THULIUM MODE-LOCKED FIBER

Thesis

Submitted to

The School of Engineering of the

UNIVERSITY OF DAYTON

In Partial Fulfillment of the Requirements for

The Degree of

Master of Science in Electro-

By

Jordan Mackenzie Adams

Dayton, Ohio

May 2019

THULIUM MODE-LOCKED FIBER LASER

Name: Adams, Jordan Mackenzie

APPROVED BY:

Andy Chong, Ph.D. Imad Agha, Ph.D. Advisory Committee Chairman Committee Member Assoc. Prof. of Physics & Electro-Optics Assistant Professor of Electro-Optics Department of Electro-Optics Department of Electro-Optics

Andrew Sarangan, Ph.D. Committee Member Professor of Electro-Optics and Department of Electro-Optics

Robert J. Wilkens, Ph.D., P.E. Eddy M. Rojas, Ph.D., M.A., P.E. Assoc. Dean of Research and Innovation Dean, School of Engineering Professor School of Engineering

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ABSTRACT

THULIUM MODE-LOCKED FIBER LASER

Name: Adams, Jordan Mackenzie University of Dayton

Advisor: Dr. Andy Chong

A Thulium fiber laser ring cavity is built and mode-locked. Four soliton spectrums are observed near 1900 nm. The pulse durations are estimated from the spectrums using a Fourier transform and from the soliton time-bandwidth product. The largest spectrum is near 550 fs in pulse duration. Kelly sidebands are noticeable on two of the spectrums and the sideband spacing is used to estimate the total dispersion of the cavity.

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Dedicated to my mother and father, Maria and David

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ACKNOWLEDGEMENTS

I would first like to thank my research advisor Andy Chong for passionately teaching and challenging students. His genuine interest in ultrafast optics and eagerness to help students learn has greatly impacted this research and my excitement for the future.

Secondly, I would like to thank Ankita Khanolkara and Chunyang Ma for investing many hours in helping me acquire the laboratory skills necessary for this research.

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TABLE OF CONTENTS

ABSTRACT ...... iii

DEDICATION ...... iv

ACKNOWLEDGMENTS ...... v

LIST OF FIGURES ...... vii

LIST OF TABLES ...... viii

CHAPTER 1 INTRODUCTION ...... 1

1.1 Ultrafast Science and Thulium-Doped Laser Applications ...... 1

1.1.1 LASER History ...... 1

1.1.2 Medical Applications ...... 2

1.2 The Theory of Mode-Locking ...... 5

1.2.1 Pulse Propagation and the Nonlinear Schrodinger Equation ...... 7

1.2.2 The Soliton Pulse ...... 11

1.2.3 The Saturable Absorber ...... 14

CHAPTER 2 BUILDING THE THULIUM RING CAVITY ...... 17

2.1 Laser Cavity Alignment ...... 17

2.2 Mode-Locking the Cavity ...... 19

CHAPTER 3 SPECTRUM ANALYSIS ...... 22

3.1 Spectrum Comparison ...... 22

3.2 Estimating Pulse Duration ...... 24

3.3 Estimating Pulse Energy ...... 27

3.4 Estimating Dispersion from the Kelly Sidebands ...... 28

REFERENCES ...... 32

vi LIST OF FIGURES

Figure 1.1. An image of blood flow in a mouse using third- from a

Ti-Sapphire laser [14] ...... 2

Figure 1.2 Thulium is classified as a lanthanide rare earth metal [16] ...... 3

Figure 1.3 The absorption spectrum water ...... 4

Figure 1.4: The erbium profile [18] ...... 5

Figure 1.5. Temporal and spectral plots of a soliton pulse [17] ...... 12

Figure 1.6. Schematic of the nonlinear polarization evolution saturable absorber ...... 15

Figure 1.7. Higher intensity light experiences more rotation ...... 15

Figure 1.8. PBS transmission as a function of intensity...... 16

Figure 2.1. Initial cavity alignment schematic ...... 17

Figure 2.2. Alignment for continuous lasing schematic ...... 18

Figure 2.3. Mode-locking schematic ...... 20

Figure 2.4. The four mode-locked spectrums ...... 21

Figure 3.1. The four mode-locked spectrums ...... 22

Figure 3.2. The spectrum relationship ...... 23

Figure 3.3. A soliton spectrum from a cavity with 8.4 m of Tm-doped fiber [21] ...... 24

Figure 3.4. Fourier Transform pulse duration estimate ...... 26

Figure 3.5. Kelly sidebands spacing of a soliton spectrum...... 30

Figure 3.6. Dispersion estimates using Kelly sidebands...... 32

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LIST OF TABLES

Table 3.1. Time-bandwidth pulse duration estimates ...... 25

Table 3.2. Fourier transform compared to time-bandwidth pulse duration estimates ...... 26

Table 3.3. Estimated total dispersion values from Kelly sideband spacing, listed with the the common values for SMF, Tm and Ho at 1900 nm ...... 30

viii CHAPTER 1

INTRODUCTION

1.1. Ultrafast Science and Thulium-Doped Laser Applications

1.1.1. LASER History

One year after Albert Einstein published his theory of general relativity in 1916, he proposed yet another groundbreaking idea—stimulated emission [1]. The idea was formed in the wake of his 1905 demonstration of the photoelectric effect. Einstein’s

Nobel prize winning demonstration established the particle nature of light and began the field of quantum mechanics. Realizing the photon—a ‘quantum’ of an oscillating electric field—lead to the hypothesis that a photon incident on an excited atom could stimulate the emission of an identical photon. 37 years later, Charles Townes confirmed Einstein’s concept by successfully creating a device that used stimulated emission [2].The device used a mirrored feedback loop with charged ammonia to amplify microwaves to approximately 10 nW of continuous power. By 1960, Theodore Maimon fabricated a mirrored ruby cylinder that emitted stimulated visible light—a process coined light amplification by stimulated emission of radiation—the first laser [3] . Within 6 months,

American scientists developed Uranium and He-Ne [4].The 1.15 µm He-Ne and

0.694 µm Ruby were quickly commercialized and sparked the exponential growth of laser technology and research in the following decade.

