Ultrafast Fiber Lasers Enabled by Highly Nonlinear Pulse Evolutions

ULTRAFAST FIBER LASERS ENABLED BY HIGHLY NONLINEAR PULSE EVOLUTIONS

A Dissertation Presented to the Faculty of the Graduate School

of Cornell University in Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy

Cornell University 2019

Ultrafast lasers have had tremendous impact on both science and applications, far beyond what their inventors could have imagined. Commercially-available solid-state lasers can readily generate coherent pulses lasting only a few tens of femtoseconds. The availability of such short pulses, and the huge peak intensities they enable, has allowed scientists and engineers to probe and manip- ulate materials to an unprecedented degree. Nevertheless, the scope of these advances has been curtailed by the complexity, size, and unreliability of such devices. For all the progress that laser science has made, most ultrafast lasers remain bulky, solid-state systems prone to misalignments during heavy use.

The advent of fiber lasers with capabilities approaching that of traditional, solid-state lasers offers one means of solving these problems. Fiber systems can be fully integrated to be alignment-free, while their waveguide structure ensures nearly perfect beam quality. However, these advantages come at a cost: the tight confinement and long interaction lengths make both linear and nonlinear effects significant in shaping pulses. Much research over the past few decades has been devoted to harnessing and managing these effects in the pur- suit of fiber lasers with higher powers, stronger intensities, and shorter pulse durations.

This thesis focuses on less quantitative metrics of fiber laser performance, with an emphasis on furthering the versatility and practicality of ultrafast sources. Much of this work relies on the calculated use of strong fiber nonlinearities, turning conventionally-undesirable phenomena into crucial tools for enabling new capabilities. First, the generation of femtosecond-scale pulses from much slower, more robust sources is investigated, conferring not only reliabil- ity advantages but also a fundamentally greater scope for repetition rate tuning. Next, prospects for fiber lasers operating at wavelengths far from any gain media are explored. By leveraging optical parametric gain alongside chirped- pulse evolutions, energy and bandwidth generated at one wavelength can be efficiently converted to another, while keeping the pulse’s phase and compressibility intact. Both the scaling properties and the underlying theoretical considerations of this approach are discussed. Prospects for realizing optical parametric sources in birefringent step-index fibers are then studied. By using the polarization modes in a telecom-grade fiber to obtain phase-matching, new wavelengths can be generated while eschewing photonic crystal fiber and its inherent practical disadvantages. Finally, more speculative ideas for future work along these themes are discussed. BIOGRAPHICAL SKETCH

Various figures are commonly cited as a measure of the dedicated effort required to become an expert at a given task. Notwithstanding the accuracy or precision of these metrics, Walter has spent an estimated 36,000 hours of his life as a student of some kind–13,000 in grade school (first in his birth city of St. Louis, and later in Philadelphia); another 5,000 as an undergraduate, after returning to St. Louis to earn his B.A. in Physics from Washington University in St. Louis; and now another 18,000 as a graduate student at Cornell University. It is a safe assumption that being a student is now one of his most finely-honed skills. Unfortunately, upon reaching the end of his time at Cornell, Walter found studenthood not to be a particularly employable skill in and of itself. He therefore turned to his secondary area of focus: physics. He had discovered his love of physics as an undergraduate, after spending just enough time in a chemistry classroom to encounter quantum mechanics. This had eventually led him to Frank Wise’s group at Cornell’s School of Applied Physics, having become drawn to optics due to its intersection of complex physics and clearly-visible applications. He had subsequently spent his graduate years immersing himself in laser science and nonlinear wave equations, gradually coming to appreciate the myriad ways a differential equation could be instantiated into a real, physical laser. He has therefore decided to continue designing new types of laser systems at Advanced Optowave Corporation, casting aside his studentship in favor of industrial research. After over 20 years as a student, Walter is now a person.

We shall see how that works out for him.

iii To my family and friends, for supporting me throughout this thesis.

Also to the Wise group ﬂoppy drive: may your whirring never still and your

mechanism never jam.

iv ACKNOWLEDGEMENTS

First and foremost, I would like to thank Frank Wise. I owe much of my growth as a researcher to Frank–both for his patient and insightful mentorship, as well as for the habits he ingrained of always asking where the greatest impact can be had.

Special thanks go out to my committee members, Chris Schaffer and Steve Strogatz. Chris pushed me to expand my views beyond that of a typical laser scientist, both in his science policy class and during our longstanding biomedical collaborations. Steve, too, helped show me a unique appreciation for nonlinear dynamics that will continue to inform how I think about dynamical systems. I consider myself incredibly lucky to have beneﬁted from both of their teaching and guidance. I would also like to thank my colleagues, former and present alike, labmates or otherwise: Erin Lamb, Zhanwei Liu, Yuxing Tang, Logan Wright, Zimu Zhu,

Pavel Sidorenko, Michael Buttolph, Yi-Hao Chen, Hamed Pourbeyram, Michel Olivier, Dylan Heberle, and Noah Flemens, all of whom helped me grow and learn over the course of innumerable stimulating conversations.

In addition to my academic friends and colleagues, I have had the privi- lege of collaborating with a number of industrial scientists, whose technical ex- pertise, loans of equipment, and insight into how lasers can have real impact have been invaluable: Sterling Backus, from KMLabs; Robert Herda and Adam Heiniger, from Toptica Photonics; Timothy McComb, formerly from nLIGHT; and Tyson Lowder, from nLIGHT.

v TABLE OF CONTENTS

Biographical Sketch ...... iii Dedication ...... iv Acknowledgements ...... v Table of Contents ...... vi List of Tables ...... viii List of Figures ...... ix

1 Introduction 1 1.1 Organization of thesis ...... 3 1.2 Nonlinear pulse propagation in optical ﬁbers ...... 5 1.3 Four-wave-mixing in optical ﬁbers ...... 6 1.4 Modelocked oscillators ...... 8 1.5 Gain-switching ...... 10 1.6 Application spaces for ultrafast lasers ...... 11

BIBLIOGRAPHY 14

2 Femtosecond pulses from a gain-switched diode 15 2.1 Introduction ...... 15 2.2 Numerical simulations ...... 17 2.3 Experimental results ...... 20 2.4 Noise behavior ...... 23 2.4.1 Noise measurement ...... 23 2.4.2 Nature of the noise ...... 25 2.5 Repetition rate tunability ...... 27 2.6 Conclusions ...... 29

BIBLIOGRAPHY 30

3 Normal-dispersion ﬁber optical parametric chirped-pulse ampliﬁca- tion 34 3.1 Introduction ...... 34 3.2 CW-seeded FOPCPA ...... 36 3.3 Scaling FOPCPA in the time domain ...... 41 3.4 Conclusions ...... 44

BIBLIOGRAPHY 45

4 Theoretical considerations for FOPCPA 50 4.1 Introduction ...... 50 4.2 Types of FOPCPAs ...... 51 4.3 Theory of CW-seeded FOPCPA ...... 52 4.3.1 Overview ...... 52

vi 4.3.2 Pump evolution ...... 53 4.3.3 Phase dynamics ...... 55 4.3.4 Nonlinear distortions ...... 57 4.3.5 Parametric gain chirp ...... 62 4.4 Theory of continuum-seeded FOPCPA ...... 71 4.4.1 Overview ...... 71 4.4.2 Gain chirp revisited ...... 72 4.5 Conclusions ...... 77

BIBLIOGRAPHY 79

5 Parametric sources without photonic crystal ﬁber 83 5.1 Introduction ...... 83 5.2 Vector ﬁber OPOs ...... 86 5.3 Gain chirp in vector four-wave-mixing ...... 87 5.4 Other pulse propagation considerations ...... 90 5.5 Conclusions ...... 92

BIBLIOGRAPHY 93

6 Future directions 95 6.1 Alternatives to modelocked oscillators ...... 95 6.1.1 Generating pulses without modelocking ...... 95 6.1.2 Compressing pulses from non-modelocked sources . . . . 98 6.1.3 Multi-stage compression ...... 101 6.1.4 Compression using extreme dispersion engineering . . . . 104 6.1.5 Noise reduction ...... 106 6.1.6 Conclusions ...... 107 6.2 Prospects for vector ﬁber optical parametric devices ...... 108 6.2.1 Introduction ...... 108 6.2.2 In-ampliﬁer vector four-wave-mixing ...... 109 6.2.3 Extension to higher-order modes ...... 110 6.2.4 Tunable birefringence ...... 110 6.2.5 Conclusions ...... 112

BIBLIOGRAPHY 114

vii LIST OF TABLES

4.1 Simulation parameters corresponding to Figure 4.8. The gain chirp factor will be introduced later...... 63

viii LIST OF FIGURES

1.1 Cartoons illustrating typical four-wave-mixing gain for pumping in (a) anomalous dispersion and (b) normal dispersion. . . .8

2.1 Simulation results (a) at the output of the GSD, (b-c) before and after the filter, (d-e) after parabolic shaping, (f-g) after amplification, and (h) after dechirping...... 19 2.2 Schematic of the experimental system. Labels correspond to panels in Figure 2.1...... 20 2.3 (a) Spectra (scaled for visibility) and (b) autocorrelations of the pulses at the GSD output, after spectral broadening, and after filtering. Inset: logarithmic scale...... 21 2.4 (a) Spectrum and (b) interferometric autocorrelation of the output pulses (inset: measured intensity autocorrelation and its transform-limited counterpart)...... 23 2.5 Measured RF spectra at the system output using (a) a linear detector and (b) a two-photon detector. Both spectra are centered at 1 MHz, and have a resolution bandwidth of 200 Hz. Corre- sponding noise floors are shown in red...... 24 2.6 RF noise at various harmonics of the 1-MHz fundamental repetition rate, showing a quadratic behavior (solid line: fit) indicative of uncorrelated timing jitter. One point, at 448 MHz, has been omitted from the fit as a clear outlier...... 25 2.7 (a) Dispersive Fourier transform spectral measurements for 100 consecutive pulses. Red: average spectrum measured with an optical spectrum analyzer, for comparison. (b) Histogram of corresponding pulse energies...... 27 2.8 Comparison of the autocorrelations obtained at (a) 1.0 MHz and (b) 0.8 MHz...... 28 2.9 (a) Miniaturized GSD-seeded system with scaled-down performance, and (b) autocorrelations measured at various repetition rates...... 29

3.1 Spectrograms depicting the simulated pulse evolution at the (a) input and (b) output of the photonic crystal ﬁber. The dashed white line in panel (a) marks the CW seed, which is otherwise not visible on this scale. Color maps for each panel are separately normalized to the maximum intensity...... 37 3.2 Schematic of the FOPCPA system and the pulse evolution it utilizes. PBS: polarizing beam splitter. HWP: half-wave plate. PCF: photonic crystal ﬁber...... 39

ix 3.3 Measured FOPCPA output using a 30-ps stretched pump. (a) Full-field spectrum (inset: idler, linear scale). (b) Interferomet- ric autocorrelation of the dechirped idler pulses (black), and the intensity autocorrelation numerically calculated from it (orange). 40 3.4 Measured FOPCPA output using a 150-ps stretched pump. (a) Full-field spectrum (inset: idler, linear scale). (b) Intensity autocorrelation of the dechirped idler pulses (inset: dechirped pump pulses without idler generation)...... 42 3.5 Experimentally observed scaling of the idler energy vs. the stretched pump duration using FOPCPA. Red line: linear fit and extrapolation. Lower inset: close-up of the short-pump region. Upper inset: simulated 1.3-µJ, 140-fs, dechirped idler corresponding to the blue circle (pump duration = 1 ns)...... 44

4.1 Simulated pump power (solid) and frequency (dashed) at the output of an FOPCPA that is either unseeded (blue) or CW- seeded (red)...... 54 4.2 Measured FROG reconstructions of residual pump pulses at the output of a FOPCPA system with and without a seed wave present. (a) Temporal intensity and instantaneous frequency. (b) Spectral intensity. (c) Measured and retrieved FROG traces for the seeded and unseeded conditions...... 55 4.3 Simulated pump, signal, and idler from an FOPCPA using a CW seed and a linearly-chirped pump. (a) Temporal powers, (b) instantaneous frequencies...... 56 4.4 Simulated pump, signal, and idler from an FOPCPA using a CW seed and a frequency-modulated pump. (a) Temporal powers, (b) instantaneous frequencies...... 57 4.5 Illustration of adding a nonlinear chirp (solid red) or an equivalent linear chirp (dashed blue) for a chirped Gaussian pulse (solid black). B = 30π is used for the nonlinearly-chirped case. . . 59 4.6 Simulated idler (a) energy and (b) duration (dechirped and transform-limited) as a function of nonlinear phase pre-applied to the pump...... 60 4.7 Temporal Strehl ratios of the dechirped (green) initial pump and (violet) generated idler pulses as a function of nonlinear phase pre-applied to the pump...... 61 4.8 Scaling CW-seeded FOPCPA simulations in the time domain at constant pump energy. The resulting pump (solid green) and idler (dashed red) pulses are shown, along with the initial pump peak powers and the output idler energies for each case. Note the changing axes...... 64 4.9 Pump and idler pulses at the output of simulated FOPCPAs with different gain chirp factors...... 68

x 4.10 Idler energies and chirped durations obtained from simulated FOPCPAs with different gain chirp factors...... 68 4.11 Simulations of FOPCPAs with the dispersion coefﬁcients given. The CW parametric gain spectra (dashed violet lines) corresponding to the central pump frequencies vary widely in bandwidth, while the resulting pump and idler pulses remain largely unchanged...... 69 4.12 Signal and idler parameters for different gain chirp factors. In each case, the seed chirp is perfectly matched to the gain chirp. . 75 4.13 Signal and idler parameters for different relative seed chirps, αs/αp, and a constant gain chirp factor of ρ = 1. The point where the seed is matched to the gain chirp is shown as a dashed line. . 76 4.14 Signal and idler parameters for different relative seed chirps, αs/αp, and a constant gain chirp factor of ρ = 10. The point where the seed is matched to the gain chirp is shown as a dashed line. . 77

5.1 Free-body diagrams illustrating how phase-matching occurs between (a) co-polarized fields in PCF, and (b) cross-polarized fields in PM fiber...... 85 5.2 Schematic of vector OPO using four-wave-mixing in PM980 fiber. 87 5.3 Experimental output from the vector OPO shown in Figure 5.2. (a) Spectra measured at the two outputs. (b) Measured autocorrelation of the dechirped, 835-nm signal pulse (blue), and calculated autocorrelation of the transform-limited signal (green). . . . 88 5.4 (a) Spectrum and (b) dechirped signal and idler pulses from simulations of the vector OPO shown in Figure 5.2...... 88

6.1 Illustration of the major features of several non-modelocked pulse generation techniques discussed in the text...... 97 6.2 Simulation illustrating a Mamyshev regenerator’s ability to accept a long pulse with a complicated pedestal (solid green), spectrally broaden it (solid blue, and spectrogram), and convert it into a short, pedestal-free pulse (dashed red). Top panel: temporal intensity. Right panel: spectral intensity...... 101 6.3 Performance of various nonlinear compression schemes. Ovals indicate the approximate region over which each technique typically produces transform-limited, pedestal-free pulses...... 102

xi 6.4 Simulated self-similar evolution in a lossless, dispersion- decreasing FBG, including dispersive effects through third- order.(a) Seeded, 100-nJ, 40-ps pulse with a complicated intensity and phase structure (dot-dashed red), and the parabolic output (black). (b) Output pulse after dechirping to near the 5-ps transform limit. (c) Output spectrum. (d) Longitudinal evolution of the pulse spectrum and the FBG stopband. (e) Dispersion- decreasing profile calculated from the varying Bragg wavelength. 106 6.5 Schematic of vector OPO using four-wave-mixing in the pump amplification fiber...... 109 6.6 Illustration of coiling a fiber around (a) a spool with a varying radius to (b) change the fiber’s birefringence and, through it, the peak gain frequency as a function of propagation distance, thereby effecting (c) a longitudinally-shapable parametric gain spectrum...... 113

xii CHAPTER 1 INTRODUCTION

Whether or not we are aware of it, lasers have become ubiquitous in modern life. They transmit our data, scan our groceries, and infuriate our housecats.

The uses of lasers are many, and grow every year. As these application spaces have ballooned, so has the demand for smaller, brighter, or more versatile laser sources. One type of system that remains at the forefront of laser science is ultrafast lasers. Rather than the constant output of a continuous-wave laser, an ultrafast laser emits isolated pulses of light, with each pulse lasting only picoseconds or femtoseconds. The high peak intensities that arise from delivering even a modest amount of energy over such a short time interval enable new modes of interacting with materials, and are the key to many emerging applications.

Ultrafast lasers have conventionally taken the form of solid-state lasers, cap- italizing on the broad gain bandwidths of crystals such as titanium:sapphire. These highly-engineered systems represent the gold standard in short-pulse generation, readily generating pulses as short as a few optical cycles and Watt- level average powers. For all their impressive performance, however, solid-state ultrafast lasers remain impractical for many applications. Their size, cost, and lack of user-friendliness make them poorly suited for widespread uses like industrial materials processing or clinical diagnostics, where a shift in alignment or an interruption of lasing can require signiﬁcant maintenance and cause unac- ceptable levels of downtime. Furthermore, operation at higher powers typically requires cumbersome thermal control measures. As a result, the proliferation of ultrafast lasers remains far below that projected by their potential applications.

In recent years, ﬁber lasers have been explored as a means of addressing

1 these issues. Rather than traveling predominantly through free-space as in a solid-state laser, light in a fiber laser resides largely within the core of an optical fiber. Mature fiber integration techniques can be used to create nearly monolithic systems, eliminating the possibility of misalignments and improving robustness. Furthermore, due to fiber’s waveguide format and high surface-to- volume ratio, fiber lasers can maintain pristine beam quality even under high thermal loads, permitting scaling to kilowatt-scale powers while obviating complex cryostats. Nevertheless, fiber lasers face their own challenges. Ultrashort pulses in a fiber laser experience not only tight spatial confinement in the fiber core, but also long interactions lengths with the waveguiding fiber. Therefore, both linear and nonlinear propagation effects are orders of magnitude more significant in fiber lasers than in their solid-state counterparts, hindering the development of high-performance fiber lasers.

Students of physics are of course aware that light obeys Maxwell’s equations, a set of linear partial differential equations. They quickly grow accustomed to using that linearity to simplify analyses and better understand the behavior of light. At high optical intensities, however, this model breaks down, and effects that depend nonlinearly on the field strength begin to assert themselves. Through these effects, pulses of light can distort themselves, warping their spatial, temporal, and spectral profiles. Improving the performance of fiber lasers requires a thorough understanding of these numerous effects–what they do, how they interact, and how they can be managed. Traditionally, this work has focused on avoidance: using large-mode-area fibers and temporal stretching can reduce nonlinearity to a negligible level, recovering linear pulse propagation. However, as emerging applications demand ever-better performance, this standard approach is reaching its limits. It appears increasingly likely that the

2 next generation of ultrafast lasers will need to transcend this model and in- corporate new methods of tolerating ﬁber optic nonlinearity. As this school of thought is developed, nonlinearity may even come to be seen as an opportunity: as complex as it is, within that complexity lies the potential for qualitatively new capabilities beyond what a linearized model can achieve, if only they can be understood and controlled. This thesis explores several ways in which this might be done, ranging from new ways of generating ultrashort pulses to techniques for extending their wavelengths beyond existing gain media.

1.1 Organization of thesis

This thesis is organized as follows.

Chapter 1 introduces some basic principles and motivations for studying nonlinear ﬁber optics. This includes treatments of how ultrashort pulses are modeled, how they are generated, and how they are used.

Chapter 2 discusses the generation of energetic, femtosecond-scale pulses from a gain-switched diode by way of nonlinear shaping and amplification in fiber. A particular method of producing microjoule-scale, nearly transform- limited femtosecond pulses is described in depth, and the noise properties of the resulting pulses are compared with that of those from typical modelocked fiber systems. The applicability of this approach to compact and robust sources with tunable repetition rates is also explored.

