Nonlinear propagation of light pulses and water waves under the influence of damping and forcing

Thesis

Reference

Nonlinear propagation of light pulses and water waves under the influence of damping and forcing

EELTINK, Debbie

Abstract

This work reports results on the relationship between the nonlinear propagation of optical pulses and of water waves. What unites these two phenomena is that the envelope of the electric field of the light pulse, and of the water wave, can both be described by the same propagation equation: the nonlinear Schrödinger equation (NLSE). In the practical context of rogue waves on the ocean the optical system can serve as a table-top experiment. On a more fundamental level, the comparison allows cross-fertilization of insights. For water waves, the influence of viscous damping is examined with respect to the dispersion relation and the mass conservation. In addition, the effect of wind forcing on water waves is examined, specifically with respect to the spectral shift. Finally, triggering of optical filamentation by a turbulent atmosphere is demonstrated.

EELTINK, Debbie. Nonlinear propagation of light pulses and water waves under the influence of damping and forcing. Thèse de doctorat : Univ. Genève, 2019, no. Sc. 5399

DOI : 10.13097/archive-ouverte/unige:127104 URN : urn:nbn:ch:unige-1271046

Available at:

http://archive-ouverte.unige.ch/unige:127104

Disclaimer: layout of this document may differ from the published version.

Groupe of Applied Physics (GAP) Docteur Maura Brunetti

Nonlinear Propagation of Light Pulses and Water Waves under the Influence of Damping and Forcing

THÈSE

présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences, mention physique

Debbie Eeltinkpar Alkmaar (Pays-Bas)de

Thèse N° 5399

Genève

Centre d’Impression de l’Université de Genève 2019

PhD Thesis by Debbie Eeltink

Defense: Friday October 18th 2019, Geneva, Switzerland

Supervisors

Prof. Jérôme Kasparian Dr. Maura Brunetti

Jury members

Prof. John Dudley, Université de Franche-Comté, France Prof. Ton van den Bremer, University of Oxford, UK Prof. Peter Wittwer, Université de Genève, Switzerland

Publications

[1] D. Eeltink et al. “Triggering filamentation using turbulence”. Physical Review A94.3 (Sept. 2016), p. 033806.

[2] D. Eeltink et al. “Spectral up- and downshifting of Akhmediev breathers under wind forcing”.Physics of Fluids29.10 (Oct. 2017), p. 107103.

[3] A. Armaroli et al. “Nonlinear stage of Benjamin-Feir instability in forced/damped deep- water waves”.Physics of Fluids30.1 (Jan. 2018), p. 017102.

[4] A. Armaroli et al. “Viscous damping of gravity-capillary waves: Dispersion relations and nonlinear corrections”.Physical Review Fluids3.12 (May 2018), pp. 1–14.

[5] D. Eeltink et al. “Single-spectrum prediction of kurtosis of water waves in a nonconser- vative model”.Physical Review E100.1 (July 2019), p. 013102.

[6] D. Eeltink et al. “Reconciling different formulations of viscous water waves and their mass conservation”.Submitted(2019).

Ce travail présente les résultats sur la propagation non-linéaire des vagues d’eau et d’impulsions optiques. L’équation qui décrit d’une part l’enveloppe de la vague d’eau et d’autre part l’enveloppe du champ électrique est la même équation: l’équation de Schrödinger non-linéaire (NLSE). Dans le contexte de vagues scélérates sur l’océan - des vagues géantes qui semblent surgir de nulle part - les systèmes optiques pourraient servir d’expériences model et ainsi obtenir plus facilement un grand nombre de réalisations. À un niveau plus fondamental, leur comparaison permet une fertilisation croisée des idées.

Tout d’abord, afin d’apprécier leurs similitudes, nous effectuons une comparaison pas à pas à partir des équations fondamentales conduisant à la NLSE pour les deux systèmes, et nous en discutons leur comportement. Par la suite, l’essentiel de cette thèse est consacré aux ondes de gravité en eaux profondes, pour lesquelles le rôle de l’amortissement visqueux et du forçage du vent est examiné.

Deuxièmement, la prise en compte de la résistance de l’eau au cisaillement, qui définit la viscosité, complique les équations du modèle, car cela introduit un tourbillon au voisinage de la surface. Nous comparons trois voies différentes proposées dans la littérature pour obtenir un système d’équations fermé pour le problème des vagues visqueuses. En établissant la correspondance terme à terme des deux modèles et en comparant la conservation de la masse sur l’ensemble de leur domaine de définition respectif, nous sommes en mesure de déterminer que les modèles sont en effet compatibles jusqu’à leur ordre de validité. De plus, la viscosité modifie la relation de dispersion, qui est liée à l’équation de propagation et aux équations de type Euler. Nous proposons donc de modifier les équations d’Euler pour obtenir une NLSE dans laquelle des termes visqueux linéaires et non-linéaires apparaissent conjointement. Nous montrons que la partie non-linéaire ne représente que de petites corrections, justifiant les approches conventionnelles de la propagation amortie trouvées dans la littérature.

Troisièmement, la contrepartie de l’amortissement est le forçage. Le forçage par le vent est un ingrédient clé pour comprendre la dynamique des vagues océaniques. Nous étudions numériquement et expérimentalement l’effet du vent sur la dynamique spectrale. Nous dévelop- pons le modèle de vent au même ordre de raideur des vagues que la modification d’ordre plus élevé de la NLSE, c’est-à-dire l’équation de Dysthe. Il en résulte un terme de vent asymétrique à l’ordre supérieur, ce qui conduit à faire croître davantage les fréquences élevées que les basses fréquences et induit ainsi une augmentation permanente de la fréquence moyenne. Notre mod- èle est en bon accord avec les expériences de laboratoire dans les limites de la longueur du bassin expérimental.

Pour mieux comprendre la dynamique du système, nous effectuons une troncature en trois modes. Ce modèle simplifié nous permet une classification de notre équation de Dysthe modifiée dans le plan de phase. Si la viscosité domine le forçage par le vent tant au premier ordre qu’au second, un attracteur induit un déphasage du cycle récursif de l’enveloppe. De plus, le spectre se décale vers les basses fréquences. En revanche, lorsque le vent domine aux deux ordres, les solutions suivent des trajectoires sans déphasage.

