`

FIBER BASED MODE LOCKED FIBER LASER USING KERR EFFECT

Dissertation

Submitted to

The School of Engineering of the

UNIVERSITY OF DAYTON

In Partial Fulfillment of the Requirements for

The Degree of

Doctor of Philosophy in Electro-Optics

By

Long Wang

UNIVERSITY OF DAYTON

Dayton, Ohio

May, 2016

` FIBER BASED MODE LOCKED FIBER LASER USING KERR EFFECT

Name: Wang, Long

APPROVED BY:

______

Joseph W. Haus, Ph.D. Andy Chong, Ph.D. Advisor Committee Member

Committee Chairman Assistant Professor Professor Physics Department and

Electro-Optics Graduate Program Electro-Optics Graduate Program

______

Imad Agha, Ph.D. Jay Mathews, Ph.D. Committee Member Committee Member Assistant Professor Assistant Professor Physics Department and Physics Department and Electro-Optics Graduate Program Electro-Optics Graduate Program

______Muhammad Usman, Ph.D. Committee Member Associate Professor Department of Mathematics

______John G. Weber, Ph.D. Eddy M. Rojas, Ph.D., M.A., P.E. Associate Dean Dean, School of Engineering School of Engineering ii

`

© Copyright by

Long Wang

All rights reserved

2016

iii

` ABSTRACT

FIBER BASED MODE LOCKED FIBER LASER USING KERR EFFECT

Name: Wang, Long University of Dayton

Advisor: Joseph W. Haus

This dissertation reports on the research to design and build a pulsed fiber laser with the Er doped fiber based on a new mode locking technique. The numerical simulations begin by launching an optical wave in a fiber which will be amplified during propagation. The device to mode-lock the waves is outside the fiber, but connecting to fibers at both ends; it is a nonlinear optical material that can reshape the beam as it propagates using a nonlinear change of the refractive index, which is called a Kerr effect.

The device is made with a nonlinear material sandwiched between two fiber ends; it takes an optical field from one end of the fiber and propagates it to the other fiber end. In between the two ends, a nonlinear medium will be used to balance the diffraction through

Kerr effect (which can lead to Self-focusing of the optical beam). With the second fiber end working as a soft aperture, the combination of the self-focusing effect through the nonlinear medium and the aperture will act as an intensity dependent coupling loss; this effect is referred to as a fast saturable absorber which means that higher intensity

iv corresponds to higher coupling efficiency and thus the cavity modes will be gradually phase locked together to form pulses. The saturable absorber action is calculated using different nonlinear mediums (퐶푆2, 퐴푠2푆2 and 퐴푠40푆푒60) and the fibers used are assumed to be of the same size.

Whole cavity simulation is then conducted using the proposed SA design and the pulse energy produced from the laser cavity is generally below 1 nJ. In those simulations the pulse peak power is weak and the saturable absorber action is not strong.

Experiments are designed to test the mode locking idea with the chalcogenide glass plate (퐴푠40푆푒60). Firstly, a mode locked laser is constructed from a ring fiber laser cavity with an Er doped fiber as the gain fiber. Three modes from this cavity are routinely generated. Two modes have pulse durations of 220 fs and 160 fs with spectral width of about 30 nm and 40 nm, respectively. Mode 3 is more interesting since it covers a huge spectrum range (1490 to 1640 nm) and the pulse duration is estimated to be about 40 fs from the transform limited pulse calculated from the spectrum, which could be the shortest pulse ever reported from an Er doped fiber laser. Further efforts are needed to better dechirp the pulse to verify the transform limited calculation.

Due to the weak saturable absorber action from the original design, we use a telescope design to test our SA idea in experiment; the ChG 퐴푠40푆푒60 plate is placed inside a telescope which is then inserted into the laser cavity explained in previous paragraph. The telescope design is used to focus the pulse so that higher level of nonlinearity is induced. Then an iris is placed behind the glass plate to create a transmission discrimination mechanism against different powers (Kerr lens mode

v locking). Mode locking is not obtained but strong mode locking sign is identified. Q- switched pulse laser is obtained by using ChG 퐴푠40푆푒60 plate only.

vi ` ACKNOWLEDGEMENTS

I want to render thanks to my mentor, Dr. Joseph W. Haus, for being an excellent advisor. Even though he is very busy, he still gives of his precious time to me. From him,

I have learnt a many research methods, a right scientific research attitude and a wide range of communication skills. But more importantly, he sets us an extremely well example of being patient and integral. His advices and supports have been decisive in the discussions about the topics covered and the process of writing this dissertation. I will be in his debt forever.

I would like to give special thanks to Dr. Andy Chong, whose supports with the experimental equipments have been unconditional and extremely helpful. In spite of his busy schedule, he spends countless time to talk to me about the simulations, walk me through each step of building a mode locked fiber laser. His indisputable passion and strict attitude towards scientific research have been very inspiring. Also I want to thank all of the professionals of for sharing their time and insights with me.

Last but not least, my loving family who’s always loving and sincere support and constant bolster morale has been always there, despite the distance and little calls from my part.

vii

` TABLE OF CONTENTS

ABSTRACT ...... iv

ACKNOWLEDGEMENTS ...... vii

LIST OF FIGURES ...... xi

LIST OF TABLES ...... xvi

CHAPTER 1 INTRODUCTION ...... 1

CHAPTER 2 MODE LOCKING TECHNIQUE ...... 9

2.1 Active Mode Locking...... 11

2.2 Passive Mode Locking ...... 15

2.2.1 Slow saturable absorber mode locking...... 16

2.2.2 Fast saturable absorber mode locking ...... 17

2.2.3 Kerr lens mode locking ...... 18

CHAPTER 3 PULSE FORMATION AND PROPAGATION IN FIBER LASERS ...... 21

3.1 Soliton Mode Locking ...... 22

3.2 Dispersion Managed Soliton ...... 26

viii

3.3 Similariton Mode Locking ...... 27

3.4 Dissipative Soliton Mode Locking ...... 31

CHAPTER 4 NUMERICAL STUDY OF THE NEW SATURABLE DESIGN ...... 34

4.1 Numerical Study of the Fundamental Mode in Fibers ...... 38

4.2 Numerical Study of the New Mode Locker ...... 41

4.2.1 Simulation with carbon disulfide (퐶푆2) ...... 42

4.2.2 Simulation with chalcogenide glass ...... 50

4.2.3 퐶푆2 or ChG: which one to use?...... 54

CHAPTER 5 NUMERICAL STUDY OF THE FIBER LASER CAVITY USING

NEW SATURABLE ABSORBER DESIGN ...... 57

5.1 Optical Elements in Cavity ...... 57

5.1.1 Pulse propagation in fibers ...... 57

5.1.2 Pulse propagation through saturable absorber ...... 61

5.1.3 Pulse propagation through a spectral filter ...... 63

5.1.4 Pulse propagation through output coupler ...... 65

5.2 Whole Cavity Simulation ...... 66

5.2.1 Andi laser simulation ...... 66

5.2.2 Simulation with 3 mm of 퐴푠40푆푒60 ...... 70

CHAPTER 6 EXPERIMENTS ...... 74

6.1 Build a Mode Locked Er Fiber Laser...... 74 ix 6.2 Mode Locking with ChG Glass Plate...... 81

6.3 Mode Locking with a Telescope Design ...... 84

6.4 Mode Locking with Pin Hole in the Telescope ...... 86

6.5 Q-switching with the 퐴푠40푆푒60 Glass Plate ...... 87

CHAPTER 7 SUMMARY ...... 91

REFERENCES ...... 96

APPENDIX A Mode Simulation ...... 105

APPENDIX B New Saturable Absorber Design Simulation ...... 115

APPENDIX C Split Step Method ...... 118

x ` LIST OF FIGURES

Figure 1-1 Conceptual mode of a typical fiber laser. Bold black line stands for active fiber which is usually doped with rare earth element and blue line stands for the normal passive fiber. Pump is introduced into the cavity through coupler 1 and pulses are coupled out through coupler 2...... 5

Figure 2-1Actively mode-locked laser arrangement, with an intracavity modulator driven at the cavity round-trip period. Reproduced of Figure. 2.1 (a) in [10]...... 11

Figure 2-2 Pulse-shortening in slow saturable absorber mode locking. The shaded region indicates net positive gain. Reproduced of Figure. 2.7 in [10]...... 17

Figure 2-3 Gain and loss dynamics in fast saturable absorber mode locking. The shaded region corresponds to positive net gain. Reproduced of Figure 2.10 in [10]...... 18

Figure 2-4 Kerr lens mode locking with a hard aperture...... 20

Figure 3-1 Conceptual Model of typical Soliton Laser. Modified version of Figure 1.6 in [26]...... 25

Figure 3-2 Qualitative description of dispersion-managed (DM) soliton. Different rows from top to bottom represent the distance evolution of dispersion, the pulse chirp, pulse duration and pulse spectral bandwidth...... 26

xi

Figure 3-3 Illustration of similariton propagation in passive fiber (on the left ) and gain fiber (on the right ). Taken from [40]...... 31

Figure 3-4 Conceptual model of a similariton laser...... 31

Figure 3-5 Conceptual model of all-normal-dispersion laser (Andi laser)...... 33

Figure 4-1 Conceptual fiber laser cavity with the new saturable absorber design.

Coupler1 transmits wavelength nearby 972 nm and 1550 nm. Coupler 2 splits the input into two parts: one through output and one back into the fiber cavity. In the fiber cavity, the black fiber is the Er:Yb doped gain fiber and the blue fibers are single mode fibers

(SMFs). An isolator is used to ensure that the pulse is propagating clockwise...... 34

Figure 4-2 New saturable absorber design. There are four key elements that affect the transmission between fiber 1 and fiber 2: nonlinear medium, end1, end 2 and the separation 퐷 푚푚 in between them...... 35

Figure 4-3 The user interface of the Mode Solver...... 39

Figure 4-4 (a) the mode radius as a function of the fiber size; (b) the effective index as a function of fiber size...... 41

Figure 4-5 Simulation for fiber pair 20-20 with carbon disulfide (퐶푆2): (a) the coupling efficiency 휂 as a function of separation 퐷 and beam power; (b) the coupling efficiency difference between high power and low power beams. On (a), 휂 for two beams with power of 0.000287 kW and 64.02 kW are shown at 퐷 = 2.003 푚푚 and the difference between them is shown on (b). For a pulse with peak power of 64.02 kW, the optimum separation is at 퐷 = 1.865 푚푚, shown on (c)...... 47

Figure 4-6 Simulation for fiber pair 20-20 with carbon disulfide (퐶푆2): (a) Optimum separation as a function of pulse peak power; (b) Optimum coupling efficiency xii difference between the peak power and the low power region as a function the pulse peak power.Change Peak power into power...... 48

Figure 4-7 Simulation for different fiber pairs with 퐶푆2: (a) Optimum separation as a function of pulse peak power; (b) Optimum transmission difference ∆휂 between the peak power and the low power part of a pulse as a function the pulse peak power...... 48

Figure 4-8 Transmission curve for the new SA design with 퐶푆2 at 퐷 = 3 푚푚...... 50

Figure 4-9 Simulation for fiber pair 20-20 with ChG 퐴푠40푆푒60. (a) the coupling efficiency 휂 as a function of separation 퐷 and beam power; (b) the coupling efficiency difference between high power and low power beams...... 52

Figure 4-10 Simulation for fiber pair 20-20 with Chalcogenide glass 퐴푠40푆푒60: (a)

Optimum separation as a function of pulse peak power; (b) Optimum transmission difference ∆휂 between the peak power and the low power region as a function the pulse peak power...... 53

Figure 4-11 Transmission curve for 퐶ℎ퐺 퐴푠40푆푒60 at 퐷 = 3 푚푚 ...... 54

Figure 4-12 Comparison of coupling difference for fiber pair 20-20 with nonlinear medium of (a) 퐶푆2; (b) 퐴푠40푆푒60...... 55

Figure 4-13 Transmission curve of new saturable absorber design for fiber pair of 20-20 with different nonlinear mediums (a) 퐶푆2; (b) 퐴푠40푆푒60...... 56

Figure 5-1 Illustration of split step method...... 61

Figure 5-2 Transmission curve of saturable absorbers for fiber pair of 20-20 with different nonlinear mediums (a) 퐶푆2; (b) 퐴푠40푆푒60. Black lines are the original calculated values and the red dots are the interpolated values...... 62

Figure 5-3 Gaussian filter...... 65 xiii Figure 5-4 The typical Andi laser cavity. PF = passive fiber, AF = active fiber,

SA=saturable absorber, OC = output coupler and SF= spectral filter...... 66

Figure 5-5 Pulse energy after spectral filter as a function of round trip propagation...... 68

Figure 5-6 Pulse evolution inside Andi laser cavity...... 69

Figure 5-7 Pulse evolution (top row), corresponding spectrum (middle row)) and chirp

(bottom row) inside the cavity at positions of (a) before PF 1; (b) before AF; (c) before

PF 2; (d) before SA and (e) before SF...... 69

Figure 5-8 Typical Andi laser pulse evolution. Taken from [47]...... 70

Figure 5-9 The pulse energy after spectral filter along with round trip propagation for conceptual cavity 1...... 72

Figure 5-10 Pulse evolution inside the cavity 1 in stead state...... 72

Figure 5-11 The steady pulse after each element inside cavity 1 Top row is the pulse shape, center row the spectrum and the bottom row the chirp...... 73

Figure 6-1 Er fiber laser cavity...... 75

Figure 6-2 Mode 1...... 78

Figure 6-3 Mode 2...... 78

Figure 6-4 Mode 3...... 79

Figure 6-5 Pulse train for all the modes found...... 80

Figure 6-6 The Er fiber laser cavity corresponds to the design in Figure 6-1...... 80

Figure 6-7 5 mm thick 퐴푠40푆푒60 plate...... 82

Figure 6-8 Mock locking with 퐴푠40푆푒60 glass plate...... 83

Figure 6-9 Experimental setup of Figure 6-8...... 83

Figure 6-10 Mode locking with 5 mm 퐴푠40푆푒60 in a telescope design...... 85 xiv Figure 6-11 Mode locking with telescope design...... 85

Figure 6-12 Cavity with a pinhole in the telescope...... 86

Figure 6-13 Experiment realization of Figure 6-12...... 87

Figure 6-14 Laser cavity with only PBS inside the free space...... 88

Figure 6-15 (a) Spectrum and (b) CW output for cavity in Figure 6-14...... 88

Figure 6-16 Q-switch with the 퐴푠40푆푒60 glass plate. The cavity is the same the one shown in Figure 6-14...... 89

Figure 6-17 (a) Spectrum and (b) Q-switched pulses for cavity in Figure 6-16...... 90

Figure 7-1 New saturable absorber design. There are four key elements that affect the transmission between fiber 1 and fiber 2: nonlinear medium, end1, end 2 and the separation 퐷 푚푚 in between them...... 91

Figure 7-2 Transmission curve of new saturable absorber design for fiber pair of 20-20 with different nonlinear mediums (a) 퐶푆2; (b) 퐴푠40푆푒60...... 92

xv ` LIST OF TABLES

Table 4-1The mode radius and effective index as function of fiber sizes...... 40

Table 4-2 Fiber pairs used for simulation with nonlinear material 퐶푆2...... 45

Table 5-1 Polynomial coefficients of SA action for both 퐶푆2 and 퐴푠40푆푒60...... 62

Table 5-2 Andi laser cavity details...... 67

Table 5-3 Simulation details for conceptual cavity1...... 71

Table 6-1 Fiber information...... 77

xvi

` CHAPTER 1

INTRODUCTION

The laser’s history can be traced back to 1900, when Max Plank published his radiation theory, which essentially explained that electromagnetic energy could only be emitted or absorbed in discrete chunks. Inspired by Max Plank’s work, physicist Albert

Einstein published a paper in 1905, in which he proposed that the energy of light are also quantized which in turn leads to the photon concept and also earned him the Noble Prize in Physics in 1921. Taking those concepts further, Albert Einstein published another paper in 1917 in which he brought up the stimulated emission theory. He believed, besides the spontaneous absorption and emission, there existed excited states of electrons which could emit light at a certain wavelength which is the underlying physics for the lasers. It took more than 30 years for scientists to prove his theory right until Dr. Charles

Townes made the first maser which was based on the stimulated emission theory. In July

7 of 1960, the first working laser was invented at Hughes Research Institute by Theodore

Maiman[1]. Since then, lasers has undergone drastic and fast development. Ironically, it was initially described as “a solution looking for a problem” when the first laser was reported. Nowadays lasers are so widely applied in every aspects of lives that one can hardly find any industry which doesn’t make use of them. From hospitals where laser eye 1 surgeries (LASIK) are performed to grocery stores where laser barcode scanners are used for reading prices, from factories where lasers are employed for machining and welding to military fields where lasers can be used to direct energy in direct-energy weapon systems, lasers in electronic products and for internet data management and communication have become a very important part in everyone’s daily life, even when they are unaware of its presence.

Among various types of lasers, pulsed lasers are one special type of laser which are invented to produce stable and controllable energy pulses. Pulsed lasers are created by locking the longitudinal cavity modes (different colors) together so that it emits light in the form of pulses, instead of as a continuous wave. The corresponding concept is referred to mode locking which will be explained in more details in Chapter 2.

Interestingly, the very first laser, reported in 1960 by Theodore Maiman[1], was actually built with pulse operation option, even earlier than the invention of the first continuous wave laser (CW laser) reported in 1962 by Willard Boyce at Bell Labs. Today, ultrashort pulses become essential scientific tools to study dynamic behaviors of atoms and molecules. In parallel, ultrashort pulses also became important in areas with broader industrial impact. For example, ultrashort pulses are useful in many industrial applications such as precision machining, medical microscopic imaging, and medical surgeries.

Depending on the wavelength, pulse duration, pulse energy and pulse repetition rate, very different methods for pulse generation and very different types of pulsed lasers are produced. For example, in active mode locking, a rapid shutter is used to force pulse formation inside the laser and the generated pulse’s width is generally ~100 ps or longer.

2 Passive mode locking is achieved by replacing the active modulator by a nonlinear material so that the pulse can modulate itself. The pulse generated from passive mode locking is generally shorter in time. In this category solid-state lasers have been quite successful in generating pulses with durations in the picosecond and femtosecond domain, when combined with mode locking technique. Among various types of solid-state pulsed lasers, titanium-doped sapphire laser (Ti:sapphire laser) is arguably the most famous one.

The very first Ti:sapphire laser was initially reported in 1982 by Peter Moulton of MIT’s

Lincoln laboratory and then published in 1986[2]. Comparing with other competing lasing materials, Ti:sapphire is very flexible and provides many advantages. It is unmatched in its characteristics for delivering a combination of wide tunability, high average power and longtime stability. Also, sapphire has excellent thermal conductivity which alleviates the thermal effects even for high powers. More importantly, the spectral output from Ti:sapphire laser covers up to several hundred of nanometers (650 nm – 1100 nm) which allows the generation of extremely short pulses[3] and wide wavelength tunability. The maximum gain and laser efficiency can be obtained at about 800 nm and the possible tuning range is from 650 nm to 1100 nm. Due to all the merits mentioned above, Since its invention the Ti:sapphire laser has quickly replaced most dye lasers, which had previously dominated the fields of ultrafast pulse generation and widely wavelength-tunable lasers and has thereafter enabled numerous fundamental researches in physics, chemistry and biomedical industries to name a few.

Although solid-state lasers are superb tools for laboratory activities, there are several disadvantages that prevent them from wide use in industry field. First, for example, Ti:sapphire lasers are generally very expensive, partly because of the pump

3 requirements. Even there is a wide range of possible pump wavelengths for Ti:sapphire lasers, they are located in the green spectral region where powerful green laser diodes are not available and also the green pump laser are quite expensive. Furthermore, due to the complicated laser system, solid state lasers are bulky and not user friendly. A typical

Ti:sapphire laser occupies a large footprint ( ~ 1 푚2) and handling such a solid-state laser still requires a constant care of an experienced optical scientist.

More-compact, user-friendly, and cost-efficient devices are desired to proliferate industrial technologies of ultrashort pulses. Such lasers should be easy to operate stably with minimal attentions in environments such as medical clinics, manufacturing factories, mobile vehicles, airplanes, etc. Mode-locked fiber lasers offer several practical advantages over solid-state lasers and are promising candidates to replace solid-state lasers in industrial applications. Firstly, fiber systems are very cost efficient as compared to the counterpart of solid state lasers due to their large scale use in telecommunication industries. Since fiber based components are relatively cheap, fiber lasers can be built with low cost. Secondly, fiber lasers offer greater stability and compact design. Since optical fiber can be wrapped into a small volume, fiber lasers can be made in a very small size. Thirdly, a cooling device, which is necessary for solid state lasers, is not required for fiber lasers due to the high surface to volume ratio of fiber. Furthermore, the output power from fiber lasers is quite stable due to confinement of the fiber and beam quality is excellent as they operate at the fundamental transverse fiber mode. A conceptual fiber laser consists a loop of single mode fiber (SMF), where a portion of the fiber is doped with a rare earth metal and pumped by a diode laser source. Mode locking is achieved by incorporating a saturable absorber (SA) into the cavity and the pulses are introduced out

4 of the cavity by using an output coupler. A conceptual model of ring cavity is illustrated in Figure 1-1, in which bold black line stands for active fiber and blue lines are normal passive fibers. The pump is introduced into the gain fiber through coupler 1 and the pulses are coupled outside the cavity through coupler 2.

Figure 1-1 Conceptual mode of a typical fiber laser. Bold black line stands for active fiber which is usually doped with rare earth element and blue line stands for the normal passive fiber. Pump is introduced into the cavity through coupler 1 and pulses are coupled out through coupler 2. Even though fiber lasers offer many practical advantages, their performance lagged behind the solid-state lasers historically. The main drawbacks include a smaller range of tunability and a tendency to produce pulses with lower power and longer pulse width, compared to the solid state lasers. However, the performance of mode-locked fiber lasers has been improved significantly in recent years to surpass that of the solid-state counterpart. For example, by exploiting a new fiber laser design which is referred as an all normal dispersion (ANDi) design[4], the pulse energy of a mode-locked Ytterbium

(Yb) doped fiber laser surpassed those of a Ti:sapphire laser. Inspired by the pulse energy enhancement in fiber lasers, the ANDi technology was quickly adapted in the industry. In year 2014, first ANDi fiber laser commercial products were released by KMLabs. The model Y-Fi from KMLabs, which is a combination of an ANDI laser and a fiber

5 amplifier, successfully generated >400 nJ with 150 fs pulse duration. The laser performance is suitable for many applications such as nonlinear microscopies, etc.

