Development of a New Mid-Infrared Source Pumped by an Optical Parametric Chirped-Pulse Amplifier

Development of a new mid-infrared source pumped by an optical parametric chirped-pulse amplifier.

Etienne Pelletier

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto

Development of a new mid-infrared source pumped by an optical parametric

chirped-pulse ampliﬁer.

Etienne Pelletier

Doctor of Philosophy

Graduate Department of Physics

University of Toronto

2013

The mid-infrared (MIR) system presented in the thesis is based on a sub-100-fs erbium-

doped fiber laser operating at 1.55 m. The output of the laser is split in two, each arm seeding an erbium-doped fiber amplifier. The output of the first amplifier is sent to a grating-based stretcher to be stretched to 50 ps before seeding the optical parametric chirped-pulse amplifier (OPCPA). The output of the second amplifier is coupled to a highly nonlinear fiber to generate the 1 m needed to seed the a neodymium-doped yttrium lithium fluoride (Nd:YLF) system. This work represents the first time this synchronization scheme is used, and the timing jitter between the two arms at the OPCPA is reduced to 333 fs.

The pump laser for the OPCPA is a regenerative amplifier producing 1.6 W followed by a double-pass amplifier, for a final output power of 2.5 W at 1 kHz. Etalons were inserted into the cavity of the regenerative amplifier to stretch the pulses to 50 ps

The OPCPA consists of two potassium titanyl arsenate crystals in a noncollinear conﬁguration. With three passes, the gain is 3.8 106. Using a grating compressor, the pulse duration is reduced to 140 fs, with a power of 300 mW. Because of the reduction of

the timing jitter, the amplitude stability is 1 %, which is a great improvement compare

to existing systems.

To generate ultrafast light in the MIR, an optical parametric ampliﬁer is used, pumped

ii by the output of the OPCPA and seeded with its 3-m idler. Two crystals were tested, both in a single-pass configuration. For the first crystal, a 4-mm thick silver thiogallate, an efficiency of 7.4 % was reached, with 8.76 mW in the signal and 7.2 mW in the idler.

For the second crystal, a 2-mm thick lithium gallium selenide, the eﬃciency was higher, reaching 10.8 %. The power for the signal was 11.5 mW, and for the idler, 11.11 mW.

Using this new scheme, energies on par with current systems are achieved with much higher eﬃciencies.

iii Acknowledgements

At ﬁrst, I thought that the acknowledgement would be the easy part; how hard could it

be to express my gratitude to a few people? Well, harder then I was expecting; so many

people helped me in one way or another and I hope I will do a good job at giving them

the recognition they deserve.

First and foremost, I would like to thank my supervisor, Prof. R.J. Dwayne Miller, for making this opportunity happen. I am not only talking about accepting me as a student in his group and giving me the guidance to achieve my goal, but also for having an extremely contagious enthusiasm, without which none of this would have been possible.

When I ﬁrst met him, before a talk he was giving at Laval University, I was a disenchanted young man; years before the telecom bubble burst and with the job prospective in Optics still gloomy, grad school seemed like the only option. However, this ﬁrst meeting changed my mind set and I began to see grad school for what is really was; a chance to work on new and exciting science. The following year, three months into what was supposed to be a two-year master, my transformation was completed and I decided to do a Ph.D., which, even during tough times, I have never regretted.

The meeting with Prof. Miller would have never happen without the generous help of Prof. Nathalie McCarthy, a faculty member at Laval University. At a moment when

I was at a cross-road in my life, she gave appreciated advice and suggestions, without which this thesis might have never happen. And for this, I am extremely grateful to Prof.

McCarthy.

A major breakthrough in my project occurred because of Prof. Miller’s sabbatical in

Italy, in 2008. While in Europe, he found himself visiting the University of Konstanz and met up with a faculty who told him how they could generate almost any color using an erbium-doped ﬁber laser; this meeting was a turning point in my project. Already by then, we were looking into using nonlinearity in ﬁbers, but it was seen as a bit of a gamble.

However, by having a collaborator, this solution became a no-brainer and we went ahead;

iv the rest of the story can be found in this thesis. For this, I would like to express my most sincere gratitude to Prof. Alfred Leitenstorfer and his group at the University of

Konstanz; without their help, there would be no nonlinear ﬁber. I would also want to personally thank Alexander Sell, the Ph.D. student (now Doctor) who was my direct connection within the group in Konstanz, for his priceless help with the nonlinear ﬁber

(i.e. designing/making it) as well as for his numerous advice and tips on erbium-doped

ﬁber ampliﬁers.

For the last 7 years, the Miller group became a kind of adoptive family, bringing with it a mix of friendships and professional relations that is hard to describe. I owe a big thanks to all the previous group members, especially those that were there when I ﬁrst joined.

Their kindness helped ease what could have been an extremely stressful situation; joining a large group of individuals (more than 15 people at the time) is not an easy task. Of all of them, a few deserve to be singled out: Sadia Khan, our former research facilitator, who was a a great help with the administrative business as well as a willing listener; Kresimir

Franjic and Renzhong Hua, who were my ﬁrst mentors in the lab; Darren Kraemer, who was the trailblazer whose footsteps I followed as well as a constant source of advice and discussion. Over the years, the dynamic of the group changed, as well as my role in it; nonetheless, I would like to acknowledge the current members for their help and support, in particular Alexei Halpin, Christina Mueller, and Philip Johnson. As senior Ph.D. students, we had on more than one occasion discussions about our research, frustrations, and general topics regarding our life as graduate students, which helped me stay sane by keeping things in perspective.

I cannot write acknowledgments without mentioning my family. My brother and my sister who supported me throughout my Ph.D. and never once said "Shouldn’t you be done by now?", which is extremely nice of them. However, I owe more than special thanks to my parents; I would not have made it this far without their help. As a young boy, they successfully engaged me with science through weekend activities, which could be seen as

v the starting point of this long journey. During my undergrad, my parents constantly pushed me to aim for the highest grade that I could possibly achieve. Although it was not always easy (and sometimes annoying), their healthy encouragement taught me to strive for my maximum potential.

And ﬁnally, I would like to thank all of my friends who have been there for me during all those years. In particular, all of those who had to endure my complaints: Jean-

Michel Menard, my fellow Quebecer in the Physics department, who was my partner in crime (mostly telling bad jokes) from the beginning; Cristen Adams, another physicist, with whom I walked so many times from the department to the East side, especially after PGSA pub nights; Adrienne Marcotte, with whom I biked to all corners of Toronto;

Martin Parrot, an old friend since my youth, who brought the Canadiens’ fever in Toronto; and ﬁnally, Sue Hobson, to whom I owe a special thanks, as she was nice (crazy?) enough to read over my entire thesis, hunting for mistakes and poor grammar, which was not an easy task (i.e. ESL).

vi Contents

1 Introduction 1

1.1 LaserSelectiveChemistry ...... 1

1.2 Mid-infrared optical parametric ampliﬁer ...... 3

1.3 Chirped-pulse ampliﬁcation ...... 4

1.4 Thiswork-quickoverview...... 5

2 Generation of the seed:

Nonlinear propagation in ﬁbers 7

2.1 Maxwellequations ...... 7

2.2 Spatialmode ...... 11

2.3 Dispersion...... 18

2.4 Nonlinearpropagation ...... 23

2.5 Self-phasemodulation ...... 28

3 Erbium-doped ﬁber system for signal ampliﬁcation 35

3.1 Opticalproperties...... 35

3.2 Dispersioninerbium-dopedﬁbers ...... 38

3.3 Fiberlaser...... 41

3.4 Erbium-dopedﬁberampliﬁer...... 42

4 Highly-nonlinear ﬁber for all-optical synchronization 50

vii 4.1 Precompensatingﬁber ...... 50

4.2 Dispersionoptimizedﬁber ...... 52

4.3 Mechanism for wavelength tuning ...... 53

4.4 Result ...... 53

5 Nd:YLF ampliﬁcation system 58

5.1 Opticalproperties...... 58

5.2 Regenerativeampliﬁer ...... 63

5.3 Regenerativeampliﬁerdesign ...... 71

5.4 Multipassampliﬁer ...... 75

6 Optical parametric ampliﬁer system 78

6.1 Optical parametric ampliﬁcation ...... 78

6.2 Optical Parametric Chirped Pulsed Ampliﬁer ...... 91

6.3 Stretching-Compression...... 91

7 Optical Parametric Chirped-Pulse Ampliﬁer 100

7.1 OPCPA-Design ...... 100

7.2 OPCPA-Performance ...... 106

8 MIR OPA development 116

8.1 Design...... 116

8.2 Performance...... 120

9 Conclusion 129

9.1 Futurework-improvement ...... 130

9.2 Closingwords ...... 131

viii List of Tables

2.1 Coeﬃcients for the Sellmeier equation for the core, i = 1, and for the

cladding, i =2 (from[16])...... 23

3.1 The parameters for the emission and the absorption cross-section [29]. . . 39

5.1 The beam radius calculated with Paraxia at diﬀerent elements in the cav-

ity. The two non-zero focal lengths represent the thermal lens...... 72

ix List of Figures

2.1 Graphic solution for equation (2.39) for V = 8 and for (a) l = 0 and (b)

l = 1. In this case, there will be 3 modes associated to l = 0, and two to

l =1.(Adaptedfrom[12]) ...... 16

2.2 Dispersion for a single-mode ﬁber expressed using (a) the dispersion pa-

rameter and (b) the group velocity dispersion...... 23

2.3 The spectra for diﬀerent values of nonlinear phase, one the left for a Gaus-

sian pulse and on the right for an hyperbolic-secant pulse...... 30

2.4 Pulse envelope of a N=2 soliton at diﬀerent positions...... 32

3.1 On the left, the details of the energy levels of the laser transition (adapted

from [26]). A few transitions are highlighted; the arrows indicate if they

show up in the absorption or the ﬂuorescence spectra or both. On the

right, the emission and the absorption spectrum around 1550 nm for the

erbium-doped ﬁber (adapted from [27])...... 36

3.2 Absorption spectrum for an erbium-doped ﬁber (adapted from [26]). The

three common pumping bands are shown with the arrows. The level in-

volved and the wavelength associated with it are indicated...... 38

3.3 The second-order dispersion for two diﬀerent ion concentrations. The con-

centration was set to (a) 4.25 1025 ions/m3 and to (b) 14.2 1025 ions/m3. The arrows indicate the direction of increasing inversion starting with all

the ions in the lower level until the system is fully inverted...... 40

x 3.4 Layout of the ﬁber laser; WDM,wavelength-division multiplexer, OC, Out-

putcoupler...... 41

3.5 On the left: The retrieved electric ﬁeld for the output of the ﬁber laser.

On the right, the retrieved and the measured spectrum...... 42

3.6 Layoutoftheﬁberampliﬁer...... 43

3.7 On the left, the output spectrum of the ﬁrst ampliﬁer with both pump

diodes at their maximum current, 520 mA, as well as the spectrum for the

input. On the right, the spectra for diﬀerent currents of the ﬁrst diode. . 44

3.8 The spectra at the output of the first fiber amplifier with three different

currents for the ﬁrst laser diodes using a ﬁber (a) before and (b) after. . . 46

3.9 On the left: Retrieved electric field of the output of the first amplifier. On

theright,retrievedandmeasuredspectra...... 47

3.10 On the left: Retrieved electric ﬁeld at the output of the second ampliﬁer.

The FWHM is 89 fs and 65% of the energy is in the central peak. On the

right,retrievedandmeasuredspectra...... 48

3.11 The spectra, (a) to (d), and the temporal proﬁle, (e) to (h), for diﬀerent

currents in the ﬁrst diodes. Also for each current, the spectrum/pulse

shape for the operating current of 436 mA is plotted (dash curve) for

comparaison...... 49

4.1 On the left side, the dispersion curves for the two components of the

HNLF. On the right side, the index proﬁle for the dispersion-optimized

ﬁber. Typical value for the index change is around 3 % for the increase

and below 0.5 % for the reduction [36, 37]. The ratio of the two diameters

isontheorderof0.5-0.6[37]...... 52

xi 4.2 The spectrum at the output of the HNLF. On the left, the spectrum asso-

ciated with the dispersive wave and on the right, the spectrum associated

with the soliton and the unshifted beam. 3 mW is generated in the dis-

persivewave...... 54

4.3 The spectrum at the output of the HNLF. Only the peaks associated with

the dispersive wave and the red-shifted soliton are shown...... 55

4.4 The spectrum of the dispersive wave for diﬀerent temperatures of the holder. 56

5.1 On the left: the two levels associated with the main transition and their

sub-sublevels [43]. On the right: The stimulated cross-section for each

polarization for the 4F 4I transition [42]...... 60 3/2 → 11/2 5.2 On the left side, absorption spectrum for Nd:YLF [46]. On the right side,

4 a close-up of the absorption associated with the F5/2 level...... 61 5.3 The detail of the polarization state in the case of (a) the switch close and

(b) the switch open. The dash arrows show the direction of propagation.

TFP, thin-ﬁlm polarizer, QWP, quarter-wave plate...... 64

5.4 The simulated output intensity of a pulse after going through 160 times in

(a) a single 1-mm etalon, and (b) a combination of a 1-mm etalon and a

0.7-mm etalon. Two diﬀerent pulse durations were used: 3.5 ps and 10.5 ps. 67

5.5 On the left side: the transmitted pulse shape for angle from 0◦ to 4◦, with a

interval of 0.25◦ between each. On the right side, the integrated intensity

as a function of the incident angle, as well as the phase. The numbers

relatefeaturestoeachother...... 68

5.6 The normalized gain spectrum for low and high gains. In both case, the

sameatomiclinewidthwasused...... 69

5.7 The eﬀect of saturation on a square pulse. The ﬁrst curve is the input

pulse, whereas the other two are the output for two propagation lengths.

Adaptedfrom[53]...... 71

xii 5.8 Layout of the regenerative ampliﬁer. LH, laser head, HWP, half-wave

plate, QWP, quarter wave-plate, L, lens, TFP, thin-ﬁlm polarizer, E,

etalons,M,mirrors...... 72

5.9 Autocorrelation trace of the output of the regenerative ampliﬁer with no

etalon inside. The FWHM of the trace is 17 ps resulting in a pulse duration

of12ps,ifaGaussianﬁtisused...... 73

5.10 Cross-correlations traces for (a) a 1.5-mm etalon, (b) a 1-mm etalon, and

simulated and the results are shown in the dashed line...... 74

5.11 Layout of the multipass ampliﬁer; LH, laser head, HWP, half-wave plate,

L, lens, TFP, thin-ﬁlm polarizer, M, mirror...... 76

5.12 Cross-correlation traces with the multipass oﬀ (0 A) andon(18A). . . . 76

6.1 Orientation of the wavevector with respect to the principal axes of the

crystal...... 84

6.2 Diagram for a plane wave propagating in the X-Z plane. The thicker

arrows indicate the two possible polarizations...... 85

6.3 On the left: The orientations of diﬀerent ﬁelds are shown, as well as the

wavevector and the Poynting vector S. On the right: Because the Poynting

vectors for the two polarizations are diﬀerent in the crystal, although the

beam is at normal incidence, one of the two polarizations will be bent at

the interface. On the other hand, the wavevectors and the wavefronts for

thetwobeamsarestillparallel...... 87

6.4 Diagram of the diﬀerent wavevectors in the noncollinear geometry. The

(internal) noncollinear angle, α, is deﬁned in the crystal...... 87

6.5 This diagram illustrates the diﬀerent angles used to quantify the dispersion

in the idler. For its calculation, it is important to use the external angle,

γout...... 90

xiii 6.6 On the left: The design for the compressor. On the right: The layout for

thestretcher...... 92

6.7 The details of the propagation in the compressor...... 93

6.8 The two pulse trains having similar repetition rates, around 80 MHz; the

diﬀerence between the two is 0.1%. On the left, the ﬁrst four pulses are

shown; at t =0, the pulse from both trains overlap perfectly. On the right,

to amplify the eﬀect of the repetition rate mismatch, the same four pulses

are shown after 500 ns; the overlap is clearly lost. However, for the fourth

pulse in (a), the separation is already 37 ps...... 97

7.1 Phase matching curves at diﬀerent noncollinear angles for (a) KTA, and

(b) KTP. The noncollinear angle is varied from 0◦ to 4◦, going from left

to right. For KTA, the extra dashed curve is 3.2◦...... 102

7.2 The details for (a) the compressor and (b) the stretcher, with a top view

and a side view. The drawings are not to scale...... 103

7.3 A general layout of the whole OPCPA system. The speciﬁcs of each com-

ponent, e.g. Nd:YLF ampliﬁers, are oﬀered in their respective sections.

The remaining optics are: lenses for beam shaping and transport, half-

and quarter-wave plates (HWP/QWP) for polarization control, and Fara-

day optical isolators (FOI) to protect the nonlinear ﬁber against leakage

fromtheampliﬁer...... 106

7.4 The ampliﬁed and the unampliﬁed spectrum for (a) the original stretcher,

and (b) the optimized version. Although, the central wavelength diﬀered,

thespectralwidthplottedisthesame...... 109

7.5 The spectra for diﬀerent noncollinear angles. The angle quoted is the

internalone...... 110

7.6 Layout, to scale, of the KTA ampliﬁcation stages. Important distances

arealsoindicated...... 111

xiv 7.7 The beam proﬁle after (a) the second crystal, and (b) the compressor. . . 112

7.8 On the left: the retrieved electric ﬁeld intensity and the retrieved temporal

phase. On the right: the retrieved spectrum and the retrieved phase as

wellasthemeasuredspectrum...... 113

7.9 Cross-correlation trace between the seed and the pump...... 114

8.1 Phase-matching curves for diﬀerent noncollinear angles - from left to right:

0◦,1◦,2◦,3◦...... 118

8.2 By using the proper focal lengths, the collimated beam diameter for both

dispersiveelementscanbeequal...... 119

8.3 Plot showing the magniﬁcation needed for diﬀerent incident angles, for a

200-lines/mmgrating...... 120

8.4 The layout for the angular dispersion compensation setup. L1 is the 150-

mm ZnSe lens and L2 is the 75-mm CaF2 lens...... 121 8.5 The diagnostic tool for the spatial chirp. The mirror is placed a focal

length away from both the grating and CCD camera...... 122

8.6 The beam proﬁle in the diagnostic tool for (a) the compensated beam,

and (b) the uncompensated one. Note, the images are rotated by 90% to

match the actual input of the OPA. In the inlets, the beam proﬁle at the

entranceisshown...... 122

8.7 LayoutoftheMIROPA...... 123

8.8 Results of the characterization of the signal (a-c) and the idler (d-e) of

the AGS OPA. The vertical structures in both beam proﬁles arise from a

triggering issue. There is no auto-correlation trace for the idler...... 124

8.9 Results of the characterization of the signal (a-c) and the idler (d-e) for

the LGSe OPA. The vertical structures in both beam proﬁles arise from a

triggeringissue...... 127

xv Chapter 1

Introduction

The term mid-infrared (MIR) refers to the region of the electromagnetic spectrum going

from 2 m to 20 m, although these boundaries are somewhat artiﬁcial and can change depending on the ﬁeld of research. For chemistry, MIR is extremely important as the frequencies associated with atomic vibrations lie in this region. For many decades already,

MIR spectroscopy has been used to learn more about molecules and their interactions with the environment. More recently, with the advent of ultrafast technology, dynamics and structural changes can be probed in the sub-picosecond time-domain.

This is the motivation behind the work done in this thesis; not just in terms of enabling similar experiments to those already done, but also to push the boundaries for interactions between molecules and electric ﬁelds. The system developed here lays the groundwork for a scalable laser system that will deliver hundreds of microjoules in the MIR, enabling strong ﬁeld control, with laser selective chemistry (LSC) being the ultimate goal.

1.1 Laser Selective Chemistry

Chemistry, at its simplest, is the art of breaking molecular bonds to make new ones. It might sound trivial, but it is not; a great deal of eﬀort is put into ﬁnding new catalysts

1 Chapter 1. Introduction 2 to achieve speciﬁc chemical reactions.

With the invention of the laser, a dream was born: using this new tool to selectively break bonds [1, 2]. With their narrow bandwidth, high-power continuous-wave lasers could be tuned to a particular stretching mode in a molecule and be used to pump energy into it until it breaks. However, by deﬁnition, breaking a bond is a highly anharmonic process. As energy is pumped into the transition, its frequency gets red-shifted with respect to the laser and the interaction is pushed out of resonance.

Using broadband sources or multiple lasers could be an option if it was not for the fact that the energy does not stay in the targeted bond. On the most basic level, a molecule can be described as a series of balls and springs, and vibrations can be analyzed using normal modes. This works perfectly as long as the molecule stays in a low vibrational level. As energy is put into the molecule and higher levels are reached, the harmonic approximation has to break down, otherwise bonds could not be broken. Because of the anharmonicity, diﬀerent normal modes can be nonlinearly coupled together. Therefore if energy is put into a given mode, it will leak out; this is known as intramolecular vibrational-energy redistribution (IVR) [3].

The behavior of IVR varies depending on the size of molecules [3, 4]. For small molecules, there are only a few modes and the energy can ﬂow back into the initial mode. However this low number of modes makes small molecules a bad model to test laser selective chemistry (LSC). For larger ones, the number of modes is such that the molecule acts as a bath, and the energy ﬂow is irreversible, greatly complicating LSC.

The time scale for IVR in large molecules is sub-picosecond.

One way to try to beat IVR is to use high-energy ultrashort pulses in the MIR.

First, energy could be deposited into a bond before it can leak out. And secondly, the broad spectrum associated with short pulses can be matched to the spectrum of the vibrational potential, reducing the eﬀect of its anharmonicity. High pulse energy is necessary to saturate every transition. Around ten photons are needed to reach the Chapter 1. Introduction 3

top of the potential; the probability to absorb one photon has to be close to unity,

otherwise the absorption of a high number of photons will be highly improbable. For

a molecule like benzene with a cross-section of 2.2 10−19 cm2 [5], this means a photon ﬂux of 10/ (2.2 10−19) photon/cm2 is needed. For a 3-m beam focused down to a 100- m diameter, this results in a pulse energy around 250 J, which at the moment is not

achievable at kilohertz repetition rates.

1.2 Mid-infrared optical parametric ampliﬁer

So far the major limiting factor for LSC is technological. The tools needed to really ask

questions about intramolecular potentials are simply not there. To understand how the

project presented in this thesis could be part of the solution, it is important to look at

how MIR is currently generated.

Currently, most MIR systems are based on Ti:Sapphire ampliﬁers, as they can easily

produce sub-100-fs pulses with millijoules of energy at 1 kHz. However, their operating

wavelength, 800 nm, is relatively short compared to MIR, putting harsh requirements on

the crystal, especially on their transmission window. To avoid two-photon absorption, it

is important for the nonlinear crystal to be transparent to the second-harmonic of the

pump, which for Ti:Sapphire is in the ultra-violet. For this reason, MIR is generated

using two cascaded nonlinear processes. First, an optical parametric ampliﬁer (OPA) is

used to generate light in the near infrared. Then the resulting signal and idler are mixed

in a nonlinear crystal; for a diﬀerence-frequency generation (DFG) process, the output

of this second nonlinear stage can cover a wide region, 3 m to 20 m [6]. Because two

diﬀerent nonlinear stages are used, the eﬃciency is low, around 0.5 %, mostly because of

the DFG stage. With this technology, producing 250 J of MIR would therefore require

50 mJ from a Ti:Sapphire, which is far above the performance of current ampliﬁers.

By using a longer wavelength to start with, the OPA could be used to produce MIR Chapter 1. Introduction 4

directly, which would greatly improve the eﬃciency. This can be easily done by using an

Erbium-doped ﬁber laser, operating at 1.55 m. Although this only solves part of the problem: the pump for the OPA still needs to have enough pulse energy.

1.3 Chirped-pulse ampliﬁcation

Amplifying broadband laser pulses to high power can be challenging. The major problem is with their short pulse duration, even at low pulse energy, the peak intensity can be high enough to induce optical damage. For this reason, chirped-pulse amplifiers (CPAs) were introduced in 1985 [7] and are now widely used. In chirped-pulse amplification, dispersion is introduced in a controlled manner to temporally stretch the pulse and reduce the peak power. After amplification, the pulses can be recompressed, the result being short pulses with a high energy.

However, scaling CPAs to high power is still complicated; the major hurdle is heat management. The excess energy of the pump photon is dissipated as heat in the gain medium, as the pump power increases, the thermal eﬀects become more important. Ther- mal lensing, which degrades the beam quality, and thermal-induced stress, which can lead to catastrophic failure of the crystal, put limits on the amount of pumping power that can be used. Furthermore, for high-gain systems, the bandwidth is seriously reduced because of gain-narrowing.

Using nonlinear optics, in a CPA conﬁguration, is an interesting alternative to amplify broadband pulses to high energy; such systems are known as optical parametric chirped- pulse ampliﬁers (OPCPAs). Because the extra-energy of the pump photon is released in the form of a light, even in high-power nonlinear systems, heat is not a limiting factor.

Furthermore, because their operating wavelength is easily tunable, OPCPAs can be used with almost any oscillator.

The major disadvantage of OPCPAs is the harsh requirements put on the synchro- Chapter 1. Introduction 5 nization between the diﬀerent elements. With both CPAs and OPCPAs, a broadband seed is ampliﬁed by a narrowband pump, and they are obtained by using two synchronized lasers. For CPA, the degree of synchronization can be relaxed compare to the

OPCPA concept as the level of synchronization needed is determined by the upper-level lifetime, which is on the order of a microsecond. On the other hand, for OPCPA, more sophisticated synchronization schemes are needed because the gain is instantaneous; in this case the pulse duration, on the order of picosecond, sets the level of the precision required.

1.4 This work - quick overview

In this thesis, a new scheme to generate MIR light is demonstrated, the goal of which is to improve the eﬃciency and provide a new pump source with an all-optical synchronization as an enabling element for further scaling of power.

The first and important distinction with regards to currently used MIR systems is that this one is based on a Erbium-doped fiber laser. This choice was made to open up new options for the MIR OPA design, but also because fiber lasers are affordable and reliable, especially compared to Ti:Sapphire systems.

The foundation upon which this new scheme can be built is OPCPA using a potassium- titanyl-arsenate crystal pumped by a neodymium- doped lithium yttrium fluoride (Nd:YLF) amplifier and seeded by the fiber laser. Its importance resides in the simple fact that without the OPCPA, there is no long-wavelength pump for the MIR OPA.

An important contribution of this work, especially compared with previous work from this group, is the use of passive synchronization. The scheme used, which is introduced for the first time in this work, reduces the timing jitter to a usable level and simplifies the everyday use of the OPCPA. The output of the fiber laser is split in two arms, seed and pump, both having a fiber amplifier. For the pump arm, the output of the amplifier Chapter 1. Introduction 6

is coupled into a nonlinear ﬁber to generate the seed for the Nd:YLF ampliﬁer. Although

the ﬁber was provided to us by the group of Prof. Leitenstorfer, from University of

Konstanz, a great deal of work went into incorporating it to the previous system and

ensuring the stability of its output.

The pump of the OPCPA is a Nd:YLF regenerative ampliﬁer followed by a Nd:YLF

double-pass ampliﬁer. Although both are well established tools in this group, some care

and eﬀort had to be put in making them compatible with the broadband low-power seed

generated by the nonlinear ﬁber. The major modiﬁcation is the introduction of etalons

in the regenerative ampliﬁer cavity to stretch the output pulses and therefore avoiding

optical damage. However, many minor modiﬁcations were needed, especially regarding

the temporal overlap of the pump and the seed pulse.