Nous effectuons une analyse statistique avec notre modèle de propagation prenant en compte le forçage et l’amortissement afin de déterminer un indicateur qui permettrait en une mesure unique de prévoir le kurtosis (ou coefficient d’aplatissement) de la distribution de hauteur des vagues après un épisode de vent. Le kurtosis est en effet associé à un risque élevé de vagues scélérates. Cependant, comme il s’agit d’une quantité statistique, elle nécessite généralement de nombreuses mesures. Nous montrons que la largeur du spectre est fortement corrélée avec le kurtosis de l’ensemble.

Enfin, nous revenons à l’optique non-linéaire en utilisant le caractère aléatoire de la turbu- lence pour provoquer une instabilité de modulation spatiale, ce qui déclenche la filamentation à une puissance ou elle ne peut normalement pas se produire dans une atmosphère calme.

This work reports results on the relationship between the nonlinear propagation of optical pulses and of water waves. What unites these two phenomena is that the envelope of the electric field of the light pulse, and of the water wave, can both be described by the same propagation equation:

the nonlinear Schrödinger equation (NLSE). In the practical context of rogue waves on the ocean -high waves that seemingly appear out of nowhere- the optical system serve as a table-top experiment to get a high number of realizations. On a more fundamental level, the comparison allows cross-fertilization of insights.

First, in order to appreciate their parallels, we perform a step-by-step comparison from fundamental equations leading up to the NLSE for both systems, and discuss its behavior.

Subsequently, the bulk of this thesis focuses on deep-water gravity waves, for which the role of viscous damping and wind forcing are examined.

Second, taking into account the water’s resistance to shear stress, i.e. its viscosity, greatly complicates the model equations, as it introduces vorticity near the free surface. We compare three different routes of obtaining a closed system of equation for the viscous water-wave problem. By means of careful bookkeeping and comparing the mass conservation over the entire domain, we are able to conclude that the models indeed correspond to each other up to the order to which they are valid. In addition, viscosity alters the dispersion relation. The latter is linked to the propagation equation and the water wave problem. We propose to modify the water wave problem to derive an NLSE in which both linear and nonlinear viscous terms arise.

We show that the nonlinear part only represents small corrections, justifying the conventional approaches to damped propagation found in the literature.

Third, the counterpart of damping is forcing. Wind-forcing is key to understanding ocean- wave dynamics. We experimentally and numerically investigate the effect of wind forcing specifically on the spectral dynamics. We develop the wind model to the same order of steepness as the higher-order modification of the NLSE, also referred to as the Dysthe equation.

Our wind model produces an asymmetric wind term in the higher-order, which enhances higher frequencies more than lower ones and thus induces a permanent upshift of the spectral mean.

The derived model affirms laboratory experiments within the range of the facility’s length.

To gain more insight into the dynamics of the system, we perform a three-wave truncation.

This simplified model allows us to classify the solutions of the forced-damped Dysthe in the phase plane. If viscosity dominates wind forcing at both the leading and higher-order, an attractor exists that induces a phase-shift of the recursive cycle of the envelope. In addition, a downshift occurs. Conversely, if wind dominates at both orders, the solutions follow trajectories without a phase-shift.

We perform a statistical analysis with our forced-damped propagation model to determine a single-measurement precursor for wave height-kurtosis after windy conditions. Kurtosis is in turn associated with a high risk of rogue waves. However, as this is a statistical quantity, it usually requires many measurements. We find that the spectral width shows a high correlation with the kurtosis of the ensemble.

Finally, we return to nonlinear-optics by using the randomness of turbulence to spark spatial modulation instability, which in turn triggers filaments.

First and foremost I would like to thank my two supervisors, on whose shoulders I could stand in my exploration through the wonderfully complicated and surprising areas of nonlinear dynamics and hydrodynamics. Many times when I felt stuck in the project, after discussing with one of you I was injected with new ideas and motivation to find new paths. You both have genuine intentions of making your students better researchers, and you take the time and effort to teach them, even when it brings no direct benefit to you. I have seen this many times with me and others, and I appreciate this very much. I hope you continue as a team because I find it striking how complementary you are. For instance, for me it has been very useful that Jérôme likes to give physical intuitive pictures, and Maura is focused on mathematical rigor and correctness of the calculations. Jérôme, apart from your bottomless positivity that I have seen praised in a few other theses, what stands out to me is the freedom you have given me, yet being available for help where needed. I really liked this interaction with you and I think it has allowed me to grow the most. When I look around me at our group meetings, where we bring cookies and share ideas, I think you have created a very positive, and blossoming group.

Maura, I have been impressed many times by your inventiveness and your analysis capabilities.

I hope that one day I will be as good as you (also I still cannot believe you did the MMS by hand...). You have taken the time and patience to discuss things with me, sometimes for the 20^th time. I also thank you for your encouraging words from time to time. I have enjoyed my PhD so much because of the two of you, thank you.

I would like to thank the group members with whom I worked on the PhD project. Our post-doc Andrea, who is one of the funniest people I have met, who has a wealth of knowledge hidden behind a door of cryptic explanations, and who makes the best meringue. Alexis, who drinks weird tea, makes nicer figures than I do, and always has a next question. Nicolas, the theorist who showed me how to do laser experiments. Julien, who can actually make the laser work. Wahb, who with his wit and enthusiasm has inspired me to do more theory. Yves-Marie, our super bright master-student. Finally, Michel Moret, the A-Team combined in one person, is the key-ingredient to make anything work and also saved my car battery.

A big thanks to the other members of my GAP-Nonlinearity and Climate group for creating such a nice atmosphere: Stéphane, Jan-Hendrik, Liliane (with your cheerfulness and outdoor initiatives), Ariadna (the real mathematician!), Jorrit and Joan. I’d also like to thank Emmanuel Castella for giving us such a warm welcome in ‘his’ kitchen at Carl-Vogt and being so fun to have lunch with.

This PhD would not have been completed without the help of several collaborators. Hubert Branger from Aix-Marseille university: his cheerfulness and helpfulness at the giant water tank has given me very fond memories. Julien Touboul for the nice discussions we had there and after. Arthur, who showed me surfing videos in between measurements. Amin Chabchoub and Olivier Kimmoun, who were intensely involved in the same campaign. Amin has tirelessly read about 20 versions of the drafts for this paper, for which I owe him a big thanks. Finally, John Carter who I met at a conference in Philadelphia and who broke open a problem we were stuck on for a long time, and whose sense of wonder and subsequent calm and structured analysis I admire and aspire to develop myself.

