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.

v

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.

vi

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.

xi

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

xii

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

xiv

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

xv

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 L1L

. (b) The difference

in gain threshold between mode one and mode zero versus L1L

. . 44

3.7 The output power ratio from fiber ends versus L1L

. . . . . . . . . 45

3.8 Gain thresholds of the proposed DFB two section fiber laser

with different splicing phase shifts. . . . . . . . . . . . . . . . . . 48

3.9 The normalized gain thresholds, gain discrimination, and output-

power ratios of the gain-apodized DFB laser under different

splicing phase shifts, when L1L

= 0.65. . . . . . . . . . . . . . . . . 48

4.1 Configuration of the dual single-frequency fiber laser. PM is

the power meter, PD is the photodetector, ESA is the electrical

spectrum analyzer, OSA is the optical spectrum analyzer, and

FP is the Fabry-Perot spectrometer. . . . . . . . . . . . . . . . . 52

4.2 The measured transmission spectrum of the PM FBG using an

ASE source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3 The optical spectrum of the laser with 43 mW output power. . . 53

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 =2π2n7p212cλ2pρ0υaΓB

(1.4)

where n is the refractive index, p12 is the longitudinal elasto-optic coefficient

related to electrostriction effect, ρ0 is the material density, λp is the pump

wavelength, υa is the acoustic velocity in the fiber.

Extensive research effort has been put into single-frequency Brillouin

fiber lasers [57–60]. In these experiments, the pump frequency had to be

resonant with the fiber ring cavity to achieve pump intensity enhancement

sufficient to generate SBS in a short (20 m) length of fiber. A tunable coupler

or a piezo-electric controller was used to adjust the accumulated phase in

the cavity to be an integer multiple of 2π. Alternatively, a tunable laser can

be used as the Brillouin pump source in these lasers. The cavities of these

20

lasers had to be reasonablely short (

21

surrounding the engineering of gain apodization into DFB fiber lasers are

discussed.

In chapter 4, single-frequency fiber lasers based on short linear cavities

are demonstrated. First, we demonstrate a room-temperature, dual single-

frequency, linear-cavity, silica fiber laser. A polarization-maintaining (PM)

fiber Bragg grating (FBG) and a single-mode (SM) FBG are used to gen-

erate two single frequencies with two orthogonal polarizations in a linear

cavity. Second, we demonstrate a single frequency, single polarization sil-

ica fiber laser by adjusting the spectral overlap between the PM FBG and

the SM FBG using a thermal controller. The fiber laser provides a single-

frequency, single-polarization output under all pump levels. Third, dual-

frequency switching is demonstrated in a linear fiber laser cavity without any

polarization-controlling component. The laser frequency switching is caused

by pump-induced heating of the two FBGs, and can therefore be controlled

by current tuning the pump laser. This phenomenon can be used to design

dual-frequency switchable fiber lasers by carefully aligning the spectra of the

two FBGs.

In chapter 5, we demonstrate a new technique to suppress self pulsa-

tions in fiber lasers by addressing their root cause: the dynamic interaction

of the laser field and the gain. By increasing the round trip time in the laser

cavity with a long section of passive fiber, the relatively fast pumping rate

forbids the population dynamics and the self pulsations are effectively sup-

pressed. Most importantly, we demonstrate that with sufficiently long fiber,

22

the self pulsations can be completely eliminated at all pump power levels.

In chapter 6, a single-frequency, hybrid Brillouin/ytterbium fiber laser

is demonstrated in a 12-m ring cavity. The output power reaches 40 mW with

an optical signal-to-noise ratio (OSNR) greater than 50 dB. The laser works

stably without mode hopping under ambient environmental conditions. As

the Brillouin pump is increased, the laser evolves from partial injection lock-

ing to full injection locking at the Stokes wavelength. A coupled-wave model

is used to describe the partial injection locking. When the laser is fully in-

jection locked, the output power decreases as the Brillouin pump is increased

due to the gain saturation induced by Brillouin pump amplification in the yt-

terbium doped fiber. A space-dependent model including second-order SBS

is included to describe this gain saturation. Excellent agreement is achieved

between the simulation and the measurement results. To scale up the output

power, a dual-clad hybrid Brillouin/ytterbium fiber laser is proposed. Nu-

merical model including third-order SBS is included to calculate the laser

performance. Simulation shows that 5-W single-frequency laser output can

be achieved from the dual-clad hybrid Brillouin/ytterbium fiber laser with

a side-mode-suppression ratio greater than 80 dB. Experimentally, a 1 W

single-frequency fiber laser is demonstrated with an OSNR of greater than

55 dB using this dual-clad hybrid Brillouin/ytterbium laser configuration.