The compressed output of the OPCPA is used to pump the MIR OPA, for which two

crystals were tested: silver thiogallate and lithium gallium selenide. For the proof-of-

principle, the idler of the OPCPA is used to seed the OPA. This new scheme based on

a pump at 1.6 m represent an important contribution of this work and was recently published [8]. Moreover, as lithium gallium selenide is a fairly recent crystal, this work is among the ﬁrst to investigate its possible use as a replacement for silver thiogallate.

This system provides a robust platform for scaling to average powers and peak powers that will open up LSC as well as numerous other applications requiring high average-peak power pulses in the MIR. Chapter 2

Generation of the seed:

Nonlinear propagation in ﬁbers

Optical fibers play a crucial role on the source side of the optical parametric chirped pulse amplifier. The seeds for both the optical parametric amplifier and the pump laser are derived from a single fiber oscillator, and the light needed to seed the pump laser is generated in nonlinear fiber. Understanding how the electromagnetic field interacts with the fiber is the stepping stone for the next two chapters. In this chapter, the spatial profile of the field and the propagation of a pulse in a fiber will be described in detail.

2.1 Maxwell equations

The propagation and the spatial mode are both described by Maxwell’s equations. This section will lay the ground for the more in-depth discussions that will follows. As ﬁbers are generally non-conducting, σ =0, and non-magnetic, = 0, with no charge density,

7 Chapter 2. Generation of the seed: Nonlinear propagation in fibers 8

ρ =0, Maxwell’s equations can be written as [9]:

∂H (t, r) E (t, r)= , (2.1a) ∇× − 0 ∂t ∂D (t, r) H (t, r)= , (2.1b) ∇× ∂t D (t, r)=0, (2.1c) ∇ H (t, r)=0, (2.1d) ∇ where,

D (t, r)= ǫ0E (t, r)+ P (t, r) . (2.2)

The polarization, P, describes the reaction of the medium to the electric ﬁeld. For a

weak field, the polarization is a linear function of the electric field. As the field increases,

it is necessary to include a nonlinear correction.

P (t, r)= Pl (t, r)+ Pnl (t, r) , (2.3)

where Pl (t, r) is the linear polarization and Pnl (t, r) the nonlinear polarization. In free space propagation, the nonlinear part is normally omitted as the electric ﬁeld rarely gets

strong enough. However, for propagation in a fiber, the field is confined to a tight area for

a long distance, and, even for continuous wave lasers, nonlinearity can become important.

Furthermore, for short laser pulses the peak power can be orders of magnitude higher

than the average power. However, even for short pulses in ﬁbers, it is still possible to

treat the nonlinearity as a perturbation and solve Maxwell’s equations, to get the mode

proﬁle and the dispersion using only the linear polarization.

Pulses are often described by separating the overall ﬁeld in term of an oscillating wave

and an envelope. Although this approximation is in general valid, it may break down for

few-cycle pulses [10]. The pulses described in the thesis are long enough for this so-called

phasor notation to be used. In this case, the ﬁelds and the displacement can then be Chapter 2. Generation of the seed: Nonlinear propagation in fibers 9

written as [11]:

1 E (t, r)= E (t, r) e−iω0t + c.c. , (2.4) 2 0

1 − 0 H (t, r)= H (t, r) e iω t + c.c. , (2.5) 2 0

1 − 0 D (t, r)= D (t, r) e iω t + c.c. , (2.6) 2 0 where E0, H0, and D0 are the envelopes for the electric ﬁeld, magnetic ﬁeld and displacement, respectively, for a wave with a carrier frequency ω0. Using this phasor notation, Maxwell’s equations can be rewritten as:

∂ E (t, r) e−iω0t = H (t, r) e−iω0t , (2.7) ∇× 0 − 0 ∂t 0

− 0 ∂ − 0 H (t, r) e iω t = D (t, r) e iω t , (2.8) ∇× 0 ∂t 0

− 0 D (t, r) e iω t =0, (2.9) ∇ 0 H (t, r) e−iω0t =0, (2.10) ∇ 0 where only the terms that are a function of " ω " are kept. − 0 It is common to think of the medium response to the field as instantaneous, although this is not true. There can be a significant delay and the polarization is therefore a function of the field in the past. This temporality is not easy to treat in the time domain and is often ignored. This assumption can be avoided by working in the frequency domain, which is also a more natural frame to describe the propagation of broadband pulses in a dispersive media. It is common to think of the medium response to the field as instantaneous, although this is not true. There can be a significant delay and the polarization is therefore a function of the field in the past. This delayed reaction is not easy to treat in the time domain and is often ignored. However, this assumption can be avoided by working in the frequency domain, which is also a more natural frame to describe the propagation of broadband pulses in dispersive media. This is done by taking Chapter 2. Generation of the seed: Nonlinear propagation in fibers 10

the Fourier transform of the electric ﬁeld and the displacement:

∞ −iω0t 1 i(ω−ω0) E˜ (ω ω0, r)= F.T. E0 (t, r) e E0 (t, r) e dt, (2.11) − ≡ √2π −∞ D˜ (ω ω , r)= F.T. D (t, r) e−iω0t . (2.12) − 0 0 Furthermore, with the electric fields defined in equation (2.4) Using the definition of the

displacement for an isotropic medium, D˜ (ω ω , r)=˜ǫ (ω) E˜ (ω ω , r), where ǫ˜(ω) is − 0 − 0 the permittivity deﬁned in the frequency domain, the Maxwell equations in the frequency

domain are [12]:

E˜ (ω ω , r)= i ωH˜ (ω ω , r) , (2.13a) ∇× − 0 0 − 0 H˜ (ω ω , r)= iǫ˜(ω) ωE˜ (ω ω , r) , (2.13b) ∇× − 0 − − 0 E˜ (ω ω , r)=0, (2.13c) ∇ − 0 H˜ (ω ω , r)=0. (2.13d) ∇ − 0

Taking the curl of equation (2.13a) and substituting the answer in equation (2.13b) gives:

2E˜ (ω ω , r)= k2E˜ (ω ω , r); (2.14) ∇ − 0 − − 0

where,

ω ω k = ǫ˜(ω)ω = = n (ω) = n (ω) k . 0 c (ω) c 0 In both cylindrical and Cartesian coordinates, it is possible to express the Laplacian

as 2 = 2 + ∂2/∂z2, with 2 being functions of the coordinates transverse to the ∇ ∇T ∇T propagation direction, which is along the z axis, and equation (2.14) becomes:

∂2 2 + E˜ (ω ω , r)+ k2E˜ (ω ω , r)=0. (2.15) ∇T ∂z2 − 0 − 0 At this point it is useful to write the electric ﬁeld as the product of a term related to propagation, E˜ (ω ω ,z), and a term containing the polarization and the spatial mode, p − 0 Chapter 2. Generation of the seed: Nonlinear propagation in fibers 11

Es (ρ,θ). By using the method of separation of variables, equation (2.15) can be split in two diﬀerent diﬀerential equation:

2 E (ρ,θ)+ n2k2 β2 E (ρ,θ)=0, (2.16) ∇T s 0 − s ∂2 E˜ (ω ω ,z)= β2E˜ (ω ω ,z) . (2.17) ∂z2 p − 0 − p − 0

Equation (2.16) deﬁnes the spatial modes and is therefore the stepping stone for the next section. On the other hand, solving equation (2.17) is straight forward and yields:

E˜ (ω ω ,z)= A˜ (ω ω ) eiβ(ω−ω0)z, (2.18) p − 0 − 0 where the envelope is centered at a frequency ω0. This equation will be used later in section 2.3 to examine dispersion in the ﬁber.

2.2 Spatial mode

In this section the spatial modes will be derived, but first it is important to define the polarization of the fields. Purely transversal fields are, in general, only possible for plane waves; for most finite beams, there will be longitudinal fields. In general, the longitudinal fields in optical fibers cannot be ignored and are often used as a starting point for deriving the spatial mode. However, for a small contrast between the refractive indexes of the core and the cladding (n n ) , the fiber is considered to be "weakly 1 ≈ 2 guiding" and it is possible to approximate the exact polarization state to be linear [12].

Those linearly-polarized modes, called LP modes, have a small longitudinal component that can normally be ignored.

For the LP modes, the spatial component of the electric ﬁeld can be split into a radial function and azimuthal one [12]:

E (ρ,θ) R (ρ)Θ(θ)ˆa , (2.19) s ≈ x Chapter 2. Generation of the seed: Nonlinear propagation in fibers 12

where the electric ﬁeld is assumed to be linearly polarized along x. By using this deﬁnition

of the ﬁeld in equation (2.16) as well as writing the transverse Laplacian in cylindrical

coordinates, equation (2.16) is then:

ρ2 ∂2 ρ ∂ 1 ∂2 R (ρ)+ R (ρ)+ n2k2 β2 ρ2 = Θ(θ) . (2.20) R (ρ) ∂ρ2 R (ρ) ∂ρ 0 − −Θ(θ) ∂θ2 Again this can be solved by using the method of separation of variables. The azimuthal part is relatively straightforward to solve and the solution is Θ = eilθ, where l is an integer. The radial part is then:

∂2 1 ∂ l2 R (ρ)+ R (ρ) R (ρ)+ n2k2 β2 R (ρ)=0. (2.21) ∂ρ2 ρ ∂ρ 0 − − ρ2

As the refractive index changes depending on if the ﬁeld is in the core, n1, or in the cladding, n2, equation (2.21) had to be solved for two diﬀerent regions. First, for ρ smaller than the core radius, a, n k is bigger than β, and the quantity n2k2 β2 is 1 0 1 0 − positive. Equation (2.21) is then:

∂2 1 ∂ l2 R (ρ)+ R (ρ) R (ρ)+ p2 R (ρ)=0 ρ a, (2.22) ∂ρ2 ρ ∂ρ − ρ2 ≤ where p2 = n2k2 β2. The solution to equation (2.22) is: 1 0 −

R (ρ)= AJ (pρ)+ A′Y (pρ) ρ a, (2.23) l l ≤ where Jl is a Bessel function of the ﬁrst kind and Yl is a Bessel function of the second ′ kind. The solution needs to hold for ρ = 0 and as Yl diverges at the origin, A = 0. In the cladding, i.e. ρ a, β is bigger than n2k2 and n2k2 β2 is negative. Equation (2.21) ≥ 2 0 2 0 − becomes:

∂2 1 ∂ l2 R (ρ)+ R (ρ) R (ρ) q2 + R (ρ)=0 ρ a, (2.24) ∂ρ2 ρ ∂ρ − ρ2 ≥ where q2 = β2 n2k2. The solution is: − 1 0

R (ρ)= BK (qρ)+ B′I (qρ) ρ a, (2.25) l l ≥ Chapter 2. Generation of the seed: Nonlinear propagation in fibers 13

where Il is a modified Bessel function of the first kind and Kl is a modified Bessel function of the second kind. A formal definition of those two functions can be find in mathematical handbooks such as [13], but their asymptotic expansions are more useful at this point.

For large qρ, these modiﬁed Bessel functions can be expressed as [13]:

e−qρ K (qρ) , (2.26) l ≈ √2πqρ eqρ I (qρ) . (2.27) l ≈ √2πqρ

The solution needs to hold for ρ = and as I diverges as ρ increases, B′ = 0. The ∞ l electric ﬁeld is then:

AJ (pρ) eilθ ρ a, l ≤ Ex =  (2.28) BK (qρ) eilθ ρ a,  l ≥ and the magnetic ﬁeld is: 

CJ (pρ) eilθ ρ a, l ≤ Hy =  (2.29) DK (qρ) eilθ ρ a.  l ≥ However, the ﬁelds in the cladding are not independent of the ones in the core; they are linked together through the boundary conditions. In cylindrical coordinates, those conditions for the electric ﬁeld are:

Ez1 (ρ = a)= Ez2 (ρ = a) , (2.30a)

Eθ1 (ρ = a)= Eθ2 (ρ = a) , (2.30b)

2 2 n1Er1 (ρ = a)= n2Er2 (ρ = a) . (2.30c)

Under the weakly-guiding approximation, n n , the condition (2.30c) becomes E = 1 ≈ 2 r1

Er2 [12]. Because of this simpliﬁcation, the boundary conditions (2.30c) and (2.30b) can be combined and expressed as a function of the linear ﬁeld, Ex:

Ex1 (ρ = a)= Ex2 (ρ = a) . (2.31) Chapter 2. Generation of the seed: Nonlinear propagation in fibers 14

With the electric ﬁeld deﬁned in (2.28), this condition leads to:

ilθ ilθ AJl (pa) e = BKl (qa) e , J (u) A l = B, (2.32) Kl (w) where u = pa and w = qa were introduced. In a similar fashion, the same condition applied for the magnetic ﬁeld gives: J (u) C l = D. (2.33) Kl (w) The transverse ﬁelds are then:

AJ (pρ) eilθ ρ a, l ≤ Ex =  (2.34)  Jl (u) ilθ A Kl (qρ) e ρ a, Kl (w) ≥   CJ (pρ) eilθ ρ a, l ≤ Hy =  (2.35)  Jl (u) ilθ C Kl (qρ) e ρ a. Kl (w) ≥  The last condition is used to quantize the value of p and q. The major assumption behind the LP modes is that the longitudinal fields are negligible. However, no matter how small they are compared to the transverse fields, the boundary condition (2.30a) still applies. From Maxwell’s equations, the longitudinal electric field can be written as a function of the transverse magnetic field: i E = − ( H ) , (2.36) z ǫ˜(ω) ω ∇× s z where

Hs = Hy [sin(θ)ˆaρ + cos(θ)ˆaθ] . (2.37)

Using equations (2.35), the longitudinal electric ﬁeld is then:

iC ilθ ilθ − [pρJ − (pρ) lJ (pρ)] e cos(θ) ilJ (pρ) e sin(θ) ρ a, ǫ˜(ω) ω l 1 − l − l ≤ E = z   iC Jl (u) ilθ ilθ  ρ [qρKl−1 (qρ)+ lKl (qρ)] e cos(θ)+ ilKl (qρ) e sin(θ) ρ a. ǫ˜(ω) ω Kl (w) ≥   (2.38) Chapter 2. Generation of the seed: Nonlinear propagation in fibers 15

The continuity of the longitudinal electric ﬁeld at the interface implies that:

J − (u) w K − (w) l 1 = l 1 . (2.39) Jl (u) − u Kl (w)

This equation is often referred to as the eigenvalue equation. Before continuing, it is

useful to introduce the normalized frequency, an important quantity for ﬁber optics [12]:

2 2 2 2 2 2 V = √u + w = a p + q = ak0 n1 + n2. (2.40)

The normalized frequency, V , is a function of the input field, k0, and of the parameters of the fiber, a, n1, and n2, and because of this the conclusions drawn by using the normalized frequency can be transposed easily for different types of fiber used at various wavelengths. Furthermore, the normalized frequency highlights the relation between p and q, both parameters that define the spatial mode. For a given normalized frequency, p and q are obtained by solving equation (2.39).

Figure (2.1) shows how it can be done graphically [12]. Every point where the two functions cross correspond to an allowed value for p. The associated value for q is obtained by using equation (2.40). Also, each branch of the Jl−1/Jl is associated with a m-index to distinguish between the different modes. The linearly polarized modes in a fiber are designated by LPlm. It is important to note that the number of modes supported by a fiber is dictated by the normalized frequency, therefore by changing either the input wavelength or the fiber parameters the number of modes will vary.

2.2.1 Single mode ﬁber

As stated previously, the number of modes depends on the normalized frequency. As the normalized frequency decreases, the number of modes supported by the ﬁber also decreases. The frequency for which a mode stops propagating is called the cut-oﬀ frequency.

For example, for the LP11 mode, the cut-oﬀ frequency is 2.405. As the cut-oﬀ frequency for the LP is 0, this mode is always present, and for the region 0 V 2.405, the 01 ≤ ≤ Chapter 2. Generation of the seed: Nonlinear propagation in fibers 16

5 0 J /J J /J 1 0 1 0 qK /pK -pK /qK 4 1 0 -1 1 0

3 -2

2 -3

1 m=1 m=2 m=3 m=4 -4 m=1 m=2 -5 0 ap01 ap02 ap03 ap11 ap12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 ap pa

(a) (b)

Figure 2.1: Graphic solution for equation (2.39) for V = 8 and for (a) l = 0 and (b) l =1. In this case, there will be 3 modes associated to l =0, and two to l =1. (Adapted from [12])

fiber is said to be a "single-mode fiber" (SMF). In the following chapters and sections, all fibers are considered to be SMF unless stated otherwise.

In the remainder of this section, diﬀerent new variables will be introduced as they are needed to describe adequately the propagation of pulses in optical ﬁbers.

A ﬁrst important quantity is the conﬁnement factor, which corresponds to the ratio of the power within the core over the total power [12]:

P P F (V )= core = core . (2.41) Ptotal Pcore + Pcladding

The power within the core (and the cladding) is calculated by integrating the time-average

Poynting vector over the respective cross-section. The time-average Poynting vector is deﬁned as [14]:

1 ∗ S¯ = (E H ) . (2.42) 2ℜ s × s Chapter 2. Generation of the seed: Nonlinear propagation in fibers 17

For the LP01 mode, the Poynting vector is:

AC 2 J0 (pρ) ρ a, ¯ 2 ≤ Sz =  2 (2.43) AC J0 (u) 2  2 K0 (qρ) ρ a. 2 K0 (w) ≥  The power within the core is then:

a 2π AC P = J 2 (pρ) ρdρdθ core 2 0 0 0 a2 J 2 (u) = πAC J 2 (u) 1+ 1 . (2.44) 2 0 J 2 (u) 0

Similarly for the cladding:

∞ 2π AC J 2 (pa) P = 0 K2 (qρ) ρdρdθ cladding 2 K2 (qa) 0 a 0 0 a2 u2 J 2 (u) = πAC J 2 (u) 1 1 , (2.45) 2 0 w2 J 2 (u) − 0 where equation (2.39) is used to eliminate the modiﬁed Bessel functions. Using those two equations, the conﬁnement factor can be written as [12]:

w2 J 2 (u) F (V )= 1+ 0 . (2.46) V 2 J 2 (u) 1

Another useful parameter for describing dispersion (next section) is the normalized phase constant [12]:

u2 w2 β2 n2k2 b =1 = = − 2 0 . (2.47) − V 2 V 2 n2k2 n2k2 1 0 − 2 0

As the propagation constant has to be between n1k0 and n2k0, the value of b lies between 0 and 1. The conﬁnement factor can be rewritten as a function of this phase constant

[12]:

1 dbV F (V )= + b . (2.48) 2 dV Chapter 2. Generation of the seed: Nonlinear propagation in fibers 18

2.3 Dispersion

Dispersion can be described as the change in the group (or phase) velocity with frequency

and plays an important role in ultrafast optics. As a pulse propagates in a dispersive

medium, the frequency-dependent velocity will modify the pulse envelope, often resulting

in an increase of the pulse duration and/or a degradation of the pulse quality (e.g.

creation of satellite pulses). As stated previously, the starting point to get a ﬁrm deﬁnition

of dispersion is equation (2.18) which describes the propagating part of the electric ﬁeld:

E˜ (ω ω ,z)= A˜ (ω ω ) eiβ(ω)z, (2.49) p − 0 − 0

where the function A˜ (ω ω ) represents the spectral envelope of a laser pulse centered − 0

at a frequency ω0 and β (ω) is the propagation constant. Dispersion arises from the frequency dependence of the propagation constant, which is equivalent to having a non-

flat phase profile. This phase will modify the electric field in the temporal domain when

taking the Fourier transform. As the exact form of the propagation constant is arbitrary,

it is hard to make any conclusion without further approximation. This is why, when

dealing with dispersion, the propagation constant is expressed as a Taylor series centered

on the carrier frequency ω0:

1 1 β = β + β (ω )(ω ω )+ β (ω )(ω ω )2 + β (ω )(ω ω )3 + ..., (2.50) 0 1 0 − 0 2 2 0 − 0 6 3 0 − 0

n n where βn = ∂ β/∂ω . The dispersion is deﬁned through the group velocity, which is deﬁned as:

− ∂β 1 v = . (2.51) group ∂ω By combining the two equations, (2.50) and (2.51), it is possible to make some important observations. As dispersion is concerned, the zeroth-order, which is simply an oﬀset, plays no role. The ﬁrst-order term is also not contributing to dispersion as it

−1 becomes constant in the expression for the group velocity; [β1 (ω0)] is the value of the Chapter 2. Generation of the seed: Nonlinear propagation in fibers 19 group velocity at the central frequency. The higher terms are the ones responsible for dispersion as they introduce the frequency-dependence of the group velocity. The biggest contribution comes from the quadratic term, which is why the second-order dispersion, called group-velocity dispersion, GV D, is normally used to quantify how dispersive a system is:

∂2 ∂ GV D β = β (ω)= τ , (2.52) ≡ 2 ∂ω2 ∂ω group where τgroup is the group-delay (per unit length). Although this is the rigorous deﬁnition of dispersion, books and articles on ﬁber optics often use the dispersion parameter which is the derivative of the group delay with respect to the wavelength [12]:

∂ 2πc D = τ = β . (2.53) λ ∂λ group − λ2 2

For an optical fiber, there are two contributions to the dispersion of the system. One is directly related to the materials of which the fiber is made. This contribution is the change in index of refraction with frequency that results in dispersion. The second contribution comes from the fiber itself, as the guiding properties are frequency-dependent (e.g. the number of modes supported by the fiber will depend on the frequency). Although the dispersion of an optical fiber is derived as a whole, it is useful to look into the material dispersion first.

2.3.1 Material dispersion

In bulk materials, the propagation constant is [15]:

n (ω) β (ω)= ω. (2.54) c

The index of refraction is often deﬁned using a Sellmeier equation, which is a function of the wavelength; it is useful to write the wavevector in term of wavelength:

2πn (λ) β (λ)= . (2.55) λ Chapter 2. Generation of the seed: Nonlinear propagation in fibers 20

The group delay is then:

∂ λ2 ∂ τ = β (ω)= β (λ) group ∂ω −2πc ∂λ λ2 ∂ n (λ) 1 ∂n (λ) N = = n (λ) λ = , (2.56) − c ∂λ λ c − ∂λ c where N is called the group index. In this form, it is easier to calculate the dispersion parameter ﬁrst, and then use it to get the GVD:

∂ ∂ 1 ∂n (λ) λ ∂2n (λ) D = τ = n (λ) λ = − , (2.57) bulk ∂λ group ∂λ c − ∂λ c ∂λ2 and:

λ2 λ3 ∂2n (λ) β = D (λ)= . (2.58) 2 −2πc λ 2πc2 ∂λ2

2.3.2 Dispersion in optical ﬁbers

In this section the dispersion of the optical fiber will be derived. A problem or difficulty when dealing with fibers is that there are two different indices of refraction. A way to make this more manageable is to introduce a ∆-parameter such as [12]:

2 2 n1 n2 n1 n2 ∆= −2 − , (2.59) 2n2 ≈ n2 where the weakly-guiding approximation, n n , was used for the simpliﬁcation. Using 1 ≈ 2 the ∆-parameter the normalized frequency is:

2πn V = 2 √2∆, (2.60) λ and the propagation constant can be written as:

2πn β (λ)= 2 √1+2b∆. (2.61) λ

In the weakly guiding approximation, ∆ is much smaller than 1, and the propagation constant can be simpliﬁed to:

2πn 2πn 2πn β (λ) 2 (1 + b∆) = 2 + 2 b∆. (2.62) ≈ λ λ λ Chapter 2. Generation of the seed: Nonlinear propagation in fibers 21

This last expression underlines the eﬀect of the waveguide on the propagation constant;

the ﬁrst term is similar to the wavevector for a wave propagating in bulk media, and

the second term involves quantities related to the waveguide. The group delay can be

obtained by deriving the propagation constant with respect to the frequency: ∂ ∂ 2πn τ = β (λ)= 2 (1 + b∆) . (2.63) group ∂ω ∂ω λ The resulting expression can be arranged to give an intuitive form to the group delay

[12]: dβ 1 τ fiber = N F (V )+ N [1 F (V )] + N ∆[F (V ) b] . (2.64) g ≡ dω c { 1 core 2 − core 2 core − } The ﬁrst two terms have the form of the group delay caused by propagation in a bulk media of index n1 and n2 weighted by the fraction of the power within the core, F (V ), and within the cladding, 1 F (V ). The last term can be seen as a correction resulting − from the waveguide geometry. By using equation (2.64), the expression for the dispersion parameter becomes: ∂ ∂ 1 D = τ = N F (V )+ N [1 F (V )] + N ∆[F (V ) b] . (2.65) λ ∂λ group ∂λ c { 1 core 2 − core 2 core − }

The resulting expression can be divided into contributions from the material, Dm, the

waveguide, Dw, and the index proﬁle, Dp [12]:

Dλ = Dm + Dw + Dp. (2.66)

From solving equation (2.65), the diﬀerent dispersion terms are [12]:

D (λ)= D F (V )+ D [1 F (V )] , (2.67) m bulk1 core bulk2 − core 2 2 N2 ∆ d (bV ) Dw (λ)= V 2 , (2.68) − n2cλ dV 2 2 N2 ∆ y y d (bV ) d (bV ) Dp (λ)= 1+ V 2 + b , (2.69) − n2cλ 2 8 dV dV − where the proﬁle dispersion parameter y is: 2n λ d∆ y = 2 . (2.70) − N2 ∆ dλ Chapter 2. Generation of the seed: Nonlinear propagation in fibers 22

An extra term function of ∆ was omitted in equation (2.67). The material dispersion, as

the name suggests, regroups terms that involve dispersion generated by the dependence of

the index of refraction on the frequency. In equation (2.67), both terms are the dispersion

in bulk glass weighted by the power within each region. The waveguide dispersion and the

proﬁle dispersion are similar; both involve derivatives of the normalized phase constant,

which describes how the waveguide properties change with the frequency. However, the

profile dispersion is the only one containing derivatives of the ∆-parameter with respect to the wavelength. Those derivatives are related to how the index profile changes with the wavelength, hence the name, profile dispersion.

2.3.3 Speciﬁc case: SMF-28

In this section, the dispersion of a passive single mode fiber is derived. For 1560 nm, single-mode fibers are typically made of silica glass. The index difference between the core and the cladding is achieved by doping the core with germanium, which results in a increase of index of refraction (less common: the cladding can be doped with fluorine to decrease its index). The weakly-guiding fiber approximation is justified in this case as the index change is typically below 1% for passive fibers. For the calculation of the dispersion, the indexes of refraction are modeled using Sellmeier equations [16]:

2 2 Biλ Diλ ni = √ǫri = Ai + 2 + 2 , (2.71) λ Ci λ Ei − − where i =1, 2 depending if it is the index of the core or of the cladding. For the cladding,

the coeﬃcients are those measured for fused silica [16]. For the core, the coeﬃcients

are also based on fused silica, but the 3 amplitudes (A1,B1,D1) are modified to fit the specifications of the Corning SMF-28 [17], a widely used single-mode fiber for 1.55 m.