I would like to thank my office buddies and friends. Denis, who is my colleague, my neighbor and my friend, and scores a 10 on all of his titles above, and with whom I lost all of our betting money at our St. Julien casino experiment. Elise, who introduced the cultural wonders of Geneva to me (e.g. the puppet theater), who I’ve had many heart-to-hearts with at our desks, and who I still miss today. Denis and Elise, you both have colored my PhD. Gustavo, I thank you for the fun times at l’Usine and on the MTB slopes. Thomas, for our shared wonder of physics.

Mary, for all our fun lunches, Gabriel, my trustworthy neighbor. Vittorio, my office-mate, the

Tadas, Luca, Malte, Yi-Ping, Alice. And finally, Jean-Pierre, whose eloquent storytelling and physics intuition I will not forgot. And Luigi, who leads by example and from whose example I have tried to learn.

A special word of thanks to the secretaries at GAP. Isabel, Dragana and Corinne make everyone’s life so much easier and we often don’t even realize the work it takes ’behind the curtains’. Thank you very much for your help with many small things and some big things. You have always been super responsive, quick and thorough.

My jury was composed of people I admire: John Dudley, Peter Witter and Ton van den Bremer. A big thanks to all three of you for taking the time to actually read my thesis in detail (you are most likely my main audience) and for coming to my thesis defense.

Most importantly, I’d like to thank my mom and my sister for their endless support of me following my ambitions, despite these leading me far away from them sometimes. Your presence means the world to me. Dankjulliewel. Finally, I have immense gratitude to Fréderique, Femke and Josée, for providing for this thesis the not sufficient but necessary condition of keeping me alive.

Contents

1 Introduction 1

1.1 Wave motion . . . 4

1.2 Solitons and breathers . . . 5

1.3 Outline of the thesis . . . 7

2 Water waves and high intensity light: the nonlinear Schrödinger equation 9 2.1 Optics: from governing equations to NLSE . . . 9

2.1.1 Nonlinear polarization, nonlinear focusing: nonlinear Kerr effect . . . . 10

2.1.2 Maxwell Equations . . . 11

2.2 Water waves: from governing equations to NLSE . . . 20

2.2.1 Water wave problem . . . 20

2.2.2 Method of multiple scales . . . 23

2.3 Comparing the NLSEs for water waves and optics . . . 27

2.3.1 Focusing and defocusing regimes . . . 28

2.3.2 Modulation instability . . . 28

2.3.3 Typical values . . . 33

2.3.4 2D comparison . . . 34

2.3.5 Dispersion relation . . . 36

2.4 Higher-order NLSE: The Dysthe equation . . . 37

3 Viscous damping of water waves 41 3.1 Viscous model equations . . . 41

3.2 The viscous dispersion relation . . . 58

4 The forced-damped Dysthe equation 75 4.1 Wind forcing of water waves . . . 75

4.1.1 Initial stage: Phillips resonance mechanism . . . 76

4.1.2 From ripples to gravity waves: Miles resonance mechanism . . . 76

4.1.3 Boundary condition . . . 80

4.2 Forced-damped propagation equation . . . 81

4.2.1 MMS at third order: forced-damped NLSE . . . 81

4.2.2 MMS at fourth order: forced-damped Dysthe equation . . . 81

4.2.3 The downshift paradox . . . 82

4.3 Three-wave truncation of the forced-damped Dysthe equation . . . 98

4.4 Wave statistics after a wind episode . . . 109

5 Filamentation 123 5.1 Triggering filamentation using turbulence . . . 123

6 Conclusion 133

Bibliography 137

Introduction

If physics were a novel, waves would be the protagonists. There are countless domains in which the study of waves plays a vital role and the description of the problem relies on the wave nature: the wavefunction in quantum mechanics, the merging of black holes, the shock wave behaviour of traffic jams, etc. Hydrodynamics is one of the oldest and most established fields in physics. Therefore, finding analogs with this vast body of knowledge can spark new concepts and deeper understanding. In general, many major concepts in physics emerged through comparison of similar systems. For example, Fourier devised the superposition of simple sine and cosine waves to model the heat source and the solution of the heat equation.

It soon became clear that the Fourier series could be used in a variety of problems in domains such as electrical engineering, acoustics and quantum mechanics. While an old field, there are still open questions in hydrodynamics itself. The puzzle piece that we will contribute to this picture is stated in the title. In this work, we study the ‘Nonlinear propagation of light pulses and water waves under the influence of forcing and damping’.

Three concepts are brought forward in the title of the thesis. First, the analogy between nonlinear optics and water waves. Second, that water waves and optical pulses propagate nonlinearly. Third, the symmetry between damping and forcing. This introduction discusses these three concepts in order, to give a global overview of the thesis. Then, essential concepts relevant throughout the work will be discussed. Finally, a brief outline of the topics treated in each chapter is given.

(a) (b)

Figure 1.1: a) Water waves in wave-tank generated by a paddle on one end of the tank. The wave gauge placed in the water measures the surface elevation as a function of time. b) Photo of (the plasma generated by) a filament in air, integrated over thousands of shots.Photo by Alexis Gomel

How do water waves relate to nonlinear optics i.e. high-power lasers? In fig. 1.1, they are seemingly two very different systems. Their resemblance becomes more apparent when looking at fig. 1.2. Figure 1.2a depicts the time-series of waves generated in a wave tank. A wave gauge placed in water can record the water level as a function of time as a wave-train passes by: the light blue line. Its envelope is indicated by the bold line. An optical pulse, on the other hand, consists of an oscillating electric and magnetic field. Here the physically easily accessible quantity to measure is the intensity of the pulse, related to the envelope of the electric field, rather than the electric field itself. However, by measuring the frequency and the phase with

other measurement devices, the full pulse can be reconstructed: Figure 1.2b. It will become evident in chapter 2, that at first nonlinear order, the propagation of the envelope of water waves, and the envelope of the electric field of a light pulse can both be described by the nonlinear Schrödinger equation (NLSE). In fact, the NLSE describes the evolution of (weakly) nonlinear waves in various physical contexts such as plasmas, Bose-Einstein condensates, fluids, and optics. In all systems, the equation describes the dynamics of theenvelopeof a wave train, provided that this envelope varies slowly in comparison to the carrier wave. In the case of water waves the surface elevation is the carrier wave; in optics it is the electric field.

sech wave (b) optical pulse

(a)

0 5 10 15

Time (s) -10

-5 0 5 10

Surface elevation (mm)

-5 0 5

Time (fs) -1

-0.5 0 0.5 1

Electric field (V/m)

Figure 1.2: a)Water wave. Measurement of the oscillations of the water surface (thin line) by a wave gauge, and its constructed envelope (thick line). b)Optical pulse. Electric field (thin line) and its envelope (thick line).