Ultrafast optics emerged in 1963 with the mode-lock of a He-Ne laser using an acousto-optic modulator [5]. The field eventually became a prominent research area in the 1990’s with the first solid-state femtosecond Ti:sapphire laser [6]. These short pulses

1 allowed scientist to view atomic motion during a chemical reaction, build nanometer resolution structures, and perform nonlinear (Figure 1.1).

A need for operational ease and stability led to the development of semi-conductor [7] and fiber femtosecond lasers. Semi-conductor lasers are inexpensive but poor beam quality limits application. Fiber Figure 1.1: An image of blood flow in a mouse lasers are also inexpensive and using third-harmonic generation from a Ti- Sapphire laser [14] produce near perfect beam quality, have high stability and power, and are compact. Today, most ultrafast research is focused on fiber lasers.

1.1.2. Medical Applications

A common hope placed on any new technology is that it will help humans, particularly through improved medical diagnosis and treatments. Within a year and a half of the first laser, a ruby laser was used to ablate a retinol tumor. Through the next decade laser techniques emerged for blood vessel anastomosis, liver resections, and hepatic hemorrhage and gastric bleeding control (Yahr-Pelonyi). Today, laser-medical novelties are far reaching; one of these areas of interest is ultrafast optics. Femtosecond pulse lasers offer less invasive and higher precision surgery. Human tissue absorbs incident light atom-by-atom resulting in ablation area decreasing with the pulse-duration.

Thus, femtosecond pulses can be used as a high-resolution “” that minimizes side effects for treatments like radiation therapy or cyst removal. Additionally, ultrashort

2 pulses offer improved disease diagnostics through nonlinear phenomena. These nonlinear processes such as second and third harmonic, super-continuum, and frequency comb generation are only practically attainable at high intensities. With the same average power, peak intensities are magnitudes higher in femtosecond pulses than in quasi- continuous pulses. Second and third harmonics are used to obtain high-resolution tissue images without using chemical labels and can measure velocity of blood cells (Dietzel).

Additionally, supercontinuum generation can be used for coherence tomography and frequency comb generation for high-precision . Ultrafast optics has advanced medical diagnosis and treatment and promises future improvements.

Due to their ubiquity in modern communication, common ultrafast sources are ytterbium-doped and erbium-doped fiber lasers with ̴ 1000 nm and ̴ 1500 nm emission, respectively. Less common thulium-doped fiber lasers with ̴ 2000 nm emission are gaining interest in medicine because of their unique penetration depth in biological tissue.

Thulium is element 69 on the

periodic table and classified in the

lanthanide group of the rare earth

metals [16]. It is particularly rare and

found dispersed in a few specific

minerals alongside ytterbium and other

rare earth metals. Thulium absorbs Figure 1.2: Thulium is classified as a lanthanide rare earth metal [16]. around 793 nm and 1550 nm and emits

around 1900 nm.

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The 1900 nm emission makes thulium an excellent source for medical applications. Human tissue’s high-water content makes water’s absorption spectrum a good estimate for tissue penetration depth. Figure 2 Figure 1.3: The absorption spectrum water. There is sharp peak around 2 µm [15]. shows absorption increases from mid-visible up to 3-µm light. The shortest depth in the NIR is 1 µm which occurs for 3 µm light. For comparison, the size of a blood cell is around 8 µm and the combined thickness of dermis and epidermis around 4 mm. For ytterbium’s 1 µm light the penetration depths is 5 cm which is far too deep for any medical applications. However, the 2 µm absorption peak gives a depth of a few hundred micrometers—an ideal depth for many surgical procedures. 2 µm lasers are already in use for treating renal cysts, prostate hyperplasia, and kidney stones (ncbi).

Additionally, 2 µm is an optimal source for Mid-IR supercontinuum generation.

Most materials have vibration resonances in the mid-IR frequency range. Waves that match resonance frequency are absorbed and heat the material. The reflective spectrum of an object using a mid-IR supercontinuum beam accurately identifies material composition. Human breath and skin could quickly be scanned for cancer cells or other disease biomarkers, shortening time to treatment.

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1.2. The Theory of Mode-Locking

To understand mode-locking, it is helpful to build up our understanding from the basic laser. The simplest explanation for a laser assumes the amplified light’s bandwidth,

Δλ, is much smaller than its center wavelength. This is a good assumption for typical (CW) lasers. For example, NP Photonic’s, Rock™ tunable 1530-1565 nm CW laser has an emission bandwidth of about 1.6 attometers. However, the erbium gain material that emits this narrow width can amplify a much broader wavelength range.

The gain profile of erbium is shown in figure 1.4 and shows strong amplification within

1540-1560 nm.

Figure 1.4: The erbium gain profile shows possible wavelengths that can be amplified. However, the length of the laser cavity permits only certain wavelengths to survive called longitudinal modes [18]. It is first important to note that only wavelengths that divide the total cavity length into integers survive,

퐿 푁 = . (1.1) 휆

These wavelengths are termed longitudinal modes of the cavity and the total number of modes that exist within a bandwidth Δλ decreases as L decreases. Secondly, the 5 longitudinal mode which experiences the most gain and least lost per round trip will have the most amount photons to be amplified the next round trip. This positive feedback process eventually allows only the strongest longitudinal mode to survive. Given the cavity length isn’t too long, the emission of the standard laser will be one longitudinal mode. The emission will have a narrow bandwidth depending on fluctuations in the cavity length. Mode-locking takes advantage of the other longitudinal modes and synchronizes them together to constructively and deconstructivity interfere in such a way that creates a pulse. When the phases are locked in this way the modes interfere, and their energy is concentrated in a single short pulse. In fact, the larger the Δλ between the lowest and highest modes the shorter the pulse that will be created. Interference of modes to create femtosecond pulses is simple to understand, but how is this possible in a laser cavity?