In chapter 7, the primary conclusions of the thesis are presented along

with directions for future research on high-power single-frequency fiber lasers.

23

Chapter 2

Theoretical Models of Fiber Lasers

2.1 Coupled-Mode Theory in Periodic Structure

Coupled-mode theory is widely used in describing periodic wavegudes.

This section presents the derivation of coupled-mode equations in DFB struc-

tures using perturbation theory, following Yariv and Pollock’s procedures [61,

62]. Assuming that the periodic structure has a cross secion of single mode

fiber, the electrical field of the eigenmodes of the structure satisfy the wave

equation of

∇2 ~E = µ∂2 ~D

∂t2(2.1)

where µ is time-invariant.

The electrical flux in a dielectric medium can be written in terms of

polarization ~P :

~D = �0 ~E + ~P (2.2)

Therefore, the dielectric medium changes the electrical flux by a polarization

24

value ~P . Additionally, the periodic index structure leads to periodic deviation

from the average dielectric constant �, which can be described as a perturba-

tion in the polarization ~P . It can be written as

~D = �0 ~E + ~P + ~Ppert = � ~E + ~Ppert (2.3)

By putting equation 2.3 into the wave equation 2.1, the new wave equation

with the perturbation polarization as the driving term is [62]

∇2 ~E = µ�∂2 ~E

∂t2+ µ

∂2 ~Ppert∂t2

(2.4)

Standard perturbation theory technique can be used to solve equation 2.4 [61,

62]. The eigenmodes of the unperturbed fiber can be solved by setting the

driving term ~Ppert to zero. The eigenmodes of the waveguide form a complete

set. Therefore, a solution of the perturbed fiber waveguide can be written in

terms of a superposition of the eigenmodes. Assuming the polarization direc-

tion of the electrical field in fiber waveguide does not change during propa-

gation and is aligned with y axis, the perbutation term ~Ppert should have the

same polarization direction. The electrical field in the perturbed single mode

fiber can be written as

~E = ŷ[12A+(z)ε(x, y)e−j(βz−ωt) + 1

2A−(z)ε(x, y)ej(βz+ωt) + c.c.] (2.5)

where ε(x, y) is the spatial amplitude distribution of the eigenmode, A± are

the amplitudes of the forward and backward travelling waves, β is the propa-

gation constant. Putting the general solution 2.5 into the perturbation equa-

25

tion 2.4, the new perturbation equation can be written in the scalar form as

12(∂

2A+

∂z2− 2jβ ∂A+

∂z)ε(x, y)e−j(βz−ωt) + 1

2(∂

2A−

∂z2+ 2jβ ∂A

−

∂z)ε(x, y)ej(βz+ωt) + c.c. = µ ∂

2

∂t2Ppert

(2.6)

where many terms have been eliminated because the eigenmode satisfies the

unperturbed equation. In the small perturbation cases, the envelopes changes

slowly with z, therefore, the second derivative terms can be neglected. The

new equation can be written as

−jβ ∂A+∂zε(x, y)e−j(βz−ωt) + jβ ∂A

−

∂zε(x, y)ej(βz+ωt) + c.c. = µ ∂

2

∂t2Ppert (2.7)

Multiplying both sides of the equation with ε∗(x, y) and integrating over

the x, y plane yields

∂A−

∂zej(βz+ωt) − ∂A

+

∂ze−j(βz−ωt) + c.c. =

−j2ω

∂2

∂t2

∫∫xyPpert(x, y)ε

∗(x, y)dx dy (2.8)

due to the eigenmode relation that∫∫xy ε

∗(x, y)ε(x, y)dx dy = 1. To make the

forward and backward waves have maximum coupling efficiency, the right

hand driving term of equation 2.8 should have the same spatial phase and

temporal frequencies as the left hand terms. In this case, the perturbation

can be written in the form

Ppert(z, t) = ε0∆n2(z)

[A+

2ε(x, y)e−j(βz−ωt) +

A−

2ε(x, y)ej(βz+ωt) + c.c.