The main goal of this thesis is to better understand the evolution of ocean waves. To this end, the analog with nonlinear optics can help to uncover answers. However, this is by no means a one way street. An example of this exchange of knowledge between the two fields is the three-wave truncation in which the spectrum is reduced to only three modes, which can exchange energy amongst each other and as such capture the main dynamics of the entire system.

This approach was originally designed for optics, but the same method can also be applied to water waves [7, 8], as discussed in section 4.3. Another example is that ‘breather solutions’ to the NLSE that were observed in wave-tanks [9, 10] can also be created in optical fibers [11].

Therefore, this exchange between two physically very different systems is interesting in itself as cross fertilization and transfer of knowledge can occur. In addition, once the parallel between systems is established, the easiest system for a given experiment or research question can be chosen.

From a more practical point of view, one of the scientifically challenging open questions in the ocean science community, and a major research interest in recent years, is the existence of rogue waves in the ocean. These are very high waves that, according to Akhmediev’s famous quote, seemingly "appear out of nowhere and disappear without a trace"[12]. At first, these waves were disregarded as errors in buoy measurements, and considered sailor’s tales. But their existence was established beyond the shadow of a doubt on January 1^st 1995 (see fig. 1.3a), when the Draupner oil platform in the North Sea measured such a wave of 26 m in height, comparable to a 6-story building. In addition, the wave was much larger than its surrounding waves: more than twice the significant wave height of approximately 12 metres. The significant wave height is the mean wave height of the highest third of the waves.

It is clear that these huge waves appearing without warning pose a danger for ships. Pre- dictions about the probability of encountering a high wave are taken into account in defining a risk factor when planning routes. One can measure and obtain a probability distribution of wave heights. If in a long-crested sea (where waves have a long transverse extension such that it can be approximated by a 1D model) the propagation of the waves is linear, the distribution

(a) (b)

180 220 240 280 320 360

Time (s)

Surface elevation (m)

0 4 8 12 16 20

-4

25.6 m

Figure 1.3:a) Time series of the Draupner wave as measured by the oil platform on January 1st, 1995.

b) Possible rogue wave encountered by a ship in 1977, taken from [13]

will be normal or Gaussian. Currently, linear statistics are used to dimension ships and plan the routes, and a small deviation can cause casualties and costly damage. However, the distribution of wave heights on the ocean is not always Gaussian. It can also be fat-tailed. That is, high amplitude waves, and thus rogue waves, appear more often than predicted based on a linear model. Therefore, statistical analysis plays an important part in rogue waves studies, and enough measurements must be performed in order to obtain significant statistics. Because wave tank measurements are very time consuming, a laser that can send a pulse with a repetition rate of the order of MHz or KHz can serve as a quick table-top experiment alternative.

The second concept addressed in this work is nonlinearity. A function is linear if it has the properties of superposition,F(x+y)=F(x)+F(y), and homogeneity,F(αx)=αF(x). Thus an expression likeaxis linear, whilex²is nonlinear. Water waves are inherently nonlinear, as can easily be assessed by visual inspection. Whether looking at waves in a pond or in the ocean, real waves do not have the simple sine shape of fig. 1.4a. The photograph of large ocean waves (fig. 1.4c) shows that instead of a sine, the trough of the wave is rather flat, and the crest is peaked, as in fig. 1.4b.1 Figure 1.4a is termed a linear wave, because it is a solution to the linear wave equation. In contrast, the wave in fig. 1.4b is a nonlinear wave, as it is an approximate solution to the nonlinear boundary problem of the water waves and contains quadratic and cubic contributions of the sine in 1.4a. Why the propagation of light is nonlinear is less intuitive. It is due to the nonlinear response of the polarization to the incoming light, and will be discussed in more detail in chapter 2.

A nonlinear dynamical system is a system in which the derivative of the output, say a_t, wheretindicates the rate of change in time, is not proportional to the inputa. In our case, the variable for which we are trying to solve, the rate of change of the amplitude of the envelope, turns out to be a function of its amplitude (input) squared: a_t = F(a²).

According to Stanislaw Ulam, "Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals." [15]. Indeed, many variables in nature intrinsically have a nonlinear dependence on another. Just as most animals are non-elephants, most problems are inherently nonlinear. It is only in certain limits that the linear approximation is valid. A classic example of this is the pendulum. In reality, the angular acceleration of the pendulum is proportional to sinθ, where θ is the angular displacement. However, when the pendulum only makes small oscillationsθ 1, we can make the approximation sinθ ≈ θ, and the dependence is now linear. Therefore, as in zoology, nonlinearity is not a very useful

1The nomenclature here is a bit confusing because the function for a linear waveA=sin(x)itself is nonlinear, as it does not obey the superposition and homogeneity principles. However, ifAis considered the ‘building block’, the nonlinear wave contains terms∝A²and∝A³.

(a) (b)

(c) ^{Distance Distance}

Wave height Wave height

Figure 1.4:a) Linear wave. b) Nonlinear wave (Stokes wave). c) Ocean swell, displaying long troughs and peaked crests instead of a sine profile. Reproduced from [14]

distinction between areas of physics. However, a special property of nonlinear problems is that most of these cannot be readily solved, and therefore a different analysis-machinery is needed than for linear (or approximately linear) problems. Instead of solving the problem exactly, one can rely on numerics or explore the qualitative behavior of the solutions in the phase space [16].

The evolution of the solution brings us to the third concept in the title. The NLSE is well established in its conservative form, that is, without any external influence. Therefore, the natural next step is to include external forces, and study what happens to a wave packet when it undergoes damping and forcing. While it is often neglected, water is viscous, and this viscosity is responsible for the attenuation of the water waves. The wind is the very origin of waves at the ocean surface, as the waves form and grow due to the energy the wind transfers to the water surface. The vast majority of waves that we see arriving on the shore were at some point created by the wind (others can be explained by, for example, earthquakes or landslides). It is therefore essential to study the wind-wave interaction. We discuss how to include viscosity and wind in the basic physical problem, and subsequently derive a forced-damped NLSE equation.

