised index region increased the bending loss
of higher-order modes.
Another type of large-mode-area fiber was demonstrated by OFS Labora-
153
Figure A.2: An air-clad, ytterbium-doped large-mode-area fiber can producehigh beam quality and single-mode, high-power laser outputs (a). Ytterbium-doped rods form a triangularly-shaped large-mode-area core (b) [20].
tories [19]. Single, stable, higher-order mode light propagation was reported
in a fiber with core diameter of 86 µm. The light was coupled from a single
mode fiber core into the large-mode-area fiber using a long-period grating.
The large effective-index differences between the modes suppressed the mode
mixing in the fiber. The single higher order mode was sustainable to the fiber
bend radius of 4.5 cm.
Photonic-crystal fibers are manufactured with periodic structure in the
fiber core. Small NA can be easily formed in photonic-crystal fibers and there-
fore large-mode-area single mode output is possible. In one experiment, 80 W
single mode output was generated from a photonic-crystal ytterbium doped
fiber. The cross section of the active fiber is shown in figure A.2 with a core
diameter of 21 µm.
Tapered fibers can be used to select the fundamental mode from multi-
mode core fibers. The tapers function as spatial filters. In an experiment in
1999, 90 W output power with a beam quality of M2=1.4 was generated from
a dual clad fiber laser using a 2-cm tapered fiber section. One end of the taper
matched that of the dual clad ytterbium fiber, which had a clad diameter of
154
Figure A.3: SBS was suppressed by changing the doping ratio of ytterbium,germanium, and aluminum in active fiber [23].
200 µm and core diameter of 15 µm. The other end of the taper had a core
diameter of 5.25 µm and a waist diamter of 70 µm. Higher order modes in the
fiber lasers were stripped off along the taper into the cladding while only the
fundamental mode was allowed to lase in the system. The tapered fiber in-
creased the laser-beam brightness by a factor of 3.5 while reducing the output
power by 20%.
The transversal doping profile in active fibers can be tailored to select
the fundamental mode. Differential gains are provided for the fundamental
mode and higher order modes. In 1999, a fundamental-mode selection was
demonstrated in a fiber core where 65% of the cross secion was doped with
a step active-ion profile [21]. The V number of the fiber was 12. The output
beam quality of M2=1.2 was achieved when the output power was 1.7 W. The
fiber core diameter was 23 µm. This technique can be extended to the fibers
with 100 µm core diameter [22].
155
Large-mode-area fibers are designed for high power lasers not only for
the exemption of optical damage, but for suppressing the nonliear effects. For
narrowband high power fiber lasers, the suppession of Brillouin scattering is
crucial for scaling up the power. In one experiment, the SBS threshold was
increased by 6 dB by adjusting the relative doping level between different
doping ions [23]. The overlap intergal between the acoustic and optical waves
was reduced, so that the SBS threshold was increased. Figure A.3 shows the
schematic of the specialty fiber.
156
Appendix B:Scanning Fabry-Perot Spectrometer
Scanning Fabry-Perot (F-P) spectrometers are used in applications where
laser linewidths are smaller than the resolutions of optical spectrum analyz-
ers (OSA). OSAs can only provide resolutions of about 0.01 nm, while scan-
ning F-P spectrometers can resolve optical spectra up to a few MHz depending
on the finesse and free spectral range (FSR) of the instrument.
This section focuses on the alignment of the Burleigh RC-110 scanning
F-P spectrometer. This spectrometer has been used to verify the single fre-
quency operations of fiber lasers. The schematic is shown in figure B.4 [134].
Mirror A and Mirror B are coated to produce high reflectivity and form the
F-P interferometer. The screws on the side can be adjusted to change the
position and tilt angles of mirror A.
Figure B.4: Alignment of F-P spectrometer RC-110 using a laser beam.
157
Figure B.5: Use a laser beam to align the mirrors of the F-P interferometer.
