Document [original]

Dynamics and Stochastic Properties

of Passively Mode-Locked

Semiconductor Lasers Subject to

Optical Feedback

vorgelegt von

MSc. Lina Jaurigue

geb. in Ebersberg

von der Fakultät II – Mathematik und Naturwissenschaften –

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

– Dr. rer. nat. –

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. Michael Kneissl

Berichterin: Prof. Dr. Kathy Lüdge

Berichter: Prof. Dr. Eckehard Schöll, PhD

Berichter: Dr. Julien Javaloyes

Tag der wissenschaftlichen Aussprache: 26. September 2016

Berlin 2016

Abstract

Passively mode-locked semiconductor lasers produce short optical pulses at very

high repetition rates. In this thesis we investigate the inﬂuence of time-delayed

optical feedback on the dynamics and timing jitter of such a laser.

Using a delay diﬀerential equation model we investigate the dynamics and bi-

furcations of a passively mode-locked laser. When the laser is operated in the

fundamental mode-locking regime it produces a pulsed output with a repetition

rate which is determined by the length of the laser cavity. Adding optical feedback

to the laser in this regime, the dynamics depend on resonances between the period

and the feedback delay times. Feedback conditions can be selected to tune the

repetition rate of the mode-locked pulse train, to induce harmonic mode-locking

or to destabilise the periodic mode-locked dynamics, resulting in quasi-periodic or

chaotic dynamics. Surrounding each resonantly chosen delay time there is a lock-

ing range in which fundamental mode-locking is exhibited. This locking regions

becomes wider as the delay times are increased, which can lead to a large degree

of multistability between solutions locked to diﬀerent resonances. With feedback

from two external cavities the same dynamics can be exhibited as with single cavity

feedback, but the multistability of the fundamentally mode-locked solutions can be

lifted if the two external cavities are of diﬀerent lengths.

In the presence of noise, the regularity of the mode-locked dynamics is signiﬁ-

cantly reduced, since due to the absence of a restoring force the pulse positions can

drift over time. By adding resonant optical feedback correlations between tempo-

ral pulse positions are introduced, which can lead to a signiﬁcant reduction in the

timing jitter. We derive an expression for the timing jitter that shows that this re-

duction increases with the feedback delay time. However, for long feedback cavities

noise-induced modulations of the dynamics also play a role, leading to ﬂuctuations

in the pulse positions on time scales of the feedback delay time.

The noise-induced modulations that arise for long feedback delay times can be

suppressed by adding a second feedback cavity. We show that a linear stability

analysis of the mode-locked laser system allows predictions to made for the optimal

feedback conditions for this eﬀect. This is done by ﬁrst studying the suppression

of noise-induced modulations in a simple oscillatory system, the Stuart-Landau

oscillator, and relating the results for this system to the mode-locked laser system.

For the Stuart-Landau oscillator a simple characteristic equation, which provides

the dominant Floquet exponents, can be analytically derived. By comparison with

numerical results, we show that the dominant Floquet exponents of the mode-

locked laser system can be described by a simple characteristic equation of the

same form as that derived for the Stuart-Landau oscillator.

Using this characteristic equation, feedback delay times are found which eﬀec-

tively suppress the noise-induced modulations of the mode-locked pulse train. We

show that this leads to a signiﬁcant improvement in the regularity of the mode-

locked laser output, as the suppression of noise-induced modulations reduces ﬂuc-

tuations in the pulse positions on time scales of the feedback delay times, as well

as decreasing the variance of the temporal pulse positions on much longer time

scales.

Zusammenfassung

Passiv-modengekoppelte Halbleiterlaser erzeugen kurze optische Pulse bei sehr ho-

hen Wiederholungsraten. In dieser Arbeit wird der Einﬂuss von zeitverzögerter op-

tischer Rückkopplung auf die Dynamik und und den Zeitjitter eines solchen Lasers

untersucht.

Durch die Verwendung eines Diﬀerentialgleichungsmodells mit Zeitverzögerung

untersuchen wir die Dynamik und Bifurkationen von passiv-modengekoppelten Hal-

bleiterlasern. Diese Laser erzeugen im fundamentalen Modenkopplungsregime ein

gepulstes Ausgangssignal mit einer Wiederholungsrate die durch die Länge des

Lasers bestimmt ist. Wird nun zusätzlich ein Teil der optischen Ausgangsleistung

zeitverzögert in den Laser zurückgekoppelt, lässt sich die Dynamik durch Reso-

nanzeﬀekte zwischen der Periode des freilaufenden Lasers und der Zeitverzögerung

der Rückkopplung beeinﬂussen. Durch geeignete Wahl der Rückkopplungsparam-

eter kann die Wiederholungsrate der Pulsfolge geändert, harmonische Modenkop-

plung induziert oder sogar die periodische modengekoppelte Dynamik vollständig

destabilisiert werden, was zu einer quasiperiodischen oder chaotischen Dynamik

führt. Solange jedoch die Verzögerungszeit etwa einem Vielfachen der Wiederhol-

ungsrate des freilaufenden Lasers entspricht, ist die fundamentale Modenkopplung

stabil. Diese Resonanzbereiche wachsen dabei mit steigender Rückkopplungszeit.

Durch Überlapp verschiedener Resonanzbereiche mit fundamentaler Modenkop-

plung entstehen Bereiche mit hoher Multistabilität. Mit Hilfe einer weiteren optis-

chen Rückkopplung über eine zweite externe Kavität lässt sich die Dynamik in gle-

icher Weise wie mit einer einzelnen externen Kavität beeinﬂussen. Dabei kann die

Multistabilität der fundamentalen modengekoppelten Lösung jedoch aufgehoben

werden.

Die Regelmäßigkeit der modengekoppelten Dynamik ist unter Rauscheinﬂuss er-

heblich verschlechtert, da aufgrund der Abwesenheit von einer externen Referen-

zfrequenz die Pulspositionen in der Zeit stochastisch variieren. Mit Hilfe von reso-

nanter optischer Rückkopplung werden Korrelationen zwischen den Pulszeiten auf

langen Zeitskalen induziert, die zu einer deutlichen Verringerung des Zeitjitters

führen. In dieser Arbeit wird ein Ausdruck für den Zeitjitter hergeleitet, der zeigt,

dass die Regularität der Pulse sich mit längerer Rückkopplungszeit verbessert.

Für lange Rückkopplungszeiten jedoch spielen rauschinduzierte Modulationen eine

zunehmend stärkere Rolle, was zu Fluktuationen der Pulspositionen auf Zeitskalen

der externen Rückkopplung führt.

Diese rauschinduzierten Modulationen können durch Hinzufügen einer zweiten

Rückkopplungsstrecke unterdrückt werden. In dieser Arbeit wird gezeigt, dass mit

Hilfe der linearen Stabilitätsanalyse des modengekoppelten Lasersystems Vorher-

sagen für die optimalen Rückkopplungsparameter gemacht werden können. Zunächst

wird dazu die Unterdrückung rauschinduzierter Modulationen in dem einfachen

Stuart-Landau-System untersucht und die Ergebnisse dieses Systems mit dem mod-

engekoppelten Laser verglichen. Für das Stuart-Landau-System kann eine ein-

fache charakteristische Gleichung für die dominanten Floquet-Exponenten ana-

lytisch hergeleitet werden und es hat sich empirisch gezeigt, dass die dabei hergeleit-

ete Formel auch auf die Ergebnisse für das Lasersystem übertragbar ist.

Mit dieser charakteristischen Gleichung können optimale Verzögerungszeiten für

die Unterdrückung der rauschinduzierten Modulationen bestimmt werden. Wir

zeigen dabei auch, dass dies dann insgesamt zu einer signiﬁkanten Verbesserung in

der Regelmäßigkeit des modengekoppelten Laserlichts führt, da die Unterdrückung

von rauschinduzierten Modulationen Fluktuationen der Pulspositionen auf kurzen

Zeitskalen reduziert, sowie auch die Varianz der Pulspositionen auf langen Zeit-

skalen.

iii

Contents

1. Introduction 1

1.1. Semiconductor lasers ........................... 2

1.2. Mode-locking ............................... 5

1.2.1. Passive mode-locking ....................... 6

1.2.2. Timing jitter ........................... 7

1.3. Time-delayed feedback .......................... 8

1.4. Outline .................................. 9

2. Mode-locked laser model 11

2.1. Introduction ................................ 11

2.2. Derivation of the DDE model . ..................... 12

2.2.1. Dimensionless formulation of the DDE system ........ 21

2.2.2. Parameter values ......................... 22

2.3. Discussion of the DDE model ...................... 23

3. Mode-locked laser dynamics 27

3.1. Introduction ................................ 27

3.1.1. Bifurcations ............................ 27

3.2. Solitary mode-locked laser dynamics .................. 32

3.2.1. Lasing threshold ......................... 32

3.2.2. Continuous wave solutions .................... 33

3.2.3. Mode-locked solutions ...................... 36

3.3. Dynamics induced by feedback from a single external cavity ..... 50

3.3.1. Short delay ............................ 50

3.3.2. Intermediate delay ........................ 80

3.3.3. Long delay ............................ 84

3.3.4. Frequency pulling and delay-induced multistability ...... 89

Contents

3.4. Dynamics induced by feedback from two external cavities ...... 99

3.4.1. Feedback induced dynamics ................... 99

3.4.2. Frequency pulling ........................112

3.5. Summary .................................116

4. Timing jitter of the mode-locked laser 119

4.1. Introduction ................................119

4.2. Calculating the timing jitter .......................120

4.2.1. Long-term timing jitter .....................120

4.2.2. Semi-analytic timing jitter ....................125

4.2.3. Experimental methods of measuring the timing jitter .....133

4.3. Timing jitter of the solitary mode-locked laser .............137

4.3.1. Inﬂuence of the amplitude-phase coupling on the timing jitter 140

4.4. Timing jitter under the inﬂuence of feedback from a single external

cavity ...................................142

4.4.1. Feedback delay time and feedback strength dependence of

the timing jitter .........................142

4.4.2. Feedback phase dependence of the timing jitter ........150

4.4.3. Inﬂuence of amplitude-phase coupling on the timing jitter . . 151

4.5. Timing jitter under the inﬂuence of feedback from two external cavities153

4.5.1. Feedback delay time and feedback strength dependence of

the timing jitter .........................153

4.5.2. Feedback phase dependence of the timing jitter ........156

4.5.3. Comparison with experimental results . ............156

4.6. Summary .................................159

5. Noise-induced modulations 163

5.1. Introduction ................................163

5.2. Suppression of noise-induced modulations ...............165

5.2.1. Stuart-Landau oscillator .....................166

5.2.2. Mode-locked laser ........................174

5.3. Impact of noise-induced modulations on the timing jitter .......181

5.3.1. Timing jitter with noise-induced modulations .........181

5.3.2. Reduction of the timing jitter via the suppression of noise-

induced modulations . . .....................184

5.4. Summary .................................186

6. Summary and outlook 189

A. Floquet theory 193

Contents

B. Linearised mode-locked laser system 195

B.1. Linearised DDE model ..........................195

B.2. Adjoint system ..............................197

C. Suppression of noise-induced modulations 199

C.1. FitzHugh-Nagumo oscillator .......................199

List of Figures 201

List of Tables 209

List of Publications 211

Bibliography 213

Acknowledgments 227

vii

viii

Chapter 1

Introduction

Mode-locking is a technique of producing short, high-intensity optical pulses at

high repetition rates [SMI70]. Four years after the invention of the laser in 1960

[MAI60] the ﬁrst clear demonstrations of mode-locking were achieved in a loss-

modulated He-Ne laser [HAR64]. Since then the mode-locking of lasers has been a

very active ﬁeld of research [SMI70,HAU00,KEL03] and nowadays mode-locked

lasers are used or envisaged as sources of short optical pulses for a wide range

of applications including optical data communication, optical clocking, metrology,

medical imaging and surgery [TUC88,DEL91,LOE96,HOL00,AVR00,UDE02,

KEL03,SPU03,LUE11b]. Semiconductor mode-locked lasers are of particular

interest, especially for telecommunications applications [BIM06,KUN07a,RAF07,

SCH10g], as they are relatively easy to manufacture, have a small footprint, are

integrable and can be electrically pumped. The techniques for achieving mode-

locking in such devices can be classed into two groups; active mode-locking and

passive mode-locking. Active mode-locking is achieved by externally modulating

some parameter of the laser, for example the losses or the pump current, with a

driving frequency which is resonant with the modes of laser cavity [KUI70,DER92,

HAU00]. The advantage of active mode-locking is that very regular pulse trains can

be produced, however the repetition rates and the pulse widths that can be achieved

are limited by the speed of the external modulator. For passive mode-locking no

external modulation is applied. Typically, a saturable absorber is included in the

laser cavity and it is through the interplay between the dynamics in the saturable

absorber section and the amplifying section of the laser that pulses are produced. In

passively mode-locked lasers very short pulses can be generated at repetition rates

of tenths to hundreds of Gigahertz [HAU00,AVR00]. However, passively mode-

locked semiconductor lasers also have a major drawback. Due to the absence of an

external reference clock the pulse trains can be relatively irregular [SOL93], which

is detrimental for most applications. A third variant of achieving mode-locking,

1. Introduction

referred to as hybrid mode-locking, combines active and passive techniques. Hybrid

mode-locked lasers have a saturable absorber section, which provides the dominant

mechanism for mode-locking, and a section which is electrically modulated, which

improves the regularity of the pulse train [DER92,FIO10,HAB14].