](2.9)

Substituting equation 2.9 into equation 2.8, the coupling equation between

the forward and backward waves can be written as

∂A−

∂z−∂A

+

∂ze−2jβz =

jωε04

A+e−2jβz∫∫ ∞−∞

∆n2(z)ε∗εdxdy +jωε0

4A−

∫∫ ∞−∞

∆n2(z)ε∗εdxdy

(2.10)

26

In a uniform section of periodic structure, the refractive index can be written

as

∆n2(z) = ∆n01

2

[ej(

2πΛz−φ) + e−j(

2πΛz−φ)

](2.11)

where ∆n0 is the amplitude of the index modulation, Λ is the period of the

DFB structure, φ is the phase of the periodic structure at z = 0. Due to spa-

tial phase matching considerations, the coupling between the forward wave

A+ and the backward wave A− requires that the index modulation ∆n2(z)

contains the periodic terms with spatial frequencies close to 2β and −2β. If

we denote ∆β = β− πΛ

, and extract the matching terms in equation 2.10, then

the coupled equations can be written as

∂A−

∂z= jωε0

8A+e−j(2∆βz−φ)

∫∫∞−∞∆n0ε

∗(x, y)ε(x, y)dxdy

∂A+

∂z= jωε0

8A−ej(2∆βz−φ)

∫∫∞−∞∆n0ε

∗(x, y)ε(x, y)dxdy

(2.12)

If we denote the coupling coefficient as

κ =jωε0

8

∫∫ ∞−∞

∆n0ε∗(x, y)ε(x, y)dxdy (2.13)

then the coupled amplitude equations can be written as

∂A−

∂z= κA+e−j(2∆βz−φ)

∂A+

∂z= κA−ej(2∆βz−φ)

(2.14)

If the gain coefficient of the uniform fiber waveguide is g, then the coupled

equations with gain are [63]

∂A−

∂z= κA+e−j(2∆βz−φ) − gA−

∂A+

∂z= κA−ej(2∆βz−φ) + gA+

(2.15)

Equation 2.15 are the widely used coupled mode equations for DFB fiber

lasers.

27

Figure 2.1: Energy levels of a typical quasi-three level laser system.

2.2 Space-Independent Rate Equations

Ytterbium doped fiber lasers are quasi-three level systems. Figure 2.1

shows the energy level diagram of a typical quasi-three level laser [64]. The

lower laser level 1 is a sublevel of the ground level. The sublevels are assumed

to be in thermal equilibrium. When the pumping rate and population inver-

sion are uniform along the fiber axis, the ytterbium laser can be described

with a space-independent model. Assuming that the population of the ground

level and the upper level are N1 and N2, the rate equations for the population

and photons are [64]

N1 +N2 = Nt

dN2dt

= Rp − φ(BeN2 −BaN1)− N2τdφdt

= Vaφ(BeN2 −BaN1)− φτc

(2.16)

where Rp is the pumping rate, φ is the photon number, Nt is the total popu-

lation density, τ is the metastable level lifetime, Va is the volume of the gain

28

medium, τc is the photon lifetime, Be and Ba can be written as

Be =σecnV

Ba =σacnV

(2.17)

where n is the refractive index of the active fiber, V is the modal volume in

the laser cavity, σe and σa are the emission and absorption cross sections of

ytterbium doped fiber, c is the light velocity in vacuum.

Although continuous wave lasers are predominantly studied in this the-

sis, there are many cases where self pulsing occurs in CW fiber lasers. Relax-

ation oscillation is the most important physical mechanism that leads to self

pulsations.