The alignment of the scanning F-P spectrometer can be described as fol-
lows:
1. Inject a collimated laser beam in the front of the spectrometer. Put a card
with a hole in the middle of the laser and the spectrometer. Make the beam
pass through the hole.
2. Adjust the laser and the interferometer so that the reflected light from
mirror B passes through the hole.
3. Use a screen to see the transmitted beams from a distance after the inter-
ferometer.
4. Adjust the screws of mirror A to make the dots produced by the multi-
ple passes of the transmitted beam align with each other, as shown in fig-
ure B.5 [134].
A scanning voltage is applied to the piezoelectic elements after the F-P
interferometer is aligned. The mirror distance can thus be changed according
to the driving voltage. Due to the intrinsic nonlinear extension of the piezo-
electric elements, a programmed ramp voltage is used to generate a linear
piezoelectic element extension. The following steps can be followed to pro-
158
Figure B.6: The ramp waveform without correction (left) and with pro-grammed correction (right).
duce the linear extension:
1. Use a narrow band source to align the interferometer. Use an oscilloscope
and set the amplitude and bias of the ramp to observe the spectrum.
2. Switch on the programmable ramp.
3. Compare the width of a free spectral range (FSR) in the beginning and end
of the ramp scanning.
4. Adjust the programmable ramp so that the FSR has the same width at
both ends of the ramp.
A linear ramp voltage and a programmed ramp voltage are shown in
figure B.6 [134]. The right waveform shows the voltage to produce the piezo-
electric element linear extension.
The resolution of the scanning F-P spectrometer can be calcualted from
the free spectral range (FSR) and finesse of the interferometer. If the reflec-
tivities of both mirrors are R, then the finesse of the interferometer f can be
written as
f =π√R
1−R(B.1)
The resolution of the spectrometer ∆ν can be calculated from the FSR and the
159
finesse f
∆ν =FSR
f(B.2)
For example, for a scanning F-P spectrometer with a free spectral range of 1
GHz and a finesse of 150, it gives the resolution of 6.7 MHz.
160
Appendix C:Measurement of Relative Intensity Noise
Relative intensity noise (RIN) is defined as the ratio of the mean square
optical intensity fluctuation to the square of the average power [135].
RIN =< ∆P 2 >
P 2dB/Hz (C.3)
where < ∆P 2 > is the mean square optical intensity noise in a unit frequency
spacing, and P is the average power. After the laser photons are detected, the
ratio in equation C.3 can be expressed in electrical powers [135].
RIN =Nelec
PAV G(elec)
dB/Hz (C.4)
where Nelec is the power spectral density of the photocurrent at a specific fre-
quency, PAV G(elec) is the average power of the photocurrent. The system RIN
can be written as [136]
RIN(f) =SP (f)
BI2phR
dB/Hz (C.5)
where SP (f) is the spectral noise power density, B is the bandwidth in the
resolution bandwidth of the electrical spectrum analyzer (ESA), Iph is the de-
tected photocurrent, R is the resistance of the photodetector.
The electrical detected noise comes from intensity noise generated by
spontaneous emission (laser intensity noise), thermal noise and photocurrent
161
shot noise. The measured system intensity noiseNT (f) can be written as [135]
NT (f) = NL(f) +Nq +Nth(f) W/Hz (C.6)
where NL(f) is the laser intensity noise, Nq is the shot noise, Nth(f) is the
thermal noise in the detector.
The shot noise Nq can be calculated as [135]
Nq = 2qIdcBRL W/Hz (C.7)
where q is the electrical charge of 1.6×10−19 coulomb, Idc is the current on the
photodetector, B is the resolution bandwidth of the noise spectrum, usually
normalized to 1 Hz, RL is the resistance of the photodetector. For a photode-
tector with 50 ohm resistance, a photovoltage of 200 mV would generate a
shot noise level of -162 dBm/Hz.