The absence of an external modulation source means that passively mode-locked

semiconductor lasers are substantially less expensive to manufacture than active

or hybrid mode-locked lasers, which makes them very attractive for applications.

Therefore a lot of research has gone into ﬁnding methods of improving the regularity

of the pulse trains produced by such devices [SOL93,AHM96,AVR09,GRI09,

LIN10e,BRE10,REB10,DRZ13a]. One approach is to optically inject the passively

mode-locked laser with emission from a second laser [AHM96,REB10,REB11].

However, like hybrid mode-locking, this method requires additional active elements.

Another method, which requires no additional active elements, is to subject the

passively mode-locked laser to optical self-feedback. Already in the 90’s it was

shown experimentally that under certain conditions optical feedback can lead to a

reduction in the timing jitter [SOL93]. The timing jitter being a measure of the

temporal regularity of the pulse train, which can be calculated from the power

spectrum of the laser output [LIN86].

Passively mode-locked semiconductor lasers subject to optical feedback are also

interesting from a non-linear dynamics point of view. The dynamics of single

mode semiconductor lasers subject to feedback have been extensively studied both

theoretically and experimentally [LAN80b,MOR88,MOR92,ALS96,KRA00a,

ERN10b,OTT12,KIM14]. Through the addition of time-delayed feedback these

devices are found to exhibit rich dynamics ranging from continuous wave emission

to low frequency ﬂuctuations to chaos. Passively mode-locked lasers without feed-

back already exhibit complex periodic dynamics. As such, they are an ideal test

bed for the inﬂuence of time-delayed feedback on periodic dynamical systems.

In the following sections we will give brief introductions to semiconductor lasers,

mode-locking and time-delayed feedback. Then, in Section 1.4 an outline of this

thesis is given.

1.1. Semiconductor lasers

A laser1is a source of coherent light which is generated via stimulated emission.

The stimulated emission of photons is a quantum mechanical process in which an

incident photon, of the appropriate energy, interacts with an electron in an excited

state and causes it to relax to a lower energy state (ground state) by emitting a

photon [EIN17]. The emitted photon has the same frequency, polarisation, phase

and direction of travel as the incident photon. For this to occur the diﬀerence in

1The word "laser" is an acronym for "light ampliﬁcation by stimulated emission of radiation".

1.1. Semiconductor lasers

nergy

Before After

Gain medium

Mirrors

Figure 1.1.: Left: Sketch of the stimulated emission of a photon. Right: Sketch of a standing wave

in a laser cavity.

ectro

p-doped

n-doped

p-n junction

Figure 1.2.: Sketch of a semiconductor laser.

the electronic energy levels must be equal to the energy of the incident photon,

E2−E1=hν. This is illustrated in Fig. 1.1. If the medium (gain medium) in

which the stimulated emission is occurring is conﬁned to a cavity, for example by

the placement of two mirrors, then large numbers of photons can build as they

pass back and forth through the gain medium. Each time a stimulated emission

event occurs, an electron drops from the excited state to the ground state, thereby

reducing the gain of the amplifying medium. For lasing to be sustained the gain

medium must remain inverted, i.e. there must be more electrons in the excited

state than in the ground state. One of the advantages of semiconductor lasers

is that this can be achieved by electrical pumping. The alternative method is to

optically excite electrons using an external light source.

In the simplest form of a semiconductor laser the optical transitions occur at

a p-n junction and the charge carrier inversion is achieved by electrical pumping.

Figure 1.2 shows a simpliﬁed sketch of such a semiconductor laser. The active region

is indicated by the grey shading near the p-n junction. The ﬁrst semiconductor

lasers were of this type [HAL62,HOL62,NAT62]. However, they required very

high pump currents and they had to be operated at very low temperatures. To

overcome these issues the spatial conﬁnement of the charge carriers is crucial. Over

the years many conﬁnement structures have been developed. One of the earliest

examples that enabled room temperature operation used a thin layer of GaAs

between two bulk layers of AlGaAs, which has a larger band-gap, thereby forming a

1. Introduction

Energy

Density of states

3D - bulk

Energy

Density of states

Energy

Density of states

Energy

Density of states

Figure 1.3.: Sketch of the density of states for 3D, 2D, 1D and 0D structures.

quantum well [ALF68]. A quantum well is a quasi-two dimensional structure which

conﬁnes charge carriers, in one spatial direction, to a region narrower than their

de-Broglie wavelength, but allows them to move freely transverse to this direction.

This conﬁnement causes a discretisation of the energy levels the charge carriers

can occupy and leads to a lower threshold current density and lasing at higher

temperatures. Conﬁnement in two and three spatial dimensions is also possible,

leading to quantum wires and quantum dots, respectively. Figure 1.3 shows a

sketch of the density of states for diﬀerent degrees of conﬁnement. In recent years

devices based on semiconductor quantum dots have received a lot of attention owing

to their discrete states, which can lead to lower threshold currents and increased

temperature stability compared with quantum-well structures [SUG05,RAF07,

LUE11b,OTT14,LIN15].

The modelling of a semiconductor laser can be done to varying degrees of com-

plexity and depending on the structure of the gain medium diﬀerent processes need

to be included to accurately describe the charge-carrier dynamics. In a minimalis-

tic approach, a single mode semiconductor laser can be modelled using two coupled

rate equations,

dt =GN(N−Ntr)S−S

Tph

(1.1)

and

dt =J−N

T1−GN(N−Ntr)S, (1.2)

where dynamical variables are the number of photons Sand the number of charge

carriers Nin the active medium. The interaction between the charge carriers and

photons is taken into account with the gain function GN(N−Ntr), where Ntr is

the charge-carrier number at transparency. The photon lifetime Tph accounts for

the optical losses. When the number of charge carriers is large enough that the

gain can compensate for these losses, then the light can be ampliﬁed. The number

charge carriers depends on the pumping rate Jand on the non-resonant carrier

losses, which are taken into account by the charge-carrier lifetime T1.

1.2. Mode-locking

ower

Frequency

Long

cav

ty mo

Power

Frequency

Las

ng mo

Intensity

Time

se tra

Figure 1.4.: Frequency and time domain sketches of a mode-locked electric ﬁeld.

In depth introductions to general laser theory, and speciﬁcally to semiconductor

lasers, can be found in [HAK86,HAK86a]and[CHO99], respectively.

1.2. Mode-locking

The frequencies of lasing modes are determined by the gain spectrum of the am-

plifying medium and by the boundary conditions imposed by the laser cavity. The

frequencies which are resonant with the laser cavity form standing waves which

are referred to as longitudinal modes. These modes have a ﬁxed frequency spacing

which is dependent on the length of the cavity. Of these modes, only those which

fall within the bandwidth of the gain medium can be ampliﬁed. This is sketched

in Fig. 1.4. If, in addition to the mode spacing νc, the relative phases of the las-

ing modes are equal, then pulses will be formed as the lasing modes periodically

undergo constructive interference. The time between pulses, i.e. the period of the

mode-locked dynamics, will be determined by the reciprocal of the mode spacing

1/νc, and the width of the pulses will be given approximately by the reciprocal of

the width of the optical spectrum 1/Δν. Mode-locking can also occurs between

modes which are separated by multiples of the fundamental frequency of the laser

cavity νc. This is referred to as harmonic mode-locking and results periodic dy-

namics in which multiple pulses circulate in the laser cavity.

To demonstrate more clearly why pulses are formed under equal phase conditions

we consider the superposition of a set of Nlongitudinal modes at some ﬁxed point

in the laser cavity. The total ﬁeld is given by

ET(t, z)=



n=1

Enei(2πνnt+φn),(1.3)

where Enis the amplitude of the nth mode, νnis the frequency and φnis the

relative phase [SMI70]. The frequencies are all oﬀset from one another by integer

multiples of the mode spacing νc,i.e. νn=ν0+nνcwhere ν0is a fast optical

1. Introduction

Time

Tim

Time

Amplitude

Amplitude Amplitude

Equa

ases

Random phases

Figure 1.5.: Sinusoidal waveforms with frequencies of nνcfor n=1,2,3,4. In the top diagram the

phases φnare equal and the waveforms interfere constructively every 1/νc, resulting in pulses.

In the bottom diagram the phase are randomly chosen, consequently the waveforms are never

all in phase at the peaks.

frequency. Factoring out the fast optical component, ET=ETei2πν0t, the slowly

varying electric ﬁeld envelope is given by

ET(t, z)=



n=1

Enei(2πnνct+φn),(1.4)

which is periodic in 1/νc. If the relative phases are all equal, for simplicity let

φn= 0, then all modes will interfere constructively for all t=m/νc,wheremis

an integer. If the relative phases are not all equal, then at no point in time will

all modes interfere completely constructively. This is illustrated in Fig. 1.5 which

shows four sinusoidal waveforms with equal relative phases on the top and with

diﬀerent relative phases on the bottom. In the top diagram all four waveforms are

completely in phase at each time marked by the red circles, whereas on the bottom

they are never simultaneous in phase. When there is a large number of modes

which all have the same relative phases, then the total electric ﬁeld amplitude will

be large when they are all in phase and between these points in time the modes

will interfere destructively resulting in close to zero intensity between the pulses.

1.2.1. Passive mode-locking

Passively mode-locked lasers typically consist of a saturable absorber section and

a gain section.2For semiconductor passively mode-locked lasers these two sections

2Passive mode-locking techniques that do not involve a saturable absorber also exist. For example

in Kerr lens mode-locking Kerr lensing in the gain medium leads to mode-locking [BRA92], or

1.2. Mode-locking

satura

e a

sor

er ga

n sect

Figure 1.6.: Sketch of the pulse shaping that occurs in the saturable absorber and gain sections of

a passively mode-locked laser.

Tim

Amplitude

Figure 1.7.: Sketch of a perfectly regularity pulse train (blue) and an irregular pulse train in which

noise causes the time interval between the pules to vary (red).

can be made of the same material and the behaviour is characterised by the pump

conditions. The gain section is electrically pumped, thereby generating an inversion

of charge carriers, and the absorber section has a reverse bias applied to it [DER92].

To understand how mode-locking is achieved via the addition of the saturable

absorber it is helpful to consider the electric ﬁeld in the time domain. Consider

a pulse travelling in the laser cavity. As the pulse enters the saturable absorber

the front of the pulse is absorbed, which excites charge carries, until the absorber

saturates, at which point it is transparent to the rest of the pulse. In the gain section

the front of the pulse will be ampliﬁed, leaving the charge carriers depleted for the

trailing edge for the pulse, meaning that non-resonant losses are not compensated

for the trailing edge. A sketch of the pulse shaping in the absorber and gain

sections is shown in Fig. 1.6. If the absorber saturates at a lower energy than the

gain section, then this results in a narrow positive net gain window for the center

of the pulse and stable mode-locking can be achieved.

1.2.2. Timing jitter

In the ideal case a passively mode-locked laser would produce a perfectly regular

pulse train. However, due to inevitable noise sources the temporal positions of

in [JAV06] a mode-locking technique that utilises the polarisation degrees of freedom of light

is discussed.

1. Introduction

pulses can ﬂuctuate. This is illustrated in Fig. 1.7. For actively mode-locked lasers

the pulse is pulled back towards its ideal position each round trip due to the external

modulation. However, for passively mode-locked lasers there is no restoring force,

which means the pulse position can drift randomly over time. In semiconductor

passively mode-locked lasers the dominant noise source is spontaneous emission,

which leads to comparatively large timing ﬂuctuations [HAU93a,SOL93]. The

temporal ﬂuctuations of the pulse positions are referred to as the timing jitter.

1.3. Time-delayed feedback

The control of dynamical systems via time-delayed feedback is an extensively stud-

ied topic that is of relevance to a wide range of ﬁelds. Examples of feedback con-

trol are found in electronics [JUS00], quantum mechanics [BRA10a], neuroscience

[ROS04] and optics [SOL93,LAN80b,ERZ07,SLO15], among many others. Rele-

vant to passively mode-locked lasers, time-delayed feedback can be used to improve

the regularity of noisy oscillatory dynamics [SOL93,OTT12a,OTT14]. Time-

delayed feedback can also be used to induce and control chaos [PYR92,SCH07],

stabilise unstable steady states or periodic orbits [HOE05,MIC14a], or generate

complex temporal patterns [YAN14].

The introduction of time-delayed variables into a dynamical system causes the

system to be inﬁnitely dimensional as the initial conditions must be deﬁned on

the entire interval [−τ,0], where τis the delay time [HAL93]. This can generally

lead to a system exhibiting more complex dynamics than the same system without

the time-delayed feedback term, and can also lead to multistability, especially for

long delay times. For periodic systems subject to feedback a general result is that

solutions reappear at all delay times that are increased by integer multiples of the

period of the solution [YAN09]. The inﬂuence of varying the feedback delay time

between these discrete values is that the stability of the solutions is aﬀected; the

general trend being that solutions become less stable for longer delay times.

For systems in the presence of noise, feedback can be used to suppress or enhance

noise induced eﬀects [GOL03,BAL04,FLU07]. This is of relevance for the passively

mode-locked laser subject to feedback as experimental studies have shown that

noise can excite modes which are resonant to the feedback cavity [DRZ13,ARS13].