The answer to how mode-locking can occur depends on what type of dispersion the wavelengths experience. Group velocity dispersion (GVD) is a measure of the difference in speeds of different wavelengths. Single mode fiber (SMF) has zero- dispersion at 1310 nm. Below 1310 nm, the fiber has normal dispersion where longer wavelengths travel faster than shorter wavelengths and above 1310 nm, the fiber has anomalous dispersion where shorter wavelengths travel faster than longer wavelengths.

In both cases a tool called a saturable absorber is important for mode-locking, discussed in section 1.2.3. However, in normal dispersion a spectral filter is needed to achieve mode-locking. Typical normal dispersion mode-locked lasers are the ANDi laser and

Self-similar laser. In anomalous dispersion the spectral filter is not needed. A natural phenomenon draws continuous waveforms into stable pulses called solitons (section

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1.2.2). The saturable absorber is used to push the cavity to soliton formation. Soliton formation can be understood from the nonlinear Schrodinger equation discussed in section 1.2.1

Designing a laser to emit femtosecond pulses demands considerations from nonlinear dynamics and guided-wave optics. Due to short pulses, intensity dependent effects become significant in describing wave propagation in fiber waveguides. Starting from

Maxwell’s equations, short-pulse evolution is derived into the well-known Nonlinear

Schrodinger Equation (NLSE). Wave-guide parameters for stable pulses are determined and modified to maximize pulse energy and bandwidth. These parameters are actualized into a ring cavity which is then aligned and finally mode-locked.

1.2.1. Pulse Propagation and The Nonlinear Schrodinger Equation

The NLSE is derived similar to ref. [16]. The Helmholtz equation describes electromagnetic wave propagation in dielectric media and is easily derived from

2 Maxwell’s equations and the constitutive relationship, 퐷 = 휖0푛 퐸, to give:

푛2 휕2퐸 훻2퐸 − = 0. (1.2) 푐2 휕푡2

However, the strong light-matter interaction in femtosecond pulses means the displacement field is no longer linear with incident field. A nonlinear perturbation term is added

퐷 = 휖0퐸 + 푃퐿 + 푃푁퐿. (1.3)

푃퐿and 푃푁퐿 are the linear and nonlinear polarization densities, respectively. It is well known that the linear polarization density is defined as 푃퐿 = 휖0휒푒퐸, with 휒푒being the

7 medium’s electric susceptibility. Perturbation theory defines the nonlinear perturbation as

(2) 2 (3) 3 푃푁퐿= 휖0휒 퐸 + 휖0휒 퐸 + ⋯ . The electric displacement becomes

퐷 = 휖0퐸 + 휖0휒푒퐸 + 푃푁퐿 (1.4)

(2) 2 (3) 3 퐷 = (휖0 + 휖0휒푒)퐸 + 휖0휒 퐸 + 휖0휒 퐸 + ⋯ (1.5)

2 (2) 2 (3) 3 퐷 = 휖0푛0 퐸 + 휖0휒 퐸 + 휖0휒 퐸 + ⋯ (1.6)

The perturbed 퐷 now includes intensity and higher order electric field terms scaled by certain material constants—the higher order susceptibility coefficients. The new susceptibilities lead us to define a perturbed index of refraction

2 퐷 = 휖0푛 퐸 (1.7)

2 퐷 = 휖0(푛푒 + 훿푛퐿 + 훿푛푁퐿) 퐸. (1.8)

With 훿푛퐿and 훿푛푁퐿 being linear and nonlinear perturbations induced by the electric field.

Expanding and assuming small 훿푛퐿and 훿푛푁퐿 gives

2 2 2 퐷 = 휖0(푛표 + 훿푛퐿 + 훿푛푁퐿 + 2푛표훿푛퐿 + 2푛표훿푛푁퐿 (1.9) + 2훿푛퐿훿푛푁퐿)퐸

2 퐷 ≈ 휖0(푛표 + 2푛표훿푛퐿 + 2푛표훿푛푁퐿)퐸 (1.10)

2 퐷 = 휖0푛표퐸 + 푃퐿 + 푃푁퐿 (1.11)

푃푁퐿 = 2푛0휖0훿푛푁퐿퐸. (1.12)

Substituting the new current density into Maxwell’s equations, and deriving the wave equation gives

푛 2 휕2퐸 휕2푃 ∇2퐸 − 0 = −휇 푁퐿 (1.13) 푐2 휕푡2 0 휕푡2

Notice in comparison to (1.1) the wave equation for femtosecond pulses includes the second time derivative of the nonlinear polarization density.