The thermal noise average power Nth can be calculated as [135]
Nth = kBT∆f W/Hz (C.8)
where kB is the Boltzmann constant of 1.38×10−23 J/K, T is the absolute tem-
perature in kelvin, ∆f is the resolution bandwidth of the noise spectrum. At
room temperature of 25 oC (298 Kelvin), with a photodetector resistance of 50
ohm, the thermal noise level is -174 dBm/Hz.
In Chapter 4, the measured system intensity noises are much higher
than the thermal noise (-174 dBm/Hz) and the shot noise (about -162 dBm/Hz),
therefore the latter two noises can be neglected.
162
Appendix D:Single Frequency Fiber Laser Linewidth
Spontaneous-Emission-Limited Laser Linewidth
In this section the laser linewidth formula is derived based on the as-
sumption that the laser linewidth is limited by the spontaneous emission
generated phase noise, following Yariv’s description of laser linewidth [61].
Consider a laser field of the following form:
E(t) = Re[A(t)ei(ω0t+θ(t))] (D.9)
where the amplitude and phase are random variables. The autocorrelation
function of the field is
C(τ) =< E(t)E(t+ τ) > (D.10)
Where the symbol <> is the ensemble average. For a laser field, it can be
treated as a random process with ergodicity, therefore, the ensemble can be
replaced by time average of infinity. After putting equation D.9 into equa-
tion D.10, the autocorrelation function can be written as [61]
C(τ) = 14< [E(t)ei[ω0t+θ(t)] + E∗(t)e−i[ω0t+θ(t)]]
×[E(t+ τ)ei[ω0(t+τ)+θ(t+τ)] + E∗(t+ τ)e−i[ω0(t+τ)+θ(t+τ)]] >
(D.11)
163
Note that
< E(t)E(t+ τ)ei[2ω0t+θ(t)+θ(t+τ)] >= 0 (D.12)
Because the exponential term oscillates at twice the wave frequency. After
averaging over one period, it goes to zero. Therefore, only slowly varying
terms in equation D.11 are kept as [61]
C(τ) = 14< [E(t)E∗(t+ τ)e−i[ω0τ+θ(t)−θ(t+τ)] + E∗(t)E(t+ τ)ei[ω0τ−θ(t)+θ(t+τ)]] >
= 14[I(τ) + I∗(τ)]
I(τ) =< E∗(t)E(t+ τ)ei[∆θ(t,τ)+ω0τ ] >
∆θ(t, τ) ≡ θ(t+ τ)− θ(t)(D.13)
The electrical field amplitude E(t) can be treated as a constant due to the
strong gain saturation in lasers. The main contribution to laser linewidth
is the phase fluctuations θ(t). Using this approximation, I(τ) can be written
as [61]
I(τ) =< E2 > eiω0τ < ei∆θ(t,τ) > (D.14)
The expection value < ei∆θ(t,τ) > can be calculated as
< ei∆θ(t,τ) >=
∞∫−∞
ei∆θ(t,τ)g(∆θ)d(∆θ) (D.15)
where ∆θ is a random variable that has a probability distribution function of
g(∆θ). Since ∆θ is the sum of many statistically independent random vari-
ables and each random variable has the same probability distribution with
finite expection value and variance. Due to the central limit theorem in statis-
tics, g(∆θ) is a Gaussian function. The probability distribution function can
164
Figure D.7: The phasor model for a single spontaneous emission for the laserfield [61].
be written as
g(∆θ) =1√
2π < (∆θ)2 >e−(∆θ)2/2<(∆θ)2> (D.16)
Inserting equation D.16 into equation D.17 yields
< ei∆θ(t,τ) >= e−<(∆θ)2>/2 (D.17)
To obtain the value of < (∆θ)2 >, the phasor model for spontaneous
emission in laser field is used.