Analogous to the suppression of noise-induced oscillations in single mode lasers

[FLU07], an additional feedback cavity can be used to suppress these modes in the

mode-locked laser system subject to feedback [HAJ12,JAU16].

1.4. Outline

In this thesis we study the dynamics and timing jitter of a passively mode-locked

laser subject to optical feedback and the suppression of noise-induced modulations

in periodic systems with feedback. The thesis is structured as follows.

In Chapter 2the model for a passively mode-locked semiconductor laser is pre-

sented. The model is a delay diﬀerential equation system for a ring cavity laser

[VLA04,VLA05]. In Section 2.2 the model is derived. Then in Section 2.3 this

model is discussed, particularly the comparison with travelling-wave models used

for linear laser cavity geometries.

In Chapter 3we investigate the feedback induced dynamics of passively mode-

locked semiconductor lasers. First, a recap of the parameter dependence of the

solitary mode-locked laser is given in Section 3.2.3. Then we investigate the inﬂu-

ence on the dynamics of optical feedback from one external cavity in Section 3.3.

In this section the dynamics are investigated in the regimes of short, intermedi-

ate and long feedback delay times. The dynamics are investigated by both direct

numerical simulations and by using path continuation techniques. In Section 3.4

the inﬂuence of a second external feedback cavity is studied. A summary of this

chapter is presented in Section 3.5.

Chapter 4is concerned with the timing jitter of the mode-locked solutions. First,

the various methods of calculating the timing jitter are deﬁned in Section 4.2.One

of these methods is a semi-analytic approach which allows us to analytically derive

the feedback dependence of the timing jitter under certain conditions. In Section 4.3

we investigate the timing jitter of the solitary laser, then in Sections 4.4 and 4.5

the inﬂuence of one and two external feedback cavities is examined. For the dual

feedback cavity case we present a comparison with experimental results.

The results of Chapter 4will show that for long feedback delay times noise excites

weakly stable eigenmodes, which causes a modulation of the oscillatory dynamics.

In Chapter 5the suppression of these noise-induced modulations, using the second

feedback cavity, is studied. First, we investigate the suppression of noise-induced

modulations in a simple oscillatory system in Subsection 5.2.1. Then, these results

are used to study the suppression of noise-induced modulations in the mode-locked

laser system in Subsection 5.2.2. Following this, the inﬂuence of the modulation

suppression on the timing jitter is examined in Section 5.3. Finally, in Section 5.4

the results of this chapter are summarised.

A summary of this thesis is presented in Chapter 6, along with a discussion of

open questions and possible directions for future work.

Chapter 2

Mode-locked laser model

2.1. Introduction

Various models exist for passively mode-locked lasers, each with their own ad-

vantages and restrictions. One of the earliest models was a master equation ap-

proach developed by Haus which allows the mode-locked pulses to be described

analytically under the assumption of only small changes in the gain and losses per

cavity round trip [HAU75a]. This approach works well for solid state lasers, how-

ever the assumption of small changes in the gain and losses per round trip is not

applicable to semiconductor lasers. For accurate modelling of semiconductor pas-

sively mode-locked lasers, ﬁnite-diﬀerence travelling-wave models are typically used

[MUL06,AVR09,JAV10,JAV11,ROS11c,ROS11d,RAD11a,SIM12a,MOS14].

However the numerical integration of such models is computationally very expen-

sive and this makes it infeasible to use such an approach to study the inﬂuence

of feedback over a wide range of feedback delay times. In [MUL06], [AVR09]and

[SIM12a] the impact of feedback is studied, but in each case only in the regime of

short feedback cavities. An alternative modelling approach is to use a delay dif-

ferential equation (DDE) model. This was ﬁrst proposed in [VLA04,VLA05]and

extended to include optical feedback in [OTT12a]. This approach assumes a ring

cavity with unidirectional propagation and uses a lumped element method to in-

clude non-resonant losses and spectral ﬁltering. Using this model the computation-

ally cost is greatly reduced and feedback can easily be incorporated. However, the

simplifying assumptions mean that this model is strictly only applicable to ring cav-

ities, whereas linear cavity geometries are commonly used for monolithic passively

mode-locked semiconductor lasers. Despite this limitation most of the qualitative

trends, especially in dependence of feedback parameters, are very similar to ex-

perimental results for linear cavities [SAN90,BAN06,ARS13,MAR14c,OTT14b].

2. Mode-locked laser model

Due to its suitability to studying optical feedback, we will use the DDE model in

this thesis.

The modelling of the charge carriers is dependent on the choice of semiconduc-

tor material and can be done to varying degrees of complexity [CHO99,VLA05,

ROS11c,LIN13,LIN14,LIN15a]. We will use a simple model for a quantum-well

gain medium. Despite this, in subsequent chapters we will compare our results with

experiments using quantum-dot lasers and show that the qualitative dependence

on the feedback parameters is captured by the DDE model.

This chapter is structured as follows. In Section 2.2 we present the derivation of

the DDE model. Then in Section 2.3 we discuss the assumptions that underlie this

model. We also discuss key diﬀerences that arise compared with travelling wave

models, as well as extensions that have been made to this model.

2.2. Derivation of the DDE model

In this section we present the derivation of the DDE model for a passively mode-

locked semiconductor laser which is introduced in [VLA04a,VLA05]. In [OTT12a]

the model is extended to include optical feedback from one external cavity. Here

we present extension to multiple feedback cavities. The derivation follows those of

[VLA11]and[OTT12a].

The light-matter interaction needed for lasing to occur can be described using a

semi-classical theory [HAK83a,CHO99]. In this approach the light ﬁeld is described

classically, but the motion of the charge carriers is described quantum-mechanically.

Although the quantum-mechanical treatment of the carriers is necessary for lasing

to be described, the classical treatment of the light ﬁeld is suﬃcient when consider-

ing devices that involve macroscopic numbers of charge carriers and photons. The

results of the semi-classical theory are the Maxwell-Bloch equations, which describe

the evolution of the electric ﬁeld, the charge carrier inversion and the microscopic

polarisation. The derivation for a simple two-level system is given in [HAK83a].

For a semiconductor gain medium the derivation is somewhat more complicated

due to inhomogeneous broadening (distribution of charge carrier kinetic energies),

many-body Coulomb interactions and carrier-phonon interactions. A comprehen-

sive derivation of the semiconductor Bloch equations is given in [CHO99].

We will start the derivation of the DDE model directly from the standard travel-

ling wave model for semiconductor quantum-well lasers [TRO94,TAR98a,SCH88j]:

±∂E±

r(t, z)

∂z +1

∂E±

r(t, z)

∂t =grΓr

2(1 −iαr)nr(t, z)−ntr

rE±

r(t, z) (2.1)

∂nr(t, z)

∂t =ηjr(t, z)

ed −γrnr(t, z)−vgrΓrnr(t, z)−ntr

r

±|E±

r(t, z)|2.(2.2)

2.2. Derivation of the DDE model

gain

carrier density

Figure 2.1.: Sketch of the carrier dependence of the gain for a semiconductor quantum-well mate-

rial.

The dynamical variables are the left (+) and right (-) propagating slowly varying

complex electric ﬁeld amplitudes E±

rand the carrier density nr. In this model the

slowly varying envelope approximation has been made, hence the total electric ﬁeld

is given by

E(t, z)≡E+

r(t, z)e−ikz +E−

r(t, z)e+ikze−iΩ0t,(2.3)

where kis the reference wavenumber which is related to the optical frequency Ω0

via the linear dispersion relation k≡Ω0/v, with group velocity v. To arrive at

Eqs. (2.1)-(2.2) the polarisation has been adiabatically eliminated. This simpliﬁca-

tion is possible because semiconductor quantum-well lasers are generally of class B

[ARE84], meaning that the polarisation dephasing time is much faster than other

time scales of the system.

The carrier-ﬁeld interaction is described by the complex gain function

grΓr(1 −iαr)nr(t, z)−n0

r,(2.4)

which is the gain linearised about the carrier density at some operation point and

written in terms of the eﬀective transparency carrier density of this linearised func-

tion, n0

r. The diﬀerential gain is given by grand transverse modal conﬁnement

is accounted for by the factor Γr. Carrier-induced refractive index changes are

modelled by the amplitude-phase coupling factors αr(linewidth enhancement fac-

tors). The factor of 1/2 in Eq. (2.1) is included because expression (2.4) describes

the intensity gain. In Fig. 2.1 a sketch of the carrier dependence of the gain is

shown for a semiconductor quantum-well material [ASA84,LAU87a,CHO99]. If

the same material is used for the gain (r=g) and absorber (r=q) sections then

the diﬀerence in the gain comes from the diﬀerence in the carrier densities of the

two sections. Due to the reverse bias the absorber section will always be below

transparency. The operation range of the gain section will be at some higher car-

rier density due to the pump current. Therefore, for the gain and absorber section

2. Mode-locked laser model

mirror

coupler

laser cavity

spectral filtering element

gain section

passive section

aturable absorber section

fee

cav

Figure 2.2.: Schematic diagram of a ring-cavity laser subject to optical feedback from multiple

external cavities.

the gain function is linearised about diﬀerent point, leading to diﬀerent values for

the diﬀerential gain gr.

In Eq. (2.2) the injection of charge carriers is described by the pump current

density jr. Included in the pump term are the injection eﬃciency factor η,the

electron charge eand the thickness of the active region d. The lifetime of charge

carriers in the quantum-well states is limited by non-radiative scattering processes,

which are included via the carrier decay rate γr. Inspection of the carrier-ﬁeld

interaction term in Eq. (2.2) shows that the electric ﬁeld is scaled to the dimensions

(length)−3/2such that |E±

r|2gives the photon density.

A two section passively mode-locked laser consists of two active sections, a gain

section which is electrically pumped and an absorber section which is negatively

biased. Depending on the geometry of the laser, passive sections could also be

included. For the derivation of the DDE model a ring cavity geometry must be

assumed. A schematic of such a cavity is shown in Fig. 2.2. Equations (2.1)and

(2.2) are applied in each section of the laser cavity. The index r∈{g,q,p}is

used to denote the gain, absorber and passive sections, respectively. In the passive

section there are no carriers, hence the evolution of the electric ﬁeld envelope is

described solely by Eq. (2.1) with the right-hand side equal to zero. In the absorber

section jq= 0 holds, and the inﬂuence of the applied reverse bias is manifested in

the carrier decay rate γqand the gain of this section (Eq. (2.4)). A lumped-element

approach is used to include non-resonant losses at the facets between each of the

sections (κ1and κ2in Fig. 2.2). Mirror losses as well as coupling to and from the

feedback cavities is included between the gain and passive sections. Also using a

lumped-element approach, a ﬁltering element ˆ

f(ω) is included between the passive

2.2. Derivation of the DDE model

and absorber sections (red bar in Fig. 2.2). The purpose of this ﬁlter is to account

for the ﬁnite width of the gain spectrum. The z-direction is chosen along the axis

of the cavity and is the propagation direction of longitudinal modes of the laser.

Along with the ring cavity geometry, the second main assumption of the DDE

model is unidirectional propagation of the light ﬁeld. Here clockwise propagation

is chosen, i.e. E−

r= 0 and E+

r=E. The validity and implications of these two

assumptions will be discussed in Section 2.3.

Including the losses and feedback contributions as described above, the boundary

conditions between each of the sections are

Eg(t, z2)=√κ2Eq(t, z2),(2.5)

Ep(t, z3)=√rEg(t, z3)+(1−r)



n=1 ∞



l=1

r(l−1)/2kl

nrl/2

ec,nEg(t−lτn,z

3)eilCn

+(1 −r)



n=1 

n=n∞



l=1

rl/2kl

nrl/2

ec,nknr1/2

ec,nEg(t−τn−lτn,z

3)ei(Cn+lCn)

+(1 −r)



n=1 

n=n∞



l=1

r(l+1)/2kl

nrl/2

ec,nk2

nrec,nEg(t−2τn−lτn,z

3)ei(2Cn+lCn)

+··· (2.6)

and ˆ

Eq(ω,z1+L)=√κ1ˆ

f(ω)ˆ

Ep(ω,z1+L)=√κ1ˆ

f(ω)ˆ

Ep(ω,z4).(2.7)

In Eqs. (2.5)and(2.7)κ1and κ2account for the internal non-resonant loss. These

depend on the length of the active sections according to κx=e−aLx,whereais a

loss rate per unit length and Lxis the length of the active section for which the

losses are being accounted for.

In Eq. (2.6) feedback contributions from an arbitrary number Nof external

feedback cavities has been included. To experimentally implement such a feedback

scheme there are several options. These include using beam-splitters or optical cou-

plers to distribute the light among the feedback cavities [HAJ12,ARS13,NIK16].

Here we do not choose a particular scheme, but rather account for this with

the factor knwhich gives the percentage of the light coupled into feedback cav-

ity n∈{1,2, .., N}, with N

n=1 kn= 1. The sum over laccounts for feedback

contributions after multiple roundtrips in the feedback cavities. Also included

are contributions from light that has made roundtrips in diﬀerent feedback cav-

ities. The reﬂectivity of the end facets of each of the feedback cavities is given

by rec,n. This factor primarily determines the feedback strength. The fast oscil-

lating reference frequency Ω0also plays a role in the feedback terms, as due to

these fast oscillations a phase shift can accumulate with respect to the light in

2. Mode-locked laser model

the laser cavity. This can be understood by considering the delayed electric ﬁeld

E(t−τ,z3)=E+

r(t−τ,z3)e−ikz3e−iΩ0(t−τ)at the out-coupling facet (position z3).