An increase of 0.3% of the index of the core was needed to match the cut-oﬀ frequency

of the SMF-28 (1260 nm), resulting in a change of 0.6% of the amplitudes. Using those

indexes, it is possible to calculate the dispersion for a single mode ﬁber with a core Chapter 2. Generation of the seed: Nonlinear propagation in fibers 23

x Ai Bi Ci Di Ei - - m -m

2 2 2 1 1.003 A2 1.003 B2 C2 1.003 D2 E2 2 1.3121622 0.7925205 1.09967310−14 0.9116877 1 10−10

Table 2.1: Coeﬃcients for the Sellmeier equation for the core, i =1, and for the cladding, i =2 (from [16]).

50 20 Total 40 Material Waveguide 0 Profile 30

km) −20 ⋅ 20 /km) 2

10 (ps 2 −40 β D (ps/nm 0 Total −60 Material −10 Waveguide Profile −20 −80 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Wavelength (µm) Wavelength (µm)

(a) (b)

Figure 2.2: Dispersion for a single-mode ﬁber expressed using (a) the dispersion parameter and (b) the group velocity dispersion. diameter of 8.2 m (corresponding to the speciﬁcation of the SMF-28 from Corning).

Figure 2.2 shows the results for the two dispersion parameters as the two variables are commonly used.

2.4 Nonlinear propagation

Nonlinear effects in fibers play an important role in this project, either as a useful side effect in the amplifiers or as the main focus in the highly nonlinear fiber. In the following sections, the details of the nonlinear propagation will be laid out in an attempt to Chapter 2. Generation of the seed: Nonlinear propagation in fibers 24 understand what is happening, especially in the nonlinear fiber.

The starting point is the wave equation with the nonlinear polarization included: 1 ∂2E (r,t) ∂2P (r,t) ∂2P (r,t) 2E (r,t) = L + NL . (2.72) ∇ − c2 ∂t2 0 ∂t2 0 ∂t2 A few assumptions are made at this point to help with the derivation [18]:

• The nonlinear polarization will be treated as a perturbation.

• The polarization is linear and unchanged as the pulses propagate in the ﬁber, which

is technically only true for polarization maintaining ﬁbers.

• The envelope is slowly varying compared to the oscillating ﬁeld, i.e. the slow varying

envelope approximation (SVEA).

Under those assumptions, the electric ﬁeld and the polarizations can be written as: 1 E (r,t)= xˆ E (r,t) e−iω0t + c.c. , (2.73) 2

1 − 0 P (r,t)= xˆ P (r,t) e iω t + c.c. , (2.74) L 2 L

1 − 0 P (r,t)= xˆ P (r,t) e iω t + c.c. , (2.75) NL 2 NL where xˆ is the polarization vector and c.c. is the complex conjugate. As for the linear propagation, the linear polarization can be written as [18]:

t (1) ′ ′ ′ PL (r,t)= ǫ0 χ (t t ) E (r,t ) dt , (2.76) −∞ − or using equations 2.73 and 2.74

t (1) ′ ′ −iω0t ′ PL (r,t)= ǫ0 χ (t t ) E (r,t ) e dt . (2.77) −∞ − For the nonlinear polarization, only the third-order polarization is considered as ﬁbers are in general made of glass, therefore lacking the inversion symmetry needed for second order [19]: ∞ ∞ ∞ P (r,t)= ǫ dt dt dt NL 0 3 2 1× 0 0 0 E (r,t t ) E (r,t t t ) E (r,t t t t ) χ(3) (t ,t ,t ) . (2.78) − 3 − 3 − 2 − 3 − 2 − 1 xxxx 1 2 3 Chapter 2. Generation of the seed: Nonlinear propagation in fibers 25

In this form, the nonlinear response is more complex than needed to describe the nonlinear

propagation in a ﬁber. A major contribution to the polarization comes from the electronic

response, which is essentially instantaneous. If only this contribution is considered the

susceptibility can be simpliﬁed to:

χ(3) (t t ,t t ,t t )= χ δ (t t ) δ (t t ) δ (t t ) , (2.79) xxxx − 1 − 2 − 3 xxxx − 3 − 2 − 1

and the nonlinear polarization becomes:

(3) . PNL (r,t)= ǫ0χxxxx.E (r,t) E (r,t) E (r,t) . (2.80)

By substituting the equation 2.73 for the electric ﬁeld, the nonlinear polarization can be rewritten as:

1 ǫ χ(3) P (r,t)= 0 xxxx xˆ E3 (r,t) e−i3ω0t +3 E (r,t) 2 E (r,t) e−iω0t + c.c . (2.81) NL 2 4 { | | }

For an input frequency of ω0, the system will have a nonlinear response at both the fundamental frequency and its third harmonic. However, because of momentum conservation considerations, the third harmonic can only be generated for precise phase-matching conditions, which are not generally met. Then if only the fundamental terms are kept, the envelope term from equation 2.75 is:

3 P (r,t)= χ(3) E (r,t) 2 E (r,t) . (2.82) NL 4 xxxx | |

The core of the derivation is tedious and out of the range of this thesis (see [20] for more details), and therefore only the result will be shown. But before, a few variables that have to be deﬁned. As for the linear propagation, the electric ﬁeld can be separated in term of a spatial mode and a propagating term:

E˜ (r, ω)= F (x,y) A˜ (z, ω ω ) eiβ0z. (2.83) − 0

However, in this case, because of absorption and nonlinear effect, the envelope A˜ (z, ω ω ) − 0 varies as the pulse propagates in the fiber. The nonlinearity of a system is defined by the Chapter 2. Generation of the seed: Nonlinear propagation in fibers 26

nonlinear constant γ (ω):

ωn γ (ω)= 2 , (2.84) aeff c

where the eﬀective mode size is:

∞ 2 F (x,y) 2 dxdy −∞ | | aeff = ∞ , (2.85) F (x,y) 4 dxdy −∞ | |

and the nonlinear index of refractionn 2, is deﬁned by the medium. The propagation constant β (ω), the nonlinear constant γ (ω), as well as the absorption coeﬃcient α (ω) can be rewritten as a Taylor expansion center on the carrier

frequency ω0:

1 1 β (ω)= β + β (ω ω )+ β (ω ω )2 + β (ω ω )3 + ..., (2.86) 0 1 − 0 2 2 − 0 6 3 − 0 γ (ω)= γ + γ (ω ω )+ ..., (2.87) 0 1 − 0

α (ω)= α0 + ..., (2.88) where

∂nψ (ω) ψ = , (2.89) n ∂ωn ω=ω0 ψ0 = ψ (ω0) , (2.90) with ψ = β,γ, or α. For each parameter, a diﬀerent level of precision is needed. The absorption is set to be constant over the full-bandwidth, whereas for the wavevector, terms up to β2 are needed to include dispersion, and sometimes even the third-and forth- order terms if β2 is close to zero. For the nonlinear parameter, the zeroth-order would normally be enough but the ﬁrst order term is often included for added precision. With n2 and aeff being frequency independent, γ1 is reduced to:

∂γ n (ω ) ω 1 1 ∂n (ω) 1 ∂a γ = = 2 0 0 + 2 eff 1 ∂ω a c ω n (ω ) ∂ω − a ∂ω ω=ω0 eff 0 2 0 ω=ω0 eff ω=ω0 γ0 . (2.91) ≈ ω0 Chapter 2. Generation of the seed: Nonlinear propagation in fibers 27

The nonlinear propagation of a pulse can then be described by [18] ∂A (t,z) ∂ i ∂2 1 ∂3 + β β β ∂z 1 ∂t − 2 2 ∂t2 − 6 3 ∂t3 − 1 ∂ α γ A (z,t) 2 i + 0 A (t,z)=0. (2.92) − 0 | | − ω ∂t 2 0

The term β1 can be eliminated by switching to the retarded frame such as [21]:

z′ = z, T = t v−1z = t β z, (2.93) − g − 1 ∂ ∂ ∂ ∂ ∂ = β , = . (2.94) ∂z ∂z′ − 1 ∂T ∂t ∂T Then, after rearranging the terms, the nonlinear equation is: ∂A (T,z′) α i ∂2 1 ∂3 = 0 β + β A (T,z′) ∂z′ − 2 − 2 2 ∂T 2 6 3 ∂T 3 1 ∂ + γ i A (z′,T ) 2 A (T,z′) . (2.95) 0 − ω ∂T | | 0 The previous result is obtained by taking the nonlinear response to be instantaneous. A

retarded nonlinear response arises from Raman processes as they involve the displacement

of atoms. The typical timescale for those processes is on the order of 100 fs, meaning

that for long pulses, the assumption made previously is valid. However for short pulses,

the Raman eﬀects have to be treated more carefully. One way to do so is to redeﬁne the

nonlinear polarization as [18]: ∞ 3 (3) 2 PNL (r,t)= χxxxxE (r,t) E (r,t1) R (t t1) dt1. (2.96) 4 −∞ | | − The response function, R (t), contains an instantaneous electronic part, (1 f ) δ (t), − R

and a delayed Raman response fRhR (t)Θ(t), where the exact form of hR can be found

empirically [22]. The constant fR represents the "strength" of the Raman contribution to the total polarization and the Heaviside step function insures causality. Therefore, the

generalized nonlinear Schroedinger equation is [20]: ∂A (T,z′) α i ∂2 1 ∂3 = 0 β + β + A (T,z′)+ ∂z′ − 2 − 2 2 ∂T 2 6 3 ∂T 3 ∞ 1 ∂ ′ 2 γ0 i A (T,z ) A (z,t1) R (T t1) dt1 =0. (2.97) − ω ∂T −∞ | | − 0 Chapter 2. Generation of the seed: Nonlinear propagation in fibers 28

This equation can be solved numerically to get the exact solution for the propagation,

or it can be simpliﬁed further for analytic solutions.

2.5 Self-phase modulation

The goal of this section is to give a brief overview of self-phase modulation (SPM) as it is

a important phenomenon for pulses propagating in a ﬁber. The full treatment of SPM for

short pulses requires the inclusion of the dispersion parameters in the nonlinear equation;

however, the added complexity means that numerical simulation is needed. Good insight

can be obtained by neglecting those eﬀects and simply looking at the signature of the

SPM for either a Gaussian or a hyperbolic secant pulse.

The starting point is the nonlinear Schroedinger equation in which the dispersive term

and the Raman contribution are neglected [18]: ∂A (T,z) α = 0 + iγ A (T,z) 2 A (T,z) . (2.98) ∂z − 2 0 | | By introducing a normalized amplitude, U (z,T ) such as [18]:

− α0z 2 A (T,z)= P0e U (T,z) , (2.99) the equation 2.98 can be reduced to: ∂U (T,z) ie−α0z = U (T,z) 2 U (T,z) , (2.100) ∂z LNL | | −1 where the nonlinear length, LNL, is (γ0P0) . This corresponds to the characteristic length for which the nonlinear eﬀects start to be noticeable.

Equation (2.100) can be solved by replacing the normalized envelope by VeiφNL , where

V and φNL are assumed to be real. Doing so gives two equations: one for the real part and one for the imaginary part ∂V =0, (2.101) ∂z ∂φ e−α0z NL = V (T,z) 2 . (2.102) ∂z LNL | | Chapter 2. Generation of the seed: Nonlinear propagation in fibers 29

Solving the ﬁrst equation is straightforward and simply indicates that this new amplitude,

V , does not change during propagation. As V is independent of z, the second equation can be directly integrated. It is noteworthy to notice that at z =0, the nonlinear phase

has to be zero, which means that V and φNL are

V (T )= U (T, 0) , (2.103)

−α0z 1 e 2 φNL = − U (T, 0) αLNL | | = φ U (T, 0) 2 . (2.104) peak | |

The nonlinear phase can be simpliﬁed if the absorption is neglected, which is a good

assumption as the absorption is about 0.2 dB/km for a SMF28 [17] and the nonlinear

length is less than a meter for peak powers above 500 W. In this case, by expanding the

exponential, the nonlinear phase is reduced to:

z φpeak . (2.105) ≈ LNL

SPM leaves the pulse shape untouched but modiﬁes the spectrum. This eﬀect can be investigated by doing a numerical Fourier transform to look at the spectrum. Figure

2.3 shows the spectra for diﬀerent nonlinear phases, for both a Gaussian and a hyperbolic secant pulse shape. From those spectra, there are clearly two strong signatures of

SPM. One is the increase of bandwidth and the other one is the appearance of strong modulations in what was previously a featureless one.

2.5.1 Solitons

Solitons are a type of wave whose amplitude and pulse duration undergo a cyclic modulation when propagating in a dispersive media. Under the right circumstances, solitons can keep their pulse shape intact throughout the full propagation. In optics, solitons occur because of the interplay of anomalous second-order dispersion and nonlinear eﬀects.

Optical fibers offer the perfect environment to observe soliton as the light is confined over Chapter 2. Generation of the seed: Nonlinear propagation in fibers 30

1 1 0π 0π 2π 2π 0.8 4π 0.8 4π

0.6 0.6

0.4 0.4

Normalized intensity (A.U.) 0.2 Normalized intensity (A.U.) 0.2

0 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.3 0.4 0.5 0.6 0.7 Frequency (A.U.) Frequency (A.U.)

(a) (b)

Figure 2.3: The spectra for diﬀerent values of nonlinear phase, one the left for a Gaussian

pulse and on the right for an hyperbolic-secant pulse.

long distance and silica ﬁbers operate in the anomalous regime for 1.55 um pulses. As

solitons play an important role in the eﬀect of the nonlinear ﬁber, this section will do a

brief overview of the optical solitons and some of their characteristics.

Solitons arise from the combined eﬀect of dispersion and nonlinearity, therefore the starting point is the nonlinear equation simpliﬁed to keep only those terms [18]:

∂A (T,z) i ∂2A (T,z) = β + iγ A (T,z) 2 A (T,z) . (2.106) ∂z − 2 2 ∂T 2 0 | |

The derivation can be made easier by using the normalized retarded time τ = T/T0. The

pulse width, T0, is TFWHM /1.665 for a Gaussian and TFWHM /1.763 for a hyperbolic secant

pulse, where TFWHM is the full-width-half-maximum pulse duration of the intensity.

Introducing the normalized amplitude, U (z,τ)= A (z,τ) /√P0, as well, equation (2.106) becomes:

2 ∂U (τ,z) i sgn(β2) ∂ U (τ,z) i 2 = 2 + U (τ,z) U (τ,z) . (2.107) ∂z − 2LD ∂τ LNL | |

2 where the dispersion length, LD, is T0 /β2. This dispersion length can be used to generate Chapter 2. Generation of the seed: Nonlinear propagation in fibers 31

a new dimensionless variable,ζ = z/LD. Equation (2.107) can then be rewritten:

∂U (τ,ζ) sgn(β ) ∂2U (τ,ζ) i = 2 N 2 U (τ,ζ) 2 U (τ,ζ) . (2.108) ∂ζ 2 ∂τ 2 − | |

Although the parameter N seems to be introduced mostly for clarity, it is the main

parameter governing the behavior of the soliton. This is not surprising as it is composed

of both the dispersive and the nonlinear terms [18]:

L γ P T 2 N 2 = D = 0 0 0 . (2.109) L β NL | 2| When N is equal to one, dispersion and nonlinear eﬀects have equal weight, whereas

when N is greater than one, nonlinear eﬀects can be seen to be stronger than dispersion.

To solve equation (2.108), the most general (and powerful) way is to use the inverse

scattering technique. A overview of it can be found in [18]. Although this technique

allows to solve for all integers of N, it is out of the scope of this thesis.

The ﬁnal solutions for an arbitrary N are cumbersome and yield no insight. For this

reason, only the solution for N =1, also called fundamental soliton, and one solution for

N = 2, as an example of a higher-order soliton, will be presented. The solution for the

fundamental soliton is [18]:

i 2 u (τ,ζ)= η sech (ητ) e 2 η ζ , (2.110) where the parameter η is deﬁned by the initial condition. It is common to use u (0, 0)=1

leading to η = 1. For the higher-order solution, the initial conditions dictate the exact

form of the soliton. A common, and useful, initial condition is that the initial pulse has

the form of a ﬁrst order soliton, i.e. u (0,τ) = N sech (τ). Under this condition, the

second order soliton takes the form of [18]:

i 4 cosh (3τ)+3 e4iζ cosh(τ) e 2 ζ u (τ,ζ)= . (2.111) cosh (4τ)+4cosh(2τ)+3cos(4 ζ) The fundamental soliton is the only one to propagate without any change in its envelope, ζ only appears in the phase. For the N = 2 soliton, there are two terms Chapter 2. Generation of the seed: Nonlinear propagation in fibers 32

that will modify the envelope as the soliton propagates: 3 e4iζ cosh(τ) and 3cos(4ζ). However, because both are oscillating functions, the amplitude modulation is cyclic and the initial amplitude will be repeatedly recovered.

In general, the intensity is kept low enough so the fundamental soliton is created as the unchanged pulse shape is more practical. In some situations, higher-order solitons can be advantageous. As ﬁgure 2.4 shows, the pulse width for the second-order soliton can become smaller than the fundamental one. A minimum is reached for 4ζ = π, which corresponds to half of the soliton cycle. This can be advantageous if a pulse is launched in a ﬁber with enough power to excite a second-order soliton and coupled out when the pulse duration is at its minimum. This is sometimes referred to as solitonic compression.

16 ζ = 0 14 ζ = π/8 ζ = π/4 12

|Electric field| (A.U.) 4

0 −3 0 3 Normalized time, τ

Figure 2.4: Pulse envelope of a N=2 soliton at diﬀerent positions.

2.5.2 Third-order dispersion

In the previous section, to derive the expression of the soliton, only the second-order

dispersion was considered, which is in general a good approximation. However, this

breaks down when the wavelength of the soliton is getting closer to the zero-dispersion

wavelength as at this point the second-order dispersion is zero and the third-order term Chapter 2. Generation of the seed: Nonlinear propagation in fibers 33

dominates. For the case that the second-order is still present but weak enough, the

third-order dispersion (TOD) is included as a perturbation and the nonlinear equation

is:

∂u (τ,ζ) 1 ∂2u (τ,ζ) ∂3u (τ,ζ) i + + u (τ,ζ) 2 u (τ,ζ)= iǫ , (2.112) ∂ζ 2 ∂τ 2 | | ∂τ 3 where ǫ << 1. The perturbed soliton is then [23]: | | i u (τ,ζ)= η sech (ηy)exp η2ζ + iǫ 2η2τ 3tanh(ηy) , (2.113) 2 − where y = τ ǫη2z. The TOD has little impact on the envelope; it modiﬁed the group − velocity and the phase, but the width and the amplitude are unchanged. The most

important eﬀect of TOD is to couple the soliton to a dispersive wave, allowing energy

transfer between the two. If this wave is described by a plane wave, e[ikdζ+iωτ], where

ω is the frequency diﬀerence from the carrying frequency of the soliton, then the wave

equation (equation (2.112) without the nonlinear term) gives:

1 k = ω2 ǫω3. (2.114) d −2 −

For the unperturbed case, ǫ =0, then kd < 0. However, as the wavevector for the soliton

2 is ksol = η /2 > 0, there is no coupling between the dispersive wave and the soliton. For ǫ = 0, there will be values of ω for which the two wavevectors will be equal. From [24], this frequency is ω 1/2ǫ. From this the group velocity of the dispersive wave can be 0 ≈− calculated;

−1 ∂k v = d = 4ǫ. (2.115) g ∂ω − ω=ω0 As this is calculated in the retarded frame, a negative group velocity is synonymous with a slower group velocity in the lab frame. A consequence of the energy sharing is that because of conservation of momentum, the center of mass of the spectrum has to stay constant [25]. As the soliton emits energy in the dispersive wave, it will see its frequency shift deeper into the anomalous region (and the dispersive wave deeper into the normal Chapter 2. Generation of the seed: Nonlinear propagation in fibers 34 region) until it reaches a point where the energy sharing between the two is negligible.

The eﬀect of TOD is extremely important for the nonlinear ﬁber described in chapter 4. Chapter 3

Erbium-doped ﬁber system for signal

ampliﬁcation

The erbium-doped ﬁber system is a very important part of the optical parametric chirped-

pulse ampliﬁer. It not only provides the seed for the optical parametric ampliﬁer, it

also acts as the oscillator for the pump laser. This chapter is dedicated to this crucial

component, providing both some background information on erbium-doped ﬁber, and

the details of the system.

The first part of the chapter will cover the optical properties of erbium-doped fiber, and the second part will describe in details the fiber laser and the two amplifiers.

3.1 Optical properties

Before thinking about design, it is important to know what are the optical properties of erbium-doped ﬁbers. Erbium-doped glass exhibits a laser transition around 1560 nm

4 4 between the I15/2 level and the I13/2 level of the 4f electron shell [26]. This transition is part of a three-level system as the lower level is the ground state. However, the situation is somewhat more complex because the levels are split into submanifolds (see ﬁgure 3.1a) by the stark shift resulting from the electric ﬁeld generated by the charge distribution

35 Chapter 3. Erbium-doped fiber system for signal amplification 36 in the glass host. Because the energy diﬀerence between the sub-levels is comparable to

Energy(cm-1 ) 6800 6770 Absorption 6700 4 1.0 6711 I13/2 Emission 6600 6644 6540-48 6500 0.8

-1 0.6

1477nm 1527nm 1559nm 1552nm 1552nm 0.4

Energy(cm ) Intensity (A.U.) 0.2 300 268 200 201 4 I15/2 100 125-133 51-59 0 0 1420 1470 1520 1570 1620 Wavelength (nm)

(a) (b)

Figure 3.1: On the left, the details of the energy levels of the laser transition (adapted

from [26]). A few transitions are highlighted; the arrows indicate if they show up in

the absorption or the ﬂuorescence spectra or both. On the right, the emission and the

absorption spectrum around 1550 nm for the erbium-doped ﬁber (adapted from [27]).

the thermal energy at room temperature, even the "excited" sub-levels are populated.

This is true for both the ground state and the excited state, as the upper level lifetime,

10 ms, is long enough to allow thermalization of the excited electron after pumping

[28]. Those populated excited sublevels in both the lower and the upper states explain

the diﬀerences between the absorption and the emission spectrum (see ﬁgure 3.1b). A

system like erbium-doped glass that deviates from a perfect three-level system is referred

to as a quasi-three-level system.

An important aspect of erbium-doped ﬁber as a laser system is the possible pumping

transitions. There are multiple transitions that can be used to populated the upper

level of the laser transition (see ﬁgure 3.2); however there are 3 major aspects that can

be used to discriminate between them. The ﬁrst one is the energy diﬀerence between

the pump and the laser photons, known as the quantum defect. The quantum defect

reﬂects the amount of energy that has to be dissipated in the system in the form of heat. Chapter 3. Erbium-doped fiber system for signal amplification 37

Therefore, a high quantum defect means that the overall eﬃciency of the system will be low and the amount of heat generated high; both are unwanted. Another aspect that has to be considered is the excited-state absorption (ESA) which happens when an excited ion absorbs a second photon. The excited state involved is the upper-level of either the lasing transition or the pumping transition. However, ESA from the former tends to be stronger as the lifetime is much longer, e.g. 10 ms vs 7 s [26] for erbium-doped glass.

ESA is strongly dependent on the pump wavelength as it is a resonant process. The last aspect to consider is the population of the upper level of the pumping transition. For a perfect three- or four-level system, this level is assumed to be un-populated at room temperature with a rapid decaying pathway to the upper level of the lasing transition.

If those two conditions are met, the system can be fully inverted. Of all the possible pumping transitions available, only the three lying in the near-infrared are used; the other ones, in the visible, are discarded on the grounds of their high quantum defect.

The remaining transitions will be looked at in more detail.

The 810-nm transition is a weak choice for erbium-doped ﬁber. It exhibits strong ESA from the upper level of the lasing transition and the quantum defect is quite high, but the system can be fully inverted. This transition was used in the past because high-power laser diodes were ﬁrst available in this spectral range, but with the advent of high-power

980-nm laser diodes, this transition has fallen out of favor.

The 980-nm transition is commonly used to pump erbium-doped ﬁber ampliﬁers as its quantum defect is smaller than for 810-nm. There is a pathway for ESA, but the absorption of the second pump photon occurs from the upper level of the pumping transition, which minimizes its importance. The 980-nm transition, like the 810-nm one, has negligible population in the upper-level of the pumping transition and can be fully inverted.

The 1480-nm transition is used to pump erbium-doped systems because of its small quantum defect and the total absence of ESA. However, at 1480 nm, the upper level of Chapter 3. Erbium-doped fiber system for signal amplification 38

14 4 4I I11/2 13/2 980nm 1480nm 12

) 10 4 I9/2 800nm 8

Attenuation(dB 4

0 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Wavelength(ìm)

Figure 3.2: Absorption spectrum for an erbium-doped ﬁber (adapted from [26]). The three common pumping bands are shown with the arrows. The level involved and the wavelength associated with it are indicated. the transition is thermally populated reducing the maximum inversion achievable. There are laser diodes available at 1480-nm but the power is limited.

3.2 Dispersion in erbium-doped ﬁbers

Dispersion in erbium-doped fibers has contributions from the glass host and from the erbium ions. The host consists of a silica glass with some co-dopants, aluminum and germanium, to facilitate the incorporation of the erbium ions. The dispersion from the germano-aluminosilicate fiber is similar to a normal single-mode fiber (as described in section 2.3.3). For the contribution from the erbium ions, as the real and imaginary part of the susceptibility are related to each other, changes in the absorption spectrum will translate in changes in the index of refraction. Using a Lorentzian model for the absorption and the emission cross-section, it is possible to write the propagation constant Chapter 3. Erbium-doped fiber system for signal amplification 39 as follows [29]:

β (ω)= βhost (ω)+ βEr (ω) n (ω) ω F = host + σR (ω) N σR (ω) N . (3.1) c 2 e 2 − a 1 where nhost is the refractive index of the glass host, F is the fraction of the power

R R within the core, σe and σa are the real parts of the emission and of the absorption cross-section respectively due to the erbium ions, and N2/1 is the population density of the upper/lower level of the laser transition. The real part of the cross-sections can

be expressed as a function of the peak emission/absorption wavelength, λe/a and the

bandwidth of the feature, ∆λe/a. For each cross-section, there are contributions from

diﬀerent resonant features originating from diﬀerent combinations of sub-levels. From

[29], the cross-sections can be written as:

e/a e/a e/a 2Ck λ λk /∆λk R − − σe/a (λ)= 2 , (3.2) e/a e/a k 1+ 2 λ λ /∆λ − k k where the values for the diﬀerent constants can be found in table 3.1. The second-order

e e e a a a k λk Ck ∆λk λk Ck ∆λk m 10−25 m m 10−25 m

1 1.53 2.02 0.014 1.46 0.275 0.020

2 1.534 2.13 0.058 1.48 1.30 0.035

3 1.56 0.898 0.023 1.51 1.93 0.049

4 1.58 0.422 0.086 1.53 3.19 0.016

5 - - - 1.55 1.37 0.036

Table 3.1: The parameters for the emission and the absorption cross-section [29].

dispersion term can be obtained by taking the derivative of equation (3.1) twice with Chapter 3. Erbium-doped fiber system for signal amplification 40 respect to the frequency:

2 R 2 2 R ′′ d σe/a (λ) λ d λ dσe/a (λ) σe/a = 2 = dω −2πc dλ −2πc dλ 3 R 2 R λ dσe/a (λ) d σe/a (λ) = 2 2 + λ 2 . (3.3) (2πc) dλ dλ Using equation (3.1), the second-order dispersion term is:

F β = βhost + [σ′′ (ω) N σ′′ (ω) N ] . (3.4) 2 2 2 e 2 − a 1

The exact contribution from erbium ions is a function of the population density of each

40 150

20 100

50 0 0

/km) /km) 2 -20 2

(ps (ps -50

2 2

b b -40 -100

-60 -150

-200 -80 1500 1520 1540 1560 1580 1600 1500 1520 1540 1560 1580 1600 Wavelength(nm) Wavelength(nm)

(a) (b)

Figure 3.3: The second-order dispersion for two diﬀerent ion concentrations. The con-

centration was set to (a) 4.25 1025 ions/m3 and to (b) 14.2 1025 ions/m3. The arrows indicate the direction of increasing inversion starting with all the ions in the lower level until the system is fully inverted. level, and therefore of the ion concentration and the pump power. Figure 3.3 shows the dispersion parameter for two different ion densities and for each, the inversion parameter is varied. From those two graphs, it should be clear that giving a single dispersion parameter for an erbium-doped fiber is difficult but also that pump power can be used to tweak the dispersion. However, other groups have worked on similar amplifiers and Chapter 3. Erbium-doped fiber system for signal amplification 41 the dispersion can be approximated, for design purposes, to be 57 ps2/km [30], which is

roughly three times larger, in absolute value, than a normal SMF ( -20 ps2/km). ≈

3.3 Fiber laser

The oscillator is a passively mode-locked laser from Calmar Laser. The details of the

laser can be found in [31] and a summary is presented here. The oscillator is based

on a Fabry-Perot cavity with at one end a wavelength-tuning element (e.g. ﬁber Bragg

grating) and at the other end a saturable absorber (see ﬁgure 3.4). There are both

slow and fast saturation processes present in the response of the absorber to an external

electromagnetic ﬁeld.