Some of the causes that induce the field amplitude and phase fluctua-
tions can be eliminated. Examples include cavity length fluctuation, temper-
ature and acoustic fluctuations, mechanic fluctuations. These can be named
as “technique noise”. Spontaneous emission adds new power to the existing
laser field and therefore can not be eliminated. Spontaneous emission can in-
duce phase and amplitude fluctuations and is responsible for the deviation of
the laser linewidth to an ideal monochromatic field. This noise can be called
“quantum mechanical noise”.
Consider a spontaneous emission event for a laser field. As shown in
165
figure D.7, a field is rotating with an angular frequency of ω0. The amplitude
of the field is E0, which is proportional to the root square of the quanta of
the mode√n. Spontaneous emission adds one photon to the field and can be
shown as a vector of unit length in the phasor diagram. From geometry, the
change of phase of the field ∆θ can be written as [61]
∆θone emission =1√ncosφ (D.18)
where φ is a uniformly distributed random variable between 0 and 2π. Con-
sider the phase change after N random phase walk, the accumulated phase
change can be written as
< [∆θ(N)]2 >=< (∆θone emission)2 > N (D.19)
Inserting equation D.18 into equation D.19 yields
< [∆θ(N)]2 >=1
n< cos2φ > N (D.20)
where <> is ensemble average of many spontaneous emission events.
The number of spontaneous emission per unit time is N2/tspon, where N2
is the total number of atoms in the upper level and tspon is the fluorescence
lifetime of level 2. For each mode, the number of transitions is N2/(tsponp)
where p is the number of modes in the frequency range ∆ν and can be written
as [64]
p =8πν2
0∆νV n3
c3(D.21)
Where V is the gain medium volume and n is the refractive index. Further-
more, the number of transitions per unit time in each mode can be written
166
as [61]
Nspon/sec/mode = (N2
∆Nt
)(∆Nt)
tsponp(D.22)
In the above equation ∆Nt is the threshold population inversion. However,
∆Nt can be written as [61]
∆Nt =ptspontc
(D.23)
where tc is the laser photon lifetime. Therefore
Nspon/sec/mode =µ
tc(D.24)
where µ is defined as
µ =(N2)thres
(N2 −N1)thres(D.25)
where the populations are taken at threshold values. Looking back at equa-
tion D.20, considering the uniform distribution of φ, one can derive the vari-
ance of ∆θ in a time period of t as
< [∆θ(t)]2 >=1
2n
µt
tc(D.26)
Inserting equation D.26 into equation D.17, the following equation can
be derived
< ei∆θ(t,τ) >= e−µτ/(4ntc) (D.27)
Therefore, the autocorrelation of laser field can be written as [61]
C(τ) =1
4< E2 > e−µτ/4ntc(eiω0τ + e−iω0τ ) (D.28)
According to the Wiener-Khintchine theorem, the spectral density func-
tion of the laser field is the Fourier transform of the autocorrelation function.
167
Therefore the spectral density is given by
S(ω) =< E2 >
2π(
µ/4ntc(µ/4ntc)2 + (ω − ω0)2
+µ/4ntc
(µ/4ntc)2 + (ω + ω0)2) (D.29)
For ω > 0 the second term of the above equation can be neglected. There-
fore, the spectral density function can be written as [61]
S(ω) =< E2 >
2π(
µ/4ntc(µ/4ntc)2 + (ω − ω0)2
) (D.30)
This function is a Lorentzian function with a full width at half-maximum
of [61]
(∆ω)laser =µ
2ntc(D.31)
using the following relations of
P = nhν0
tc
∆ν1/2 = (2πtc)−1
(D.32)
where ∆ν1/2 is the passive resonator full width at half-maximum bandwidth.
Finally, the laser field linewidth can be written as
(∆ν)laser =2πhν0(∆ν1/2)2µ
P(D.33)
This is the famous Schawlow-Townes equation which was first derived
in 1958 [140].