Over the delay time τa phase shift of Ω0τaccumulates. For each of the feedback

cavities the phase shift per roundtrip is then given by Cn≡Ω0τn. Since the optical

frequency Ω0τis very large (THz) very small changes in the delay times τnresults

in variations in the phase that are greater then 2π. Due to this we can essentially

treat the feedback phase independently from the delay time. In Eqs. (2.5)-(2.7)

the non-resonant losses and reﬂectivities are all given with respect to the intensity,

hence the respective electric ﬁeld losses are given by the square root of these terms.

In the limit of weak feedback, i.e. small external cavity reﬂectivities rec,n,the

feedback terms in Eq. (2.6) can be simpliﬁed by making the approximation that con-

tributions after multiple roundtrips in the feedback cavities are negligible. Includ-

ing only the terms that correspond to one roundtrip in a feedback cavity Eq. (2.6)

becomes

Ep(t, z3)=√rEg(t, z3)+√r



n=1

KnEg(t−τn,z

3)eiCn,(2.8)

where

Kn≡(1 −r)kn(rec,n/r)1/2.(2.9)

In the boundary condition Eq. (2.7), ˆ

Eris the Fourier transformed electric ﬁeld

and ωis the angular frequency. In this boundary condition several assumptions

are made. Firstly, the ﬁltering element is assumed to be inﬁnitely thin, meaning

that z1=z4. Secondly, since z1+L=z4in the ring cavity geometry, the boundary

conditions are periodic, i.e.

Er(t, z)=Er(t, z +L),(2.10)

where Lis the length of the laser cavity. The laser cavity modes are therefore

restricted to

ΩL=n2πv

L(2.11)

for n=1,2,3, .... All modes fulﬁlling this condition can exist in the laser cavity.

This is shown schematically in Fig. 2.3. The grey lines in this diagram indicate

modes with a spacing of Δν=T−1=v

L,whereTis the cold cavity roundtrip time.

Without a spectral ﬁlter, according to Eqs. (2.1)and(2.2), all of these modes are

equally ampliﬁed and would have the same output power. However in real devices

only a ﬁnite number of modes will have positive net gain [HOH93,UKH04]. The

spectral ﬁlter (blue curve) is added to account for this by increasing the losses for

modes away from the maximum gain mode. In the derivation of the DDE model a

Lorentzian shaped ﬁlter is chosen;

f(ω)≡γ

γ+i(ω−ΔΩ).(2.12)

2.2. Derivation of the DDE model

Power

Frequency (THz)

Figure 2.3.: Schematic diagram of the optical spectrum of a semiconductor mode-locked laser. The

grey lines indicate the cavity modes and the red lines indicate the modes that can lase. The

blue curve represents the spectral ﬁlter that is applied to account for the frequency dependence

of the gain.

This expression is written in the frame of the slowly varying amplitude, i.e. the

optical frequencies are given by ω=Ω

0+ω.ΔΩ≡Ω0−Ωmax is the detuning

between the frequency of the maximum of the ﬁlter Ωmax and the reference optical

frequency. The full-width at half maximum of the ﬁlter is given by γ. This param-

eter determines the number of modes that participate in the mode-locking, which

is roughly given by γT[VLA05]. The Lorentzian ﬁlter shape is chosen as it allows

the derivation of a system of delay diﬀerential equations.

Before applying the boundary conditions it is useful to make a change of coordi-

nates to a frame co-moving with the electric ﬁeld: (t, z)→ (t,z) with t=t−z/v

and z=z/v. In this coordinate system Eqs. (2.1)-(2.2), applied to each section of

the laser, are given by

∂Ar

∂z=1

2(1 −iαr)Nrt,zArt,z,(2.13)

∂Ng

∂t=Jgt,z−γgNgt,z−Ngt,z|Agt,z|2(2.14)

and ∂Nq

∂t=−Jqt,z−γqNqt,z−˜rsNqt,z|Aqt,z|2,(2.15)

with Np(t,z)≡0. Here the dynamical variables have been rescaled such that Ar≡

vggΓgErand Nr≡vgrΓr[nr−n0

r]. Accordingly, the rescaled pump parameters

are Jg≡vggΓgjg−γgn0

gand Jq≡vgqΓqγqn0

q, and the saturation energy ratio

˜rs≡gqΓq/ggΓghas been introduced [HAU00,OTT14]. The saturation energies

describe how much energy must be absorbed or emitted for the absorber and gain

sections to become transparency. The ratio of these energies is a crucial parameter;

2. Mode-locked laser model

for mode-locking to occur the absorber must have a smaller saturation energy than

the gain section, i.e. ˜rs>1 as the saturation energies are proportional to (grΓr)−1.

In this co-moving frame the evolution of the electric ﬁeld amplitude along the

sections of the laser is now described by ordinary diﬀerential equations (Eq. (2.13)).

Integrating these equations over each section yields

Aqt,z

2=e−1

2(1−iαq)Q(t)Aqt,z

1,(2.16)

Agt,z

3=e1

2(1−iαg)G(t)Agt,z

2(2.17)

and

Apt,z

3=Apt,z

4,(2.18)

where

Qt≡−z

z

Nqt,zdzand Gt≡z

z

Ngt,zdz

are the dimensionless carrier densities integrated over the absorber and gain sec-

tions, which we shall refer to as the gain and losses, respectively. Next, the bound-

ary conditions can be applied to obtain the evolution of the electric ﬁeld amplitude

over one roundtrip in the laser cavity. Using Eqs. (2.5)and(2.8), transformed to

the co-moving frame, the evolution from Aq(t,z

1)toAp(t,z

4) is given by

Apt,z

4=√rκ2e1

2(1−iαg)G(t)−1

2(1−iαq)Q(t)Aqt,z

1

+√rκ2



n=1

KneiCne1

2(1−iαg)G(t−τn)−1

2(1−iαq)Q(t−τn)Aqt−τn,z

1.(2.19)

In order to apply Eq. (2.7), it must ﬁrst be transformed to the time domain and

the co-moving frame. The transformation to the time domain is done using the

convolution theorem which states

F[g∗h(t)] = F[g(t)]F[h(t)],(2.20)

where gand hare functions of t,Fis the Fourier transform and (∗) denotes a

convolution product. In the co-moving frame, and in terms of the rescaled ﬁeld,

Eq. (2.7) can now be rewritten as

Aqt−T,z

1+T=√κ1ft−T∗Apt−T,z

4

=√κ1t−T

−∞

ft−T−θApθ,z

4dθ, (2.21)

once again Tis the cold cavity roundtrip time given by L/v. The upper limit of

the integral has been set to t−T, instead of inﬁnity, in order to preserve causality.

2.2. Derivation of the DDE model

This is done under the assumption that Aq(θ,z

4)=0forθ>t

−T. Substituting

Eq. (2.19) into Eq. (2.21) yields

Aqt,z

1=t−T

−∞

ft−T−θR(θ)Aqθ,z

1dθ



n=1

KneiCnt−T

−∞

ft−T−θR(θ−τn)Aqθ−τn,z

1dθ, (2.22)

where we have deﬁned

R(θ)≡√κe1

2(1−iαg)G(θ)−1

2(1−iαq)Q(θ)(2.23)

which describes the gain and losses accumulated over one roundtrip in the laser

cavity. Here we also deﬁne κ≡κ1κ2r. This parameter accounts for all the non-

resonant and mirror losses per roundtrip. To obtain the left hand side of Eq. (2.22)

the periodic boundary condition has been applied, which in the co-moving coordi-

nates is expressed as

Aqt,z

1=Aqt−T,z

1+T.(2.24)

The form of which can be derived by substituting z+Lfor zin the transformation,

i.e. (t, z +L)→ t−(z+L)

v,(z+L)

v=(t−T,z+T).

The evolution of the ﬁeld over one roundtrip is now expressed entirely in terms of

the electric ﬁeld amplitude in the saturable absorber section at position z

1,which

we relabel A(t)≡Aq(t,z

1). In order to obtain a diﬀerential equation for the time

evolution of A(t) Eq. (2.22) is diﬀerentiated with respect to t.Todothisweﬁrst

substitute in the ﬁltering function, which in the time domain is written as

ft≡γe(−γ+iΔΩ)t.(2.25)

With some slight rearrangement this yields

Ate(γ−iΔΩ)t=t−T

−∞

γe(γ−iΔΩ)(T+θ)R(θ)A(θ)dθ



n=1

KneiCnt−T

−∞

γe(γ−iΔΩ)(T+θ)R(θ−τn)A(θ−τn)dθ, (2.26)

Diﬀerentiating Eq. (2.26) then results in a DDE for the time evolution of A(t);

dt+(γ−iΔΩ) At=γRt−TAt−T

+γ



n=1

KneiCnRt−T−τnAt−T−τn.(2.27)

2. Mode-locked laser model

To obtain diﬀerential equations for the time evolution of the integrated carrier

densities G(t)andQ(t), we integrate Eqs. (2.14)and(2.15)overthegainand

absorber sections, respectively:

∂G

∂t=Jgt−γgGt−z3

Ngt,z|Agt,z|2dz,(2.28)

∂Q

∂t=Jqt−γqQt+˜rsz2

Nqt,z|Aqt,z|2dz.(2.29)

Here we have introduced the unsaturated gain and absorption parameters

Jgt≡z3

z2Jgt,zdz(2.30)

and

Jqt≡z2

z1Jqt,zdz,(2.31)

respectively. To express the integrals in Eqs. (2.28)and(2.29) in terms of G(t),

Q(t)andA(t) we must make use of the equations describing the evolution of

the ﬁeld along the gain and absorber sections (Eqs. (2.13), (2.16)and(2.17), and

boundary condition Eq. (2.5)). From Eq. (2.13) we obtain

A∗

∂Ar

∂z+Ar

∂A∗

∂z=1

2(1 −iαr)NrA∗

rAr+1

2(1+iαr)NrArA∗

r(2.32)

by multiplying by the complex conjugate of the ﬁeld amplitude A∗

rand adding the

complex conjugate of Eq. (2.13) multiplied by Ar. This expression gives

∂|Ar|2

∂z=Nrt,z|Art,z|2.(2.33)

Integrating over the gain and absorber sections then gives

|Ag,q t,z

3,2|2−|Ag,q t,z

2,1|2=z

3,2

z

2,1

Ng,q t,z|Ag,q t,z|2dz.(2.34)

Using Eqs.(2.16), (2.17)and (2.5) the left hand side can be expressed in terms

of G(t), Q(t)andA(t), which we then substitute into Eqs. (2.28)and(2.29)to

obtain ∂G

∂t=Jgt−γgGt−κ2e−Q(t)eG(t)−1|At|2(2.35)

and ∂Q

∂t=Jqt−γqQt−˜rse−Q(t)eQ(t)−1|At|2.(2.36)

Equations (2.27), (2.35)and(2.36) now describe the evolution of the electric

ﬁeld amplitude at position z1and the gain and absorption per roundtrip in the

2.2. Derivation of the DDE model

laser cavity. As of yet the inﬂuence of spontaneous emission is not included. We

treat this eﬀect in a phenomenological manner by adding a complex Gaussian white

noise term to the electric ﬁeld equation (Eq. (2.27)). Adding this noise term and

making the transformation A(t)≡1

√κ2A(t)eiΔΩtwe obtain the ﬁnal system of

coupled DDEs;

dt=−γAt+γRt−Te−iΔΩTAt−T+Rspξt

+γ



n=1

KneiCnRt−T−τne−iΔΩ(T+τn)At−T−τn,(2.37)

∂G

∂t=Jgt−γgGt−e−Q(t)eG(t)−1|At|2(2.38)

and ∂Q

∂t=Jqt−γqQt−rse−Q(t)eQ(t)−1|At|2.(2.39)

Here the ratio of excitation energies has been rescaled, rs≡˜rs/κ2. The noise

strength is given by Rsp and ξ(t)=ξR+iξIhas the properties

ξit= 0 (2.40)

and

ξitξjt=δi,jδt−t,(2.41)

for i, j ∈{R, I}.

2.2.1. Dimensionless formulation of the DDE system

For numerical simulations it is convenient to rescale the DDE system such that the

parameters are all dimensionless. In [VLA05] the authors do this by rescaling by

the absorber recovery rate γq. We shall take the same approach as in [OTT12a]

and use the cold cavity roundtrip time T. This is a convenient time scale of the

system to use when studying the system with feedback as features in the feedback

delay time dependence vary on this time scale, as will be shown in the next chapter.

We write the dimensionless form of Eqs. (2.37)-(2.39)as

dt =−γE(t)+γR(t−T)e−iΔΩTE(t−T)+Rspξ(t)

+γ



n=1

KneiCnR(t−T−τn)e−iΔΩ(T+τn)E(t−T−τn),(2.42)

∂G

∂t =Jg(t)−γgG(t)−e−Q(t)eG(t)−1|E(t)|2(2.43)

2. Mode-locked laser model

and ∂Q

∂t =Jq(t)−γqQ(t)−rse−Q(t)eQ(t)−1|E(t)|2,(2.44)

where time has been rescaled as t=t/T, the cold cavity roundtrip time is now

T=T/T= 1 and E=√TA. For the other parameters we use the same symbols

as in Eqs. (2.37)-(2.37), but they now represent the rescaled quantities. All times

are rescaled by dividing by Tand all rates are rescaled by multiplying by T.R(t)

is as deﬁned in Eq. (2.23).