1.1 Wave motion

We now give some general insight into wave motion that will be useful for the rest of the thesis. It is important to realize that when we observe a surface wave passing, it is the energy that is propagating forward, not the water particles. Water waves involve a combination of longitudinal (like a pressure wave) and transverse (vertical, like an oscillating string fixed at both ends) motion. If we track a water particle in the passing of a deep water wave, fig. 1.5 it performs a combination of longitudinal (moving back and forth) and transverse (bobbing up and down) wave motion, namely a circular trajectory. The radius of the circle decreases with water depth. This is a leading-order approximation, which holds for the linear wave in fig. 1.4a. However, for the higher-order wave in fig. 1.4b (and in reality), the trajectory of the water particle is an open loop instead of a closed circle. This causes a small drift, called the Stokes drift [17].

When tracking the wave instead of the water particle, fig. 1.6 shows that there is a difference in velocity of the individual wave called thephase velocity (the red dot) and velocity of the packet as a whole, thegroup velocity(the green dot). The reason for this difference in velocity is that both water and the medium through which the light pulse moves in optics are dispersive:

wave direction

Figure 1.5: Particles follow a circular trajectory as the wave passes by. Therefore, instead of moving with the phase velocity, the particles are not transported from their position. In reality, the trajectory is not a closed circle, therefore there is in fact a small drift much smaller than the phase velocity.

different wavelengths travel at different velocities.

Consider a single traveling sine wave as in fig. 1.5. When looking at a fixed position in space, it oscillates in time with a frequencyω = ²_T^π, whereT is the wave period. Taking a snapshot at a fixed time, fig. 1.5 shows an oscillation with a wavenumberk = ²_λ^π, whereλis the wavelength. The dispersion relation connects these two quantities: ω = F(k). Imagine the next frame in time of fig. 1.5, with a time interval of∆t =T = 1/frequency= ²_ω^π, that is, one period. The distance traveled by a given point is then also the length of one spatial period

∆x =λ= ²_k^π. Therefore, the velocity of the wavec_p = ^∆x_∆t = _T^λ = ^ω_k.

Now consider a superposition of two sinusoidal waves as in fig. 1.6 They have the same amplitude but a slightly different frequency. Their interference results in a beat pattern, or wave group. In the dispersion relationship for deep water wavesω =p

gk, the phase velocity c_p = ^ω_k = p

g/k that tracks an individual oscillation (red dot) is twice the group velocity c_g = ^dω_dk = ¹₂p

g/k (green dot). Indeed, the red dot moves twice as fast as the green dot. The group velocity is the velocity of the energy transport.

Time

Distance

Figure 1.6: Snapshot of a wave group at different times. The red dot tracks a single wave and as such the dashed line represents the phase velocity in thex,tplane. The green dot tracks the wave group. Its corresponding dashed line represents the group velocity. Image adapted from [18]

1.2 Solitons and breathers

From section 1.1 we understand that due to dispersion, a wave packet composed of different frequencies will spread apart. Due to the nonlinear nature of water waves, it is possible to have

a ‘solitary wave’ that does not spread as it moves forward, and that these types of spectacular waves can be seen as building blocks for rogue waves.

The earliest demonstration of a soliton2 is also its most illustrative. In 1834, Scottish scientist John Scott Russell wrote the following: "I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped - not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed". Russell was fascinated and built an experimental tank in his garden to continue his studies of his ‘Wave of Translation’. Understandably, he described the day of the discovery as the happiest day of his life [19]. Figure 1.7 depicts Russell’s drawing (a) and a photo (b) showing a recreation of this soliton in 1995 at a nonlinear waves conference at the Heriot-Watt University in Edinburgh.

(a) (b)

Figure 1.7: a) Drawing of the wave of translation by Russel [20]. b) Photo of the recreation at the Herriot-Watt university in 1995 [19].

The soliton can exist because the dispersion is compensated by the nonlinear dependence on the amplitude. It therefore cannot exist in a linear system. Russell’s wave of translation is an example of a soliton of the surface elevation itself. Envelope equations such as the NLSE also exhibit solitons. However, these are envelope-solitons, in which the envelope does not deform (stays focused), but the carrier wave does oscillate.

Similar to the focusing effect in the temporal domain, a focusing effect can be observed in spatial form, and explains the existence of high-intensity self-propagating ‘light-bullets’ called filaments. When shining a flash light or a laser pen, the beam will not reach infinitely far because the light will disperse, as discussed in section 1.1. Simultaneously, the light will diffract: it will move in different directions, again breaking apart the ‘packet’. For very high light intensities, it turns out that the refractive index in the beam depends on the square of the amplitude of the beam itself, i.e., on its intensityI: n= n₀+n₂I. Here, n₀andn₂are the linear and nonlinear refractive index respectively. This phenomenon is known as the Kerr effect.

Figure 1.8a shows that because the outer regions have a lower intensity than the inner, a lens that focuses the beam is created. This lensing-effect makes the center part even more intense.

At a certain point, this high intensity at the center is so strong that it knocks electrons out of their orbits and ionizes the atoms, thus creating a plasma. This plasma has a defocusing effect as∆n < 0 and there is a radial gradient in the density of the plasma. The dynamic balance of the Kerr effect and the plasma defocusing causes the pulse to ‘guide itself’.

2Here we use ‘solitary’ wave and ‘soliton’ interchangeably. Strictly speaking, a soliton does not change amplitude when it crosses another soliton, unlike two solitary waves crossing.

Other examples of solitons in optics are pulses in optical fibers in which the dispersion of the wave packet is compensated by the temporal consequence of the Kerr effect, called self- phase modulation. Another example is spatial solitons in a planar wave guide, where dispersion is neglected and the spatial pulse stays collimated because the diffraction in the non-guided dimension is compensated by the Kerr effect, as in the filament. We shall see in section 2.1.2 that technically, a filament is not a pure soliton, as there is dispersion in time over the filament region.

plasma channel kerr effect filament

t₁ t₂ t₃

(a) (b)

Figure 1.8:a) The Kerr effect amplifies the beam profile steepness, leading the beam to focus more and more. b) The beam collapses, creating a plasma channel.

What these types of solitons share is that they have the shape of a hyperbolic secant (sech), as in fig. 1.2, which is a solution of the NLSE. Another fundamental group of solutions of the NLSE equation are breathers. The term breather originates from the characteristic that they are localized in space but oscillate or ‘breathe’ in time, or vice versa. Examples of these solutions of the NLSE are given in Figure 1.9 in adimesional coordinatesζ (spacelike) andτ(timelike) (see section 4.3 for definitions). Figure 1.9a shows a sech solution, and fig. 1.9b an Akhmediev breather (AB) [21] that oscillates in the transverse coordinateτ. A special case of the AB is the Peregrine breather [22], which has a period between the peaks inτ that tends to infinity, displayed in fig. 1.9c. While there are other mechanisms to explain rogue waves, the energy concentration (the increase in local amplitude) of the Peregrine breather illustrates quite clearly why solitons are interesting in this respect.