Starting from the space-independent laser rate equation 2.16, an analyt-

ical form of the self-pulsing condition can be derived. If we use the notation

that f = σaσe

, N = N2 − fN1, then equation 2.16 can be written as [64]

dNdt

= Rp(1 + f)− (σe+σa)cnV φN −fNt+N

τ

dφdt

= VaσecnV

Nφ− φτc

(2.18)

For any pulsing behavior starting from small perturbations, the popula-

tion inversion and photon number can be written as

N(t) = N0 + δN(t)

φ(t) = φ0 + δφ(t)

(2.19)

where δN � N0, δφ� φ0. Substituting equation 2.19 into equation 2.18, after

the very small product δNδφ is ignored, the equation takes the linear form of

dδN(t)dt

= − (σa+σe)cnV

(φ0δN(t) +N0δφ(t))− δN(t)τdδφ(t)dt

= VaσecnV

φ0δN(t)

(2.20)

29

Differentiating the photon population equation and substituting in the

inversion population yields a single equation for δφ

d2δφ

dt2+ (φ0

c

nV(σa + σe) +

1

τ)dδφ

dt+σe(σa + σe)c

2Van2V 2

N0φ0δφ = 0 (2.21)

This equation has the solution of the form

δφ = δφ0 exp(pt) (2.22)

After substitution into the equation of 2.21, a simple equation of p can be

written as [64]

p2 +2

t0p+ ω2 = 0 (2.23)

where2t0

= φ0cnV

(σa + σe) +1τ

ω2 = σe(σa+σe)c2VaN0φ0

n2V 2

(2.24)

p has the solution of

p = − 1t0±√

1

t20− ω2 (2.25)

If p is real, i.e, equation 2.21 has two solutions of exponential decays, there

will be no pulsing for the laser. The following condition must hold

1

t0> ω (2.26)

To write the condition in a more explicit form, equations 2.24 are used

with the note that in quasi-three level fiber lasers φ0 can be written as [64]

φ0 =nV

N0(σe + σa)c

fNt +N0τ

(x− 1) (2.27)

30

Figure 2.2: Schematic diagram of laser power amplification.

where x = RpRcp

is the pumping rate. Therefore, the condition for a quasi-three

level fiber laser to be free from self-pulsations is

τcτ>

4(x− 1)x2

(1 +fNtN0

) (2.28)

In the case where p is complex, the relaxation oscillation angular fre-

quency ω can be extracted and written as

ω =

[x− 1τcτ

(1 +fNtN0

)

]1/2(2.29)

Equation 2.28, 2.29 govern the relaxation oscillation dynamics in ytterbium-

doped fiber lasers, which behave as quasi-three level systems.

2.3 Space-Dependent Laser Model

A model can be applied to fiber lasers that describes the spatial depen-

dence of the pump power and population inversion. To derive the space-

dependent model, a section of active gain medium dz is investigated. Fig-

ure 2.2 shows the schematic diagram of the laser power amplification along

a section of gain medium [65]. If we consider a laser signal wavefront with

31

power P (z, t) travelling along the +z direction in the population inverted gain

medium of length dz, the equation of signal amplification can be derived as

follows. If the energy density in the dz section is ρ(z, t), due to the energy

conservation law, the rate of stored energy is the injected energy flux minus

the output energy flux, plus the stimulated emitted energy flux. The relation

can be written as [64]

∂

∂t[ρ(z, t)dz] = P (z, t)− P (z + dz, t) + Γ(σeN2 − σaN1)P (z, t)dz (2.30)

where σe and σa are the stimulated emission cross-section and the stimulated

absorption cross-section, N2 and N1 are the populations of the upper level and

the lower level, Γ is the overlap factor between the active ions and the signal

mode. Considering that P (z, t) = υgρ(z, t) where υg is the signal group velocity,

the z dependent power amplification equation can be written as

∂P (z, t)

∂t+ υg

∂P (z, t)

∂z= υgΓ(σeN2 − σaN1)P (z, t) (2.31)

Incorporating scattering loss and spontaneous emission, the laser power

equations can be written as

1

υg

∂P (z, t)

∂t+∂P (z, t)

∂z= Γ(σeN2 − σaN1)P (z, t)− αP (z, t) + 2σeN2hνδν (2.32)

where the term of 2σeN2hνδν represents spontaneous emission at the signal

frequency ν in two orthogonal polarizations, δν is the signal bandwidth, α is

the scattering loss coefficient of the laser medium.