In Chapter 3, ﬁber optical parametric chirped-pulse ampliﬁcation with a continuous-wave seed is investigated as a means of generating femtosecond pulses at new wavelengths. Several such systems operating near the bio-

3 imaging window at 1300 nm are demonstrated, and the scalability of this approach through stretching is veriﬁed numerically and experimentally. Finally, the potential for such systems to generate microjoule-scale, femtosecond pulses at new wavelengths is discussed.

In Chapter 4, the theoretical underpinnings of fiber optical parametric chirped-pulse amplification are studied in more depth. Systems seeded by continuous-wave beams are discussed first, followed by those seeded by broadband, chirped pulses. The phase and gain dynamics in such lasers are analyzed, with an emphasis on how the parametric gain chirp dominates the interactions between the pump and seed pulses. The implications for designing and optimizing fiber optical parametric chirped-pulse amplifiers for a variety of pur- poses are discussed.

Chapter 5 explores the use of vector four-wave-mixing in polarization- maintaining fiber as a means of generating new wavelengths without the use of photonic crystal fiber. The advantages and disadvantages of this approach are enumerated, both from a practical point of view and based on the underlying nonlinear wave physics. Proof of concept is demonstrated in the form of a fiber optical parametric oscillator constructed entirely of standard, step-index fibers which generates ultrashort pulses at 840 nm and 1340 nm. Routes for scaling up and improving upon this result are discussed.

Finally, in Chapter 6, a more speculative assessment of ultrafast sources that lack modelocking is given. Various approaches for generating picosecond-scale pulses from non-modelocked lasers are compared, along with possible techniques for compressing such pulses to femtosecond durations and improving their coherence–potentially to the point where they could replace modelocked

4 systems. Emphasis is placed on the potential impact stemming from the simplicity and robustness that such systems could one day provide. Additional avenues of research for vector optical parametric devices are also explored.

These include alternative formats and architectures, as well as prospects for wavelength-tuning such systems and the new nonlinear dynamics that may result from doing so.

1.2 Nonlinear pulse propagation in optical ﬁbers

Light in an optical ﬁber, just like all other light, obeys Maxwell’s equations, and in particular, the electromagnetic wave equation:

2 2 2 1 ∂ E~ ∂ P~ L ∂ P~ NL ∇2E~ − = µ + µ (1.1) c2 ∂t2 0 ∂t2 0 ∂t2 where E~ is the electric field, and P~ L and P~ NL are the linear and nonlinear parts, respectively, of the induced polarization in the fiber. Starting from this formalism, a standard derivation [1] yields the nonlinear Schrodinger¨ equation (NLSE), a key tool for modeling pulse propagation in fiber:

∂A 1 ∂2A = − iβ + iγ|A|2A (1.2) ∂z 2 2 ∂t2 where A = A(z, t) describes the slowly-varying envelope of the complex electric

ﬁeld as it propagates along the z axis of a ﬁber, and β2 and γ give the strengths of group-velocity dispersion (GVD) and self-phase modulation (SPM), respectively.

The NLSE makes a number of simplifying assumptions. For instance, it assumes a scalar field propagates in a single-mode fiber, where the mode’s spatial profile has been factored out; it assumes the field envelope A varies much more

5 slowly than the optical carrier wave, i.e., the pulse spans many optical cycles; and it assumes many complicating or higher-order effects are negligible. Much of the NLSE’s versatility stems from the lifting of its various assumptions as needed: gain and loss can be readily accounted for, as can higher-order dispersive and nonlinear effects involving one or more polarization or spatial modes. In this manner, the NLSE can be generalized to encapsulate the manifold possi- bilities of optical ﬁber itself.

1.3 Four-wave-mixing in optical ﬁbers

One particular nonlinear process merits special attention here due to its cen- trality to this thesis. Four-wave-mixing is a third-order nonlinear process whereupon two photons are annihilated, and two different photons are created (termed a signal and an idler). The most common type of four-wave-mixing in optical ﬁbers is partially-degenerate four-wave-mixing, where the two original photons come from the same pump wave while the new photons belong to the signal and idler, respectively. This process must satisfy conservation of energy:

2ωp = ωs + ωi (1.3) where ωp, ωs, and ωi are the pump, signal, and idler frequencies, respectively. Between the signal and the idler, the redshifted wave is alternatively referred to as the Stokes wave, while the blueshifted wave is referred to as the anti-Stokes wave.

Four-wave-mixing can occur spontaneously if neither the Stokes nor the anti-Stokes wave is seeded. In this case, the process is seeded by shot noise, and is consequently initially weak. If this process is allowed to continue, a

6 Stokes/anti-Stokes pair is generated, but the resulting waves are incoherent. Coherent four-wave-mixing can be achieved by seeding the signal with another, externally-provided wave. Optical parametric ampliﬁcation can then occur, with the signal being ampliﬁed and an idler wave being newly generated. While equal numbers of signal and idler photons are always generated, the dif- ference in photon energies at different wavelengths means the signal and idler waves will not experience the same increase in power. This asymmetry between the changes in the pump, signal, and idler powers (Pp, Ps, and Pi, respectively) is encapsulated by the Manley-Rowe relations: ! ! ! d Pp d Ps d Pi = −2 = −2 (1.4) dz ωp dz ωs dz ωi

For signiﬁcant conversion from the pump to the signal and idler waves to occur, the four-wave-mixing process must be phase-matched. In the low-power limit, this amounts to satisfying:

∆β ≡ βs + βi − 2βp (1.5)

where βx ≡ β(ωx) = n(ωx)ωx/c is the wavevector at frequency ωx. In general, equation 1.5 is a vector equation; however, in a ﬁber, the waveguide dispersion included in n(ωx) accounts for this, permitting the scalar form of the equation to be used. By explicitly accounting for the nonlinear contribution to phase- matching, the unit power gain g can be written in the small-signal approximation as [1]: ! g2 = −∆β ∆β + 4γP (1.6) where, by Taylor expanding β(ω = ωp + Ω) about some pump frequency ωp,

∞ X β n ∆β = 2 2 Ω2n (1.7) (2n)! n=1

7 From equation 1.6, it can be seen that the peak parametric gain occurs at ∆β(Ω)+ 2γP = 0.

The gain spectrum in equation 1.6 takes different forms depending on the sign of the group-velocity dispersion. In anomalous dispersion, very broadband parametric gain can be achieved, but only in a single spectral region con- tiguous with and localized near the pump wavelength (Fig. 1.1a). In normal dispersion, the individual gain windows become narrower, but separate from the pump wavelength (Fig. 1.1b). Thus, this regime permits the generation of light in brand new spectral regions far from the pump, albeit with more modest bandwidths. In practice, a compromise is often reached by pumping just to the normal side of the zero-dispersion wavelength. While this is at times difﬁcult to precisely achieve, systems that do so can achieve the best of both worlds, exhibiting generously broad bandwidths at wavelengths far from the pump.

(a) (b)

Figure 1.1: Cartoons illustrating typical four-wave-mixing gain for pumping in (a) anomalous dispersion and (b) normal dispersion.

1.4 Modelocked oscillators

A natural environment for encountering and studying nonlinear pulse propagation is a modelocked oscillator. An optical resonator is said to be modelocked

8 when its cavity modes maintain a ﬁxed phase relation over a long time interval. Following from Fourier theory, this corresponds to a periodic train of short pulses in the time domain. The number of participating modes can easily be on the order of 105, spanning multiple THz; as a result, modelocked lasers can generate extremely short (< 100 fs) pulses with high (>kW) peak powers, leading to dispersion, nonlinearity, and their manifold interactions all playing signiﬁcant roles in pulse-shaping. A variety of different modelocking regimes have been discovered, each distinguished by the parameters of the cavity and the way a pulse evolves within it [2, 3]. Despite this diversity, certain features are common to all modelocked lasers. In particular, some mechanism for initiating the pulse from noise (so-called self-starting) is required. This can be achieved actively, by acousto-optically or electro-optically modulating the cavity, or passively, by using an intra-cavity saturable absorber or effective saturable absorber. In the latter case, the saturable absorber preferentially attenuates lower-intensity light, allowing a pulse to effectively modulate itself much more quickly and strongly than is possible using active techniques. Passive modelocking consequently results in the highest-energy, shortest-duration, and lowest-noise oscillators.

In a typical modelocked oscillator, a single pulse continually circulates and evolves through the cavity (i.e., fundamental modelocking). For stable operation, the pulse evolution should be self-consistent: the pulse should be unchanged after each roundtrip through the cavity, modulo overall phase factors. The various dispersive and dissipative processes in the cavity must balance to make this possible, with the precise balance needed depending on the particular pulse evolution used. More thorough reviews of modelocking techniques can be found in [4] and [5]. The perfect self-consistency of a good modelocked state has further implications: a stably-circulating pulse exhibits low intensity, phase,

9 and timing jitter, resulting in a highly coherent output at a consistent repetition rate. In the time domain, this corresponds to a stable, reliable source of short pulses; in the frequency domain, as per the Fourier relations, it corresponds to a comb containing hundreds of thousands of narrow lines, all spaced perfectly uniformly in frequency and maintaining a ﬁxed phase relation with respect to one another. Depending on the application, either or both of these characteristics can be a valuable feature.

1.5 Gain-switching

Another means of short-pulse generation is gain-switching. In this technique, an active medium without feedback is sharply modulated [6]. Quickly and strongly pumping the medium results in a population inversion that quickly exceeds the lasing threshold; provided that the upper-state lifetime is sufﬁciently long, stimulated emission will be minimal, and the inversion can continue to grow. Eventually, stimulated emission will increase, and the built-up inversion will be rapidly depleted by the emission of one or more gain relaxation oscilla- tions. Proper system design can limit the emission to a single, sharp oscillation with a duration on the order of the photon lifetime, which may be much shorter than either the gain lifetime or the modulation speed. For instance, one format of particular interest is a gain-switched diode, which can produce pulses as short as tens of picoseconds. Such systems can be much simpler and more robust than modelocked oscillators, which require a resonant cavity supporting a stable pulse propagation. Furthermore, gain-switched diodes can be electronically triggered, allowing them to operate at either ﬁxed repetition rates or more complicated, even aperiodic, patterns. Nevertheless, gain-switching also has

10 a number of disadvantages compared to modelocking. Perhaps the biggest of these is the incoherent nature of the pulses generated: each pulse is seeded by whatever spontaneous emissions are present when the active medium reaches threshold, and even small changes in the number and phase of these initial photons can lead to noticeable amplitude, duration, or timing jitter in the emitted pulse [7]. The degree to which this noise will prove harmful is highly application-speciﬁc.

1.6 Application spaces for ultrafast lasers

Much of the motivation for studying ultrafast lasers stems from their broad applicability throughout both science and industry. Ultrafast science beneﬁts greatly from the precision offered by modelocked lasers: either in the time domain, where pulses can probe very fast dynamics in a system of interest, or in the frequency domain, where the uniformly-spaced spectral comb enables optical metrology with unprecedented precision.

Industrial applications take advantage of ultrafast lasers in a different way: high-energy (e.g., microjoule- or millijoule-scale) picosecond- or femtosecond- scale pulses impacting a small target can ablate it more quickly than that energy can diffuse away. Operating in the so-called cold ablation regime, a short-pulse laser can chip away at a material without giving it a chance to heat up, preventing thermal effects from reducing the precision of machining. Ultrafast micromachining can furthermore be applied to a wide range of materials, making ultrafast lasers a versatile workhorse for cutting metals, polymers, glass, and sapphire with high throughputs. By making use of multiphoton absorption pro-

11 cesses, pulses with sufﬁciently high peak powers can be used to machine even transparent media.

The precision and efﬁcacy of ultrafast laser machining can be readily trans- lated to surgical applications–most commonly, in safely and reliably making corneal incisions for laser corrective surgery. A wider range of biomedical applications can be found at lower, less destructive, pulse energies. For instance, nonlinear microscopy can be performed with nanojoule-scale pulses. Here, the goal is typically to localize the excitation volume within the three-dimensional focal volume where multiphoton effects are appreciable, enabling high-resolution imaging of in vitro or in vivo biological systems. With an appropriate choice of the excitation wavelength relative to the sample’s scattering and absorption spectra, a focal volume can be intensely illuminated at depths exceeding a millimeter. Such deep-imaging capabilities are particularly valuable in neuroimaging, as much insight can be derived from studying the densely-connected, millimeter-scale cerebral cortex. Emerging techniques such as functional neuroimaging might involve performing an initial, structural scan of the region under study, and subsequently using precisely-placed pulses to monitor the neurons of interest under minimal illumination. With additional work, biomedical techniques using ultrafast lasers may move from research labs to hospitals to form the basis of non-invasive clinical diagnostics.

These and other application spaces, spanning multiple diverse aspects of society, form a compelling argument for the continued research and development of ultrafast lasers. While improving the powers, pulse energies, and durations obtainable from such systems is likely to expand their range of applications, less quantitative improvements are also needed to maximize their utility. For ultra-

12 fast lasers to be used by non-specialists, they must be robust under temperature changes, mechanical perturbations, and long-term use–conditions that might be encountered as they are wheeled about doctors’ offices or operated constantly on factory floors. Improving their flexibility will also be valuable. Optimiz- ing the interaction between a laser pulse and a sample often includes tuning the pulse’s wavelength or arrival rate, both of which are generally configurable only to a low degree in the laser systems of today. For instance, the excitation wavelength can have a dramatic effect on the imaging depth in microscopy systems [8], while fine control over the pulse rate can enable efficient illumination patterns in functional neuroimaging [9] or novel micromachining regimes [10]. Ultrafast sources that can address these issues will be a major focus of this thesis.

13 BIBLIOGRAPHY

[1] Govind P Agrawal. Nonlinear Fiber Optics. Academic Press, New York, 3 edition, 2001.

[2] F.W. Wise, Andy Chong, and W.H. Renninger. High-energy femtosecond ﬁber lasers based on pulse propagation at normal dispersion. Laser & Pho- tonics Review, 2(1-2):58–73, 2008.

[3] Wise Group. Guide to Modelocked Pulse Evolutions.

[4] Erich P Ippen. Principles of passive mode locking. Applied Physics B Laser and Optics, 58(3):159–170, 1994.

[5] Hermann A Haus. Mode-locking of lasers. IEEE Journal of Selected Topics in Quantum Electronics, 6(6):1173–1185, 2000.

[6] K. Y. Lau. Gain switching of semiconductor injection lasers. Applied Physics Letters, 52(4):257–259, 1988.

[7] M. Schell, W. Utz, D. Huhse, J. Kassner,¨ and D. Bimberg. Low jitter single- mode pulse generation by a self-seeded, gain-switched Fabry–Perot´ semiconductor laser. Applied Physics Letters, 65(24):3045–3047, 1994.

[8] Nicholas G. Horton, Ke Wang, Demirhan Kobat, Catharine G Clark, Frank. W. Wise, Chris B Schaefer, and Chris Xu. In vivo three-photon microscopy of subcortical structures within an intact mouse brain. Nature Photonics, 7(3):205–209, 2013.

[9] Bo Li, Mengran Wang, Chunyan Wu, Kriti Charan, and Chris Xu. An adap- tive excitation source for multiphoton imaging. In Conference on Lasers and Electro-Optics, page JTh5C.5, Washington, D.C., 2018. OSA.

[10] Can Kerse, Hamit Kalaycıoglu,˘ Parviz Elahi, Barbaros C¸etin, Denizhan K Kesim, Onder¨ Akc¸aalan, Seydi Yavas¸, Mehmet D. As¸ık, Bulent¨ Oktem,¨ Heinar Hoogland, Ronald Holzwarth, and Fatih Omer¨ Ilday. Ablation- cooled material removal with ultrafast bursts of pulses. Nature, 537(7618):84–88, 2016.

14 CHAPTER 2 FEMTOSECOND PULSES FROM A GAIN-SWITCHED DIODE 1

2.1 Introduction

Fiber lasers are becoming increasingly widespread in scientific and industrial settings. In contrast with their solid-state counterparts, fiber systems benefit from a compact, robust format and diffraction-limited beam quality even at high average powers. However, the majority of their impact has to-date been in long-pulse or continuous-wave applications. While few fiber systems can generate millijoule-scale, ultrafast pulses [2], microjoule-level pulses obtainable by amplifying a modelocked fiber oscillator hold advantages for imaging, precision machining, and ultrafast measurement. Nevertheless, such systems have yet to find widespread adoption outside of research labs. Their limited impact stems from several persistent, practical obstacles. Maintaining stable modelocking in the face of environmental perturbations is an ongoing engineering problem. Separately from this issue, modelocked oscillators are constrained to operate at their fundamental repetition rate. This loss of flexibility can be limiting for applications where pulses must be synchronized with scanning optics or other components, or where both average and peak power need to be optimized in tandem. Key examples include nonlinear microscopy and micromachining. In both cases, tailoring the temporal pulse pattern permits not only greater throughput, but also entirely new imaging and material processing modalities [3, 4].

Gain-switched diodes (GSDs) offer a way to overcome these challenges. As

1Much of this work was presented in Optica [1].

15 integrated optoelectronic devices, they are robust against environmental ﬂuctu- ations. They also can be electronically triggered to produce arbitrary repetition rates or more complicated pulse trains. From an applications standpoint, this is a key advantage over modelocked oscillators. However, GSDs also have a number of disadvantages. Ultrafast processes in passively modelocked laser cavities produce highly coherent pulses with fs-scale durations. By contrast, GSDs form pulses from noise using far slower processes. The pulses therefore exhibit much greater interpulse amplitude and phase noise, as well as higher intrapulse phase noise; consequently, pulses are much longer (10-100 ps) and cannot be directly compressed to the transform limit. As a result, despite offering several unique capabilities, GSDs have thus far had almost no impact in ﬁelds demanding high peak powers and femtosecond-scale pulses.

Much research has focused on making GSDs more attractive as ultrafast sources. Some of the earliest work in this area used soliton compression to obtain pulse durations of hundreds of femtoseconds [5] or even shorter [6], often in conjunction with ampliﬁcation or nonlinear pulse cleaning [7]. How- ever, soliton effects ultimately limited these pulses to picojoule-scale energies.

Bypassing these constraints through chirped-pulse amplification enabled one early, standout result to achieve 2-nJ, 230-fs pulses, but required a specially engineered diode structure [8]. In the past few years, amplification of GSDs to microjoule energies has been demonstrated, but with picosecond-scale durations [9, 10]. Only recently have GSDs been amplified to such high energies at sub-picosecond durations [11]. A key component in this system was a nonlinear spectral-temporal filtering process [12] similar to the well-known Mamyshev regenerator [13]. While this stage was primarily intended to suppress amplified spontaneous emission (ASE), it had the side effect of shortening the pulses by

16 a modest factor. Subsequent ampliﬁcation in the normally-dispersive regime permitted pulse compression to 0.6 ps with peak powers exceeding 1 MW. Al- though this represented a new record for GSDs, the pulses remain too long for many ultrafast applications, and their large deviation from the transform- limited duration (>4x) indicates a lack of control over the pulse. In addition, the system relies on a specially engineered diode, which emits high-quality seed pulses.

Here, we present a scheme for compressing and amplifying pulses from a GSD. Starting with partially coherent, ≈10-ps pulses from a commercially- available GSD, we generate 140-fs pulses with energies of 2.4 µJ. These represent the shortest pulses from a GSD at comparable energies, and the highest peak power by an order of magnitude. We achieve these results using a multi- stage system. We first use a Mamyshev regenerator to shape the pulses from the GSD, at once compressing them and improving their coherence. The clean pulses that we thereby obtain are then amenable to parabolic pre-shaping, followed by amplification and compression to femtosecond durations [14]. By using propagation regimes that effectively leverage nonlinear effects, we are able to compress the amplified pulses to near the transform limit.