Sech Akhmediev Peregrine

(a) (b) (c)

Figure 1.9: a) Sech soliton. This solution does not change shape as it propagates. b) Akhmediev breather. This solution is periodic , or "breathes", in time and is localized in space. c) Peregrine breather.

This solution is a limit of the Akhmediev breather where the period between oscillations tends to infinity, creating a solution localized in both space and time, and is a prototype for rogue waves.

1.3 Outline of the thesis

The outline of the thesis is as follows:

• In chapter 2, we start from the basic equations for deep water waves and light pulses, and show that the propagation of the envelope can indeed be described by the NLSE. While the NLSE is a good first approximation capturing the main physics of the system, for

water waves a more precise description, in particular with respect to the asymmetry of the spectrum, is given by the higher-order NLSE, also termed the Dysthe equation. At the end of the chapter we compare the NLSE in the physical contexts of optical fibers, planar waveguides, and deep water waves.

• In chapter 3, we dive deeper into the fundamentals of the water wave problem. This chapter looks at the effect of viscosity on the basic equations. The addition of viscosity causes a vortical boundary layer below the surface, which significantly complicates the mathematical description of the problem. We contrast and compare different methods for dealing with this complication and examine the mass conservation of the problem.

Furthermore, we examine the viscous dispersion relationship and its relation to the model equations and the propagation equation.

• In chapter 4, the effect of the wind is included in the basic equations and the forced- damped Dysthe equation is derived. This higher-order NLSE equation allows a realistic comparison to wind-forced waves in a wavetank, and shows the effect of wind and viscosity on the spectrum. Additionally, we make a three-wave truncation of the spectrum in order to explore the phase space of the problem. Finally, our studies are extended by performing a statistical analysis in order to find a measurable predictor for rogue waves.

• In chapter 5, we return to nonlinear optics and discuss the spatial counterpart of the modulation instability, which allows us to trigger filaments using turbulence.

• In chapter 6 we summarize our results.

Water waves and high intensity light:

the nonlinear Schrödinger equation

In this chapter, we compare the nonlinear physics of water waves and optics, starting from the basic principles of each system, and ending with the propagation equation for the field of interest: the Nonlinear Schrödinger equation (NLSE).

For optics (section 2.1), the relevant field is the envelope of the electric field of the light pulse.

We start at the Maxwell equations that govern the dynamics of electromagnetic fields, from which we can derive the wave equation. Making the relevant approximations and simplifications, we obtain an NLSE for different situations. For water waves (section 2.2), the field of interest is the envelope of the surface elevation. Here, we start at the laws of conservation of mass and conservation of momentum (the Navier-Stokes equation) which for the inviscid case reduce to the Euler equations. Together with the boundary conditions, these form the water wave problem, or the water wave equations. Performing an expansion that assumes a slow timescale for the envelope as compared to the surface elevation (slowly varying envelope approximation, SVEA), the NLSE can be obtained for this envelope. The NLSE can be written as

i∂a

∂ζ +P∂²a

∂ξ² +Qa|a|²=0 (2.0.1)

The propagation of the envelopeain some coordinateζ, whether this be time or space, is affected by:

• a dispersive or diffractive termP^∂_∂ξ^2a₂, which causes the speed of the different wavelengths to spread ifξis in time (dispersion), or causes a directional spread ifξ is the transverse spatial coordinate (diffraction).1

• a nonlinear termQa|a|², inducing a phase that depends nonlinearly on the amplitude of the envelopea, and allows interaction between modes in the spectrum.

The two terms will be discussed in more detail as the equation is derived for the light pulses and water waves, after which we can compare the various properties for these two very different systems in section 2.3.

2.1 Optics: from governing equations to NLSE

If anything, a link between nonlinear water waves and nonlinear optics is that a Scottish scientist was at its origin. In 1875, the Scottish physicist John Kerr observed a change in the refractive index of organic liquids and glasses in the presence of an electric field. It turned out that this change is due to the nonlinear dependence of the material polarization on the electric field.

1If the initial condition is specially tweaked this term can actually function to compress (or focus) instead of spread (or, defocus).

It causes a Gaussian beam profile to focus itself -termed nonlinear focusing- as mentioned in chapter 1. This is often seen as the birth of nonlinear optics. Therefore, before starting from the Maxwell equations, we derive the Kerr effect from basic principles, and along the way set up the notation used in the remainder of this section.

2.1.1 Nonlinear polarization, nonlinear focusing: nonlinear Kerr effect

First, let us make clear what is meant by nonlinear polarization, the origin of the Kerr effect.

When an external electric field (i.e. the light pulse) E®_ext, is applied to a dielectric material, charges do not flow from one side of the material to the other as a current in a conductor, but only slightly shift position, while staying bound to their atom. As such, they create microscopic electric dipoles, see fig. 2.1.

- - - -

- - -

- - - -

-

+ + + +

+

+

+

+

- - - -

- - -

- - - -

-

+ + + +

+

+

+

+

- - - -

- - -

- - - -

-

+ + + +

+

+

+

+

- - - -

- - -

- - - -

-

+ + + +

+

+

+

+

E_ext

Figure 2.1:Left: Material at rest. Right: The applied electric field polarizes the atoms, turning them into dipoles. The top surface is now charged negatively while the bottom surface is charged positively.

The microscopic dipoles add up to a macroscopic polarization. In the example of fig. 2.1, a layer of positive bound charge is formed on one side of the material, and a layer of negative bound charge on the other side; the material is polarized at the surface. 2 In general, the microscopic polarizability is not harmonic for large electric fields, thus it is not simply linearly proportional to the incoming electric field. The macroscopic polarization P can therefore be Taylor expanded around the linear polarization response, provided the latter is dominant.