However, the mode-locked fiber lasers still fall significantly behind solid-state lasers for the shortest pulse duration. As a shortest mode-locked pulse source, a

Ti:sapphire laser is well known for its capability of generating extremely short pulse durations and shortest pulses obtained in laboratories have durations around 5.5 fs[3]. In contrast, the shortest pulse duration from mode-locked fiber lasers has stagnated around

~30 fs for last 10 years[5][6]. After many years of research effort, finally a new unique mode locking technique, which is referred as an amplifier similariton (AS) [7], was successfully implemented to generate ~20 fs[8].

Although the pulse duration of fiber lasers are still longer than the solid-state lasers and also fibers lasers are still subject to the instabilities of temperature variations, pressure and the inherent fragility of the glass, extensive researches in pulsed fiber lasers are being done nowadays. The main reason to justify the ongoing research on fiber lasers is that when the above mentioned difficulties are solved they will be able to outperform the semiconductor counterparts with the advantage of much more lower cost of fabrication and greater portability.

To generate the ultrashort pulses within a fiber laser cavity, people nowadays proposed many different types of saturable absorbers. One common type, usually used in a linear cavity, is a semiconductor saturable absorber mirror (SESAM). The modulation depth of a SESAM can be controlled by adjusting the length of the absorber material and the design of a SASEM; it is very compact. But the price for such a device is usually high

(generally more than 500 Euros for 1550 nm) and the relaxation time is long, usually

6 more than 2 pico-seconds. Second type is the nonlinear optical loop mirror (NOLM) or nonlinear amplifying loop mirror (NALM). These devices usually have high modulation depth. However, the fibers needed for passive fiber in the loop tends to be very long

(easily 100 meters or longer in length). Also, since the NOLM or NALM depends on the nonlinear polarization inside the fibers, these devices are generally sensitive to the environment and their properties drift with time. People also use nonlinear polarization rotation effect in ring fiber cavities as the saturable absorber to generate ultrashort pulses, which is also environmentally sensitive due to presence of fiber spans. Thus a new type of saturable absorber would be useful if the design is compact and manufacturing price is low. Especially, it would be much better for pulse generation in simulation if the SA can be exactly characterized by a well-defined function, and environment independent, which is our goal in this dissertation.

This dissertation primarily focuses on the development of a wavelength-tunable

Erbium doped pulsed fiber lasers with high peak power based on a new mode locking technique based on Kerr effect. This document follows in Chapter 2 with a review on the theory of the mode locking. At the end this chapter, the concept of Kerr lens mode locking is explained. In Chapter 3 the pulse formation and propagation in fiber are explained. An introduction of various types of mode-locked fiber lasers is briefly illustrated and the corresponding pulse shaping mechanism in a nutshell for each type is also touched. Since the new mode locking design provided in this thesis depends on the fiber mode distribution, a thorough study of fiber itself is shown in Chapter 4.

Particularly, we numerically explore how the mode changes in various types of fiber and how the change helps in developing our mode locker. Different materials are used in the

7 simulation to optimize the mode locker design. Then in Chapter 5 simulations for different fiber laser cavities are performed, using the new mode locker design obtained in

Chapter 4. Experiments are performed to confirm our mode locker design and detailed in

Chapter 6. Also, a follow up discussion is presented by the end of Chapter 5 which comes with a closing conclusion to finalize this dissertation.

8 ` CHAPTER 2

MODE LOCKING TECHNIQUE

Mode locking is a technique where many frequency modes in a laser cavity are phased together to form a short pulse in the laser cavity. The phasing between the modes is self-organized within the cavity; it is analogous to adaptation of a biological species to survive under adverse conditions. In the case of mode locking the cavity modes have to cooperate to overcome the high losses in the cavity. The element that creates the mode locking has a high loss at low intensity of the cavity modes and saturates to a low loss at high intensities. The laser cavity is designed so that no single frequency mode can overcome the loss. In this case “survival of the fittest” is the result of many modes cooperating to reduce the cavity loss enough to establish gain.

The history of laser mode locking is an evolving progression of improving the understanding the mode locking mechanisms, and at the same time of generation the shorter and shorter laser pulses. The discovery of different mode locking techniques is well organized by H. A. Haus in a review paper published in the year of 2000[9]. We will briefly introduce the mechanism of several mode locking techniques that are of historical and current values in the ultra-short pulses generation and the organization will closely follow the material in Chapter 2 of [10]. 9

To generate ultrafast pulses, many factors need to be considered: second and higher order material dispersions, gain and gain saturation, gain bandwidth, self-phase modulation (SPM) and modulation. Modulation is the key elements to initiate and maintain the pulse operation. Both phase modulation and amplitude modulation are possible and in the following section we will only concentrate on the illustration of amplitude modulation. There are generally two ways to achieve amplitude modulation: active or passive. Among the first invented mode locking techniques, active modulation is achieved by inserting a modulator in the cavity driven by external sources. The modulation frequency is usually chosen to be the same as the pulse round trip rate.

However, as will be demonstrated, pulse width generated by active modulation is limited by the modulation frequency and pulse shortening process based on such a method becomes ineffective when the pulse becomes too short. However, this drawback can be overcome by utilizing passive mode locking techniques, such as self-amplitude modulation (SAM), where the modulator is replaced by a nonlinear optical medium and the external driving source is replaced by the beam itself. The transmission through the nonlinear medium depends on the intensity of the beam; the higher the intensity of the beam, the higher the transmission rate. Thus when the pulse propagating in the cavity passes through the nonlinear medium, the center part of the pulse with higher intensity will have higher transmission while the wings of the pulse will be clipped due to the low intensity which in turn results in a shorter pulse after the nonlinear medium. In the following section various types of mode locking techniques will be briefly illustrated.

10 2.1 Active Mode Locking

Active mode locking was first illustrated by Hargrove group[11] in 1964 and then

Yariv[12] in 1965. The generic active modulation laser arrangement is schematically shown in Figure 2-1. Simply speaking, a typical active mode locking cavity consists three parts: the linear cavity loss and delay, the gain medium and the modulator. As shown in the figure, the modulator inserted in the cavity is externally driven at frequency 휔푚 =

2휋/푇, where 푇 is the round-trip time of the cavity. Due to the periodic feature of the driving signal, the transmission rate for the modulator is also periodical; the modulator acts like an open door and has peak transmission once per pulse round-trip time, and the transmission exponentially decreases with time further from round-trip time.

Figure 2-1Actively mode-locked laser arrangement, with an intracavity modulator driven at the cavity round-trip period. Reproduced of Figure. 2.1 (a) in [10]. We assume the initial beam to be a Gaussian pulse as the following:

2 푗휔0푡 −Γ푡 푗휔0푡 퐸(푡) = 푅푒{퐴(푡)푒 } = 푅푒{퐴0푒 푒 }, ( 2-1 ) where the pulse center frequency is assumed to be 휔0 and the pulse width is characterized by pulse width parameter Γ . To relate the full width half maximum

(FWHM) pulse width to the pulse width parameter, one can use the following equation:

11 ln (2) 푇 = 2√ , ( 2-2 ) 0 Γ where 푇0 is denoted as the FWHM pulse width. The Fourier transform of the pulse is:

1 퐸̃(휔) = [퐴̃(휔 − 휔 ) + 퐴̃∗(−휔 − 휔 )], ( 2-3 ) 2 0 0 where the Fourier transform of the envelope function 퐴(푡) is:

휋 2 퐴̃(휔) = 퐴 √ 푒−휔 /4Γ. ( 2-4 ) 0 Γ

Travelling through the cavity without considering the gain and modulator, the pulse experiences a linear time independent loss and pulse delay as the following:

푗휔퐿 − 푒−푗휔푇 ∗ 푒−푙0 = 푒 푐 ∗ 푒−푙0, ( 2-5 ) where effective optical cavity length is denoted as 퐿 in which one should take into consideration the length of the free space, gain medium and the modulator. 푙0 is the linear cavity loss and 푐 is the speed of light.

Assuming that the gain medium is homogeneously broadened, the gain spectrum can be best expressed in the spectral domain as the following:

푔 푔̃(휔) = 2, 4(휔 − 휔0) ( 2-6 ) 1 + 2 휔퐺 and

푔0 푔 = , ( 2-7 ) 1 + 푃/푃푠푎푡 where g is the saturated gain, 휔0 is the spectrum center and assumed to be the same as the pulse spectrum center frequency, 휔퐺 is the spectral full-width-half-maximum, 푃 is the instant pulse power and 푃푠푎푡 is the saturation power for the gain medium. If the 12 spectrum of the pulse is narrow compared to the center frequency, then the gain can be approximated by Taylor’s expansion:

2 4(휔 − 휔0) 푔̃(휔) = 푔 (1 − 2 ). ( 2-8 ) 휔퐺

After passing through the gain medium, the pulse becomes:

퐿 휔̃ 2 휋 −푗휔̃ 퐶 − 퐴̃(휔̃) = 퐴 √ 푒−푗휔0퐿퐶/푐 ∗ 푒푔−푙0 ∗ 푒 푐 ∗ 푒 4Γ′, ( 2-9 ) 0 Γ

1 1 16푔 where 휔̃ = 휔 − 휔0 , ′ = + 2 and 퐿퐶 is the amplifier length, the gain 푔 is the Γ Γ 휔퐺 proportional to 퐿퐶.

In the steady state, the change per round trip is very small and the pulse width parameter after gain medium can be approximated by the following:

2 ′ 16푔Γ Γ = Γ − 2 . ( 2-10 ) 휔퐺

Then the beam is transformed back to the time domain as:

′ 퐿 2 Γ −Γ′(푡− 퐶) 퐴(푡) = 퐴 푒푔−푙√ 푒 푐 . ( 2-11 ) 0 Γ

We can see that the pulse width parameter Γ′ is decreased after passing through the gain medium which increases the pulse width in time domain.

Then the last element in the cavity is the modulator which can modeled as the following:

휂(푡) = exp(−∆푚(1 − cos(휔푚푡))), ( 2-12 ) where ∆푚 is the modulation depth. Near by the round trip time, the modulation can be expanded by Taylor’s theory as the following:

13 1 퐿 2 − ∆ 휔2 (푡− ) 휂(푡) = 푒 2 푚 푚 푐 . ( 2-13 )

Thus the beam after passing through the modulator becomes:

′ 퐿 2 Γ −Γ′′(푡− ) 퐴(푡) = 퐴 푒푔−푙√ 푒 푐 , ( 2-14 ) 0 Γ

2 ′′ 16푔Γ 1 2 where Γ = Γ − 2 + ∆푚휔푚. 휔퐺 2

As deducted from the analysis above, the pulse width parameter changes along propagation in the cavity. The gain medium increases the pulse duration while the modulator decreases it. To achieve the steady state, the increase in pulse width from the gain medium should be balanced by the modulator which can be mathematically expressed as the following:

Γ′′ = Γ, ( 2-15 ) which leads to the following equation:

2 16푔Γ 1 2 − 2 + ∆푚휔푚 = 0, ( 2-16 ) 휔퐺 2

Which can be further simplified into the following equation:

1 ∆푚 Γ = √ 휔푚휔퐺. ( 2-17 ) 4√2 푔

From the equation above we can see that there are many factors which can affect the pulse width. Particularly, the output pulse width from active mode locking is limited by the speed of the modulator. Also, it is interesting to see that increasing modulation frequency can increase the pulse width parameter Γ and in turn decrease the pulse width.

14 2.2 Passive Mode Locking

As discussed in the last section, the pulse shortening in active mode locking is limited by the speed of externally driving force and become ineffective for very short pulses. This drawback can be overcome by using passive mode locking. Passive mode locking techniques are those that utilize the wave in the cavity itself, instead of an external driven force, to cause a change in some factors through an element which in turn changes the pulse inside. Depending on the different element, the passive mode locking can occur quite differently. However, generally speaking, the fundamental principle for all passive mode locking techniques are the same: the beam inside the cavity self- modulates itself which is faster than any kind of active modulation, which in turn results in shorter pulses, compared to those generated by active mode locking. The key ingredient in pulse formulation in this case is the self-modulation effect, which introduces a saturable absorber action into the cavity and causes the modes to be locked in phase. In this section we summarize passive mode locking into two groups: fast saturable absorber

(SA) mode locking and slow one. By the end of this section, an artificial super-fast saturable absorber through Kerr effect will be explained in details.

The foundation of the fast and slow saturable absorber was developed by the H.

A. Haus in [13] [14] and summarized by A. M. Weiner in [10], H. A. Haus in [15] and E.

P. Ippen in [16]. The difference between these two types of mode locking is the relative magnitude of the material response time 휏퐴 and the pulse width 푇푝. When 휏퐴 ≪ 푇푝, we refer to fast saturable absorber mode locking and for 휏퐴 ≫ 푇푝 we refer to the slow one.

Both fast and slow saturable absorber mode locking depend on the dynamic change of the time dependent loss and gain, and thus the net gain as a function of time is 15 the key role to understand both cases. As in last section, the cavity linear loss is denoted as 푙0, the time dependent loss and gain are denoted here as 푙(푡) and 푔(푡). The net gain can be calculated as the following:

푔푛푒푡 = 푔(푡) − 푙(푡) − 푙0. ( 2-18 )

The time dependent loss and gain can be solved by applying rate equations to the absorber and gain medium, respectively as in[10].

2.2.1 Slow saturable absorber mode locking

A pulse passing through a slow saturable absorber experiences lower loss at its higher power part and higher loss in the low power region and thus becomes shorter in time after the absorber. Pulses generated from slow saturable absorber are generally longer, compared to that from fast saturable absorber. From the rate equations, one can derive the time dependent loss and gain (refer to more details in [10]) which are

(푖) (푖) schematically shown in Figure 2-2. 푔 and 푙 are initial saturable gain and loss, 푙0 is the time independent linear cavity loss, 푙(푡) and 푔(푡) are the overall time dependent loss and gain. As shown in Figure 2-2, the total loss is larger than the gain before the pulse enters the absorber. When the pulse enters the slow saturable absorber, the loss saturates so that it drops below the gain. Then later on the gain suffers even more saturation so that the gain becomes smaller than the loss. In between there is a time window where the pulse experiences net gain and thus gets amplified.

16

Figure 2-2 Pulse-shortening in slow saturable absorber mode locking. The shaded region indicates net positive gain. Reproduced of Figure. 2.7 in [10]. 2.2.2 Fast saturable absorber mode locking

The recovery time of a fast saturable absorber is relatively short compared to the pulse duration and thus it responds to the beam intensity almost in an instantaneous fashion. To analyze the function of the fast SA, we further assume the gain recovery time is large enough so that the gain saturation during the pulse is very small and thus can be considered as a constant. As shown in Figure 2-3 , the center region of a pulse with higher intensity saturates the absorber and transmits through the absorber while the outer edges of the pulses suffer higher losses. Also note, the gain does not help to shape the pulse.

17

Figure 2-3 Gain and loss dynamics in fast saturable absorber mode locking. The shaded region corresponds to positive net gain. Reproduced of Figure 2.10 in [10]. Even though fast SA are able to achieve extremely short pulses by definition, it is hard to find real absorbers which recovers with recovery time on the femtosecond level.

Thus ultra-short pulses are often generated utilizing the so called artificial saturable absorber. There are many ways to achieve the artificial fast SA effect. For example, combing the self-focus effect with a hard aperture, one can create an intensity dependent amplitude modulation which leads to Kerr lens mode locking[17][18] and additive pulse mode locking (APM)[19][20]. Since the purpose of this dissertation is to develop a new way to achieve the fast saturable absorber effect based on the nonlinear refractive index, we will explore more on the Kerr Effect in the following section.

2.2.3 Kerr lens mode locking

A laser beam can be self-focused due to the change of nonlinear refractive index, also called Kerr effect, which is a function of the beam intensity. The light passing through a medium interacts with the atoms or particles inside, which causes higher order nonlinearity. In the so called central-symmetric medium, the second order nonlinearity,

18 called 휒(2) process, doesn’t exist and the third order nonlinearity, called 휒(3) process, plays the main role of nonlinearity. While all 휒(3) processes are possible, they often require that a phase-matching condition is satisfied, which demands additional design efforts. However, the self-induced effect can happen naturally since the phase-matching condition is automatically satisfied, often referred to as Kerr Effect[21].

In the nonlinear medium, the Kerr effect leads to a phase delay proportional to the intensity profile which is usually the largest on the optical axis and smaller outside the axis. Thus this effect is parallel to the lens function which deforms the phase profile to converge or diverge the beam. The Kerr effect can be expressed mathematically by the following function:

퐼 푛 = 푛0 + 푛2퐼, ( 2-19 ) where 푛 is the refractive index under the influence of the beam, 푛0 is the constant

퐼 refractive index, 푛2 is called the nonlinear index and 퐼 is the light intensity.

The application of self-focusing as a passive device for pulse formation was first reported for Ti:sapphire lasers in 1991 by Spence group[22]. It was called Kerr-lens mode locked laser and achieved pulses as short as 60 fs pulses. Since that time there has been rapid progress in generation of ultra-short pulses based on the same technique with various cavity designs[23]. The reason ultra-short pulses can be generated with Kerr effect is due to the short response time of this effect, which is usually estimated to be 1 to

2 fs. However, the real material response time is typically greater than the normal estimate and the corresponding effects on the pulse duration will be explained in details in Chapter 4. Mode locking action using a self-focusing 휒(3) nonlinearity in conjunction with a hard aperture is schematically shown in Figure 2-4; this is called Kerr lens mode 19 locking with a hard aperture. At low intensities the optical beam is severely clipped by the hard aperture. However, an intense optical beam going through a Kerr medium will be spatially contracted and maintain its temporal profile. The degree of the beam’s contraction depends on the instantaneous intensity of the pulse, which means higher intensity experiences higher degree of beam contraction at the aperture. When the pulse passes through a hard aperture, it experiences lower loss at high intensities due to the nonlinear focusing effect and higher loss at low intensities where the aperture clips the more of the beam. Thus the pulse after the aperture becomes shorter in time as the low intensities are filtered out of the cavity. When the pulse shortening due to the aperture is balanced by broadening due to gain and the material dispersion in the cavity the laser may exhibit stable operation. The combination of the Kerr medium plus the aperture has an effect that is called saturable absorber action; in other words the losses are lower at higher intensity.

Figure 2-4 Kerr lens mode locking with a hard aperture. The new design of mode locking in this dissertation is based on the concept illustrated here and will be examined in greater details through numerical simulation later in Chapter 4.

20 ` CHAPTER 3

PULSE FORMATION AND PROPAGATION IN FIBER LASERS

The pulse propagation in fiber is governed by pulse propagation equation and the lengthy derivation of it from Maxwell equations can be found in Chapter 2 of [24] :

∞ 휕퐴 푔̂ 훼 𝑖푛훽 휕푛퐴 = 퐴 − 퐴 + 𝑖 ∑ 푛 휕푧 2 2 푛! 휕푇푛 푛=2 ( 3-1 ) | |2 2 𝑖 휕 2 휕 퐴 + 𝑖훾 (|퐴| 퐴 + (|퐴| 퐴) − 푇푅 퐴), 휔0 휕푇 휕푇 where A is the slowly varying amplitude of the pulse envelope, z the propagation coordinate, and T the retarded time. 훽푛’s are the material dispersion parameters which are obtained from the Taylor expansion of the propagation constant 훽(휔) around the carrier frequency 휔0, as shown in the following equation:

1 훽(휔) = 훽 + (휔 − 휔 )훽 + (휔 − 휔 )2훽 0 0 1 2 0 2 ( 3-2 ) 1 + (휔 − 휔 )3훽 + ⋯. 6 0 3

훽2 is the group velocity dispersion (GVD) and 훽3 is the third order dispersion (TOD).

Since the pulse equation is transformed to line with the co-moving frame T (also called retarted frame), 훽1 doesn’t show up in the equation. 훾 is the nonlinearity parameter given 21 by 훾 = 푛2휔0/푐퐴푒푓푓, where 푛2 is the nonlinear index, 휔0 the central angular frequency, 푐

푖 휕 2 the velocity of light in vacuum, and 퐴푒푓푓 the effective mode area. (|퐴| 퐴) and 휔0 휕푇

휕|퐴|2 푇 퐴 are higher order terms related to the self-steepening and intrapulse Raman 푅 휕푇 scattering. For pulses whose durations are in the 0.1 picoseconds range, TOD, self- steepening and Raman Effect can be simply ignored. 훼 is the linear fiber loss coefficient which is generally a small number. 푔̂ is the active fiber gain per unit length which theoretically is a function of position in the fiber. The gain spectrum relates to the rare earth dopant introduced into the fiber core and generally covers a range of wavelength.

Usually the gain effect is considered in the frequency domain by multiplication and the exact treatment for gain will be discussed in details in the numerical simulation part in next chapter.

As can be seen from the pulse propagation equation, many factors in the fiber laser cavity influence the pulse formation, among which GVD and third order nonlinearity are the dominating ones. Mode locking is achieved by introducing a saturable absorber (SA) into the cavity which provides the self-amplitude modulation action to promote the mode from the noise and stabilize the pulses once they are generated.

3.1 Soliton Mode Locking

When a pulse is propagating in a passive fiber with anomalous GVD, gain is taken as zero and loss is negligible. If the propagation equation is limited to include only

GVD and third order nonlinearity, it can be reduced to the well-known Nonlinear

Schrodinger equation (NLSE):

22 휕퐴 훽 휕퐴2 = −𝑖 2 + 𝑖훾|퐴|2퐴. ( 3-3 ) 휕푧 2 휕푇2

Then three dimensionless variables are then introduced as the following:

퐴 푧 푇 푈 = , 휉 = , 휏 = , ( 3-4 ) √푃0 퐿퐷 푇0 where 푃0 is the peak power of the pulse, the 푇0 the pulse duration of the incident pulse and 퐿퐷 the dispersion length. Substituting equation ( 3-4 ) back in to equation ( 3-3 ) we will have the normalized version:

2 휕푈 푠푔푛(훽2) 휕푈 = −𝑖 + 𝑖푁2|푈|2푈. ( 3-5 ) 휕휉 2 휕휏2

푁 is an integre parameter which is a function of dispersion length and nonlinear length, expressed in the following equation:

2 퐿퐷 훾푃0푇0 푁2 = = , ( 3-6 ) 퐿푁퐿 |훽2| where 퐿퐷 and 퐿푁퐿 are the dispersion length and nonlinear length, respectively. Equation (

3-3 ) or equation ( 3-5 ) belongs to one special class of equations that can be solved using inverse scattering method which was first introduced by Zakharov and Shabat in

1972[25]. Many solutions to NLSE exist for different values of integer parameter 푁.