Laser Diode

Erbium-dopedfiber

OC WDM Saturable Wavelength-selective Absorber Reflector

Figure 3.4: Layout of the ﬁber laser; WDM,wavelength-division multiplexer, OC, Output

coupler.

The slow component of the saturation is responsible for the self-starting of the modelocking; no outside active element is needed to induce modelocking. The fast component helps to reduce the pulse duration but it is not the limiting factor. By controlling the dispersion, e.g. by tweaking the length of single-mode fiber in the cavity, the laser can operate in a solitonic regime. In this case, the saturable absorber interacts with the soliton and the final pulse duration is a function of the dispersion in the cavity and the modulation depth induced by the saturable absorber [32]. The spectrum of the fiber laser is Chapter 3. Erbium-doped fiber system for signal amplification 42

1 5 1 Retrieved E. field Retrieved spectrum Retrieved phase Retrieved phase Measured spectrum 0.8 4

2 0.6 3

0.5 0.4 2 Phase (rad) Phase (rad) Intensity (A.U.) Intensity (A.U.)

0.2 1

0 0 0 0 −1000 −500 0 500 1000 1540 1560 1580 1600 1620 Time (fs) Wavelength (nm)

(a) (b)

Figure 3.5: On the left: The retrieved electric ﬁeld for the output of the ﬁber laser. On the right, the retrieved and the measured spectrum. centered around 1560 nm with a 50-nm bandwidth and a corresponding pulse duration of

85 fs (see ﬁgure 3.5). The pulses were characterized using a homemade second-harmonic frequency resolved optical gating (FROG). The output power is 8.91 mW for a repetition rate of 19.8 MHz at pump diode current of 315 mA.

3.4 Erbium-doped ﬁber ampliﬁer

The design of the amplifiers is based on two papers [30, 33], both related to spectral broadening of an erbium-doped fiber system. The first issue to deal with is the choice of the fiber. To avoid excessive nonlinearity, which would deteriorate the quality of the pulse, it is important to keep the length of the fiber as short as possible. On the other hand, sufficient pump power has to be absorbed to achieve the desired gain. The target of these amplifiers is to boost the pulse energy to the nJ level; therefore the gain has to be on the order of 10. By using highly-doped fibers, it is possible to use short fibers with good pump absorption. The fiber chosen for the amplifiers is a highly-doped fiber from Liekki that exhibits an attenuation of 80 dB per meter at 1530 nm, compared to Chapter 3. Erbium-doped fiber system for signal amplification 43

30 dB per meter for more standard fibers. Spectral broadening by self-phase modulation in the fiber can be beneficial if it is kept relatively low. From previous work [30, 33], choosing a smaller core for the amplifier can generate sufficient self-phase modulation to broaden the spectrum without degrading the pulses. For this reason, a 4-m core fiber was chosen over a 8-m core. For the pump, although the 980-nm pumping band exhibits

ESA (see previous section) and has a bigger quantum defect, 980-nm lasers diodes have been preferred over 1480-nm for simplicity and power availability.

3.4.1 First ampliﬁer

For the first amplifier, a 2-m fiber was used based on previous work [30]. Two 270- mW 980-nm laser diodes were used in a bi-directional configuration (see fig 3.6). The laser diodes exhibit a linear dependence between the output power and the pumping current, with a slope of 0.54 mW per mA, with the maximum power achieved at 520 mA.

Wavelength-division multiplexers (WDMs) are used to combine and separate the pump and the seed. The losses associated with the WDM and the splices are about 13% of the input power. The power going through the erbium ﬁber, measured at the 980-nm port

Laser Laser diode1 diode2

Erbium-dopedfiber

Ouputport Inputport WDM WDM

Figure 3.6: Layout of the ﬁber ampliﬁer. of the second coupler, was on the order of 50 W. This corresponds to an attenuation of 18 dB per meter compared to an expected attenuation of 50 dB per meter (from the Chapter 3. Erbium-doped fiber system for signal amplification 44 manufacturer data [27] and the ratio of the absorption at 1530 mn and 980 nm (from

ﬁgure 3.2). The discrepancy between the two can be attributed to the bleaching of the erbium ions caused by the high-pump power and the long-lived metastable state (10 ms).

This high absorption justifies the use of a bi-directional pumping, as to avoid attenuation of the signal, population inversion has to be achieved over the whole amplifier. To avoid feedback from one diode to the other, the operating wavelength of one diode is shifted ; the first diode wavelength is 980 nm 1 nm versus 977 1 nm for the second one. The ± ± amplified spontaneous emission (ASE) when the amplifier is unseeded is 20 mW. When

seeded with 1.988 mW, the maximum output is 38.1 mW, corresponding to a gain of 20.

The input and the output spectra are showed in ﬁgure 3.7a. Modifying the pumping

1 2 Amplifier: output 520 mA Amplifier: input 450 mA 0.8 350 mA 1.5

0.6 1 0.4 Intensity (A.U.) Intensity (A.U.) 0.5 0.2

0 0 1540 1560 1580 1600 1620 1540 1560 1580 1600 1620 Wavelength (nm) Wavelength (nm)

(a) (b)

Figure 3.7: On the left, the output spectrum of the ﬁrst ampliﬁer with both pump diodes

at their maximum current, 520 mA, as well as the spectrum for the input. On the right,

the spectra for diﬀerent currents of the ﬁrst diode.

condition should aﬀect not only the gain but also the spectrum. Because the spectrum

is broadened by self-phase modulation, a change in the peak intensity would modify the

spectrum. As was shown in the previous section, the exact dispersion of the ampliﬁer

will be a function of the pump power. Therefore by modifying the pumping condition,

not only is the pulse energy changed but also its pulse duration. For the first amplifier, Chapter 3. Erbium-doped fiber system for signal amplification 45 changing the pumping current has little effect on the spectrum. Tweaking the current of the first diode has shown more pronounced, but still small, changes. Figure 3.7b shows the spectrum for different currents of the first diode. The major difference of going to 520 mA from 350 mA, is a decrease of the peak around 1550 nm and an increase in the peak at 1600 nm. This relative insensitivity could be due to the fact that as the inversion is increased, the gain is higher, but the dispersion parameter gets closer to the anomalous region. One of the important ideas behind the design is that the amplifier will cancel the anomalous dispersion of the SMF and compress the pulses as they propagate. Therefore, by reducing the dispersion of the amplifier, the compression is not as important, and the gain in energy is offset by the longer pulse duration. Also, the drop in the output power of the amplifier does not correspond to the change in the power of the first diode as the current is dialed down. At 450 mA, the output power of the amplifier goes to 36.0 mW, a drop of 6%, and for 350 mA, it goes down to 33.5 mW, a drop of 12 %, whereas the pumping power drops by 15% and 35% respectively. This seems to indicate that the amplifier is operating near saturation.

The output spectrum can be modified by pre-chirping the input pulses by adding a single-mode fiber before the amplifier. As the dispersion from erbium-doped fibers and single-mode fibers can have opposite sign, chirp acquired in the single-mode fiber would be compensated as the pulses propagate through the amplifier. As nonlinearities are a function of the intensity, depending on the phase profile of the input pulses, the spectrum will be more or less affected. Because of the 50-50 splitter, there is a minimum of 60-cm of fiber before the amplifier. A 1-m single-mode fiber was added before the amplifier to see if the spectrum at the output of the amplifier could be improved. As a side note, adding the fiber does not modify the spectrum at the input of the amplifier. However, the spectrum at the output is greatly modified; figure 3.8a shows the result for two different diode currents. For both currents, the spectrum has similar features; the bandwidth is reduced and there are sharper modulations. As neither is an improvement, no pre-chirp Chapter 3. Erbium-doped fiber system for signal amplification 46

2 2 520 mA 520 mA 350 mA 350 mA Ref Ref 1.5 1.5

1 1 Intensity (A.U.) Intensity (A.U.) 0.5 0.5

0 0 1540 1560 1580 1600 1620 1540 1560 1580 1600 1620 Wavelength (nm) Wavelength (nm)

(a) (b)

Figure 3.8: The spectra at the output of the first fiber amplifier with three different currents for the first laser diodes using a fiber (a) before and (b) after.

ﬁber was used.

Because the tip of the output fiber of the amplifier got damaged during the early stage of its operation, it was decided to add a short fiber, 400 mm, after the amplifier to make it easier to replace the output tip in case of damage. The output power went down by nearly 50% due to different losses in the fiber - mostly from a defective splice. The output spectra for two different currents are shown in figure 3.8b; the addition of the

fiber does not strongly modify the spectrum. The final pulse characteristics are showed in figure 3.9. Interesting enough, the chirp is strong enough that the spectral feature can be resolved in the time domain.

3.4.2 Second ampliﬁer

The second fiber amplifier is similar to the first amplifier although there are two main differences. The second amplifier was to be used with the highly nonlinear fiber (chapter

4). To facilitate the connection between the two, it was decided to add a 1-m long single mode fiber after the amplifier. To avoid over-stretching the pulses, the erbium amplifier Chapter 3. Erbium-doped fiber system for signal amplification 47

1 50 1 Retrieved E. field Retrieved spectrum Retrieved phase Retrieved phase 70 Measured spectrum

0.5 0.5

Phase (rad) 50 Phase (rad) Intensity (A.U.) Intensity (A.U.)

0 0 0 −1500 −1000 −500 0 500 1000 1500 1540 1560 1580 1600 1620 Time (fs) Wavelength (nm)

(a) (b)

Figure 3.9: On the left: Retrieved electric field of the output of the first amplifier. On

the right, retrieved and measured spectra.

was built with a 3-m long ﬁber. To compensate for a possible increase in losses, more

powerful diodes were used. The diodes used have a maximum power of 370 mW at 560

mA with a slope of 0.66 mW per mA. However, the losses in the WDM and the splices are

close to 40%. The output power of the ampliﬁer at 1560 nm is 38 mW, comparable to the

one of the ﬁrst ampliﬁer, and the ASE is 24 mW. The operating current is determined

using the output of the nonlinear ﬁber (chapter 4) and has to be adjusted from day-

to-day depending on the conditions in the lab. Figure 3.10 shows the output of the

second ampliﬁer at one of those set points (current in the ﬁrst diode, 436 mA; in the

second, 560 mA). The electric ﬁeld recovered from this speciﬁc data set is typical; for all

optimal set points, there is a sharp peak containing most of the power, with some pre-

and post-pulses.

Figure 3.11 shows the temporal proﬁles at 4 diﬀerent pumping conditions. For currents other than the operating one, the pulses are getting longer. For the extreme cases,

(see ﬁgures 3.11e and 3.11h), the pulses developed strong satellites. At the same time, the spectra are changing, but unlike the pulse duration, the spectrum is not "optimal" at the operating condition; it is broader at 350 mA than it is at 436 mA. From those Chapter 3. Erbium-doped fiber system for signal amplification 48

1 Retrieved spectrum Retrieved phase Measured spectrum

0.5 Phase (rad) Intensity (A.U.)

0 0 1540 1560 1580 1600 1620 Wavelength (nm)

(a) (b)

Figure 3.10: On the left: Retrieved electric ﬁeld at the output of the second ampliﬁer.

The FWHM is 89 fs and 65% of the energy is in the central peak. On the right, retrieved and measured spectra. two observations, it can be suggested that by modifying the pumping conditions, the dispersion, and therefore the pulse duration, are modified through the influence of the excited erbium ions. The spectra suggest that a slight under pumping over compressed the pulses as it would lead to more SPM, i.e. a broader spectrum, but also a longer pulse duration. For the over-pumped amplifier and the one pumped at 250 mA, the smaller bandwidth indicates the effect of SPM is reduced. As the pumping is increased, the dispersion is decreasing, which for the over-pumped amplifier would result in an incom- plete pulse compression that would explain the reduced SPM. On the other hand, for the amplifier pumped at 250 mA, the pulses would be compressed and stretched too quickly, reducing the effective length of the fiber, resulting also in a reduced SPM.

This part of the thesis work was central to the proper amplification of the seed pulses and for stable generation of the frequency shifted input to the pump amplifier chain. One can appreciate the sensitivity of the amplification process with respect to the spectrum, time profile, and dispersion of the seed pulse from this study. This part of the thesis work provided the foundation for the critical design of the subsequent optical system and Chapter 3. Erbium-doped fiber system for signal amplification 49 passive time synchronization.

1 1 1 1 250 mA 300 mA 500 mA 560 mA 436 mA 436 mA 436 mA 436 mA 0.8 0.8 0.8 0.8

0.6 0.6 0.6 0.6

0.4 0.4 0.4 0.4 Intensity (A.U.) Intensity (A.U.) Intensity (A.U.) Intensity (A.U.) 0.2 0.2 0.2 0.2

0 0 0 0 1520 1540 1560 1580 1600 1620 1520 1540 1560 1580 1600 1620 1520 1540 1560 1580 1600 1620 1520 1540 1560 1580 1600 1620 Wavelength (nm) Wavelength (nm) Wavelength (nm) Wavelength (nm)

(a) 250 mA (b) 300 mA (c) 500 mA (d) 560 mA

1 1 1 1 250 mA 300 mA 500 mA 560 mA 436 mA 436 mA 436 mA 436 mA 0.8 0.8 0.8 0.8

0.6 0.6 0.6 0.6

0.4 0.4 0.4 0.4 Intensity (A.U.) Intensity (A.U.) Intensity (A.U.) Intensity (A.U.) 0.2 0.2 0.2 0.2

0 0 0 0 −1000 −500 0 500 1000 −1000 −500 0 500 1000 −1000 −500 0 500 1000 −1000 −500 0 500 1000 Time (fs) Time (fs) Time (fs) Time (fs)

(e) 250 mA (f) 300 mA (g) 500 mA (h) 560 mA

Figure 3.11: The spectra, (a) to (d), and the temporal profile, (e) to (h), for different currents in the first diodes. Also for each current, the spectrum/pulse shape for the operating current of 436 mA is plotted (dash curve) for comparaison. Chapter 4

Highly-nonlinear ﬁber for all-optical synchronization

The highly nonlinear ﬁber (HNLF) is the central part of the synchronization system.

The ﬁber was provided by the group of Prof. Alfred Leitenstorfer from the University of Konstanz and was carefully designed using numerical simulation by Dr. Alexander

Sell. The HNLF is made of two fibers spliced together, a precompensating fiber and a dispersion-optimized fiber. In first part this chapter, each of those two components will be described in detail. In the second part, the characteristic of the output will be addressed. Numerical simulations for the nonlinear fiber were by our collaborators in

Germany but were not included in the thesis; instead the emphasis is put on the a simple treatment of the physics and how it can or cannot explain the complex HNLF.

4.1 Precompensating ﬁber

The precompensating fiber (PF) is a regular single-mode fiber used to compress the pulses via solitonic compression (section 2.5.1) before they enter the nonlinear fiber. As was shown previously, for solitonic compression, the length of the fiber is crucial to obtain the minimal pulse duration (see section 2.5.1).

50 Chapter 4. Highly-nonlinear fiber for all-optical synchronization 51

Although numerical simulation is needed to fully capture the behavior of the pulses in the PF, the simple solitonic description can still be used to describe the evolution in the pulses in the PF. In section 2.4, the nonlinear coeﬃcient was deﬁned as:

ω0n2 2πn2 γ0 = = . (4.1) aeff c aeff λ For a single mode fiber (SMF-28), the nonlinear refractive index, n , is 2.7 10−16 cm2/W 2 [34] and the mode diameter at 1.56 m is 10.4 m at a wavelength of 1.56 m resulting in a nonlinear coefficient of 1.28 (kW m)−1 . The input pulses are 89 fs long (FWHM) with an energy of 1.9 nJ resulting in a peak power of 19 kW. From those two parameters, the nonlinear length (LNL = 1/γ0P0) is 41 mm. At 1.56 m, the GVD for a silica fiber

2 is about -25 ps /km. For a Gaussian beam with a FWHM pulse duration of 89 fs, T0 is

2 53 fs and the dispersion length (LD = T0 /β2) is 112 mm. From those two characteristic length, the soliton order is (N = LD/LNL) is 1.65. As the soliton launched in the ﬁber is close to N =2, its behavior can be explained, to ﬁrst order, by using the N =2 soliton

[18]. In section 2.5.1, it was shown that the pulse duration of a second-order soliton

reaches a minimum half-way through its cycle. From the expression for the envelope of a

second-soliton (equation (2.111)), this corresponds to 4ζ =4z/LD = π. For a dispersion

length LD of 112 mm, this gives a propagating distance of 87.6 mm. The PF used in this setup was optimized (by our collaborators) solving the full nonlinear Schroedinger equation, and its length is 80 mm. The simple solitonic description used here was for a N=2 soliton with no higher order dispersion, which is not entirely true for the PF, and could explain noted discrepancies. For the estimation of the compression factor, the simple picture breaks down completely. By using the second-order soliton without any modification, the ratio of the duration at the input over the pulse duration at the output is 4.5. However from simulation, the same ratio is estimated to be 2.8 [35]. The reason behind the different propagation lengths can also explain the difference in compression factor. Furthermore, as dispersion is involved in setting the pulse duration, it is not surprising that the estimation of the compression is off. Chapter 4. Highly-nonlinear fiber for all-optical synchronization 52

4.2 Dispersion optimized ﬁber

The second part of the nonlinear fiber is a dispersion-optimized fiber (DOF). The dispersion of this fiber was optimized using simulation to give the best conversion at 1 m.

The dispersion curve of the ﬁber used here is shown in ﬁgure 4.1a. Although the exact

Precompression fiber 40 Dispersion−optimized fiber

/km) 0 2 (ps 2 β −20

−40 n2 2a −60 1 1.2 1.4 1.6 1.8 2 2b Wavelength (µm)

(a) (b)

Figure 4.1: On the left side, the dispersion curves for the two components of the HNLF.

On the right side, the index proﬁle for the dispersion-optimized ﬁber. Typical value for

the index change is around 3 % for the increase and below 0.5 % for the reduction [36, 37].

The ratio of the two diameters is on the order of 0.5-0.6 [37].

physical properties of the fiber were not specified, those types of fibers are achieved by

modifying the refractive index proﬁle and reducing the core dimensions. Figure 4.1b

shows a typical proﬁle for a nonlinear ﬁber [36]; the region of high refractive index is

doped with germanium whereas the regions of lower index are doped with ﬂuorine. To

keep the ﬁber single mode and to increase the nonlinearity the core of the ﬁber is reduced

to less than 4 m. This core reduction coupled with the increase of the nonlinear index

of refraction due to the augmentation of germanium ions leads to a higher nonlinear

coeﬃcient, 9 (W km)−1 compared to 1.3 (W km)−1 for a normal silica ﬁber. For an input pulse of 31 fs (i.e. using a compression factor of 2.8) and with an energy of 1.5 nJ, the Chapter 4. Highly-nonlinear fiber for all-optical synchronization 53

2 nonlinear length is 2.6 mm. The dispersion length is 65 mm, where a β2 of -4.4 ps /km was used (see ﬁgure 4.1a). For this input pulse, the ﬁber is able to support solitons up to N=5 order.

4.3 Mechanism for wavelength tuning

The basic mechanism has been attributed [35] to similar processes leading to soliton

ﬁssion [38]. The DOF is optimized to have its zero-dispersion wavelength (ZDW) closer to the spectral region associated with the input spectrum as well as being less dispersive in this region, resulting in an increased inﬂuence of the third order dispersion (TOD).

The eﬀect of TOD is discussed in more detail in section 2.5.2, but in summary, because of TOD the soliton is coupled to a dispersive wave propagating in the normal dispersion region (β2 > 0). As the soliton propagates, its central wavelength moves away from the ZDW as energy is transferred to the dispersive wave until the soliton is stabilized. For

the ﬁber used in this project the ZDW is 1.44 m, meaning that the 1.56-m soliton will get red-shifted. It has been pointed out by [35] that, although the eﬀect of TOD is responsible for some of the spectral broadening, further spectral separation results from four-wave mixing between the soliton and the dispersive wave. Because of the mixing, the group-velocity mismatch between the two waves is the limiting factor in the amount of spectral broadening achievable as it leads to the loss of temporal overlap.

4.4 Result

The spectrum of the output of the HNLF is shown in ﬁgure 4.2. For the dispersive wave,

figure 4.2a, the center of the spectrum is around 1060 nm with a bandwidth of 50 nm, which is sufficient to cover the operating wavelength of the Nd:YLF amplifier, 1053 nm.

On the red-side, there are two clearly distinct peaks. The ﬁrst and most intense one corresponds to the spectrum of the unshifted pulse, i.e. same spectrum as the input. Chapter 4. Highly-nonlinear fiber for all-optical synchronization 54

The second and smaller peak, around 1680 nm, is associated with the red-shifted soliton.

The presence of the unshifted peak results from the fact that the soliton launched is not a fundamental soliton (a more detailed analysis can be found in [39])

1 1

0.9 0.8 0.8

0.6 0.7 0.6

0.4 0.5 Intensity (A.U.) Intensity (A.U.) 0.4 0.2 0.3

0 0.2 1000 1050 1100 1150 1450 1500 1550 1600 1650 1700 1750 1800 Wavelength (nm) Wavelength (nm)

(a) (b)

Figure 4.2: The spectrum at the output of the HNLF. On the left, the spectrum associated

with the dispersive wave and on the right, the spectrum associated with the soliton and

the unshifted beam. 3 mW is generated in the dispersive wave.

4.4.1 Sensitivity to the pumping conditions

It was pointed out in section 3.4.2 that tweaking the current in the diodes modiﬁes the

profile of the pulses coming out of the amplifier. Not surprisingly, this also affects the

performance of the HNLF. From ﬁgures 3.11 and 4.3, it is possible to see that, for the

dispersive wave, the shortest pulse is associated with the highest conversion eﬃciency

and the shortest peak wavelength. Any modiﬁcation of the pumping conditions results

in a decrease of the intensity coupled with a small red-shifting of the spectrum (ﬁgure

4.3a). This general behavior is similar whether the current is increased or decreased,

but the output is more sensitive to an increase of current, as can be seen in ﬁgure 4.3a.

This asymmetrical sensitivity is consistent with what has been observed in the output Chapter 4. Highly-nonlinear fiber for all-optical synchronization 55 of the ﬁber ampliﬁer (see section 3.4.2). The behavior of the soliton associated with the

4 x 10 2000 2.4 380 mA 380 mA 400 mA 2.2 400 mA 425 mA 425 mA 1500 460 mA 2 460 mA

1.8

1000 1.6

1.4 Intensity (A.U.) Intensity (A.U.) 500 1.2

0 0.8 1000 1050 1100 1150 1600 1650 1700 1750 1800 Wavelength (nm) Wavelength (nm)

(a) (b)

Figure 4.3: The spectrum at the output of the HNLF. Only the peaks associated with

the dispersive wave and the red-shifted soliton are shown.

dispersive wave is quite diﬀerent. The intensity of the peak corresponding to its spectrum

is fairly insensitive to a change in pumping conditions but the central wavelength moves

as the current is changed; the solitonic peak is the most red-shifted at the operating

current. As the current is turned down, the peak is initially unchanged then moves

towards the main peak. The eﬀect is similar for the over-pumped situation, but it is

more drastic; for 460 mA, the red-shifted peak is partially overlapping with the main

peak.

4.4.2 Stability

The role of the HNLF is to passively synchronize two laser systems by seeding a Nd:YLF

ampliﬁer and as a consequence the stability of the output is crucial. On short timescales,

up to hours, the output is extremely stable, both in terms of amplitude and spectral

quality (peak wavelength, bandwidth). However, throughout the day but mostly from

day to day, there is some drift in the output that can be compensated by changing the Chapter 4. Highly-nonlinear fiber for all-optical synchronization 56 pumping conditions. The timescale of these changes seems to indicate that they are due to drifts in the temperature of the lab, which can aﬀect both the HNLF and the Erbium ampliﬁer.

The HNLF is sensitive to temperature changes because it modiﬁes the dispersion. As the dispersion of the ﬁber was optimized to maximize the output at 1 m, any change in

the index of refraction will reﬂect on the performance. The variation of dispersion with

temperature was computed for two diﬀerent types of ﬁber [16, 40], and in both cases the

change was on the order of 1.5 10−3 ps/nm km◦C, which is small compared to the net dispersion. The inﬂuence of the temperature on the HNLF was tested by using a heated

fiber holder. In an effort to stabilize the output spectrum of the HNLF, the fiber was mounted on an aluminum holder equipped with a thermo-electric cooler (TEC). Using this TEC, it is possible to change the holder temperature. To test the sensitivity of the HNLF, the holder was set to 23.7◦C and the spectrum was optimized by tweaking the current in the first diode. After, the temperature of the holder was changed and the spectrum recorded without reoptimization. The result is shown in figure 4.4. There are clear changes of amplitude with changes in temperature. Although the temperature jumps used for the testing correspond to changes that could be seen in the lab, jumps of a few degrees are not common enough to fully explain the sensitivity of the output, especially as the holder is temperature controlled.

1000 26.1 °C 24.6 °C 800 23.7 °C 23.1 °C 22.3 °C 600

400 Intensity (A.U.)

200

0 1000 1050 1100 1150 Wavelength (nm)

Figure 4.4: The spectrum of the dispersive wave for diﬀerent temperatures of the holder. Chapter 4. Highly-nonlinear fiber for all-optical synchronization 57

Another element that could be sensitive to temperature change is the amplifier, which consists consists of 3-meter of active fiber that is cooled by conduction through the base-plate and by natural convection. Neither the temperature of the base-plate of the amplifier or of its enclosure are actively controlled; the cooling relies solely on the thermal inertia of the lab. Therefore, any change in the room temperature (and to a certain extent, the humidity) will change the operating temperature of the amplifier, which will change the dispersion and the pulse characteristics. Those changes would be small because the effect of temperature on the dispersion of the erbium-doped fiber should be on the same scale as a single-mode fiber because the host glass is similar. However, because the output of the HNLF was shown to be quite sensitive on the characteristic of the input pulse, it is possible that even changes in the amplifier dispersion are enough to modify the output spectrum of the HNLF. Because the amplifier is passively cooled, testing this hypothesis is quite difficult. However, by changing the base-plate from plexiglas to aluminum and therefore allowing efficient conductive cooling as well as coupling the temperature of the base-plate to the temperature of the optical table, the long term stability of the HNLF is improved.