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The field and polarization density can be represented as the multiple of slow and fast varying terms as

퐸 = 푎(푧, 푡)푒푖(훽0푧−휔푡) (1.14)

−푖휔푡 푃푁퐿 = 푝푁퐿(푧, 푡)푒 (1.15)

푖훽0푧 With 푝푁퐿(푧, 푡) = 2푛0휖0훿푛푁퐿푎(푧, 푡)푒 . Plugging into the wave equation gives

휕2푎(푧, 푡) 휕푎(푧, 푡) 푛 2 휕2푎(푧, 푡) 휕푎(푧, 푡) [ + 𝑖2훽 − 훽2푎(푧, 푡)] − 0 [ − 𝑖2휔 − 휔2푎(푧, 푡)] 휕푧2 0 휕푧 0 푐2 휕푡2 휕푡

2 휕 푝푁퐿 휕푝푁퐿 = −휇 [ − 𝑖2휔 − 휔2푝 ] 푒−푖(훽0푧−(휔−휔)푡) (1.16) 0 휕푡2 휕푡 푁퐿

휕2푎(푧,푡) 휕푎(푧,푡) 휕2푎(푧,푡) 휕푎(푧,푡) Assuming a slowly varying envelope ≪ 2훽 , ≪ 2휔 ≪ 휕푧2 0 휕푧 휕푡2 휕푡

휕2푝 휕푝 휔2푎(푧, 푡) and 푁퐿 ≪ 2휔 푁퐿 ≪ 휔2푝 which reduces (1.15) to 휕푡2 휕푡 푁퐿

2 휕푎(푧, 푡) 푛0 𝑖2훽 + (훽2 − 휔2) 푎(푧, 푡) = 휇 휔2푝 푒−푖(훽0푧) (1.17) 0 휕푧 0 푐2 0 푁퐿

2 휕푎(푧, 푡) (훽0 − 휔)(훽0 + 휔)푎(푧, 푡) 휇0휔 푝푁퐿 − 𝑖 = −𝑖 푒−푖(훽0푧) (1.18) 휕푧 2훽0 2훽0

With 훽0 + 휔 ≈ 2훽0 this gives

2 휕푎(푧, 푡) 𝑖 푛0 휇0휔 푝푁퐿 −푖(훽0푧) − (훽0 − 휔)푎(푧, 푡) = −𝑖 푒 (1.19) 휕푧 2 푐 2훽0

푛 (휔) where 훽 is the free space propagation constant while 0 휔 describes propagation in 0 푐 media. The medium propagation constant is defined and expanded in 휔 for convivence

푛 (휔) 훽(휔) ≡ 0 휔 (1.20) 푐

1 휕푛 (휔)휔 휕2푛 (휔)휔 훽(휔) = [푛 (휔 )휔 + 0 | (휔 − 휔 ) + 0 | (휔 − 휔 )2 + ⋯] 푐 0 0 휕휔 0 휕휔2 0 (1.21) 휔0 휔0

휕푛푛 (휔)휔 Letting 휔̃ = 휔 − 휔 and 훽 = 0 | the equation becomes 0 푛 휕휔푛 휔0

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2 훽(휔) = 훽0 + 훽1휔̃ + 훽2휔̃ ⋯ (1.22)

Fourier transforming the wave-equation and inserting 훽(휔) gives

2 휕푎(푧, 휔) 𝑖 휇0휔 푝푁퐿(휔) 2 −푖(훽0푧) − (훽0 − 훽0 + 훽1휔̃ + 훽2휔̃ ⋯ )푎(푧, 휔) = −𝑖 푒 (1.23) 휕푧 2 2훽0

For typical mode-locked pulses ∆휔 ≪ 휔 and the 훽3 and higher-order terms are negligible. Higher-order terms become significant when pulses fall below 50 fs.

Additionally, at wavelengths near the waveguide’s zero-dispersion point, 훽2 approaches zero and 훽3 becomes significant.

The inverse Fourier transform is taken, 푝푁퐿 inserted and the higher order terms dropped

2 2 휕푎(푧, 푡) 𝑖 휕푎(푧, 푡) 휕 푎(푧, 푡) 휇0휔 2푛0휖0훿푛푁퐿퐸 −푖(훽0푧) − (훽1 + 훽2 2 ) = −𝑖 푒 휕푧 2 휕푡 휕푡 2푛0휔/푐 (1.24)

휕푎(푧, 푡) 𝑖 휕푎(푧, 푡) 휕2푎(푧, 푡) 𝑖휔훿푛 − (훽 + 훽 ) = − 푁퐿 푎(푧, 푡). (1.25) 휕푧 2 1 휕푡 2 휕푡2 푐

For fiber waveguides, only the Kerr nonlinearity is significant above 50 fs and is

푚2 defined as 훿푛 = 푛 |푎(푧, 푡)|2, where 푛 is the nonlinear index in [ ]. 푁퐿 2 2 푊

2 휕푎(푧, 푡) 𝑖 휕푎(푧, 푡) 휕 푎(푧, 푡) 휔0푛2 − (훽 + 훽 ) = −𝑖 |푎(푧, 푡)|2푎(푧, 푡)푒−푖휔0푡 (1.26) 휕푧 2 1 휕푡 2 휕푡2 푐

휔푛 The self-phase modulation coefficient is defined as 훾 ≡ 2. Letting 푡′ = 푡 − 푡 with 푐 0

푡0 = 푧/푣푔 sets the time origin at the pulse center and eliminates the group velocity term.

This gives the typical NLSE which describes propagation of ultrafast pulses in fibers:

휕푎(푧, 푡′) 𝑖 휕2푎(푧, 푡′) − 훽 = −𝑖훾|푎(푧, 푡′)|2푎(푧, 푡′). (1.27) 휕푧 2 2 휕푡′2

′ Equation (1.26) can be normalized by letting 푧 = 휉퐿퐷, 푡 = 휏푇표 and 푎 = 푢√푃0,

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2 1 휕푢 𝑖 1 휕 푢 2 − 훽2 2 2 = −𝑖훾푃0|푢| 푢 (1.28) 퐿퐷 휕휉 2 푇0 휕휏

2 푇0 where 퐿퐷 = is the dispersion length, 푃0 pulse duration, and 푇표 is the pulse width. By |훽2|

1 defining a characteristic nonlinear length 퐿푁퐿 = , (1.27) becomes 훾푃0

휕푢 𝑖 휕2푢 퐿 퐷 2 (1.29) + 푠𝑖푔푛(훽2) 2 = −𝑖 |푢| 푢. 휕휉 2 휕휏 퐿푁퐿

퐿 where the ratio 푁2 ≡ 퐷 describes which phenomenon, dispersion or nonlinearity, 퐿푁퐿 dominates pulse evolution. For 퐿퐷 < 퐿푁퐿, 푁 < 1 and dispersion effects are stronger and for 퐿퐷 > 퐿푁퐿, 푁 > 1 and nonlinearity dominates. When both effects are nearly equal in strength 푁 ≈ 1 and a soliton pulse can form.