When 푁 = 1 for negative GVD, the phase modulation effect by GVD is exactly balanced by third order nonlinearity through self-phase modulation (SPM) and the solution is referred as the fundamental soliton:

푇 퐿퐷 퐴(푧, 푇) = √푃0 sech ( ) exp (𝑖푧 ∗ ), ( 3-7 ) 푇0 2 where the amplitude 푃0 can be expressed as the following:

23 |훽2| 푃0 = 2. ( 3-8 ) 훾푇0

Equation ( 3-8 ) is often referred as soliton area theorem, in which the pulse peak power and duration are inter-related. Given the pulse duration, the peak power of the fundamental soliton is fixed by the fiber system and higher |훽2| results in solitons with higher energy. Similarly, the pulse peak power becomes higher if the pulse has a shorter pulse duration.

When N equals to an integer larger than one, the solutions to NLSE are referred to higher order solitons, which are also called breather solitons since the amplitudes of them are periodic along the propagation distance. While all the solitons discussed so far are for fibers with anomalous GVD, solutions to NLSE exist for fiber in the normal dispersion regime and the corresponding solutions are called dark solitons. Since it is most likely to have bright solitons in the soliton-laser system, we simply neglect the discussion about the dark solitons here in this thesis. From equation ( 3-7 ) we can see that the fundamental soliton preserves its shape during propagation, which makes it a great candidate for fiber laser because the cavity periodic boundary condition can be readily satisfied.

A typical soliton fiber laser cavity is schematically shown in Figure 3-1. The fiber section in the cavity is in the anomalous dispersion region but with positive nonlinearity and it consists at least one segment of gain fiber. SA and OC in the figure represents saturable absorber and output coupler.

24

Figure 3-1 Conceptual Model of typical Soliton Laser. Modified version of Figure 1.6 in [26]. The interplay between the anomalous dispersion and nonlinearity is dominant in soliton formation. Once suitable balance between them is achieved, the soliton duration is almost irrelevant to other parameters such as gain, gain bandwidth and other filtering effects. A saturable absorber (SA) is still needed for initializing and stabilizing the mode locking. However, the gain, gain bandwidth and SA have little effects on the pulse shaping once the pulses are stable.

Simple as it is, soliton lasers dominated the mode-locked fiber laser field for over two decades[27][28][29][30]. However, since the soliton generation is clamped by the dispersion and nonlinearity (which can be seen from the soliton area theorem), the pulse duration is usually in the pico-second range and the pulse energy is limited up to about

~0.1 푛퐽. Once the accumulated nonlinear phase can’t be balanced by the dispersion, the pulse starts to break into multi-pulses. Also, the pulse circulating in the cavity is not a perfect soliton in a strict sense since it undergoes the perturbation of gain and loss and that’s why soliton lasers are often referred as quasi-soliton lasers. This periodic perturbation causes the soliton couple into some dispersive waves which could result in

Kelly side bands [31]. 25 3.2 Dispersion Managed Soliton

Instead of using fibers with only anomalous dispersion as in soliton mode locking, dispersion managed solitons are generated by utilizing a dispersion map where normal and anomalous fibers are periodically displaced. The utilization of dispersion map is usually referred as dispersion management (DM) and the corresponding solutions are called DM solitons. A dispersion map and its corresponding DM soliton are illustrated in

Figure 3-2, where the dispersion parameter 퐷 is related to the GVD ( 훽2 ) by the following relation:

2휋푐 퐷 = − 훽 . ( 3-9 ) 휆2 2

Figure 3-2 Qualitative description of dispersion-managed (DM) soliton. Different rows from top to bottom represent the distance evolution of dispersion, the pulse chirp, pulse duration and pulse spectral bandwidth. Since the dispersion is a function of propagation distance, the pulse experiences periodic net balance between dispersion and nonlinearity at different positions and thus

26 the pulse duration and chirp oscillate periodically along the propagation, which is why

DM solitons are also called stretched pulses[32] and can be graphically verified from

Figure 3-2. On average the pulse in the dispersion map broadens due to net anomalous dispersion which should be exactly balanced by nonlinearity and thus it goes back to its initial pulse width and spectrum bandwidth after each roundtrip propagation. Thus these breathing pulses can still be viewed as solitons on the average sense and are good candidate for mode locked fiber lasers since the periodic cavity boundary condition can be easily satisfied. Interestingly, stretched pulses can also be generated from a cavity with small net normal dispersion.

DM solitons have drawn significant attentions since the alternating sign of dispersion helps suppressing the nonlinear phase accumulation which results in higher energies, usually by one order of magnitude, in comparison to the conventional soliton lasers. In addition, the breathing property of the DM solitons helps reducing the Gordon-

Haus timing jitter in optical communications [33][34].

The stretched pulse propagation can be still be modeled by the NLSE with varying coefficients:

휕퐴 훽 (푧) 휕퐴2 = −𝑖 2 + 𝑖훾|퐴|2퐴, ( 3-10 ) 휕푧 2 휕푇2 which can’t be solved directly and usually is simulated numerically[35].

3.3 Similariton Mode Locking

Besides dark soliton, another solution to NLSE with normal dispersion was found by Anderson et al. in 1993[36], with the amplitude in the form of:

푇2 2 2 ( 3-11 ) 퐴 (0, 푇) = 퐴0 (1 − 2) , 푇 ≤ 푇0. 푇0 27 The suggested pulse is highly chirped with parabolic temporal and spectral profile. It propagates in the normal dispersive medium self-similarly, which means the pulse is form-invariant during propagation and is always a scaled version of itself. Thus the solution is sometimes termed as ‘similariton’. However, in the passive fiber, the similariton is not a nonlinear attractor; without parabolic seeding, it will never become parabolic during propagation by itself.

Then self-similar propagation of parabolic pulses in optical fiber amplifiers were theoretically and experimentally tested by Fermann[38] and theoretically and numerically studied by Kruglov et al. in [37]. Specifically, similariton can be studied by adding gain into NLSE:

휕퐴 훽 (푧) 휕퐴2 푔(푧) = −𝑖 2 + 𝑖훾|퐴|2퐴 + 퐴, ( 3-12 ) 휕푧 2 휕푇2 2 where 푔(푧) is the gain distribution along propagation in the amplifier. If the GVD, nonlinearity and gain are taken to be constants, equation ( 3-12 ) can be analyzed in the normal dispersion regime using symmetry reduction and the solution in the limit 푧 → ∞ is[38]:

푇 2 √ 퐴(푧, 푇) = {퐴0(푧) 1 − [ ] exp[𝑖휑(푧, 푇)] , |푇| ≤ 푇0(푧), 푇0(푧) ( 3-13 ) 0, |푇| > 푇0(푧).

The pulse peak amplitude 퐴0(푧), the phase 휑(푧, 푇) and pulse width 푇0(푧) are given by the following equations:

−1 2 −1 2 휑(푧, 푇) = 휑0 + 3훾(2푔) 퐴0(푧) − 푔(6훽2) 푇 , ( 3-14 )

28 1/3 −1/6 퐴0(푧) = 0.5(푔퐸푖푛) (훾훽2/2) exp (푔푧/3) , ( 3-15 )

−2/3 1/3 1/3 ( 3-16 ) 푇0(푧) = 3푔 (훾훽2/2) 퐸푖푛 exp (푔푧/3), where 퐸푖푛 is the energy of the input pulse to the amplifier, 휑0 is an arbitrary phase constant. As shown in equation ( 3-13 ), the pulse amplitude is parabolic, and from equation ( 3-14 ) the temporal phase is quadratic. The corresponding linear chirp can be derived by taking the first derivative of the temporal phase:

휕휑(푧, 푇) 훿휔(푇) = − = 푔(3훽 )−1푇. ( 3-17 ) 휕푇 2

Even though a similariton in passive normal dispersive fibers can’t form and stabilize by itself, it is demonstrated that it is a strong nonlinear attractor in fiber amplifiers. Thus similaritons are often referred as amplifier similaritons in the laser system and passive similaritons in the passive system, which are shown in Figure 3-3. In contrast to solitons, amplifier similaritons propagate in fiber amplifiers self-similarly; they can tolerate high pulse energy without wave-breaking which makes them attracting potentials in generating high energy pulses. First observation of similariton in laser cavity was made by Ilday and colleges in 2004[7] and a chirped-pulse with energy up to 10 nJ was generated which is 3 times higher than that from a DM soliton laser[39].

As explained before, both soliton lasers and DM soliton lasers are based on the soliton-like pulse shaping mechanism, which is the results of the phase compensation between net anomalous dispersion and third order nonlinearity. The pulse is indeed amplified by the gain medium but compensated by satuable absorber (SA) action. Once the pulses are generated and stabilized, SA and gain only becomes small perturbations to

29 the pulse evolution and the general pulse still preserves the characteristics of a soliton which makes them almost automatically satisfying the roundtrip boundary condition.

However, when the net dispersion becomes largely normal, soliton-like pulse shaping is not practical anymore. Although similariton exists in such a cavity, the underlying pulse shaping mechanism is quite different. As shown in Figure 3-3, similaritons propagate in gain fibers self-similarly, which means it can’t meet the periodic boundary condition by itself. In a similariton laser, the SA is necessary for initiating the pulse formation, as in soliton lasers. However, in order to stabilize the pulses, a spectral filter is needed to cut the spectrum of the pulses back to its starting point and an output coupler brings the shape of the pulse back to the initial amplitude.

The conceptual model of a similariton laser is as shown in Figure 3-4.

Similariton in fibers can be modeled using the pulse propagation equation which is repeated here:

∞ 휕퐴 푔̂ 훼 𝑖푛훽 휕푛퐴 = 퐴 − 퐴 + 𝑖 ∑ 푛 휕푧 2 2 푛! 휕푇푛 푛=2 ( 3-18 ) | |2 2 𝑖 휕 2 휕 퐴 + 𝑖훾 (|퐴| 퐴 + (|퐴| 퐴) − 푇푅 퐴), 휔0 휕푇 휕푇 with SA and extra filter being considered separately.

30

Figure 3-3 Illustration of similariton propagation in passive fiber (on the left ) and gain fiber (on the right ). Taken from [40].

Figure 3-4 Conceptual model of a similariton laser. 3.4 Dissipative Soliton Mode Locking

Previously we showed that similaritons can be analytically analyzed using NLSE in the gain fiber if the gain, dispersion and nonlinearity are taken to be constants.

However, in a laser cavity, there are more elements besides gain fiber such as saturable absorber (SA), output coupler (OC) and spectrum filter (SF). Thus in the cavity there are not only phase modulation terms (SPM, GVD) but also amplitude modulations (gain, SA).

If we take into consideration these terms on an ‘averaged’ sense, cubic-quintic Ginzburg-

Landau equation (CQGLE) is formed as the following:

31 휕푈(푧, 푡) 1 퐷 휕2푈(푧, 푡) = 푔푈(푧, 푡) + ( − 𝑖 ) 휕푧 Ω 2 휕푡2

+ (훼 + 𝑖훾)|푈(푧, 푡)|2푈(푧, 푡) ( 3-19 )

+ 훿|푈(푧, 푡)|4푈(푧, 푡), where 푈(푧, 푡) is the slowly varying envelope of the electrical filed, 푡 the retarted time, 푧 the propagation distance, 푔 the linear gain, Ω the spetral filtering effect, 퐷 the GVD, 훾 the third order nonlinear coefficient, 훼 and 훿 the cubic and quintic saturable absorber terms. This equation without quintic term is also known as the master equation and is valid to express the pulse propagation as long as the intra-cavity fluctuations are small.

Even the real cavity could be far away from this condition, the solutions to this equation still provide great insights on the qualitative performance of different lasers. Especially the work of scientists has been successfully guided by the analytical results in search of high energy pulses with net large normal dispersion[19][41][42].

Even though the general solution to equation ( 3-19 ) is unknown, one exact pulse solution (also called dissipative soliton) exists[42][43][44][45]:

푖훽 푡 퐴 (− ln[cosh( )+퐵]+푖휃푧) 2 휏 푈(푧, 푡) = √ 푡 푒 , ( 3-20 ) cosh (휏) + 퐵 where A, B, 휏, 훽 and 휃 are real constants. Inserting solution ( 3-20 ) back into equation

( 3-19 ) and equating the real and imaginary parts, one can get six equations which reveal the inter-relation between the system parameters ( 푔, Ω, 퐷, 훼, 훾 푎푛푑 훿 ) and solution parameters (A, B, 휏, 훽 and 휃)[41][42].

As seen from ( 3-20 ), this solution is highly chirped in an all-normal laser cavity which was first experimentally demonstrated by Andy Chong [4][46][47]. Thus this all-

32 normal-dispersion fiber laser is often referred as ‘Andi laser’. A typical Andi laser cavity is shown in Figure 3-5.

Figure 3-5 Conceptual model of all-normal-dispersion laser (Andi laser). Andi lasers inherently depends on lossy and dissipative pulse shaping process, which is facilitated by a spectral filter (SF); since the pulse is highly chirped, the SF cuts out the frequency and at the same time shrinks the pulse duration which ensures the periodic boundary condition can be satisfied. The Andi laser can routinely generate pulses with duration ~100 fs and a few Nano-Joules (nJ) and the energy has been scaled up to 84 nJ and duration down to 31 fs [48] which is consistent with the prediction in [6].

33 ` CHAPTER 4

NUMERICAL STUDY OF THE NEW SATURABLE DESIGN

This project is designed to study a new approach to achieving a mode locked fiber laser by harnessing the Kerr effect on the transverse beam shape, i.e. self-focusing effect.

The conceptual laser cavity under study is shown in Figure 4-1. The black line in the figure stands for the Er:Yb doped fiber and the blue lines are normal single mode fibers.

An isolator is used to ensure that the pulse is propagating clockwise.

Figure 4-1 Conceptual fiber laser cavity with the new saturable absorber design. Coupler1 transmits wavelength nearby 972 nm and 1550 nm. Coupler 2 splits the input into two parts: one through output and one back into the fiber cavity. In the fiber cavity, the black fiber is the Er:Yb doped gain fiber and the blue fibers are single mode fibers (SMFs). An isolator is used to ensure that the pulse is propagating clockwise. There is a free space region in the dashed rectangle where the saturable absorber function is realized. In that region, both transmitting and receiving fibers are tapered to modify the mode sizes and are placed face to face with a certain separation in between 34 them. The gap is filled with a nonlinear medium (for example 퐶푆2 or chalcogenide glass).

The new mode locker design is better shown in Figure 4-2. The transmitting fiber is denoted as fiber 1, the receiving fiber is denoted as fiber 2 and the separation is denoted as 퐷. In this thesis we assume that fiber 1 and fiber 2 are placed face to face without any angular tilt and lateral shift.

In Figure 4-2, the beam coming out from fiber 1 is reshaped in the nonlinear medium due to the net balance of diffraction and Kerr effect; the beam coupling into fiber

2 is affected by the beam size. Comparing Figure 4-2 to Figure 2-4, the similarity between the end of fiber 2 and the previously discussed hard aperture is apparent, and the analogy is also closely connected to the pulse shortening process in the fiber cavity with a design illustrated in Figure 2-4. The transmission between fiber 1 to fiber 2 depends on the initial and final fiber mode radii, the separation between the fibers, the nonlinear material and the pulse power carried in the wave. Note that the transmission and coupling efficiency are interchangeable in this paper and denoted as 휂.

Figure 4-2 New saturable absorber design. There are four key elements that affect the transmission between fiber 1 and fiber 2: nonlinear medium, end1, end 2 and the separation 퐷 푚푚 in between them. To understand the pulse transmission through the free space region, let’s first assume that the beam transmitting through the free space is a continuous wave. In this paper we study the saturable absorber device and its action; for simplicity we assume fiber 1 and fiber 2 have fiber modes with the same radii at the ends. The coupling or

35 transmission from fiber 1 to fiber 2 can be maximized if the diffraction in the nonlinear medium can be exactly balanced by the nonlinearity and the corresponding beam power is called critical power [49][50], which is better explained in section 4.2.1.3 and section

4.2.2.3. With lower or higher power, the coupling efficiency deteriorates. In this paper, all the powers used are assumed to be smaller than the critical power, since beyond the critical power the simulation becomes unstable as the beam collapses and beam filamentation can occur [51]. Below the critical power, the higher power the beam contains, the more nonlinearity it experiences, thus more the diffraction can be balanced and the transmitted beam more closely matches the mode inside fiber 2. In other words, beam carrying higher power is going through the new saturable absorber device with higher transmission rate. This is analogous to the free space region for the fast saturable absorber (SA) [13], [14], [16], [19].

The beam from fiber 1 is assumed to be the mode inside multiplied by a constant and it can be calculated through Finite Difference Method (FDM)[52]. The transmission rate through the free space region as a function of beam power can be calculated through beam propagation method (BPM)[21] and the terminologies of free space and SA will refer to the same thing in this dissertation. Simulation for various configurations can be easily made by adjusting any one of the four elements. Still, the exact transmission rate should take into consideration beam transitioning through the taper region and the reflection on the surfaces, even though in the numerical study we don’t have to under the assumption that the transition is 100% in the tapered region and the surfaces are coated with anti-reflection layers. The details of simulation on the transmission between fiber 1 and fiber 2 can be found in APPENDIX A and APPENDIX B.

36 The gain in the laser cavity is set below the threshold for continuous wave lasing.

A pulse circulating inside the laser cavity can be sustained since it lowers the losses in the saturable absorber. Since the nonlinear medium is assumed to react instantaneously to the pulse which is explained in section 4.2.1 and section 4.2.2, we slice the pulse into many temporal segments, one segment of the pulse can be viewed as a continuous wave and the analysis of transmission through the SA for each slice is similar to the case of a continuous wave; the dispersion added to the pulse in this short section is negligible. Due to dispersion the pulse width is broadened when it is propagating in the laser cavity; when the pulse passes through the SA, we consider the pulse as a series of continuous wave; different slices of the pulse are going through the saturable absorber (SA) independently. Similar to the continuous wave analysis, different slices of the pulse containing varying powers will pass through the SA with different transmission amplitudes; the center part of the pulse with higher power experiences higher transmission rate than that in the wings of the pulse. The pulse will be shortened by the

SA action. Once the shortening process through the SA is balanced by the broadening effects from the rest of the cavity, the laser may approach the steady state.

To make the simulation of the pulse propagation inside the cavity easier, we divide the propagation into two parts: the propagation of a pulse through the saturable absorber and the propagation in the gain and normal fibers. Pulse propagation in the gain and normal fibers can be simulated using standard Split-Step-Method (SSM)[24] which will be explained in details in Chapter 5. The pulse transmission through the mode locker is done by viewing the pulse as a series of continuous wave. Each different slice from the pulse is going through SA independently with a different transmission rate, which is

37 calculated by using a continuous wave as discussed in previous paragraph. Thus the key to understand the function of the free space region is how the transmission relates to different continuous waves and this part is going to be covered in this chapter.

In Section 4.1 we will study the first two parameters of the SA (fiber 1 and fiber

2); we will study various types of fibers and explore the mode distribution inside through

ModeSolver. In Section 4.2, we will simulate the SA with two nonlinear mediums in place (carbon disulfide 퐶푆2 and chalcogenide glass 퐴푠40푆푒60). The effect of the fiber sizes, separation and nonlinear medium on the transmission rate will be studied simultaneously. Transmission curves, e.g. the saturable absorber action, will be provided for both materials, which are going be used in the whole cavity pulse simulation in

Chapter 5.

4.1 Numerical Study of the Fundamental Mode in Fibers

As explained before, the transmission of the new saturable absorber (SA) design is a function of the fiber sizes. To ensure a high coupling efficiency, the beam entering the fiber 2 should be matched to the mode inside as much as possible. In this section, we will explore the fundamental mode distribution in different fibers using the home-cooked mode solver.

The mode solver is based on the Finite difference Method (FDM) [52] and the user interface is shown in Figure 4-3. On the panel, fiber information as input is shown in the up-right corner which is characterized by parameters: 푛푐표푟푒, 푛푐푙푎푑, 푛푎푖푟 representing the refractive indices for each layer of material, respectively; 푟푐표푟푒 , 푟푐푙푎푑 , 푟푎푖푟 representing the radius of each layer from the center of the fiber; 푁1 being the number of points between – 푟푐표푟푒 to 푟푐표푟푒 , 푁2 between 푟푐표푟푒 and 푟푐푙푎푑 , 푁3 between 푟푐푙푎푑 to 푟푎푖푟 . 38 Thus the total number of points on the 2D fiber cross section in simulation is (푁1 + 2 ∗

푁2 + 2 ∗ 푁3)2. The wavelength is the center wavelength emitted by the gain fiber. The simulation is carried out iteratively using Inverse Scattering Method (ISM). Each round of calculation gives out a mode distribution and an effective index, and the convergence is checked by comparing results from each two adjacent iterations. Once the difference between each two rounds of calculation is smaller than the pre-set tolerance, the code is stopped, the final results are shown on the user-interface and the mode distribution is also graphically illustrated on graph panel.

Figure 4-3 The user interface of the Mode Solver. Since in the laser cavity we use Er doped fiber as the gain medium, wavelength under simulation is fixed at 1.55 휇푚. The mode radius is defined to be the one where the intensity has dropped to 1/푒2 of the fiber center intensity. The effective index and mode radius as functions of fiber size are recorded in Table 4-1. As a comparison, the results from the well-known software Comsol are also included.

39 The mode radius and effective index as functions of fiber size are plotted in

Figure 4-4 in red asterisks. Also plotted in black circles are the results calculated from the software Comsol. As we can see from the figures, the calculation from the ModeSolver closely follows the results from Comsol.

Then the next problem to ask ourselves is: how to taper the fibers to maximize the coupling efficiency between them? Since the coupling efficiency depends on not only the fiber sizes but also the nonlinear medium, the separation and the instantaneous power of the beam, there is no established equation through which calculation can be conducted directly. For each fiber pair, the beam has its own dynamic of diffraction and nonlinearity which can only be numerically studied. In next section we will choose several fiber pairs, calculate the fundamental mode distributions inside and calculate the coupling efficiency by taking into consideration of all the other factors.

Table 4-1The mode radius and effective index as function of fiber sizes.

r_mod (m) r_mod(m) n_eff n_eff r_core(m) r_clad(m) (ModeSolver) (Comsol) (ModeSolver) (Comsol) 0.328 5 3.67863 3.7 1.45059 1.45065 0.656 10 6.99489 7.0337 1.45397 1.454014 0.984 15 9.43586 9.57 1.45463 1.454671 1.312 20 10.4684 10.1027 1.45488 1.4549 1.64 25 8.48851 8.2195 1.45504 1.45507 1.968 30 6.11308 6.1164 1.4552 1.45239 2.296 35 5.16477 5.1502 1.45543 1.455466 2.624 40 4.72125 4.7274 1.45572 1.455758 2.952 45 4.55303 4.5937 1.45605 1.45608 3.28 50 4.59078 4.5901 1.45637 1.4564 4.1 62.5 4.8022 4.8516 1.45709 1.457117

40

Figure 4-4 (a) the mode radius as a function of the fiber size; (b) the effective index as a function of fiber size. 4.2 Numerical Study of the New Mode Locker

In this section, we are going to calculate the transmission curve of the new SA design with different configurations. In each configuration, the fibers and the nonlinear medium are fixed and the separation between fiber ends are adjustable. For one separation, we can propagate a continuous wave through the free space region and calculate the corresponding coupling efficiency using Beam Propagation Method (BPM)

[21]. By varying the power of the continuous beam, we can generate a transmission rate as a function of power. The calculation is repeated for different separations and a 2-D plot of coupling efficiency 휂 can be generated as a function of the beam power and separation. Then the same procedure can be done to different SA configurations where sizes for fiber1 and fiber2 are different and the 2-D transmission curves are generated.