In conclusion, as stability is concerned, even with the slow drift, the output is stable enough to allow the system to be used for hours without tweaking. The use of the HNLF to provide two colours, in a fully synchronized manner, for the amplifier chain was a key development in this thesis work. The degree of synchronization will be discussed in chapter 7 where it is shown to be well sufficient for stable amplification. The stability and robustness of this all-optical approach of synchronization is a significant improvement over the previous approach that employed electronic synchronization as the system had to be tweaked many times during a hour and the degree of synchronization was insufficient for stable amplification using parametric amplification. Chapter 5

Nd:YLF ampliﬁcation system

3+ Neodymium-doped yttrium lithium fluoride (Nd :LiYF4 - Nd:YLF) lasers and amplifiers have been and still are a major work horse in many laser labs. Using either lamps or laser diodes, Nd:YLF systems can deliver millijoules of energy at different repetition rates. However, with their narrow bandwidth, they cannot break the ps barrier for pulsed operation. For this reason, they are often used as pump lasers for Ti:Sapphire lasers or, as in this case, optical parametric chirped-pulse amplifiers (OPCPAs).

In this chapter, the design of the Nd:YLF ampliﬁer will be detailed, starting with the laser properties of Nd:YLF, followed by an overview of regenerative ampliﬁers. In the last part of the chapter, the performance of the current Nd:YLF system is discussed.

5.1 Optical properties

In this section, the optical properties of Nd:YLF will be discussed. First and foremost,

YLF is a birefringent crystal, and therefore the optical properties depend on the orientation of the polarization with respect to the major axes of the crystal, labeled a, b, and c. Because the crystal is uniaxial, the b-axis is the same as the a-axis; only the latter will be used. The c-axis, being unique, acts as a reference for the polarization. The polarization perpendicular to the c-axis is called σ-polarization and the one parallel is

58 Chapter 5. Nd:YLF amplification system 59 called π-polarization. For the rest of this chapter, the electric ﬁeld will be assumed to be

σ-polarized unless otherwise stated.

For two given levels, because of the Stark shift, there are a multitude of possible

transitions. However, only the one with the highest cross-section is considered, and be-

cause the cross-section varies with the polarization, this dominant transition will change

accordingly. It is common to label the transition after the wavelength of the photon

emitted.

Like many of the other rare-earth ions (e.g. erbium), the diﬀerent lasing transitions in

Nd:YLF involve electrons in the 4f shell. There are two main lasing transitions in Nd:YLF

4 both having the F3/2 state as the upper level, which has a lifetime of 500 s. For the

4 ﬁrst transition, the lower level is the I11/2 state. For the σ-polarization, the wavelength associated with the dominant transition is 1053 nm and for the π-polarization, it is 1047

4 nm [41]. For the second transition, the lower level is the I13/2 state and the wavelengths are 1313 nm (σ) and 1321 nm (π) [41]. The 4F 4I transition is normally used 3/2 → 11/2 because its stimulated cross-section is more than ﬁve times bigger than the cross-section

for the 4F 4I transition [42]. 3/2 → 13/2 Another important property to consider is the thermal lensing. Because cooling can

only be done through the surface of the gain medium and the energy is deposited over

the whole volume, this generates a temperature and a stress gradient. Both aﬀect the

index of refraction and the result is the creation of an eﬀective lens. It is important

to note that for a birefringent crystal, the focal length of this lens depends on both

the crystal axes and on the polarization. Figure 5.1 shows the detail of the energy

levels for the 4F 4I transition and the stimulated-emission cross-section for the 3/2 → 11/2 two polarizations. For regenerative ampliﬁers, the σ-polarization 1053-nm transition

is normally used, even though the cross-section is higher for the π-polarization 1047- nm transition. One reason is that Nd:YLF exhibits a weaker thermal lens for the σ-

polarization, which allows for simpler designs for the ampliﬁers (See section 5.2 for more Chapter 5. Nd:YLF amplification system 60

4F 11,597 3/2 11,538 20

-20 2 1053nm 1047nm 10

2,264 4 2,228 5 I11/2

2,079 Cross-Section(10 cm ) 2,042 1,998

1030 1040 1050 1060 1070 1080

Wavelength(nm)

(a) (b)

Figure 5.1: On the left: the two levels associated with the main transition and their sub-sublevels [43]. On the right: The stimulated cross-section for each polarization for the 4F 4I transition [42]. 3/2 → 11/2 detail on design issues). Another advantage of the 1053-nm transition is the energy storage capacity. Parasitic oscillations due to ampliﬁed spontaneous emission (ASE) are the limiting factor for how strong the gain can be, and thereby how much energy can be stored. The strength of the ASE is a function of the stimulated-emission cross-section and the gain; therefore for a given level of ASE, the gain will be higher for a lower cross-section.

The spectrum of Nd:YLF is qualitatively different from that of Er:glass. The spectrum for the latter is mostly featureless with a bandwidth of roughly 55 nm. For Nd:YLF, as shown in figure 5.1b, there are numerous well-resolved, few-nm-wide, peaks. This difference is due to the nature of the host matrix. For Nd:YLF, because the host is a crystal, the static-electric field background is more uniform from site to site, resulting in a smaller inhomogeneous broadening compared to a glass host. Figure 5.2a shows the absorption spectra for YLF for a π-polarized pump. For narrow-band pumps, the main

4 pumping band is located around 800 nm and is associated with the F5/2 level (ﬁgure Chapter 5. Nd:YLF amplification system 61

5.2b). Although the absorption is stronger at 792 nm, the line at 806 nm is normally used, the main reason being that more powerful AlGaAs/GaAs laser diodes operate at this wavelength. The 806-nm peak is also more attractive because of its broader bandwidth. The output wavelength of a laser diode is a function of its temperature; a broader feature will be less sensitive to the temperature ﬂuctuation of the diodes. In recent years, there has been a regain of interest [44, 45], ﬁrst for neodymium-doped

3+ yttrium aluminum garnet (Nd :Y3Al5O12 - Nd:YAG) and neodymium-doped yttrium

3+ vanadate (Nd :YVO4), two other common Nd-doped crystals, to pump directly to the upper level of the lasing transition at 880 nm. This reduces the heat load and improves the

light-to-light eﬃciency as the quantum defect is smaller. However, for applications where

those two features are not crucial, the advantages of the 806-nm transition, highlighted

previously, still hold against the 880-nm transition.

1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2 Absorption coefficient (A.U.) Absorption coefficient (A.U.)

0 0 400 500 600 700 800 770 780 790 800 810 820 830 Wavelength (nm) Wavelength (nm)

(a) (b)

Figure 5.2: On the left side, absorption spectrum for Nd:YLF [46]. On the right side, a

4 close-up of the absorption associated with the F5/2 level. Chapter 5. Nd:YLF amplification system 62

5.1.1 Pump source

There are three common types of optical pump sources available for Nd:YLF ampliﬁers:

flashlamps, continuous-wave arc lamps, and laser diodes. Flashlamps and arc lamps are both based on tungsten lamps filled with either xenon or krypton. Because they have a broad emission spectrum, the overall electric-to-optical efficiency is low and a lot of heat is deposited in the gain medium. Furthermore, there are strong fluctuations in the output power of the lamps limiting the stability of the amplifier. For these reasons, both types of lamps are now supplanted by laser diodes in most amplifiers, except for low-repetition rate high-pulse energy systems. For those, stability is not as much of an issue and the high power output of the flashlamps can be fully exploited. Laser diodes have imposed themselves as the main pump source, partially because of the downsides of lamps, but also because of their attractive features. Because of their narrow spectrum, energy can be put efficiently in a low-lying pumping level, reducing the heat load and improving the overall efficiency. Furthermore, because of their directional output and their smaller size, laser diodes can be used in more compact and versatile pumping schemes. However, diodes also have their drawbacks. For one, their wavelength is a function of their temperature.

This by itself is not too problematic. Typical changes are in the order of 0.3 nm/◦C, but because laser diodes are normally used to access narrow features in the absorption spectrum, even small changes in wavelength can impact greatly on the amount of energy absorbed. Because of this, laser diodes need more sophisticated cooling systems than lamps. Another drawback is that diodes are prone to catastrophic failures. A sharp rise in temperature or an electrostatic discharge can be enough to destroy or seriously damage a laser diode, and extra-care is needed when handling and operating diodes. Chapter 5. Nd:YLF amplification system 63

5.2 Regenerative ampliﬁer

Regenerative ampliﬁers (regens) have been used with Nd-doped materials since the early

1980s [47] and are still widely used. The main feature of a regen is the cavity surrounding its gain medium, which allows the overlap of the seed with the gain region to be optimal for a high number of passes. Because of this amplification of low-energy pulses can be achieved with a good energy extraction and low noise. The coupling in and out of the cavity is achieved with minimal losses by using an optical switch, which also imposes limitations on the gain. As they propagate in the amplifier, the pulses accumulate a nonlinear phase due to the Kerr effect. In the temporal domain this leads to self-phase modulation (see section 2.5). For the spatial profile, because the center of the beam has higher intensity, it will pick up this nonlinear phase faster than the wings. For a strong enough phase, this self-focusing becomes important and can lead to optical damage. The nonlinear phase is accounted for by calculating the so-called B-integral defined as:

2π L B = n (z) I (z) dz. (5.1) λ 2 0 The spatial dependence of the nonlinear index is to account for the diﬀerent materials

in the cavity. The value of B should be kept below 3 [48] to avoid optical damage.

By adding material to the beam path, the limit for the B-integral is reached for lower

intensity. For a given gain material, there will be a maximum extractable pulse energy

that depends on the pulse duration.

5.2.1 Optical switch

An important element of the regenerative ampliﬁer is the optical switch. A common

design for a switch is based on a folded conﬁguration and consists of a Pockels cell (PC),

a quarter-wave plate, a polarizer, and a folding mirror, as shown is ﬁgure 5.3. The main

component of the PC is a crystal showing signiﬁcant change of index of refraction when

a voltage is applied. Typical crystals used are often nonlinear crystals such as rubidium Chapter 5. Nd:YLF amplification system 64

Output

V = Vλ/4 V = 0 Input Input TFP QWP Crystal Mirror TFP QWP Crystal Mirror

(a) Switch close (b) Switch open

Figure 5.3: The detail of the polarization state in the case of (a) the switch close and

(b) the switch open. The dash arrows show the direction of propagation. TFP, thin-ﬁlm

polarizer, QWP, quarter-wave plate.

titanyl phosphate, (RbTiOPO4 - RTP) and beta-barium borate, (β-BaB2O4 - BBO). If the polarization of the input pulse is properly aligned with respect to the crystal axes,

◦ applying the "half-wave" voltage, Vλ/2, will rotate the linear polarization by 90 ; the PC

then behaves like a half-wave plate (HWP). By applying a "quarter-wave" voltage, Vλ/4, the PC acts as a quarter-wave plate (QWP) and a linear polarization becomes circular, and vice-versa. Figure 5.3 shows how by applying the right voltage in the PC, it is possible to change the output port of the switch.

The maximum repetition rate of a PC is limited by two factors. The ﬁrst one is related to the crystal itself. For some crystals, the piezo-electric eﬀect is strong enough, and the deformations induced by the train of high-voltage pulses can excite resonances.

This modulates the index, reducing the eﬃciency of the PC [49].

The second factor is the electronics needed to drive the PC. For a typical BBO PC

, the quarter-wave voltage will be on the order of 5 kV [49]. With rise times around 10 ns, the charging/ discharging of the crystal generates high currents and, through ohmic losses, heat. As the repetition rate increases so does the heat generated, and eventually dissipating this heat becomes impossible. Regens working at 100 kHz are quite common and by using clever tricks, repetition rates up to 250 kHz can be achieved [50]. However, to maximize the output pulse energy, it is better to run regen at lower rates. For Nd:YLF, Chapter 5. Nd:YLF amplification system 65 as the upper-level lifetime is 500 s, a 1-kHz repetition rate oﬀers the optimal combination of pulse energy and average power.

5.2.2 Cavity stability

By using a cavity, a regen can easily amplify pJ pulses to the mJ level, but to achieve this level of amplification without being swamped by noise, different factors need to be considered. First, to maximize the energy extraction while reducing the ASE, the TEM00 mode of the cavity and the gain region must overlap closely. For a side-pumped laser with a cylindrical gain medium, this requires maximal filling of the aperture. A common design benchmark is to aim for a mode diameter measuring 2/3 of the aperture diameter; for this ratio, the transmission of the aperture is 99% for a Gaussian beam [51]. Secondly, ≈ the cavity has to be robust against thermal lensing. The changes in parameters of the cavity mode should be small over the range of thermal lens power experienced by the regen. Also, as the thermal lens is astigmatic for birefringent crystals, the design of the cavity should minimize the effect of this asymmetry on the output beam quality.

5.2.3 Regenerative pulse shaping

By trapping a pulse in a cavity, not only can it make many more trips in the gain medium, but also it will interact more strongly with other components. This increased interaction can be used to modify either the shape of the pulse or its spectrum; this is referred to as regenerative pulse shaping. A simple use of regenerative pulse shaping is the stretching of the pulse duration by using an intra-cavity etalon. As stated previously a limiting factor in the energy extraction of an ampliﬁer is optical damage caused by self-focusing. For a given system, by increasing the pulse duration, and therefore reducing the intensity, it is possible to amplify the pulses to higher energy. For broadband pulses, gratings can be used to stretch the pulses in time, but for narrowband gain media, such as Nd:YLF, this technique is hard to apply. For this reason, regenerative pulse shaping is used. Chapter 5. Nd:YLF amplification system 66

The eﬀect of the etalon on the pulse duration is easy to explain. As a pulse reaches the

etalon, most of the power goes through, but a small fraction is reﬂected back. Because

the etalon surfaces are parallel to such a high degree, some of the back reﬂected light

is reﬂected in the forward direction by the second surface, creating a small tail to the

pulse. For an uncoated etalon, the eﬀect of single pass is weak, but for a high number of

round-trips in the cavity this leads to signiﬁcant broadening.

Formally, the interaction between the etalon and a pulse can be described as a con-

volution between the transfer function of the etalon, h (t), and the electric ﬁeld of the laser pulse [52]:

∞

Eout (t)= h (τ) Ein (t τ) dτ. (5.2) −∞ − There are a few parameters important to describe the transfer function of an etalon:

the thickness of the etalon, d, its index of refraction, n, the product of the amplitude

reﬂectivity of each surface, r1 and r2 denoted by R, and the etalon transmitivity, T = 1 R. With those parameters, the transfer function of the etalon can be built from the − simple description of the pulse-etalon interaction made in the previous paragraph [52]:

∞ h (t)= T Rm exp(iφ ) δ (t mτ) , (5.3) m − 0 where τ is the transit time, 2nd/c, and φm is the extra phase resulting of m round-

trips in the etalon, 2mndk0, with k0 being the wavevector. The convolution of this transfer function with the input electric ﬁeld results in the generation of a pulse train

with decreasing amplitude:

m E (t)= T Rm exp(iφ ) E (t mτ) . (5.4) out m in − 0 It is important to notice that there is an appreciable diﬀerence in behavior depending on

the transit time. For transit times longer than the pulse duration, the reﬂected pulses do

not overlap much and the resulting electric ﬁeld is highly structured. As the transit time

gets closer to the pulse duration, the modulations become less important and eventually Chapter 5. Nd:YLF amplification system 67 disappear. Figure 5.4a shows the simulated output intensity for a 1-mm etalon (7-ps transit time for a round trip) for two diﬀerent input pulse durations. Although there is a

1 1 FWHM: 3.5 ps FWHM: 3.5 ps FWHM: 10.5 ps FWHM: 10.5 ps 0.8 0.8

0.6 0.6

0.4 0.4 Intensity (A.U.) Intensity (A.U.)

0.2 0.2

0 0 −150 −100 −50 0 50 100 −150 −100 −50 0 50 100 Time (ps) Time (ps)

(a) (b)

Figure 5.4: The simulated output intensity of a pulse after going through 160 times in

(a) a single 1-mm etalon, and (b) a combination of a 1-mm etalon and a 0.7-mm etalon.

Two diﬀerent pulse durations were used: 3.5 ps and 10.5 ps.

clear diﬀerence in the overall structure of the pulse, both pulse shapes share a common

feature: a overall asymmetry due to a small trailing edge. The pulse shape can be made

less sensitive to the pulse duration by using two diﬀerent etalons, as both would induce

modulations in the output with diﬀerent periods. The net eﬀect is a smoothing of the

pulse shape. Figure 5.4b shows the result of a simulation for a 0.7-mm etalon and a

1-mm etalon. Even for an input pulse of 3.5 ps, which is smaller than the transit time of

both etalons, there is no sign of modulation

Another point to take into consideration is the inﬂuence of the phase on the pulse

shape. For the previous simulation, the phase was assumed to be 2π, but this is not always true. One would expect that like any other coherent superposition a phase jump of π between two subsequent pulses would result in destructive interference and the pulse shape would be the most affected. Although the phase can be modified by changing the thickness of the etalon, in practice this is done by tweaking the angle of the beam with Chapter 5. Nd:YLF amplification system 68 the etalon. To determine the sensitivity of the transmitted pulse shape on the incident angle, the process was simulated for several angles and the transmitted intensity was integrated. Also as reference, the phase-shift for every angle was calculated; the results are shown in figure 5.5. From the graph of the integrated intensity, it can be seen that the transmission of the etalon is not too sensitive at near normal incidence and that there is a sharp drop for a π-phase, which also modulate the intensity (see figure 5.5a).

1 2 3

(a) (b)

Figure 5.5: On the left side: the transmitted pulse shape for angle from 0◦ to 4◦, with a

interval of 0.25◦ between each. On the right side, the integrated intensity as a function

of the incident angle, as well as the phase. The numbers relate features to each other.

5.2.4 Eﬀect of gain

In the previous section, the eﬀect of an etalon on a pulse was described for a passive

cavity. However, for a regenerative ampliﬁer, distortions introduced by the ampliﬁcation

itself have to be considered at the same time as the shaping done by the etalon.

The most important effect of an amplifier is to act as a filter. Because the gain

medium has a ﬁnite linewidth, any excess bandwidth in the pulse spectrum will be cut,

resulting in an increase of the pulse duration (in the case of a transform-limited pulse). Chapter 5. Nd:YLF amplification system 69

Furthermore, this effect is amplified by gain narrowing, which describes the reduction of the bandwidth of an amplifier compared to the atomic linewidth as the gain is increased.

The reason behind this narrowing is that the gain spectrum is not ﬂat over its bandwidth; for example, laser transitions are often described by a Lorentzian linewidth. Because of

1 Low gain High gain 0.8

0.6

0.4 Normalized gain (A.U.) 0.2

0 −100 −50 0 50 100 Frequency (THz)

Figure 5.6: The normalized gain spectrum for low and high gains. In both case, the same

atomic linewidth was used.

this, for frequencies near the center of the linewidth, the gain will be higher than for

frequencies in the wing. This causes the center of the spectrum to grow much faster,

resulting in a narrower spectrum than the linewidth should dictate. Figure 5.6 shows two

Lorentzian gain curves with the same linewidth but with diﬀerent gains. The two curves

were normalized to highlight the reduction of bandwidth associated with the increase of

gain.

This bandwidth reduction can be formally derived by ﬁrst writing the gain as [48]:

g G (ω) = exp  0  , (5.5) 2(ω ω ) 2 1+ − 0   ∆ω   a    with:

g0 =∆NσmaxL, (5.6) Chapter 5. Nd:YLF amplification system 70

where ∆N is the population inversion density, σmax is peak value of the cross-section and L is the propagation length in the amplifier. However, because the population inversion is difficult to measure, the small-signal gain, g0, is better defined as a function of the gain

at the central frequency, G0:

g0 = ln(G0) . (5.7)

From there, the FWHM bandwidth of an ampliﬁer, ∆ω, can be determined by solving equation (5.5) for G (∆ω/2) = G0/2 resulting in:

ln(2) ∆ω =∆ωa . (5.8) ln(G0/2) As regenerative ampliﬁers can easily reach gains as high as 109, bandwidth reductions of

5 times can occur. For Nd:YLF, the bandwidth would be 0.27 nm from a linewidth of

1.35 nm for a minimum pulse duration of 7 ps.

The finite bandwidth of the amplifier is not the only way gain shapes the pulse envelope; saturation also introduces distortions. Saturation occurs when the intensity of a laser pulse is strong enough to significantly reduce the population inversion. A subsequent pulse would therefore experience less gain. However, this can also be seen during the amplification of a single pulse. In the saturation regime, the trailing edge of a pulse experiences a lower gain than the leading edge because the central peak depopulated the upper level of the transition. The amplification of a square pulse (see figure 5.7) is a simple way to illustrate this; the front part of the pulse always gets more gain than the back resulting in asymmetry in the pulse shape. For more realistic pulse shapes, the effect of saturation is more complex to predict. As pointed out in [53], the characteristic of the leading edge plays a central role. With Gaussian pulses, the leading edge rises faster than saturation such that there will be a steepening of the pulse temporal profile.

On the other hand, with hyperbolic-secant pulses, because its slope is not steep enough, the leading edge is unchanged and after an initial increase, the pulse duration is constant under saturated ampliﬁcation. Chapter 5. Nd:YLF amplification system 71

3 Power (A.U)Power 2

0 t0 Time (A.U.)

Figure 5.7: The eﬀect of saturation on a square pulse. The ﬁrst curve is the input pulse, whereas the other two are the output for two propagation lengths. Adapted from [53].

5.3 Regenerative ampliﬁer design

Figure 5.8 shows the details of our regenerative ampliﬁer, which is based on a Cutting-

Edge Optronics laser head consisting of a 63-mm-long 3-mm-diameter 0.9%-Nd-doped

YLF rod pumped by nine 20-W diode bars (AlGaAs/GaAs) for a total power of 180 W.

The diodes are positioned around the rod in a side-pumping configuration. The coupling in and out of the cavity is achieved using a 4-mm-aperture BBO Pockels cell. The length of the cavity is 1.64 m, which corresponds to a round-trip time of 11 ns. Both end-mirrors are flat and are combined with a 1.5-m lens to make a stable cavity. The lens is placed just in front of the laser head to reduce the effect of the thermal lens. From the manufacturer of the laser head, the thermal lens has a focal length of -5 meters along the a-axis and of 7.5 meters along the c-axis at a pumping current of 18 A. Using specialized software for laser design (Paraxia-plus from Sciopt) it is possible to simulate the behavior of the cavity of the regenerative amplifier and to find the beam radius at different positions in Chapter 5. Nd:YLF amplification system 72

From HNLF To Multipass M4

QWP 220 mm L3 M1 L 500 mm 2 E2 HWP2 240 mm

TFP LH 1 L

M3 M2

HWP1 E1 230 mm

290 mm

680 mm

Figure 5.8: Layout of the regenerative ampliﬁer. LH, laser head, HWP, half-wave plate,

QWP, quarter wave-plate, L, lens, TFP, thin-ﬁlm polarizer, E, etalons, M, mirrors. the cavity for diﬀerent powers of the thermal lens (see table 5.1).

f ﬁrst mirror second mirror laser head

m m m m

0 650 680 770

-5 750 750 820

7.5 600 660 760

Table 5.1: The beam radius calculated with Paraxia at diﬀerent elements in the cavity.

The two non-zero focal lengths represent the thermal lens.

Because the thermal lens has two diﬀerent focal lengths, the cavity mode is astigmatic.

In an attempt to compensate for this, a more sophisticated cavity using cylindrical lenses could be used, but the current design was successfully seeded with low-energy pulses, therefore this option was not pursued.

This amplifier was originally seeded with a Nd:YLF modelocked laser with an output pulse duration of 70 ps. The new input beam is the output of the HNLF, with a bandwidth of 50 nm centered at 1060 nm, and a power of 3 mW, after filtering out the components at longer wavelength. However, because of Nd:YLF narrow bandwidth, the regenerative Chapter 5. Nd:YLF amplification system 73 amplifier is effectively seeded with roughly 1 pJ. The output of the HNLF is broad enough to exploit the full bandwidth of the amplifier, meaning that the pulses could be as short as 7 ps. As the previous version of the laser system [54] was already operating at a

B-integral close to 3, it was decided to insert an etalon in the cavity to stretch the pulse.

Three scenarios were tested: a 1.5-mm etalon, a 1-mm etalon, and two etalons.

5.3.1 Performance - no etalon

Before testing the etalons, the output of the regen without etalon was tested. A pumping current of 15 A was used to avoid over-amplifying the pulse and damaging the regen.

At this pumping current, the output power is 0.7 W and the switching time is 1.498 s,

which corresponds to 136 round-trips in the cavity. Figure 5.9 shows the auto-correlation

of the output pulse as well as two diﬀerent ﬁts, a Gaussian and hyperbolic secant squared.

Both ﬁts capture the behavior around the peak, and neither does extremely well for the

wings. From the Gaussian ﬁt, the pulse duration can easily be extracted and is 12 ps,

1 AC: Signal Fit: Gaussian 0.8 Fit: Sech2

0.6

0.4 Intensity (A.U.)

0.2

0 −40 −20 0 20 40 Time (ps)

Figure 5.9: Autocorrelation trace of the output of the regenerative ampliﬁer with no

etalon inside. The FWHM of the trace is 17 ps resulting in a pulse duration of 12 ps, if

a Gaussian ﬁt is used.

signiﬁcantly bigger than the 7 ps associated with a gain-limited ampliﬁer. The excess

pulse duration can be attributed to gain saturation as explained in the previous section. Chapter 5. Nd:YLF amplification system 74

5.3.2 Performance - etalon

For the regenerative ampliﬁer with etalons, the pulses were characterized by the cross- correlation of the un-stretch 1.55 m pulses from the seed arm of the OPCPA with the

1 m pulses from the regenerative ampliﬁer. This characterization method was used because, for a short "probe", it directly yields the pulse shape as well as the pulse duration. Figure 5.10a shows the trace for the 1.5-mm etalon. The predominant feature

1 1 XC XC Sim Sim 0.8 0.8

0.6 0.6

0.4 0.4 Intensity (A.U.) Intensity (A.U.)