2.2 Numerical simulations

To illustrate the nonlinear processes we exploit, we numerically simulate a rep- resentative system. The results are shown in Figure 2.1, corresponding to the lo- cations labeled in Figure 2.2. We start with an experimentally-motivated model of a pulse from a GSD, comprising a 2.5-nJ, 10-ps, coherent, Gaussian core su-

17 perimposed on a 2-nJ, broad, incoherent pedestal (Fig. 2.1a). Various real or artificial saturable absorbers can differentiate between these two pulse components to some degree. However, using a Mamyshev regenerator offers a number of particular advantages, including strong and uniform suppression of the incoherent pedestal and a relatively flat (in the ideal case) transmittance for higher- energy pulses. This stands in contrast to alternatives such as nonlinear optical loop mirrors and nonlinear polarization evolution, which feature oscillatory transfer functions, or semiconductor saturable absorbers, which offer limited modulation. In addition to its uses as a pulse regenerator [13], Mamyshev regeneration has also been demonstrated experimentally as a modelocking mechanism [15, 16, 17, 18] and numerically to increase the coherence of a field [19]. The key advance we report here is to use a Mamyshev regenerator as a simultaneous pulse compressor and coherence discriminator, and to subsequently use these properties to enable further nonlinear pulse shaping.

In our simulated Mamyshev regenerator, the pulse first propagates through 63 meters of passive fiber (Nufern PM980-XP). Self-phase modulation and optical wavebreaking act on the coherent, high-intensity core to generate a series of new spectral components. Meanwhile, the less-intense pedestal spectrally broadens negligibly. Filtering a portion of the newly-created bandwidth therefore isolates part of the coherent component, producing a 0.1-nJ, 3-ps pulse within 40% of its transform-limited duration (Fig. 2.1b-c). Importantly, the pulse is clean and more coherent, making it a good candidate for amplification by parabolic pre-shaping [14]. In this technique, an optimized length of normally-dispersive, passive fiber is used to parabolically shape the pulse [20], which is then amplified in the highly nonlinear regime. The parabolic pulse shape converts nonlinear phase into a linear frequency sweep, permitting com-

300

(a)

Diode

150 Power(W)

-80 0 80

Time (ps)

1 Before Before 160

(b) (c)

After After

Filter

80 Power(W) Intensity (arb.)Intensity

0 0

-40 0 40 1027 1031 1035

Time (ps) W avelength (nm)

1 16

(d) (e)

Shaper

8 Power(W) Intensity (arb.)Intensity

0 0

-20 -10 0 1031 1034

Time (ps) W avelength (nm)

(f) (g)

400

Amp

200 Power(kW) Intensity (arb.)Intensity

0 0

-20 -10 0 1020 1030 1040

Time (ps) W avelength (nm)

Pulse

16 (h)

Output

8 Power(MW)

-8400 -7900 -7400

Time (fs)

Figure 2.1: Simulation results (a) at the output of the GSD, (b-c) before and after the filter, (d-e) after parabolic shaping, (f-g) after amplification, and (h) after dechirping. pression ratios of 30 or more without compromising the pulse quality [14, 21, 1]. Here, we simulate shaping the pulse in 33 meters of passive fiber (Nufern PM980-XP; Fig. 2.1d-e), before amplifying it first in a core-pumped fiber amplifier (1.4 m Nufern PM-YSF-HI) and then in a 2.3-m double-clad fiber amplifier. The latter is modeled after chirally coupled core (3C R ) fiber, which uses a large, 34-µm core diameter to manage high-energy amplification while a helical,

19 Mamyshev regenerator (a) gain-switched passive fiber bandpass diode filter (b-c) passive fiber (h) (f-g) grating parabolic dechirper Yb-doped Yb-doped (d-e) pre-shaper fiber fiber amplifier Figure 2.2: Schematic of the experimental system. Labels correspond to panels in Figure 2.1. secondary core maintains single-mode behavior [22]. While this is an attractive feature, our system is equally compatible with conventional ﬁber ampliﬁers.

The pulse can be amplified to 3 µJ (Fig. 2.1f-g) before distortions from gain narrowing and stimulated Raman scattering become noticeable, and dechirps to 140 fs in a realistic grating compressor (Fig. 2.1h). It is worth remarking that if amplification by parabolic pre-shaping is applied directly to the GSD pulse, the pedestal will remain incompressible, causing only a fraction of the total energy to contribute to the peak power. Furthermore, studies indicate that Raman scattering will prevent a 10-ps pulse from being compressed to significantly less than 300 fs [21]. Thus, the combination of Mamyshev regeneration and amplification by parabolic pre-shaping is critical to obtaining such short, clean pulses from a typical GSD.

2.3 Experimental results

Encouraged by these results, we design an equivalent experimental system along the lines of Figure 2.2. Our starting point is a commercial GSD (OneFive

20 Katana-10 LP), which produces pulses similar to those numerically modeled. The repetition rate is continuously tunable from 50 kHz to 10 MHz, and is set at 1 MHz for most of what follows. The autocorrelation at the diode output re- veals a prominent, 14-ps peak on top of a broad, complicated background. As in simulations, we use a Mamyshev regenerator with a tunable bandpass filter to isolate part of the coherent component. The filter characteristics are experimentally determined, and will be explained later. Figure 2.3 depicts the resulting effects on the pulse. As expected, the Mamyshev regenerator strips away the incoherent pedestal, and we observe primarily a clean, Gaussian autocorrelation. The 0.1-nJ filtered pulses are 3 ps long, within 40% of the transform limit.

1.0

GSD Output

Bef ore Filter

After Filter

0.5

(a) Intensity (arb.)

0.0

1024 1028 1032 1036

W avelength (nm)

1.0

0 GSD Output

Bef ore Filter -10

After Filter

-20

-30 AC Signal (dB) Signal AC

-80 0 80

0.5

Delay (ps)

(b) AC Signal (norm.)

0.0

-100 0 100

Delay (ps)

Figure 2.3: (a) Spectra (scaled for visibility) and (b) autocorrelations of the pulses at the GSD output, after spectral broadening, and after ﬁltering. Inset: logarithmic scale.

Following ampliﬁcation by parabolic pre-shaping and dechirping, we measure the pulses shown in Figure 2.4. After dechirping the 3.2-µJ pulses, we ob-

21 tain a pulse energy of 2.4 µJ. The modulated and greatly-broadened spectrum re- flects the prominence of nonlinearity in the amplifier. Nevertheless, it is evident that our system efficiently converts nonlinear phase to a cleanly-dechirpable, quadratic phase. Autocorrelations (Fig. 2.4b) show a well-compressed pulse with an inferred duration of 140 fs (using a calculated deconvolution factor of 1.35). The intensity autocorrelation (inset) confirms this result, and shows that our pulses are very near the 135-fs transform limit (red trace). Despite the partially coherent nature of the GSD output, the 8:1 ratio of the interferometric autocorrelation indicates a mostly coherent pulse, affirming the role of the Mamy- shev regenerator in promoting coherence. When we launch a small fraction of the dechirped pulses into passive fiber, we observe spectral broadening consistent with simulations where the peak power of the full pulse is taken to be 13

MW. This eliminates the possibility that the pulses contain a significant pedestal or noise burst component. In our system, we find that the pulse energy is limited by stimulated Raman scattering: at higher powers, the rapid growth of the incoherent Stokes wave coincides with a reduction of the signal energy, and noticeable increases in the noise (see section 2.4). Hybrid systems employing GSDs, fiber nonlinearities, and solid-state or thin disk nonlinear amplification might yield significantly higher energies [23, 24].

22 1.0

(a)

0.5 Intensity (norm.)

0.0

1010 1020 1030 1040 1050

W avelength (nm)

Meas.

(b)

T L

4 AC Signal (norm.) Signal AC

-2000 0 2000

Delay (fs)

2 AC Signal (norm.)

-800 -400 0 400 800

Delay (fs)

Figure 2.4: (a) Spectrum and (b) interferometric autocorrelation of the output pulses (inset: measured intensity autocorrelation and its transform-limited counterpart).

2.4 Noise behavior

2.4.1 Noise measurement

One disadvantage of our system is that the output shows significant pulse-to- pulse fluctuations. A GSD generates independent pulses from spontaneous emission, making each pulse slightly different. Subjecting these pulses to cascaded, nonlinear processes magnifies the differences. Careful design of the Mamyshev regenerator can produce a locally flat transfer function, reducing this effect [25]. It is this consideration which guides our choice of the filter characteristics, leading to the filters shown in Figures 2.1(b-c) and 2.3.

We assess the noise quantitatively using RF spectral measurements. Figures

23 2.5(a) and 2.5(b) depict the RF spectra observed using a one-photon detector and a two-photon detector, respectively, each measured at the 1-MHz fundamental with a resolution bandwidth of 200 Hz. In both cases, the spectra consist primarily of the expected comb line and a uniform background. Following the formalism presented by [26], we interpret the latter predominantly as uncorrelated, pulse-to-pulse amplitude noise, which is consistent with theoretical ex- pectations for a gain-switched diode-based source [27]. The relative magnitude of this noise can be derived from the ratio between the peak of the comb line and the uniform background, and is 2% for the one-photon case (corresponding to energy jitter) and 6% for the two-photon case (corresponding to jitter of the two-photon photocurrent, which is proportional to the product of energy and peak power). Our system also displays non-negligible uncorrelated timing jitter

(40 ps rms; Fig. 2.6), as is typical of GSD-based systems and is often exacerbated by Mamyshev regeneration [25].

-40 -40 (a) (b) -60 -60

-80 -80

-100 -100

-120 -120 RF Power (dB) RF Power (dB)

-140 -140

-20 -10 0 10 20 -20 -10 0 10 20 Frequency (kHz) Frequency (kHz)

Figure 2.5: Measured RF spectra at the system output using (a) a linear detector and (b) a two-photon detector. Both spectra are centered at 1 MHz, and have a resolution bandwidth of 200 Hz. Corresponding noise ﬂoors are shown in red.

24 0.02

0.01 Relative RF Background 0.00 0 200 400 600 800 RF Frequency (MHz)

Figure 2.6: RF noise at various harmonics of the 1-MHz fundamental repetition rate, showing a quadratic behavior (solid line: ﬁt) indicative of uncorrelated timing jitter. One point, at 448 MHz, has been omitted from the ﬁt as a clear outlier.

2.4.2 Nature of the noise

Although these pulse-to-pulse ﬂuctuations will be too high for some applications, some discussion of the noise is warranted. Firstly, the timing jitter in a modelocked oscillator differs in character from that in a GSD. In the former, the pulse arrival time is highly correlated between subsequent roundtrips, and the timing jitter manifests as a gradual, yet unbounded, drift over some observation period [28]. Over sufﬁciently long observation periods, the timing jitter can reach picosecond scales in the absence of stabilization techniques [29, 30, 28]. Timing jitter in modelocked oscillators therefore depends on the observation period (or, equivalently, on the range of RF frequencies over which the noise is integrated). By contrast, timing jitter in a gain-switched diode is uncorrelated from shot to shot. Each new pulse in a gain-switched diode forms from spontaneous emission noise, and so the arrival times of any two pulses are independent of one another. Following from the statistics of the initial noise, the

25 arrival times do nonetheless have a statistical distribution. The measured timing jitter represents the width of this distribution, and is independent of the observation time. Over some observation time, the accumulated timing jitter from our source and from a typical modelocked oscillator will become comparable, with the gain-switched diode actually accumulating less timing jitter over even longer observation times. Ultimately, since the underlying natures of the arrival time ﬂuctuations are fundamentally different, direct comparison based on the single metric of timing jitter is insufﬁcient and potentially misleading.

There certainly exist applications for which a 40-ps uncertainty in the arrival time of a 150-fs pulse will be problematic. These applications are those that require carrier-envelope offset stabilization (i.e., most frequency comb applications), those that require synchronization of the pulsed source to some other device with sub-40-ps precision, and those that require interactions between subsequent pulses in a pulse train. Nevertheless, virtually all other mature applications are insensitive to this, including many of the most popular applications of high-power, ultrafast ﬁber lasers. Numerous systems only require synchronization to scanning optics on nanosecond or microsecond time scales. This is the case in most types of machining and multiphoton microscopy, where there is only one beam present. Still other applications use a single, master oscillator, and derive multiple pulse trains from a beam splitter (e.g., multi-wavelength sources or optical parametric ampliﬁers). In these instances, recombination of the same pulse later in the system can render inter-pulse timing jitter a non- issue.

Secondly, the estimated 13-MW peak power represents a pulse ensemble average, making it a valid performance metric for noise-tolerant applications.

26 We furthermore measure the single-pulse spectra using the dispersive Fourier transform technique [31], and conﬁrm that the distributions of the pulses and their energies are not heavy-tailed, nor are the averages strongly skewed by extreme outliers (Fig. 2.7). Of the 100 pulses shown, only two deviate markedly from the mean spectrum, and this outlier status is not reﬂected in their energies. Additional experiments suggest that such outliers originate from the GSD itself, which displays similar statistics (not shown).

W a v e l e n g t h ( n m ) 1 0 5 0 1 0 4 0 1 0 3 0 1 0 2 0 1 0 1 0 1 . 0 ) . b

r ( a ) a (

a 0 . 5 n g i S

0 . 0 0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 T i m e ( p s )

y c n

e a v g . = 2 4 1 5 n J u

q 0 . 2 ( b ) s t . d e v . = 5 4 n J e r F

v 0 . 1 i t a l e

R 0 . 0 2 2 0 0 2 3 0 0 2 4 0 0 2 5 0 0 2 6 0 0 2 7 0 0 P u l s e E n e r g y ( n J )

Figure 2.7: (a) Dispersive Fourier transform spectral measurements for 100 consecutive pulses. Red: average spectrum measured with an optical spectrum analyzer, for comparison. (b) Histogram of corresponding pulse energies.

2.5 Repetition rate tunability

An advantage of a GSD is that it can be electronically triggered at arbitrary repetition rates. Provided the ampliﬁers are adjusted to maintain the same pulse energy in each stage, the nonlinear evolution remains unchanged, and the same

27 level of performance can be maintained. As an example of the ﬁne control afforded by such devices, Figure 2.8 depicts the pulses obtained at 0.8 MHz. Slight differences with the 1 MHz results presented in the main text can be observed, most likely due to the ampliﬁer powers being imperfectly adjusted. However, the main features remain unchanged: the pulse energy at 0.8 MHz is still 2.4-µJ, and the pulses still dechirp to 150 fs (roughly 10% longer than the transform limit).

Additional experiments were using a similar system. By removing the power amplifier and extending the preamplifier, the system was scaled down for improved portability at the expense of reducing the energy to 130 nJ. Us- ing this system, we reconfirmed that adjusting the pump power to maintain the same pulse energy allows the repetition rate to be tuned over a larger range without noticeably affecting the output pulses (Fig. 2.9).

8 (a) 6 1.0 MHz 4

AC Signal (norm.) Signal AC -800 -400 0 400 800 Delay (fs)

8 (b) 6 0.8 MHz 4

AC Signal (norm.) Signal AC -800 -400 0 400 800 Delay (fs)

Figure 2.8: Comparison of the autocorrelations obtained at (a) 1.0 MHz and (b) 0.8 MHz.

28 1 . 0 0 M H z , ) 0 . 8 5 M H z , . ) .

2 0 1 6 5 f s m 2 0 1 7 0 f s r m r o o n ( n

( l

l a a 0 . 9 5 M H z , n 0 . 8 0 M H z , n g i g

i 1 6 4 f s 1 7 2 f s S

S 1 0 1 0

C C A A 0 . 9 0 M H z , 0 . 5 0 M H z , 1 5 6 f s 1 6 7 f s 0 0 - 1 0 0 0 0 1 0 0 0 - 1 0 0 0 0 1 0 0 0 D e l a y ( f s ) D e l a y ( f s ) (a) (b)

Figure 2.9: (a) Miniaturized GSD-seeded system with scaled-down performance, and (b) autocorrelations measured at various repetition rates.

2.6 Conclusions

In conclusion, we have investigated generating energetic, femtosecond-scale pulses without the use of modelocking. In particular, we have demonstrated a system which generates microjoule-scale, 140-fs pulses from a gain-switched diode. We estimate the peak power to be 13 MW, an order of magnitude higher than previous systems based on gain-switched diodes. Attaining this level of performance has historically only been possible by amplifying a modelocked oscillator; by breaking with this trend, our system offers applications an unprecedented level of control over the delivered repetition rate in a highly robust format. While pulse-to-pulse amplitude ﬂuctuations on the order of a few percent remain a concern, additional work may ameliorate this, potentially allowing similar systems in certain ﬁelds to rival or even surpass modelocked oscillator technology.

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33 CHAPTER 3 NORMAL-DISPERSION FIBER OPTICAL PARAMETRIC CHIRPED-PULSE AMPLIFICATION 1

3.1 Introduction

Although ultrafast fiber lasers have made great strides in terms of the energies, durations, and powers they can generate, fiber sources remain conspicuously lacking in wavelength tunability. Available gain media restrict the vast majority of fiber lasers to spectral regions near 1 µm, 1.55 µm, and 1.9 µm. These restrictions can be limiting for popular applications such as nonlinear microscopy, where the laser’s wavelength can be just as important as its power. For instance, it has been shown that in deep-tissue bio-imaging, microjoule-level pulses with wavelengths near 1.3 µm or 1.7 µm can penetrate further into a sample [2], and bio-imaging as a whole already benefits from significant investments into flu- orophores active at specific wavelengths [3]. While work is ongoing to fill the spectral gaps using new fiber dopants [4, 5, 6, 7, 8, 9], these efforts have yet to produce results on par with established ultrafast sources.

Nonlinear wavelength conversion offers a way to leverage existing technology to generate pulses at new wavelengths. Techniques such as the soliton self-frequency shift [10, 11, 12, 13], soliton self-mode conversion [14], and self-phase-modulation-enabled spectral shifting [15, 16] have performed im- pressively in this regard, but their reliance on dechirped pulses limits their scalability due to intensity-related damage thresholds. State-of-the-art solid- state optical parametric systems offer unparalleled performance [17], but suf-

1Much of this work was presented in Optics Letters [1].

34 fer from their complexity, cost, and alignment sensitivity. Fiber optical parametric amplification offers another alternative. Through degenerate four-wave- mixing, two pump photons can be converted to a blueshifted signal photon and a redshifted idler photon, with the magnitude of the shift determined by energy conservation and phase-matching. The latter depends on the dispersive properties of the fiber. Pumping in the anomalous-dispersion regime results in broadband phase-matching, but only over a frequency range localized near the pump wavelength; by contrast, pumping in the weakly-normal regime can make use of higher-order dispersion to phase-match widely-separated spectral bands, making this regime more directly applicable to the generation of brand new frequencies. Fiber optical parametric oscillators (FOPOs) and amplifiers (FOPAs) based on four-wave-mixing have been demonstrated over a variety of formats and wavelengths, but tend to produce pulses that are either energetic but long [18, 19, 20, 21], or short but with lower energy [22, 23, 24, 25, 26]. Thus, the performance of fiber optical parametric systems remains below what is needed by emerging applications.

Fiber optical parametric chirped-pulse amplification (FOPCPA) has been proposed as a means of overcoming this trade-off [27]. The technique com- bines the spectral flexibility of fiber optical parametric amplification with the energy and peak power of chirped-pulse amplification (CPA). Temporally stretching the pulses prior to parametric amplification and subsequently recompress- ing them helps reduce unwanted nonlinear distortions, while the use of larger stretching factors offers tremendous scalability. Furthermore, chirping the pump gives rise to a chirped parametric gain profile where different processes are phase-matched at each point in time, permitting the overall gain spectrum to be very broad [28, 29, 30]. Since FOPCPA was proposed, several groups

35 have experimentally demonstrated FOPCPA systems [28, 31, 32, 33, 34, 29, 30]. However, only recently have such systems have produced femtosecond pulses with microjoule-level energies [35]. Furthermore, all experimental FOPCPA systems to-date have used anomalous-dispersion pumping, and as such, none have generated new wavelengths beyond what is already obtainable using standard doped ﬁbers.

Here, we demonstrate an FOPCPA system pumped, for the ﬁrst time, in the normally-dispersive regime. This approach allows us to partially transfer both the energy and the bandwidth of femtosecond pulses to new wavelengths, which, due to the normal dispersion, may be widely-separated from the pump. The FOPCPA is seeded with a continuous-wave signal beam, eliminating the need for complex seed pulse generation or precise pump-seed synchronization. As a demonstration of the method, we present results of femtosecond pulse generation near the important bio-imaging window at 1.3 µm. Finally, we experimentally and numerically investigate prospects for scaling up the energy and peak power of the generated pulses.