Comparison between different configurations will be made so that the best one for mode locking will be selected for next stage simulation.

In order to obtain strong saturable absorber action, we need to find a material with a nonlinearity as high as possible and at the same time the relaxation time as small as 41 possible. Carbon disulfide (퐶푆2) naturally becomes one of our interests since extensive researches have been conducted with it due to its high nonlinearity. Besides, as a liquid,

퐶푆2 is easy to operate and be incorporated into our mode locker design and details about

퐶푆2 will be discussed in the section 4.2.1. Among the glasses, chalcogenide glass group has been a rising star due to its extremely high third order nonlinearity and thus will be included in our search and its properties will be touched in section 4.2.2.

4.2.1 Simulation with carbon disulfide (푪푺ퟐ)

To characterize the nonlinear optical parameters of a specific material, several experimental detection techniques are usually implemented, such as Z-scan, optical Kerr gate (OKG)/optical Kerr effect (OKE), degenerate four-wave mixing (DFWM) and optical heterodyne detection of optical Kerr effect (OHD-OKE). However, the measurements are often neither easy nor accurate to be obtained directly. Thus the nonlinear parameters are often determined against a reference material, with carbon disulfide (퐶푆2) being one of the most frequently used. 퐶푆2 has a strong nonlinearity among the liquids and is one of the most widely studied nonlinear material.

4.2.1.1 Relaxation time of 푪푺ퟐ

To mode lock the fiber laser, the relaxation time of the nonlinear medium should be at least shorter than the steady pulse. Otherwise, the pulse may be severely distorted.

There are several mechanisms that contribute to the change of third order nonlinearity of

퐶푆2, each with its own characteristic response time and polarization dependence [53].

Roughly speaking, those mechanisms can be divided into two categories: those from bound electrons and those from nuclei. Response time from bound electrons is on the order of 1 fs, which can be viewed as instantaneous compared to most pulses. As a result 42 of the large mass of nuclei, response time from nuclear mechanisms to the change of nonlinearity are generally longer, up to hundreds of femtoseconds. The overall change in nonlinearity can be modeled by adding up contributions from each individual mechanism and it is demonstrated that for pulses with different durations relative contribution from each mechanism is different, as shown in Fig. 4 of [53]. Particularly, for pulses with duration below ~200푓푠, the change of nonlinearity is dominated by bound electrons and the response time can be considered as instantaneous. For pulses with durations beyond ~200푓푠, other nuclear mechanisms starts to contribute to the nonlinearity and the corresponding response time is much longer, generally hundreds of fs[53]. It is reported in [54] that the longest component in 퐶푆2 is due to the decay of optically induced birefringence ( also called molecular re-orientation and on the order of ~ 1.6 - 2.2 ps ).

Thus it is reasonable to consider the response time of 퐶푆2 to be ‘fast’ as long as the pulse is either in the 10’s ps range or below ~200푓푠.

4.2.1.2 Nonlinearity of 푪푺ퟐ

As mentioned in the last section, the nonlinearity in 퐶푆2 is induced by many factors: part from bound electrons and part from nuclei and contribution in the change of nonlinearity from each mechanism is different, depending on pulse duration and center wavelength. To obtain the exact value of the third order nonlinearity, one should know the exact pulse beforehand, which is generally unknown before laser cavity is built.

However, based on experience, we know the pulse in the cavity is about a few picoseconds (which is true for a typical Andi Laser cavity) and the center wavelength is

1.55 um. Thus we assume that both the electrons and nuclei contribute to the change of nonlinearity to the maximum degree. The corresponding nonlinear index reported from 43 [53] is roughly between 25 × 10−19 푚2/푊 and 34 × 10−19 푚2/푊, which is consistent with the value provided in Table 4.1.2 of [50]. In this thesis, we use 푛2 = 32 ×

10−19 푚2/푊 (refer to Table 4.1.2 of [50]) for simulation.

4.2.1.3 Critical power of 푪푺ퟐ

Beam propagating in the nonlinear medium encounters diffraction and self- focusing effects. With exact balance between those two effects, the beam can propagate without being distorted which is also called self-trapping effect. Analysis in Chapter 7 of

[50] shows that self-trapping occurs only if the power carried by the beam is exactly equal to the so-called critical power, shown as following:

2 2 휋(0.61) 휆0 푃푐푟 = , ( 4-1 ) 8푛0푛2 where 휆0 is wavelength for continuous wave or the center wavelength for the pulse, 푛0 is the linear refractive index and 푛2 stands for the nonlinear index. As discussed in last section, carbon disulfide has the largest third order nonlinearity among the liquids Table

4.1.2 of [50]:

2 −14 푐푚 푛2_퐶푆 = 3.2 × 10 ( ), ( 4-2 ) 2 푊 and the corresponding critical power in 퐶푆2 at 휆0 = 1.55 μ푚 is calculated to be:

푃푐푟_퐶푆2 = 67.3 푘푊, ( 4-3 )

4.2.1.4 Saturable absorber based on 푪푺ퟐ

In this section, the saturable absorber design is based on carbon disulfide (퐶푆2) and the fibers in SA are chosen as a pair. For example, if the fiber pair of 10-10 is used in simulation, it means both the emitting and accepting fibers are tapered to be of 10 휇푚 as

44 the outer radius. Note that both fibers are initially the same normal single mode fiber with an outer radius of 62.5 휇푚 and a core radius of 4.1 휇푚. All the fiber pairs used for simulation are shown in Table 4-2.

Table 4-2 Fiber pairs used for simulation with nonlinear material 퐶푆2. Fiber Pairs 62.5—62.5 40—40 30—30 20—20 10—10 (휇푚 − 휇푚)

Let’s take the fiber pair 20—20 as an example where both fiber 1 and fiber 2 are chosen to have 20 휇푚 as the outer radius. The coupling efficiency as a function of continuous beam power and separation between fiber1 and fiber2 is plotted in Figure

4-5(a). Simulation beyond the critical power can’t be trusted due to filamentation and thus is ignored. As we can see for each separation, the transmission gets higher with increased beam power, up to roughly 67.3푘푊, which is the critical power and calculated using the equation given in [50]. To achieve mode locking, we care not only the absolute coupling efficiency but also the transmission difference between high and low powers which is shown in Figure 4-5(b). For each separation, the transmission difference is obtained by subtracting the coupling efficiency of the lowest power (0.000287 푘푊 in our simulation) from the absolute transmission rate. The higher the transmission difference is, the stronger SA action would be achieved. For example, with 퐷 =

2.003 푚푚, the continuous beam with power of 64.02 푘푊 propagates through the SA with a transmission rate of 83.01% (data tip in Figure 4-5 (a)) and the transmission rate for continuous beam with power of 0.00287 푘푊 is 15.48% (data tip in Figure 4-5 (a)) as shown on Figure 4-5 (a). The difference between this two continuous beams is 67.53% which is shown on Figure 4-5 (b). Similarly, at 퐷 = 2.003 푚푚, the transmission rate

45 difference can be calculated for beams with other powers by subtracting the transmission rate of lowest power (0.00287 푘푊) from that of the specified beam. Similar calculation can be carried out for other separations, which is shown Figure 4-5 (b).

Even though the transmission curves so far are calculated using continuous beams, they can be used for pulses. For instance, let’s assume that a pulse oscillating in the laser cavity has peak instantaneous power of 64.02 푘푊. Transmission 휂 through the new SA design for the lowest power part of the pulse can be assumed to be the same as 휂 of CW with power of 0.00287 푘푊. Then transmission difference ∆휂 between the center slice (high power part) and the low power slice (0.00287 푘푊) is the same as the one between the two continuous beams (67.53%).

Even for the same pulse, transmission curve is different with a different separation. Thus the value of 퐷 should be adjusted so that the best mode locking operation can be achieved. For example, using the same pulse with peak instantaneous power of 64.02 푘푊, we can check on Figure 4-5(b) to look for the separation at which the ∆휂 between peak power and the lowest power is the largest. In this case, the largest difference occurs at 퐷 = 1.865 푚푚 and the corresponding ∆휂 is 67.79%.

For each different cavity, the stable pulse that can be supported is different and thus the peak instantaneous power of the pulse is also different. From the analysis above, for pulses with different peak instantaneous powers, optimum mode locking separations are different. By setting both fiber 1 and fiber 2 to be 20 휇푚 as the outer radius, and using 퐶푆2 as the nonlinear medium, the best separation for different pulses can be found from Figure 4-5(b) and is plotted in Figure 4-6(a). As shown, as the peak instantaneous power of a pulse goes up, the optimum separation between fiber ends increases 46 accordingly. The ∆휂 between the peak power and the low power of a pulse at optimum separation is plotted in Figure 4-6(b) from which we can see that the coupling difference for a pulse is also increasing with increased pulse peak power. For example, if the pulse is assumed to be the same as before (peak power is 55.33 푘푊 and the far wings has 휂 almost same as 0.00287 푘푊 ), the optimum separation can be obtained from Figure

4-6(a) (퐷 = 1.36 푚푚 shown as datatip in Figure 4-6(a)) and the corresponding optimum

∆휂 is shown in Figure 4-6(b) (∆휂 = 46.96% shown as datatip in Figure 4-6(b)).

Figure 4-5 Simulation for fiber pair 20-20 with carbon disulfide (퐶푆2): (a) the coupling efficiency 휂 as a function of separation 퐷 and beam power; (b) the coupling efficiency difference between high power and low power beams. On (a), 휂 for two beams with power of 0.000287 kW and 64.02 kW are shown at 퐷 = 2.003 푚푚 and the difference between them is shown on (b). For a pulse with peak power of 64.02 kW, the optimum separation is at 퐷 = 1.865 푚푚, shown on (c). Note that Figure 4-6 is only for the fiber pair (20um-20um). We extract the similar information for other fiber pairs and plot them together in Figure 4-7. From

Figure 4-7(b), we can see that for a certain pulse (with a certain peak power), the 47 optimum transmission difference ∆휂 is almost the same no matter which fiber pair is used. However, as shown in Figure 4-7(a), the optimum separations 퐷 are different for different fiber pairs. The fiber pair that allows largest separation is the pair of 20-20.

While the transmission curve is insensitive to the fiber pairs, it is better to conduct the experiment with larger separation in between fiber ends because it brings in more room of controlling.

Figure 4-6 Simulation for fiber pair 20-20 with carbon disulfide (퐶푆2): (a) Optimum separation as a function of pulse peak power; (b) Optimum coupling efficiency difference between the peak power and the low power region as a function the pulse peak power.Change Peak power into power.

Figure 4-7 Simulation for different fiber pairs with 퐶푆2: (a) Optimum separation as a function of pulse peak power; (b) Optimum transmission difference ∆휂 between the peak power and the low power part of a pulse as a function the pulse peak power.

48 4.2.1.5 Saturable absorber action with 푪푺ퟐ

From the analysis in previous sections, we know that for each pulse there is an optimum mode locking separation where the pulse center and wings transmission rate difference ∆휂 is maximized. However, the steady pulse remains unknown until we run the simulation for the whole cavity. If we adjust the separation towards the optimum point after simulation, the steady pulse would adjust accordingly which in turn would result in a new optimum point for the new pulse. Thus in practice, it is an evolving process in which design of SA has to keep adjusting. However in this paper we fix the design of SA and adjust it later in laser simulation which will be published in future. It can be seen that for 퐶푆2 the optimum separation 퐷 is close to 3 푚푚 when the pulse peak power is close to the critical power. Thus we will use fiber pair of 20-20 with 퐷 = 3 푚푚 in the simulation for the new saturable absorber design and the corresponding transmission (saturable absorber action) is plotted in Figure 4-8.

49

Figure 4-8 Transmission curve for the new SA design with 퐶푆2 at 퐷 = 3 푚푚. 4.2.2 Simulation with chalcogenide glass

Combined with net-working elements 퐴푠, 퐺푒, 푃, 푆푏 푎푏푑 푆𝑖l, Chalcogenide glasses

(ChG) contain chalcogens (푆, 푆푒, 푇푒). It is well known that ChG transmits into the far infrared region and have drawn substantial attention from researchers owning to its extremely high nonlinearity among the crystals and its fast response time. It has become a promising candidate in all-optical-switching and processing devices, which is the key element in high-speed optical communications.

4.2.2.1 Relaxation time of chalcogenide glass

The origin of intensity-dependent nonlinearity in Chalcogenide glass is from both electronic and nuclear. The response time from electrons and nuclei are generally below

50 fs and a few hundreds of fs (500 fs in [55] and 100 fs in [56]), respectively. Specially, the nuclei contribution to the nonlinearity is found to be on the order of ~15% in heavy 50 metal oxide and sulfide glasses [56], which is consistent with the result of 12% − 13% for sulfide based glasses provided in [57]. Thus with ChG, as long as the steady pulse under study is within the pico-second range, it would be safe to view the material response time as instantaneous.

4.2.2.2 Nonlinearity of chalcogenide glass

As with the 퐶푆2, the third order nonlinearity of ChG strongly depends on the wavelength. Furthermore, different ChGs are made of different component atoms and correspondingly exhibit varying nonlinearity. Particularly, at tele-communication wavelength of 1.55 푢푚, 퐴푠40푆푒60 has been found to enjoy both high nonlinearity and figure of merit (FOM), 2.3 × 10−17푚2/푊 and 11 , respectively[58][59]. It is worth to mentioning to the reader that extremely large third order nonlinearity has been achieved

−16 2 near 1 푢푚 ; 9.0 ± 1.4 × 10 푚 /푊 is realized in a 4.8 휇푚 thick 푇푒20퐴푠30푆푒50 thin film [60] at 1064 푛푚 and 2000 to 27000 times as large as that of fused silica

−20 2 (푛2_푓푢푠푒푑_푠푖푙푖푐푎 = 3.0 × 10 푚 /푊) at 1.05 휇푚 in 퐴푔 doped 퐴푠40푆푒60[61]. Specially in

[61], higher dopant of silver generally results in higher nonlinearity. However, high percentage of dopant of silver inevitably increases the linear absorption rate.

4.2.2.3 Critical power of chalcogenide glass

−17 2 From [58], nonlinear index is 푛2 = 2.3 × 10 푚 /푊 and the refractive index at

휆 = 1.55 휇푚 is 푛0 = 2.81. Thus based on equation ( 4-1 ), the critical power of ChG glass 퐴푠40푆푒60 is calculated to be:

푃푐푟_퐶ℎ퐺 = 5.43 푘푊. ( 4-4 )

51 4.2.2.4 Mode locker based on chalcogenide glass

From simulation results for 퐶푆2 we know that the optimum transmission 휂 between fiber 1 and fiber 2 (refer to Figure 4-2) is almost irrelevant to the fiber pairs and optimum separation 퐷 between fiber 1 and fiber 2 is larger with fiber pairs of 20-20.

Thus in this section we only use the fiber pair of 20-20 in our simulation. The transmission and the corresponding transmission difference map are plotted in Figure 4-9.

From critical power calculation in section 4.2.2.3, we know that chalcogenide glass

퐴푠40푆푒60 has much lower critical power, compared to 퐶푆2 , which can be seen by comparing Figure 4-9 to Figure 4-5. Low critical power means that the same coupling efficiency difference ∆휂 can be achieved with ChG 퐴푠40푆푒60 at much lower power. This translates to a much stronger saturable action for ChG 퐴푠40푆푒60, which makes ChG

퐴푠40푆푒60 much easier to use in practical experiment since strong SA helps initiate mode locking process as explained in Chapter 3.

Figure 4-9 Simulation for fiber pair 20-20 with ChG 퐴푠40푆푒60. (a) the coupling efficiency 휂 as a function of separation 퐷 and beam power; (b) the coupling efficiency difference between high power and low power beams.

52 Similar to the case of 퐶푆2, the optimum coupling difference and separation for

ChG 퐴푠40푆푒60 as functions of power can be extracted from Figure 4-9 (b) and are plotted in Figure 4-10. While saturable absorber (SA) with ChG enjoys higher transmission for the same pulse, it can also be made thicker by comparing Figure 4-10 (a) to Figure 4-6(a).

Figure 4-10 Simulation for fiber pair 20-20 with Chalcogenide glass 퐴푠40푆푒60: (a) Optimum separation as a function of pulse peak power; (b) Optimum transmission difference ∆휂 between the peak power and the low power region as a function the pulse peak power. 4.2.2.5 Saturable absorber action with chalcogenide glass

For the same reason as explained in section 4.2.1.5, we choose to set the thickness of the ChG 퐴푠40푆푒60 plate to be 3 푚푚 thick and the corresponding coupling efficiency is plotted in

53

Figure 4-11 Transmission curve for 퐶ℎ퐺 퐴푠40푆푒60 at 퐷 = 3 푚푚

4.2.3 푪푺ퟐ or ChG: which one to use?

From the analysis in previous sections, we know that for each pulse there is an optimum mode locking separation where the pulse center and wings transmission rate difference is maximized, no matter which material to be chosen as the nonlinear medium.

Then it comes to the question: which nonlinear medium to use? The answer depends on the estimate of the steady pulse power. To successfully achieve the mode locking function, we should try to maximize the difference of transmission rate between the high and low power region of the pulse. To accomplish this, the peak instantaneous power of the pulse should be as close as possible to the critical power. Thus to determine which nonlinear medium to use in the practical experiment, we should estimate the power

54 of the pulse that can be sustained in the cavity and then choose the right material accordingly. For example, if we need to produce a pulse with peak power of 5.734 푘푊, it wouldn’t be a good idea to use 퐶푆2 as the nonlinear medium, since transmission difference ∆휂 between peak power and low power region for 3 mm 퐶푆2 is only 0.6731%, shown as the datatip on Figure 4-12 (a). However, chalcogenide glass 퐴푠40푆푒60 would be a better idea, since the corresponding ∆휂 for the same pulse with 3 mm 퐴푠40푆푒60 is

74.47%, shown as the datatip in Figure 4-12 (b).

Figure 4-12 Comparison of coupling difference for fiber pair 20-20 with nonlinear medium of (a) 퐶푆2; (b) 퐴푠40푆푒60. Thus depending on the estimate of the laser pulse duration and energy within the cavity, different nonlinear mediums should be picked. If the one estimates the peak instantaneous power reaches close to the critical power of 퐶푆2, one should use 퐶푆2 as the nonlinear medium. On the other hand, if the peak instantaneous power of the pulse is expected to be lower than critical power of ChG 퐴푠40푆푒60, one should use CHG instead as the nonlinear medium.

As a comparison, the transmission 휂 for both 퐶푆2 and 퐴푠40푆푒60 (20-20 fiber pair 퐷 = 3 푚푚) are plotted together in Figure 4-13. Note that the saturable action in

55 Figure 4-13 only covers powers lower than critical power. Transmission beyond critical power are not shown, since filementation can occur and both experiment and simulation become unstable. Also note that the transmission curve corresponds to power, not amplitude. In the simulation we will try to limit the pulse so that the peak power doesn’t exceed the critical power of the nonlinear medium used. Exact numerical treatment for the SA action based on Figure 4-13 will be explained in details in Chapter 5.

Furthermore, due to the refractive index change, reflection always occurs at the interface between two different materials, which would further reduce the theoretical coupling efficiency between fiber 1 and fiber 2 in the mode locker design (see Figure 4-2).

However, this effect isn’t taken into consideration in the simulation, assuming that it can be negated to a negligible level by index matching the nonlinear medium to the fibers.

Figure 4-13 Transmission curve of new saturable absorber design for fiber pair of 20-20 with different nonlinear mediums (a) 퐶푆2; (b) 퐴푠40푆푒60.

56 ` CHAPTER 5

NUMERICAL STUDY OF THE FIBER LASER CAVITY USING

NEW SATURABLE ABSORBER DESIGN

The general conceptual laser cavity model under study is shown in Figure 4-1.

Since the transmission from fiber 1 to fiber 2 is already obtained, we simply replace the

Kerr medium by a symbol of saturable absorber. Additionally, we will use extra spectral filter for pulse shaping. In the following sections, we will first introduce how different elements are handled in simulation and then whole cavity simulation will be performed for different cavities.

5.1 Optical Elements in Cavity

5.1.1 Pulse propagation in fibers

The pulse simulation in fibers is carried out using the pulse propagation equation

( 3-1 ). Self-steepening and intrapulse Raman scattering can be ignored since we expect the pulses from our study are wide enough and contributions from those process are very minimum. Considering only up to the third order dispersion, we rewrite the pulse propagation equation as the following:

휕퐴 푔̂ 훼 훽 휕2퐴 훽 휕3퐴 = 퐴 − 퐴 − 𝑖 2 + 3 + 𝑖훾|퐴|2퐴, ( 5-1 ) 휕푧 2 2 2 휕푇2 6 휕푇3 57 where A is the slowly varying amplitude of the pulse envelope, z the propagation coordinate, and T the retarded time. 훽2 is the group velocity dispersion (GVD) and 훽3 the third order dispersion (TOD).훾 is the nonlinearity parameter given by 훾 = 푛2휔0/푐퐴푒푓푓, where 푛2 is the nonlinear index, 휔0 the central angular frequency, 푐 the velocity of light in vacuum, and 퐴푒푓푓 the effective mode area inside fiber. 훼 is the linear fiber loss coefficient which is generally a small number and 푔̂ the active fiber gain coefficient for active fibers. For passive fibers, the gain is simply set to be zero.

The gain spectrum relates to the rare earth dopant introduced into the fiber core and generally covers a range of wavelength. In this work, fiber gain is considered in the frequency domain by multiplication. Specifically, the gain spectrum in this work is assumed to have Gaussian shape centered at the pulse center wavelength, which is expressed in the following equation:

푔 휔 − 휔 2 푔(휔) = 0 푒푥푝 (−2.7726 ( 0) ), 퐸(푧) ∆휔 ( 5-2 ) 1 + 퐸푠푎푡 where 휔 is the angular frequency, 휔0 the center frequency ( assuming the same as the pulse center frequency), ∆휔 the gain full-width-half- maximum (FWHM) bandwidth.