If the laser signal has a narrow bandwidth and generates SBS waves,

the above equation must be modified to correctly describe the power propa-

32

gation along the active fiber. In these cases, the spontaneous scattering is

the mechanism that leads to the multiple order Stokes waves, therefore, the

spontaneous emission is normally negligible compared to the sponetaneous

scattering. If the laser operates in the continuous wave regime, the laser

signal and multiple order SBS power propagation equations can be written

as [66]

dP±idz

= ±[σeiN2 − σaiN1]ΓiP±i ∓ αiP±i ± gB1

AeffP±i (P

∓i−1 − P∓i+1)∓ gSB(P∓i−1 − P±i )

(2.33)

where ± and ∓ stand for the wave propagation directions, i stands for the ith

optical wave, Aeff is the effective mode area of gain medium, gB is the SBS

gain coefficient, and gSB is the spontaneous scattering gain coefficient which

can be written as

gSB = gB1

Aeffhν∆νi (2.34)

where ∆νi is the optical bandwidth of the ith optical wave.

The population inversion in equations 2.32 and 2.33 can be written as

n2 =

∑iσai Γi(P

+i + P

−i )(Ahc/λi)

−1

1τ2

+∑i

(σei + σai )Γi(P

+i + P

−i )(Ahc/λi)

−1 (2.35)

where n2 = N2/(N1 +N2) and τ2 is the metastable level lifetime.

Equations 2.32 and 2.33 can be solved with finite difference method to-

gether with the equation 2.35 to obtain the longitudinal power profiles of the

waves and population inversion in the laser cavity.

33

2.4 Chapter Summary

In this chapter, various models for fiber lasers have been reviewed and

derived. First, coupled mode equations in distributed feedback fiber lasers

were derived with perturbation theory. Second, a space-independent rate

equation model for quasi-three level fiber lasers was reviewed. The relax-

ation oscillation frequency was derived from the rate equations. Finally, a

space-dependent laser model was reviewed for fiber lasers, including stimu-

lated Brillouin scattering.

34

Chapter 3

Gain Apodized Single Frequency DFB

Fiber Lasers

3.1 Introduction

DFB fiber lasers show the advantage of high stability with relative struc-

ture among the various ways of generating single frequency fiber laser sources

[67–69]. In this chapter, the effects of axial gain apodization on the perfor-

mance of DFB fiber lasers are investigated for the first time. In particular,

the impact of gain apodization on threshold behavior is explored along with

its effect on output power and mode discrimination. First, the physics of gain

apodization in DFB lasers are explored and compared to conventional config-

urations. Secondly, the impact of gain apodization on phase shifted DFB fiber

lasers is investigated. Finally, issues surrounding the engineering of gain

apodization into DFB fiber lasers are discussed. The investigation shows that

35

Figure 3.1: Schematic diagram of a periodic active waveguide.

if properly tailored, ideally the lasing threshold can be reduced by 21% with-

out sacrificing modal discrimination, while simultaneously increasing the dif-

ferential output power between both ends of the laser [35].

3.2 Fundamental Matrix Model

Although DFB lasers are widely used for single-mode operation, their

mode spectrum is more complicated. In a uniform index-coupled DFB fiber

laser without phase shift or end mirrors, DFB lasers can operate in one of two

degenerate longitudinal modes, symmetrically located along the Bragg fre-

quency of the grating. Nominally, only a single mode runs due to fabrication

imperfections that cause slight asymmetry.

The coupled-mode theory can be used to analyze the threshold behav-

ior in simple DFB lasers. Figure 3.1 illustrates the schematic of the coupling

between forward and backward waves in a DFB structure. To derive the fun-

damental matrix model, the coupled mode equations 2.15 are rewritten

∂A−

∂z= κA+e−j(2∆βz−φ) − gA−

∂A+

∂z= κA−ej(2∆βz−φ) + gA+

(3.1)

36

To solve the above equations, the following notations are used:

EA(z) = A+ exp(−jβz)

EB(z) = A− exp(+jβz)

(3.2)

where β is the propagation constant of the optical wave in the laser medium.