3.2 CW-seeded FOPCPA

The operating principles of our system are illustrated by the spectrograms in

Figure 3.1. At the system’s input (Fig. 3.1a), a broadband, stretched pump pulse is launched with a continuous-wave (CW) seed. As an example, a 400-nJ pump pulse centered at 1.03 µm with a chirped duration of 30 ps and a bandwidth of

≈7 nm is shown, alongside a 40-mW seed at 0.85 µm (marked with a dashed line to aid visibility). We numerically simulate the propagation of these two waves

36 in a photonic crystal ﬁber (PCF) with a zero-dispersion wavelength near 1.04 µm (NKT SC-5.0-1040). The ﬁber length is chosen to be short enough to support the desired pulse evolution (see below), yet long enough to physically handle with ease; in the simulation shown, a length of 6.7 cm is chosen to match experiments. In the PCF, the pump experiences weakly normal dispersion, leading to phase-matching at a signal band near 0.85 µm and an idler band near 1.3

µm. As a result, the simulations show signiﬁcant energy conversion into these two spectral bands, manifesting respectively as a narrowband, ampliﬁed signal component and a broadband, 70-nJ idler pulse (Fig. 3.1b). The latter exhibits a clearly linear chirp, allowing it to be cleanly compressed to its 260-fs transform limit.

1300 1300

(a) (b)

1200 1200

1100 1100

1000 1000 Wavelength (nm) Wavelength (nm) Wavelength

900 900

-40 -20 0 20 40 -40 -20 0 20 40

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Figure 3.1: Spectrograms depicting the simulated pulse evolution at the (a) input and (b) output of the photonic crystal ﬁber. The dashed white line in panel (a) marks the CW seed, which is otherwise not visible on this scale. Color maps for each panel are separately normalized to the maximum intensity.

The nonlinear pulse evolution in the PCF can be easily understood if temporal effects such as group-velocity dispersion (GVD) and group-velocity mismatch (GVM) are neglected. This situation can be engineered by keeping the PCF length well below the dispersion and temporal walk-off lengths of the stretched pulses, which also helps curtail the fabrication-induced, longitudinal

37 ﬁber ﬂuctuations that might otherwise impede wavelength conversion [36, 37]. Under these conditions, each point in time can be thought of independently as a separate, quasi-CW, four-wave-mixing process driven by a different temporal- spectral component of the chirped pump. Because each such process must independently conserve energy, the energy conservation relation between the pump at frequency ωp, the signal at frequency ωs, and the idler at frequency ωi can be written explicitly as a function of time:

ωi(t) = 2ωp(t) − ωs (3.1) where the constant ωs reﬂects the seed’s CW nature. Thus, seeding a CW signal results in a broadband idler that mimics the pump’s phase up to a factor of two. This behavior has been previously observed and studied in ﬁber optical parametric sources [38, 39, 28, 29], and is furthermore evident in our simulated FOPCPA (Fig. 3.1), validating the assumption of negligible GVD and GVM.

This transfer of the pump’s phase to the idler is a crucial feature of CW-seeded FOPCPA: not only does the factor of two in equation 3.1 help buffer the idler against parametric gain narrowing, but the preservation of the pump’s chirp allows the idler to be dechirped as easily as the pump, retaining the advantages of existing CPA technology. Note that while the idler is background-free, the signal retains a pedestal in the form of the unampliﬁed, inter-pulse seed light (not visible on the spectrogram’s linear scale). Our described system bears similarities to that presented in [18]; however, where that system deliberately targeted narrowband pulse generation, ours uses a broadband, highly-chirped pump, resulting in qualitatively different pulse propagation dynamics and a femtosecond-scale output.

We experimentally validate our simulations by constructing a system along the lines of Figure 3.2. A CPA system based on ytterbium-doped ﬁber generates

38 linearly-polarized pump pulses at 1.03 µm with ≈7 nm of bandwidth, stretched in fiber to 30 ps and amplified to up to ≈500 nJ at a repetition rate of 1.2 MHz. A polarizing beam splitter is sandwiched between two half-wave plates, permitting precise control of the incident power and polarization. The CW seed wave is supplied by a linearly-polarized, 0.85-µm diode with a 1-MHz linewidth, and is also equipped with a half-wave plate to align its polarization with that of the pump. Finally, the pump and signal beams are combined at a dichroic mirror and simultaneously coupled into a 6.7-cm piece of the PCF specified above. At the output of the PCF, the beams are collimated, and another dichroic mirror isolates the idler for characterization.

Figure 3.2: Schematic of the FOPCPA system and the pulse evolution it utilizes. PBS: polarizing beam splitter. HWP: half-wave plate. PCF: photonic crystal ﬁber.

With 390 nJ of pump energy and 40 mW of seed power coupled into the PCF, we obtain the results shown in Figure 3.3. The full-ﬁeld spectrum (Fig. 3.3a) shows three distinct waves: the broadband pump at 1.03 µm; the narrowband, ampliﬁed signal at 0.85 µm; and the newly-generated, broadband idler at 1.3 µm. Note that due to chromatic aberrations in coupling to the optical spectrum analyzer, relative intensities are only accurate within each distinct spectral band, and should not be compared quantitatively between different regions.

39 When we isolate the idler using a dichroic mirror, we measure its energy to be 47 nJ, corresponding to 12% power conversion. Accounting for the signal as well, the total pump depletion is therefore 30%. Compressing the idler using a standard grating compressor (Lightsmyth T-1200-1310) yields the autocorrelation in Figure 3.3b. We infer from this a pulse duration of 210 fs, very near the 200-fs transform limit calculated from the spectrum. Coupling a fraction of the dechirped pulses into a length of passive fiber results in spectral broadening in good agreement with simulations, confirming the coherent and transform- limited nature of the generated pulses. The idler spectrum exhibits some structure (Fig. 3.3a, inset), which may be related to the structure or depletion of the pump pulse [40]; this does not appear to significantly affect the dechirped pulse quality.

(a)

0.01

-10

0.00 Intensity (arb.) Intensity

1300 1320

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-40

-50

800 900 1000 1100 1200 1300

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8 (b)

2 AC Signal (norm.) Signal AC

-1200 -600 0 600 1200

Delay (fs)

Figure 3.3: Measured FOPCPA output using a 30-ps stretched pump. (a) Full- ﬁeld spectrum (inset: idler, linear scale). (b) Interferometric autocorrelation of the dechirped idler pulses (black), and the intensity autocorrelation numerically calculated from it (orange).

It is worth emphasizing that, although we refer to our system as an ampliﬁer,

40 the wavelength we aim to generate is not actually the one we seed. While the seeded wave is indeed ampliﬁed, it is of more limited use due to its small bandwidth and its long pedestal of unampliﬁed, interstitial, CW light. Rather, the desired, broadband pulse is generated from scratch as a byproduct of amplifying the ultimately-discarded seed. Not only does the seed’s presence galvanize this process [41], but it also ensures (provided that its coherence time is much longer than the pump duration) that the newly-generated wave is coherent despite it not itself being seeded: every idler photon created is associated with a conjugate signal photon, leading to a shared coherence between the two waves.

This stands in contrast with wholly unseeded systems, in which noisy and incoherent pulses form from amplified vacuum fluctuations. While we have chosen here to seed the 0.85-µm signal and reap the 1.3-µm idler, that choice might be easily reversed without significantly altering the system. In that case, a narrowband wave at 1.3 µm would be obtained alongside a train of broadband, dechirpable, 0.85-µm pulses. Simulations and theoretical considerations alike suggest that such a system would perform comparably to ours.

3.3 Scaling FOPCPA in the time domain

Although FOPCPA is but one means of generating pulses at new wavelengths, it possesses a key advantage in its potential for scalability. Because the pump’s various temporal components evolve largely independently of one another in the PCF, stretching the pump further while maintaining a constant peak power preserves the nonlinear optical dynamics while scaling them in time. The result is an increase in the generated pulse’s energy independently of its bandwidth or phase, limited only by the compressor’s subsequent ability to compensate

41 the additional stretching. One estimate of the ultimate performance of FOPCPA can be reached by considering a supergaussian pump stretched to 1 ns, corresponding roughly to the practical limit of high-performance CPA systems [42].

Numerical simulations of such a pulse reveal that, given adequate management of third-order dispersion from the stretcher and compressor, 1.3-µJ, 140-fs pulses at 1.3 µm might be within reach (see below).

10 (a) 0.01

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0.00 Intensity (arb.) Intensity

1300 1320

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800 900 1000 1100 1200 1300

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1 (b)

0 0.5 AC Signal (norm.) Signal AC

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Figure 3.4: Measured FOPCPA output using a 150-ps stretched pump. (a) Full- ﬁeld spectrum (inset: idler, linear scale). (b) Intensity autocorrelation of the dechirped idler pulses (inset: dechirped pump pulses without idler generation).

As a step towards realizing this scalability, we construct a different 1.03-µm pump source, this one generating up to 4-µJ pulses with a 7-nm bandwidth, which we stretch to 150 ps using a grating stretcher (Lightsmyth T-1000-1040).

Coupling 1.9 µJ into the PCF, we are able to generate idlers with energies up to 180 nJ (Fig. 3.4). The quality of the compressed pulses is noticeably diminished, as evidenced by the 600-fs inferred pulse duration, the low-intensity pedestal, and the appearance of secondary temporal structures; however, the generated

42 idler remains broadband enough to support 220-fs pulses, indicating that it has still inherited a signiﬁcant fraction of the pump’s bandwidth. Furthermore, the loss of pulse quality appears to be a direct consequence of this bandwidth inher- itance: when we dechirp and characterize the pump pulses themselves without idler generation, we observe a comparable duration (680 fs) and similar temporal features (inset of Fig. 3.4b). We therefore believe that the idler’s poor compressibility stems from irregularities of unknown provenance in the pump phase. We expect that using a pump source with a more linear chirp will eliminate this problem, yielding high-energy idlers that can be compressed to their femtosecond-scale transform limit.

In addition to the two experimental results presented here, we have performed four more experiments spanning a range of pump durations. Figure 3.5 summarizes the idler energies obtained using each pump duration, evincing the expected, linear relationship (red line). Furthermore, extrapolating this trend to a 1-ns pump duration–roughly the stretching limit imposed by grating size constraints in ordinary chirped-pulse amplification [42]–shows reasonable agreement with the numerical simulations previously discussed (blue circle and inset). This is a significant extrapolation, and one that should be interpreted with caution. While the observed agreement between experiments and simulations lends credence to it, realizing it may require additional experimental prac- ticalities to be overcome, such as control of third-order dispersion in the pulse stretcher/compressor. Although we expect the peak pump power to remain below the threshold for fiber end-facet damage [43, 44], coreless endcaps may be useful for extending the margin of error. Successfully achieving the predicted limit would bring all-fiber sources into a new performance class, allowing them to fulfill a wide array of new application demands as cheaper, more robust al-

43 ternatives to existing solid-state systems.

1200

5 Power (MW) Power

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Time (fs)

25 400 Idler Energy (nJ) Energy Idler Idler Energy (nJ) Energy Idler

0 15 30

Pump Duration (ps)

0 250 500 750 1000

Pump Duration (ps)

Figure 3.5: Experimentally observed scaling of the idler energy vs. the stretched pump duration using FOPCPA. Red line: linear ﬁt and extrapolation. Lower inset: close-up of the short-pump region. Upper inset: simulated 1.3-µJ, 140-fs, dechirped idler corresponding to the blue circle (pump duration = 1 ns).

3.4 Conclusions

In summary, we have demonstrated FOPCPA pumped in the normally- dispersive regime. Using highly-chirped pump pulses and a CW seed, our system generates femtosecond-scale pulses at wavelengths far from the pump, and can be scaled up to microjoule-level energies by stretching in the time domain. In preliminary experiments, we use a 1.03-µm pump and a 0.85-µm seed to generate 1.3-µm pulses with energies as high as 180 nJ that can be dechirped to a few hundred femtoseconds. Our results support the theoretically-predicted scalability of our approach, indicating a route for wavelength-ﬂexible ﬁber lasers to reach unprecedented performance levels.

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49 CHAPTER 4 THEORETICAL CONSIDERATIONS FOR FOPCPA

4.1 Introduction

As discussed in Chapter 3, fiber optical parametric chirped-pulse amplification systems have the potential to shore up a key weakness of fiber sources. If designed properly, an FOPCPA can generate ultrashort pulses at in-demand wavelengths with performance on par with or exceeding that of solid-state optical parametric sources. These powerful capabilities motivate further research into the dynamics of FOPCPA systems: which factors are most important to FOPC- PAs, and how can they be arranged to maximize the performance and versatility of such sources? Here, we present theoretical results aimed at shedding light on these questions. We consider systems seeded by either narrowband or broadband sources, and explore the dynamics of phase transfer between the pump, signal, and idler waves. A major property of this relationship is encapsulated in the parametric gain chirp factor, which plays a role in determining both the energy and the bandwidth of the generated waves. Finally, we show how properly matching the pump, seed, and fiber parameters can result in energetic, ultrashort pulses being generated without the need for compromise between these metrics—a route to all-fiber systems with unprecedented combinations of pulse properties.

50 4.2 Types of FOPCPAs

We can identify several variants of FOPCPA, distinguished by the relative bandwidths of the pump and seed. In the ﬁrst case, both the pump and the seed are narrowband; this regime, as it does not generally produce ultrashort pulses, falls outside the scope of this work. In a second case, the pump is broadband while the seed is narrowband or CW; for brevity, we will refer to this as the CW- seeded case, and it has been discussed at length in Chapter 3. Finally, the seed might be broadband, while the pump might be either narrowband or broadband; we refer to this approach as continuum-seeded FOPCPA. CW-seeded and continuum-seeded FOPCPA are the two broad approaches for generating ultrashort pulses in this manner, and each comes with its own set of advantages and disadvantages. CW-seeded FOPCPA involves a simpler system due to the lack of a need for a separate continuum-generation stage, and the trivial ease of syn- chronizing the pump pulses with the CW seed. On the other hand, an external laser is required to generate the CW seed, and for reasons that will be elaborated on below, such systems only parametrically generate a single new pulse. By contrast, in a continuum-seeded FOPCPA, not only must a broadband seed be stably generated, but it must also be optically synchronized with the pump pulses. Nevertheless, this can readily be done without resorting to a second laser. This, along with its simultaneous generation of two new colors, may give continuum-seeded FOPCPA a complexity advantage according to some criteria. As will be discussed, the additional degree of freedom provided through the seed can furthermore enable levels of performance beyond the capabilities of CW-seeded FOPCPA.

51 4.3 Theory of CW-seeded FOPCPA

4.3.1 Overview

The pulse dynamics in CW-seeded FOPCPA have been elaborated on in Chap- ter 3. In brief, a broadband, highly-chirped pump pulse interacts with a narrowband, CW seed, which we will take to be the signal. In an appropriately- chosen fiber, the four-wave-mixing interaction between these two waves can be phase-matched. If the pump and seed frequencies are widely-separated, as is often desired for applications, this will require the group-velocity dispersion to be weakly normal, implying only a small group-velocity mismatch between the waves. This assumption of negligible group-velocity mismatch can be re- inforced by using a pump with a moderate (i.e., kW-scale) peak power and a highly-chirped (i.e., > tens of ps) duration, which is readily obtainable using standard amplification techniques. Under these conditions, four-wave-mixing occurs to good approximation independently at each point in time: different temporal/spectral components of the pump drive separate, quasi-CW four- wave-mixing processes seeded by the fixed-wavelength seed. These interactions can be described by the time-resolved energy conservation equation:

2ωp(t) = ωs(t) + ωi(t) (4.1) where the time-dependence of the pump frequency ωp reﬂects the chirped pump, and the CW nature of the seed will eliminate the time-dependence of the signal frequency ωs. This separation of different pulse components can be likened to frequency-domain nonlinear optics [1, 2], where the spatial rather than the temporal domain is used to maintain the independence of the components.

52 It is clear that for equation 4.1 to hold, the idler frequency ωi must also vary in time in a manner that mirrors ωp up to a factor of two [3, 4]. Thus, this mixing of a broadband pump and a CW signal produces not only a broadband, coherent idler, but one that is chirped in the same way as the pump. Provided that the pump is linearly-chirped (as can often be arranged using existing pulsed amplification techniques), this implies a linearly-chirped idler that can subsequently be spectrally isolated and dechirped to its femtosecond-scale transform limit. The generation of this idler is accompanied by the amplification of the seed; however, because this wave retains both its narrowband nature and an interstitial pedestal of unamplified, CW light, it is typically of limited use. Mod- ulating the seed intensity in sync with the pump pulses may provide a way to circumvent this restriction, allowing CW-seeded FOPCPA systems to generate dual-wavelength, femtosecond-picosecond pulse trains for applications like coherent Raman spectroscopy [5]. In general, though, the CW seed wave acts merely as a facilitator of the desired nonlinear process: it coherently stimulates frequency conversion to the idler, while the pump drives the process and supplies the idler’s energy and bandwidth.

4.3.2 Pump evolution

In general, the analysis of ﬁber optical parametric systems is complicated by the presence of multiple waves and the large number of third-order interaction terms between them. To avoid this, solid-state parametric systems often take advantage of the undepleted pump approximation, which takes the pump to be ﬁxed and permits solution of the reduced-dimensionality system. As will be seen, FOPCPA systems generally need not operate in this regime, and can in fact

53 perform optimally under appreciable pump depletion. Nevertheless, FOPCPA systems can be considerably simplified by the dominance of four-wave-mixing over other propagation phenomena. This is reflected in the effect of the signal and idler on the pump pulse. In simulations, the pump at the output of a typical, seeded FOPCPA (Figure 4.1, red) is virtually identical to that when the FOPCPA is unseeded, in which case no signal/idler pair is generated (Fig- ure 4.1, blue). Despite the significant pump depletion, propagation through the FOPCPA can be treated nearly perturbatively, with the signal and idler serving only to deplete the pump intensity quasi-independently at each point in time.

U n s e e d e d 1 . 0 S e e d e d 1 . 5 ) z ) . H U T . (

A ( y

c r 0 . 0 n e 0 . 5 e w u o q P e r F - 1 . 5 0 . 0 - 1 0 - 5 0 5 1 0 T i m e ( p s )

Figure 4.1: Simulated pump power (solid) and frequency (dashed) at the output of an FOPCPA that is either unseeded (blue) or CW-seeded (red).

Experiments conﬁrm these ﬁndings. In Figure 4.2, FROG is used to compare the pump pulse at the output of an FOPCPA when the seed is present to that when the seed is blocked. By and large, introducing the seed and thereby allowing the signal/idler to grow only carves a noticeable hole out of the pump’s spectral and temporal center, leaving it otherwise unaffected. This lack of a mu- tual interaction is underlaid by the rapid, exponential growth of the signal and idler waves—potentially exceeding 50 dB—and their resultingly short nonlinear interaction lengths.

54 F U n s e e d e d 1 . 0 U n s e e d e d 1 . 5 ) 1 . 0 r . e ) S e e d e d . S e e d e d U q . U u . A e ( A

n ( y

c t r

0 . 0 i y e

0 . 5 s 0 . 5

( n w T e o t H P n z I ( a ) - 1 . 5 ) ( b ) 0 . 0 0 . 0 - 1 0 - 5 0 5 1 0 1 0 2 5 1 0 3 0 1 0 3 5 1 0 4 0 T i m e ( p s ) W a v e l e n g t h ( n m ) ( c ) U n s e e d e d U n s e e d e d y M e a s u r e d y R e t r i e v e d c c n n e e u u q q e e r r F F

T i m e T i m e S e e d e d S e e d e d

y M e a s u r e d y R e t r i e v e d c c n n e e u u q q e e r r F F

T i m e T i m e Figure 4.2: Measured FROG reconstructions of residual pump pulses at the output of a FOPCPA system with and without a seed wave present. (a) Temporal intensity and instantaneous frequency. (b) Spectral intensity. (c) Measured and retrieved FROG traces for the seeded and unseeded conditions.