Laser Linewidth Enhancement Factor
Experiments show that the measured linewidth of He-Ne laser is close to
the theoretical linewidth as calculated by equation D.33. However, the mea-
sured laser linewidth of a semiconductor can be 70 times more than that of the
168
theoretical calculation. This disagreement is due to the neglect of the modu-
lation of refractive index in semiconductor laser medium, which is generated
by spontaneous emission due to electron density fluctuation.
To describe this linewidth broadening effect, a coefficient of linewidth
enhancement factor is introduced by Henry as [141]
α =∆n
′
∆n′′(D.34)
where n′ is the real part of refractive index and n′′ is the imaginary part of
refractive index. The linewidth enhancement factor describes the coupling
between intensity and phase fluctuations. It was shown that the coupling
between the intensity and phase fluctuations broadens the laser linewidth by
(1+α2) [141].
A number of investigators have studied the linewidth enhancement fac-
tor [141–146]. Agrawal revealed the intensity dependent property of linewidth
enhancement factor in semiconductor lasers [144]. Olofosson and Brown [145,
146] demonstrated that a longitudinal modulation of modal gain will change
α while a longitudinal modulation of modal refractive index does not influence
it. Specifically, the gain coupling in a gain coupled laser strongly enhances or
reduces the linewidth enhancement factor α.
While linewidth enhancement factor plays an important role in semicon-
dutor lasers, for single frequency fiber lasers, it turns out to be small. In a
recent experiment, a single frequency fiber laser was measured to have a laser
linewidth close to the Schawlow-Townes equation prediction [38]. In this case
169
the α coefficient is close to zero. This is because the spontaneous emission in
fiber lasers does not generate large changes to the refractive index as that of
semiconductor lasers does.
Laser Linewidth Measurement
There are a few methods for laser linewidth measurements. A regular
optical spectrum analyzer (OSA) can measure laser linewidths of greater than
0.01 nm. A scanning Fabry-Perot spectrometer can give a resolution of a few
MHz. For a better resolution, a Fourier transform spectrometer or a delayed
self-heterodyning detection is required.
Consider the detection of an interfered field by using a Michelson inter-
ferometer. Assuming the light power is splitted equally in the beam splitter,
the total field in the detector plane can be written as
Ed(t) = E(t) + E(t+ τ) (D.35)
where τ is the time delay between the two arms.
The photocurrent of the detector is [61]
id = aE2d(t) (D.36)
where a is some coefficient related to the detector. The photocurrent is pro-
portional to the time averaging of square of the detected field.
Inserting equation D.35 into equation D.36, the photocurrent can be
170
Figure D.8: Schematic of delayed self-heterodyning measurement of laserlinewidth [38].
written as [61]
id = a[E2(t) + E2(t+ τ) + 2E(t)E(t+ τ)] = 2a[E2(t) + E(t)E(t+ τ)] (D.37)
The ratio between the τ dependent term and the τ independent term is pro-
portional to the autocorrelation function of the field C(τ). The laser field spec-
trum S(ω) can be obtained by taking the Fourier transform of the autocorre-
lation function.
S(ω) =1
π
∫ +∞
−∞C(τ)e−iωτdτ (D.38)
This is the principle of Fourier transform spectroscopy. Specifically, to achieve
a resolution of ∆ω, a minimum time delay of π∆ω
is required.
For a laser field, the phase fluctuation (as opposite to amplitude vari-
ation) is the predominant reason for finite linewidth. Therefore, a simple
technique of delayed self-heterodyning can be applied to do a laser linewidth
measurement.
The schematic for a delayed self-heterodyning measurement of laser
linewidth is shown in figure D.8.
The main laser fluctuation comes from the laser phase variation, there-
171
Figure D.9: Phasor of total optical field at the detector [61].
fore, the detected electrical field can be written as
Etotal =1
4E0e
iθ(t) +1
4E0e
i[ω0td+θ(t+td)] (D.39)
where E0 is the amplitude of the field in each arm, ω0 is the angular frequency
of the field, td is the optical delay in the longer arm. The laser field is illus-
trated in figure D.9. If the time delay is much less than the coherent time of
the laser field, i.e. tdτc, the angle α does not change since θ(t + td) ≈ θ(t).