0.2 0.2

0 0 −60 −40 −20 0 20 40 60 −60 −40 −20 0 20 40 60 Time (ps) Time (ps)

(a) (b)

1 XC Sim 0.8

0.6

0.4 Intensity (A.U.)

0.2

0 −60 −40 −20 0 20 40 60 Time (ps)

(c)

Figure 5.10: Cross-correlations traces for (a) a 1.5-mm etalon, (b) a 1-mm etalon, and

results are shown in the dashed line. Chapter 5. Nd:YLF amplification system 75 in the pulse is the series of sub-peaks. The peaks are 15-ps apart, which corresponds to the transit time of the pulses in the etalon. The width of the peaks is hard to define properly but it seems consistent with the limitation introduced by gain narrowing. Compared to the simulated output of the etalon, there is a strong attenuation of the trailing edge, which could be resulting from gain saturation. The "macro" pulse has a FWHM duration on the order of 50 ps, but because of the strong structure, this number is hard to define properly. Figure 5.10b shows the trace for the amplifier with the 1-mm etalon. The pulse is featureless and has a FWHM of 27 ps. Compared to the "expected" pulse shape, the experimental one is shorter and exhibits damping of its trailing edge. Figure 5.10c shows the result for the combination of a 1-mm etalon and a 0.7-mm etalon. The pulse shape is shown in figure 5.10c. The resulting pulse duration is about 47 ps and the pulse shape also shows some suppression of the trailing edge. It was decided to use the combination of two etalons for the rest of the project because the longer pulse duration helps reduce the effect of timing jitter. With two etalons in the cavity, the output power for a pumping current of 19.4 A is 1.6 W, corresponding to a pulse energy of 1.6 mJ at 1 kHz. The shot-to-shot stability was measured using a silicon photodiode and a DAQ card and is on the order of 0.5%

5.4 Multipass ampliﬁer

A double-pass amplifier, illustrated in figure 5.11, is used to further increase the pulse energy. The amplifier is based on the same laser as the regenerative amplifier, and in the two-pass configuration the final power is 2.5 W for a double-pass gain of 1.56. The dimensions of the laser head make it necessary to use a small angle between the two passes, which limits the number of passes that can be achieved. Attempts to increase the number of passes using a Faraday rotator were made, but the additional gain was offset by an increase in losses. At the multipass position, the diameter of the beam is 1.5 Chapter 5. Nd:YLF amplification system 76

290 mm 290 mm 290 mm

HWP2 M2 TFP M1

M LH 3

To HWP1 OPCPA L2 L1 From regenerative amplifier

Figure 5.11: Layout of the multipass amplifier; LH, laser head, HWP, half-wave plate, L, lens, TFP, thin-film polarizer, M, mirror. mm to optimize the fill factor without increasing the diffraction losses. The effect of the multipass on the pulse duration was investigated. Figure 5.12 shows the cross-correlation traces for the amplifier off (0 A) and on (18 A). The amplification does not affect the pulse significantly; the difference seen in figure 5.12 could be caused by experimental uncertainty.

1 0 A 18 A 0.8

0.6

0.4 Intensity (A.U.)

0.2

0 −60 −40 −20 0 20 40 60 Time (ps)

Figure 5.12: Cross-correlation traces with the multipass oﬀ (0 A) and on (18 A).

The above study provided the essential amplification of the pump pulse that serves to drive the OPCPA process. The most significant accomplishment in this regard was the demonstration that the output of the HNLF was sufficient to overcome competing Chapter 5. Nd:YLF amplification system 77

ASE in conventional Nd:YLF ampliﬁers. This ampliﬁer concept is scalable up to the KW average power level and completely eliminates cryocooling and other problems associated in scaling other laser concepts for high power MIR generation. Chapter 6

Optical parametric ampliﬁer system

Optical parametric amplifiers (OPAs) play a vital role in this project as they provide the gain at 3.2 m and, in the chirped-pulse configuration, at 1.6 m. To optimize their efficiency, it is important to understand the different parameters governing their operation.

The goal of this chapter is to provide a general overview of OPAs as an introduction to the following chapters. Also, because optical parametric chirped-pulse ampliﬁers

(OPCPAs) oﬀer diﬀerent challenges, the last part of this chapter is dedicated to them.

6.1 Optical parametric ampliﬁcation

With optical parametric amplification, gain is achieved through an off-resonant nonlinear effect, making it possible to amplify light at arbitrary wavelengths for bandwidths able to support ultrashort pulses. However, to fully exploit this versatility, there are many requirements that need to be fulfilled.

This section will browse over the parameters important for optical parametric ampli-

ﬁcation, starting from a basic plane-wave description of the phenomenon.

78 Chapter 6. Optical parametric amplifier system 79

6.1.1 Theory

In optical parametric ampliﬁcation, a high energy photon is used to amplify two photons of lower energies. It is insightful to look at optical parametric ampliﬁcation using the plane-wave approximation, starting from the wave equation with the nonlinear polarization included:

ǫ ∂2 1 ∂2 2E r r E r PNL r (t, ) 2 2 (t, )= 2 2 (t, ) , (6.1) ∇ − c ∂t ǫ0c ∂t where ǫr is the relative permittivity. For second-order nonlinear processes, there are three electric ﬁelds interacting together, and therefore both the electric ﬁeld and the polarization can be expressed as a sum of monochromatic waves having a frequency ωq:

−iωqt E (t, r)= Eq (r) e + c.c., (6.2) q NL NL −iωqt P (t, r)= Pq (r) e + c.c.. (6.3) q Introducing these deﬁnitions in equation (6.1) results in several equations of the form:

ǫ ω2 ω2 2E r r q E r q PNL r q ( )+ 2 q ( )= 2 q ( ) , (6.4) ∇ c −ǫ0c for which the electric field and the polarization have the same frequency. For most second- order nonlinear effects, the nonlinear polarization can be considered to be instantaneous as only the contribution from the electrons is important. In this case, the different components of the nonlinear polarization can be written as:

NL (2) Pi (t, r)= ǫ0χijkE (t, r)j E (t, r)k (6.5)

(2) where χijk is the second-order nonlinear susceptibility tensor and the indices indicate the polarization. In the literature, nonlinear crystals are normally described according to a

1 (2) contracted susceptibility tensor, dijk = 2 χijk, which has interesting symmetry properties (for more details see [11]). Although the ﬁelds and the nonlinear polarization can be arbitrarily aligned with respect to axes of the nonlinear crystal, for second order nonlinear Chapter 6. Optical parametric amplifier system 80 processes only a given set of polarizations will result in substantial nonlinearity. For this reason, an eﬀective susceptibility, deff , is used, which contains all the information related

to the polarization of the ﬁelds and the geometry of the problem. By using deff , the nonlinear polarization and the electric ﬁelds can be treated as scalar quantities:

NL 2 P (t, r)=2ǫ0deff E (t, r) . (6.6)

Introducing in equation (6.6) the electric ﬁelds as deﬁned in equation (6.2) generate a

multitude of terms. For optical parametric ampliﬁcation, only the terms for which the

frequency of the polarization matches the frequency of one of the input ﬁelds (pump - p,

signal - s, and idler - i) are relevant. As frequencies are related to each other, ωp = ωs +ωi, the only terms needed are:

∗ ∗ NL −iωpt −iωst −iωit P (t, r)=4ǫ0deff E (r)s E (r)i e + E (r)p E (r)i e + E (r)p E (r)s e + c.c. , (6.7)

which leads to three coupled wave equations:

2 2 ǫrω 4deff ω 2E (r) + p E (r) = p E (r) E (r) , (6.8) ∇ p c2 p − c2 s i 2 2 ǫ ω 4d ω ∗ 2E (r) + r s E (r) = eff s E (r) E (r) , (6.9) ∇ s c2 s − c2 p i 2 2 ǫ ω 4d ω ∗ 2E (r) + r i E (r) = eff i E (r) E (r) . (6.10) ∇ i c2 i − c2 p s

For plane waves, the ﬁelds are E (r) = A (z) eikqz, with k = (ǫ )ω /c n (ω ) ω /c, q q q r q ≡ q q and the previous equations are reduced to:

2 2 d d 4deff ω − − A (z)+2ik A (z)= p A (z) A (z) e i(ks+ki kp)z, (6.11) dz2 p p dz p − c2 s i 2 2 d d 4d ω ∗ − − − A (z)+2ik A (z)= eff s A (z) A (z) e i(kp ki ks)z, (6.12) dz2 s s dz s − c2 p i 2 2 d d 4d ω ∗ − − − A (z)+2ik A (z)= eff i A (z) A (z) e i(kp ks ki)z. (6.13) dz2 i i dz i − c2 p s Chapter 6. Optical parametric amplifier system 81

For a slowly varying envelope, the second derivative can be omitted, and the ﬁnal set of coupled equations is:

d A (z)=2id k A (z) A (z) ei∆kz, (6.14) dz p eff p s i d ∗ A (z)=2id k A (z) A (z) e−i∆kz, (6.15) dz s eff s p i d ∗ A (z)=2id k A (z) A (z) e−i∆kz, (6.16) dz i eff i p s

where n (ω) ω2/c2 = k2 and ∆k = k k k were used. Two important comments can p − i − s be made at this point. First from equation (6.14), unless the amplitude of the signal and the idler are high enough, ∂Ap/∂z =0, and therefore the amplitude of the pump can be considered constant; this is referred to as the undepleted-pump approximation. As the signal and the idler are ampliﬁed, their intensities reach a point where this approximation is no longer valid, leading to gain saturation. Secondly from equation (6.15), the rate of change of the signal is proportional to the amplitude of the idler, meaning that absorption of the idler in the nonlinear crystal is detrimental to ampliﬁcation of the signal. Because of this, nonlinear crystals need to have a transparency window extending up to the idler.

Under the undepleted-pump approximation, A3 = cte and dA3/dz =0, it is straightforward (but tedious) to solve the remaining two coupled equations. For optical parametric ampliﬁcation, the idler is generated during the ampliﬁcation, therefore the initial conditions are As (0) = As0 and Ai (0) = 0, and the solutions are [11]:

i∆k i∆k A = cosh(gz) sinh(gz) A e 2 z, (6.17) s − 2g s0 κi ∗ i∆k A = sinh(gz) A e 2 z, (6.18) i g s0

where g = κ κ∗ (∆k/2)2 and κ = 2id k A (z) /n2 . The growth of both s i − s/i eff s/i p s/i ﬁelds is approximately exponential and governed by the g-parameter, which peaks for

the phase-matching conditions where ∆k is zero. As the absolute value of the phase-

mismatch increases, the g-parameter goes down, which results in a lower gain for a

crystal of a ﬁxed length. For a perfectly phase-matched process, the previous equations Chapter 6. Optical parametric amplifier system 82 are reduced to:

As = cosh(gz) As0, (6.19) κ A = i sinh(gz) A∗ , (6.20) i g s0

and

2 2 2 4π deff A (z)p g = κ κ∗ = s i 2 nsniλsλi 2 2 8π d Ip = eff , (6.21) cǫ0npnsniλsλi where the deﬁnition of the time-average Poynting for a plane wave was used:

nǫ c S = I = 0 A (z) 2 . (6.22) 2 | |

The small-signal gain is then deﬁned as

2 2 2 2 Ps As exp 8π deff Ip G = = 2 z . (6.23) Ps0 As0 ≈  2 cǫ0npnsniλsλi     6.1.2 Phase-matching

For most optical materials, the index of refraction decreases with increasing wavelength, which makes it impossible to satisfy the phase-matching condition. This requires that the index of refraction for the signal is either higher or lower than the one for the pump and the idler. The most common way to solve this problem is through critical phase- matching, which takes advantage of the birefringence that all second-order nonlinear crystals exhibit.

Index of refraction for birefringent crystal

As critical phase-matching relying on birefringence, it is important to be able to express the index of refraction for a plane wave propagating in an arbitrary direction. For a Chapter 6. Optical parametric amplifier system 83 birefringent crystal, the general form of the electric displacement is:

Di = ǫi,jEj, (6.24) j where the indexes i and j represent the diﬀerent polarizations. In general, the permittivity tensor has non-zero cross-terms. However, to simplify the problem without losing generality, the laboratory axes can be aligned on the principal axes of the nonlinear crystal such that ǫi,j is diagonal. In such a situation, the electric displacement can be written as:

2 Di = ǫi,jδi,jEi = ǫonPiEi, (6.25)

where nPi are the indexes of refraction associated with the principal axes.

The propagation is described by Maxwell’s equations [55]:

k E = ωH, (6.26) × o k H = ωD, (6.27) × − where the wavevector is:

2nπ k = nk sˆ = s,ˆ (6.28) 0 λ

and:

sˆ = sin (θ) cos (φ)ˆx + sin (θ) sin (φ)ˆy + cos (θ)ˆz. (6.29)

The angles define the direction of the k-vector as shown in figure 6.1. The magnetic field is eliminated by multiplying equation (6.26) by k from the left and combining it with × equation (6.27). The resulting expression can be simplified to:

(nk )2 [ˆs(ˆs E) E]= ω2D. (6.30) 0 − − 0 Chapter 6. Optical parametric amplifier system 84

φ y x

Figure 6.1: Orientation of the wavevector with respect to the principal axes of the crystal.

The next step is to rewrite this equation in term of the polarization components of the

ﬁelds:

(nk )2 [s (ˆs E) E )] = ω2D = ǫ ω2n2E 0 i − i − 0 i − 0 0 i i n2 [s (ˆs E) E )] = n2E i − i − i i n2 s (ˆs E)= E , (6.31) (n2 n2) i i − i where i = x, y, or z. The electric ﬁeld can be taken out of the equation by multiplying by si on each side and summing over all i:

n2 2 E E 2 2 si (ˆs )= siEi =s ˆ (n ni ) − 2 n 2 2 2 si =1. (6.32) (n ni ) − After some algebra, the ﬁnal expression for the index of refraction of a wave in a birefringent crystal is:

1 sin2(θ)cos2(φ) sin2(θ)sin2(φ) cos2(θ) s2 + + =0. (6.33) −2 −2 i −2 −2 −2 −2 −2 −2 n ni ≡ n nx n ny n nz − − − − For a given set of angles, the indexes of refraction for both polarizations are obtained by

solving equation (6.33), which yields 4 solutions. Two are negative and can be discarded

whereas the positive two can be assigned to each polarization. To facilitate the deter-

mination of the phase-matching angle, one of the two angles is normally ﬁxed; either

φ =0◦ or 90◦ or θ = 90◦ (The case where θ =0◦ is omitted because it oﬀers no tweaking parameter as n is either nx or ny). This sets the refractive index for one polarization and Chapter 6. Optical parametric amplifier system 85 the other index can be tuned using the remaining angle, then called the phase-matching angle. For example, if the direction of propagation is in the X Z plane (ﬁgure 6.2), the − z k θ

x y

Figure 6.2: Diagram for a plane wave propagating in the X-Z plane. The thicker arrows

indicate the two possible polarizations.

angle φ is zero, and the expression for the indexes becomes:

n−2 n−2 n−2 n−2 sin2(θ)+ n−2 n−2 cos2(θ) =0. (6.34) − y − z − x For the polarization that is perpendicular to the X Z plane, the index is insensitive − to the angle and is equal to ny. For the polarization lying in the plane, the index of refraction is obtained by solving:

n−2 n−2 sin2(θ)+ n−2 n−2 cos2(θ)=0, (6.35) − z − x or, by using sin2(θ)+cos2(θ)=1:

1 sin2(θ) cos2(θ) 2 = 2 + 2 . (6.36) n nz nx

For the two extreme cases, θ =0◦ or 90◦, the wavevector and the remaining polarization are aligned along the principal axes and the index is nx or nz. Phase-matching is achieved by selecting the proper polarization for each ﬁeld and by adjusting their index of refraction using the phase-matching angle. As a side note, there are three types of critical phase-matching based on which ﬁelds share the same polarization. In type I phase-matching, the signal and the idler have the same polarization, in type II, the pump and the idler, and in type III, the pump and the signal. Type III Chapter 6. Optical parametric amplifier system 86 phase-matching is often omitted because in the degenerate case, the idler and the signal are indistinguishable, and type III is the same as type II.

Walk-oﬀ angle

Beam walk-off is an important phenomenon for critical phase-matching, because it leads to a loss of spatial overlap between the different fields, limiting the efficiency of nonlinear processes. Walk-off occurs because, for a wave having a wavevector with an arbitrary direction in the birefringent crystal, the electric field, E, and the electric displacement,

D, are no longer necessarily parallel. In the case that they are not, there will be a small angle between the wavevector, D, and the Poynting vector, E (see ﬁgure 6.3a). This ⊥ ⊥ angle is known as the walk-oﬀ angle and is given by [56]:

−1/2 sin(θ)cos(φ) 2 sin(θ)sin(φ) 2 cos(θ) 2 tan(ρ)= n + + . (6.37) n−2 n−2 n−2 n−2 n−2 n−2 − x − y − z For a beam propagating in the X-Z plane, the polarization along the Y-axis suﬀers no

walk-oﬀ, whereas the angle for the other polarization is given by:

−1/2 sin(θ) 2 cos(θ) 2 tan(ρ)= n + . (6.38) n−2 n−2 n−2 n−2 − x − z Figure 6.3b depicts what happens for two collinear beams with orthogonal polarizations

entering a birefringent crystal; because of the walk-oﬀ angle, the spatial overlap is even-

tually lost, limiting the useful length of nonlinear crystals.

Non-collinear geometry

Critical phase-matching works extremely well for quasi-monochromatic ﬁelds, such as

long laser pulses, as the index of refraction does not change much over the bandwidth of

the ﬁeld. For short pulses, the variation of the refractive index over the larger bandwidth

is considerable and phase-matching cannot be achieved over the whole spectrum. By

taking advantage of the vectorial nature of the wavevector, the bandwidth of OPAs can Chapter 6. Optical parametric amplifier system 87

x z k y S ρ D H E

(a) (b)

Figure 6.3: On the left: The orientations of diﬀerent ﬁelds are shown, as well as the

wavevector and the Poynting vector S. On the right: Because the Poynting vectors for the

two polarizations are diﬀerent in the crystal, although the beam is at normal incidence,

one of the two polarizations will be bent at the interface. On the other hand, the

wavevectors and the wavefronts for the two beams are still parallel.

be improved. By sending the signal at an angle with respect to the pump, the bandwidth

can be increased as the orientation of the idler can vary to accommodate the change in

the refractive index. However, determining the phase-matching angle is somewhat more

complicated in the noncollinear geometry.

ks ki

α β

Figure 6.4: Diagram of the diﬀerent wavevectors in the noncollinear geometry. The

(internal) noncollinear angle, α, is deﬁned in the crystal.

In the rest of this section, the details of the Type II phase-matching will be worked out for a process occurring in the X-Z plane of a biaxial birefringent crystal having nz >ny >nx. Furthermore, the polarizations of the pump and the idler will be parallel to Chapter 6. Optical parametric amplifier system 88 the Y-axis, and therefore their indexes of refraction will be independent of the angle. This speciﬁc example was chosen because it is representative of the situation for potassium titanyl arsenate, a crystal used in this work. For the other two crystals used, silver thiogallate and lithium gallium selenide, there is no analytical solution and the phase- matching angle has to be found numerically.

From ﬁgure 6.4 and the deﬁnition of the wavevector, it is possible to write:

n (λ ,θ) n (λ ) n (λ ) s s = y p cos(α) y i cos(α + β) , (6.39) λs λp − λi where α is the noncollinear angle and β can be deﬁned as:

n (λ ) λ β = sin−1 y p i sin(α) α. (6.40) n (λ ) λ − y i p Phase-matching is achieved by tweaking the index of refraction of the signal, given by:

1 sin2 (θ) cos2(θ) sin2 (θ) 1 sin2 (θ) 2 = 2 + 2 = 2 + −2 ns (λs,θ) nz (λs) nx (λs, ) nz (λs) nx (λs, ) 2 2 nx (λs) nz (λs) 2 = 2 − 2 sin (θ). (6.41) nx (λs) nz (λs)

Equations (6.39) and (6.41) can be combined to give:

2 2 1 1 nx (λs) nz (λs) 2 2 = 2 + 2 − 2 sin (θ). (6.42) ny(λp) ny(λi) n (λ , ) n (λ ) n (λ ) λ2 cos(α) cos(α + β) x s x s z s s λp − λi 2 After subtracting 1/nx (λs, ) on both sides, the expression can be written as:

2 2 2 ny(λp) ny(λi) nx (λs, ) λs cos(α) cos(α + β) 2 2 − λp − λi nx (λs) nz (λs) 2 2 = 2 − 2 sin (θ). (6.43) 2 2 ny(λp) ny(λi) nx (λs) nz (λs) n (λs, ) λ cos(α) cos(α + β) x s λp − λi The phase-matching angle is obtained by isolating sin(θ):

nz (λs) sin(θpm)= ny(λp) cos(α) ny(λi) cos(α + β)× λp − λi 2 2 1/2 nx(λs) ny(λp) ny(λi) 2 cos(α) cos(α + β) λ λp λ s − − i . (6.44)  2 2  nx (λs) nz (λs)  − 

  Chapter 6. Optical parametric amplifier system 89

The bandwidth of the amplifier can be estimated by finding the range of frequencies for which the phase-matching angle is constant. This is repeated for different noncollinear angles to find the optimal one. This optimization is often done graphically; an example can be found in chapter 7.

A more formal way to determine the bandwidth of an OPA, ∆λ, was introduced for the collinear case by [57] and extended for the noncollinear case by [58], the result being:

λ2 u ∆λ = s | si|, (6.45) c leff

where leff is the eﬀective length of the ampliﬁer. The quantity usi is the group velocity mismatch given by:

− 1 1 1 u = , (6.46) si v cos(α + β) − v i s where vs/i is the group velocity of the signal/idler. For the case where the group-velocity mismatch is zero, the bandwidth is determined by the group-dispersion mismatch, GDM, deﬁned as:

1 λ λ GDM = tan(α + β)tan(β) s + i cos(α + β) (GV D + GV D ) , (6.47) si 2πv2 n n − s i s s i 2 where GV Ds/i is ∂ks/i/∂ω . The bandwidth is then:

0.8λ2 1 ∆λ = s . (6.48) c g l | si| eff Angular dispersion of the idler

One disadvantage of the noncollinear geometry is that the idler produced is angularly dispersed; each frequency propagates in a slightly different direction. Although this has no effect on the quality of the signal, it makes the idler somewhat unusable. In collinear geometries it is quite common to use the idler because of its longer wavelength. However, this angular dispersion in noncollinear geometries can be compensated (see section 8.2.1), but this needs to be quantified. Chapter 6. Optical parametric amplifier system 90

Continuing to simplify the problem, the signal beam is taken normal to the surface instead of the pump. Because of the small angle between the two, the error introduced is negligible. From equation (6.40), the angle γ, deﬁned in ﬁgure 6.5, is:

n (λ ) λ γ β + α = sin−1 y p i sin(α) , (6.49) in ≡ n (λ ) λ y i p

where ny is the index of refraction along the y-axis, λp/i the wavelength of the OPCPA pump/idler, and α the noncollinear angle. The external angle is therefore given by

out

x y

Figure 6.5: This diagram illustrates the diﬀerent angles used to quantify the dispersion

in the idler. For its calculation, it is important to use the external angle, γout.

λ γ = sin−1 [n (λ )sin(γ )] = sin−1 n (λ ) i sin(α) . (6.50) out y i in y p λ p The angular dispersion is obtained by diﬀerentiating the external angle with respect to

the wavelength:

dγout d −1 Didler = = sin (C1λi) dλi dλi C1 = 1 , (6.51) (1 C2λ2) 2 − 1 i where

ny (λp) C1 = sin(α) . (6.52) λp Chapter 6. Optical parametric amplifier system 91

6.2 Optical Parametric Chirped Pulsed Ampliﬁer

In this section, the two particularities of OPCPAs will be addressed. The ﬁrst, and obvious, one is the temporal stretching of the seed pulse. The second one is the timing jitter, to which OPCPAs are sensitive.

6.3 Stretching - Compression

The main characteristic of OPCPAs is the use of a narrowband high-average-power laser to amplify a broadband pulse. This diﬀerence in bandwidth necessarily leads to a mismatch of the pulse durations; a typical pump pulse is longer than 1 ps whereas the signal pulse can be less than 100 fs. Unlike for gain media, there is no population transfer in optical parametric ampliﬁcation; the gain and the energy are "where" the pump pulse is.

Because of this, for a signal pulse much shorter than the pump pulse, the energy transfer between the two will be ineﬃcient. However, because of its broad bandwidth, the seed can easily be stretched to match the pump duration by using dispersion; this is known as chirped-pulse ampliﬁcation (CPA).

The idea of CPA was introduced for conventional laser amplifiers for different reasons than for OPAs; by stretching the pulses, the peak intensity is reduced and optical damage can be avoided, allowing amplification to high energy. However, the techniques developed for conventional systems can also be used with OPCPA. In this work, chirping schemes based on a grating pair will be used for the compressor and the stretcher. The idea behind using a grating pair is that the incoming beam is angularly dispersed by the first grating and after some propagation, the second grating is used to recollimate the beam.

The result is that the path length for each frequency is diﬀerent, introducing dispersion.

Although the explanation is the same for the compression and the stretching, the setups used for each are diﬀerent because the dispersion of the compressor needs to have the opposite sign as the one from the stretcher, so the two can cancel each other. In this case, Chapter 6. Optical parametric amplifier system 92 a transform-limited pulse at the entrance of the system will still be transform-limited at the exit.

The most common design for a compressor, introduced by Treacy in 1969 [59] and shown in ﬁgure 6.6a, consists of two parallel gratings and has an anomalous dispersion.

Because there are no optics apart from the gratings, this design has the highest efficiency for grating-based system and is normally used after the amplification. By using only two gratings, it is not possible to generate normal dispersions; this can only be achieved by using extra optics. In this work, the stretcher used is based on the Martinez design [60], shown in figure 6.6b. By using a telescope, it is possible to inverse the dispersion of the grating pair without changing its amplitude, allowing for perfect compensation for the compressor.

X1 S1 S2 X2

f1 f1 f2 f2

(a) (b)

Figure 6.6: On the left: The design for the compressor. On the right: The layout for the

stretcher.

In the rest of this section, the equations for dispersion of the stretcher/compressor

will be rederived. There are already some papers [61, 62] summarizing these expressions,

but there are some discrepancies between the two and neither presents their derivation.

Figure 6.7 shows the details of the path of a single frequency in the compressor. Not

only will the distance AB change with the diﬀraction angle, but so will the distance BC. Chapter 6. Optical parametric amplifier system 93

For a given frequency, the optical path length is then: L p = g [1+cos(γ θ)] , (6.53) cos(θ) −

where Lg is the straight distance between the two gratings and γ is the incident angle. The frequency dependence of the path length is contained in the diﬀraction angle, θ: λ θ = sin−1 sin(γ) , (6.54) d − where d is the groove spacing. For design purposes, the dispersion introduced by the

B C θ γ A

P Lg

Figure 6.7: The details of the propagation in the compressor.

gratings is better described using the expansion in quadratic and higher-order terms,

as introduced in section 2.3. Although, those terms are normally calculated from the

expression of the phase, in the case of a grating pair the situation is more complicated,

as the phase does not only arise from the optical path diﬀerence but also from the gratings

themselves (for details, see [59]). However, the group delay, τ = ∂φ (ω) /∂ω, can easily

be expressed as a function of the path length [59]: p τ = . (6.55) c The quadratic phase is then given by: ∂τ 1 ∂p 2π L ∂θ φ(2) = = = g , (6.56) ∂ω c ∂ω ωd cos2 (θ) ∂ω where

∂θ ∂ − 2πc 1 2πc 2πc = sin 1 sin(γ) = − = − . (6.57) ∂ω ∂ω ωd − 1 sin(θ) ω2d ω2d cos(θ) − Chapter 6. Optical parametric amplifier system 94

The ﬁnal expression for the quadratic phase is:

4π2cL φ(2) = g , (6.58) −ω3d2 cos3 (θ) or in terms of wavelength:

L λ3 φ(2) = g . (6.59) −2πd2c2 cos3 (θ)

The cubic phase is obtained by diﬀerentiating the quadratic phase which gives:

12π2cL 2πc sin(θ) φ(3) = g 1+ , (6.60) ω4d2 cos3 (θ) ωd cos2 (θ) or:

3L λ4 λ sin(θ) φ(3) = g 1+ . (6.61) 4π2c3d2 cos3 (θ) d cos2 (θ) An important fact should be pointed out: the second and third order dispersions evolve diﬀerently with a change in the groove spacing. The quadratic phase depends only on d2

whereas the cubic phase also has a term function of d3. Therefore as the groove density is increased, i.e. d decreasing, the ratio between the cubic and the quadratic phase increases; in other words, the third order becomes more important. This behavior, which is also true for the higher-order terms, is important for the design of the stretcher-compressor pair.