4.3.3 Phase dynamics

A well-known feature of solid-state optical parametric ampliﬁers is the transfer of the pump’s phase onto the idler, while leaving the signal phase largely undis- torted [6]. A similar phenomenon occurs in FOPCPA systems, with the phase dynamics being primarily dictated by the energy conservation equation (Eqn. 4.1). Figure 4.3 illustrates the quantitative accuracy of this effect. The intensity proﬁles of the pump, signal, and idler are shown in panel (a), while panel (b) shows the instantaneous frequencies in the time domain. With the signal and idler frequencies properly offset and the pump frequency doubled, the trivial behavior predicted by equation 4.1 is evident.

55 1 0 ( a ) P u m p S i g n a l

) I d l e r W k (

e 5 w o P

0 - 1 0 - 5 0 5 1 0 T i m e ( p s )

P u m p ( 2 x ) )

z 2 S i g n a l ( - 6 2 . 3 T H z ) H I d l e r ( + 6 2 . 3 T H z ) T (

y c

n 0 e u q e r

F - 2 ( b )

- 1 0 - 5 0 5 1 0 T i m e ( p s )

Figure 4.3: Simulated pump, signal, and idler from an FOPCPA using a CW seed and a linearly-chirped pump. (a) Temporal powers, (b) instantaneous frequencies.

While the transfer of a linear pump chirp is a natural means of generating cleanly compressible pulses at new wavelengths, the direct mapping of the pump phase onto the idler is not limited to linear chirps. As an example, using a supergaussian pump pulse with a sinusoidally-modulated linear chirp to pump an FOPCPA results in an idler with a similarly-modulated chirp (Figure 4.4). Provided the pump profile remains relatively uniform, the idler’s mirror- ing of the pump phase may be of use in nonlinear amplifiers where preserving some self-phase modulation plays a role in improving dechirping [7, 8, 9], or in advanced applications that can benefit from pulse-shaping at disparate wavelengths [10].

56 P u m p S i g n a l ( a ) 1 0 ) I d l e r W k (

r e

w 5 o P

0 - 4 0 - 2 0 0 2 0 4 0 T i m e ( p s )

P u m p ( 2 x ) )

z S i g n a l ( - 6 2 . 2 T H z ) 2 H I d l e r ( + 6 2 . 2 T H z ) T (

y c

n 0 e u q e r - 2 F ( b )

- 4 0 - 2 0 0 2 0 4 0 T i m e ( p s )

Figure 4.4: Simulated pump, signal, and idler from an FOPCPA using a CW seed and a frequency-modulated pump. (a) Temporal powers, (b) instantaneous frequencies.

4.3.4 Nonlinear distortions

Our discussion of the pump phase transferring onto the idler has thus far largely assumed a supergaussian-like pump pulse, where the temporal intensity proﬁle is relatively ﬂat. Under these conditions, the FOPCPA behavior is largely driven by the phase dynamics, with intensity-induced, temporal gain-shaping effects being minimal. While this model has proven a useful aid for understanding these underlying phase dynamics, it is perhaps of greater practical relevance to consider a Gaussian-like pump pulse. In particular, we explore the dynamics of a Gaussian pump pulse that has been nonlinearly chirped by self-phase modulation prior to the FOPCPA. This is a scenario which might easily come about in the straightforward case of an FOPCPA pumped by a CPA system, where

57 self-phase modulation in the pump amplifier is often a major limitation. More generally, high pump powers can be useful in optimizing the performance of an FOPCPA system (see section 4.3.5); furthermore, even when this is not the case, pump MOPAs with smaller core sizes may couple more efficiently to the small core of a PCF at the expense of incurring nonlinear effects during amplification. Here, we investigate the effect of a nonlinearly-chirped pump on a CW-seeded

FOPCPA system, and ﬁnd that the generated idler is surprisingly robust to even large amounts of pump nonlinearity.

It is well-known that self-phase modulation produces a nonlinear phase that is proportional to the temporal intensity of a pulse; thus, the chirp acquired by a Gaussian-like pulse will be nonlinear as the whole, yet locally linear near its temporal center. A potential concern is that, because CW-FOPCPA can map the pump’s chirp onto that of the generated wave, the output of a CW-FOPCPA might feature a similarly nonlinear chirp, making the idler difﬁcult to dechirp.

To assess the extent of these distortions, we simulate a series of FOPCPAs with various pumping conditions. All of the pump pulses are Gaussian with the same energy and duration. In each different simulation, the linearly-chirped pump is additionally pre-conditioned with some amount of self-phase modulation, with the peak nonlinear phase shift characterized by B. Doing so imposes a nonlinear chirp on the pump, increasing its bandwidth in the process; to isolate the effect of the nonlinear chirp from this change in bandwidth, we repeat this battery of simulations with equivalent, linearly-chirped pump pulses, where the magnitudes of the linear chirps are chosen to locally match those of the nonlinear chirps near t = 0. An example of this, for B = 30π, is shown in Figure 4.5.

58 N o n l i n e a r C h i r p L i n e a r E q u i v a l e n t

1 2 6 ) z H ) T ( W

k y ( c

n y t i 0 e s u n 6 q e e t r n F I

. t s n - 6 I

0 - 2 5 0 2 5 T i m e ( p s )

Figure 4.5: Illustration of adding a nonlinear chirp (solid red) or an equivalent linear chirp (dashed blue) for a chirped Gaussian pulse (solid black). B = 30π is used for the nonlinearly-chirped case.

At the output of these simulations, we characterize the generated idlers by their dechirped durations (as calculated assuming an ideal compressor with no third-order dispersion) and their transform-limited durations (Figure 4.6). We

ﬁnd that increasing the pump’s chirp either linearly or nonlinearly increases the idler’s bandwidth while reducing the converted energy. This can be understood intuitively: with a larger pump bandwidth and a reasonably wide phase- matching bandwidth, a greater number of spectral components will convert to some degree, yet a smaller fraction of the pump’s energy will correspond to those components that experience maximally efﬁcient conversion. More strik- ing are the scant differences between the results using the nonlinearly- and linearly-chirped pumps. The corresponding idlers’ energies and transform- limited durations are almost indistinguishable, with the only notable sign of the pump’s nonlinear chirp being an increase in the dechirped idler duration of no more than 11%.

We further illustrate this effect by conﬁning our attention to the case of the

59 2 8 P u m p C h i r p : 2 2 0 ( a ) ( b ) P u m p C h i r p : N o n l i n e a r N o n l i n e a r ( d e c h i r p e d ) L i n e a r 2 0 0 N o n l i n e a r ( t r a n s f o r m l i m i t ) )

2 4 ) J

s L i n e a r ( d e c h i r p e d ) f n ( (

L i n e a r ( t r a n s f o r m l i m i t ) n y 1 8 0 o g i t r 2 0 a e r n u

E 1 6 0 D

r r e e l 1 6 l d d I I 1 4 0

1 2 1 2 0

0 2 0 4 0 6 0 8 0 0 2 0 4 0 6 0 8 0 N o n l i n e a r P h a s e , B ( / π) N o n l i n e a r P h a s e , B ( / π) (a) (b)

Figure 4.6: Simulated idler (a) energy and (b) duration (dechirped and transform-limited) as a function of nonlinear phase pre-applied to the pump. nonlinearly-chirped pump. Figure 4.7 depicts the temporal Strehl ratios (de-

fined as the ratio of the dechirped peak power to the transform-limited peak power) of the initial pumps and the generated idlers. As expected, imposing a nonlinear chirp where B exceeds π causes the Strehl ratio of the initial pump to degrade sharply, reflecting the anticipated loss of compressibility. In spite of this, the idler’s Strehl ratio remains high, and only ever drops as far as 85%. The idler remains highly compressible even up to B ≈ 90π; that this is an unre- alistically high number far beyond the Raman threshold in a typical amplifier only serves to underscore the degree to which FOPCPA preserves the idler’s compressibility even in the face of a Gaussian pump strongly modulated by nonlinearity.

This pulse-cleaning effect is a direct consequence of the temporally peaked pump pulse, and can be explained by a simple, analytical model. Previously, we showed that a temporally ﬂat pump will cause the pump’s phase to map onto that of the idler. While that transfer is in principle still in effect here, it is largely masked by the differential gain from the pump’s temporal proﬁle. In this case,

60 1 . 0

0 . 8 o i t

a 0 . 6 R

l h e r

t 0 . 4 S

0 . 2 I d l e r P u m p 0 . 0 0 2 0 4 0 6 0 8 0 N o n l i n e a r P h a s e , B ( / π)

Figure 4.7: Temporal Strehl ratios of the dechirped (green) initial pump and (violet) generated idler pulses as a function of nonlinear phase pre-applied to the pump. the pump power is given by:

2 2 Pp(t) = P0 exp (−t /τ ) (4.2)

Assuming a baseline linear chirp α, and following a nonlinear pre-chirp, the instantaneous frequency and its derivative are, respectively:

2 2 2 ωp(t) = αt + (2Bt/τ ) exp (−t /τ ) (4.3)

2 2 dωp 2B 2t 2B 3t = α + 1 − exp (−t2/τ2) ≈ α + 1 − (t τ) (4.4) dt τ2 τ2 τ2 τ2 √ Deviations from a linear chirp become apparent near t ≈ τ/ 3, at which point

Pp ≈ P0 exp (−1/3) ≈ 0.72P0. In the undepleted pump approximation, the exponential gain coefﬁcient is proportional to Pp; therefore, considering that the peak gain in CW-FOPCPA systems can be on the order of 60 dB, the local con- √ version at t ≈ τ/ 3 can be suppressed by over 17 dB relative to the peak. While a more complete analysis requires accounting for pump depletion, the conclusion remains largely unchanged: strong gain shaping in the time domain

61 means the nonlinearly chirped regions of an SPM-pre-chirped pump never experience strong conversion, resulting in an effective pulse cleaning of the idler. Thus, even pump pulses that are too severely nonlinearly distorted to be directly useful in applications can drive FOPCPA systems that generate cleanly- compressible, ultrashort pulses.

4.3.5 Parametric gain chirp

Much of the interest in FOPCPA stems from its potential for scalability, unique among ﬁber wavelength conversion methods. This follows directly from the independence of the various spectral/temporal pulse components previously discussed. Once an FOPCPA system has been successfully designed with a given pump pulse, its performance can be improved by stretching the pump pulses further before amplifying them to the original peak power, as demonstrated in Chapter 3.

Having discussed the task of scaling in the time domain at constant pump peak power, it is natural to wonder whether FOPCPA systems might be similarly scalable in time at constant pump energy. One issue immediately presents itself: when pumping near the zero-dispersion wavelength, the nonlinear contribution to phase-matching will be non-negligible, leading to the gain spectrum shifting with the pump power. That said, regimes can be found where this effect is minimal (e.g., when pumping an appreciable distance on the normally- dispersive side of the zero-dispersion wavelength), or where a tunable seed compensates for this effect. It is therefore of interest to consider the pulse dynamics of constant-energy scaling in the absence of this shift. As we will see,

62 this line of inquiry will ultimately help clarify an important aspect of optimizing FOPCPA systems in general.

We begin by simulating a CW-seeded FOPCPA system with varying pump parameters. The pump is centered at 1030 nm, while the seed is placed at 950 nm. The pump’s energy and transform-limited duration are held constant at 20 µJ and 140 fs, respectively, while its linearly-chirped duration and peak power are allowed to vary. A high-order supergaussian pump shape is used to ensure that XPM effects are uniform across most of the pulse. The ﬁber length is set to

1 cm for a pump with 75 kW of peak power, and is scaled inversely with the pump power so as to compare like points in the pulse propagation. The PCF has a nonlinear parameter of γ = 11.1/kW/m, a group-velocity dispersion of

200 fs2/mm, and a third-order dispersion of -7475 fs3/mm in all cases, while the fourth-order dispersion is varied to keep the peak gain at 950 nm at different pump powers (see Table 4.1). We emphasize that these parameters are not intended to be realistic; rather, they are chosen so as to isolate and clearly illustrate a facet of the pulse propagation dynamics. In the same vein, Raman scattering and self-steepening are neglected here. The relevance of the illustrated effect to more typical systems will be discussed later.

Pump Pump Fiber Idler Gain β Power Duration Length 4 Energy Chirp (fs4/mm) (kW) (ps) (cm) (nJ) Factor 50 409 1.5 -1.249e5 1790 -0.96 75 273 1.0 -1.368e5 3320 -0.69 100 205 0.75 -1.487e5 4400 -0.48 200 102 0.38 -1.962e5 4910 0.00 400 51.2 0.19 -2.912e5 4790 0.39 1600 12.8 0.047 -8.613e5 4480 0.82

Table 4.1: Simulation parameters corresponding to Figure 4.8. The gain chirp factor will be introduced later.

63 5 0 k W ( a ) P u m p 7 5 k W ( b ) P u m p 1 0 0 k W ( c ) P u m p 5 0 1 7 9 0 n J I d l e r 7 5 3 3 2 0 n J I d l e r 1 0 0 4 4 0 0 n J I d l e r ) ) ) W W W k k k ( ( (

r r r e e e w w w o o o P P P

0 0 0 - 3 0 0 0 3 0 0 - 2 0 0 0 2 0 0 - 1 5 0 0 1 5 0 T i m e ( p s ) T i m e ( p s ) T i m e ( p s ) 2 0 0 k W ( d ) P u m p 4 0 0 k W ( e ) P u m p 1 6 0 0 k W ( f ) P u m p 2 0 0 4 9 1 0 n J I d l e r 4 0 0 4 7 9 0 n J I d l e r 1 6 0 0 4 4 8 0 n J I d l e r ) ) ) W W W k k k ( ( (

r r r e e e w w w o o o P P P

0 0 0 - 7 0 0 7 0 - 3 5 0 3 5 - 1 0 0 1 0 T i m e ( p s ) T i m e ( p s ) T i m e ( p s )

Figure 4.8: Scaling CW-seeded FOPCPA simulations in the time domain at constant pump energy. The resulting pump (solid green) and idler (dashed red) pulses are shown, along with the initial pump peak powers and the output idler energies for each case. Note the changing axes.

The results of these simulations are shown in Figure 4.8. As the pump power increases from 50 kW to 200 kW, the efficiency of the conversion to the idler at 1125 nm initially improves. There are several potential explanations for this, such as the increase of the four-wave-mixing bandwidth with power [11], or the lessening of walk-off effects as the fiber length is scaled accordingly (with the total walk-off decreasing from 2 ps to 0.3 ps over the range of conditions shown). However, none of these hypotheses can explain what happens next: as the power continues to increase beyond 200 kW, the conversion efficiency reaches a maximum and then begins to wane once more. In the process, the locus of conversion shifts towards the trailing edge of the pulse. Clearly, the peak pump power has a non-trivial effect on the underlying pulse dynamics.

To clarify this mechanism, we note that broadband FOPCPA seeded by a CW seed fundamentally differs from typical ampliﬁcation schemes in that the desired wave is not seeded. Rather, a signal is seeded at some other wavelength, and it is the ampliﬁcation of that wave that results in the desired pulse being

64 generated as a byproduct. This is, in a sense, the inverse of the typical broadband ampliﬁcation problem: instead of using a single pump wavelength to amplify an array of different signal wavelengths, efﬁcient CW-seeded FOPCPA requires using many different pump wavelengths to amplify one single, common signal wavelength. The more pump wavelengths that participate in this process, the more broadband the generated byproduct will be, up to an idler bandwidth of twice the pump bandwidth (as per Eqn. 4.1). The effective gain spectrum obtained in the idler region is determined by a sum over the pump frequencies, weighted not by their individual four-wave-mixing gain spectra as one might intuitively expect, but by the overlap of those spectra with the seed wavelength.

We can further our understanding of this effect by considering the parametric gain chirp. Our treatment of this parameter systematically extends that presented in [12]. The CW analysis of four-wave-mixing (section 1.3; [13]) gives the offset between the pump and signal/idler frequencies as a function of the dispersion coefficients as evaluated at the pump frequency. If we allow the pump frequency to vary slightly, not only will the reference point for that offset change, but so will the offset’s magnitude as the dispersion coefficients are reevaluated at the new pump frequency. Thus, varying the pump frequency will in general produce some different, nonlinear sweep in the peak gain frequency. We define this rate of change of the peak parametric gain frequency with respect to the pump frequency as the gain chirp factor. If the gain chirp factor were identically zero, then every spectral component of a broadband pump would produce peak parametric gain at the same signal wavelength, leading to the maximally efficient scenario for CW-seeded FOPCPA. For a CW seed, this is the chirp-matched case. This idea of chirp-matching has been explored in fiber parametric amplifiers [14, 15, 12, 16], optical parametric oscillators [17], and Ra-

65 man fiber amplifiers [18]; here, we formalize it in terms of the pump and fiber parameters for the case of four-wave-mixing to a fixed sideband frequency.

Mathematically, let a fiber have second-, third-, and fourth-order dispersion coefficients β2, β3, and β4, defined at some fixed frequency ω0. Define the pump, signal, and idler frequencies respectively as:

ωp ≡ ω0 + Ωp (4.5)

ωs ≡ ω0 + Ω (4.6)

ωi ≡ 2ωp − ωs = ω0 + 2Ωp − Ω (4.7)

Bearing in mind that the relevant dispersion coefﬁcients for phase-matching will vary as a function of Ωp, the linear wavevector mismatch can be written as:

1 1 ∆β ≡ β + β − 2β = β Ω2 + β Ω4 − 2β Ω − β Ω2 + β Ω3 Ω (4.8) s i p 2 12 4 2 3 3 4 p where, assuming that Ωp Ω, we have neglected terms higher than ﬁrst-order in Ωp. It is well-known that maximal parametric gain occurs at values of Ω that satisfy ∆β + 2γP = 0 [11]; by differentiating this expression with respect to Ωp, we arrive at an expression for the gain chirp factor, ρ:

∂ωs ∂Ω 3β Ω ρ ≡ = = 1 − 3 (4.9) ∂ω ∂Ω 6β + β Ω2 p |ωp=ω0 p Ωp=0 2 4

Alternatively, rewriting β4 in terms of the peak power for a given value of Ω:

β Ω3 ρ 3 = 1 + 2 (4.10) 2β2Ω + 8γP

Given some (linear) pump chirp ∂ωp/∂t = αp, the peak of the parametric gain will follow a chirp ∂ωgain/∂t = ραp in the vicinity of the signal band. Equation 4.10 makes clear the role of the peak power, and the trend observed in Figure 4.8.

As shown in Table 4.1, the gain chirp factor vanishes for a peak power of 200 kW

66 in the simulated system. Hence, this peak power gives the maximally efﬁcient, uniform conversion shown in Figure 4.8(d), while for other values of the peak power, the non-zero gain chirp leads to some pump components contributing imperfectly to the ampliﬁcation process.

An interesting feature of equation 4.10 which has been noted before [12] is the appearance of the third-order dispersion coefﬁcient, which is typically dis- missed as irrelevant to the intensity dynamics in ﬁber optical parametric systems. In the standard, CW analysis of four-wave-mixing, the odd-order dispersion terms cancel out and have no effect on the phase-matching relation; however, when the pump is broadband, the third-order dispersion turns out to control the leading-order term in the gain chirp factor. A useful consequence of this is that the third-order dispersion can be used to vary the gain chirp factor largely independently of the other system parameters. Figure 4.9 depicts the simulated output of several FOPCPA systems which are identical but for the third-order dispersion, which is chosen to obtain gain chirp factors between 0 and 1.8. At ρ = 0, conversion to the idler takes place nearly uniformly over a broad duration (and therefore, due to the linearly-chirped pump, bandwidth).

As ρ increases, different temporal and spectral components experience different degrees of conversion, reducing the chirped duration, bandwidth, and energy of the generated idler. Similar results are observed for ρ < 0, with the time axis simply being reversed. It can be seen in Figure 4.10 that the energy and chirped duration (and consequently, bandwidth) are approximately Lorentzian in |ρ|. Note that for a CW seed, the gain chirp factor has no direct effect on the idler’s chirp; rather, it helps determine the temporal/spectral window over which parametric conversion occurs, with the idler frequency obeying Eqn. 4.1 within this window.