For all the numerical simulations we use the dimensionless form of the DDE

system and all for results that will be presented in the subsequent chapters, we will

refer to the dimensionless parameter values.

2.2.2. Parameter values

In this subsection we will brieﬂy discuss some of the ﬁnal parameters of the DDE

model (Eqs. (2.42)-(2.44)) and the values that will be used.

In an experiment the tunable parameters that determine the light output of a

passively mode-locked laser are the pump current that is injected into the gain

section and the reverse bias which is applied to the absorber section. In our ﬁnal

system of equations these control parameters are related to Jgin Eq. (2.43)and

Jqand γqin Eq. (2.44). Jgis related to the pump current, but not directly as it

referenced to the current needed to achieve transparency in the gain section, i.e. it

is proportional to the excess pump current. In the absorber section the applied bias

modiﬁes the energy barrier which needs to be overcome for carriers to escape the

quantum-well [MAL06d]. This inﬂuences the carrier recovery rate γqand hence the

carrier density about which the gain should be linearised. The carrier recovery rate

γq, the diﬀerential gain gqand the eﬀective transparency carrier density n0

qall enter

into the unsaturated absorption parameter Jq, meaning that the bias dependence

is manifested in Jqand γq.

In the subsequent chapters will we see that the amplitude-phase coupling factors,

αgand αq, can strongly inﬂuence the dynamics. These parameters are included

to account for carrier-induced refractive index changes and are proportional to

∂(δn)/∂N

∂g/∂N ,whereδn is the carrier-induced refractive index change and ∂g/∂N is the

diﬀerential gain [CHO99]. For quantum-well based gain media δn and gdepend

non-linearly on the carrier densities leading to carrier dependent α-factors. In the

model presented here these factors are assumed to be constant. We will partially

account for the carrier dependence of the α-factors by investigating scenarios where

the values are diﬀerent in the gain and absorber sections. However, our default

values for αgand αqwill be zero.

For the charge-carrier recovery rates we will use values that are typical for semi-

conductor quantum-well materials. In the gain section the carrier lifetimes can

2.3. Discussion of the DDE model

typically range from 0.1-1 ns [GOE83,HAD04,JON95b]. In the absorber section

the carrier lifetimes are lower due to the reverse bias which sweeps out the carri-

ers. Typical values which can be found in the literature are in the range 5-50ps

[KAR94]. However, in both cases the recovery times strongly depend on the struc-

ture of the gain material.

The parameter values that we will use throughout this thesis, unless stated oth-

erwise, are those given in Table 2.1. We choose a cold-cavity roundtrip time of

25 ps, which results in a repetition rate of about 40 GHz in the fundamental mode-

locking regime. For a linear cavity, this roundtrip time corresponds to a length of

about 1 mm, which is a typical length for a passively mode-locked semiconductor

laser [FIO11,FIO11a,ARS13].

parameter value dimensionless parameter value dimensionless

T25 ps 1 γ2.66 ps−166.5

γg1ns

−10.025 γq75 ns−11.875

Jg0.12 ps−13.0 Jq0.3ps

−17.5

rs25.025.0Cm00

κ0.1 0.1 ΔΩ 0 0

αg00 αq00

Table 2.1.: Parameter values used in numerical simulations, unless stated otherwise.

2.3. Discussion of the DDE model

The main assumptions made to be able to derive the DDE system (Eqs. (2.37)-

(2.39)) are that the laser has a ring cavity geometry and that the propagation

of light is unidirectional. This approach has been widely, and successfully, em-

ployed to study the dynamics of various laser systems. These include hybrid mode-

locked lasers [FIO10,ARK13], passively mode-locked lasers with optical injection

[REB11,PIM14,HAB14] or optical feedback [OTT12a,OTT14b,SIM14,JAU16]

and Fourier domain mode-locking [SLE13]. In many of these cases experimental

comparison is made with devices that have a linear cavity geometry and bidirec-

tional propagation. Despite this diﬀerence the DDE models have been shown to

reproduce the qualitative behaviour of the dynamics very well [PIM14,ARS13,

FIO10,NIK16]. There are however still some qualitative diﬀerences that will be

discussed in this section.

In [VLA09] the authors numerically study a travelling-wave model for a monolithic-

semiconductor passively mode-locked laser with a linear cavity geometry. The car-

rier and ﬁeld equations are the same as those used in the DDE model and spectral

ﬁltering is also included in a lumped element approach, i.e. the essential diﬀerence

2. Mode-locked laser model

between the DDE system of [VLA05] and the travelling-wave system of [VLA09]is

the laser-cavity geometry and hence that in the latter case there are two counter-

propagating ﬁelds. The bifurcation diagrams for these two systems are qualitatively

very similar. The main diﬀerence that arises is in the 2nd harmonic mode-locking

regime. In this regime there are two pulses travelling in the laser cavity. In the

linear-cavity model these two pulses collide, which leads to faster saturation of the

medium at the point of the collision. For the cavity model used in [VLA09], this

collision takes place in the gain section. This is unfavourable for mode-locking and

consequently leads to a break-up of the symmetry of the pulses (period doubling

bifurcation). The heights and separation of the pulses change such that the colli-

sion only occurs in the gain section every second roundtrip, thereby reducing the

saturation of the gain section. This is not observed in the DDE model as the two

pulses cannot interact.

In a linear cavity pulse interaction can also occur in the fundamental mode-

locking regime. As the pulse is reﬂected at the end facets there is an interaction

between the forward and backward moving parts of the pulse, this again leads to

faster saturation of the medium at the point of interaction. This eﬀect is utilised

in the so-called self-colliding-pulse mode-locking, where the saturable absorber is

placed near the high reﬂectivity facet to increase its modulation [JON95b]. Such

interaction eﬀects in the fundamental mode-locking regime are also not present in

the DDE model. There are however other factors related to the geometry of the

cavity, which have been shown to have a greater inﬂuence on the modulation of the

absorber section, that can be observed in the DDE model. In [JAV11] the eﬀect

of the positioning of the saturable absorber in a two section laser, i.e. whether or

not the absorber is located at the end of the cavity with the anti-reﬂection coated

facet, is studied using a travelling-wave model. Here the authors predict that the

output power of the pulses is increased, the pulse widths are decreased and the

amplitude and timing jitter is reduced, when the absorber is placed in front of

the anti-reﬂection coated facet. The reasoning for this is that despite a reduction

in the interaction of the counter-propagating ﬁelds due to the reduced reﬂectivity,

the power of the pulse impinging on the absorber from the gain section is now

much greater since there are less mirror losses at the end of the gain section. This

leads to faster saturation of the absorber [SIM13], which is known to improve the

mode-locking properties [HAU75a]. This dependence on the absorber position has

subsequently been demonstrated experimentally [MOS15,ZHU15a]. In the DDE

model similar eﬀects can be observed by placing the out-coupling losses before or

after the absorber section [ROS11d].

Depending on the heterostructure of the laser under investigation, reﬁnements

can be made to the DDE model by choosing more sophisticated models for the

carriers. In [VLA10], [ROS11d]and[SIM14] the respective authors have used

the DDE approach with charge-carrier equations tailored to quantum-dot lasers.

2.3. Discussion of the DDE model

However, comparisons between the results that will be presented in the subsequent

sections, and those present in [SIM14], show that the qualitative trends of the

feedback dependence are the same for the simple DDE we use (Eqs. (2.42)-(2.44))

and the more complicated quantum-dot based DDE models.

Aside from the ring-cavity geometry and the unidirectional propagation, there

are other simplifying assumptions that mean that the DDE model can only be

used to make qualitative not quantitative predicts for experiments. One of these

is the lumped-element approach to including the internal non-resonant losses. By

including the internal losses only at the ends of the gain and absorber sections the

electric ﬁeld is larger than it should be as it propagates through these sections,

which aﬀects the dynamics of the charge carriers. This can be improved upon by

dividing the gain and absorber in to multiple sections and including internal losses

between each section. The authors of [ROS11d] have done this for a quantum-dot

mode-locked laser and have shown that by adding more sections to the ring cavity

they obtained better agreement with a travelling-wave model for an equivalent

linear cavity. This is however at the expense of a greater computational cost, as

each added section requires an additional carrier equation.

Chapter 3

Mode-locked laser dynamics

3.1. Introduction

In this chapter the dynamics of a passively mode-locked semiconductor laser will

be studied using the model presented in Chapter 2. The dynamics are stud-

ied via direct numerical simulations and by performing numerical path continu-

ation to analyse the bifurcations and stability of the solutions of the DDE model

(Eqs. (2.42)-(2.44)). The path continuation is carried out using the software pack-

age DDE-BIFTOOL [ENG01]. Throughout this chapter we will be considering the

underlying deterministic dynamics of the DDE system, i.e. Rsp =0.

This chapter is structured as follows. First a brief introduction to the types of

bifurcations that will be encountered will be given in Subsection 3.1.1.Thenthe

dynamics of the solitary laser will be reviewed in Section 3.2. In Sections 3.3 and

3.4 the dynamics arising due to optical feedback from one and two external cavities,

respectively, will be investigated.

3.1.1. Bifurcations

A bifurcation occurs when there is a sudden qualitative change in the behaviour of

a dynamical system, which is brought on by the smooth change of some parameter

of that system. There are two classes of bifurcations, local bifurcations and global

bifurcations. In the following analysis of the mode-locked laser system (Eqs. (2.42)-

(2.44) we will only encounter local bifurcations. These can be characterised by

investigating the stability of solutions near the bifurcation point. The stability of

a solution is in turn characterised by the eigenvalues of the linearised system1,or

1A DDE system is linearised as demonstrated for the following two dimensional example with

one delay time. Let us consider a system of the form

x(t)=F(x(t),x(t−τ)) ,

3. Mode-locked laser dynamics

(a)

Supercritical Hopf

amplitude

Subcritical Hopf

amplitude

(b)

(c)

unstable

equilibrium

stable

equilibrium

unstable limit cycle

stable limit cycle

stable

equilibrium

unstable

equilibrium

Figure 3.1.: Hopf bifurcation: (a) Complex conjugate eigenvalues λpassing through the real

axis in a Hopf bifurcation of a steady state solution. Bifurcation diagram of a supercritical (b)

and subcritical (c) Hopf bifurcation in dependence of some bifurcation parameter k.

in case of periodic systems by the Floquet multiplers or Floquet exponents (see

Appendix A).

Here we give a brief overview of the bifurcations that will be encountered in the

DDE model (Eqs. (2.42)-(2.44)). For a rigorous introduction to bifurcation theory

see, for example, [KUZ95,GUC86].

Hopf and torus bifurcations

In a Hopf (Andronov-Hopf) bifurcation a limit cycle (periodic solution) is generated

out of a steady state (equilibrium) solution of a dynamical system. At the bifurca-

tion point the equilibrium changes stability via a pair of purely imaginary complex

conjugate eigenvalues (Fig. 3.1a). The limit cycle that is created can be either

stable or unstable depending on the criticality of the bifurcation. In a supercritical

Hopf bifurcation a stable equilibrium becomes unstable and a stable limit cycle is

generated (Fig. 3.1b). In a subcritical Hopf bifurcation an unstable equilibrium

stabilises and an unstable limit cycle is generated (Fig. 3.1c). The frequency of the

limit cycle generated in the Hopf bifurcation is given by the imaginary eigenvalues

at the bifurcation point.

where x=(x1,x

2)Tare the dynamical variables and F=(F1,F

2)Tare functions of the

dynamical variables xat time tand at the delayed time t−τ. Then the linearised system,

about a ﬁxed point x0, is given by

δx(t)=Aδx(t)+Bδx(t−τ),

where δx=x−x0and the Jacobi matrices are deﬁned as Aij =∂Fi

∂xj(t)and Bij =∂Fi

∂xj(t−τ)

evaluated at x0, for i, j ={1,2}.

3.1. Introduction

(

a) (b)

Figure 3.2.: Torus bifurcation: (a) Complex conjugate characteristic multipliers μpassing

through the unit circle (|μ|= 1) in a torus bifurcation of a limit cycle. (b) Motion on a

two-dimensional invariant torus.

A torus bifurcation, also referred to as a secondary Hopf bifurcation, is the

bifurcation of a limit cycle in which a two-dimensional invariant torus is created.

At the bifurcation point the limit cycle changes stability via two complex conjugate

characteristic (Floquet) multipliers passing through the unit circle (Fig. 3.2a). Like

the Hopf bifurcation, the torus bifurcation can be supercritical or subcritical. The

periodicity of the motion on the torus depends on the frequency of the original

limit cycle (ω1) and the frequency that is introduced in the torus bifurcation (ω2)

(Fig. 3.2b). If the ratio of these two frequency is rational then the motion on

the torus will be periodic. However, if the frequencies are incommensurable the

motion can be quasi-periodic, which means that the trajectory completely ﬁlls in

the surface of the torus.