Neglecting terms ofO(4), it can be written as P(t)=₀

χ⁽¹⁾E+ χ⁽²⁾E²+ χ⁽³⁾E³+O(4)

=P_L +P₂+P₃

=P_L+P_NL

where χ^(k) is the k^th order susceptibility, i.e. the constant of proportionality between the polarization and the electric field for each order, ₀ is the permittivity of the vacuum and E the amplitude of the electric field. The polarization P and electrical field E are considered as scalar for simplicity. In general, χ is a tensor that represents the polarization-dependent nature of the response and the symmetries of the material. In the case of a centro-symmetric material, χ⁽²⁾= 0, and thereforeP_NL= P₃. This is true for the media we will be considering, such as argon, air, and water. Considering an electromagnetic plane wave with one frequency (monochromatic), we can write the electric field as

E(t)= E₀cos(ωt)= 1

2Eˆ₀e^iωt+c.c. (2.1.1)

2The polarization of the medium is not to be confused with the polarization plane of the light pulse.

where c.c. stands for complex conjugate. The third order polarization as a function of the electric field is

P_NL=₀χ⁽³⁾E³ =₀χ⁽³⁾ 3

8Eˆ₀|Eˆ₀|²e^iωt+ 1

8Eˆ₀³e^i3ωt

+c.c. (2.1.2)

The first term pertains to the main frequencyω, and shows there is a nonlinear dependence of the polarization on the electric field. On a side note, the second term shows there is generation of a wave with frequency 3ω, known as third harmonic generation (THG). The total polarization for the fundamental frequency of the wave is given by

P(t)=₀χ⁽¹⁾1

2Eˆ₀e^iωt +₀χ⁽³⁾3

8Eˆ₀|Eˆ₀|²e^iωt+c.c. (2.1.3)

=₀

χ⁽¹⁾+3

4χ⁽³⁾|Eˆ₀|² 1

2Eˆ₀e^iωt+c.c.

=₀χ1

2Eˆ₀e^iωt+c.c.

where we define χ= χ⁽¹⁾+³₄χ⁽³⁾|Eˆ₀|².

We have now obtained the key ingredient to understand nonlinear focusing. The refractive index of a material describes the light velocity in the mediumvas compared to the velocity of light in vacuumc. For a dielectric medium

n= c v =

r ₀ =p

1+ χ (2.1.4)

where is the permittivity and ₀ the permittivity of the vacuum. As the light propagates through the medium, we know now that the polarization causes the energy from the electric field to be transiently stored in the medium as the electrons in the material are temporarily offset, and the dipole is aligned to the field. While this energy is re-emitted, the beam travel is slowed by this interaction with the material. Writing χin its nonlinear and linear part:

n=

1+ χ⁽¹⁾+3

4χ⁽³⁾|Eˆ₀|² ₁/2

(2.1.5)

=

n²₀+3

4χ⁽³⁾|Eˆ₀|² ₁/2

(2.1.6)

≈n₀+ 1 2n₀

3

4χ⁽³⁾|Eˆ₀|²+h.o. (2.1.7)

=n₀+n₂I (2.1.8)

where the definitionn₀ =p

1+ χ⁽¹⁾is used for the linear refractive index, an expansion around χ⁽³⁾ ≈ 0 is used for the approximation step, and h.o. stands for higher-order terms. Using I =2n₀₀c|E|²= ¹₂n₀₀c|Eˆ₀|², we obtainn₂ = ³₄ ^χ(3)

cn²₀₀. 2.1.2 Maxwell Equations

The starting point to obtain the NLSE for the light pulse are the macroscopic Maxwell equations for the electric fieldE®and magnetic flux densityB®, in which the total charge in the macroscopic material is separated in a free and a bound part: ρ = ρ_f+ρ_b. This derivation largely follows [23], which can be consulted for details. As we are only concerned with the bound charges as explained in the previous section, we neglect the free charges, the free current J_f, and

charge densityρ_f. As the bound charges give rise to the macroscopic polarization discussed in section 2.1.1, two auxiliary fields are introduced: the displacement fieldD® and the magnetic field intensityH®, that for non-magnetic materials are defined as:

H® = 1

µ₀B® (2.1.9)

D® =₀E®+P® (2.1.10)

whereµ₀is the permeability of free space, which is related to₀as₀µ₀c²=1. The macroscopic Maxwell equations can be written as:

∇ · ®D=ρ_f Gauss’s law (2.1.11)

∇ · ®B=0 Magnetic Gauss’s law (2.1.12)

∇ × ®E=−∂B®

∂t Faraday’s law (2.1.13)

∇ × ®H=J®_f + ∂D®

∂t Ampère’s law (2.1.14)

where∇ = ∂

∂x,_∂y^∂,_∂z^∂

. Taking the curl of eq. (2.1.13) and using eqs. (2.1.9) and (2.1.14) to eliminateB®gives

∇

∇ · ®E

− ∇²E® =− 1 ₀c²

∂²

∂t²

D® (2.1.15)

While the first term in nonlinear optics is non-vanishing, it can usually be dropped for cases of interest [23, p.71]. This can be intuitively understood by the fact that at the relevant space scale to our studies, no regions of net positive or negative charge exist in the material. In general, if there are no localized charges, there is no divergence. Following section 2.1.1, we can express

D® =₀E®+P®_L+P®_NL=D®_L+P®_NL (2.1.16) whereD®_L=₀E®+P®_L. In the case of a lossless, isotropic, dispersion-less medium, eq. (2.1.15) reduces to a simple relation D®_L = _0LE®, where _L, the dielectric tensor, is a scalar, as the medium is isotropic. In this case eq. (2.1.15) becomes

∇²E®−_L c²

∂²

∂t²E®= 1 ₀c²

∂²

∂t²P®_NL (2.1.17)

For a dispersive medium –in which we are interested– eq. (2.1.17) applied to each spectral component separately. Consequently, the dielectric tensor is a complex quantity that is frequency dependent:L →L(ω)and eq. (2.1.17) becomes

∇²E®− _L(ω) c²

∂²

∂t²E® = 1 ₀c²

∂²

∂t²P®_NL (2.1.18)

When the right hand side of eq. (2.1.18) is set to zero, this is the regular wave equation for free waves that propagate with velocityc/n. Therefore, the termP®_NLcan be seen as a forcing term in this equation. Again, this is a demonstration of how the nonlinear response of the medium can act as a source. This equation is the basis for the nonlinear propagation of light.

From hereon out, different versions of the NLSE can be derived which are suitable in different situations.