67 ( a ) P u m p ( b ) P u m p I d l e r I d l e r ) 1 0 ρ = 0 . 0 ) 1 0 ρ = 0 . 3 W W k k ( (

r r e e

w 5 w 5 o o P P

0 0 - 5 0 0 5 0 - 5 0 0 5 0 T i m e ( p s ) T i m e ( p s )

( c ) P u m p ( d ) P u m p I d l e r I d l e r ) 1 0 ρ = 0 . 9 ) 1 0 ρ = 1 . 8 W W k k ( (

r r e e

w 5 w 5 o o P P

0 0 - 5 0 0 5 0 - 5 0 0 5 0 T i m e ( p s ) T i m e ( p s )

Figure 4.9: Pump and idler pulses at the output of simulated FOPCPAs with different gain chirp factors.

L o r e n t z i a n f i t L o r e n t z i a n f i t

1 2 0 ) ) s J p n ( 4 0

(

n y o i g t r 8 0 a e r n u E

r r

e 2 0 l e l d I 4 0 d I

- 3 - 2 - 1 0 1 2 3 - 3 - 2 - 1 0 1 2 3 G a i n c h i r p f a c t o r , ρ G a i n c h i r p f a c t o r , ρ

Figure 4.10: Idler energies and chirped durations obtained from simulated FOPCPAs with different gain chirp factors.

The widths of the Lorentzians shown in Figure 4.10–i.e., the sensitivity of a system to the gain chirp factor–depends on the phase-matching bandwidth of any given, quasi-CW component. Provided that the gain chirp is perfectly matched, however, the phase-matching bandwidth is essentially irrelevant. For instance, Figure 4.11 depicts simulations of three different FOPCPA systems, all with ρ = 0, but with β2 and β4 adjusted to vary the phase-matching bandwidth at a constant frequency offset, and β3 adjusted to maintain perfect gain chirp matching. The parametric gain spectrum of the central pump frequency is shown in violet in panels (a.ii), (b.ii), and (c.ii), with a bandwidth ranging from

68 much larger than to much smaller than the idler bandwidth. Nevertheless, the generated idlers are almost identical in energy and bandwidth, evincing the central importance of gain chirp matching in FOPCPA systems.

3 0 8 ( a . i ) P u m p S i g n a l I d l e r ( a . i i ) 1 0 ) ) )

B 0 m d W c ( / k

( B y

t - 3 0 r i d ( e s 4 5 n w n i e

o - 6 0 t a P n I - 9 0 G 0 0 - 4 0 - 2 0 0 2 0 4 0 8 0 0 9 0 0 1 0 0 0 1 1 0 0 1 2 0 0 1 3 0 0 T i m e ( p s ) W a v e l e n g t h ( n m ) 2 3 4 (a) β2 = 1 fs /mm, β3 = −20.5 fs /mm, β4 = −198.8 fs /mm. 3 0 8 ( b . i ) P u m p S i g n a l I d l e r ( b . i i ) 1 0 ) ) )

B 0 m d W c ( / k

( B y

t - 3 0 r i d ( e s 4 5 n w n i e

o - 6 0 t a P n I - 9 0 G 0 0 - 4 0 - 2 0 0 2 0 4 0 8 0 0 9 0 0 1 0 0 0 1 1 0 0 1 2 0 0 1 3 0 0 T i m e ( p s ) W a v e l e n g t h ( n m ) 2 3 4 (b) β2 = 20 fs /mm, β3 = −118.6 fs /mm, β4 = −1719 fs /mm. 3 0 8 ( c . i ) P u m p S i g n a l I d l e r ( c . i i ) 1 0 ) ) )

B 0 m d W c ( / k

( B y

t - 3 0 r i d ( e s 4 5 n w n i e

o - 6 0 t a P n I - 9 0 G 0 0 - 4 0 - 2 0 0 2 0 4 0 8 0 0 9 0 0 1 0 0 0 1 1 0 0 1 2 0 0 1 3 0 0 T i m e ( p s ) W a v e l e n g t h ( n m ) 2 3 4 (c) β2 = 80 fs /mm, β3 = −428.5 fs /mm, β4 = −6520 fs /mm.

Figure 4.11: Simulations of FOPCPAs with the dispersion coefﬁcients given. The CW parametric gain spectra (dashed violet lines) corresponding to the central pump frequencies vary widely in bandwidth, while the resulting pump and idler pulses remain largely unchanged.

Although varying the third-order dispersion is a convenient way of isolating changes in the gain chirp for studying fundamental FOPCPA dynamics, it is rarely a degree of freedom afforded in experiments. In practice, the wave-

69 length and peak power of the pump are much more readily adjustable parameters. While practical or physical considerations may prevent perfect gain chirp matching from being achieved, a proper choice of these parameters can still help reduce the gain chirp mismatch and improve the efficiency of an FOPCPA system. Another feature that may improve performance is the sign asymmetry inherent to equation 4.10: if a specific wavelength is not necessarily needed, choosing to seed the Stokes vs. the anti-Stokes wavelength (i.e., changing the sign of Ω) will flip the sign of the second term in equation 4.10, potentially bring- ing the gain chirp for that process closer to zero.

Another effect of relevance here is the shift in the gain spectrum as the pump power depletes, and as the signal/idler powers become comparable to the remaining pump power. A full analysis that includes this phenomenon is quite complicated, and has been discussed in depth elsewhere [19, 20]. The inﬂu- ence of a varying pump power on four-wave-mixing has also been studied as a source of undesirable structure in FOPCPA systems [21, 22]. Here, we will limit ourselves to the observation that differentiating the phase-matching relation (∆β + 2γP = 0, with ∆β given by equation 4.8) by γP gives:

∂Ω Ω = 2 (4.11) ∂(γP) β2Ω + 4γP the absolute value of which, for β2 > 0, decreases monotonically with γP. Thus, pump depletion has a weaker effect on the gain spectrum when the pump power is large initially. This may explain why, in Figure 4.8, the performance falls off less quickly for high pump powers than for low ones.

70 4.4 Theory of continuum-seeded FOPCPA

4.4.1 Overview

A second class of FOPCPA systems involves replacing the narrowband, CW seed with a broadband, stretched pulse. From a practical point of view, this approach has much to recommend it: in contrast with the CW-seeded case, the system can be entirely driven by a single oscillator, with no second, narrowband source needed. Broadband seed generation can be accomplished through any number of standard fiber techniques, such as the soliton self-frequency shift [23, 24], dispersive wave generation [25], or coherent supercontinuum generation [26, 27], with the FOPCPA nicely compensating for the inherently low- energy nature of these fiber processes. Nevertheless, seeding with a broadband pulse adds its own set of complications. The generated seed must be coherent and highly stable, with low jitter in its energy, timing, and wavelength. The pump and seed pulses must also be both synchronized and roughly matched in duration, a process which can become technically involved when pulses at widely-separated wavelengths are involved. The degree to which these challenges will yield a net advantage over the CW-seeded case will depend on the needs of individual systems and applications. Provided these conditions can be fulfilled, however, continuum-seeded FOPCPA systems can exhibit impressive performance [28, 29], motivating a closer look at the underlying pulse propagation dynamics. Here, we consider continuum-seeded FOPCPA as a logical extension of the CW-seeded case. We find that under optimal conditions, continuum-seeded FOPCPA has the potential to significantly outperform its CW-seeded counterpart, producing pulses with a broad range of colors and

71 bandwidths.

4.4.2 Gain chirp revisited

In section 4.3.5, we investigated how the parametric gain chirp affects the pulse dynamics in a CW-seeded FOPCPA. Here, we extend that analysis to the continuum-seeded case. Recall that when the seed is CW, a gain chirp factor near zero is desirable so that many pump frequencies can collectively amplify the same, ﬁxed wavelength; this allows each pump frequency to contribute to the conversion process and efﬁciently generate an idler component at a different frequency. With a chirped seed, however, this is no longer necessary: even if the gain chirp is far from zero, the seed wavelength can vary in time to track the moving gain peak, maintaining optimal conversion.

A simple, analytical model serves to illustrate the modiﬁed situation. Let the pump and seed pulses both be linearly chirped:

ωp(t) = ωp0 + αpt (4.12)

ωs(t) = ωs0 + αst (4.13)

where we have now lifted the CW assumption of αs = 0. The idler frequency follows accordingly:

ωi(t) = (2ωp0 − ωs0) + (2αp − αs)t ≡ ωi0 + αit (4.14)

Deﬁne the gain chirp factor ρ ≡ ∂ωs/∂ωp as before. The moving gain peak in the region of the seed can then be written:

ωgain(t) = ωgain(0) + ρ · [ωp(t) − ωp0] = ωgain(0) + ραpt (4.15)

72 Matching the seed to the gain chirp implies ωs = ωgain, from which we obtain the seed and idler chirps:

αs = ραp (4.16)

αi = (2 − ρ)αp (4.17)

Since the signal and idler will have comparable durations, αs and αi are furthermore proportional to the two pulses’ bandwidths. Thus, we see that under perfect gain chirp matching, the pump bandwidth is magnified by a factor of ρ during transfer to the signal, and (2 − ρ) during transfer to the idler (less some factor related to the imperfect conversion in the lower-intensity wings of the pump). If both of these magnification factors are far from zero, highly broadband pulses can be generated from even a narrowband pump, provided that a sufficiently broadband seed can be supplied. This stands in contrast with the CW-seeded case, where the idler’s maximum bandwidth is limited to twice that of the pump—not surprising in light of Eqn. 4.17, given that a CW seed is optimally matched by a gain chirp factor of ρ = 0.

To verify this result numerically, we simulate a series of FOPCPAs using a linearly-chirped pump and seed. The pump’s transform limit is 200 fs, and that of the seed is varied while the seed’s chirped duration remains fixed. In each case, we vary the gain chirp factor via the third-order dispersion as in section 4.3.5, while adjusting the seed chirp to match the gain chirp. Propagation through a 3-cm fiber is simulated; this allows the various frequency components to be appreciably amplified, while simplifying the parametric gain dynamics by preventing the 10-kW pump from markedly depleting.

Figure 4.12 summarizes the results of these simulations. As expected, the gain chirp matching means the pump always experiences efﬁcient, uniform con-

73 version, with the energy and chirped signal/idler durations remaining constant (Fig. 4.12a). However, the bandwidths of the signal and idler, plotted as the inverse of their transform-limited durations, follow absolute-value trends centered at ρ = 0 and ρ = 2, respectively. This follows naturally from equations 4.16 and 4.17, both in the functional form and in the particular values of ρ at which the signal or idler bandwidth vanishes regardless of the pump bandwidth. These cases can be viewed as a parametric analogue to coherent Raman scattering with spectral focusing [30, 31]. Note that even when one pulse’s bandwidth goes to zero, the other’s can remain appreciable, offering the possibility of generating synchronized pairs of femtosecond and picosecond pulses.

To gain a fuller picture of these gain dynamics, we consider two cases in greater depth: one with a moderate gain chirp factor (ρ = 1), and another with a large one (ρ = 10). For each of these gain chirp factors, we vary the seed chirp relative to the pump chirp (αs/αp) while holding the pulse durations ﬁxed. Because the stretched seed duration is always much longer than the pump, this is functionally equivalent to ﬁne-tuning the stretching factor of an arbitrarily broadband seed pulse at constant peak power.

The energies and bandwidths of the signal and idler pulses for ρ = 1 are shown in Figure 4.13. The converted energy peaks at a relative seed chirp of 1, indicative of the perfect gain chirp matching at this point. Meanwhile, the generated signal and idler bandwidths decrease to zero at relative seed chirps of 0 and 2, respectively. While the lack of perfect gain chirp matching at these points means equations 4.16 and 4.17 do not strictly hold, the same qualitative behavior applies: a relative seed chirp of 0 means there is no signal bandwidth to begin with (i.e., the CW-seeded case), whereas a relative seed chirp of 2 results

74 3 0 ( a ) S i g n a l

) I d l e r J n (

2 0 y g r e

n 1 0 E

0 - 8 - 4 0 4 8 G a i n C h i r p F a c t o r , ρ )

s 3 5

p ( b ) S i g n a l (

n I d l e r o i t 3 0 a r

u D

d 2 5 e p r i h

C 2 0 - 8 - 4 0 4 8 G a i n C h i r p F a c t o r , ρ ) z

H S i g n a l ( c )

T 4 0 (

I d l e r 1 - ) t i

m i L

2 0 m r o f P u m p s n a

r 0 T

( - 8 - 4 0 4 8 G a i n C h i r p F a c t o r , ρ Figure 4.12: Signal and idler parameters for different gain chirp factors. In each case, the seed chirp is perfectly matched to the gain chirp. in a narrowband idler as per equation 1. Importantly, the two zero-bandwidth points occur near the relative gain chirp that maximizes the converted energy. Thus, for this value of ρ, it is impossible to obtain pulses that are simultaneously energetic and broadband.

A different picture emerges when ρ is increased to 10 (Figure 4.14). The generated bandwidths behave qualitatively the same as before, with the signal and idler becoming narrowband at relative seed chirps of 0 and 2, respectively. In

75 S i g n a l ) S i g n a l 2 0 ( a ) z x = 1 ( b ) I d l e r H 2 0 I d l e r T (

1 ) - J ) t i n (

m i y L g

r 1 0

e 1 0 m r n o E f s n a

x = 1 r T ( 0 0 - 8 - 4 0 4 8 - 8 - 4 0 4 8 R e l a t i v e S e e d C h i r p R e l a t i v e S e e d C h i r p

Figure 4.13: Signal and idler parameters for different relative seed chirps, αs/αp, and a constant gain chirp factor of ρ = 1. The point where the seed is matched to the gain chirp is shown as a dashed line. this case, however, the gain chirp is perfectly matched at a relative seed chirp of 10. As a result, the maximum energy point shifts to a higher relative seed chirp, away from the narrowband regions. The resulting pulses boast both high conversion efﬁciencies and short durations, with simulations producing ∼30-fs signal and idler pulses despite the pump transform limit being only 200 fs. Even larger gain chirp factors serve to enhance this effect by further magnifying the pump bandwidth: provided a seed with the requisite chirp can be generated, ultrashort continuum-seeded FOPCPA sources beneﬁt from |ρ| far from 0 and

2, such that |αs|, |αi| |αp| (Eqns. 4.16-4.17). As discussed in section 4.3.5, this holds even if the phase-matching bandwidth of any given pump component is narrow, as long as the gain-chirp-matching condition is satisfied. This conclusion should be compared to the CW-seeded case, where the lack of freedom in the seed chirp pins the optimal gain chirp factor to ρ = 0 and more modest performance. Once broadband and efficient amplification is achieved, the energy can be scaled as usual by stretching the pump and seed together, taking care to preserve the relative chirp between them.

In principle, each part of the FOPCPA system need only fulﬁll one role well:

76 S i g n a l ) S i g n a l

2 0 z

I d l e r H 4 0 I d l e r x = 1 0 T (

1 ) - J )

( a ) t ( b ) i n (

m i y L g

r 1 0

e 2 0 m r n o E f s n a r

x = 1 0 T ( 0 0 - 1 0 0 1 0 2 0 - 1 0 0 1 0 2 0 R e l a t i v e S e e d C h i r p R e l a t i v e S e e d C h i r p

Figure 4.14: Signal and idler parameters for different relative seed chirps, αs/αp, and a constant gain chirp factor of ρ = 10. The point where the seed is matched to the gain chirp is shown as a dashed line. the pump supplies energy, the seed provides bandwidth, and the fiber properties magnify the pump bandwidth as needed. With only singular demands on each individual component, the potential exists for such systems to be optimized modularly, facilitating their engineering. This division of labor underlies the possibility of continuum-seeded FOPCPA enabling fiber sources with unprecedented performance. Continued research into coherent, broadband seed generation [32, 33] and the design of photonic crystal fibers with large gain chirp factors will be important for taking advantage of this capability.

4.5 Conclusions

In summary, we have investigated a number of properties of FOPCPA systems, primarily via numerical means. We have studied the effect of the pump phase on the generated pulses, as well as how the pump, seed, and ﬁber properties interact. In particular, we have developed expressions describing the parametric gain chirp and its effect on both CW-seeded and continuum-seeded FOPC-

PAs. In the CW-seeded case, the system depends on an inverse gain bandwidth,

77 with a vanishing gain chirp yielding optimal conversion to the idler with up to twice the pump bandwidth. With a continuum seed, on the other hand, the gain chirp factor magniﬁes the pump bandwidth, allowing different combinations of signal and idler bandwidths to be achieved with a given pump. In this case, provided a matched seed can be generated, larger gain chirp factors yield the best conjunction of high energies and broad bandwidths. Taking these properties into account will help enable the design of FOPCPA systems with a wide-ranging set of parameters, each tailor-made for a given application.

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82 CHAPTER 5 PARAMETRIC SOURCES WITHOUT PHOTONIC CRYSTAL FIBER1

5.1 Introduction

Photonic crystal fiber (PCF) has been a key component in almost all fiber OPOs and OPAs to-date. Much of the success of PCF stems from the unmatched design flexibility it allows: light in PCF is guided by a complex microstructure of air and silica, which can be customized to produce drastically different dispersion characteristics. In this way, even the higher-order waveguide dispersion can be tuned during the fabrication process, allowing fibers to be drawn that satisfy phase-matching for a desired four-wave-mixing process. That said, PCF is not without its disadvantages. The air-silica microstructures that give it its attractive properties can be easily damaged or contaminated by improper handling or cleaning, making PCF more difficult to work with. These features also make it difficult to splice PCF: specialized splicing programs are required to achieve losses on the order of ≈ 1 dB [2, 3], and even that amount of loss has been shown to limit device performance [4]. This issue is compounded by the modal mismatch between PCF and most step-index fibers, especially if bridge fibers are not used during splicing. The difficulty in splicing PCF to standard fibers is a marked practical disadvantage, as a key benefit of fiber technology is the prospect of creating fully monolithic systems. Furthermore, the very flexibility that lends PCF its widespread appeal can act against it, as even tiny manufactur- ing imperfections can significantly change important parameters, often within a single fiber draw [5, 6]. Finally, the most useful PCFs for most fiber parametric

1Parts of this work was presented at the Conference on Lasers and Electro-Optics [1].

83 systems invariably have small mode areas. This is a fundamental waveguiding property: making the tailored waveguide dispersion strong enough to overcome the natural, material dispersion requires tight spatial conﬁnement. As a result, dispersion-engineered PCFs feature lower nonlinear and damage thresholds, potentially hindering power-scaling.

An alternate means of phase-matching widely-separated wavelengths relies on the interactions between different modes in a ﬁber. From a practical point of view, the simplest embodiment of this is a polarization-maintaining (PM)

fiber. Such fibers can be single-mode and fully compatible with standard fiber- handling techniques, facilitating adoption and integration. In a PM fiber, strong linear birefringence breaks the polarization mode degeneracy, producing a slow and a fast axis separated by the birefringence, δβ ≈ δn ω/c (with δn generally taken to be dispersionless). The dispersion of each individual mode is typically dominated by group-velocity dispersion, which, if positive, would normally preclude phase-matching. The birefringence offers a crucial means of compensating for this: by pumping various combinations of these two axes, a variety of different processes can be phase-matched [7]. In particular, pumping the slow

ﬁber axis maximizes the frequency offset of phase-matching to a signal and idler wave: s 2δβ + (2/3)γPp Ω = (5.1) β2 where we have assumed β2 and δβ are both positive. Importantly, while the pump is polarized along the slow axis, the signal and idler are both generated in the fast axis. It is this process, with birefringence compensating positive group- velocity dispersion, that we will refer to as vector four-wave-mixing [8, 9]. This interaction of physical processes stands in contrast with phase-matching in PCF, where nonlinearity and slightly positive group-velocity dispersion are counter-

84 balanced by significant, negative, higher-order dispersion [5]. Free-body diagrams qualitatively depicting the phase balances in these two cases are shown in Figure 5.1. Importantly, because phase-matching in the vector case relies on birefringence rather than carefully-tailored higher-order dispersion, it is largely independent of the modal confinement, potentially enabling the use of large- mode-area fibers with very high damage thresholds.