The equations 3.1 can be solved analytically as [63]

EA(z) = [c1 exp(Γ1z) + c2 exp(Γ2z)] exp[(g − jβ)z]

EB(z) = {exp(−j(2∆β′z − φ)]/κ}[c1Γ1 exp(Γ1z) + c2Γ2 exp(Γ2z)] exp[−(g − jβ)z](3.3)

where c1 and c2 are some constants and ∆β′ and Γ1,2 are written as

∆β′ = ∆β + jg

Γ1 = j∆β − γ

Γ2 = j∆β + γ

γ2 = k2 − (∆β′)2

(3.4)

For some gain media that are not uniformly periodic, they can be seg-

mented into many different sections, each of which is uniform. For the ith

uniform section, according to the notations in figure 3.1, the electrical fields

are related through a fundamental matrix [63] EA (zi+1)EB (zi+1)

= F

i11 F

i12

F i21 Fi22

EA (zi)EB (zi)

(3.5)

37

where the matrix elements are written as

F i11 = [cosh (γiLi) + j∆β′iLi sinh (γiLi)/(γiLi)] exp (jβ

iBLi)

F i12 = −κiLi sinh (γiLi) exp [−j (βiBLi + φi)]/(γiLi)

F i21 = −κiLi sinh (γiLi) exp [j (βiBLi + φi)]/(γiLi)

F i22 = [cosh (γiLi)− j∆β′iLi sinh (γiLi)/(γiLi)] exp [−j (βiBLi)]

(3.6)

where ∆β′i = ∆βi + jgi, γ2i = k2i − (∆β′i)2, βiB = π/Λi, Λi is the period of the

ith section and Li is the length of the ith section. The matrix form provides

a convenient and powerful tool for studying DFB laser behaviors. Many key

parameters including the gain thresholds of all longitudinal modes, output

power ratio from both ends of a DFB laser can be calculated with the fun-

damental matrix model. With the above fundamental matrix formalism, the

active gratings can be split into N sections, where the total matrix will be

Ft = FNFN−1...F2F1. For a nonuniform DFB fiber laser, the coupling coeffi-

cient κ and gain coefficient g can change with the position z. For DFB fiber

lasers without a phase shift, the phase terms in equation 3.6 can be writ-

ten as φi = φi−1 + 2βiBLi−1 where i = 1, 2, 3, ...N . For phase-shifted DFB fiber

lasers, the phase terms in equation 3.6 is φi = φi−1 + 2βiBLi−1 + ∆φi where

i = 1, 2, 3, ...N . Adding the boundary conditions A+(0) = A−(L) = 0, the gain-

threshold condition can be obtained from the relation Ft11 = 0. Nominally,

this relation will produce a mode spectrum with different modes appearing at

different frequencies ∆β.

For high-power operation, it is desirable not only to have a low threshold,

but also to have most of the light coming out of only one side of the cavity. By

38

Figure 3.2: Schematic of (a) a gain-apodized DFB fiber laser, (b) a uniform

DFB fiber laser, and (c) a uniform DFB fiber laser with end reflector R2 =

tanh2(κL2).

using the total matrix Ft, the output-power ratio from both ends of the fiber

can be written as

P1P2

=

∣∣∣∣∣A−(0)A+(L)∣∣∣∣∣2

= |F21|2 (3.7)

where P1P2

presents the ratio of the power coupling out at z = 0 compared to

z = L.

3.3 Gain Apodization Physics

To understand the physics introduced by gain apodization, we apply

the formalism in the former section to three cases. In all cases, the grating

strength κ and period Λ are kept constant and no phase shift will be included.