The equation is easier to simplify in Fourier space, so we make the following transformation for all variables:

X(®® r,t)= 1 2π

∫ X(®ˆ r, ω)e^−iωtdω (2.1.19)

yielding

∇²Eˆ+_L(ω)ω²

c²Eˆ =− ω²

₀c²Pˆ_NL (2.1.20)

Now, instead of solving for the carrier wave, we are interested in the envelope, which should vary slowly with respect to its carrier:

E(®® r,t)= A(®r,t)e^{i(k0x−ω0t)}+c.c. (2.1.21) ˆ

E(®r, ω) ' A(®ˆ r, ω−ω₀)e^ik0x+c.c. (2.1.22) whereω₀is the carrier wave frequency, andk₀is the linear part of the wavevector at the carrier wave frequency. We can also writek²(ω)=(ω)^ω2

c², giving:

∇²_⊥Aeˆ ^ik0x+ ∂²Aeˆ ^ik0x

∂x² +k²(ω)Aeˆ ^ik0x =− ω²

₀c²Pˆ_NL (2.1.23) where ∇_⊥² stands for ∂²/∂y²+∂²/∂z². Expanding ^∂2Ae_∂x^{ˆ i k}2^0x, the term ^∂_∂x^2A2^ˆ can be neglected since the envelope varies slowly

∇²_⊥Aˆ+ ∂²Aˆ

∂x² +2ik₀∂Aˆ

∂x +[k²(ω) −k₀²]Aˆ =− ω²

₀c²Pˆ_NLe^−ik0x (2.1.24) As we assume a narrow bandwidth,∆ω/ω₀ 1, thusk(ω) = p

_L(ω)ω/cis only slightly different from k₀, and we can approximate [k²(ω) − k₀²] ≈ 2k₀(k(ω) −k₀). In addition, we expandk(ω)around k₀.

k(ω)=k₀+ dk dω

ω0

(ω−ω₀)+ 1 2

d²k dω² ω0

(ω−ω₀)²+... (2.1.25)

=k₀+ 1 c_g

ω0

(ω−ω₀) − 1 2c²_g

dc_g dω ω0

(ω−ω₀)²+.... (2.1.26)

=k₀+k₁(ω−ω₀)+1

2k₂(ω−ω₀)²+... (2.1.27) where k₁ is the inverse of the group velocity and k₂ the dispersion of the group velocity.

Consequently, using

2k₀(k(ω) −k₀)=2k₀k₁(ω−ω₀)+k₀k₂(ω−ω₀)² (2.1.28) in eq. (2.1.24) gives:

∇²_⊥Aˆ+2ik₀∂Aˆ

∂x +2k₀k₁(ω−ω₀)Aˆ+k₀k₂(ω−ω₀)²Aˆ =− ω²

₀c²Pˆ^NLe^−ik0x (2.1.29) Performing an inverse Fourier transform on the frequency domain defined by ¯ω =ω−ω₀, to return back to the time domain gives

∇²_⊥A+2ik₀ ∂A

∂x +k₁∂A

∂t

−k₀k₂∂²A

∂t² = 1 ₀c²

∂²P_NL

∂t² e^{−(ik0x−ω0t)} (2.1.30) To include the nonlinear susceptibility of section 2.1.1, a slowly varying envelope approxi- mation is also considered for the polarization:

P_NL(®r,t)= p(®r,t)_NLe^{i(k0x−ω0t)} (2.1.31)

For an instantaneous 3^{r d}order response, p(®r,t)_NL = 3₀χ⁽³⁾|A|²A.3 If a material displays a higher-order nonlinearity, or a delayed response like in stimulated Raman scattering, this can also be included. In addition, expanding the nonlinear term by including its time derivative, can account for self-steepening. Here, we do not include these correction factors to the nonlinear source term so that the envelope equation reads:

2ik₀ ∂A

∂x +k₁∂A

∂t

+∇²_⊥A−k₀k₂∂²A

∂t² =−ω₀²

c²3χ⁽³⁾|A|²A (2.1.32) To simplify the first two terms, we can choose a frame moving at the group velocity of the pulse, i.e. that follows the envelope. To this end, we make the following change of coordinates (recall thatk₁=1/c_g):

x⁰= x τ=t− 1

c_gz (2.1.33)

∂

∂x = ∂

∂x⁰ − 1 c_g

∂

∂τ

∂

∂t = ∂

∂τ (2.1.34)

yielding:

i∂A

∂x⁰ + 1

2k₀∇_⊥²A− 1 2k₂∂²A

∂τ² + 3ω²₀

2k₀c²χ⁽³⁾|A|²A=0 (2.1.35) Usingn₀=c/c_p =c/(ω₀/k₀)andk₂=− ¹

c²g dcg

dω we can write the final result of this section:

the general propagation equation for a nonlinear optical pulse

i∂A

∂x⁰ + 1 2k₀

∂²A

∂y² + ∂²A

∂z²

+ 1 2

1 c_g²

dc_g dω

∂²A

∂τ² + 3ω₀

2n₀cχ⁽³⁾|A|²A=0 (2.1.36) which contains both a dispersion term¹_2c¹2

g dcg

dω ∂²A

∂τ², and a diffraction term₂¹_k0∇²_⊥A.

When comparing with the NLSE as defined in eq. (2.0.1), also called the 1D+1 NLSE, the number of terms is different. The 1D+1 NLSE is integrable and contains the derivative with respect to the propagation coordinate, a term with a second order derivative with respect to another coordinate, and a nonlinear term. Therefore, to have an integrable NLSE, either the dispersion term or the diffraction term has to disappear in eq. (2.1.36). In the following sections we look at the different limits in which we can neglect one or the other.

Temporal solitons: pulses in optical fibers

An optical fiber is a 1D structure in which the pulse is guided along the propagation directionx, and the transverse directionsy,zof the electric field are limited. When decomposing the field into its transverse profile and an x,tdependent-envelope, the transverse modes will appear in the Fourier space in eq. (2.1.23). When limiting the transverse dimension, it can be shown that the diffraction term can be discarded. In addition, the nonlinear term becomes dependent on the cross-section and can be written as_{c A}^ωn_eff², where A_effis the effective surface of the fiber, and n₂is the Kerr index. As a consequence, eq. (2.1.36) can be written as:

i∂A

∂x⁰ + 1 2

1 c²_g

dc_g dω

∂²A

∂τ² + ωn₂

c A_eff|A|²A=0 (2.1.37) The second assumption in the case of optical fibers is that the bandwidth is narrow. Con- sequently, again we can apply the SVEA, so that we can expand the dispersion relation around

3Note thatA=¹₂Eˆ₀when we compare to eq. (2.1.1)