(a) Scalar (PCF): nonlinearity higher-order GVD > 0 dispersion < 0

(b) Vector (PM fiber): nonlinearity birefringence GVD > 0

Figure 5.1: Free-body diagrams illustrating how phase-matching occurs between (a) co-polarized fields in PCF, and (b) cross-polarized fields in PM fiber.

Here, we demonstrate the first ultrafast fiber optical parametric oscillator to make use of vector four-wave-mixing. The system is constructed entirely of standard step-index fibers, allowing it to transcend the challenges associated with PCF. We show that this system is capable of generating nanojoule-scale, femtosecond pulses at 835 nm and 1340 nm, with simulations suggesting performance increases on the horizon. Finally, we present an overview of how pulse propagation in a vector fiber OPO or OPA differs qualitatively from that in a typical, PCF-based device, and the new effects and interactions that must be taken into account.

85 5.2 Vector ﬁber OPOs

As a step towards demonstrating ultrafast vector four-wave-mixing, we build a system as shown in Figure 5.2. A segment of single-mode, telecom-grade, PM

ﬁber (Nufern PM980, L = 20 cm) is used as the parametric gain medium. With a

2 −4 group-velocity dispersion of β2 ≈ 24 fs /mm and a birefringence of δn ≈ 3.6×10 , this standard fiber is suited to generating light near 835 nm and 1340 nm when pumped at 1030 nm. In principle, this fiber could form the basis for a CW- seeded vector OPA similar to the system discussed in Chapter 3; however, as a narrowband diode at one of these sideband wavelengths was not available, a vector OPO configuration was chosen here. The system is synchronously pumped with an ANDi oscillator [10], directly amplified to several hundred nJ in a Yb-doped fiber amplifier with a 25-µm core diameter. These 6-ps chirped pump pulses pass through a dichroic mirror and enter the PM980 polarized along its slow axis. The residual pump light, along with the idler wave at 1340 nm, are subsequently output-coupled through a short-pass filter, while the 835- nm signal is partially output-coupled, then fed back into the cavity through a calculated fiber delay line. Because this fiber is non-PM for reasons of conve- nience, a fiber polarization controller is used to align the feedback with the fast axis of the PM980.

Adjusting the pump power, the feedback polarization, the signal output- coupling ratio, and the optical delay produces the modelocked results shown in Figure 5.3. Signal pulses are 835 nm are produced with 7 nJ of energy, and can be dechirped using a grating pair to 200 fs (near the 180-fs transform limit). While the idler pulses were not directly measured in this experiment, the Manley- Rowe relations can be used to estimate their energy at ≈4 nJ, with enough band-

signal v

idler v outv out residual dichroic

pump out mirror pump MOPA optical () () ANDi delay PM980 oscillator PBS HWP (passive fiber) 25 μm filter HWP YDFA

fiber polarization () () feedback fiber controller (non-PM)

Figure 5.2: Schematic of vector OPO using four-wave-mixing in PM980 fiber. width to support 200-fs durations. These numbers are already on the order of what is routinely achievable using PCF-based fiber OPOs, and may only be on the low side of what is possible: simulations of the experimentally-assessed cavity predict signal and idler energies of ≈20-30 nJ (Figure 5.4). It is left as a goal of future work to explain, and if possible, close this discrepancy. A leading sus- pect is the non-PM feedback fiber, which may be responsible for some amount of signal depolarization. Replacing this with all-PM feedback fiber will be an important step in determining the extent of this problem. Alternatively, the problem of feedback depolarization could be rendered moot by redesigning the system in an OPA configuration, either by acquiring a narrowband, CW, seed diode at 835 nm or 1340 nm or by using the same pump source to synchronously generate a continuum seed extending to one of these wavelengths.

5.3 Gain chirp in vector four-wave-mixing

We can analyze the vector four-wave-mixing process in terms of the parametric gain chirp, introduced in section 4.3.5. The larger value of β2 as compared to the scalar (PCF) case means we can neglect the contribution of the fourth-order

87 2 0 S i g n a l M e a s . ( 2 0 0 f s ) ( a ) ) 1 . 0 ( b ) P u m p + I d l e r . T L ( 1 8 0 f s ) m )

0 r B o d n ( (

l y

t - 2 0 a i s n 0 . 5 n g i e t - 4 0 S

n I C

- 6 0 A 0 . 0 8 0 0 9 0 0 1 0 0 0 1 1 0 0 1 2 0 0 1 3 0 0 1 4 0 0 - 3 0 0 0 - 1 5 0 0 0 1 5 0 0 3 0 0 0 W a v e l e n g t h ( n m ) D e l a y ( f s )

Figure 5.3: Experimental output from the vector OPO shown in Figure 5.2. (a) Spectra measured at the two outputs. (b) Measured autocorrelation of the dechirped, 835-nm signal pulse (blue), and calculated autocorrelation of the transform-limited signal (green).

3 0 ( a ) 1 2 0 ( b ) S i g n a l I d l e r ) 0 ) B W d k (

( 8 0

- 3 0 y r t i e s w n - 6 0 o e t

P 4 0 n I - 9 0

- 1 2 0 0 8 0 0 9 0 0 1 0 0 0 1 1 0 0 1 2 0 0 1 3 0 0 1 4 0 0 - 1 0 0 0 0 1 0 0 0 W a v e l e n g t h ( n m ) T i m e ( f s )

Figure 5.4: (a) Spectrum and (b) dechirped signal and idler pulses from simulations of the vector OPO shown in Figure 5.2. dispersion to phase-matching. The relevant equations become:

1 1 ω0 + Ωp β = β + β Ω + β Ω2 + δn (5.2) p 0 1 p 2 2 p 2 c 1 1 1 ω + Ω β = β + β Ω + β Ω2 + β Ω3 − δn 0 (5.3) s 0 1 2 2 6 3 2 c 1 β = β + β (2Ω − Ω) + β (2Ω − Ω)2 i 0 1 p 2 2 p 1 1 ω0 + 2Ωp − Ω + β (2Ω − Ω)3 − δn (5.4) 6 3 p 2 c

88 Noting the change in the nonlinear term due to the differing strengths of co- and counter-polarized cross-phase modulation:

∆β = βs + βi − 2βp 1 1 1 1 = β Ω2 + β (2Ω − Ω)2 + β Ω3 + β (2Ω − Ω)3 2 2 2 2 p 6 3 6 3 p ω Ωp − β Ω2 − 2δn 0 − 2δn 2 p c c 2 = γP for perfect phase-matching (5.5) 3

Differentiating with respect to Ωp at Ωp = 0, and then using equation 5.1 to eliminate δn:

β Ω2 − 2δn/c ρ = 1 − 3 (5.6) 2β2Ω 1 β Ω 1 Ω 1 γP = 1 − 3 + − (5.7) 2 β2 2 ω0 3 β2ω0Ω

For typical step-index ﬁbers pumped near 1 µm, the second term on the right- hand side of equation 5.7 is of order 1, while the third and especially the fourth terms are much smaller; thus, the gain chirp can be approximated as ρ ≈ 1 −

(β3Ω)/(2β2). Thus, because standard ﬁbers have β3 > 0 at this wavelength, CW- seeding is expected to work better with an anti-Stokes seed (Ω > 0, ρ closer to zero), while continuum-seeding is expected to beneﬁt from a Stokes seed (Ω < 0,

ρ further from zero). In addition to the direct relevance of these conclusions to vector OPA systems, they may also help inform the design of future vector OPO systems, where the seed wavelength and chirp can be chosen through the particulars of the oscillator’s feedback [11].

89 5.4 Other pulse propagation considerations

The practical advantages of vector four-wave-mixing must be weighed against the shortcomings from a pulse propagation standpoint. For one, cross-polarized four-wave-mixing has a unit gain of (2/3)γP for perfect phase-matching, as op- posed to 2γP for the co-polarized, scalar case. Compensating this decrease in unit gain requires thrice higher pump powers. The problem is compounded by the larger mode areas of PM fiber: while this aids in power-scaling if an adequate pump can be supplied, it also increases the demands on such a pump. When comparing PCF to single-mode PM fiber for optical parametric generation, the mode area increases by a factor of three, resulting in a ninefold total required power increase. This can quickly stretch the limits of what is obtainable from standard, linear, CPA systems, and may require new amplification regimes that can produce chirped pulses with much higher peak powers [12, 13, 14].

Once this gain deficit is closed by boosting the pump source, other unique facets of the nonlinear pulse propagation must be contended with. Compen- sating group-velocity dispersion with birefringence instead of higher-order dispersion produces a more delicate balance of effects, leading to a much narrower phase-matching bandwidth in the CW analysis. By itself, this is not a major limitation–in section 4.3.5, we showed that properly matching a seed to the gain chirp can yield broadband, stretched-pulse amplification even when the parametric gain of any given CW pump component is narrow. However, the greater pump powers required to achieve sufficient cross-polarized gain means cross-phase modulation of the signal and idler will be significant. A seed that is initially chirp-matched will find its frequencies shifting over the course of the propagation, changing its chirp. Slight shifts in the parametric gain spectrum

90 as the pump depletes may add to this effect. Temporal walk-off effects, discussed below, may also shift the seed out of the region of optimal gain, even in the chirp-matched case. The relatively narrow parametric gain bandwidths associated with vector four-wave-mixing make such deviations all the more important, complicating the design of vector OPOs/OPAs.

Another factor that must be accounted for in vector parametric systems is the non-negligible group-velocity mismatch between the pump, signal, and idler. Surprisingly, the fact that the waves inhabit different polarization modes tends not to be a major issue: the polarization-mode walk-off goes as δn/c, which is typically on the order of 1 ps/m. Chromatic dispersion is much more significant. In a PCF, phase-matching widely-separated wavelengths is achieved by pumping at slightly positive group-velocity dispersion, resulting in a small group-velocity mismatch as a matter of course (often ∼ps/m). The practical advantage of using standard, PM fiber, however, rests on no special effort having gone into suppressing the fiber’s natural group-velocity dispersion. The group-velocity mismatch between the signal and the idler can consequently easily reach ∼10 ps/m. The effects of this are complicated, and lie beyond the scope of this thesis. Simulations show a variety of behaviors, ranging from parametric gain being smeared across the spectro-temporal domains to temporal regions of inhibited and enhanced conversion. A more thorough study of four-wave-mixing under the simultaneous conditions of narrowband parametric gain, strong cross-phase modulation, significant pump depletion, and fast temporal walk-off promises to be fascinating from the point of view of nonlinear pulse propagation, and may hold the key to demonstrating high-performance, PCF-free, fiber optical parametric systems.

91 5.5 Conclusions

In summary, we consider the generation of energetic, ultrashort pulses at new wavelengths using vector four-wave-mixing in polarization-maintaining ﬁber.

By using linear birefringence, rather than higher-order dispersion, to enact phase-matching, we show that photonic crystal fiber can be eschewed in favor of standard, step-index fiber. This leads to a number of advantages, both for practicality and for power-scaling. As a first demonstration of this approach, we demonstrate a fiber OPO using entirely standard, step-index fiber, which generates nanojoule-level, femtosecond-scale pulses at 835 nm and 1340 nm. We

ﬁnd that a number of factors complicate systems such as this relative to standard, scalar ﬁber OPOs, offering both challenges and opportunities for future research.

92 BIBLIOGRAPHY

[1] Walter Fu and Frank W. Wise. Femtosecond Optical Parametric Oscillator Based on Vector Four-Wave-Mixing in Step-Index Fiber. In Conference on Lasers and Electro-Optics, page SF1L.1, Washington, D.C., 2019. OSA.

[2] Limin Xiao, M. S. Demokan, Wei Jin, Yiping Wang, and Chun-Liu Zhao. Fusion Splicing Photonic Crystal Fibers and Conventional Single- Mode Fibers: Microhole Collapse Effect. Journal of Lightwave Technology, 25(11):3563–3574, nov 2007.

[3] Limin Xiao, Wei Jin, and MS Demokan. Fusion splicing small-core photonic crystal ﬁbers and single-mode ﬁbers by repeated arc discharges. Optics Letters, 32(2):115, 2007.

[4] Dmitriy E Churin, R J Olson, N Peyghambarian, and Khanh Kieu. High power, widely tunable synchronously pumped ﬁber-based optical parametric oscillator. In Conference on Lasers and Electro-Optics, page SW4P.2, 2016.

[5] Stephane´ Pitois and Guy Millot. Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical ﬁber. Optics Communications, 226(1-6):415–422, oct 2003.

[6] J. S. Y. Chen, S. G. Murdoch, R. Leonhardt, and J. D. Harvey. Effect of dispersion fluctuations on widely tunable optical parametric amplification in photonic crystal fibers. Optics Express, 14(20):9491, 2006.

[7] R. K. Jain and K. Stenersen. Phase-matched four-photon mixing processes in birefringent ﬁbers. Applied Physics B Photophysics and Laser Chemistry, 35(2):49–57, 1984.

[8] Govind P Agrawal. Nonlinear Fiber Optics. Academic Press, New York, 3 edition, 2001.

[9] Stuart G Murdoch, R Leonhardt, and John D Harvey. Polarization modulation instability in weakly birefringent ﬁbers. Optics letters, 20(8):866–8, 1995.

[10] Andy Chong, Joel Buckley, Will Renninger, and Frank Wise. All-normal- dispersion femtosecond ﬁber laser. Optics Express, 14(21):10095, 2006.

93 [11] Maximilian Brinkmann, Tim Hellwig, and Carsten Fallnich. Optical parametric chirped pulse oscillation. Optics Express, 25(11):12884, 2017.

[12] Lawrence Shah, Zhenlin Liu, Ingmar Hartl, Gennady Imeshev, Gyu C Cho, and Martin E Fermann. High energy femtosecond Yb cubicon ﬁber ampli- ﬁer. Optics Express, 13(12):4717–4722, 2005.

[13] Shian Zhou, Lyubov Pavlovna Kuznetsova, Andy Chong, and Frank. W. Wise. Compensation of nonlinear phase shifts with third-order dispersion in short-pulse ﬁber ampliﬁers. Optics express, 13(13):4869–4877, 2005.

[14] Lyubov Pavlovna Kuznetsova and Frank. W. Wise. Scaling of femtosecond Yb-doped fiber amplifiers to tens of microjoule pulse energy via nonlinear chirped pulse amplification. Optics Letters, 32(18):2671–2673, 2007.

94 CHAPTER 6 FUTURE DIRECTIONS

6.1 Alternatives to modelocked oscillators1

For some applications, having a modelocked oscillator is essential. The coherence and noise properties of such devices are unrivaled by any other technology, and their raw performance will only improve as pulse propagation physics continues to develop. As has been discussed, the Mamyshev oscillator stands to address many longstanding issues with modelocked fiber oscillators. How- ever, modelocked oscillators remain limited in certain ways. All lasers require a means of initiating modelocking and restoring it if it is lost, and even in reliably self-starting lasers, a given modelocked state must be maintained during operation. For these reasons, even the best modelocked fiber lasers may sometimes prove lacking. In this chapter, we discuss alternatives: fiber systems that generate ultrashort pulses without starting from a modelocked oscillator. We first review the major methods of producing pulses without modelocking. We then turn our attention to how useful pulses can be obtained from such sources with the aid of strong nonlinearity, building off of the work presented in Chapter 2.

6.1.1 Generating pulses without modelocking

One straightforward method of generating short pulses is by launching long (nanosecond or longer) pulses into anomalously dispersive ﬁber, and ﬁltering

1Much of this work was presented in Optics Express [1].

95 the ultrashort solitons that result from modulation instability [2]. Perhaps surprisingly, even the noisy, chaotic pulse bursts obtained via this approach have been successfully used for multiphoton imaging [3]. While evidently-controlled pulses are still preferable, this result reﬂects a common trend in ultrafast laser applications, where practical considerations spur unexpected laser source de- velopments. This can sometimes prompt new features (e.g., tunability), and other times simply justify a performance reduction in exchange for a lower-cost or more reliable device. More deliberate methods that can deterministically generate individual, high-quality pulses can only increase the appeal of these cheap, but ill-controlled, sources (Fig. 6.1). For instance, a high-bandwidth intensity modulator can carve pulses out of a continuous-wave (CW) beam by attenuat- ing the beam in a desired pulse sequence. Techniques with greater efﬁciency

(up to unity in some cases) have been demonstrated using rapid amplitude or phase modulation schemes based on time-lenses [4, 5, 6], temporal zone plates [7], temporal holograms [8, 9], or the temporal Talbot effect [10], followed by a dispersive delay line that compresses the modulated beam into a pulse train. The improved efficiency can reduce the number of amplification stages needed; however, these techniques generate a repetitive pulse train, rather than individual pulses, and are therefore less versatile than techniques such as direct intensity modulation. Furthermore, they can leave a residual CW background. Short pulses can also be generated without an expensive, high-speed modulator. Some of the first work on non-modelocked sources used seeded modulation instability, either by periodically modulating a CW beam or by beating two spectrally-separated CW beams against one another, and using soliton effects to convert the modulation to a high-repetition-rate, high-duty-cycle pulse train [11, 12, 13, 14]. Self-similar propagation [15, 16] and active pulse shaping [17]

96 have been applied to similar systems. Gain-switched diodes offer another alternative: even a relatively slow current modulator can take advantage of the fast gain dynamics in an optical diode to generate ultrashort pulses on demand

[18], although the pulses typically exhibit random ﬂuctuations and incomplete coherence [19].

Figure 6.1: Illustration of the major features of several non-modelocked pulse generation techniques discussed in the text.

A primary shortcoming of modelocking-free laser sources is the achievable pulse duration. Systems based on modulated CW light are limited by the bandwidth and power of electro-optic modulators and their drivers. Rarely do they produce pulses shorter than a few picoseconds, and more economical systems are limited to tens of picoseconds. Enhancement loops that compound the effect of a single modulator can improve this to the sub-picosecond regime, but they add complexity and sacriﬁce control over the pulse pattern and repetition rate [20]. While gain-switched diodes are not bound by the speed of an optical modulator, the natural rate of their internal photon-carrier dynamics is such that they, too, emit picosecond-scale pulses. Extending non-oscillator sources to the short pulse durations commonly associated with modelocked oscillators (e.g., sub-300 fs) stands to greatly broaden their impact, and will allow a broad range of ultrafast applications to exploit their robustness and ﬂexibility. The

97 salient challenge, therefore, is to nonlinearly compress picosecond-scale pulses by multiple orders of magnitude in a manner that permits scaling to useful energies (e.g., microjoules or many nanojoules).

Simultaneously, the generated pulses must display low noise, although often not to the level displayed by typical modelocked lasers. This poses another challenge, as nonlinear techniques generally magnify small, pulse-to-pulse ﬂuctu- ations. One might wonder whether non-modelocked sources can control these ﬂuctuations in the long run without the optical feedback that stabilizes such processes in a modelocked oscillator. The relative novelty of non-modelocked sources that achieve comparable pulse parameters to their modelocked brethren makes this an open question. As we will see, however, proper system design can help mitigate this concern and, in some cases, actually reduce noise through nonlinearity.

6.1.2 Compressing pulses from non-modelocked sources

In anomalous-dispersion ﬁber, pulses can be adiabatically compressed as fundamental solitons. Following from the soliton area theorem, evolution in

fibers incorporating amplification [21, 22] or longitudinally-decreasing dispersion [23, 24, 25] produces transform-limited, sub-picosecond pulses. However, for standard fiber parameters, the soliton area theorem and higher-order effects limit this approach to sub-nanojoule energies. The reliance on anomalous dispersion also typically restricts soliton compression to wavelengths longer than ≈1.3 µm, although this spectral range can be extended in photonic crystal fibers

[26], hollow-core photonic bandgap ﬁbers [27], and higher-order-mode ﬁbers

98 [28, 29].