퐸푠푎푡, the saturation energy, is the pulse energy of an incident pulse which leads to a

1 reduction of the gain to ⁄푒 of its small signal gain. Generally, the saturation energy 퐸푠푎푡 depends on the wavelength, mode area, the overlap-inversion factor, absorption and emission cross-section, and so on. Exact treatment of such a value can be found in

[62][63]. Since in the simulation we want to explore the potential output laser characteristics as functions of cavity design, saturation energy, as one of these parameters, can be adjusted numerically and the exact calculation is not necessary for the 58 simulation. Usually we specify the FWHM wavelength bandwidth and the converting into the frequency domain takes the following expression:

2휋푐 ∆휔 = − 2 ∆휆, ( 5-3 ) 휆0 where ∆휆 is the FWHM wavelength bandwidth and 휆0 is the center wavelength. 퐸(푧) is the pulse energy, which is calculated using the following equation:

퐸 = ∫|퐴|2푑푇. ( 5-4 )

It is worth directing the reader’s attention to three other gain spectrum models that are widely used in fiber laser simulations. One is called Lorentzian mode and shown here

[64]:

푔 1 푔(휔) = 0 ∗ . 퐸(푧) 휔 − 휔 2 ( 5-5 ) 1 + 1 + ( 0) 퐸푠푎푡 ∆휔/2

The second model is variance of Lorentzian model which can be found in[65]:

푔 푔(휔) = 0 . 퐸(푧) 휔 − 휔 2 ( 5-6 ) 1 + + ( 0) 퐸푠푎푡 ∆휔/2

The third one is called parabolic model which is the first order of Taylor expansion of equation ( 5-5 ):

푔 휔 − 휔 2 푔(휔) = 0 (1 − ( 0) ), 퐸(푧) ∆휔/2 ( 5-7 ) 1 + 퐸푠푎푡

To propagate the pulse through the fiber, we follow the standard split-step method[24]. The general description is briefed as the following. Sorting the elements on the right side of ( 5-1 ) into two group leads us to the following equations:

59 휕퐴 = (푁̂ + 퐷̂)퐴, ( 5-8 ) 휕푧 where, 푁̂ 푎푛푑 퐷̂ are propagation operators and their expressions are shown as the following:

푁̂퐴 = 𝑖훾|퐴|2퐴, ( 5-9 ) and

푔̂ 훼 훽 휕2퐴 훽 휕3퐴 퐷̂퐴 = 퐴 − 퐴 − 𝑖 2 + 3 . ( 5-10 ) 2 2 2 휕푇2 6 휕푇3

Then operators 푁̂ 푎푛푑 퐷̂ will be carried out for each step of propagation separately. More specifically, the pulse will propagate half step under operator 퐷̂, then full step under operator 푁̂ and then another half step under operator 퐷̂ to finish one step of propagation, which can be symbolically shown as the following:

퐴(푧 + ℎ) ≈ 푒푥푝(ℎ/2퐷̂)푒푥푝(ℎ푁̂)푒푥푝(ℎ/2퐷̂)퐴, ( 5-11 ) where, 푧 is the current position in the fiber, ℎ is predefined step size. The split step method is also graphically shown in Figure 5-1.

To numerically implement the split step method (SSM), operator 퐷̂ is taken into frequency domain so that the time derivative terms can be easily treated and operator 푁̂ is handled in the real time and space domain. Thus operator 퐷̂ can be expanded into the following expression:

ℎ 푒푥푝(ℎ/2퐷̂)퐴 = 𝑖푓푓푡 (푒푥푝 ( ∗ 퐷̂(−𝑖휔)) ∗ 푓푓푡(퐴)). ( 5-12 ) 2

60

Figure 5-1 Illustration of split step method. 5.1.2 Pulse propagation through saturable absorber

A saturable absorber is implemented in a laser cavity to obtain a self-amplitude modulation of the light. Such an absorber introduces losses to pulses within the cavity, which is relatively large for low intensities but significantly smaller for higher intensities.

Thus, a pulse becomes shorter in time after passing through the saturable absorber since the high intensity at the peak of the pulse saturates the absorber more strongly than its low intensity wings. Thus, with the use of such a saturable absorber, pulses will be lifted from noise fluctuations in the cavity and self-starting mode locking can be achieved.

There have been various saturable absorber mechanisms that are used to achieve the desired intensity discrimination. Saturable absorbers based on Optical Kerr effect have been developed and are referred as artificial saturable absorber. These include, but are not limited to, the Kerr lens mode locking[18] and nonlinear polarization rotation[66][67].

Non-artificial saturable absorbers includes semiconductor saturable absorber mirror

(SESAM)[68], carbon based absorbers such as single-walled carbon nanotubes

(SWCNTs)[69], graphene based absorbers[70] and so on.

In this work, we focus on Kerr lens mode locking with both carbon disulfide

(퐶푆2 ) and Chalcogenide glass (ChG) 퐴푠40푆푒60 . The saturable absorber action as a 61 function of power for both 퐶푆2 and 퐴푠40푆푒60 are shown in Figure 4-13. The interpolated value for both 퐶푆2 and 퐴푠40푆푒60 are calculated and plotted on the same figure as in

Figure 5-2. The black lines are the originally simulated value and red dots are the interpolated values.

Figure 5-2 Transmission curve of saturable absorbers for fiber pair of 20-20 with different nonlinear mediums (a) 퐶푆2; (b) 퐴푠40푆푒60. Black lines are the original calculated values and the red dots are the interpolated values. The interpolations are performed using five and four degree polynomial fitting for

퐶푆2 and 퐴푠40푆푒60, respectively. The polynomial expression follows the format below:

5 4 3 2 1 휂 = 푐5푃 + 푐4푃 + 푐3푃 + 푐2푃 + 푐1푃 + 푐0, ( 5-13 ) where the polynomial coefficients are shown in Table 5-1.

Table 5-1 Polynomial coefficients of SA action for both 퐶푆2 and 퐴푠40푆푒60.

Polynomial Coefficients 푪푺ퟐ 푨풔ퟒퟎ푺풆ퟔퟎ 푐5 5.372083958874943e-9 0 푐4 -6.692714040763931e-7 0.000542238845372 푐3 3.017330504997214e-5 0.000083644028993 푐2 -5.246404544450205e-4 0.000970522101039 푐1 0.00441633860163199 0.038705066387964 푐0 0.070195950007893 0.192207923824026

62 For some cavities, simulation will be performed with other types of saturable absorber for comparison and the corresponding SA action will be descripted by a simplified transfer function:

푞 푇(푡) = 1 − 0 , |퐴(푡)|2 1 + ( 5-14 ) 푃푠푎푡 where 푇(푡) is transmission for the corresponding theoretical SA, 푞0 the modulation

2 depth, |퐴(푡)| the instantaneous pulse power and 푃푠푎푡 the saturation power for the SA.

Value for each parameter will be clearly defined in each section where this theoretical model of SA is applied.

5.1.3 Pulse propagation through a spectral filter

As briefly discussed in Chapter 3, spectral filter is an essential element in dissipative soliton lasers and amplified similariton lasers because a proper spectral filter can reshape the pulses inside the cavity and help stabilize the lasers. More specifically, passing through a spectral filter, the side lobes on the pulse spectrum will be removed and the pulse will be brought back to its shape after one round trip so that periodic cavity boundary condition can be met and laser output is stabilized.

There are many different types of spectral filters that are commonly used in fiber laser systems, such as birefringence filters, interference filters, Gaussian filters and so on.

A birefringence filter usually comprises a birefringence plate combined with a polarizer and the underlying working principle is similar to that of a Lyot filter[71]. Going through a birefringent plate, different spectral components in a pulse will experience different phase shift and thus different polarization states. When a polarizer is placed behind the birefringent plate, the pulse going through the polarizer will then have a polarization

63 dependent loss. The interference filter (also called a dichroic filter) usually consists of multilayers of dielectric material with different refractive indices so that it can transmit one or more bands and reflect other. It can be a band-pass, band-rejection, long-pass or low-pass filter. The filter is usually optimized for a normal incident angle and the central wavelength of filter usually decreases if the angle of incidence is increased from zero.

The previous two types of filters may have multi-bands transmission structures which may have negative effect on mode locking process. To eliminate such an effect, one can use a Gaussian filter which has only single peak without periodic or secondary structures[72]. In [72] the author talks about one way to make a Gaussian filter: redirect the incoming collimated beam using a grating and put the collimator on one order of reflection from the grating to receive light, as shown in Figure 5-3. However, this method suffers big loss since only one diffraction order of light is usually used and the energy on other orders are simply wasted.

64

Figure 5-3 Gaussian filter. In this work, we will use a Gaussian filter in our simulation, assuming it follows the expression below:

2 (휔 − 휔0) 푆퐹(휔) = exp (−2.7726 ∗ ), ( 5-15 ) ∆휔2 where 휔0 is the central frequency, 휔 is the angular frequency, ∆휔 the FWHM spectral bandwidth. One can follow equation ( 5-3 ) to convert the wavelength bandwidth to frequency bandwidth.

5.1.4 Pulse propagation through output coupler

The output coupler in this work simply follows the expression below:

퐴표푢푡 = √푅 ∗ 퐴푖푛, ( 5-16 ) where 푅 is the power output coupling ratio.

65 5.2 Whole Cavity Simulation

Before jumping right into the simulation with the new saturable absorber design, it worth a while to test our code with the existing publications.

5.2.1 Andi laser simulation

Andi laser has been extensively studied in recent years and the typical Andi laser cavity is shown in Figure 5-4. The details for each segment is shown in Table 5-2. The active fiber used here is Yb doped gain fiber and the corresponding working wavelength is 1030 nm.

Figure 5-4 The typical Andi laser cavity. PF = passive fiber, AF = active fiber, SA=saturable absorber, OC = output coupler and SF= spectral filter. This cavity consists of six elements in total: two pieces passive fibers (PF), one piece of active fiber (AF), one saturable absorber (SA), one output coupler (OC) and one spectral filter (SF). The details for each element is shown in Table 5-3, where the initial seeding pulse is also included. The initial pulse is assumed to follow a super-Gaussian profile:

1 푡 2푚0 − (1+퐶 )( ) 2 0 푇 ( 5-17 ) 퐴(푡) = √푃0 ∗ 푒 0 , where, 푃0 is initial pulse power, 퐶0 the chirp, 푚0 the beam shape factor and 푇0 the pulse width. Pulse full-width-half-maximum (FWHM) duration to 푇0 is related by the following expression:

퐹0 푇0 = . (2 ∗ √ln(2)) ( 5-18 )

66 Several parameters are defined for the initial seed: full time window (푇푚푎푥), number of data points (푛푡), initial seed peak power (푃0), pulse full width half maximum duration

(퐹0), beam shape factor (푚0) and the chirp parameter (퐶0).

Table 5-2 Andi laser cavity details.

푃퐹1 푙푒푛푔푡ℎ (푚) 3 2 훽2(푓푠 /푚푚) 23 2 훽3(푓푠 /푚푚) 0 훾(1/푊푚) 0.0047 훼(1/푚) 0 퐴퐹 푙푒푛푔푡ℎ (푚) 0.6 2 훽2(푓푠 /푚푚) 23 2 훽3(푓푠 /푚푚) 0 훾(1/푊푚) 0.0047 훼(1/푚) 0 푔0(1/푚) 11.5128 푔푎𝑖푛 푏푎푛푑푤𝑖푑푡ℎ (푛푚) 40 퐸푠푎푡(푛퐽) 1 푃퐹2 푙푒푛푔푡ℎ (푚) 1 2 훽2(푓푠 /푚푚) 23 2 훽3(푓푠 /푚푚) 0 훾(1/푊푚) 0.0047 훼(1/푚) 0 푆퐴 푞0 0.7 푃푠푎푡(푊) 1000 푂퐶 푅 0.7 푆퐹 푑휆(푛푚) 10 퐼푛𝑖푡𝑖푎푙 푃푢푙푠푒 푇푚푎푥(푠) 6.1497퐸 − 11 14 푛푡 2 푃0(푊) 200 퐹0(푠) 100퐸 − 15 푚0 1 퐶0 0

As a function of round trip propagation, the pulse energy after spectral filter is plotted in

Figure 5-5. As shown, the pulse quickly converges to a steady state. The pulse evolution inside the cavity is shown in Figure 5-6, with pulses at some particular positions

67 illustrated in Figure 5-7. Specifically in Figure 5-7 column (a) represents the pulse before the first section of passive fiber (PF1), (b) before the gain fiber (AF), (c) before the second section of passive fiber (PF2), (d) before the saturable absorber (SA) and (e) before the spectral filer (SF). For each column the top row is the pulse intensity profile, middle row is the spectrum of the pulse and the bottom row is the corresponding chirp.

As a comparison, pulse evolution from a typical Andi laser cavity is shown in Figure 5-8

[47]. Due to the fact that the exact simulation parameters used in this dissertation are different from that in the paper, the exact the shape of the spectrum at each position is slightly different; however, the simulation reflects the characteristic pulse evolution. As shown from the spectrum, we see the characteristic ‘batman’ shape of spectrum at the out-put coupler, which is consistent with the results from [4][46][47]. Thus our simulation code is validated through the comparison. The details of Andi laser cavity simulation can be found in APPENDIX C.

Figure 5-5 Pulse energy after spectral filter as a function of round trip propagation.

68

Figure 5-6 Pulse evolution inside Andi laser cavity.

Figure 5-7 Pulse evolution (top row), corresponding spectrum (middle row)) and chirp (bottom row) inside the cavity at positions of (a) before PF 1; (b) before AF; (c) before PF 2; (d) before SA and (e) before SF.

69

Figure 5-8 Typical Andi laser pulse evolution. Taken from [47].

5.2.2 Simulation with 3 mm of 푨풔ퟒퟎ푺풆ퟔퟎ

In this section, we simulate a conceptual cavity which is the same as shown inFigure 5-4. The saturable absorber here is replace by our SA design shown in Figure

4-11.

70 Table 5-3 Simulation details for conceptual cavity1.

푃퐹1 푙푒푛푔푡ℎ (푚) 3 2 훽2(푓푠 /푚푚) 50 2 훽3(푓푠 /푚푚) 0 훾(1/푊푚) 0.0035 훼(1/푚) 0 퐴퐹 푙푒푛푔푡ℎ (푚) 0.6 2 훽2(푓푠 /푚푚) 16.5 2 훽3(푓푠 /푚푚) 0 훾(1/푊푚) 0.0035 훼(1/푚) 0 푔0(1/푚) 11.5128 푔푎𝑖푛 푏푎푛푑푤𝑖푑푡ℎ (푛푚) 40 퐸푠푎푡(푛퐽) 0.00396 푃퐹2 푙푒푛푔푡ℎ (푚) 3 2 훽2(푓푠 /푚푚) 50 2 훽3(푓푠 /푚푚) 0 훾(1/푊푚) 0.0035 훼(1/푚) 0 푂퐶 푅 0.5 푆퐹 푑휆(푛푚) 10 퐼푛𝑖푡𝑖푎푙 푃푢푙푠푒 푇푚푎푥(푠) 6.1497퐸 − 11 14 푛푡 2 푃0(푊) 100 퐹0(푠) 10퐸 − 15 푚0 1 퐶0 0

The total dispersion of cavity 1 is ~ 0.3099 푝푠2 and the repetition rate is

~30 푀퐻푧. The pulse energy inside the cavity after the spectral filter is potted along the round trip propagation in Figure 5-9. As shown, the pulse converges to a steady state after around 60 round trips of propagation and the pulse energy of the output laser is around 2 pJ. In the stead state, the pulse evolution inside the cavity is plotted in Figure

5-10. Also the pulses after each element in cavity 1 are potted in Figure 5-11, with the corresponding spectrum and chirp shown together on the same graph.

71

Figure 5-9 The pulse energy after spectral filter along with round trip propagation for conceptual cavity 1.

Figure 5-10 Pulse evolution inside the cavity 1 in stead state.

72

Figure 5-11 The steady pulse after each element inside cavity 1 Top row is the pulse shape, center row the spectrum and the bottom row the chirp.

The pulse energy generated from conceptual cavity 1 is very low (~2푝퐽) and the

SA doesn’t have a noticeable intensity discrimination effect on the pulse. Further investigation should be conducted to optimize the output laser energy.

73 ` CHAPTER 6

EXPERIMENTS

To experimentally test the mode locking with ChG glass plate, a series of experiments are arranged. As suggested by the simulation results, the instantaneous power level required by the original mode locker design is too high so that strong saturable absorber action couldn’t be obtained. Thus original saturable absorber design with ChG 퐴푠40푆푒60 is experimentally realized with new methods and the details are described later in the corresponding sections.

6.1 Build a Mode Locked Er Fiber Laser.

In this step we build a mode locked Er doped fiber laser. The laser cavity is shown in Figure 6-1.

74

Figure 6-1 Er fiber laser cavity. In the laser cavity the gain fiber is Liekki Er110-4/125, denoted as a bold black line. Collimators are ordered from Oz-Optics company, designed for 1550 nm. The pigtails of collimators are Corning HI1060 fibers, which are shown as blue lines. WDM stands for wavelength division multiplexer, with OFS980 fiber as the lead fibers. The red line is DCF38 fiber obtained from Thorlabs, used as a dispersion control section. An isolator is placed inside the free space to ensure that the pulses are only rotating anticlockwise. Going through the fibers, pulses experience nonlinear polarization rotation effect (NPR), due to the nonlinear birefringence of the fiber. Coming out from the top right collimator, the polarization state of the pulses will be adjusted by the quarter wave plate (QWP) and half wave plate (HWP) so that higher power part of the pulse will experience lower output coupling rate than that of the low power part of pulse when it goes through the polarized beam splitter (PBS). Thus combined with NPR, the partial cavity from the top right collimator to PBS serves as the saturable absorber. After passing

75 through the birefringent plate (bire-plate), the polarization state of the pulse will be further adjusted so that different wavelengths have different polarization states and thus pass the first polarizer in the isolator with different transmission rate. Thus PBS, bire- plate and polarizer in the isolator, as a whole, serves a spectral filter. A QWP placed behind the isolator is to scramble the polarization state entering into the fiber and ensure that the nonlinear polarization rotation effect through the fiber section is induced and suitable for mode locking. Details for each type of fiber are shown in Table 6-1. A fiber coupled diode at 976 nm is used here as the pump, which is obtained from Gooch &

Housego with module serial number of D1405085. The pump diode is spliced to the

WDM with an isolator in between them (not shown) to prevent back flowing of the laser light into the pump diode. Before completing the laser cavity, the pump current is tuned to 900 mA and a total power of 408 mW @ 976 nm is tested with a power meter placed after the WDM. Then after constructing the laser cavity, highest continuous wave (CW) laser output is about 55 mW with pump current of 900 mA (408 mW into the cavity), which corresponds to 13.4% efficiency from pump to output laser. The fiber sections add up to 5.83 푚 and the free space takes roughly 40 to 45 푐푚. Assuming the refractive index of the fibers at 1550푛푚 is about 1.447, the roundtrip time is estimated to be 29.45 ns to 29.62 ns. From the second order dispersion information shown in Table 6-1, total cavity dispersion is on the order of ~0.04 푝푠2.

76

Table 6-1 Fiber information.

Device HI1060 OFS980 Er110-4/125 DCF38 Length (m) 2.45 1.48 0.9 1 Cute-off 920 960 800-980 <1520 wavelength (휆 𝑖푛 푛푚) Numerical 0.14 0.16 0.2 0.14 Aperture (NA) Mode Field 9.5 7.5 6.5 6.01 Diameter (MFD um) GVD Parameter -0.00939 0.0045 0.0135 0.048433502 (β2) (ps2/푚)

By adjusting the orientation of the wave plates and bire-plate, different modes of pulses are obtained. The typical three modes that can be routinely generated are discussed here. The most readily achieved pulses is shown in Figure 6-2, which is called mode 1.

The corresponding FWHM spectral bandwidth is about 30 nm as shown in Figure 6-2(a).

The autocorrelation result from the dechirped pulse is shown in Figure 6-2(b), from which temporal FWHM pulse duration is estimated to be about 210 푓푠. As a comparison, the transform limited pulse, reconstructed from the spectrum in Figure 6-2(a), is plotted in Figure 6-2(c), from which the FWHM pulse duration is about 160 푓푠. There is a difference between the values from the autocorrelation and transform limited pulse, which originates from imperfection of the dechirped pulse. The secondly readily available pulses is shown in Figure 6-4 and called mode 2. Mode 2 has roughly 40 nm

FWHM spectral bandwidth with fine spectral side band features outside the bandwidth.

The corresponding pulse duration from autocorrelation result and transform limited pulse is about 167 푓푠 and 140 푓푠, respectively. Thus compared to mode 1, pulses of mode 2 have generally smaller pulse duration. The average pulse power for both mode 1 and 2 77 are in the range of 42.8 mW to 46.8 mW with 900 mA pump current and the pulse energies for both mode 1 and 2 are on the order of 1.4 nJ.

Figure 6-2 Mode 1.

Figure 6-3 Mode 2. The most interesting mode obtained is mode 3, shown in Figure 6-4. Compared to mode 1 and 2, mode 3 is harder to obtain. As shown in Figure 6-4 (a), the spectrum is not a nice Gaussian shaped function and thus the spectral width shouldn’t be characterized by

FWHM. As seen, the spectrum of mode 3 covers a huge range, from 1490 nm to 1640 nm. The autocorrelation of mode 3 is shown in Figure 6-4 (b), which indicates a pulse duration of 123 푓푠. However, as can be seen from the shape of the autocorrelation result,

78 the dechirped pulse is not close to transformed limited pulse in any sense. We speculate that the pulse of mode 3 directly out from the cavity is close to transform limited pulse and gets over-compensated going through the dechirping stage. For comparison, the transform limited pulse for mode 3 is reconstructed and plotted in Figure 6-4 (c), from which we predict a FWHM pulse duration of 40 푓푠, which could be the shortest pulse ever reported from a 퐸푟 doped fiber laser so far[73] and definitely deserves further investigation. The pulse trains of all the three modes described here are similar and the pulse train for mode 3 shown in the Figure 6-5. The pulse period from Figure 6-5 reads as

29.4 ns (corresponding to a repetition rate of ~33.7 푀퐻푧), which is consistent with the previous estimation. The experimental setup is shown in Figure 6-6. Note that autocorrelation results for all the three modes are normalized to a value of 8. However, since the pulses used for autocorrelation are not perfectly transform limited, the background line doesn’t perfectly line up with the value of 1.

Figure 6-4 Mode 3.

79

Figure 6-5 Pulse train for all the modes found.

Figure 6-6 The Er fiber laser cavity corresponds to the design in Figure 6-1.