Saddle-node bifurcation

In a saddle-node bifurcation two equilibria (or limit cycles) of a dynamical system

collide and disappear. The bifurcation is characterised by one real eigenvalue pass-

ing through zero, or in the case of a saddle-node bifurcation of limit cycles, one

real characteristic multiplier passing through unity (Fig. 3.3). A typical scenario

involving a saddle-node bifurcation is depicted in Fig. 3.4. Here an unstable limit

cycle is ﬁrst created in a subcritical Hopf bifurcation. Following the limit cycle

solution from the Hopf bifurcation point, the bifurcation parameter kdecreases

until the solution becomes stable in a saddle-node bifurcation and folds back in k

to continue to exist for larger kvalues. Due to this folding of the solution as a

function of the bifurcation parameter, saddle-node bifurcations are also referred to

as fold bifurcations.

3. Mode-locked laser dynamics

(a)

(b)

Figure 3.3.: Saddle-node bifurcation: (a) One real eigenvalue λpassing through zero in a

saddle-node bifurcation of equilibria. (b) One real characteristic multiplier μpassing through

unity (Re μ= 1) in a saddle-node bifurcation of limit cycles.

mplitude

stable limit cycle

stable

equilibrium

unstable

equilibrium

unstable

limit cycle

subcritical Hopf

saddle-node

Figure 3.4.: Bifurcation diagram showing a subcritical Hopf bifurcation followed by a saddle-node

bifurcation of limit cycles.

Period-doubling bifurcation

A period-doubling bifurcation of a limit cycle involves the creation of a second limit

cycle which has approximately twice the period of the original limit cycle. This

occurs by a change in the height of the maxima of the limit cycle, as illustrated in

Fig. 3.5. This bifurcation can be characterised by one real characteristic multiplier

passing though the negative side of the unit circle, i.e. μ=−1 (see Fig. 3.6a).

Period-doubling bifurcations also come in subcritical and supercritical variants.

In a supercritical period-doubling bifurcation a stable limit becomes unstable and

a second stable limit cycle is born. This scenario is depicted in Fig. 3.6b. In

the subcritical period-doubling bifurcation an unstable limit cycle becomes stable

and a second unstable limit cycle is created, as depicted in Fig. 3.6c. When the

bifurcation point is approached from the side of limit cycle with twice the period,

3.1. Introduction

amplitude

time time

amplitude

Figure 3.5.: Illustration of a limit cycle before and after a period-doubling bifurcation.

(

Supercr

per

bli

maxima

Subcritical period-doubling

maxima

(b)

(c)

unstable limit cycle

stable limit cycle

unstable limit cycle

Figure 3.6.: Period-doubling bifurcation: (a) One real characteristic multipliers passing

through Re μ=−1 in a period-doubling bifurcation. Bifurcation diagram of a supercritical

(b) and subcritical (c) period-doubling bifurcation in dependence of some bifurcation parameter

which then disappears at the bifurcation point, then this bifurcation is also referred

to as a period-halving bifurcation.

Pitchfork bifurcation

In a pitchfork bifurcation one steady state solutions changes stability and two

additional solutions are created. A pitchfork bifurcation can be subcritical or su-

percritical and the bifurcation diagram has the same form as those depicted in

Fig. 3.6b-c. The diﬀerence being that in this case the two bifurcating branches

belong to diﬀerent steady state solutions instead of one limit cycle solution.

3. Mode-locked laser dynamics

3.2. Solitary mode-locked laser dynamics

In this section we will explore the dynamics of the passively mode-locked laser

in the absence of feedback. We will largely be recapping the results of [VLA05,

VLA11] which are necessary for understanding the dynamics of the system with

feedback in the subsequent sections. We will be addressing the bifurcations of the

continuous wave solutions of Eqs. (2.42)-(2.44) (with Kn= 0 and Rsp = 0), as well

as the bifurcations of the mode-locked solutions. Further, we will investigate the

dependence of the mode-locked solutions on several key parameters by numerical

integration of the DDE system.2

3.2.1. Lasing threshold

For a semiconductor mode-locked laser to start lasing, a suﬃciently large pump

current must be applied such that all losses can be overcome. Below this threshold

pump current, the laser is in the oﬀ state. The oﬀ state is a steady state solution

of Eqs. (2.42)-(2.44) and is given by

E=0,G=Jg/γgand Q=Jq/γq,(3.1)

where the solutions for Gand Qare obtained by setting the derivatives in Eqs. (2.43)-

(2.44) to zero. At the threshold pump current the oﬀ state becomes unstable and

lasing begins. The simplest lasing states are continuous wave solutions of the form

E(t)=E0eiωt,whereωis referenced to the optical frequency Ω0. Substituting this

continuous wave solution into Eq. (2.42) yields

ω2+γ2=γ2κeG−Q(3.2)

and ω

γ=−tan 1

2(αgG−αqQ) + (ΔΩ + ω)T(3.3)

[VLA05]. Here Gand Qare steady state solutions of Eqs. (2.43)and(2.44), and

are given by the equations

0=Jg−γgG−e−QeG−1|E0|2(3.4)

and

0=Jq−γqQ−rse−QeQ−1|E0|2.(3.5)

This set of transcendental equations has multiple solutions for the frequency ω

and the intensity |E0|2, which correspond to diﬀerent lasing modes. Modes with

2In certain limits the DDE model can also be analysed analytically, this was done in [VLA05,

VLA11]. Also present in these works are comparisons with the models of Haus [HAU75a] and

New [NEW74].

3.2. Solitary mode-locked laser dynamics

increasing detuning from the center of the spectral ﬁlter, i.e. the gain maximum of

the lasing medium, have a smaller eﬀective gain and hence the threshold current

is mode dependent. At the lasing threshold of each mode, Eqs. (3.2)-(3.5)must

be fulﬁlled with E0= 0. From Eqs. (3.2), (3.5)and(3.4) the mode dependent

threshold current is

Jth

g(ω)=γgJq

γq−ln (κ)+ln

1+ω2

γ2 (3.6)

where ωmust still fulﬁl Eq. (3.3).

The mode with the smallest absolute detuning from the center of the spectral

ﬁlter, ωmin, has the lowest threshold current, and it is at this threshold current that

the oﬀ state becomes unstable. All other modes bifurcate from the oﬀ state when it

is already unstable, and are themselves also unstable. For the bifurcation analysis

that follows we choose ΔΩ, αgand αqequal to zero. In this case the maximum

gain continuous wave solution has ω=ωmin = 0. The mode-locked solutions that

are of interest bifurcate from this continuous wave solution.

3.2.2. Continuous wave solutions

Using the path continuation software package DDE-BIFTOOL [ENG01]wein-

vestigate the pump current dependence of the continuous wave solution that is

generated at the lasing threshold (Eq. (3.6) with ω=ωmin). Figure 3.7 shows the

results for this solution. In Fig. 3.7a the electric ﬁeld amplitude E0(red line), the

gain G(green line) and the losses Q(blue line) are plotted. The electric ﬁeld in-

tensity I=|E0|2depends linearly on the pump current, hence E0has a square-root

dependence. The stability of the solutions is indicated by the thickness of the lines

in Fig. 3.7a; stable solutions are plotted with thick lines. A close-up of the current

region near threshold, which is Jth

g=0.153 for the parameters used here, is shown

Fig. 3.7b. The grey region indicates the current range for which the oﬀ state is sta-

ble. Here we see that the continuous wave solution is initially stable (as it is born

in a supercritical Hopf bifurcation), but loses stability slightly above the threshold

current. At relatively large currents (Jg>60Jth

g) the continuous wave solution

restabilises. Within the current range where the solution is unstable, the number

of unstable eigenmodes varies. In Fig. 3.7c-d the real parts of eigenvalues are plot-

ted. Since this is a DDE system there are inﬁnitely many eigenmodes, however

only a ﬁnite number of these can have eigenvalues above any given ﬁnite threshold

value. Here we only depicted those with Re[λ] close to zero. Those above zero are

plotted in red to indicate that these correspond to unstable eigenmodes. In this

case each line that crosses Re[λ] = 0 corresponds to a Hopf bifurcation, meaning

that a pair of complex conjugate eigenvalues crosses the real axis. In Fig. 3.8 this

is shown for the Hopf bifurcation (H1) between the vertical dashed blue lines in

3. Mode-locked laser dynamics

0 2 4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

|E0|,G/4,Q/4

(a)

|E0|G/4Q/4

0 2 4 6 8 10 12

Pump current Jg

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

Re[λ]

(c)

0.15 0.25

(b)

0.15 0.25

(d)

Figure 3.7.: (a) Electric amplitude E0(red line), gain G(green line) and losses Q(blue line) of

the maximum gain continuous wave solution in dependence of the pump current Jg. A close-up

of the current region near threshold is shown in (b). Thick lines indicate where the continuous

wave solution is stable and the vertical black lines indicate the pump current values where the

solution changes stability. In the grey region the oﬀ state (Eq. (3.1)) is stable. (c) Real parts

of the eigenvalues λcorresponding to the solutions plotted in (a). Eigenvalues with Re[λ]>0

are plotted in red. In (d) a close-up of the eigenvalues near threshold is shown. Parameters:

κ=0.121, ω=ωmin = 0, all other parameters as in Table 2.1.

−0.06 −0.04 −0.02 0.00 0.02

Re[λ](2π/T)

−10

−5

Im[λ](2π/T)

(a)

−0.06 −0.04 −0.02 0.00 0.02

Re[λ](2π/T)

(b)

Figure 3.8.: Eigenvalues of the maximum gain continuous wave solution for the pump current

values indicated by the left (a) and right (b) vertical dashed blue lines in Fig.3.7c. Parameters:

ω=ωmin = 0, (a) Jg=5.9, (b) Jg=6.1, κ=0.121, all other parameters as in Table 2.1.

Fig. 3.7c. Plotted in Fig. 3.8a-b are the real and imaginary parts of the eigenvalues

3.2. Solitary mode-locked laser dynamics

0 2 4 6 8 10 12

stable cw

(a)

0.1 0.2 0.3

stable

off

state

(b)

Figure 3.9.: (a) Hopf bifurcation curves in the Jg-Jqplane. A close-up of the region near the

threshold pump current is shown in (b). In the light grey regions the continuous wave (cw)

solution is stable and in the dark grey region the oﬀ state (Eq. (3.1)) is stable. Parameters:

ω=ωmin =0,κ=0.121, all other parameters as in Table 2.1.

before and after the Hopf bifurcation. Comparing subplots (a) and (b) of Fig. 3.8

it can be seen that one pair of eigenvalues changes stability (stable eigenvalues are

plotted in green and unstable ones in red).

Post fabrication, the tunable parameters of a passively mode-locked semicon-

ductor laser are the pump current and the bias voltage. As discussed in Subsec-

tion 2.2.2, the inﬂuence of the bias voltage is largely manifested in the unsaturated

absorption Jq. Hence Jqcrucially inﬂuences the operation of the laser. In Fig. 3.9

the stability and bifurcations of the maximum gain continuous wave solution are

shown in the Jg-Jqplane. In the light grey areas the continuous wave solution is

stable and in the dark grey area the oﬀ state is stable. The blue curves indicate the

Hopf bifurcations. The solutions presented in Fig. 3.7 correspond to the position

of the horizontal dashed line in Fig. 3.9a. The period of the limit cycle that is

generated in each of the Hopf bifurcations is determined by the purely imaginary

eigenvalues at the bifurcation point. The Hopf curve labelled H1generates a limit

cycle with a frequency approximately equal to the fundamental frequency of the

laser cavity Ωf=2π/T (frequency separation of the cavity modes, i.e. Ωf=2πΔν

in Fig. 2.3). This can be seen in Fig. 3.8, which shows eigenvalues for solutions be-

fore and after the H1Hopf bifurcation. Comparing subplots (a) and (b) of Fig. 3.8,

it can be seen that Im[λ]≈2π/T for the eigenvalues that cross the real axis. It

is out of H1Hopf bifurcation that the fundamental mode-locked solution develops.

The Hopf curves labeled Hn,forn=2,3,4, generate limit cycles with frequencies

given approximately by nΩf. These limit cycles evolve into harmonically mode-

3. Mode-locked laser dynamics

locked solutions. Lastly, the Hopf curve labeled HQhas a much lower frequency,

approximately an order of magnitude lower than Ωffor the chosen parameters.

This bifurcation is related to the Q-switching found in single-mode lasers with a

saturable absorber [YAM93,DUB99a,LUE11b].

The general trends shown in Fig. 3.9 are: For increasing unsaturated absorption

Jqthe stability range of the continuous wave solution decreases and the number

of Hopf bifurcations the system goes through increases. The additional Hopf bi-

furcations that are introduced at higher Jqvalues produce higher frequency limit

cycles. The stability of the limit cycles that are produced in the Hopf bifurcations

depend on the criticality of the bifurcation and on the stability of the solution it

is bifurcating from. The Hopf curves in Fig. 3.9 do not contain information about

the criticality, however this can be obtained by investigating the stability and bi-

furcations of the emerging limit cycles. For the fundamental Hopf curve H1this is

done in the next section.

3.2.3. Mode-locked solutions

As indicated by the multitude of Hopf bifurcations depicted in Fig. 3.9, the DDE

system for the mode-locked laser has several periodic solutions. In this section we

will focus on the fundamental mode-locked solution. The stability and bifurcations

of this solution in the Jg-Jqplane are depicted in Fig. 3.10. Torus and saddle-node

bifurcations of the fundamental periodic solution are indicated by blue and green

lines, respectively. The H1Hopf curve out of which the fundamental mode-locked

solution emerges is plotted in black. This is the same curve that is shown in Fig. 3.9,

however now the criticality is also shown. In the dashed regions the H1bifurcation

is subcritical and in the solid range it is supercritical. The change in criticality

of the H1curve occurs at the point where the H1and saddle-node branches meet

(green circle). The supercritical part of H1and the saddle-node branch enclose the

parameter region where the fundamental periodic solution exists, and the green

shaded part of this region is where this solution is stable. The Hopf curves HQ

and H2-H4are depicted by the thin grey lines. The points where these branches

intersect the H1curve are Hopf-Hopf points and are indicated by the red circles.