Normal-dispersion compression techniques have more success in scaling to high energies. By balancing nonlinearity and normal dispersion, a dispersive self-phase modulation (DSPM) compressor [30] can linearize the chirp across much of a pulse, allowing the pulse to be dechirped cleanly. How- ever, the length of fiber needed to achieve the proper balance increases with the initial pulse duration. Thus, systems that start from long (≈10-100 ps) pulses often must compromise on pulse quality for practicality’s sake [31]. A better-controlled method is self-similar amplification, whereby pulses subject to gain, self-phase modulation, and normal dispersion asymptotically become parabolic, linearly-chirped pulses–so-called similaritons [32, 33]. While this regime is most commonly used for amplification, the exponential bandwidth growth and linear chirp makes it ideal for pulse compression as well. Further- more, the asymptotic similariton acts as a global nonlinear attractor, making this technique applicable in principle to arbitrary seed pulses and potentially reducing noise [34, 35]. Dispersion-decreasing fibers can replicate this phenomenon, exchanging the benefits of amplification for greater bandwidth and spectral flexibility [36]. Of course, self-similar evolution has limitations in practice. Al- though the similariton’s basin of attraction is infinite, the speed of the attraction depends on the seed parameters [37, 38, 39]. In typical fibers, a long or ill-shaped seed pulse might not reach the attractor within the finite fiber length. An arbitrarily long amplifier would resolve this in principle, but in practice, the onset of spontaneous Raman scattering and gain narrowing limit the maximum fiber length, and therefore the effective range of the attractor. These effects reduce the usefulness of this technique for the picosecond-scale durations and varied pulse shapes that modelocking-free sources can generate.

99 Amplification following parabolic pre-shaping is related to self-similar amplification, but has several important differences [40]. Pulses in normally- dispersive fiber can transiently evolve to a nearly-parabolic state even in the absence of gain [41], and can be subsequently amplified in another fiber. The pulses need not be near the amplifier’s similariton attractor to benefit from their parabolic shape’s linearization of nonlinear phase: following amplification in the highly nonlinear regime, they can be substantially and cleanly compressed to the transform limit. While this is a powerful technique, the lack of a nonlinear attractor means that only pulses that are initially well-behaved (for instance,

Gaussian or hyperbolic secant pulses near the transform limit) exhibit the necessary parabolic evolution. As a result, this technique can be unsuitable for the more complicated pulses that may be formed from non-modelocked systems.

The Mamyshev regenerator from telecommunications can be a versatile tool for compressing and reshaping pulses [42]. As Figure 6.2 illustrates, launching a pulse into a nonlinear fiber and then isolating newly-created spectral components with an off-center bandpass filter has a number of effects on the pulse, including reducing amplitude jitter [42, 43], stripping amplified spontaneous emission [44, 45], suppressing pedestals or background light [46, 47, 48], and enhancing intra-pulse coherence [49, 50, 51]. These effects are also accompanied by pulse compression by a modest factor (typically ≈2-5), even for input pulse durations ranging from sub-nanosecond [52] to femtosecond timescales [53]. With an appropriate choice of bandpass filter, the output can be made transform-limited or linearly-compressible to the transform limit [43].

100 Figure 6.2: Simulation illustrating a Mamyshev regenerator’s ability to accept a long pulse with a complicated pedestal (solid green), spectrally broaden it (solid blue, and spectrogram), and convert it into a short, pedestal-free pulse (dashed red). Top panel: temporal intensity. Right panel: spectral intensity.

6.1.3 Multi-stage compression

The problem of compressing pulses from non-modelocked sources is compounded by the long initial pulse durations. The most powerful compression techniques require a balance between dispersion and nonlinearity (Fig. 6.3). Applying them directly to ≈10-100-ps pulses can necessitate many kilometers of fiber (e.g., [15, 54]). While such fiber lengths are common in telecommunications, they can have significant consequences for a high-energy pulsed laser system’s cost, footprint, and loss (particularly at wavelengths where fiber losses are higher). These restrictions can be particularly limiting for DSPM compressors, where using a more practical yet sub-optimally-short fiber length leads to pulses with a low-intensity pedestal rather than high-quality pulses [31]. This can be seen in Fig. 6.3, where, barring standout results, only compressors operating near the shaded regions tend to produce transform limited pulses.

101 Figure 6.3: Performance of various nonlinear compression schemes. Ovals indicate the approximate region over which each technique typically produces transform-limited, pedestal-free pulses. Data are from [51, 45, 40, 55, 56, 57, 14, 15, 16, 24, 27, 52, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68].

If a more complex system is tolerable, using multiple different compressors in sequence can ameliorate the problem. For instance, the Mamyshev regenerator relies mainly on nonlinearity alone, which can be strengthened through ampliﬁcation to permit spectral broadening in more reasonable ﬁber lengths

[52]. Pulses may thereby be compressed and even amplified, while the Mamy- shev regenerator’s normalizing effect on the pulse shape enables further shaping via more sophisticated techniques. The diverse properties of Mamyshev regenerators make them powerful front-line pulse shapers even beyond their compression capabilities: in systems seeded by gain-switched diodes, they can improve the intrapulse coherence and reduce amplitude fluctuations, especially when cascaded; in systems founded on intensity modulators, they can amplify the low-power pulse trains while curbing amplified spontaneous emission; and in systems based on amplitude/phase modulation of CW light, they can suppress the residual inter-pulse background even at non-uniform repetition rates.

102 A Mamyshev regenerator following a DSPM compressor can also attenuate the pedestal produced by the latter, although interposing another nonlinear process before the Mamyshev regenerator will reduce its ability to dampen noise from the seed laser. An example of this approach is the work that was presented in Chapter 2, based on a gain-switched diode. There, the combination of Mamyshev regeneration and amplification following parabolic pre-shaping led to pulse parameters that previously required an amplified, modelocked oscillator. This result proved for the first time that, by taking advantage of fiber nonlinearities, non-modelocked sources stand a chance of competing with modelocked systems as application-relevant ultrafast lasers. While the demonstrated source exhibited pulse-to-pulse fluctuations as per the nature of the gain- switched diode seed, the observed energy jitter was already low enough to be acceptable for many applications, and incorporating additional techniques may produce further reductions in noise (see section 6.1.5). While we are optimistic that sources based on gain-switched diodes may improve to eventually rival modelocked lasers, it should be noted that many applications–for instance, micromachining [136] and multiphoton microscopy [3]–can be performed with significantly noisier pulses than even the source dicsussed in Chapter 2.

It bears noting that the compression and ampliﬁcation schemes described here for gain-switched diodes could be naturally adapted to modulated CW sources [69]. Indeed, this may be a preferable alternative in some cases: pulses generated from a sufﬁciently quiet CW diode will inherit the diode’s long coherence time and low noise as a matter of course. However, this approach is not without its disadvantages. The coherence and noise will still depend on the linewidth and stability of the CW diode, which will ultimately correlate strongly with the diode’s cost. In addition, the degree of compression required will de-

103 pend on the speed at which the diode can be modulated, necessitating a trade- off between the cost of fast driving electronics and the complexity of additional compression stages. As non-modelocked systems with performance comparable to modelocked lasers have only recently been demonstrated, it remains to be seen which approaches, if any, will have enduring and widespread impact. The decision to favor one method over another, along with the judgment of whether the simplicity and advantages of using a non-modelocked seed outweigh the complexity of compressing the pulses, will ultimately come down to the needs of individual applications, and the resources available to them.

6.1.4 Compression using extreme dispersion engineering

As an alternative to the multi-stage compression schemes thus far discussed, the difficulties of shaping long pulses from non-modelocked sources may be resolvable through appropriate dispersion engineering. Under the right conditions, the dispersion in a certain spectral range can be made extremely large, permitting soliton and self-similar evolutions in more practical fiber lengths. An example of this is near the mode cutoff in a higher-order-mode fiber, where dispersion nearly two orders of magnitude higher than that of typical fibers has been reported [70]. Even more powerful dispersion engineering can be performed near (but outside) the reflection band of a fiber Bragg grating (FBG) [71], where inscribed structures up to a meter long can boost the dispersion by more than six orders of magnitude [72, 73, 74, 75]. Pulse compressors based on this technology have been proposed and demonstrated, including those using soliton and similariton evolutions [76, 77, 78, 79, 80], although the experimental demonstration of Bragg similaritons remains inconclusive [81]. Intriguingly, be-

104 cause the dispersion induced by the grating far exceeds the material or waveguide dispersion, this approach can be transposed to any wavelength of interest where ﬁbers are available. This ﬂexibility synergizes well with pulsed sources

(modelocked or otherwise) that can be designed at a variety of wavelengths in the near-infrared [82, 83] or mid-infrared [84, 85]. Concurrent development of spectrally-ﬂexible gain media based on Raman scattering [86, 87, 88, 89], four- wave-mixing [90, 91, 92], or semiconductor optical ampliﬁers [93, 94] will help exploit this potential for applications where the source wavelength is critical.

Despite the power of dispersion engineering in a FBG, pulse propagation in such devices is complicated by several factors. Firstly, the strong group-velocity dispersion is accompanied by disproportionately stronger higher-order dispersion [95, 96], which can cause soliton or similariton propagation to break down [77, 79, 97, 98, 99, 100]. Secondly, the pulse evolution will be constrained by the nearby reflection band of the FBG. As per the Kramers-Kronig relations, the strength of dispersion increases with proximity to a sharp transmission or reflection feature, forcing a compromise between the induced dispersion and the pulse bandwidth. Finally, maintaining the same strength of nonlinearity (i.e., the same peak power) with much longer pulses necessitates proportionally high pulse energies, resulting in strong gain saturation during self-similar amplification [79, 81]. Using a chirped FBG to effect a dispersion-decreasing fiber may alleviate this issue, as the varying dispersion behaves like amplification where the virtual gain is saturation-free [36]. One can envision such a device acting as a compact, passive pulse shaper, drawing long pulses from a simple pulse generator towards the similariton attractor for reshaping (Fig. 6.4). Once a parabolic pulse is generated in this manner, compression can be performed immediately (as in Fig. 6.4), or nonlinear amplification can be interposed to further increase

105 the bandwidth, as in ampliﬁcation by parabolic pre-shaping [40].

T i m e ( p s ) T i m e ( p s ) W a v e l e n g t h ( n m ) - 4 0 0 4 0 - 1 0 1 0 3 0 1 0 2 9 . 6 1 0 3 0 . 0 1 0 3 0 . 4 3 1

I n p u t ) ( a ) ( b ) ( c ) . ) )

O u t p u t m r W W o

k 2 k n ( (

1 0 (

r r y t e e i s w w n o 1 o e P P t n I 0 0 0 1 0 0 1 0 0 ( d ) ( e ) ) P r m o

c 8 0 8 0 p (

a e g c a n t i o a

t 6 0 6 0 n s

i D D i

s n t a o i 4 0 4 0 n t c a F B G S t o p b a n d e g

( a c p m

o 2 0 2 0 ) r P

0 0 1 0 2 8 . 0 1 0 2 8 . 5 1 0 2 9 . 0 1 0 2 9 . 5 1 0 3 0 . 0 0 1 0 0 2 0 0 W a v e l e n g t h ( n m ) G V D ( p s 2 / m )

Figure 6.4: Simulated self-similar evolution in a lossless, dispersion-decreasing FBG, including dispersive effects through third-order.(a) Seeded, 100-nJ, 40-ps pulse with a complicated intensity and phase structure (dot-dashed red), and the parabolic output (black). (b) Output pulse after dechirping to near the 5-ps transform limit. (c) Output spectrum. (d) Longitudinal evolution of the pulse spectrum and the FBG stopband. (e) Dispersion-decreasing proﬁle calculated from the varying Bragg wavelength.

6.1.5 Noise reduction

Given the premium that some specialized applications place on noise, it is worth considering how the jitter in non-modelocked might be reduced further. Ad- ditional nonlinear optical processes might accomplish this, at the expense of adding complexity. For instance, it has been shown that a long sequence of Mamyshev regenerators can effect an asymptotically stepwise transfer function and funnel incoherent pulses towards a uniform eigenstate [49, 50], with shorter, more practically attainable chains still exhibiting intermediate nonlinear attraction.

106 Alternatively, further intensity and timing jitter reductions might be accomplished by self-seeding [101, 19, 102] or injection seeding [103, 104] a gain- switched diode. The former sacrifices fine control over the diode’s repetition rate, but has the advantage that a second light source is not needed. Although gain-switched diodes naturally lack inter-pulse coherence–a hallmark trait of modelocked oscillators, and a necessary one in comb-based applications–there is some evidence that even that feature might not be out of reach. Self-seeded gain-switched diodes have been shown to exhibit inter-pulse coherence [105], and an interesting question is whether injection seeding can accomplish the same. Given that gain-switched diodes have been shown to inherit coherence from an externally-seeding, pulsed, master laser, it seems plausible that an external CW seed might likewise confer inter-pulse coherence [106]. However, this remains to be experimentally verified for injection-seeded gain-switched diodes to compete with modelocked sources in optical comb applications while retaining their pulse-on-demand capabilities. If successful, implementing these approaches could make cheap, robust, non-modelocked systems viable alternatives to modelocked oscillators even for noise- or coherence-sensitive applications.

6.1.6 Conclusions

Generating ultrafast pulses without modelocked lasers has the potential to enable new applications of femtosecond pulses, driven by qualitatively new kinds of nonlinear ﬁber light sources. To be sure, non-modelocked lasers are unlikely to ever match modelocked oscillators in every respect. Despite tremendous advances in optical and electro-optic pulse shaping, it is difﬁcult to imagine what

107 kinds of advances could replicate the pristine, broadband coherence of a finely- tuned oscillator. Nevertheless, for a significant number of applications, the full capabilities of a modelocked oscillator are unnecessary, or may yet fall short. It is here where non-modelocked sources will have a chance to shine: cases where simplicity and robustness are of primary importance, or where pulses need to be delivered at precise, aperiodic intervals. By taking advantage of a wide array of nonlinear pulse evolutions, non-modelocked laser systems stand a chance of filling this niche with high-energy, ultrashort pulses in an unprecedentedly versatile format.

6.2 Prospects for vector ﬁber optical parametric devices

6.2.1 Introduction

In addition to the technical-level improvements suggested in Chapter 5, numerous avenues of research remain for extending vector fiber optical parametric devices to new levels of performance and practicality. Here, we discuss several open questions that may bear on the future of such devices. We first consider alternative system architectures, ranging from in-amplifier designs to the use of higher-order-modes. We then address how a measure of tunability might be achieved, before concluding with a discussion of how this tunability might be used to engineer new kinds of fiber systems with highly-optimized phase- matching properties.

108 6.2.2 In-ampliﬁer vector four-wave-mixing

As another embodiment of this approach altogether, one might imagine com- bining the pump amplifier and the PM fiber into a single device, as shown in Figure 6.5. Here, a low-power train of pump pulses is launched into a PM fiber gain medium. There, their energies increase quasi-exponentially, resulting in the bulk of their nonlinear propagation (and, hence, the vector four-wave- mixing) occurring in the last ∼tens of cm of the fiber. This has practical advantages in that focusing a high-power pump beam into a bare PM fiber is never needed, reducing any resulting thermal instabilities, and that coupling losses from the pump amplifier to the PM fiber are made trivially zero. Care must be taken in this arrangement that neither the signal nor the idler faces significant linear absorption due to the gain medium. As a surprising bonus, some authors have suggested that performing four-wave-mixing in an active medium could effect a type of quasi-phase-matching, improving the bandwidth and conversion efficiency of the parametric process [107]. More research is needed to gauge

the strength and impact of this speculative, yet intriguing, possibility.

signal v

idler v outv residual 976 nm out pump dichroic pump out mirror

v optical () x () oscillator delay PM YDF combiner PBS HWP filter () ()

feedback fiber

Figure 6.5: Schematic of vector OPO using four-wave-mixing in the pump am- pliﬁcation ﬁber.

109 6.2.3 Extension to higher-order modes

We have already discussed scaling the performance of vector optical parametric devices via the mode area. Taking this approach to its natural limit, however, may introduce new degrees of freedom yet. It is well-known, and often- lamented, that unless special care is taken in the design of a ﬁber, increasing the core size will eventually lead to multimode behavior. While the existence of many different transverse modes may have consequences for the quality of the emitted beam, it also creates an opportunity for a much wider range of four- wave-mixing interactions, each with different phase-matching conditions. Re- cent work has already begun exploring this approach in the sub-nanosecond pulse regime, and it will be of great interest to extend this approach to chirped, ultrafast pulses [108, 109, 110]. Ensuring the environmental robustness of the multimode ﬁber, as well as that of the spatial beam shaping that will be necessary at the input and output ends, will be key to making such systems viable.

6.2.4 Tunable birefringence

A benefit of using photonic crystal fiber is that the broad phase-matching bandwidths, brought about through careful dispersion engineering, permit wavelength tuning away from the parametric gain peak. This is much harder to achieve using vector four-wave-mixing, due to the more narrowband phase- matching. While the birefringence of polarization-maintaining fibers is typically specified to greater precision than the dispersion of a photonic crystal

ﬁber, care must still be taken to choose pump and seed wavelengths that are appropriately phase-matched. The demand becomes particularly stringent for

110 CW-seeded vector OPAs, as tunable, narrowband diodes can be prohibitively expensive.

A degree of tunability can nevertheless be achieved through mechanical or thermal means. It has long been known that bending or coiling an optical ﬁber induces a birefringence that goes as δn ≈ α(r/R)2, where r is the cladding radius, R is the bending/coiling radius, and α is a constant determined by the ﬁber’s mechanical properties (typically ≈ −0.1) [111, 112]. If the bending is performed

0 under tension, an additional, constructively-adding birefringence δn ≈ α (r/R)z

0 is produced, where α ≈ −0.5 and z is the relative strain (typically limited to . 0.01 to avoid breaking the ﬁber) [113]. Squeezing the ﬁber in a V-groove is another option, producing a birefringence of δn ≈ h · [1 − cos(2γ) sin(γ)](F0/r)

−12 2 where h ≈ 2.4 × 10 m /N, F0 is the linear force density applied (typically . 10 N/m), and γ is the half-angle of the V-groove [114].

The relative magnitude of these effects is generally small, but can be non- negligible next to the built-in birefringence of typical PM fibers. Using esti- mates of mechanical fiber damage as an upper limit, the birefringence produced by free bending and bending under tension may each be on the order of 10−5, while squeezing appears limited to 10−7. This is unlikely to suffice for shifting the parametric gain to substantially different spectral regions, but may be useful for fine-tuning the gain relative to a proscribed seed. Indeed, several of these process have been used to tune [115] or even quasi-phase-match [116, 117] vector fiber optical parametric devices, although they have yet to be applied to high pulse energies, femtosecond-scale durations, or large wavelength separations.

Because these adjustments can be imposed externally and in situ, an interesting possibility also exists for longitudinally-tuning the parametric gain spec-

111 trum in this manner. Extending the spooling approach demonstrated by Mur- doch et al. [117] to a specially shaped spool with a continuously-varying radius could enable tailorable control over δn(z), and through it, the parametric gain spectrum (Figure 6.6). This might be employed to compensate for the cross- phase modulation or pump depletion effects mentioned in section 5.4, or perhaps by sweeping the gain through the seed spectrum to phase-match different components at different longitudinal positions. At its most basic, the latter could extend the overall parametric gain spectrum; however, more sophisticated schemes could yield richer benefits. For instance, the dynamic interactions between group-velocity dispersion and the longitudinally-chirped frequency conversion might be used to help enhance or shape the generated pulses. With an appropriately-shaped phase-matching distribution, adiabatic frequency conversion techniques could even be utilized to obtain highly-efficient broadband conversion, similar to those being demonstrated in tapered fibers [118]. The ability to sharply and longitudinally tailor the parametric gain through mechanical means is an unanticipated advantage of vector fiber optical parametric systems over PCF-based lasers, and one that may lead to brand new areas of research into nonlinear pulse propagation.

6.2.5 Conclusions

We ﬁnd that a number of factors complicate vector ﬁber OPOs and OPAs relative to standard, scalar, PCF-based OPOs, offering both challenges and opportunities for future research. While vector four-wave-mixing imposes more system design restrictions than PCF, it also opens the door for new types of nonlinear pulse evolutions, with the potential for capabilities beyond what is possible in

112 (a) (c)

(b)

Figure 6.6: Illustration of coiling a ﬁber around (a) a spool with a varying radius to (b) change the ﬁber’s birefringence and, through it, the peak gain frequency as a function of propagation distance, thereby effecting (c) a longitudinally- shapable parametric gain spectrum.

PCF.

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