80

6.2 Mode Locking with ChG Glass Plate.

Based on the mode locked laser described in last section, we add in the ChG glass plate. The 퐴푠40푆푒60 glass plate is made by Irradiance Glass company and has a thickness of 5 mm (the thinnest available), shown in Figure 6-7. As mentioned at the beginning of this chapter, the original design of the mode locker, shown in Figure 4-2 with the SA function shown in Figure 5-2 (b), couldn’t serve as a strong saturable absorber. That’s because the pulses generated with ChG glass plate have weak peak powers so that high power and low power part of the pulse basically experience the same transmission, which means the power discrimination induced by the ChG plate is weak. Furthermore, the thickness of 5 mm is too thick for the original design to function properly. Thus in experiment, instead of putting the 퐴푠40푆푒60 glass plate in between two tapered fiber ends,

we put it in the free space in the laser cavity shown in Figure 6-1. The experimental setup is shown in Figure 6-8, with the corresponding experimental setup shown in Figure 6-9.

However, this cavity can only generates Q-switched pulses and no mode locking sign is found with ChG glass plate in place, no matter how the wave plates and birefringent plate are adjusted.

81

Figure 6-7 5 mm thick 퐴푠40푆푒60 푝푙푎푡푒.

82

Figure 6-8 Mock locking with 퐴푠40푆푒60 glass plate.

Figure 6-9 Experimental setup of Figure 6-8.

83

6.3 Mode Locking with a Telescope Design

Since mode locking with the bare 퐴푠40푆푒60 glass plate can’t be achieved, we come up with another design – telescope design. We use two lenses to make a telescope and the lenses have similar focal lengths (roughly 3 cm). Then we insert the

퐴푠40푆퐸60 glass plate inside the telescope and replace the 퐴푠40푆퐸60 glass plate with the telescope design, which is shown in Figure 6-10. As shown, pulses from collimator on the top right corner will be focused by lens 1. The 퐴푠40푆푒60 glass plate is placed at the focus of the beam so that the pulse will experience higher level of nonlinearity compared to shinning laser through the glass plate directly. Lens 2 is placed after the 퐴푠40푆푒60 glass plate to bring the beam back. The telescope is designed so that the higher intensity part of the pulse experiences higher nonlinearity than that of the lower power part. Lens 2 is placed at a position after the 퐴푠40푆푒60 glass plate so that eventually the higher intensity part of pulse couples back to the other collimator better than the low power part of the pulse. Since the saturable absorber action induced by 퐴푠40푆푒60 is small, we keep the original SA (NPR+collimator+HWP+QWP+PBS) in the laser cavity and test the effect of the additional SA action (the telescope design). The experiment corresponding to cavity in Figure 6-10 is shown in Figure 6-11. Mode locking is obtained with the telescope design, as shown in Figure 6-11.

84

Figure 6-10 Mode locking with 5 mm 퐴푠40푆푒60 in a telescope design.

Figure 6-11 Mode locking with telescope design. Comparing the results from cavities in Figure 6-8 and Figure 6-10 we think the

퐴푠40푆푒60 helps mode locking to some degree. However, due to the existence of nonlinear 85 polarization effect, it is still possible that mode locking is fully due to the NPR, not the

퐴푠40푆푒60 glass plate.

6.4 Mode Locking with Pin Hole in the Telescope

To eliminate the NPR effect, we take HWP and QWP out of the laser cavity of

Figure 6-10. However, there is no mode locking sign since contribution from the

퐴푠40푆푒60 glass plate is too small. We then modify the laser cavity of Figure 6-10 by putting an iris inside the telescope after the glass plate, as shown in Figure 6-12. With the iris fully open, we adjust the wave plates and birefringent plate until mode locking is obtained and then slightly adjust the half wave plate so that the mode locking is gone.

Then we close the iris to its smallest size. Even though the mode locking still can’t be obtained, we find strong mode locking sign since we see many spikes dancing all over the spectrum by checking the spectrum analyzer. We further adjust the pump to its highest level but still can’t get the cavity mode locked.

Figure 6-12 Cavity with a pinhole in the telescope. 86

Figure 6-13 Experiment realization of Figure 6-12.

6.5 Q-switching with the 푨풔ퟒퟎ푺풆ퟔퟎ Glass Plate

Since we couldn’t get the 퐴푠40푆푒60 glass plate to mode lock the laser the way we would like it to, we need to find decisive proofs to demonstrate that the nonlinear glass plate helps shaping the laser output. One proof happens with the Q-switched laser. We build a laser cavity as shown in Figure 6-14. The free space section in the cavity is adjusted be 21 cm long and pump current is tuned to 715 mA so that only continuous wave is generated out of the cavity. The CW spectrum corresponding to cavity in Figure

6-14 is shown in Figure 6-15.

87

Figure 6-14 Laser cavity with only PBS inside the free space.

Figure 6-15 (a) Spectrum and (b) CW output for cavity in Figure 6-14. Then the glass plate is put inside the free space, as shown in Figure 6-16. By adjusting the relative glass orientation, we generate Q-switched pulsed laser. The corresponding spectrum and Q-switch pulse is shown in Figure 6-17.

Comparing laser cavities shown in Figure 6-14 and Figure 6-16, the only difference in Figure 6-16 is the extra 퐴푠40푆푒60 glass plate. Since only with 퐴푠40푆푒60 glass plate we can obtain Q-switch laser output, it is safe to make a conclusion that the

88

퐴푠40푆푒60 glass plate is the only driver for Q-switching in this case. The reason 퐴푠40푆푒60 glass plate causes Q-switching can be thought of in following way. The loss causes the energy in the laser cavity to build up. Once the gain surpasses the loss, lasing action occurs. When the beam goes through the 퐴푠40푆푒60 glass plate, higher instantaneous power experiences higher level of nonlinearity and thus lower level of net diffraction effect. In other words, the 퐴푠40푆푒60 glass plate helps collimate the pulses going through it but it has better collimating effect for the higher power part than low power part. Then the pulse experiences better coupling efficiency for the high power part when entering into the receiving collimator, which means the whole free space effectively serves as passive saturable absorber.

Figure 6-16 Q-switch with the 퐴푠40푆푒60 glass plate. The cavity is the same the one shown in Figure 6-14.

89

Figure 6-17 (a) Spectrum and (b) Q-switched pulses for cavity in Figure 6-16.

90

` CHAPTER 7

SUMMARY

In this dissertation, we have studied a new saturable absorber design, which is based on the Kerr lens principle. The proposed SA is shown in Figure 4-2, which is replaced here:

Figure 7-1 New saturable absorber design. There are four key elements that affect the transmission between fiber 1 and fiber 2: nonlinear medium, end1, end 2 and the separation 퐷 푚푚 in between them. We conduct extensive numerical studies to simulate the transmission rate from

Fiber 1 to Fiber 2 which is then used as the SA action in the whole lase cavity simulation.

To increase the chance of success, we select two materials as the Kerr medium: 퐶푆2 and

퐴푠40푆푒60.

Chalcogenide glass (ChG) group has been proven to have very high nonlinearity among the solids. From the [58] 퐴푠40푆푒60 is reported to have a refractive index of 푛0 =

−17 2 2.81@ 휆 = 1.55 휇푚 and a nonlinear index of 푛2 = 2.3 × 10 푚 /푊 , which is the highest nonlinearity among the glasses so far. At the same time, carbon disulfide (퐶푆2) 91

−19 2 has the highest nonlinearity among the liquids (푛2 = 32 × 10 푚 /푊). Plugging the selected nonlinear medium into the SA design, we have the corresponding SA transmission curve shown below:

Figure 7-2 Transmission curve of new saturable absorber design for fiber pair of 20-20 with different nonlinear mediums (a) 퐶푆2; (b) 퐴푠40푆푒60. The SA curve is then plugged into the laser cavity with details of each element shown in Table 5-3. The simulation generates an output laser with a pulse energy of 5 pJ, pulse width of ~ 5 ps and almost a linear up chirp. The pulse energy can be scaled up to a few hundreds of pJ by increasing the value of saturation energy 퐸푠푎푡 in gain fiber.

However, beyond a certain saturation energy (on the order of ~0.1 푛퐽), pulses start to break inside the cavity and no stable pulse has been generated from the simulation.

Further investigation into the cavity design is needed in future to optimize the laser output.

One reason for the low pulse energy in simulation is due to the weak saturable absorber action of the new SA design. Fitting the SA action of Figure 7-2 (b) using the theoretical SA model in equation ( 5-14 ), the SA action of the new design has

92 modulation depth and saturation power of 80% and 20kW, respectively. The modulation depth of 80% is not very low. However, as mentioned in section 4.2.3, the SA transmission calculated for both 퐶푆2 and 퐴푠40푆푒60 corresponds to power or intensity, not amplitude. Thus for pulse propagation through our saturable absorber design (see section

5.1.2) we need to use the square root of the transmission for the pulse amplitude, which brings the modulation depth down significantly.

Experiments are conducted to test the SA action of the new mode locker design in a laser cavity shown in Figure 6-1. Since the minimum thickness of 퐴푠40푆푒0 that we can order is 5 mm, which is too thick to fit in our SA design, we decide to test the SA action of the ChG glass plate differently. Specifically, we mode-lock the laser cavity shown in

Figure 6-1. Then we put the 퐴푠40푆푒60 plate into the cavity as shown in Figure 6-8. Since the glass plate introduces extra loss into the cavity and doesn’t have much self-focusing effect due to the low power of the pulse inside the cavity, we can’t mode lock the cavity, no matter how we adjust wave plates. As a comparison, we put the glass plate inside a telescope, which is then inserted into the cavity, as shown in Figure 6-10. Mode locking is successfully obtained as shown Figure 6-11. Even though the original saturable absorber due to the nonlinear polarization rotation is still inside the cavity, it is believed that the 퐴푠40푆푒60 glass plate is helping to a degree in mode locking. To demonstrate this point, we conduct another experiment as shown in Figure 6-12. In this experiment, we put an iris inside the telescope behind the ChG plate and adjust the cavity into a non- mode-locked state. By closing the iris to its smallest size, suddenly spikes appeared jumping all over the place on the optical spectrum analyzer, which is a strong indication

93 that the laser is mode locking. Since the SA action due to the ChG plate is not very strong itself and closing iris introduces additional loss, which further increases the threshold for mode locking; as a consequence, we were not successful in mode locking the laser.

However, this experiment clearly demonstrates that the telescope design is helping the mode locking process.

Although we didn’t observe mode locking in a laser cavity by using only the ChG glass plate, we have successfully obtained a Q-switched pulsed laser with ChG glass plate the only saturable absorber possible inside the cavity. The Q-switched laser cavity is shown in Figure 6-16.

To better utilize the Kerr lens mode locking technique with 퐴푠40푆푒60 plate, the author has a few suggestions for future work. First numerical simulation should be conducted for the SA design shown in Figure 7-1 with asymmetric fibers. In this work, the fibers are assumed to be the same size. However, an asymmetric design may result in a better saturable absorber action. Second lenses with shorter focal length are suggested for the telescope design, which would further focus the pulse, increase the local intensity and correspondingly enhance the self-focusing effect due to the nonlinearity. Thus the chance of mode locking the cavity with only SA action from ChG glass would be higher.

Last but not least important, more whole cavity simulations should be conducted to optimize the cavity design so that the output pulse energy can be maximized while keeping the pulse duration relatively short. We have been very frustrated when doing the whole cavity simulations. In the simulation different values for a specific parameter could result in a totally different output. Thus each parameter should be clearly clarified before

94 simulation is carried out. For example, different pulses can be generated or may not be generated at all if an initial pulse seed is different. By saying the initial pulse is characterized by Gaussian noise, the reader has no idea how the exact initial seed is constructed and thus simulations for the same cavity could be very different. It seems to us that many key aspects of the simulations in some papers are not clearly clarified, which makes it hard for the readers to reproduce the results inside. Specifically, in Fig.4 of [64] the author claimed that a smooth pulse was generated for a 32 meters laser cavity with a white noise as the initial distribution. However, the ‘white noise’ defined by the author was not white at all and was not clearly defined in the paper, which leads in an unreproducible result. To make things clear and well defined, we include our codes for each calculation in the corresponding appendix and ensure that the readers would understand how exactly simulations are performed using the cods.

One future project that would extend the findings of this dissertation is the development of a double-clad, co-doped Erbium:Ytterbium gain fiber for a ultrashort pulse fiber laser with high-peak power using the Kerr lens focusing effect. The laser would be developed as a pump wave for a nonlinear crystal to generate wide bandwidth

Terahertz (THz) radiation. Alternatively we plan to use longer-pulses (ns pulse width),wavelength tunable mode-locked laser to generate narrowband THz waves using the nonlinear optical parametric conversion effect called three wave mixing. Due to the wavelength tuning of the output spectral characteristics of the fiber lasers, this new design will be capable of generating narrow band tunable over ten’s of THz.

95

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104

`

APPENDIX A

Mode Simulation

The simulation of the design shown in Figure 4-2 can be divided into two parts: (a) simulate the modes inside the fiber and (b) calculate the transmission between fiber 1 and fiber 2. The mode inside a fiber is calculated using Finite Difference Method (FDM)[52] and is shown in this section and the propagation through the free space is explained in

Appendix B. The main routine to calculate the mode simulation shown as below. It uses two pre-defined functions: “fiber()” and “SVMODES()”. “fiber()” is defined to discretize the fiber cross section and “SVMODES()” is a mode solve algorithm based on semi- vectorial method. Both sub-functions are shown behind the main routine.

Here is the main routine is:

% Main routine to solve the mode in a fiber close all;clear;clc;tic; %fiber information lambda=1.55; % wavelength in microns n_core=1.46; % refrative index of fiber core NA=0.12; % Numerical Aperture n_clad=sqrt(n_core^2-NA^2); % refrative index of fiber cladding n_air = 1; % define the material outside the fiber. Here we assume the material outside the boundary is air. r_clad=25; % cladding radius from the center to the cladding/air boundary (um) r_core=2.952; % core radius from the center to the core/cladding interface (um)

105 r_air=50; % radius from the center of fiber to the outer most boundary (um)

N1=40; % number of points between -r_core and r_core N2=200; % number of points between r_core and r_clad N3=40; % Number of points between r_clad and r_air

%Analysis Window Discretization % If you want to define your own waveguide, make sure all the dimensions % are in microns (um) and also make sure that the waveguide is surrounded % by an outer material. For example, here we want to calculated the fiber mode. % The fiber index profile defined here has to include one layer of air or % water or something else. vx=[-r_air N3 -r_clad N2 -r_core N1 r_core N2 r_clad N3 r_air]; vy=vx; index= [n_core,n_clad,n_air]; radius=[r_core,r_clad,r_air]; [X,Y,Index]=fiber(vx,vy,index,radius); % fiber is a self-defined function to generate the index and dimension matrice Note the X and Y matrices have units of microns.

% plot index profile figure h=surf(X,Y,Index); set(h,'edgecolor','none') set(gcf,'renderer','painters'); view(0,90) colorbar; axis equal;

%Initial guess of wavefunction distribution r=sqrt(X.^2+Y.^2); V=2*pi*r_core*NA/lambda; w0=5; E0=exp(-r.^2/w0^2); % initial guess of the field distribution inside the fiber.

% plot initial guess of the field distribution inside the fiber figure h=surf(X,Y,abs(E0).^2); set(h,'edgecolor','none') set(gcf,'renderer','painters'); view(0,0)

% Parameters definitions for mode calculation % BCs -- Boundary conditions (1 Dirichlet 2 Newmann 3 Analytical 4 Transparent) % field -- define the polarization of the mode ('ex', 'ey' or 'scalar') % tolerance -- define how accurate the calculated effective index will be. % This number is calculated by tolerance = % (max(|En+1|)-max(|En|))/max(|En|) % n_out: define the refractive index outside the boundary. Since we assume % the fiber is all in air, thus n_out is set to be 1. n_eff=(n_clad+n_air)/2; % Initial guess of effective index E=E0; % Initial guess of the filed distribution inside the fiber BCs=4; % Boundray condition on the cladding/air bourndary (1 Dirichlet 2 Newmann 3 Anaytical 4 Transparent) field='ex'; % The polarization of the light. 'ex', 'ey' or 'scalar' tolerance=1e-4; % How accuate you want the effective index to be? The smaller the tolerance, the more accurate the answer would be. 106 n_out=1; % the index outside the boundary % calculate the mode and the corresponding effective index in the % waveguide. [n_eff,E]=SVMODES(lambda,n_eff,X,Y,Index,E,BCs,field,tolerance);% SVMODES()is the self-defined function based on semi-vectorial method. surf(X,Y,abs(E).^2); view(0,0); shading interp;

Function “fiber()” is: function [X,Y,Index]=fiber(vx,vy,index,radius) % This function generates the mesh, index, and discretization of the cross section of 3D waveguide % % The inputs: % % vx, vy: input vector for discretizing the analysis window. % Taking the fiber as an example, the input vector vx can be used as:

% [-r_air N1 -r_clad N2 -r_core N3 r_core N4 r_clad N5 r_air].

% N1 is the number of points between -r_air and -r_clad. By assigning different % points for different section, we generate non-uniform mesh. % All radii are in micons (um). % index : [n_core n_clad n_air] % radius: [r_core r_clad r_air] % % The outputs: % % X,Y : mesh using function [X,Y]=meshgrid(x,y). % dx, dy : discretization width. Both are matrices. lx=(length(vx)-1)/2; ly=(length(vy)-1)/2; % x vector x=[]; for m=1:lx; x1=vx(2*m-1); x2=vx(2*m+1); num=vx(2*m); v=linspace(x1,x2,num); if m>1 v=v(2:end); end x=[x,v]; end

% y vector y=[]; for m=1:ly; x1=vy(2*m-1); x2=vy(2*m+1); num=vy(2*m); v=linspace(x1,x2,num); if m>1 v=v(2:end); end y=[y,v]; 107 end

% mesh gird [X,Y]=meshgrid(x,y);

% % dx Matrix % lx=length(x); % ly=length(y); % dx=X(:,2:lx)-X(:,1:(lx-1)); % d=dx(:,lx-1); % dx=[dx,d]; % % dy Matrix % dy=Y(2:ly,:)-Y(1:(ly-1),:); % d=dy(ly-1,:); % dy=[dy;d];

% Index profile n_core=index(1); r_core=radius(1); n_clad=index(2); r_clad=radius(2); n_air=index(3); % r_air=radius(3); r=sqrt(X.^2+Y.^2); num1=size(r,1); num2=size(r,2); Index=zeros(num1,num2); for ii=1:num1 for jj=1:num2 if r(ii,jj)<=r_core Index(ii,jj)=n_core;

elseif r(ii,jj)<=r_clad Index(ii,jj)=n_clad;

else Index(ii,jj)=n_air; end end end

Function “SVMODES()” is: function [n_eff,E]=SVMODES(lambda,n_eff,X,Y,Index,E,BCs,field,tolerance) % This is the mode-solver core routine based on semivectorial method. % For more details, please refer % to the handout of 'Integrated Optics' class by Andrew Sarangan and the book % of 'Introduction to Optical Waveguide Analysis: Solving Maxwell's Equations and the Schrodinger Equation'. % % The coordinates system is shown in the figure below. The z-direction is % pointing into the screen. % 1 2 3 4 5 6 7 % ------> (x) m % 1 | % 2 | % 3 | 108

% 4 | % 5 | % 6 | % \|/ (y) % Indexing is from left to right and from top to bottom. For example, for % (n,m) position in the matrix is (n-1)*M+m % The meshed grid boundary are defined in the following figure: % top % ------> x % | | % left | | right % | | % | | % | | % \|/ y | % ------% bottom % There are five points in each grid equation as shown in the following: % O Ayn(n,m)*E2(n-1,m) (Ayn=An) % | % | % | % | % O------O ------O Axp(n,m)*E2(n,m+1) (Axp=Ae) % Axn(n,m)*E2(n,m-1) A00(n,m)*E2(n,m) % (Axn=Aw) | % | % | % | % O Ayp(n,m)*E2(n+1,m) (Ayp=As) % % % Inputs: % lambda: wavelength % n_eff: initial guess of the effective index. % dx,dy: discretization width along x and y respectively. Both of dx and dy are matrices. % Index: refractive index profile of the analysis window. % E: initial guess of the wave distribution % BCs: The Boundary Condition number from 1 to 4 % 1 Dirichlet Boundary; The filed outside analysis window is set to be % zero. % 2 Neumann Boundary; The field outside the boundary is set to be the % same as the field on the boundary % 3 Analytical boundary: the analytical wave function outside the analysis window % to be connected with a wave function at the % edge of the analysis window is assummed to % decay exponentially wit hthe decay constant % -k0*sqrt(abs(n_eff^2)-Index(p,q)^2): % exp(-k0*sqrt(abs(n_eff^2)-Index(p,q)^2)*delta) % where k0 is the wave number in a vaccum, n_eff is the % effective index, delta is the discretization % width at the edge of the analysis window, % and the Index(p,q) is the index on the edge. % 4 transparent boundary: On the left edge of the meshed grid where m=1, then E2(n,m-1)=E2(n,0) is % following outside the grid. However, based on the transparent condition, 109

% the value of E2(n,0) can be calculated by using: % E2(n,1)/E2(n,0)=E1(n,2)/E1(n,1)=TL(n) so that we have :E2(n,0)= % E2(n,1)/TL(n).Thus Axn(n,m)*E2(n,0)=Axn(n,m)*E2(n,1)/TL(n). Thus the % original five points grid becomes four points because E2(n,0) is % converted into E2(n,1) by multiplying Axn(n,m)/TL(n). % On the right edge of the meshed grid where m=M, then E2(n,m+1)=E2(n,M+1) is % following outside the grid. However, based on the transparent condition, % the value of E2(n,M) can be calculated by using: % E2(n,M+1)/E2(n,M)=E1(n,M)/E1(n,M-1)=TR(n) so that we have :E2(n,M+1)= % E2(n,M)*TR(n).Thus Axp(n,m)*E2(n,M+1)=Axp(n,m)*E2(n,M)*TR(n). Thus the % original five points grid becomes four points because E2(n,M+1) is % converted into E2(n,M) by multiplying Axp(n,m)*TR(n). % On the top edge of the meshed grid where n=1, then E2(n-1,m)=E2(0,m) is % following outside the grid. However, based on the transparent condition, % the value of E2(0,m) can be calculated by using: % E2(1,m)/E2(0,m)=E1(2,m)/E1(1,m)=TT(m) so that we have :E2(0,m)= % E2(1,m)/TT(m).Thus Ayn(n,m)*E2(0,m)=Ayn(n,m)*E2(1,m)/TT(m). Thus the % original five points grid becomes four points because E2(0,m) is % converted into E2(1,m) by multiplying Ayn(n,m)/TT(m). % On the bottom edge of the meshed grid where n=N, then E2(n+1,m)=E2(N+1,m) is % following outside the grid. However, based on the transparent condition, % the value of E2(N=1,m) can be calculated by using: % E2(N+1,m)/E2(N,m)=E1(N,m)/E1(N-1,m)=TB(m) so that we have :E2(N+1,m)= % E2(N,m)*TB(m).Thus Ayp(n,m)*E2(N+1,m)=Ayp(n,m)*E2(N,m)*TB(m). Thus the % original five points grid becomes four points because E2(N+1,m) is % converted into E2(N,m) by multiplying Ayp(n,m)*TB(m). % field: must be 'EX', 'EY', or 'scalar' % tolerance: sets up a gauge to measure how accurate the result is. % n_out: the index of refraction outside the boundary. There are four % values in n_out: [north south west east]. % Outputs: % % n_eff : effective refractive index % E : The mode distribution % % The penta matrix is solved by using Shifted Inverse Power Method.