The formulas for the dispersions are valid for both the compressor and the stretcher.

However, for the stretcher, the distance between the gratings is deﬁned as [62]:

L = (f + f s s ) , (6.62) g − 1 2 − 1 − 2 where f1/2 is the focal length of the first/second lens, s1/2 is the distance between the first/second grating and the first/second lens (see figure 6.6b). With this design, the sign of the dispersion can be flipped by moving the grating across the focal point of the lens. In the stretcher configuration, the two gratings are positioned closer to the lenses, so s

6.3.1 Pulse duration

Knowing the second and the third order dispersion is not enough to properly design the stretcher and compressor; it is important to be able to estimate the stretched pulse duration so it can be matched to the pump pulse. This can be done easily if two assumptions are made: the pulses can be described by a Gaussian, and the increase of the pulse duration is due to the second-order dispersion. Both assumptions are reasonable for the case of a passively modelocked laser. Because dispersion is deﬁned in the frequency domain, the ﬁrst step is to take the Fourier-transform of the pulse:

t2 T 2 F.T. exp exp( iω t) = T exp 0 (ω ω )2 , (6.63) −2T 2 − 0 0 − 2 − 0 0

where T0 is the pulse duration and ω0 is the frequency of the carrier wave. By propagating in the stretcher (or compressor), the pulse picks up an extra phase. The resulting ﬁeld, keeping only the second-order term, is:

T 2 i E (ω) exp 0 (ω ω )2 exp φ(2) (ω ω )2 ∝ − 2 − 0 2 − 0 1 exp T 2 + iφ(2) (ω ω )2 . (6.64) ∝ −2 0 − 0 The new pulse duration is obtained by taking the inverse Fourier transform of the ﬁeld:

t2 E (t) exp − . (6.65) ∝ 2(T 2 + iφ(2)) 0 Separating the argument of the exponential in terms of a real and an imaginary part

gives:

t2 T 2 iφ(2) T 2 iφ(2) t2 0 − = 0 − . (6.66) 2 (2) 2 (2) 2 2(T0 + iφ ) (T0 iφ ) 4 (2) − 2 T0 +(φ ) Only the real part is important for the pulse duration:

2 2 2 T0 t t = 2 . (6.67) 2 T 4 +(φ(2))2 φ(2) 0 2T 2 1+ 0 T 2 Chapter 6. Optical parametric amplifier system 96

By comparing this expression to the argument of the exponential on the left-hand side of equation (6.63), it is straightforward to deﬁne a new pulse duration:

(2) 2 ′ φ T = T0 1+ . (6.68) 0 T 2

However, in the lab, the FWHM of the pulse intensity, TFWHM , is easier to measure, and therefore it would be more convenient to express equation (6.68) as a function of

TFWHM . The relation between T0 and TFWHM can easily be derived:

2 2 (TFWHM /2) (TFWHM ) 1 exp 2 = exp 2 = , (6.69) − T0 − (2T0) 2 therefore:

TFWHM =2 ln(2)T0. (6.70) Equation (6.68) then becomes:

(2) 2 ′ 4ln(2) φ T = TFWHM 1+ . (6.71) FWHM T 2 FWHM By knowing the initial and the ﬁnal pulse duration it is possible to calculate the amount of dispersion needed and from there the parameters of the stretcher and compressor.

6.3.2 Timing jitter

Timing jitter refers to the ﬂuctuations in the arrival time of laser pulses, which can be caused by changes in the repetition rate of the laser or the path length of the system.

Absolute timing jitter is the noise at the output of a single laser; for passively mode- locked lasers it can be quite small [63]. Relative timing jitter describes the fluctuations of one laser compared to a clock, either a crystal oscillator, or another laser as it is the case with OPCPAs. For two independent lasers, the relative timing jitter can be much higher than the absolute one, as a small change in the repetition rate of one laser will slowly change its arrival time. This effect is illustrated in figure 6.8. Chapter 6. Optical parametric amplifier system 97

−10 0 10 20 30 40 490 500 510 520 530 540 Time (ns) Time (ns)

(a) (b)

Figure 6.8: The two pulse trains having similar repetition rates, around 80 MHz; the

difference between the two is 0.1%. On the left, the first four pulses are shown; at t =0, the pulse from both trains overlap perfectly. On the right, to amplify the effect of the repetition rate mismatch, the same four pulses are shown after 500 ns; the overlap is clearly lost. However, for the fourth pulse in (a), the separation is already 37 ps.

Many ampliﬁers have two synchronized lasers, one being the pump source. For conventional laser systems, timing jitter is not a serious problem because of the nature of the gain. The interaction between the pump and seed is mediated through a population transfer in the gain medium. The seed does not need to overlap in time with the pump and the meaningful timescale is the upper level lifetime. As long as the timing ﬂuctua- tions are much smaller than the lifetime, the seed is insensitive to the arrival time of the pump. For solid-state lasers, the upperlevel lifetime goes from 3.2 s for Ti:Sapphire to

10 ms for Erbium-doped ﬁbers, and therefore simple electronic triggering is good enough.

On the other hand, OPAs and OPCPAs are more sensitive to timing jitters. In optical parametric ampliﬁcation, there are no real energy levels involved hence no population transfer; the gain is essentially instantaneous. The seed has to overlap with the pump to experience any gain and the exact form of the gain proﬁle will depend on this overlap.

Furthermore, for good conversion efficiency, the seed and the pump durations need to be Chapter 6. Optical parametric amplifier system 98 on the same order. Put together, those two requirements mean that even timing jitter of a few percent of the pulse duration can greatly influence the amplification process.

In general for OPAs, the timing jitter is not a problem because it is low enough not to interfere. The reasons for this are that the pump and the seed are produced by the same ampliﬁed laser pulse and that the path lengths are relatively short.

For OPCPAs, timing jitter is a serious issue because of the use of a narrowband ampliﬁer. One of the problems is the seeding of this ampliﬁer. This is typically done using a second oscillator synchronized to the seed laser of the OPCPA. By using phase- locked loops to actively stabilize either or both lasers, their repetition rates can be kept in sync. With a proper design, this scheme can reduce the jitter to less than a fs [64].

However, active synchronization is inherently an unstable process and makes the everyday use of the system more complex. An alternative to this is to use a single oscillator to generate the seed for the OPCPA and for the narrowband amplifier. This is referred to as passive synchronization. Compared to active synchronization, setting up a passive system can be harder, as generating two completely different wavelengths with a single oscillator is not trivial. However, once the system is operational, the everyday utilization is much simpler. Neither passive nor active synchronization can eliminate all the jitter because there is a second source of timing jitter: path length fluctuations in the amplifier.

During its amplification, the pump pulse can travel a much longer distance than the seed pulse, especially if a regenerative amplifier is used, and even a small relative fluctuation in the path length can generate a significant jitter.

So far, the importance of timing jitter and its possible sources were discussed; the question of how to measure it will now be addressed. For OPCPAs, only the relative jitter between the seed and the pump is important, which can easily be deduced from the cross- correlation (XC) between them. The XC is obtained by mixing the pump and the signal in a nonlinear crystal and measuring the resulting nonlinear signal as a function of the time delay between the two pulses. The nonlinear interaction can be any second-order Chapter 6. Optical parametric amplifier system 99 process but in the case of two near-infrared pulses, sum-frequency generation has the advantage of generating an output in the visible. The relative timing jitter is measured by ﬁxing the timing delay between the two pulses and observing the noise in the signal.

If the two lasers are extremely stable, the noise in the XC is solely due to the timing jitter. By choosing a time delay where the slope of the XC signal is linear, the amplitude noise can easily be translated in timing jitter because of the linear relation between the two. However, the situation becomes more complicated if one of the input has amplitude noise, which is often the case of the pump laser. In this case, the analysis is not as straightforward, but if the noise can be described by a Gaussian distribution, then the diﬀerent noises can be added in quadrature, and the timing jitter can still be extracted.

This aspect of of the problem is characterized in chapter 7.

The above analysis gives the background for the proper development of the OPCPA for optimal ampliﬁcation and maximum stability of the output parameters of interest, gain, wavelength, and pulse duration. Chapter 7

Optical Parametric Chirped-Pulse

Ampliﬁer

Since their introduction in 1992 by Dubetis [65], optical parametric chirped-pulse am- pliﬁers (OPCPAs) have been more and more common. A major reason is their ability to amplify broadband pulses to high power, and this at arbitrary wavelength. Their functionality and eﬃciency are determined by the nonlinear crystal used and the proper design of the compressor/stretcher system.

In the ﬁrst part of this chapter, the choices made for those important components are explained, and in the remainder of the chapter, the details of the OPCPA as well as the performances of each element of the system are presented.

7.1 OPCPA - Design

7.1.1 Nonlinear Crystal

Because optical parametric ampliﬁcation is not a resonant process, for a given pump and signal there is a multitude of nonlinear crystals that can be used. In the following section, the options available for an optical parametric ampliﬁer (OPA) pumped at 1.053 m and

100 Chapter 7. Optical Parametric Chirped-Pulse Amplifier 101

seeded at 1.55 m will be examined.

β-barium borate (BaB2O4 - BBO) has been the major workhorse for Ti:Sapphire- based OPAs operating in the visible, because its indexes of refraction allow for phase- matching to be achieved over a broad bandwidth. Furthermore, the damage threshold of

BBO is among the highest for nonlinear crystal and its transmission window goes down to 189 nm. However, for an OPA operating in the near infrared and pumped at 1.053

m, BBO is no longer the best solution as there is no broad phase-matching, and the transmission of BBO around 3 m, where the idler lies, is on the order of only 20%.

Pumping at 1.053 m opens the door for less standard crystals. Potassium titanyl arsenate (KTiOAsO4 - KTA) and potassium titanyl phosphate (KTiOPO4 - KTP) are particularly interesting because they oﬀer a signiﬁcant gain bandwidth around 1.55 m

(see figure 7.1). Both are non-hygroscopic and have an effective nonlinear coefficient around 2 pm/V (2.01 for KTA and 2.27 for KTP) for optical parametric amplification at 1.55 m. KTA has an enhanced transmittance in the MIR compared to KTP [66].

This improved transmission window goes from 3 m to 5 m for ﬁelds parallel to the x- and z-axis, and from 3.6 m to 5 m for ﬁelds parallel to the y-axis. However, for KTA the group-velocity mismatch is minimized when the idler, at 3.24 m, is parallel to the y-axis, and therefore the enhanced transmission in the MIR is not a major factor.

In reference [67], it is shown that the damage threshold of KTA is roughly twice that of KTP, which is a signiﬁcant advantage as it allows for higher pump intensities, and therefore for higher gain. The safe level for KTA was set to 1.2 GW/cm2 (for 1200 shots and measured with a 8-ns 20-Hz 1064-nm Q-switch laser). Because the damage threshold intensity is a function of ∆t−1/2 [41], where ∆t is the pulse duration, the threshold for

50 ps pulse at 1053 nm is 15 GW/cm2.

The two crystals might have similar bandwidth, but the phase-matching curve for

KTP is centered on 1.6-1.65 m which is red-shifted compared to the output of an

Erbium-doped ﬁber laser. On the other hand, from ﬁgure 7.1a, it can be seen that for Chapter 7. Optical Parametric Chirped-Pulse Amplifier 102

1.65 1.65

1.6 1.6 m) m) µ µ

1.55 1.55 Wavelength ( Wavelength ( 1.5 1.5

1.45 40 45 50 55 45 50 55 60 Phase−Matching angle (°) Phase−Matching angle (°)

(a) (b)

Figure 7.1: Phase matching curves at diﬀerent noncollinear angles for (a) KTA, and (b)

KTP. The noncollinear angle is varied from 0◦ to 4◦, going from left to right. For KTA, the extra dashed curve is 3.2◦.

KTA a nonlinear angle between 3 and 3.5 degrees oﬀers a wide phase-matching region centered at 1.55 m.

For its diﬀerent advantages over KTP, KTA was chosen for the OPCPA gain medium.

7.1.2 Stretcher-Compressor

In this section, the diﬀerent parameters for the stretcher and the compressor will be set.

The apparatus are treated together because decisions made for one will aﬀect the other one. Figure 7.2 shows the details of the compressor and the stretcher used in this work.

The general designs are described in the previous chapter, but for the stretcher, a curved mirror in a folded conﬁguration is used instead of lenses. The major advantage of mirrors over lenses is the lack of chromatic aberration. Furthermore, as the diameter of a lens increases so does its thickness, which introduces signiﬁcant material dispersion, unbal- ancing the compressor-stretching pair. Using mirrors avoids this problem and arbitrarily large optics can be used. Chapter 7. Optical Parametric Chirped-Pulse Amplifier 103

Top

Side

Ouput

Input

(a) (b)

Figure 7.2: The details for (a) the compressor and (b) the stretcher, with a top view and

a side view. The drawings are not to scale.

A folding conﬁguration is used to reduce the foot-print of the stretcher and facilitate

the alignment of the grating. To avoid any residual spatial chirp, the entrance and the

exit gratings have to be perfectly antiparallel; in a folded geometry this alignment is

made quite easily by aligning all optics along the central axis.

The stretcher and compressor are doubly passed because as the beam enters either

apparatus, it undergoes a lateral shift that is a function of the wavelength. This intro-

duces spatial chirp, which can be removed by doing a second pass. The dispersion of a

double-pass system is simply twice that of a single-pass.

There are three parameters shared by the stretcher and compressor: the spacing between the gratings, the incident angle, and the groove density of the grating. For a desired pulse duration, only two of these are independent. The groove density and incidence angle are normally selected ﬁrst, and the grating spacing is adjusted to achieve the proper pulse duration.

There are two aspects to consider when choosing the groove density. For one, the eﬃciency is better for higher densities [62]. On the other hand, as the groove density Chapter 7. Optical Parametric Chirped-Pulse Amplifier 104

increases, the inﬂuence of the higher-order terms becomes stronger. For a perfectly bal-

anced stretcher-compressor pair, this is not an issue. However, when material dispersion

is added, the compressor has to be detuned to compensate for the extra dispersion and

strong higher terms can be problematic [62], giving an advantage to low-density gratings.

For OPCPAs, material dispersion is not as important because high gain can be obtained

within a few passes in relatively short crystals. Eﬃciency is then the main concern and

a density of 750 lines/mm was chosen to optimize it. The gratings used are gold-coated

holographic gratings with an eﬃciency above 90% for 1.55 m.

There are a few factors guiding the choice of the incident angle. For holographic gratings, the diffraction efficiency is higher when only the first order is present. The region of high-efficiency is the broadest for the Littrow configuration, i.e. when the first order is back-reflected [68]. However, working at the Littrow angle complicates the design and the alignment of both the compressor and the stretcher, and a lot of effort can be saved by using a slightly bigger angle. For a grating with 750 lines/mm, the Littrow angle at 1.55 m is 35.5◦ and the incident angle is chosen to be 43◦ resulting in a diffraction angle of 29◦. At this angle, the region of high-efficiency goes from 1.12 m to 2.25 m, which is much larger than the bandwidth of the input pulses.

As stated previously, the seed pulses are stretched to match the pulse duration of the pump, which is 47 ps. From equation (6.71), the amount of dispersion needed is given by:

1 2 T 2 T ′ 2 φ(2) = FWHM FWHM 1 . (7.1) 4ln(2) T − FWHM

′ For an initial pulse duration, TFWHM , of 100 fs, and a ﬁnal pulse duration, TFWHM , of 47 ps the resulting dispersion is 1.69 10−24 s2. From equation (6.59), the spacing between the gratings is then given by:

2πc2d2 cos3 (θ) L = φ(2)∗. (7.2) G λ3 Chapter 7. Optical Parametric Chirped-Pulse Amplifier 105 where in a single-pass conﬁguration, φ(2)∗ = φ(2) and in double pass, φ(2)∗ = φ(2)/2. For a double-pass stretcher using a 750 lines/mm grating with a diﬀraction angle of 29◦, a spacing between the gratings of 122.5 mm will result in 47-ps pulses.

So far, the emphasis was put on the stretcher because to properly compensate its dispersion, the specifications of the compressor should be identical. However, because the output of the erbium-doped amplifier is not transform-limited and the nonlinear crystal introduced extra dispersion, the exact operating conditions for the compressor are different and determined experimentally.

The last parameter to determine is the strength of the mirror. Although it does not influence the dispersion, the focal length of the mirror will affect the overall efficiency of the stretcher. One of the problems with the folded geometry is that for a single- pass the input and the output beam are separated by introducing a small vertical angle.

By doing so, the normal axis of the incident plane on the grating is no longer parallel to the grooves, resulting in a diffraction cone [69]. This conical diffraction introduces distortions, and although they cannot be compensated for, they can be minimized by using a mirror with a long focal length to reduce the vertical angle. The focal length cannot be made arbitrarily large because the diffracted beam has to fit on the mirror otherwise the spectrum will be clipped. For this stretcher, the mirror used is a 6-inch- diameter (152.4 mm) gold-coated curved mirror with a focal length of 24 inches (609.6 mm). The bandwidth of the stretcher can be calculated, the first step being to obtain the dispersion of the grating:

1 D = . (7.3) d cos(θ)

For a 750 lines/mm grating with a diﬀraction angle, θ, of 29◦, the dispersion is 0.86 mrad/nm, resulting in a bandwidth for the stretcher of 144 nm, which is more than suﬃcient for the current input pulse. Chapter 7. Optical Parametric Chirped-Pulse Amplifier 106

7.2 OPCPA - Performance

In this section, the diﬀerent components of the OPCPA, ﬁgure 7.3, will be described in more detail. In parallel, a description of their performance is given.

Figure 7.3: A general layout of the whole OPCPA system. The specifics of each component, e.g. Nd:YLF amplifiers, are offered in their respective sections. The remaining optics are: lenses for beam shaping and transport, half- and quarter-wave plates

(HWP/QWP) for polarization control, and Faraday optical isolators (FOI) to protect the nonlinear ﬁber against leakage from the ampliﬁer.

7.2.1 Stretcher

As stated previously, the first fiber amplifier provides the seed for the signal arm of the

OPCPA. After being coupled out of the fiber, the pulses go to the stretcher. For an input power of 24 mW, the power after the stretcher goes down to 5 mW, for a total efficiency of only 20 %. In the double-pass configuration, the beam bounces four time on the grating, but for a specified efficiency above 90%, the transmittance of the stretcher should be around 65%. Two factors can explain this discrepancy. First, the polarization of the output of the fiber amplifier is in a complicated state; even by using a half-wave Chapter 7. Optical Parametric Chirped-Pulse Amplifier 107 plate and a quarter-wave plate, the polarization is never linear. Because the efficiency of the grating drops to 45% for the polarization parallel to the grooves, the stretcher is equivalent to a polarizer. Secondly, the grating used is already a few years old and its efficiency was 77%, which puts the expected efficiency close to 35%. The performance of the stretcher could be improved by using polarization-maintaining (PM) fibers in the erbium amplifier and by changing the grating. However, PM fibers are harder to work with, and as the output of the stretcher is amplified, the efficiency of the stretcher is not crucial.

7.2.2 Delay line and temporal overlap

Following the stretcher, a delay line is used to tweak the overlap of the seed and the pump in the OPA. For active synchronization, the overlap can be adjusted with the electronics, whereas for passive synchronization, the path lengths of the pump and the seed have to be carefully matched, which is challenging as the time between two seed pulses is 50 ns.

The first step is to obtain a rough overlap of the two pulses using the switching time of the regenerative amplifier. The optimal pulse in the regen pulse train is determined by the energy of the seed and the gain of the amplifier; by playing with either or both it is possible to modify which pulse is switched out. Because the round-trip time of the regen is 11 ns, this is similar to a delay line with 11-ns step. By using a photo-diode, it is possible to monitor the seed and the pump as the timing delay is changed. Once this "electronic" optimization is done, a second coarse alignment is done by changing the path of the pump. Using a long mechanical delay line, 6.7 ns, located before the regen, it is possible to bring the pulses to within 1 ns of each other. For the final optimization, a delay line in the seed arm, consisting of a 25-mm translation stage on a 300-mm rail, is used. Still using the photodiode, the coarse alignment is done using the rail system.

Then, by monitoring the generation of a sum-frequency signal by mixing the seed and the pump in a BBO crystal, the ﬁne alignment is achieved using the translation stage. Chapter 7. Optical Parametric Chirped-Pulse Amplifier 108

7.2.3 Beam shaping

Before getting to the OPA, both the seed and the pump have to be shaped to the desired

dimensions. From previous works on a similar OPCPA, a pump diameter of 1.5 mm was

found to be optimal; for smaller diameters, the eﬃciency is lower because of the lost of

spatial overlap. For the seed, a diameter slightly bigger than the pump oﬀers a better

conversion eﬃciency. However it was decided to use only two telescopes to keep the setup

compact and simple, meaning that sacriﬁces had to be made. The decision was taken to

keep the seed smaller than the pump in the last crystal to preserve the beam quality.

For the seed, the telescope consists of a 250-mm and a 100-mm lens separated by

310 mm, and the dimensions of the beam are 1.53 1.46 mm2 (D D ) at the ﬁrst × X × Y crystal position and 1.18 1.16 mm2 at the second crystal. For the pump, a 125-mm × lens combined with a concave 100-mm lens with a separation of 26 mm give a beam with dimensions of 1.42 1.38 mm2(D D ) at the ﬁrst crystal position and of 1.85 1.64 × X × Y × mm2 at the second crystal.

7.2.4 Optimization - Stretcher length

Following the guidelines from the design section, the separation of the grating in the stretcher was set to 123 mm. As can be seen on ﬁgure 7.4a, the ampliﬁed spectrum, after two passes, is much narrower than the input spectrum. There are two possible reasons behind this. For one, the calculation for the dispersion of the stretcher was based on the initial pulse duration. However, the pulse duration is only truly representative of the bandwidth for a Gaussian spectrum, which is clearly not true here. In this case, the dispersion is over-estimated, and the spectrum is spread over more than the expected

47 ps. Secondly, the gain in the OPA grows exponentially with the pump intensity; if the seed and the pump are about the same duration, the central part of the seed will be ampliﬁed more, and in the case of a chirped pulse, the bandwidth will be reduced. Chapter 7. Optical Parametric Chirped-Pulse Amplifier 109

1 1 Amplified Amplified Unamplified Unamplified 0.8 0.8

0.6 0.6

0.4 0.4 Intensity (A.U.) Intensity (A.U.)

0.2 0.2

0 0 1530 1540 1550 1560 1570 1580 1590 1600 1540 1550 1560 1570 1580 1590 1600 1610 Wavelength (nm) Wavelength (nm)

(a) (b)

Figure 7.4: The amplified and the unamplified spectrum for (a) the original stretcher, and (b) the optimized version. Although, the central wavelength differed, the spectral width plotted is the same.

To improve the bandwidth, the spacing between the gratings was reduced to 70 mm, as small as allowed by the opto-mechanics. The new ampliﬁed spectrum, ﬁgure 7.4b, shows a clear improvement of the bandwidth.

7.2.5 Optimization - Noncollinear angle

The noncollinear angle was first set to 3.2◦. Figure 7.4b shows the spectrum of the amplified output after two passes. As the bandwidth should be slightly broader for larger angles, a few more noncollinear angles were tested. Figure 7.5 shows the result of this optimization. All three spectra are noticeably broader than the initial one. However, the differences between them are marginal. The angle was set at 3.3◦ because the output power after two passes was the highest. Chapter 7. Optical Parametric Chirped-Pulse Amplifier 110

3 3.45° 3.38° 2.5 3.30° 3.2° 2

1.5

Intensity (A.U.) 1

0.5

0 1550 1560 1570 1580 1590 1600 1610 1620 Wavelength (nm)

Figure 7.5: The spectra for diﬀerent noncollinear angles. The angle quoted is the internal one.

7.2.6 Optimization - Compressor

As stated previously, the compressor has to be detuned compared to the stretcher to account for the initial phase of the seed and the extra dispersion introduced by the KTA crystals. With the current design, the incident angle is not easily modiﬁed as the gratings are rotated independently, making it challenging to conserve their parallelism. For this reason, the spacing between the gratings is the main tweaking parameter, although a few incident angles need to be tested to obtain the optimal pulse duration. The coarse optimization of the compressor was done using a real-time auto-correlator whereas for

fine tweaking and the final characterization, a SHG-FROG was used. The optimal compression was obtained for an incident angle of 45◦ and a spacing on 60 mm (the results of the characterization are presented in the next section). It is important to notice that the compressor settings are sensitive to the phase profile at the output of the fiber amplifier.

For example, changing the power of the pump diodes modiﬁes the pulse characteristics, and therefore the optimal settings for the compressor. However, the general features are always the same: a larger incident angle and a smaller spacing between the gratings compared to the stretcher. Chapter 7. Optical Parametric Chirped-Pulse Amplifier 111

7.2.7 KTA OPA - Performance

The detailed layout of the KTA OPAs is shown in ﬁgure 7.6. The ﬁrst OPA consists

of a 15-mm KTA crystal in a double-pass conﬁguration. The power of the unampliﬁed

seed is 2.8 mW at a 20-MHz repetition rate, resulting in a pulse energy of 140 pJ. For

a pump power of 2.3 W (corresponding to a peak intensity of 5.3 GW/cm2), the seed power after the first pass is 4.2 mW. The extra power is then 1.4 mW, but at a repetition rate of 1 kHz, for an amplified pulse energy of 1.4 J and a first pass gain of 104. For

a ﬂat-top pump, both spatially and temporally and with a perfect overlap for the whole

length of the crystal, the optimal gain, calculated from equation (6.23), would be 28 104. The diﬀerence between the calculated and measured gain can be explained by less-than- optimal pumping conditions. The output power after the second pass is 320 mW for a gain of 228. Although the spot sizes for the second pass were not measured, the peak intensity of the pump is assumed to be similar to the ﬁrst pass. The strong reduction of the gain is due to saturation; 320 mW at 1.55 m corresponds to a pump depletion of

471 mW, 21% of the initial pump power.

From the Scale: stretcher / 10 mm delay line

f = 250 mm 300 mm f = 100 mm 230 mm 160 mm From the 18 mm multipass

KTA1 f = 125 mm f = -100 mm

40 mm KTA2 To the compressor

Figure 7.6: Layout, to scale, of the KTA ampliﬁcation stages. Important distances are

also indicated.

The second OPA consists of a 10-mm long crystal in a single-pass conﬁguration. The Chapter 7. Optical Parametric Chirped-Pulse Amplifier 112

gain in the second crystal is only 1.65 for a ﬁnal output power of 520 mW. The low gain

can be explained by saturation, a greater mismatch between the pump and the seed size,

and the reduction in the pump peak intensity (only 4.2 GW/cm2).