1.2.2 The Soliton Pulse

A soliton is a pulse whose shape doesn’t change upon propagation—meaning

휕푢 = 0. This happens only when the group velocity dispersion (GVD) term and (self- 휕푧 phase modulation) SPM nonlinearity term balance each other so that 푁 ≈ 1 and

𝑖 휕2푢 = 𝑖|푢|2푢. (1.30) 2 휕휏2 the solution for (1.29) is found using the inverse scatter method to be

푖휉 푎(푧, 푡) = √푃표푠푒푐ℎ(휏/푇표)푒 (1.31) which is a hyperbolic secant pulse envelope. Equation 1.29 is only valid for the fundamental soliton where N = 1. As the pulse power is increased, higher order solitons form when N = 2, 3,⋯.However, these higher order solitons change shape while propagating meaning (1.29) is invalid. Starting with (1.28) the envelope 푢(휉, 휏) can be derived and shown to cyclically repeat its shape after a certain propagation distance

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2 휋 푇표 푧0 = . (1.32) 2 |훽2|

However, the higher orders easily break up into the fundamental order.

For N = 1, the power in the pulse is constrained to

|훽2| 푃0 = 2 (1.33) 훾푇0 which is known as the soliton area theorem. The GVD and SPM cancel all phase across the pulse, so the pulse duration 푇0 can be given by the Fourier transform relation for an unchirped hyperbolic secant pulse,

푇0Δ휈 = 0.315. (1.34)

Δ휆 Plugging in Δ휈 = 푐 2 gives the duration in terms of wavelength, 휆푐

휆2 푇 = 푐 0.315. (1.35) 0 cΔ휆

Therefore, the full width at half maximum (FWHM) pulse duration as well as the pulse energy can be calculated from the spectral FWHM.

An example of a soliton pulse and spectrum from an erbium laser is shown in figure 1.4. It is interesting that a soliton pulse train forms out of a continuous signal in anomalous dispersive media [16].

Small perturbations in a continuous waveform create gain for center-shifted frequencies. The gain allows the spectrum to broaden while propagating and at a certain Figure 1.5. Temporal and spectral plots of a soliton distance dispersion breaks the continuous signal into pulse [17].

12 pulses. As the pulse continues to travel, it shortens and become more intense. The intensity changes the fiber index of refraction and alters the pulse propagation, the nonlinear process known as self-focusing which results SPM. The intensity is concentrated in the center of the pulse and drops monotonically both temporally and spatially in SMF. This intensity gradient creates an index of refraction gradient which mimics that of a focusing lens. If self-focusing was the only processes involved, the self- induced lens would spatially focus the pulse smaller and smaller until the large intensity ionized the fiber material. However, self-focusing also broadens the pulse spectrum which allows dispersion to simultaneously spread it temporally, weakening the intensity and reducing the self-focusing. As discussed above, a certain wave-shape balances and stabilizes the self-focusing effect and the dispersive effect—a secant squared pulse. This stability acts as a node, to which many initial waveforms converge, which allows a

“soliton” pulse to maintain shape over large distances. With typical fiber absorption of approximately 0.1 nJ / km, pulses can propagate 100 km before falling below threshold power required for a fundamental soliton and distinguishing. Pulses can travel farther than 100 km by using an Erbium-Doped Fiber Amplifier (EDFA) every 50 km. Today,

EDFA’s are a vital part of data communication.

Wavelengths larger than 1315 nm fall within the anomalous dispersion region in standard single mode fibers (SMF). Thulium (Tm) emits around 1900 nm to 2000 nm light. A cavity made of SMF and Tm-doped SMF will have net anomalous GVD and allow soliton formation. Despite solitons being able to form from a continuous wave from, adding a saturable absorber (SA) helps pushes the cavity into emitting ultrafast

13 soliton pulses. The SA passes only high intensity light which occurs when many longitudinal cavity modes constructively interfere.

1.2.3 The Saturable Absorber

A saturable absorber is used to a push CW laser into forming soliton pulses.

Saturable absorbers are tools that absorb low intensity light and pass high intensity light.

A common saturable absorber is a semiconductor saturable absorber mirrors (SESAMs).

A SESAM works by taking advantage of an electron’s relaxation time from the conductions to the valence band which is a few 100 picoseconds [2]. The semiconductor will absorb until the conduction band is saturated. If an intense femtosecond pulse encounters the semiconductor, the first section of the pulse will be absorbed, but after saturation the remaining part will pass. Another common saturable absorber is nonlinear polarization evolution (NPE). Unlike the SESAM, NPE has no refresh time and is versatile, only needing a nonlinear material to take place. NPE is an intensity dependent birefringence,

푛̅2 2 ∆푛 = (|퐴 |2 − |퐴 | ) (1.36) 푁퐿 3 푥 푦 where the amount of birefringence is proportional to the difference in orthogonal

2 2 polarization intensities, |퐴푥| and |퐴푦| , and the nonlinear coefficient 푛̅2. While propagation the birefringence brings a phase accumulation which rotates the polarization.