110 k0=2*pi/lambda; beta=n_eff*k0; N=size(E,1); % number of rows(elements along y direction) M=size(E,2); % number of colums(elements along x direction) % n_air = n_out; % assuming the material surrounding the waveguide is all the same. n_air=Index(1,1); % assuming the material surrounding the waveguide is air. % n_airn=n_out(1); % n_airs=n_out(2); % n_airw=n_out(3); % n_aire=n_out(4);

% discretization dx=X(:,2:end)-X(:,1:(M-1)); dx=[dx,dx(:,(M-1))]; dy=Y(2:end,:)-Y(1:(N-1),:); dy=[dy;dy((N-1),:)]; e=dx; w=[dx(:,1),dx(:,1:(M-1))]; s=dy; n=[dy(1,:);dy(1:(N-1),:)]; switch lower(field) case 'ey' %calculating matrices Aw, Ae, An, As and A00 Index_n=[n_air*ones(1,M);Index(1:(N-1),:)]; Index_p=[Index(2:N,:);n_air*ones(1,M)]; % Index_n=[n_airn*ones(1,M);Index(1:(N-1),:)]; % Index_p=[Index(2:N,:);n_airs*ones(1,M)]; Ind_n=(Index.^2+Index_n.^2)/2; Ind_p=(Index.^2+Index_p.^2)/2;

Aw=2./w./(e+w); Ae=2./e./(e+w); An=(2./n./(n+s)).*(Index_n.^2./Ind_n); As=(2./s./(n+s)).*(Index_p.^2./Ind_p); Ax=-Ae-Aw; Ay=An+As-4./n./s; A00=Index.^2*(k0^2)+Ax+Ay;

case 'ex' Index_n=[n_air*ones(N,1),Index(:,1:(M-1))]; Index_p=[Index(:,2:end),n_air*ones(N,1)]; % Index_n=[n_airw*ones(N,1),Index(:,1:(M-1))]; % Index_p=[Index(:,2:end),n_aire*ones(N,1)]; Ind_n=(Index.^2+Index_n.^2)/2; Ind_p=(Index.^2+Index_p.^2)/2;

Aw=(2./w./(e+w)).*(Index_n.^2./Ind_n); Ae=(2./e./(e+w)).*(Index_p.^2./Ind_p); An=2./n./(n+s); As=2./s./(n+s); Ax=Ae+Aw-4./e./w; Ay=-An-As; A00=Index.^2*(k0^2)+Ax+Ay;

case 'scalar' Aw=2./w./(e+w); Ae=2./e./(e+w); 111

An=(2./n./(n+s)); As=(2./s./(n+s)); Ax=-Ae-Aw; Ay=-An-As; A00=Index.^2*(k0^2)+Ax+Ay; end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%% Taking care of the boundary conditions. %%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% The electrical field on the plane l is denoted as E1(n,m) and the on plane l+1 E2(n,m) delta=1; ii=0; c(1)=0; while delta>tolerance ii=ii+1; if ii>500 error('The iteration is over 500!!') end

if BCs==1 % Dirichlet Boundary Condition A00=A00; elseif BCs==2 % Neumann Boundary Condition A00(:,1)=A00(:,1)+Aw(:,1); % on the left edge(x_min) A00(:,M)=A00(:,M)+Ae(:,M); % on the right edge (x_max) A00(1,:)=A00(1,:)+An(1,:); % on the top edge (y_min) A00(N,:)=A00(N,:)+As(N,:); % on the bottom edge (y_max)

elseif BCs==3 % Analytical Boundary Condition

TL=exp(-k0*dx(:,1).*sqrt(abs(n_eff^2-Index(:,1).^2))); % left edge (x_min) A00(1:N,1)=A00(1:N,1)+Aw(1:N,1).*TL;

TR=exp(-k0*dx(1:N,M).*sqrt(abs(n_eff^2-Index(1:N,M).^2)));% right edge (x_max) A00(1:N,M)=A00(1:N,M)+Ae(1:N,M).*TR;

TT=exp(-k0*dy(1,1:M).*sqrt(abs(n_eff^2-Index(1,1:M).^2))); % top edge (y_min) A00(1,1:M)=A00(1,1:M)+An(1,1:M).*TT;

TB=exp(-k0*dy(N,1:M).*sqrt(abs(n_eff^2-Index(N,1:M).^2)));% top edge (y_max) A00(N,1:M)=A00(N,1:M)+As(N,1:M).*TB;

elseif BCs==4 % Transparent Boundary condition % On the left side of the meshed grid, we assume % E2(n,1)/E2(n,0)=E1(n,2)/E1(n,1)=TL(n) TL=E(1:N,2)./E(1:N,1); mark=((imag(TL)<0)~=0); TL(mark)=real(TL(mark)); % If the imaginary part of the TL elements is negative,set the imaginary parts to be zeros. (exp(jk_x*x))

% On the right side of the meshed grid, we assume % E2(n,M+1)/E2(n,M)=E1(n,M)/E1(n,M-1)=TR(n) TR=E(1:N,M)./E(1:N,M-1); mark=((imag(TR)>0)~=0); 112

TR(mark)=real(TR(mark)); % If the imaginary part of the TR elements is positive,set the imagainary parts to be zeros (exp(-jk_x*x))

% On the top side of the meshed grid, we assume % E2(1,m)/E2(0,m)=E1(2,m)/E1(1,m)=TT(m) TT=E(2,1:M)./E(1,1:M); mark=((imag(TT)<0)~=0); TT(mark)=real(TT(mark)); % If the imaginary part of the TT elements is negative,set the imaginary parts to be zeros(exp(jk_y*y))

% On the bottom side of the meshed grid, we assume % E2(N+1,m)/E2(N,m)=E1(N,m)/E1(N-1,m)=TB(m) TB=E(N,1:M)./E(N-1,1:M); mark=((imag(TB)>0)~=0); TB(mark)=real(TB(mark)); % If the imaginary part of the TB elements is positive,set the imaginary parts to be zeros (exp(-jk_y*y))

A00(1:N,1)=A00(1:N,1)+Aw(1:N,1)./TL; % on the left edge (x_min) A00(1:N,M)=A00(1:N,M)+Ae(1:N,M).*TR; % on the right edge (x_max) A00(1,1:M)=A00(1,1:M)+An(1,1:M)./TT; % on the top edge (y_min) A00(N,1:M)=A00(N,1:M)+As(N,1:M).*TB; % on the bottom edge (y_max) else error('Wrong Boundary Conditon!! Please choose a number from 1 to 4.'); end

% construct A matrix for [A]{Phi}=beta^2*{Phi} where beta is the eigen % value and the Phi is the eigen_vector

% calculating the negative most off diagonal vector. The the negative % most off diagonal vector comes from the point Ayn(n,m)*E2(n-1,m). On the % top edge for the first row points, Ayn(n,m)*E2(n-1,m) disappear from the % five points system and thus it is zeros for all the top row elements. An(1,:)=zeros(1,M); an=reshape(transpose(An),[],1); % the negative most diagonal an=[an((M+1):end);zeros(M,1)];

% Calculating the adjacent negative diagonal. The adjacent negative % diagonal comes from the point Axn(n,m)*E2(n,m-1). On the left edge of the % dicretized region, Axn(n,m)*E2(n,m-1)=Axn(n,1)*E2(n,0) disappear and thus % it is zeros for all the left column points. Aw(:,1)=zeros(N,1); aw=reshape(transpose(Aw),[],1); % the negatice adjacent diagonal aw=[aw(2:M*N);0];

% Calculating the adjacent positive diagonal. The adjacent positive % diagonal comes from the point Axp(n,m)*E2(n,m+1). On the right edge of the % dicretized region, Axp(n,m)*E2(n,m+1)=Axp(n,M)*E2(n,M+1) disappear and thus % it is zeros for all the right column points. Ae(:,M)=zeros(N,1); ae=reshape(transpose(Ae),[],1); % the positve adjacent diagonal ae=[0;ae(1:(M*N-1))];

% calculating the positive most off diagonal. The the positive % most off diagonal comes from the point Ayp(n,m)*E2(n+1,m). On the % bottom edge for the bottom row points, Ayp(n,m)*E2(n+1,m)=Ayp(N,m)*E2(N+1,m)disappear from the % five points system and thus it is zeros for all the bottom row elements.

113

% Ma=[Ayp(1:(N-1),1:M);zeros(1,1:M)];% This is right physicallly. However, to construct the penta diagonal matrix, we have to do things differently. As(N,:)=zeros(1,M); as=reshape(transpose(As),[],1); % positive most diagonal as=[zeros(M,1);as(1:(N-1)*M)];

% main diagonal a00=reshape(transpose(A00),[],1); % the main diagonal

% constructing the penta matrix A B=[an aw a00 ae as]; d=[-M -1 0 1 M]; A=spdiags(B,d,N*M,N*M); % constructing the matrix A-beta^2*I A=A-beta^2*speye(size(A));

% solve the equation (A-beta^2*I)E2=E1/eta by using Shifted Inverse % Power Method E_vector=reshape(transpose(E),[],1); eta1=norm(E_vector,inf); b=E_vector./eta1; E_vector=A\b; c(ii+1)=E_vector'*b/(b'*b) delta=abs(c(ii+1)-c(ii)); E=vec2mat(E_vector,M); end [C,I]=max(abs(E_vector)); eta=E_vector(I); % eta=norm(E_vector,inf); E=E/eta; n_eff=sqrt(beta^2+1/c(ii+1))/k0;

114

` APPENDIX B

New Saturable Absorber Design Simulation

Transmission through fiber 1 to fiber 2 in Figure 4-2 depends on the modes in fiber 1 and 2, the nonlinear medium and the distance between end 1 and end 2. Once the fiber modes are already obtained, propagation through the nonlinear medium in the free space can be done using continuous wave split step method[21]. The matlab code for this method is shown in this appendix.

%propagation through a Kerr medium. All units are converted into SI %units. Carbon disulfide is used as example. The index n0=1.67, and the %ki_3=2.2e-12 (esu=cm^2/statvolt^2)=2.2e-12*1.4e-8 (m^2/V^2). Refer to %equation A(14) in appendix A from "Nonlinear Optics" by Robert W. Boyd. clear;clc;close all;tic; %load the mode distribution from the ModeSolverGUI: first three lines are %for the beam from the first fiber end, and the next three lines are the %mode in the second fiber end. If there is no next three lines, we assume %the two fiber ends are the same. load('first_fiber_Mode.mat','Mode'); load('first_fiber_X.mat','X'); load('first_fiber_Y.mat','Y'); E1=abs(Mode); X1=X; Y1=Y; load('second_fiber_Mode.mat','Mode'); load('second_fiber_X.mat','X'); load('second_fiber_Y.mat','Y'); E2=abs(Mode); X2=X; Y2=Y;

% dz is the propagation step size. iteration is the number of propagation steps. dz=1e-6; iteration=2000;

115

% lambda is the wavelength, c is the speed of light.eps0 is the permittivity. ki_3 is the % third order susceptibility. n0 is the refractive index of the light in the % Carbon Disulfide. lambda=1.550e-6; % in meter c=3e8; w0=2*pi*c/lambda; % the central frequency eps0=8.854e-12; % ki_3=2.2e-12*1.4e-8; % third order nonlinearity of CS2 % n0=1.63; % refractive index of CS2 n0=2.81; % refractive index: chalcogenide As40Se60 n2 = 2.3e-17; % third order nonlinear index of As40Se60 gamma = w0*eps0*n0*n2/2; operator

% the following two lines define the 2-d grid in space. Num=1024; x1=linspace(-250,250,Num); x2=x1; [xq,yq]=meshgrid(x1,x2); mode1 = gridfit(X1,Y1,E1,x1,x2); mode2 = gridfit(X2,Y2,E2,x1,x2); xq=xq*1e-6; yq=yq*1e-6; figure, mesh(xq,yq,abs(mode1)), shading interp figure, mesh(xq,yq,abs(mode2)), shading interp %% Center_Amp_Vector=linspace(0.00001,1,100r)*1e8;% define the center amplitude of the beam from the first fiber.

% % assume the x and y dimensions are the same. N=length(x1); M=N; % The following line defines the spatial frequency step. dx=(x1(2)-x1(1))*1e-6; dy=dx; dfx=1/N/dx; dfy=1/M/dy; % % The following line defines the sampling frequency. % fsx=1/dx; % fsy=1/dy; % The following two lines define the 2-d grid in spatial frequency. fx=[-N/2:N/2-1]*dfx; % fx=linspace(-1/2*fsx,1/2*fsx,N) fy=[-M/2:M/2-1]*dfy; % The following line defines the diffraction factor over half step of propagation. k0=2*pi*n0/lambda; % Diff=(exp(1i*2*pi^2/k0*fx.^2*dz/2))'*exp(-1i*2*pi^2/k0*fy.^2*dz/2); [Fx,Fy]=meshgrid(fx,fy); Diff=exp(-1i*2*pi^2/k0*(Fx.^2+Fy.^2)*dz/2); % Shifting is required to properly align the FFT of A Diff=fftshift(Diff);

% zr=pi*(waist0)^2/lambda*n0 % reilay range gap=10; coupling=zeros(length(Center_Amp_Vector),round(iteration/gap)); for kk=1:length(Center_Amp_Vector) Center_Amp=Center_Amp_Vector(kk); A=Center_Amp*mode1; 116

power(kk)=n0*eps0*c/2*sum(sum(abs(A1).^2))*dx^2/1000; % in kW ii=0; for k=1:iteration % The following two lines represent the diffraction over half propagation step. Afft2=fft2(A).*Diff; A=ifft2(Afft2); % The following two lines represents the Kerr effect. (Third oder nonlinearity). NON=exp(1i*gamma*abs(A).^2*dz); A=A.*NON; % The following two lines is another half step of diffraction. Afft2=fft2(A).*Diff; A=ifft2(Afft2); if mod(k,gap)==0 ii=ii+1; propagation(ii)=k*dz*1e6; coupling(kk,ii)=abs(sum(sum(conj(A).*mode2)))^2/sum(sum(abs(A).^2))/sum(sum(abs (mode2).^2)); % calculate the transmission between the beam propagating from the fiber 1 through the nonlinear medium and the mode in fiber 2. end end end figure, mesh(propagation,power, coupling);xlabel(‘Propagation (um)’),ylabel(‘power (kW)’);

117

` APPENDIX C

Split Step Method

The whole fiber laser cavity simulation in this dissertation is written in Matlab.

One illustration of how simulation is done is shown here for the cavity shown in Figure

5-4. There are six elements inside the cavity: three sections of fiber, a saturable absorber, an output coupler and a spectral filter. Propagation in the fiber is simulated using split step method (SSM). Numerical realization for the other elements are explained in details in Chapter 5. Each element in the cavity is written as a sub function.

The main routine for simulation is shown here: clear all; close all; clc;tic; % Cavity: SMF - Active - SMF - SA -OC - SF fprintf('The cavity is: PF-AF-PF-SA-OC-SF\n'); %% ------% Cavity Constants %------% c=0.03; % m/s, light speed lambda_center=1.030e-4; % pulse center wavelength in meters center_Omega = 2*pi*c/lambda_center; %------% normal fiber parameters ( passive fiber = pf) %------% Lpf1=300; % passive fiber length in meters beta2pf1=23e-5; % passive fiber second order dispersion in s^2/m beta3pf1=0; % passive fiber third order dispersion in s^3/m gammapf1=0.000047; % passive fiber third order nonlinearity in 1/W/m alphapf1=0; betapf1=[beta2pf1,beta3pf1]; % used for self-defined functions % ------% gain fiber parameters (Active Fiber = af) %------% Laf=60; % meters, active fiber length beta2af=23e-5; % s^2/m, second order dispersion beta3af=0; % s^3/m, third order dispersion gammaaf=0.000047; % 1/W/m, third order nonlinearity paramter

118 g0=30/Laf/4.343; % small signal gain for power in 1/m, converted from 30dB/m. Pout = Pin* exp(g0*L) and g0*L (dB) = 30 dB => g0 (1/m)=(30dB/L)/(10*log10e) BW_gain=40e-7; Esat=1000; % J n0=1.5; % Refractive Index @ center wavenlength 1.55 um. This value is the same for both active and passive fiber. alphaaf=0; % fiber loss in 1/m. The value is the same for passive fiber. This has to be converted to 1/m during the calculation. GainModel = 1; GainInfo=[g0,Esat,GainModel,BW_gain]; % used for self-defined functions, the number 1 means use Lorentzian gain spectrum betaaf=[beta2af,beta3af]; % used for self-defined functions

%------% normal fiber parameters ( passive fiber = pf) %------% Lpf2=100; % passive fiber length in meters beta2pf2=23e-5; % passive fiber second order dispersion in s^2/m beta3pf2=0; % passive fiber third order dispersion in s^3/m gammapf2=0.000047; % passive fiber third order nonlinearity in 1/W/m alphapf2=0; betapf2=[beta2pf2,beta3pf2]; % used for self-defined functions

% ------% Saturable absorber parameters %------% q0=0.7; % Modulation depth Psat=1000; % Saturation power (watts)

% ------%% spectral filter %------% BW_sf = 10e-7;

% ------% Output coupler %------% Dump_Rate=0.7; % Out-coupling parameter

%% ------% Input Pulse Information %------% Tmax=61.497; % full time window nt=2^14; P0=200; % peak power (watts) FWHM=0.1; mshape=1; % mshap = 'wn' for white Gaussian noise; 0 for sech; 1 or greater for gaussian chirp0=0;

% generate a pulse [ A,t,dt,lambda,omega] = PulseGenerator( c,Tmax,nt,FWHM,P0,mshape,chirp0,lambda_center); initial_pulse=A;

%% ------% Disretize propagation distance in fiber %------% dzaf=1; % propagation step size in gain fiber dzpf1=1;% propagation step size in first section of passive fiber dzpf2=1;% propagation step size in second section of passive fiber % calculating the convergence parameters; eps1=1; eps2=1; eps3=1; counter=1;

% Tracking the pulse with iteration. Energy1(1)=sum(abs(A).^2)*dt*1e-3; % energy of the pulse in nJ 119

A_loop=ones(1,nt); %% ring fiber simulation loop. ii = 1; % for saving pulse along loop save_field_loop = 1; % switch to save pulse evolution along loop save_field_inside = 20; % save pulse evolution inside cavity while (eps1>1e-5)||(counter<50)%||(eps2>1e-3)||(eps3>1e-2) % for counter=1:40 counter=counter+1; fprintf('=== start roundtrip # %d ===\nPeak Power (W) =\t',counter-1); A1=A; % % ------% Fiber 1: Propagation in the passive fiber %------% [ A,Aw1] = PropagateFiber_fourier(c,save_field_inside,A,lambda_center,dt,Lpf1,dzpf1,betapf 1,gammapf1,alphapf1,0,0); % [ A,Aw1] = PropagateFiber_runge_kutta(c,save_field_inside,A,lambda_center,dt,Lpf1,dzpf1,be tapf1,gammapf1,alphapf1,0,0); A2=A; % % ------% Fiber 2: Propagation in the active fiber %------% [ A,Aw2] = PropagateFiber_fourier(c,save_field_inside,A,lambda_center,dt,Laf,dzaf,betaaf,g ammaaf,alphaaf,0,GainInfo); % [ A,Aw2] = PropagateFiber_runge_kutta(c,save_field_inside,A,lambda_center,dt,Laf,dzaf,beta af,gammaaf,alphaaf,0,GainInfo); A3=A; % % ------% Fiber 2: Propagation in the passive fiber %------% [ A,Aw3] = PropagateFiber_fourier(c,save_field_inside,A,lambda_center,dt,Lpf2,dzpf2,betapf 2,gammapf2,alphapf2,0,0); % [ A,Aw3] = PropagateFiber_runge_kutta(c,save_field_inside,A,lambda_center,dt,Lpf2,dzpf2,be tapf2,gammapf2,alphapf2,0,0); A4=A; % ------%% Saturable Absorber %------% [ A ] = PropagateSA( A,q0,Psat ); fprintf('A4 = %f nJ\n', sum(abs(A).^2)*dt*1e-3);

% ------%% Out-put coupler % ------% [ Aout,A ] = PropagateOC( A,Dump_Rate); % fprintf('A5 = %f nJ\n', sum(abs(A).^2)*dt*1e-3); A5=A; % ------%% spectral filter %------% [ A ] = PropagateSF( c,A,t,lambda_center,BW_sf); % fprintf('A6 = %f nJ\n', sum(abs(A).^2)*dt*1e-3); A6=A; % calculating the convergence parameters; Energy1(counter)=sum(abs(A).^2)*dt; eps1=abs(Energy1(counter)-Energy1(counter-1))/Energy1(counter-1); if mod(counter,save_field_loop)==0 A_loop(ii,1:nt)=A; ii = ii + 1; end fprintf('\n'); fprintf('Pulse energy In cavity after SF = %f nJ\n', sum(abs(A).^2)*dt*1e- 3); end 120

fprintf('Pulse energy Out cavity at OC = %f nJ\n', sum(abs(Aout).^2)*dt*1e-3); final_pulse=A; A_in = [Aw1.field;Aw2.field;Aw3.field;A4;A5;A6]; % pulse evolution inside the cavity