Superfluorescence (SF) is a source of noise in the amplifier resulting from the OPA signal arising from spontaneous down-conversion. SF is measured by blocking the seed before the first stage so that each stage is seeded by the SF from the previous ones. For the first two stages, the SF is negligible, whereas for the last one there is 5 mW of SF.

This sets the upper limit for the SF of the seeded OPA to less than 1% as the noise is expected to go down when the seed is present.

The pulse-to-pulse stability was measured using using an InGaAs photodiode and a

DAQ card. For a stability of the regen around 0.5%, the stability of the OPCPA is 1%, which is quite good. Figure 7.7 shows the beam proﬁle of the output of the OPCPA after

(a) (b)

Figure 7.7: The beam proﬁle after (a) the second crystal, and (b) the compressor.

the last crystal and after the compressor; the images were obtained using two-photon

absorption in a Si CCD camera, which explains the noise in the pictures. In both cases, Chapter 7. Optical Parametric Chirped-Pulse Amplifier 113

1 40 1 20 Retrieved E. field R. spectrum Retrieved phase R. phase M. spectrum

0.5 30 0.5 Phase (rad) Phase (rad) Intensity (A.U.) Intensity (A.U.)

0 20 0 0 −500 0 500 1000 1540 1550 1560 1570 1580 1590 1600 1610 1620 Time (fs) Wavelength (nm)

(a) (b)

Figure 7.8: On the left: the retrieved electric ﬁeld intensity and the retrieved temporal

phase. On the right: the retrieved spectrum and the retrieved phase as well as the

measured spectrum.

there is clearly astigmatism, but the general quality is good. The M 2 is 1.74 along the

x-axis and 3.38 along the y-axis. The M 2 was measured using a series of knife-edge

measurements after a 400-mm focusing lens.

The power after the compressor is 300 mW for a transmission eﬃciency of 58%, which

is compatible with the 90% eﬃciency of the gratings. Figure 7.8a shows the retrieved electric ﬁeld after optimization of the compressor. The FWHM of the pulse is 130 fs with close to 70% of the energy within the central peak. The structures in the pulse

can be attributed to the modulations in the spectrum and, to some extent, the residual

phase. The retrieved power spectrum and phase can be found in ﬁgure 7.8b as well as

the measured spectrum.

7.2.8 Timing jitter

One of the major goals of this project was to use passive synchronization to reduce the

timing jitter between the seed and the pump. Although the simple fact that the amplitude Chapter 7. Optical Parametric Chirped-Pulse Amplifier 114

noise in the OPCPA is low indicates that passive synchronization is working, quantifying

the timing jitter is essential to understand the importance of this work. As explained

previously, the timing jitter is extracted from the cross-correlation (XC) between the

pump and the seed in a BBO crystal. As the XC was also used to characterize the shape

of the pump pulse, the grating in the stretcher was replaced by a ﬂat mirror to keep the

seed relatively short. Figure 7.9 shows the XC trace; around 40 ps the signal is roughly

linear. At this point, the relative noise of the XC is 0.83% for a pump noise of 0.35%.

Removing the noise of the pump by subtracting it in quadrature results in a timing noise

Intensity (mV) 30

10 0 20 40 60 80 100 Time delay (ps)

Figure 7.9: Cross-correlation trace between the seed and the pump.

of 0.75% or 0.4 mV. From the slope at 40 ps, the amplitude noise can be translated into timing noise. For a slope of 1.2 mV/ps, 0.4-mV fluctuations give a jitter of 333 fs. The remaining jitter can be attributed to fluctuations in the path length of the regenerative amplifier. With pulses trapped for 870 ns in the regenerative amplifier, any fluctuations in the cavity length will be amplified. The only true solution for this problem would be to replace the regenerative amplifier, which is not trivial. However, the jitter is less than

1% of the current pump pulse duration, which was the target. The residual timing jitter could be a problem if the pump duration was to be reduced.

The above result represents a considerable advance in ultrafast pulse ampliﬁcation.

Effectively, this all-optical approach has solved the time synchronization problem inher- Chapter 7. Optical Parametric Chirped-Pulse Amplifier 115 ent to the OPCPA approach. Relatively minor engineering modifications with respect to improving the length stability of the regenerative amplifier with better temperature regu- lation will improve the amplitude stability. However, the current stability is already close to that of the input seed and pump sources. This concept can now be scaled to orders of magnitude higher powers using proven scaling principles for Nd-based amplifiers. Chapter 8

MIR OPA development

For most ultrafast applications in the mid-infrared (MIR), the laser pulses are generated

using optical parametric ampliﬁcation. Typical systems based on Ti:Sapphire lasers and

using two nonlinear stages can generate microjoules from 3 m to 20 m. In this work, a novel configuration based on a single nonlinear stage and pump at 1.6 m is pursued as it could be scaled up to higher pulse energy. For the proof-of-principle, the optical parametric amplifier (OPA) is seeded with the idler of the optical parametric chirped- pulse amplifier (OPCPA), for which the angular chirp was compensated.

The nature of the two crystals tested for the OPA will be covered in the ﬁrst part of this chapter as well as the details of the angular compensation setup. In the second part, the performance of the chirp compensator and the two OPAs will be discussed.

8.1 Design

8.1.1 Nonlinear Crystals

The major reason to use 1.6 m instead of the standard 800 nm of Ti:Sapphire, is that for longer pump wavelengths, there is a whole new set of crystals that can be used. In addition, the problem of multiphoton absorption and induced optical damage in these

116 Chapter 8. MIR OPA development 117

new materials is largely mitigated. The main crystal used for any OPA pumped in the

near-infrared is silver thiogallate (AGS) because of its transparency range that goes from

0.5 m to 13.2 m [70] and of its nonlinear coeﬃcient that is around 14 pm/V for Type II phase matching, which is about an order of magnitude higher than typical values for BBO or KTA. There are other crystals with higher nonlinear coeﬃcient, e.g. silver gallium selenide (AgGaSe2 - AGSe), but they are not transparent for the second harmonic of the pump, making them susceptible to two-photon absorption. A major shortcoming of AGS is its low surface damage threshold, around 20 MW/cm2 (single short, 1064 nm, 17.5 ns

[71]); in comparison, KTA can be used safely up to 1.2 GW/cm2 [67].

For this reason, it was decided to try a second crystal: Lithium gallium selenide

(LGSe). LGSe is a fairly novel nonlinear crystal and has advantageous properties over

AGS, mainly a higher thermal conductivity and a broader transparency window, 0.37 m to 13.2 m [72]. However, for an OPA pumped at 1.55 m it is the possible higher damage threshold that is interesting. Although there are no studies on LGSe damage threshold, a similar crystal, lithium gallium sulﬁde (LiGaS2 - LGS), exhibits no surface damage and a damage threshold on the order of 285 MW/cm2 [73]. As LGSe has a similar nonlinear coeﬃcient to AGS, around 10 pm/V, an increase of the damage threshold would improve the scalability, as higher pump energy could be used for the same clear aperture.

Figure 8.1 shows the phase-matching curves for the two crystals. For both, using a noncollinear angle does not improve the phase-matching bandwidth. To facilitate the separation of the idler and signal after ampliﬁcation, a noncollinear angle is still introduced but kept as small as possible to avoid excessive angular chirp in the idler.

8.1.2 Angular chirp compensation

The MIR OPA is seeded with the MIR idler of the OPCPA. As pointed out in numerous occasions, in a noncollinear geometry, the idler is angularly dispersed. To make the beam somewhat usable, this angular dispersion can be transformed into spatial chirp by using Chapter 8. MIR OPA development 118

4 4

3.5 3.5

3 3 Wavelength (um) Wavelength (um)

2.5 2.5 45 50 55 60 65 70 75 40 45 50 55 60 65 70 Phase−matching angle (°) Phase−matching angle (°)

(a) AGS (b) LGSe

Figure 8.1: Phase-matching curves for diﬀerent noncollinear angles - from left to right:

0◦,1◦,2◦,3◦. a collimating lens. Because in the MIR OPA the seed, 50 ps, is much longer than the pump pulse, 130 fs, and is highly chirped, the ampliﬁcation in the OPA would remove most of the chirp. However, compensating for the chirp before insures good beam quality at the OPA position and makes the alignment easier.

The angular chirp can be removed with the help of a grating. By using a telescope to match the dispersion of the grating to the one of the idler (see figure 8.2) it is possible to get a beam free of chirp. The dispersion of a grating can be obtained by differentiating the diffraction angle with respect to the wavelength:

dθ d λ D = = sin−1 sin(γ) (8.1) grating dλ dλ d − − 1 λ 2 d λ = sin(γ) sin(γ) (8.2) d − dλ d − 1 = . (8.3) d cos(θ)

On the other hand the dispersion in the idler is given by equation (6.51):

C1 Didler = 1 , (8.4) (1 C2λ2) 2 − 1 i Chapter 8. MIR OPA development 119 where

ny (λp) C1 = sin(α) . (8.5) λp

f1 f1 + f2 f2

Figure 8.2: By using the proper focal lengths, the collimated beam diameter for both dispersive elements can be equal.

For the dispersion to cancel, the following equality has to be satisﬁed:

f1 tan(φidler/2) = f2 tan(φgrating/2) , (8.6) with

φ = D ∆λ, (8.7) idler/grating idler/grating where ∆λ being the bandwidth of the idler. Under the small angle approximation, i.e. tan(φ/2) φ/2, the previous equation can be reduced to: ≈

f1Didler = f2Dgrating, (8.8)

or in terms of the magniﬁcation of the telescope, M = f2/f1,

D M = idler . (8.9) Dgrating

At this point, there are three variables that can be used to make this equality hold: the magnification (by selecting the proper lenses), the diffraction angle (through the incident Chapter 8. MIR OPA development 120 angle), and the grating groove density. Because the choice of grating is fairly limited, the groove density is set first, imposing constraints on the other two parameters. A groove density of 200 lines/mm was chosen as the grating was readily available in the lab. Of the remaining two, the diffraction angle offers more flexibility as it can be tuned continuously. On the other hand, the magnification is more of a discrete variable because, in terms of time and finance, it is much easier to work with standard focal lengths. By plotting equation (8.9), it is possible to find an incident angle that is associated to achievable magnification. From figure 8.3, an incident angle around 35◦ is associated with a magnification of 0.5, which is easily accessible.

0.52

0.5

0.48

0.46

0.44 Magnification 0.42

0.4

0.38 0 5 10 15 20 25 30 35 40 45 Incident angle (°)

Figure 8.3: Plot showing the magniﬁcation needed for diﬀerent incident angles, for a

200-lines/mm grating.

8.2 Performance

8.2.1 Angular dispersion

Figure 8.4 shows the setup for the angular dispersion compensation. It should be noted that the idler used is coming from the second pass instead of the third because it showed better beam quality. The compensation setup consists of a 150-mm ZnSe lens to collimate Chapter 8. MIR OPA development 121

Scale: 10 mm

50mm Grating 75 mm KTA

L1 To MIR OPA L2 To second KTA crystal 112 mm

190mm 25 mm

Figure 8.4: The layout for the angular dispersion compensation setup. L1 is the 150-mm

ZnSe lens and L2 is the 75-mm CaF2 lens.

the angular dispersion of the idler and, 217 mm after, a 75-mm CaF2 lens to focus the beam at a 35◦ angle on a grating, as calculated in the previous section.

To optimize the setup, an easy way to visualize the chirp is needed. This is achieved by using a simple spectrometer setup, a grating dispersing the beam followed by a lens, but where a cylindrical mirror is used instead. The different spectral components can be resolved in the focusing plane of the optics as a function of their position along the non-focusing direction. However, it is more convenient and safer to have the diffraction plane horizontal with the spatial chirp along the vertical axis, which is typically not the case for the idler. By using a periscope at a 90◦ angle, it is possible to rotate the beam profile to satisfy this condition. Figure 8.5 shows the diagnostic setup and figure 8.6 the results for both the compensated and uncompensated beams. Although not perfect, the compensator removed most of the chirp as the spectrally resolved beam profile is straighter. The power at the input of the compensator is 80 mW and 15 mW at the output. The low efficiency, only 18%, is the consequence of using a grating not optimized for 3 m combined to a beam with a polarization parallel to the groove. Chapter 8. MIR OPA development 122

Cylindrical Periscope A' From the OPCPA mirror

B A Beam profile A

B' B

Grating A' CCD camera

Figure 8.5: The diagnostic tool for the spatial chirp. The mirror is placed a focal length away from both the grating and CCD camera.

(a) (b)

Figure 8.6: The beam proﬁle in the diagnostic tool for (a) the compensated beam, and

(b) the uncompensated one. Note, the images are rotated by 90% to match the actual input of the OPA. In the inlets, the beam proﬁle at the entrance is shown.

8.2.2 OPA setup

The OPA setup is the same for the two nonlinear crystals; only their position changes. In the seed arm, the main components are a delay line to adjust the overlap and a 500-mm

CaF2 lens to focus the seed at the OPA position. There is also a relaying telescope, two 150-mm ZnSe lenses, to keep the beam size from blowing up. For the pump, only

500-mm lens is needed. Figure 8.7 shows the detailed setup. Chapter 8. MIR OPA development 123

Scale: 10 mm AGS (LGSe) 250 mm From compressor Crystal (350 mm) 1.6 μm idler

signal To diagnotic 300 mm (400 mm) 230 mm

310 mm

From OPCPA ~3 μm

630 mm

Figure 8.7: Layout of the MIR OPA.

8.2.3 AGS OPA

The first crystal used was 4-mm thick AGS cut at 55.5◦ for Type II phase-matching for an effective nonlinear coefficient of 13.7 pm/V . The crystal is located at 300 mm from the lens in the pump arm. At this position the seed spot size is 1.32 mm along the X-direction and 3.75 mm along the Y-direction. For the pump, the X-diameter is 1.56 mm and the

Y-diameter, 2.53 mm. Before testing the OPA, the pump pulses were characterized with a SHG-FROG and the pulse duration was 138 fs with about 90% of the energy within the central peak. For these pumping conditions and considering the losses due to reﬂection, a pump power of 214 mW results in a peak intensity of 69 GW/cm2. Taking into account the ∆t−0.5 scaling laws for damage threshold [41] (∆t being the pulse duration), from the 20 MW/cm2 measured for 17.5-ns pulse, the damage threshold for a 130-fs pulse should be 7 GW/cm2. This is an order of magnitude lower than the pulse peak intensity and, still, the crystal shows no sign of damage. This is attributed to a breakdown in the scaling law for pulses lower than 20 ps, as reported in [74].

The seed input power is 3.40 mW for a 50-ps pulse duration and the energy of the seed overlapping the 138-fs pump pulse is only 9.4 nJ. After the amplification, the total output power is 14.60 mW, out of which 2.44 mW is leakage from the pump. The total amplified power is then 8.76 mW, for an amplified pulse energy of 8.76 J. The gain, defined as the energy of the amplified pulse divided by the energy of the seed overlapping Chapter 8. MIR OPA development 124

Signal:

1 AC trace 0.9 Gaussian fit 0.8

0.7

0.6

0.5

0.4 Intensity (A.U.) 0.3

0.2

0.1

0 −1 −0.5 0 0.5 1 Time(ps)

(a) Beam proﬁle (b) Auto-correlation (c) Spectrum

Idler:

(d) Beam proﬁle (e) Spectrum

Figure 8.8: Results of the characterization of the signal (a-c) and the idler (d-e) of the

AGS OPA. The vertical structures in both beam proﬁles arise from a triggering issue.

There is no auto-correlation trace for the idler.

the pump, is 933. The gain is much lower than the expected value of 2.7 109. However, with such a high small-signal gain, the undepleted pump approximation is unrealistic and

saturation plays an important role. For the idler, the total power is 22.5 mW with 15.3

mW of pump leakage resulting in 7.2 mW of ampliﬁed power. The combined power for

the signal and the idler is 15.96 mW for a total efficiency of 7.4%. Figures 8.8a and 8.8d show the beam profiles for the idler and the signal; both beams have featureless profile with strong astigmatism caused by the ellipticity of the pump beam.

The spectra for the idler and the signal were recorded. However, because of the lack of spectrometer covering the MIR, the pulses were ﬁrst frequency-doubled. The spectrum for the fundamental is then reconstructed by taking spectra of the SHG signal for diﬀerent Chapter 8. MIR OPA development 125

phase-matching angles. From those reconstructed spectra, ﬁgure 8.8c for the signal and

8.8e for the idler, the approximated bandwidth can be extracted: 125 nm for the signal

and 300 nm for the idler. The strong diﬀerence in bandwidth can be explained by the

use of a broadband pump pulse.

The pulse duration of the signal was measured using an autocorrelator based on two-

photon absorption in an InGaAs photo-diode; ﬁgure 8.8b shows the resulting trace. A

Gaussian function with FWHM of 250 fs was used to ﬁt the signal. There are clearly some

wings that the Gaussian cannot capture indicating that there is a second "component”,

such as a close prepulse or a pedestal. However, the Gaussian ﬁt can still be used to

extract the pulse duration for the central feature; from the FMWM of the autocorrelation,

using a deconvolution factor of √2, the pulse duration is 177 fs. The duration for a

transform-limited pulse with a 125-nm bandwidth is roughly 100 fs. The discrepancy

between the two values can be explained by two factors. First, because the seed is much

longer than the pump, the ampliﬁed signal cannot be shorter than the 130-fs pump.

Furthermore, because of the group velocity mismatch, the pump moves with respect to

the initial overlap during the ampliﬁcation, eﬀectively stretching the signal. For AGS,

group index, deﬁned as the ratio of the speed of light in vacuum over the group velocity,

is 2.434 for the pump and 2.430 for the signal, resulting in relative displacement of the

pump of 53 fs. The second factor is an artifact of the characterization setup. As the

autocorrelator uses an InGaAs photodiode, it is extremely sensitive to the 1.6-m pump.

Therefore a ﬁlter has to be put in front of the detector otherwise the small nonlinear signal is swamped by the strong 1.6-m background. For the measurements presented here, a

3-mm thick piece of germanium was used because of its strong absorption of the pump.

However, germanium has an indirect band gap of 0.66 eV causing a sharp absorption feature at 1.88 m. This results in rapid variation of the index of refraction between 2 and 6 m, going from 4.1 to 4.01, meaning that germanium is a highly dispersive medium.

The propagation through 3-mm of germanium will stretch a 100-fs pulse to 161 fs. Chapter 8. MIR OPA development 126

Attempts to measure the pulse duration of the idler were made but without success.

There are two diﬀerences between the idler and the signal that could explain the lack of

autocorrelation signal. The group velocity mismatch is more important for the idler as

its group index is 2.392, resulting in a displacement of 560 fs for the 4-mm crystal. Also,

the bandwidth of the idler is able to support a 55-fs pulse, meaning that the 3-mm Ge

plate could stretch the pulse to 260 fs. These eﬀects will contribute to reduce the peak

intensity, resulting in a diminution/disappearance of the AC signal.

The stability of the signal was measured using 2-photon absorption in an InGaAs

photo-diode. For a shot-to-shot stability of the regen of 1%, ﬂuctuations on the order of

2.6% were measured in the nonlinear signal in the diode. If the noise can be added in quadrature, then the noise on the 3 m would be 1.8%. The noise on the 1.55-m was

measured for comparison and was 1.8%.

8.2.4 LGSe OPA

The second crystal used was 2-mm thick LGSe cut at 46.9◦ in the X-Y plane for type II

phase matching for an eﬀective nonlinear coeﬃcient of 9 pm/V. The crystal was placed

400 mm away from the focusing lens in the pump arm. At this position the signal spot

size is 1.38 mm along x and 3.01 mm along y, whereas the pump is 1.40 mm by 2.06 mm.

Again, before working on the OPA, the pulse duration of the pump was measured to be

140 fs with 85% of the energy within the central peak with a resulting peak intensity of

90 GW/cm2 for an input power of 210 mW.

The input power of the signal is 4.6 mW for a 50-ps pulse, with one 12.9 nJ overlapping

with the 140-fs pump. The total output power is 16.14 mW with no leakage from the

pump, resulting in 11.54 J of ampliﬁed energy and a gain of 895. For the same reason as

with the AGS crystal, the gain is much lower than the theoretical value of 5.7 107. For the idler, the output power is 11.11 mW with no pump leakage. The combined ampliﬁed

power is 22.65 for an overall eﬃciency of 10.8% which is higher than the AGS OPA. The Chapter 8. MIR OPA development 127

Signal:

1 AC: Signal Fit 0.8

0.6

0.4 Intensity (A.U.)

0.2

0 −1 −0.5 0 0.5 1 Time (ps)

(a) Beam proﬁle (b) Auto-correlation (c) Spectrum

Idler:

1 AC: Idler Fit 0.8

0.6

0.4 Intensity (A.U.)

0.2

0 −1 −0.5 0 0.5 1 Time (ps)

(d) Beam proﬁle (e) Auto-correlation (f) Spectrum

Figure 8.9: Results of the characterization of the signal (a-c) and the idler (d-e) for the

LGSe OPA. The vertical structures in both beam proﬁles arise from a triggering issue.

proﬁles of the two MIR beams are shown in ﬁgures 8.9a and 8.9d. Similarly to the AGS, the two beams are astigmatic as the pump is elliptic .

The reconstructed spectrum of the signal is shown in ﬁgure 8.9c and of the idler, in

ﬁgure 8.9f. The FWHM bandwidth of the signal is 275 nm, which is enough to support a 55-fs pulse, whereas for the idler, the bandwidth is just 150 nm, with an associated transform-limited pulse duration of 107 fs.

The pulse duration of the signal was characterized using a two-photon-absorption photodiode. The autocorrelation trace is shown in ﬁgure 8.9b and its FWHM is 416 fs resulting in a pulse duration of 291 fs. Dispersion alone cannot explain this discrepancy as after propagating in 3-mm of germanium, a 50-fs pulse would be stretched close to 260 fs. As the group-velocity mismatch plays no role as both the signal and the pump have Chapter 8. MIR OPA development 128 a group index of 2.303, the extra stretching could be explained by the initial stretching due to the pump duration.

Unlike for the AGS OPA, it was possible to make an auto-correlation of the idler.

This strengthens the idea that the group-velocity mismatch was the problem. For LGSe crystal, the difference in group index between the pump and the idler is 0.031, resulting in a displacement of the pump of 206 fs for a 2-mm crystal, which is more than half the value for the 4-mm AGS. The autocorrelation trace for the idler, shown in figure 8.9e, has a FWHM of 333 fs for a pulse duration of 235 fs. From its spectral bandwidth, the pulse duration of the idler should be around 160 fs, if dispersion is solely responsible for the stretching. Clearly, the effect of group-velocity mismatch has to be taken into account to explain the autocorrelation.

The stabilities of the pump and of the signal were both 4.4%, which is higher than for the AGS OPA, which can be explained by the noisier regen, with its output having

2% ﬂuctuations. Chapter 9

Conclusion

In this thesis, a new type of mid-infrared optical parametric ampliﬁers (MIR OPAs) was demonstrated. Although ultrafast MIR OPAs are available with great tunability [6], there is still room for improvement. The main problem tackled in this work was the scalability in power of MIR sources, which is primordial if laser selective chemistry is to be achieved.

For the MIR OPA, two crystals were tested: silver thiogallate, AGS, and lithium gallium selenide, LGSe. In both case, the pump beam was the compressed 1.6-m output of the optical parametric chirped-pulse amplifier (OPCPA), with a power around 210 mW and a pulse duration of 140 fs. For the seed, the idler of the OPCPA was used, but only after the angular dispersion was compensated. With the AGS crystal, the amplified power in the signal was 8.76 mW and 7.2 mW in the idler, for a combined efficiency of

7.4%. With the LGSe crystal, the efficiency was slightly higher at 10.8 %, with 11.54 mW of amplified power in the signal, and 11.11 mW in the idler. Although the OPA was far from being optimal, the effficiencies obtained are already an order of magnitude higher than the what can be achieved in a two-step system pumped by Ti:Sapphire [6].

With this increase of eﬃciency, hundreds of microjoule in the MIR should be achievable.

The MIR system is pumped by a KTA OPCPA operating at 1.6 m. Based on an

129 Chapter 9. Conclusion 130

Erbium-doped ﬁber oscillator and pumped by a Nd:YLF ampliﬁer, the OPCPA produces

300 mW of 130-fs pulses at a repetition rate of 1 kHz. Because of a better synchronization,

the stability of the OPCPA is down to 1 % from the previous 3 %. The OPCPA is the

core of this project as the increase in eﬃciency in the MIR generation is solely due to the

use of a longer pump wavelength, which is made possible by the OPCPA. Furthermore,

because of the nature of the gain, the output power can easily be scaled up, which would

trickle down to the MIR.

If the OPCPA is the foundation of this project, the passive-synchronization scheme is

the enabling element. Through propagation in a highly nonlinear ﬁber, the 1 m needed to seed the ampliﬁer is generated with the same Erbium-doped oscillator used for the

1.56 m, eliminating the timing jitter associated with the mismatch in repetition rates

for two independent lasers. The remaining jitter, 333 fs, is attributed to the path length

ﬂuctuations in the pump arm.

9.1 Future work - improvement

Although major milestones have been reached at this point, there are still a few obvious

improvements that can be made to push this laser system into a whole diﬀerent category

and compete head-to-head with free-electron lasers.

On the source side, the long term stability of the HNLF could be improved by switch-

ing to polarization-maintaining ﬁbers and by actively cooling the ampliﬁer and its en-

closure. A more interesting modiﬁcation would be to introduce a HNLF in the seed arm

to add tunability to the OPCPA, as well as improving the pulse quality. However this

would require rebuilding the ﬁrst ampliﬁer to shorten the duration of its output.

A major step forward in terms of power scalability would be to change the gain

medium to neodymium-doped yttrium vanadate (Nd:YVO4) and neodymium-doped yt-

trium aluminum garnet (Nd:Y3Al5O12 - Nd:YAG); these have the same lasing wavelength Chapter 9. Conclusion 131

and can therefore be used in tandem to exploit each of their qualities. Nd:YVO4 is well suited for a pre-amplifier because of its larger cross-section, resulting in higher gain for given pump power. It is then possible to amplify a weak seed to moderate power with a multipass amplifier, eliminating the need for a regenerative amplifier, and therefore reducing the path length of the pump arm and the complexity the system. This comes with a price: a new way to stretch the pump pulses would have to be used as it is currently done via the regen. A straightforward solution would be to use chirped fiber Bragg gratings as they can easily introduce the amount of dispersion needed. However, some thought needs to be put on the coupling between the HNLF and the stretching fiber, which is not trivial. On the other hand, Nd:YAG is more suited for high-power amplifiers because of its higher thermal conductivity and better mechanical properties. Furthermore, as an isotropic crystal, it can be made into ceramic, facilitating the manufacturing of large crystals, which help in handling high-power pulses.

Although the MIR OPA has a better eﬃciency than MIR systems based on Ti:Sapphire and cascaded nonlinear processes, when compared to near infrared (NIR) OPAs, whose ef-

ficiency can reach 30% [6], there is clearly room for improvement. However these systems are used in a double-pass configuration and it can be expected that similar efficiencies could be achieved with the MIR OPA if a second pass was added. Another interesting feature from these NIR OPAs that could be copied is the seeding with white-light.

With this type of seed, the output of the OPA can be easily tuned over a broad range, increasing its versatility.

9.2 Closing words

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