NPE takes place in nonlinear material, which in the mode-locked laser is single

−20 2 mode fiber (SMF). SMF has nonlinear coefficient , 푛̅2 = 2.2 − 3.4 ∗ 10 m /W. In addition to the SMF, two quarter waveplates (QWP), a half waveplate (HWP) and polarizing beam splitter (PBS) are used to create an NPE saturable absorber as shown in

14

Figure 1.6. Schematic of the nonlinear polarization evolution saturable absorber in figure 1.5. The linear polarizing beam splitter has a cosine squared transmission with respect to the angle of incident polarization, while the waveplates are used to adjust polarization angle and resulting PBS transmission. The polarization rotation high intensity light experiences can create a saturable absorbing effect as shown in the figure

1.6 drawing. At certain orientations of the waveplate, higher intensity light’s polarization will have a smaller incident angle with respect to the PBS polarizing angle and thus experience higher transmission.

Figure 1.7. Elliptical polarization will be incident on the polarizing beam splitter at a certain angle, depending on the waveplate orientation. The drawing shows that light at a higher intensity (red) rotate more the lower intensity light (black). This can result in increased transmission through the beam splitter.

Figure 1.7 shows a simulation of PBS transmission versus wavelength and polarization angle at two different radiant powers. Around 300 W, 1900 nm light transmission is %10, but at 900 W transmission is 75%. The increased intensity rotates

15 the polarization which increases transmission. This effect makes it so low intensity light has more loss compared to high intensity light and thus only high intensity pulses will survive.

Figure 1.8. Plots of PBS transmission versus wavelength, polarization angle, and PBS transmission at 1900nm light shown for (top) 300 W and (bottom) 900 W. Increasing power rotates polarization and increases transmission through the PBS.

16 CHAPTER 2

BUILDING THE THULIUM RING CAVITY

2.1 Laser Cavity Alignment

The Tm-Ho laser schematic is shown in figure #a. 30 m of SMF and 2 m of Tm-

Ho-doped SMF are used and give a total cavity GVD of -20.4. Notice the length of SMF.

Lengthening the cavity increases pulse energy but reduces bandwidth, which allows easier mode-locking.2 A 1550 nm signal is amplified using an EDFA and free-space coupled into the WDM to pump the cavity. The 1550 nm continuous-wave pump excites the Tm-Ho ion and these excited electrons relax and emit 1900 and 2000 nm light—a process called amplified spontaneous emission (ASE). ASE is bidirectional and is used to

Figure 2.1 The output of the forward direction collimator is reflected back into the same collimator. When the reflecting mirror is properly aligned, the backward collimator emits a few micro-joules of pump light which is faintly visible with an IR-card in a dark room.

17 align ring laser cavities into lasing. Moderately pumped ytterbium cavities will have a few milliwatts of ASE and are aligned by eye using an IR card. Unlike an ytterbium- doped fiber, the Tm-Ho-doped fiber does not emit strong ASE. In fact, this ASE is on the order of nanowatts making traditional alignment techniques impossible. The forward collimator emits a few milliwatts of unabsorbed 1550 nm pump light. The problem arises from the backward direction, which emits no discernable power. Figure 2.1 shows the alignment method used to achieve lasing. A beam splitter deflects the pump onto a mirror which reflects the beam back into the forward collimator. The light travels back through the gain, the WDM, and out through the backwards collimator. The mirror is adjusted until the backward collimator emission is maximized, reaching only a few

Figure 2.2 The beam from F is focused into the FTIR spectrometer and the 1550 nm pump is detected. The collimators are adjusted until another line—from the tm-ho gain emission—appears near 1900 nm.

18 micro-watts of pump light. The few micro-watts of pump light show up faintly on an IR- card in a dark room. Now, both collimators emit visible beams and can be coarsely aligned. Part of the mirror’s reflection is sent into a Fourier transform infrared (FTIR) spectrometer with an HgCdTe detector (Figure 2.2). Unaligned, only 1550 nm light is detected. Notice, ASE is not even measurable; the holmium-thulium emission will only be measurable when the ring cavity is in near perfect alignment. A simple iterative process is used to align the cavity into continuous lasing. First, collimator B is adjusted to overlap both beams at the lens of collimator F. Next, collimator F is adjusted to the overlap both beams at the lens of collimator B. During this adjustment, the reflection of F loses alignment and B loses emission. Finally, the mirror is re-adjusted.

The collimators and mirror are adjusted until continuous lasing at 1900/2000 nm appears on the spectrometer. Lasing power is maximized with further minute adjustments, making the cavity more efficient and easier to mode-lock.

2.2 Mode-Locking the Cavity

An isolator, half-wave plate, and two quarter-wave plates are added as a nonlinear-polarization evolution (NPE) saturable absorber (SA) to achieve passive mode- locking. The waveplates are adjusted until only the high intensity pulses pass through the isolator.

19

The cavity will only allow wide bandwidth pulses, where longitudinal modes constructively interfere. Additionally, the pulse evolves into a secant squared shape, a

Figure 2.3 (1) The waveplates are rotated until the oscilloscope displays pulses. (2) The detector is switched to the FTIR and the spectrum recorded. The other mode-locked spectrums can easily be found by slight rotations of the waveplates. soliton with balanced dispersion and nonlinearity. Thus, the cavity emits a high-intensity, wide-bandwidth soliton pulse. The time-bandwidth product for a secant pulse is

휏 ∗ 퐵 = 0.315 (2.1)

휏 = 0.315/퐵. (2.2)

20

Notice, to achieve femtosecond pulses a THz level bandwidth is required. This corresponds to around 4 nm bandwidth for 1900 nm light. Attempting to mode-lock, the spectrometer is typically used as instantaneous feedback for wave-plate adjustment.