The peak reflectivity of the grating is determined by R = tanh2(κL) and, to

not lose generality, typical values for κ and L are chosen. In all the following

sections, the coupling coefficient of the fiber grating is κ = 1 cm−1. The grating

39

length is 3 cm in most cases. Since the length under which the gain will drop

from its maximum value to zero is very small, the gain apodization along the

z axis will be approximated by a step function. The gain-apodized DFB fiber

laser is schematically shown in figure 3.2(a), where the L1 section is highly

doped with uniform gain coefficient g, and L2 has no gain. This case will be

compared to two other cases. The first, a DFB fiber laser of length L1 and

uniform gain but no unpumped section, is shown in figure 3.2(b). The sec-

ond case, shown in figure 3.2(c), is the same laser as shown in figure 3.2(b),

but with a reflector at the end of the cavity where the grating would be in

the apodized case. The reflectivity value is chosen to be the peak reflectivity

of the unpumped fiber grating of case figure 3.2(a), namely, R2 = tanh2(κL2).

This value was chosen to directly compare to the apodized case figure 3.2 (a).

The gain thresholds for these cases, where L1=2.5 cm and L2=0.5 cm

are shown in figure 3.3. The horizontal axis is the normalized frequency

∆βL (L = L1 + L2), while the vertical axis is the normalized gain thresh-

old gthL1. The gain is normalized with L1 since the value of gL1 relates to the

absorbed pump power at threshold. The mode spectra of the three different

lasers is nearly identical, since the lasing cavities are of nearly equal length.

When compared to the short DFB laser, the gain-apodized DFB lasers show

nearly a 30% reduction in lasing threshold due to its passive grating section.

The DFB with the reflector similarly shows a reduction in lasing threshold for

its first-order mode. However, the threshold reduction applies significantly to

all modes since the reflector is spectrally uniform. For the gain-apodized DFB

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 L1L

is close to 0.7; the gain threshold is 17.9% less

compared to the uniform DFB fiber laser (L1L

= 1). For gain lengths L1L

less

than unity, the longitudinal distribution of light extends into the unpumped

region, creating an effectively higher reflectivity. Since no gain is extracted

43

from this region, the effective grating strength is increased, thus creating a

lower gain threshold. For values of L1L

that are too small (less than 0.7 in this

case), the grating-length product becomes too small to produce sufficient re-

flection, effectively increasing the laser threshold via reduced feedback. Fig-

ure 3.5 demonstrates that gain apodization can decrease the laser threshold

if properly tailored.

3.4 Gain Apodization in Phase Shifted DFB Lasers

It is convenient to avoid mode degeneracy by introducing a phase shift

in the middle of the grating. As is well known, the π phase shift will enable

a narrowband filter in the grating forbidden band, thereby allowing the 0th

order mode to have a low lasing threshold [26]. Considering the influence of

this geometry, it is instructive to understand the role of gain apodization on

phase shifted DFB fiber lasers.

Figures 3.6 (a) and 3.6 (b) show the lowest mode gain threshold and the

mode discrimination of the uniform gain, phase shifted DFB fiber lasers. As

before, the total cavity length L is 3 cm and the coupling coefficient is 1 cm−1.

The results show that the apodization with the lowest gain threshold also has

nearly the largest mode discrimination. Slightly different to the optimum L1L

of 0.7 for a normal DFB laser in figure 3.5, the optimum gain apodization

profile will be where L1L

is close to 0.6. From figure 3.6(a), the gain threshold

can be reduced 21.2% compared to the normal phase-shifted DFB fiber laser,

44

Figure 3.6: (a) The lowest-mode gain threshold versus L1L

. (b) The difference

in gain threshold between mode one and mode zero versus L1L

.

45

Figure 3.7: The output power ratio from fiber ends versus L1L

.

with nearly the same modal discrimination, as shown in figure 3.6 (b).

Since the gain apodization has introduced system asymmetry, the output-

power ratio from both ends of the laser will also be modified. To investigate

these characteristics, the output-power ratio of equation 3.7 is plotted against

the apodized gain length L1L

in figure 3.7. The power ratio from both ends of

the fiber changes monotonically with the apodization gain length L1L

. Higher

output power from the pumped end of the cavity can be obtained at the opti-

mum pumped length L1L

for the minimum threshold shown in figure 3.6 (a);

the power ratio can be increased by 12.4%. This asymmetry, combined with

the 21.2% threshold reduction, can lead to a substantial increase in output

power due solely to gain apodization.

46

3.5 Thermal and Splicing Phase Effects

It was shown in the former section that gain apodizati