% adjust the time and lambda lambda = lambda*1e7; % convert wavelength into nm tvector=[-10 10]; fvector=[1000 1060]; filename = 'C:\Users\Wang Long\Desktop\kerr kens\PhD_dissertation\simulations\Fiber laser cavity simuation\PhD thesis ring cavity simulation\SA_theoretical\ANDi laser\simulation results'; filepath = [filename,'\data.mat'];save(filepath); %% plot pulse_energy after spectral filter after each roundtrip to see the % convergence. pulse_energy_loop = sum(abs(A_loop).^2,2)*dt*1e-3; roundtrip = [1:length(pulse_energy_loop)]; figure('color','w'), hFig=plot(roundtrip,pulse_energy_loop,'r*');xlabel('Round trip number');ylabel('Pulse energy after the spectral filter (nJ)'); xt=[0 10 20 30]; for i=1:length(xt) xl{i}=num2str(xt(i)); end yt=[0.1 0.2 0.3 0.4]; for i=1:length(yt) yl{i}=num2str(yt(i)); end hP=get(hFig,'parent');set(hP,'xtick',xt,'xticklabel',xl,'ytick',yt,'yticklabel' ,yl,'linewidth',2,'Fontsize',11); % filepath = [filename,'\Pulse_energy_loop.fig']; %% plot pulse evolution inside cavity for steady state figure('color','w'),hFig=waterfall(t,[1:size(A_in,1)],abs(A_in).^2);xlabel('Tim e (ps)'), ylabel('Postition'); %xlim([-30 30]); yt = [1 size(Aw1.field,1),size(Aw1.field,1)+size(Aw2.field,1),size(Aw1.field,1)+size(Aw 2.field,1)+size(Aw3.field,1),size(Aw1.field,1)+size(Aw2.field,1)+size(Aw3.field ,1)+1,size(Aw1.field,1)+size(Aw2.field,1)+size(Aw3.field,1)+2,size(Aw1.field,1) +size(Aw2.field,1)+size(Aw3.field,1)+3]; yl = {'Start','PF1','AF','PF2','SA','OC','SF'}; xt=[-30 -15 0 15 30]; for i=1:length(xt) xl{i}=num2str(xt(i)); end zt=[0 0.6 1.2]; for i=1:length(zt) zl{i}=num2str(zt(i)); end hP = get(hFig,'Parent');set(hP,'Fontsize',7,'linewidth',2,'YTick',yt,'Yticklabel',yl ,'xtick',xt,'xticklabel',xl,'yticklabelrotation',45,'ztick',zt,'zticklabel',zl) ; view(125,50)

%% plot the pulse after each element h=figure('Color','w');set(h,'position',[ 101 159 1065 525]); subplot(3,5,1),plot(t,abs(A1./max(abs(A1))).^2,'k');ylabel('Pulse'), xlim(tvector); subplot(3,5,2),plot(t,abs(A2/max(abs(A2))).^2,'k'); xlim(tvector); 121 subplot(3,5,3),plot(t,abs(A3/max(abs(A3))).^2,'k'); xlim(tvector); subplot(3,5,4),plot(t,abs(A4/max(abs(A4))).^2,'k'); xlim(tvector); subplot(3,5,5),plot(t,abs(A5/max(abs(A5))).^2,'k'); xlim(tvector); a=fftshift(abs(fft(A1))); a=a./max(abs(a)); b=fftshift(abs(fft(A2))); b=b./max(abs(b)); c=fftshift(abs(fft(A3))); c=c./max(abs(c)); d=fftshift(abs(fft(A4))); d=d./max(abs(d)); e=fftshift(abs(fft(A5))); e=e/max(abs(e)); subplot(3,5,6),plot(lambda,a.^2,'k');xlabel('(a)'),ylabel('Spectrum');xlim(fvec tor); subplot(3,5,7),plot(lambda,b.^2,'k');xlabel('(b)'),xlim(fvector); subplot(3,5,8),plot(lambda,c.^2,'k');xlabel('(c)'),xlim(fvector); subplot(3,5,9),plot(lambda,d.^2,'k');xlabel('(d)'),xlim(fvector); subplot(3,5,10),plot(lambda,e.^2,'k');xlabel('(e)'),xlim(fvector); subplot(3,5,11),plot(t,-gradient(unwrap(angle(A1))),'k'); ylabel('Chirp');xlim(tvector); subplot(3,5,12),plot(t,- gradient(unwrap(angle(A2))),'k');xlabel('(b)'),xlim(tvector); subplot(3,5,13),plot(t,- gradient(unwrap(angle(A3))),'k');xlabel('(c)'),xlim(tvector); subplot(3,5,14),plot(t,- gradient(unwrap(angle(A4))),'k');xlabel('(d)'),xlim(tvector); subplot(3,5,15),plot(t,- gradient(unwrap(angle(A5))),'k');xlabel('(e)'),xlim(tvector); toc; The sub-function “PropagateFiber_fourier” is shown as the following. There are two parameters that should be adjusted before running the code: the num_step and gate opening ‘mask’. “num_step” determines how many sections the fiber will be sliced into based on the nonlinear length. “mask” helps converging from the initial guess of the pulse to a steady pulse by filtering out the noise on the boundary of the time windows.

Note that there is a hidden function for the propagation inside the fiber called

‘PropagateFiber_runge_kutta’. This function is basically the same as

‘PropagateFiber_fourier’; it also propagates pulse through the fiber section but the operator 푁̂ is calculated using the fourth order Runge-Kutta method. Here is the code of

‘PropagateFiber_fourier’:

122 function [ A,PulseEvolution] = PropagateFiber_fourier(c,save_field,A,lambda_center,dt,L,dz,beta,gamma,alpha,Ra man,GainInfo);

% This function propagate a pulse through a piece of fiber using Fourier transform method. The fiber can be % either passive fiber or active fiber, which can be reflected by the % number of inputs. This function ultilize adaptive step size based on the % power of the pulse. The nonlinear step is done using Fourier transform. % % Inputs: % c: the speed of light in vacuum. % save_field: = 0 do not save field; ~= 0, save the pulse if % mod(iteration, save_field)==1; % A - Input pulse; absolute pulse without normalization. % lambda_center - the center wavelength of the pulse % dt - time resolution % L - length of the fiber % dz - the default propagation step size. The real step size will be the % smaller one among (Lnl/step, default_dz, L-prop_length), where Lnl is the % nonlinear length,, default_dz the input dz and prop_length the propagated % length in the fiber. % beta - material dispersions. [beta2, beta3, ...] all in SI units % gamma - third order nonlinearity coefficient % alpha - linear fiber power loss % Raman - related to the slope of the Raman gain spectrum % GainInfo - the gain and gain saturation information for gain fiber. % GainInfo(1) - the small signal gain for power, g0, in 1/m % GainInfo(2) - the saturation energy for gain fiber, Esat, in J % GainInfo(3) - the gain spectrum type. % 1 - Lorentzian model 1: g(omega) = g0/(1+PulseEnergy/Esat)/(1+omega^2/(BW)^2) % 2 - Lorentzian model 2: g(omega) = g0/(1+PulseEnergy/Esat+omega.^2/(BW)^2) % 3 - parabolic model: g(omega)=g0./(1+PulseEnergy/Esat)*(1- omega.^2/BW^2) % 4 - Gaussian model: g(omega)=g0./(1+PulseEnergy/Esat).*exp(- omega.^2/2/BW^2) % Note, BW here is gain FWHM in Hz, specific to each own model and % could be different for differnet models. % % GainInfo(4) - FWHM gain bandwidth in meters,dlambda. It should % be converted into BW in Hz % - for model 1, 2 and 3: BW = 2*pi*c*dlambda/lambda_center^2/2, c is % speed of light % - for model 4: BW=dlambda*2*pi*c/lambda_center.^2/2/sqrt(2*log(2)) % % Outputs: % A - the pulse after the fiber % PulseEvolution - a struture with saved field, propagation and corresponding time vector in the propagated fiber.

PulseEvolution.field = []; PulseEvolution.position = []; savefield_counter = 1; % Note all units are IS standardard. Dimentions in meters, time in seconds, 123

% energy in Joules, and so on. step_num = 50; % divide the nonlinear length Lnl into 'step_num' sections nt = length(A); % number of data points in time t = dt*[-nt/2:(nt/2-1)];% absolute time vector, from small to big, centered at zero omega = 2*pi*[(0:nt/2-1),(-nt/2:-1)]/(t(end)-t(1)); % swaped absolute frquency around the center frequency frequency = omega + 2*pi*c/lambda_center; lambda=2*pi*c./frequency;

% constructing a edgefilter mask = 200; Mask = zeros(1,nt); for nnn=1:nt if nnn

% choose initial propagation step size prop_length = 0; % the length that has already been propagated.This parameter will be adjusted after each step of simulation and indicate how far the fiber has been simulated. counter =0; % iteration counter Power = max(abs(A).^2); % peak power of the input pulse. Lnl = (gamma*Power)^-1; % calculate the nonlinear length for the corresponding peak power default_z = dz; if save_field == 0; dz = min([Lnl/step_num,L-prop_length,default_z]); % choose the step size else dz = default_z; end

% Initial Dispersion operator for passive fiber and should take gain spectrum into consideration for gain % fiber. It will be updated according to the adaptive step size. XX = -alpha/2; for ii = 2:length(beta)+1; XX = XX + 1i*beta(ii-1)*omega.^(ii)/factorial(ii); end if (GainInfo == 0) % if it is Passive Fiber Afft = fft(A); while (prop_length

% Full step of Nonlinear: Fourier Method Ramman = ifft(-1i*omega.*fft(abs(A).^2))*Raman; Non=exp(1i*gamma*dz*(abs(A).^2-Ramman)); A=Non.*A; 124

% half step of Dispersion Afft = exp(XX*dz/2).*fft(A); A=ifft(Afft);

% attenuate the values on the boundaries A = Mask.*A;

prop_length = prop_length+dz; if save_field == 0 % Adaptive dispersion operator Power = max(abs(A).^2); Lnl = (gamma*Power)^-1; dz = min([Lnl/step_num,L-prop_length,default_z]); % choose the step size to be either 1/50 of the nonlinear length elseif (save_field >= 0)&&(mod(counter,save_field)==1) PulseEvolution.field(savefield_counter,1:nt)=A; PulseEvolution.position(savefield_counter) = prop_length; % Aw(savefield_counter,1:nt)=A; % propagation(savefield_counter) = prop_length; savefield_counter = savefield_counter + 1; end end % if save_field ~=0 % PulseEvolution.field = Aw; % PulseEvolution.position = propagation; % end elseif (GainInfo ~= 0) %If it is active fiber g0 = GainInfo(1); % small signal gain for power Esat = GainInfo(2); % saturation energy for gain fiber Gain_model = GainInfo(3); Afft = fft(A); switch Gain_model case 0 % model 0: Consider gain saturation but without gain filter BW = 2*pi*c*GainInfo(4)/lambda_center^2; % convert the gain FWHM band width from meters into Hz gain_spectrum_core = 1-omega.^2/BW^2; while (prop_length

% half step of dispersion PulseEnergy=sum(abs(A).^2)*dt; gain_spectrum=g0./(1+PulseEnergy/Esat).*gain_spectrum_core; Dispersion = exp((0.5*gain_spectrum+XX)*dz/2); % dz here is the adaptive step which is adjusted at the end of each step of propagation A=ifft(Dispersion.*Afft);

% Full step of Nonlinear: Fourier Method Ramman = ifft(-1i*omega.*fft(abs(A).^2))*Raman; Non=exp(1i*gamma*dz*(abs(A).^2-Ramman)); A=Non.*A;

% half step of dispersion PulseEnergy=sum(abs(A).^2)*dt; gain_spectrum=g0./(1+PulseEnergy/Esat).*gain_spectrum_core; Dispersion = exp((0.5*gain_spectrum+XX)*dz/2); % dz here is the adaptive step which is adjusted at the end of each step of propagation

Afft = Dispersion.*fft(A); 125

A=ifft(Afft);

% attenuate the values on the boundaries A = Mask.*A;

% addapting the propagation step size prop_length = prop_length+dz; % calculate the length in the fiber of how long has been propagated if save_field == 0 Power = max(abs(A).^2); % calculate the peak power of the pulse at the end of each step of propagation Lnl = (gamma*Power)^-1; % calculate the nonlinear length based on the new pulse peak power dz = min([Lnl/step_num,L-prop_length,default_z]); % choose the step size to be either 1/50 of the nonlinear length elseif (save_field >= 0)&&(mod(counter,save_field)==1) PulseEvolution.field(savefield_counter,1:nt)=A; PulseEvolution.position(savefield_counter) = prop_length; savefield_counter = savefield_counter + 1; end end case 1 % model 1: lorentizan model 1 BW = 2*pi*c*GainInfo(4)/lambda_center^2; % convert the gain FWHM band width from meters into Hz gain_spectrum_core = 1./(1+omega.^2/BW^2); % the shape of the gain spectrum % BW = GainInfo(4); % gain bandwidth in meters % gain_spectrum_core = 1./(1+(lambda-lambda_center).^2/BW^2); while (prop_length

% half step of dispersion PulseEnergy=sum(abs(A).^2)*dt; gain_spectrum=g0./(1+PulseEnergy/Esat).*gain_spectrum_core; Dispersion = exp((0.5*gain_spectrum+XX)*dz/2); % dz here is the adaptive step which is adjusted at the end of each step of propagation A=ifft(Dispersion.*Afft);

% Full step of Nonlinear: Fourier Method Ramman = ifft(-1i*omega.*fft(abs(A).^2))*Raman; Non=exp(1i*gamma*dz*(abs(A).^2-Ramman)); A=Non.*A;

% half step of dispersion PulseEnergy=sum(abs(A).^2)*dt; gain_spectrum=g0./(1+PulseEnergy/Esat).*gain_spectrum_core; Dispersion = exp((0.5*gain_spectrum+XX)*dz/2); Afft = Dispersion.*fft(A); A=ifft(Afft);

% attenuate the values on the boundaries A = Mask.*A;

% addapting the propagation step size prop_length = prop_length+dz; % calculate the length in the fiber of how long has been propagated if save_field == 0 Power = max(abs(A).^2); % calculate the peak power of the pulse at the end of each step of propagation 126

Lnl = (gamma*Power)^-1; % calculate the nonlinear length based on the new pulse peak power dz = min([Lnl/step_num,L-prop_length,default_z]); % choose the step size to be either 1/50 of the nonlinear length elseif (save_field >= 0)&&(mod(counter,save_field)==1) PulseEvolution.field(savefield_counter,1:nt)=A; PulseEvolution.position(savefield_counter) = prop_length; savefield_counter = savefield_counter + 1; end end case 2 % model 2: lorentzian model 2 BW = 2*pi*c*GainInfo(4)/lambda_center^2; % convert the FWHM band width from meters into Hz gain_spectrum_core = omega.^2/BW^2; % the shape of the gain spectrum while (prop_length

% half step of dispersion PulseEnergy=sum(abs(A).^2)*dt; gain_spectrum=g0./(1+PulseEnergy/Esat+gain_spectrum_core); Dispersion = exp((0.5*gain_spectrum+XX)*dz/2); A=ifft(Dispersion.*Afft);

% Full step of Nonlinear: Fourier Method Ramman = ifft(-1i*omega.*fft(abs(A).^2))*Raman; Non=exp(1i*gamma*dz*(abs(A).^2-Ramman)); A=Non.*A;

% half step of dispersion PulseEnergy=sum(abs(A).^2)*dt; gain_spectrum=g0./(1+PulseEnergy/Esat+gain_spectrum_core); % Lorentzian model 2 Dispersion = exp((0.5*gain_spectrum+XX)*dz/2); Afft = Dispersion.*fft(A); A=ifft(Afft);

% attenuate the values on the boundaries A = Mask.*A;

% addapting the propagation step size prop_length = prop_length+dz; if save_field == 0 Power = max(abs(A).^2); Lnl = (gamma*Power)^-1; dz = min([Lnl/step_num,L-prop_length,default_z]); % choose the step size to be either 1/50 of the nonlinear length elseif (save_field >= 0)&&(mod(counter,save_field)==1) PulseEvolution.field(savefield_counter,1:nt)=A; PulseEvolution.position(savefield_counter) = prop_length; savefield_counter = savefield_counter + 1; end end case 3 % model 3: parabolic model BW = 2*pi*c*GainInfo(4)/lambda_center^2; % convert the FWHM band width from meters into Hz gain_spectrum_core = 1-omega.^2/BW^2; % gain spectrum shape 127

while (prop_length

% half step of dispersion PulseEnergy=sum(abs(A).^2)*dt; gain_spectrum=g0./(1+PulseEnergy/Esat).*gain_spectrum_core; % parabolic model Dispersion = exp((0.5*gain_spectrum+XX)*dz/2); A=ifft(Dispersion.*Afft);

% Full step of Nonlinear: Fourier Method Ramman = ifft(-1i*omega.*fft(abs(A).^2))*Raman; Non=exp(1i*gamma*dz*(abs(A).^2-Ramman)); A=Non.*A;

% half step of dispersion PulseEnergy=sum(abs(A).^2)*dt; gain_spectrum=g0./(1+PulseEnergy/Esat).*gain_spectrum_core; % parabolic model Dispersion = exp((0.5*gain_spectrum+XX)*dz/2); Afft = Dispersion.*fft(A); A=ifft(Afft);

% attenuate the values on the boundaries A = Mask.*A;

% addapting the propagation step size prop_length = prop_length+dz; if save_field == 0 Power = max(abs(A).^2); Lnl = (gamma*Power)^-1; dz = min([Lnl/step_num,L-prop_length,default_z]); % choose the step size to be either 1/50 of the nonlinear length elseif (save_field >= 0)&&(mod(counter,save_field)==1) PulseEvolution.field(savefield_counter,1:nt)=A; PulseEvolution.position(savefield_counter) = prop_length; savefield_counter = savefield_counter + 1; end end case 4 % model 4: Gaussian model % BW=2*pi*c*GainInfo(4)/lambda_center.^2/sqrt(2*log(2))/2;% convert the FWHM band width from meters into Hz % gain_spectrum_core = exp(-omega.^2/2/BW^2); BW=GainInfo(4)/sqrt(2*log(2))/2; gain_spectrum_core = exp(-(lambda-lambda_center).^2/2/BW^2); while (prop_length

% Full step of Nonlinear: Fourier Method Ramman = ifft(-1i*omega.*fft(abs(A).^2))*Raman; Non=exp(1i*gamma*dz*(abs(A).^2-Ramman)); A=Non.*A;

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% half step of dispersion PulseEnergy=sum(abs(A).^2)*dt; gain_spectrum = g0./(1+PulseEnergy/Esat).*gain_spectrum_core; Dispersion = exp((0.5*gain_spectrum+XX)*dz/2); Afft = Dispersion.*fft(A); A=ifft(Afft);

% attenuate the values on the boundaries A = Mask.*A;

% addapting the propagation step size prop_length = prop_length+dz; if save_field == 0 Power = max(abs(A).^2); Lnl = (gamma*Power)^-1; dz = min([Lnl/step_num,L-prop_length,default_z]); % choose the step size to be either 1/50 of the nonlinear length elseif (save_field >= 0)&&(mod(counter,save_field)==1) PulseEvolution.field(savefield_counter,1:nt)=A; PulseEvolution.position(savefield_counter) = prop_length; savefield_counter = savefield_counter + 1; end end end % if save_field ~=0 % PulseEvolution.field = Aw; % PulseEvolution.position = propagation; % end elseif (nargin >11)||(nargin<10) warning('Wrong number of Inputs for fiber. # of inputs should be either 10 (for passive fiber) or 11 (for active fiber)!!'); end Power = max(abs(A).^2); fprintf('%6.2f\t',Power) end The sub-function “PropagateSA” is: function [ A ] = PropagateSA( varargin );

% This function propagates a pulse through a saturable absorber.

% use % varargin = [A,t,dt] for self-defined SA % varargin = [A,t,dt,q0,Psat] for standard model

% input: % A - input pulse % q0 - modulation depth % Psat - saturation power

% Outputs: % A - the pulse after the fiber % Energy - the energy along the fiber. Energy = sum(abs(A).^2)*dt % Prms - RMS Power width. Prms=sum(abs(A).^4)/sum(abs(A).^2) % Trms - RMS time width. Trms = sqrt(sum(t.^2.*abs(A).^2)/sum(abs(A).^2)) % Pulse information:PulseInfo = { A,t,dt,lambda,omega,lambda_center}; A = varargin{1};

129 if (nargin == 3) q0 = varargin{2}; Psat = varargin{3}; T = 1-q0./(1+abs(A).^2/Psat); A=sqrt(T).*A; elseif (nargin == 1) % 4th degree polynomial for chalcogenide glass As40Se60 with thickness % of 3 mm % first piece is from power between [0, 5.589] R4 = 0.000542238845372; R3 = 0.000083644028993; R2 = 0.000970522101039; R1 = 0.038705066387964; R0 = 0.192207923824026;

P_SA=abs(A).^2*1e-3; % convert the power from watts into kW % max(P_SA) for i = 1:length(P_SA) Power_SA = P_SA(i); if (P_SA(i)<5.589) T(i)=R4*Power_SA.^4+R3*Power_SA.^3+R2*Power_SA.^2+R1*Power_SA+R0; else T(i) = 0.000138; end end A = sqrt(T).*A;

end Power = max(abs(A).^2); fprintf('%6.2f\t',Power)

The sub-function “PropagateOC” is: function [ Aout,Ain ] = PropagateOC( A,Dump_Rate);

% This function propagates a pulse through a output coupler

% Inputs: % A - input pulse % t - time sequence. seconds % dt - time resolution % Dump_rate - the ratio of energy that will be dumped out of the cavity

% Outputs: % A - the pulse after the fiber % Energy - the energy along the fiber. Energy = sum(abs(A).^2)*dt % Prms - RMS Power width. Prms=sum(abs(A).^4)/sum(abs(A).^2) % Trms - RMS time width. Trms = sqrt(sum(t.^2.*abs(A).^2)/sum(abs(A).^2))

Aout=sqrt(Dump_Rate)*A; Ain=sqrt(1-Dump_Rate)*A; Power = max(abs(Ain).^2); fprintf('%6.2f\t',Power) end

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The sub-function “PropagateSF” is: function [ A ] = PropagateSF( c,A,t,lambda_center,BW);

% This function propagates a pulse through a spectrum filter.

% Inputs: % c - speed of light in vacuum % A - input pulse % t - time sequence. seconds % dt - time resolution % BW - FWHM bandwidth for gain in meters. for example, 3 nm = 3e-9 m. % lambda_center - pulse center wavelength in meters. % Outputs: % A - the pulse after the fiber % Energy - the energy along the fiber. Energy = sum(abs(A).^2)*dt % Prms - RMS Power width. Prms=sum(abs(A).^4)/sum(abs(A).^2) % Trms - RMS time width. Trms = % sqrt(sum(t.^2.*abs(A).^2)/sum(abs(A).^2)) nt = length(A); % number of data points in time omega = 2*pi*[(0:nt/2-1),(-nt/2:-1)]/(t(end)-t(1)); % shifted absolute frquency around the center frequency % gaussian filter BW = (BW*2*pi*c/lambda_center.^2)/2/sqrt(2*log(2)); % convert the BW from meters to Hz. FWHM SF = exp(-omega.^2/2/BW^2); % gaussian filter A=ifft(SF.*fft(A)); Power = max(abs(A).^2); fprintf('%6.2f\t',Power) end

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