However, the FTIR spectrometer has an unusable 2-3 second refresh rate at the necessary

0.1 nm resolution. Instead of the spectrometer, a pulse train oscilloscope is used to give instantaneous feedback. The oscilloscope displays a flat line with continuous lasing and a stable pulse train when mode-locked. The three wave-plates are rotated quickly through many possibilities, effectively channeling the cavity through different modulation depths, until pulses form. Once mode-locked, the spectrum is recorded. At this point, small adjustments in the waveplates will reveal different mode- locked spectrums. Five mode- lock spectrums were observed, shown in figure 2.4.

Figure 2.4. Four soliton spectrums were found. The largest (3) is about 10 nm in width. Notice (2) and (4) have Kelly sidebands.

21

CHAPTER 3

SPECTRUM ANALYSIS

3.1 Spectrum Comparison

The recorded soliton spectrums have interesting features and similarities. First, notice spectrums (2) and (4) appear parabolic with sharp peaks near the edges. These sharp peaks are known as Kelly sidebands discussed in section 3.3. These sidebands are spectral components that broke off from the soliton due to a small waveguide defect.

They will either travel faster or slower than the soliton but must reach cavity resonance with the pulse.

Figure 3.1. Four soliton spectrums were found. The largest (3) is about 10 nm in width. Notice (2) and (4) have Kelly sidebands.

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Secondly, looking at figure 3.2, spectrums (1), (2) and (3) all share similarities in certain wavelength sections. In fact, (1), (2), and (3) all share the same valley just above

1903 nm and (2) and (3) at 1905 nm. This can be explained by air absorption during the free space propagation. Also, (2) and (3) look identical in the center between 1903 and

1905 nm, but (3) is smooth and (2) has sidebands at the edges. The reason for this is that spectrum (2) had only a small perturbation to the saturable absorber. Physical speaking, spectrum (2) was found from spectrum (3) by turning the waveplates a fraction of a degree. This small perturbation caused a part of the soliton energy to break off and form sidebands. In fact, all spectrums were found within the same few degrees and appeared in order of wavelength. That is, if spectrum (1), centered at 1903 nm, was the first to be

Figure 3.2. The soliton spectrum jumps into different states when the waveplates are rotated. Starting in (1) the measured spectrum’s center wavelength increases while transitioning to (2), (3), and (4) with each waveplate rotation. 23 found, slightly rotating the waveplates gave spectrum (2) centered at 1905 nm. Rotating more would give spectrum (3), and the finally (4) centered at 1909 nm.

For comparison, an example of a soliton spectrum is shown in figure 3.3. The laser had 8.4 m of Tm-doped fiber, and 12 m of SMF [21]. The gain is about 7 m longer which allowed a significantly broader spectrum.

Figure 3.3. A soliton spectrum from a cavity with 8.4 m of Tm-doped fiber [21]. The spectrum is broader and has many Kelly sidebands.

3.2 Estimating Pulse Duration

Typically, the pulse is sent through an auto-correlator to measure its duration.

However, for a soliton pulse this is not necessary. Because GVD and SPM cancel phase across the pulse, there is no chirp and the pulse is at the transform limited pulse duration.

Because of this, two alternative methods can be used to estimate the pulse duration. First, the pulse duration of an unchirped pulse can be estimated using the time-bandwidth product. The time-bandwidth product for a secant pulse is

24

휏 ∗ Δ푣 = 0.315 (3.1)

휏 = 0.315/Δ푣 (3.2)

Spectrum 3 has wavelength bandwidth around 8 nm. Using the relationship Δ푣 =

2 푐Δ휆/휆푐 gives a frequency bandwidth of Δ푣 = 0.664 푇퐻푧 for Δ휆퐹푊퐻푀 = 8 푛푚.

Plugging this into the time-bandwidth product gives 휏퐹푊퐻푀 = 0.470 푓푠. The estimated pulse durations for all spectrums, except spectrum 2, are listed in Table 3.1. Spectrum 2 is convoluted with no clear FWHM.

Table 3.1. Time-bandwidth pulse duration estimates

푆푝푒푐푡푟푢푚 1 2 3 4

Δ휆퐹푊퐻푀 2.8 푛푚 ? 5.4 푛푚 3.5 푛푚

휆푐 1901.7 푛푚 1905.0 푛푚 1905.0 푛푚 1909.6 푛푚

휏퐹푊퐻푀 1.4 푝푠 ? 0.706 푝푠 1.09 푝푠

Secondly, the pulse duration was estimated by taking the Fourier transform of the intensity spectrum. Taking the Fourier transform of an intensity spectrum is only possible because the phase across the pulse is zero. The pulses are shown in respective order in the time domain in Figure 3.3 with the pulse width shown in Table 3.2. The supposed shortest pulse, spectrum 3, has a pulse duration of 600 fs. However, spectrum 2 has a shorter estimated pulse duration of 550 fs.

25

Figure 3.4. Taking the Fourier transform of the mode locked spectrum gives the temporal pulse. The FWHM duration is displayed in the upper right-hand corner of each subplot

Table 3.2. Fourier transform compared to time-bandwidth pulse duration estimates

푆푝푒푐푡푟푢푚 1 2 3 4

퐹푇: 휏퐹푊퐻푀 1.55 푝푠 0.55 푝푠 0.6 푝푠 1.15 푝푠

푇퐵: 휏퐹푊퐻푀 1.4 푝푠 ? 0.706 푝푠 1.09 푝푠

휏퐹푇/휏푇퐵 0.875 ? 1.01 0.95

26

3.3 Estimating Pulse Energy

Another feature of a soliton is that its pulse energy is limited by the soliton area theorem,

2|훽 | 퐸 = 2 (1.37) 푝 훾휏 where

2휋 훾 = 푛̅ 퐴 . 휆 2 푒푓푓

푓푠2 For SMF at 1900 nm, the group velocity dispersion is 훽 = −95800 , the nonlinear 2 푚

−20 2 2 coefficient is 푛̅2 = 2.5 ∗ 10 푚 /푊, and the mode field area is 퐴푒푓푓 = 휋(5.25) 휇푚.