A Hopf-Hopf point is where two branches of Hopf bifurcations intersect. Generally

two branches of torus bifurcations emanate from a Hopf-Hopf point. Here we only

show the torus bifurcations of the periodic solution generated in the H1bifurcation.

Hence there are four branches of torus bifurcations depicted in Fig. 3.10,which

emanate from the Hopf-Hopf points marked by the red circles.

To better understand this bifurcation diagram it is helpful to consider cuts along

Jg. Starting at the low current part of the H1curve, the fundamental periodic

solution is generated in a supercritical Hopf bifurcation. If Jqis below the Hopf-

Hopf point where the HQcurve intersects H1, then the solution that is generated is

3.2. Solitary mode-locked laser dynamics

0246810

stable FML

stable cw

Figure 3.10.: Bifurcations of the fundamental mode-locked solution in the Jg-Jqplane. Blue lines

is indicate Torus bifurcations (T1-T4) and the green line is a saddle-node bifurcation (SN). The

black line is the fundamental Hopf bifurcation curve of the continuous wave solution (H1),

with the solid segment being supercritical and the dashed segment subcritical. The green circle

marks the point where the H1curve changes criticality. The thin grey lines show the remaining

Hopf bifurcations of the continuous wave solution (compare with Fig. 3.9). Hopf-Hopf points

involving the H1curve are indicated by red circles. Shaded in green is the stability region of

the fundamental mode-locked (FML) solution. In the light grey regions the continuous wave

solution is stable and in the dark grey region the oﬀ state (Eq. (3.1)) is stable. The black cross

indicates the position of the parameters corresponding to Table 2.1. Parameters: κ=0.121,

ω=ωmin = 0, all other parameters as in Table 2.1.

stable. However if Jqis above this point then the solution is unstable, even though

the bifurcation is supercritical, because the continuous wave solution is already

unstable when the H1bifurcation occurs (see Fig. 3.9 (b)). In the latter case, if

Jgis increased, the solution ﬁrst becomes stable after the torus bifurcation T4.

Increasing the pump current further, the solution remains stable until the saddle-

node bifurcation. At this point the solution coalesces with the unstable periodic

solution that is generated in the subcritical part of the H1curve. This is depicted in

Fig. 3.11 for the unsaturated absorption Jqvalue indicated by the horizontal dotted

black line in Fig. 3.10. In Fig. 3.11a the electric ﬁeld maxima are plotted and in

Fig. 3.11b the period of the solutions is shown. Stable solutions are indicated by

the green symbols in Fig. 3.11a and by the solid line in Fig. 3.11b. Also indicated

in Fig. 3.11b are the positions of the bifurcations, which have the same labelling

as in Fig. 3.10. The solution generated in the subcritical H1bifurcation can have

3. Mode-locked laser dynamics

0.0

1.0

2.0

3.0

4.0

5.0

|E|max

(a)

0 1 2 3 4 5 6 7 8 9

Pump Current Jg

0.92

0.96

1.00

1.04

Period T0

(b) H1H1

Figure 3.11.: Branch of periodic solutions continued from the H1curve, corresponding to the

dotted black line in Fig.3.10. Electric ﬁeld maxima |Emax|are plotted in (a) and the period T0is

plotted in (b). Stable section of the branch are plotted in green (solid lines) and unstable in red

(dotted lines). In (b) the positions of the Hopf, torus and saddle-node bifurcations are indicated

by the black, blue and green circles, respectively. The red crosses correspond to Fig. 3.12a-c.

Parameters: Jq=9.58, κ=0.121, all other parameters as in Table 2.1.

multiple unstable directions, the number of which depend on Jq. These added

unstable directions are related to the stability of the continuous wave solution at

the point of the H1bifurcation (e.g. see Fig. 3.8). Starting from the subcritical H1

bifurcation at about Jg=6.9 in Fig. 3.11, the solution is unstable and has relatively

small maxima. First there are three pairs of complex conjugate multipliers with

|μ|>0, because the continuous wave solution has an additional three pairs of

unstable eigenvalues at the bifurcation point. This is shown in Fig. 3.12a. As

the periodic solution is continued one real multiplier, which was equal to one at

the bifurcation, becomes larger than one (Fig. 3.12b). Then, as Jgis increased

further the height of the maxima increases and the solution goes through a series

of torus bifurcations (T1-T3) in which unstable pairs of multipliers enter the unit

circle. Figure 3.12c shows the multipliers for a solution between T2and T3at

Jg=8.45 and compared with Fig. 3.12b two pairs of multipliers have entered the

unit circle. After the T3bifurcation there is only one real characteristic multiplier

with |μ|>1. Following the three torus bifurcations, as the solution is continued

further this branch folds over in Jgand the solutions becomes stable after the

saddle-node bifurcation. As the branch is continued further, Jgdecreases and

eventually the solutions lose stability after the torus bifurcation T4. Finally, the

solution disappears in the H1bifurcation near Jg=0.2.

Near the H1bifurcations, in which the fundamental periodic solution is gener-

ated, the oscillations are initially sinusoidal with a very small amplitude. As the

3.2. Solitary mode-locked laser dynamics

−1.0−0.5 0.0 0.5 1.0

Re[μ]

−1.0

−0.5

0.0

0.5

1.0

Im[μ]

(a)

−1.0−0.5 0.0 0.5 1.0

Re[μ]

(b)

−1.0−0.5 0.0 0.5 1.0

Re[μ]

(c)

Figure 3.12.: Characteristic multipliers of the fundamental periodic solution bifurcating from the

subcritical part of H1at Jq=9.58. The solutions correspond to the red crosses in Fig. 3.11b.

Parameters: (a) Jg=6.9, (b) Jg=7.47, (c) Jg=8.45, κ=0.121, all other parameters as in

Table 2.1.

0.0 0.2 0.4 0.6 0.8 1.0

Time (T0)

|E| Proﬁle

Figure 3.13.: Electric ﬁeld proﬁles for the stable mode-locked solutions of the branch depicted in

Fig. 3.11. The time axis spans one period T0of the solutions. Parameters: Jq=9.58, κ=0.121,

all other parameters as in Table 2.1.

solution is continued it then evolves into the mode-locked solution. The pulse pro-

ﬁles, for the stable section of the branch depicted in Fig. 3.11, are shown in Fig. 3.13.

As the current is increased the pulse amplitude increases (also see Fig. 3.11a), the

pulses become wider and the period T0decreases (Fig. 3.11b). Between pulses

the electric ﬁeld amplitude drops to zero. Physically these pulsed solutions, and

their dependence on the pump current, can be understood by considering the gain

and absorber dynamics. Figure 3.14 shows an example of the dynamics of the

gain, losses and electric ﬁeld amplitude in the mode-locked regime. Shown here

are time traces of the electric ﬁeld amplitude (blue), the gain (green) and the total

losses Qtot =Q−ln (κ) (red), which are the sum of the absorber losses and the

non-resonant internal and mirror losses. Also shown is the net gain G=G−Qtot

(black). From this time trace it is evident that the dynamics evolve in two stages;

3. Mode-locked laser dynamics

0 5 10 15 20 25 30 35 40

Time (ps)

−1

|E|,G/2,Q

tot/2,G/2

(a)

|E| G/2Qtot/2G/2

5.2 5.6 6.0

Time (ps)

0.0

0.5

1.0

1.5

2.0

2.5

(b)

Figure 3.14.: (a) Time trace of the electric ﬁeld amplitude |E| (blue), gain G(green), total losses

Qtot =Q−ln (κ) (red) and the net gain G=G−Qtot (black). (b) Close-up of the region enclosed

in the grey box in (a). This solution corresponds to the black cross in Fig. 3.10. Parameters:

κ=0.121, all other parameters as in Table 2.1.

0 50 100 150 200 250 300

Spectral Filter Width γ

1.00

1.03

1.04

1.06

(1) (2) (3) (4) (5)

(a)

(b)

0.0

2.0

4.0

6.0

8.0

|E|

(1) (2)

Time (T0)

(3) (4) (5)

Figure 3.15.: (a) Period T0and (b) pulse proﬁles of the fundamental mode-locked solution as a

function of the Lorentzian spectral ﬁlter width. The pulse proﬁles are plotted over one period.

Parameters: Jg=5.01, Jq=9.58, κ=0.121, all other parameters as in Table 2.1.

a slow stage and a fast stage. The fast stage occurs when the pulse enters the

gain and absorber media. As the leading edge of the pulse enters the gain section

the inverted charge carriers are depleted, this causes Gto decrease. In the ab-

sorber section the leading edge of the pulse is absorbed which causes the creation

of electron-hole pairs and drives the absorber towards transparency, meaning that

the losses Qtot decrease. The rates at which the gain and absorber sections saturate

is determined by the linear gain coeﬃcients ggand gq, respectively. Since gq>g

the absorber saturates faster than the gain section, which leads to a window of

positive net gain (shaded region in Fig. 3.14b). This net gain window allows the

center of the pulse to be ampliﬁed. After the pulse has passed, Gand Qrecover

3.2. Solitary mode-locked laser dynamics

0 5 10 15 20 25 30 35 40

Time (ps)

−1

|E|,G/2,Q

tot/2,G/2

(a) |E| G/2Qtot/2G/2

2 4 6 8 10

Time (ps)

0.0

0.5

1.0

1.5

2.0

2.5

(b)

Figure 3.16.: (a) Time trace of the electric ﬁeld amplitude |E| (blue), gain G(green), total losses

Qtot =Q−ln (κ) (red) and the net gain G=G−Qtot (black). (b) Close-up of the region

enclosed in the grey box in (a). Parameters: Jq=3,κ=0.121, all other parameters as in

Table 2.1.

in the slow stage, at time scales given by γgand γq. The absorber recovery rate

is faster than that of the gain section, which leads to the losses dominating during

the slow stage and hence to the electric ﬁeld amplitude being close to zero between

pulses. The slow and fast stages can, under certain approximations, be described

analytically due to the separation of the dominating time scales [VLA05].

The pulse repetition period is slightly longer than the cold cavity roundtrip

time; in the example depicted in Fig. 3.14 the period is T0=1.014T.Thisisa

consequence of the response time of the spectral ﬁlter, and the time it takes for

the absorber to saturate, i.e. the leading edge of the pulse is absorbed causing

the eﬀective pulse group velocity to be smaller than v. If the leading edge of the

pulse has a higher intensity then the absorber takes less time to saturate, causing a

decrease in T0. The intensity of the leading edge of the pulse is determined by the

pulse energy and the pulse width. As the pump current is increased the pulse energy

increases and hence the period decreases (stable section of Fig. 3.11b). The pulse

width is dependent on the number of modes participating in the mode-locking,

which is determined by the width of the spectral ﬁlter. The laser cavity mode

spacing is T−1, therefore the number of lasing modes can be estimated as γT−1.

If γT−1is large the pulse width is given approximately by γ−1[VLA05,VLA11].

Therefore, for increasing γ, the period decreases due to the decreased ﬁlter response

time and due to the increased peak intensity. The dependence of the pulse proﬁle

and the period T0on the spectral ﬁlter width is illustrated in Fig. 3.15.

In the example depicted in Fig. 3.14 the net gain is negative during the entire

slow stage. According to New’s stability criterion, this is a necessary condition for

3. Mode-locked laser dynamics

0 2 4 6 8 10

Pump Current Jg

0.0

1.0

2.0

3.0

4.0

5.0

|E|max

up-sweep

down-sweep

FML 2ndHML

3rdHML

Figure 3.17.: Numerical bifurcation diagram showing the pump current dependence of the mode-

locked laser output. Plotted in blue (red) are electric ﬁeld maxima |E|max for an up-sweep

(down-sweep) in the initial conditions. The dotted white line indicates the stable fundamental

mode-locked (FML) solution. The dotted grey and black lines indicated 2nd and 3rd order

harmonic mode-locking (HML), respectively. Parameters: κ=0.121, all other parameters as in

Table 2.1.

the background to be stable against perturbations [NEW74].3There are however

other mode-locked solutions for which New’s criterion is not fulﬁlled. Figure 3.16

shows such an example for Jq= 3. Here the positive net gain window is open before

the arrival of the pulse. In this case, this behaviour is caused by the relatively low

unsaturated absorption Jq. Because Jqis small the absorber is fully recovered

before the pulse returns, whereas the gain is still recovering. Within this region

of positive net gain the system can be very sensitive to perturbations as these

can be ampliﬁed. However, the authors of [VLA11] have shown that mode-locked

solutions are not necessarily unstable against noise perturbations if New’s criterion

is violated. The relative group velocities of the pulse and the perturbations also

play a role in determining the stability of the background. Background stability

will be important for the investigation of the DDE system subject to noise, which

is presented in Chapter 4.