Therefore, the total field amplitude is a constant. The detected signal does not
change with time and no information about laser linewidth can be obtained.
However, if tdτc, the detected signal contains laser linewidth information.
The photocurrent id is proportional to E2total. After time averaging, the cur-
rent is proportional to the modulus of the square of the field. Therefore, the
detected current can be written as [61]
id = αE20(eiθ + ei[ω0td+θ(t+td)])× (e−iθ(t) + e−i[ω0td+θ(t+td)])
= αE20(2 + ei[θ(t)−ω0td−θ(t+td)] + e−i[θ(t)−ω0td−θ(t+td)])
(D.40)
where α is a coefficient determined by the detector.
172
To derive the laser spectral density function, the autocorrelation func-
tion needs to be derived. The phase fluctuation in τ time period is [61]
∆θ(t, τ) ≡ θ(t+ τ)− θ(t) (D.41)
From central limit theorem we know that ∆θ(t, τ) is a Gaussian random
variable. Similarly, ∆θ(t, τ)−∆θ(t+ td, τ) is also a Gaussian random variable.
Following a process that is similar to the section D.1, one can derive [61]
< ei[∆θ(t,τ)−∆θ(t+td,τ)] >= e−1/2<[∆θ(t,τ)−∆θ(t+td,τ)]2> (D.42)
Follow the mathematics used in the former section, one can derive
< [∆θ(t, τ)−∆θ(t+ td, τ)]2 >= 2 < [∆θ(τ)]2 > −2 < ∆θ(t, τ)∆θ(t+ td, τ) >
< [∆θ(τ)]2 >= µτ2ntc
= 2ττc
(D.43)
Therefore, the autocorrelation function can be derived by using equa-
tion D.40 as [61]
C(τ) = α2E40 [4 + 2e−
τ(tc/2) e<∆θ(t,τ)∆θ(t+td,τ)>] (D.44)
In the limit when tdτc, the term < ∆θ(t, τ)∆θ(t + td, τ) > goes to zero.
Therefore, the autocorrelation can be written as [61]
C(τ) = α2E40 [4 + 2e−
τ(tc/2) ] (D.45)
By doing a Fourier transform of the autocorrelation function, one can obtain
the spectral density function of id as [61]
S(ω) =2α2E4
0
π[
( 4τc
)
( 2τc
)2 + ω2+ 4πδ(ω)] (D.46)
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Therefore, the full width at half maximum of the spectral density function of
id is
(∆ω)id =4
τc= 2(∆ω)laser (D.47)
Using the equation D.47 laser field linewidths can be measured by using the
delayed self-heterodyning method.
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Appendix E:Numerical Methods
Numerical Solution to Coupled Equations in HybridBrillouin/Ytterbium Fiber Lasers
The equation of 6.7 describes the interactions of the 915 nm pump power,
Brillouin pump power and multiple order SBS. To solve this equation, appro-
priate boundary condition of equation 6.8 is applied. The differential equation
does not have an analytical solution and has to be solved with a computer. To
do that, first, the equation of 6.7 has to be discretized. This can be done with
the finite difference method [132]. The power values of the waves along the
fiber corresponds to arrays of numbers. To find the steady state solution, a
relaxation method is used. This method finds the solution by iterating suf-
ficient times. However, without a close start number, the relaxation method
does not lead to the solution due to the difficulty of convergence.
To find out the solution satisfying the boundary conditions, a shooting
method is used [133]. For the equation of 6.7, a trial number of P1(0) is used,
with the condition of P2(0) = 0 and P4(0) = 0, a very small value of P3(0) is
used. Equation 6.7 is used to calculate P1(L), PP (L). These should match
175
equation 6.8 if the right trial solution of P1(0) is used. Finally, P3(0) can be
calculated with the relaxation method using the solved Pi(0)s.