Plugging these values into (1.37) gives 퐸푝 = 398 푝퐽.

Additionally, the pulse energy can be calculated from the average cavity power and pulse repetition rate with 푃푎푣푔 = 30 푚푊 ,

퐸 = 푃푎푣푔휏푐푎푣푖푡푦

퐸 = 푃푎푣푔퐿/푐

30푚 퐸 = 푃 푎푣푔 3 ∗ 10−8푚/푠

퐸 = (30푚푊)10−7푠

퐸 = 3 푛퐽

Which is almost a magnitude in difference from the soliton area theorem calculation. This is because most of the average power is unused 1550 nm pump light. A filter could be used to block the pump accurately measure the 1900 nm light, to confirm the soliton area theorem calculation.

27

Interestingly the mode-locking occurred when the pump power was much higher the lasing power. Without waveplates the 1900 nm continuous line is much higher than the pump power. When the waveplates are added and rotated the 1900 nm shrinks and rises. At the same time the 1550 nm shrinks and rises, but lags behind the 1900 nm light because polarization rotation angle through the waveplates are a function of angle. When the pump light has high transmission and the 1900nm continuous line has low transmission is where mode-locking occurs. Additionally, because there is so much pump power left over, more Tm-dope gain fiber gain can be added to the cavity. Larger bandwidth and shorter pulses can be expected when adding more gain fiber, similar to the results shown in figure 3.3.

3.4 Estimating Dispersion from the Kelly Sidebands

The spectral spacing of the Kelly sideband noticeable in spectrums 2 and 4 can be used to estimate the dispersion in the cavity. The derivation for the dispersion and spectral relationship follows [17]. Small geometrical perturbations in the fiber waveguide cause the soliton to shed energy every round trip. The sidebands form when the dispersive light’s propagation constant 훽푑 is in resonance with the soliton pulse propagation

2휋푁 = (훽푝 − 훽푑)퐿. (3.3)

This means the previous dispersive wave meets the soliton as it sheds the next wave.

The dispersive wave propagates similar to the soliton pulse without the nonlinear effects. The propagation constant is given by a Taylor’s series expansion

1 훽 = 훽 + 훽 Δ휔 + 훽 (Δ휔)2 (3.4) 푑 표 1 2 2 while the soliton pulse of width 휏표 is given by

28

1 훽 = 훽 + 훽 Δ휔 − 훽 τ−2. (3.5) 푝 표 1 2 2 o

Plugging the propagation constants into the resonance condition gives

1 1 2휋푁 = (훽 Δ휔 − 훽 τ−2 − (훽 + 훽 Δ휔 + 훽 (Δ휔)2))퐿 (3.6) 1 2 2 o 표 1 2 2

1 2휋푁 = − 훽 ((Δ휔)2 + τ−2)퐿 (3.7) 2 2 o

퐷휆2 훽 = − and Δ휔 = 2휋푐Δ휆/휆2 are inserted and the equation is solved for Δ휆 2 2휋푐 푐

퐷휆2 Δ휆 2 푐 −2 (3.8) 2휋푁 = ((2휋푐 2 ) + τo ) 퐿 4휋푐 휆푐

Δ휆 2 8휋2푐푁 −2 (3.9) (2휋푐 2 ) = 2 − τo 휆푐 퐷휆푐퐿

2푁 휆2 √ 푐 (3.10) Δ휆 = 휆푐 − 2. 푐퐷퐿 (2πc휏표)

Letting the sign of N determine the sign of Δ휆 and relating the full pulse width to the full- width-half-maximum pulse width 휏표 = 휏/2ln (1 + √2) gives

2|푁| 휆2 √ 푐 (3.11) Δ휆 = 푠푔푛(푁)휆푐 − 0.0787 2 . 푐퐷퐿 (c휏표)

The spectral spacing data from spectrum 2 and 4 are plotted in figure 3.5 and the dispersion 퐷 was chosen to best fit the data. Spectrum 2 dispersion is estimated to be

퐷 = 80 푝푠/(푛푚 ∙ 푘푚) while for spectrum 4, 퐷 = 70 푝푠/(푛푚 ∙ 푘푚). These estimates are slightly different and would likely be more similar and precise as we increased the cavity length L. Increasing the length would allow more sidebands to form so that there could be more data to fit a better estimate. Table 3.3 compares these estimated values

29 with common single mode fiber which has 퐷 = 50 푝푠/(푛푚 ∙ 푘푚). One reason the estimates are different could be because of the thulium doped-fiber’s slight differences in numerical aperture NA = 0.16 and core diameter 8.0 µm compared to standard SMF with

NA = 0.14 and core diameter 8.2 µm.

2Δ휆1

2Δ휆2

Figure 3.5. The Kelly sideband were assumed symmetric and spacing from the center was estimated by taking half the spacing between the corresponding bands.

Table 3.3. Estimated dispersion values from Kelly sideband spacing, listed with the common values for SMF, Tm and Ho at 1900 nm

푆푝푒푐푡푟푢푚 2 80 푝푠 퐷 [ ] 푆푝푒푐푡푟푢푚 4 70 푛푚 ∙ 푘푚 푆푀퐹 − 28 50

30

Figure 3.6. The dispersion values are estimated for spectrum 2 and 4. The estimates for the two spectrums are slightly different, and 20-30 ps/nmkm different than SMF at 1900 nm

31

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