Outside the current range where the fundamental mode-locked solution is sta-

ble, a mode-locked laser can display various dynamics. We investigate these via

direct numerical integration of Eqs. (2.43)-(2.44).4Figure 3.17 showsanumer-

ical bifurcation diagram of the pump current dependence. Plotted are maxima

3What we mean by "stable against perturbations" is whether the solutions still exist in the

presence of noise, or if the dynamics are signiﬁcantly changed when a small noise term is

added.

4For the numerical simulation of the DDE system (Eqs. (2.42)-(2.44)) we use the Euler inte-

gration method with a time step of h=10

−4. For the initial conditions we generally use

(ER,EI,G,Q)=(0.4,0,4,1) over the entire time interval [−τ,0], where ERand EIare the real

3.2. Solitary mode-locked laser dynamics

970 975 980 985 990 995 1000

0.0

1.0

2.0

3.0

|E|

(a)

990 992 994 996 998 1000

(b)

990 992 994 996 998 1000

0.0

1.0

2.0

3.0

|E|

(c)

990 992 994 996 998 1000

(d)

970 975 980 985 990 995 1000

Time (T)

0.0

1.0

2.0

3.0

|E|

(e)

970 975 980 985 990 995 1000

Time (T)

(f)

Figure 3.18.: Time traces of the mode-locked laser output for various pump currents. Parameters:

(a) Jg= 1, (b) Jg= 3, (c) Jg= 6, (d) Jg=7.5, (e) Jg=8,(f)Jg=9,κ=0.121, all other

parameters as in Table 2.1.

of the electric ﬁeld amplitude which are collected over a time span of one hun-

dred cold-cavity roundtrip times. The results plotted in blue were obtained by

performing an up-sweep in the initial conditions. This means that the history of

the previous current step was used as the initial conditions for the next, as the

current was incrementally increased. In the down-sweep (red symbols) the cur-

rent was incrementally decreased. The idea behind such an approach is to stay on

one solution branch as long as possible in regions of multistability. In regions of

Fig. 3.17 where the red and blue symbols do not overlap the system has landed on

diﬀerent solutions for the up- and down-sweeps, i.e. these are regions of bistability

or multistability. For the current values where only one maxima is plotted the

system is exhibiting mode-locked dynamics. The regions of fundamental, 2nd and

3rd order harmonic mode-locking are indicated by the white, grey and black dotted

lines, respectively. At low currents, and at high currents, there are solutions with

varying pulse heights. Examples of the electric ﬁeld amplitude dynamics in each of

the qualitatively diﬀerent regions of Fig. 3.17 are shown in Fig. 3.18. Starting at

low pump currents the system is in a regime of Q-switched mode-locking. These

dynamics arise out of the T4torus bifurcation of the fundamentally mode-locked

solution (see Fi.g 3.10). In this regime the amplitude of the mode-locked pulses is

and imaginary components of the complex electric ﬁeld amplitude Eand τis the longest total

delay time.

3. Mode-locked laser dynamics

strongly modulated on time scales of the Q-switching frequency (frequency of the

HQHopf bifurcation) and the pulses appear in bunches, with a near zero electric

ﬁeld amplitude for several laser cavity roundtrips between the pulse bunches (see

Fig. 3.18a). Such Q-switched solutions arise at low currents because they are en-

ergetically favourable. As the current is increase past the torus bifurcation (near

Jg= 2), fundamental mode-locking becomes stable (Fig. 3.18b). In agreement with

these results, Q-switched mode-locking and the transition to fundamental mode-

locking has been observed experimentally in both quantum well and quantum dot

semiconductor passively mode-locked lasers [PAL91,BAN06,KUN07a,VLA10].

As the current is increased further, 2nd and 3rd harmonic mode-locking also be-

come stable. These are mode-locked regimes with repetition frequencies that are

integer multiples of the fundamental repetition rate, i.e. in the 2nd and 3rd har-

monic cases there are two (see Fig. 3.18c) and three (see Fig. 3.18d) peaks per

laser cavity roundtrip, respectively. Multistability between diﬀerent orders of har-

monic mode-locking has also been observed experimentally, particularly in devices

with relatively long passive sections [SAN90,MAR14c]. An example of harmonic

mode-locking in a semiconductor ring-cavity laser was demonstrated in [HOH93].

At relatively high currents the harmonic mode-locked solutions break-up and the

system exhibits chaotic oscillations of the peak intensity. The break-up of harmonic

mode-locking occurs at high currents because the gain and absorber do not have

time to recover between pulses, causing an interaction between subsequent pulses

[NIZ06,VLA11].

Experimentally it is typically the power spectrum of the laser output that is

measured. For easier comparison with experimental results, the power spectra

corresponding to the time traces of Fig. 3.18 are shown in Fig. 3.19. The funda-

mental repetition frequency of the simulated laser is approximately νf= 40GHz

((T0·25ps)−1), accordingly for fundamental mode-locking there is a peak in the

power spectra near 40 GHz and at higher harmonics (Fig. 3.19b). For Q-switched

mode-locking additional peaks are present with a frequency separation correspond-

ing to the modulation frequency (Fig. 3.19a). The power spectra of the harmoni-

cally mode-locked solutions show only peaks at multiples of nνf,wheren=2,3is

the harmonic order (see Fig. 3.19c-d).

Inﬂuence of amplitude-phase coupling

So far we have investigated the dynamics in the absence of amplitude-phase cou-

pling. However in semiconductor quantum-well materials α-factor values are typi-

cally in the range 2-6 [HAR83,OSI87,FOR07]. Furthermore, it is know from other

laser systems in which the phase of the light plays a role, that the amplitude-phase

coupling can strongly inﬂuence the behaviour, generally leading to more complex

dynamics. For example in optically injected lasers the α-factor aﬀects the extent

3.2. Solitary mode-locked laser dynamics

0 20 40 60 80 100 120

−80

−60

−40

−20

S|E|2(dBc)

(a)

0 20 40 60 80 100 120

(b)

0 20 40 60 80 100 120

−80

−60

−40

−20

S|E|2(dBc)

(c)

0 20 40 60 80 100 120

(d)

0 20 40 60 80 100 120

Frequency (GHz)

−80

−60

−40

−20

S|E|2(dBc)

(e)

0 20 40 60 80 100 120

Frequency (GHz)

(f)

Figure 3.19.: Power spectra of the electric ﬁeld amplitude, S|E|2, of the mode-locked laser output

corresponding to the dynamics depicted in Fig. 3.18. Parameters: (a) Jg= 1, (b) Jg= 3, (c)

Jg= 6, (d) Jg=7.5, (e) Jg=8,(f)Jg=9,κ=0.121, all other parameters as in Table 2.1.

0 50 100 150 200

Time (ps)

0.0

0.5

1.0

1.5

2.0

2.5

|E|

(b)

−2

−1

Re[E],Im[E]

(a) Re[E] Im[E]

0 1 2 3 4 5 6

Time (25 ps)

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

|E|,φ

(c)

αg=αq=0

αg=2,αq=1.5

Figure 3.20.: (a) Time trace of the real (Re[E]) and imaginary (Im[E]) parts of the complex electric

ﬁeld amplitude for αg= 2 and αq=1.5. (b) Corresponding time trace of electric ﬁeld amplitude

|E|. (c) Pulse proﬁle (blue) and electric ﬁeld phase φ(black) for αg= 2 and αq=1.5, pulse

proﬁle for αg=αq= 0 (green). Parameters: κ=0.121, all other parameters as in Table 2.1.

of the locking cone and in semiconductor lasers with feedback the α-factor is re-

lated the stability range of continuous wave solutions [MOR92,ROT07,ERN10b,

OTT12,LIN15]. Therefore, in this section the inﬂuence of amplitude-phase cou-

pling on the mode-locked laser dynamics will be addressed.

3. Mode-locked laser dynamics

From Eqs. (3.3)and(3.6) it is known that the lasing threshold is dependent

on the amplitude-phase coupling. For non-zero α-factors the lasing modes are

detuned from the center of the spectral ﬁlter. Hence the gain of the central mode

is reduced and therefore the lasing threshold is increased. The shift of the lasing

modes with respect to the spectral ﬁlter also means that the modes are detuned

from the reference rotating frame (Ω0). In the simulations this leads to an added

rotation of the complex electric ﬁeld amplitude, as shown in Fig. 3.20awhich

depicts a time trace of the real (Re[E]) and imaginary (Im[E]) parts of the complex

electric ﬁeld amplitude for αg= 2 and αq=1.5. In this example the rotation

frequency of the complex electric ﬁeld is incommensurable with the pulse repetition

frequency, meaning the Re[E] and Im[E] are quasi-periodic, but |E| is periodic. The

quasi-periodic dynamics of the electric ﬁeld components can be compensated for

by shifting the rotating frame of the system. However, in addition to the shift of

the modes, non-zero amplitude-phase coupling also causes the pulses to become

chirped. Meaning that the instantaneous frequency dφ

dt varies within the pulse,

where φis the phase of the electric ﬁeld. Figure 3.20c shows an example of the non-

linear variation of the phase for αg= 2 and αq=1.5 (black line), the corresponding

electric ﬁeld amplitude is depicted in blue. For comparison the pulse proﬁle for

zero amplitude-phase coupling is also plotted (green line). In this case the addition

of amplitude-phase coupling leads to narrower pulses. Experimental studies have

shown that the self-phase modulation caused by the amplitude-phase coupling leads

to a wider optical spectrum [DER92], however this does not necessarily mean that

the pulses become narrower in the time domain. If the lasing modes are not all

phase-locked then, despite the wider optical spectrum, the pulse can also become

wider [KUI70,JON95b].

In the fundamental mode-locked regime the pulse repetition frequency, i.e 1/T0,

varies only slightly with the amplitude-phase coupling, however the shift of the

optical frequencies is relatively pronounced. This can be seen in Fig. 3.21 in which

optical spectra are shown for various values of αg=αq. The frequency is given in

reference to the center of the spectral ﬁlter Ω0. Due to the shift of the lasing modes

a phase-shift accumulates as a pulse travels around the ring cavity, i.e. there is

a phase diﬀerence between the electric ﬁeld at time tand delayed ﬁeld from time

t−T. This will be important for understanding the dependence of the dynamics

under the inﬂuence of optical feedback, which will be presented in the next section.

In Fig. 3.22 the impact of non-zero amplitude-phase coupling on the pump cur-

rent dependence of the mode-locked laser output is shown. In real devices the

α-factor in the gain and absorber sections should be diﬀerent due to the diﬀer-

ence in the charge-carrier densities, however, here we start our analysis with case

that αg=αq. Plotted in Fig. 3.22 are the maxima in the electric ﬁeld ampli-

tude for up- and down-sweeps in the initial conditions for α-factor values of 1-5.

The results were obtained in the same manner as Fig. 3.17. For low values of the

3.2. Solitary mode-locked laser dynamics

|SE|(dB)

αg=αq=0

(a)

|SE|(dB)

αg=αq=1

(b)

|SE|(dB)

αg=αq=2

(c)

−100 −50 0 50 100

Frequency offset from Ω0(GHz)

|SE|(dB)

αg=αq=3

(d)

Figure 3.21.: Optical spectra |SE|of the mode-locked laser output for various α-factor values.

Parameters: κ=0.121, all other parameters as in Table 2.1.

amplitude-phase coupling the current dependence is very similar to that present

in Fig. 3.17 for αg=αq= 0. At low currents there is a regime of Q-switched

mode-locking, which can be identiﬁed in Fig. 3.22 by the low current ranges with

many maxima of varying heights. Past the regime of Q-switched mode-locking,

fundamental mode-locking becomes stable. As the pump current is increased fur-

ther there are ranges of 2nd and 3rd order harmonic mode-locked. For increasing

values of the amplitude-phase coupling the ranges of fundamental and harmonic

mode-locking decrease and the onset of fundamental mode-locking is pushed to

higher pump currents. These results are qualitatively similar to those presented in

[VLA05,VLA11], where αg=αqwas used as the bifurcation parameter for a ﬁxed

pump current and it was found that for increasing amplitude-phase coupling there

is a sudden transition from fundamental mode-locking to chaotic dynamics.

The type of dynamics exhibited for large amplitude-phase coupling can vary

slightly from those presented previously for the case where amplitude-phase cou-

pling is absent (Fig. 3.18). Examples of the electric ﬁeld dynamics are shown

for αg=αq= 4 in Fig. 3.23. The Q-switched mode-locking regime develops

into a mode-locked regime with modulated pulse heights, as shown for Jg=3

in Fig. 3.23a. The harmonic mode-locked solutions can also exhibit pulse height

modulation, as depicted in Fig. 3.23bforJg=4.5. In this case there are two pulse

height envelopes corresponding to the two pulses in the laser cavity. In Fig. 3.23c

3. Mode-locked laser dynamics

0.0

1.0

2.0

3.0

4.0

|E|max

(a)

αg=αq=1 up-sweep

0.0

1.0

2.0

3.0

4.0

|E|max

(b)

αg=αq=2

down-sweep

0.0

1.0

2.0

3.0

4.0

|E|max

(c)

αg=αq=3

0.0

1.0

2.0

3.0

4.0

|E|max

(d)

αg=αq=4

01234567

Pump Current Jg

0.0

1.0

2.0

3.0

4.0

|E|max

(e)

αg=αq=5

Figure 3.22.: Numerical bifurcation diagram showing the pump current dependence of the mode-

locked laser output for various α-factor values. Plotted in blue (red) are electric ﬁeld maxima

|E|max for an up-sweep (down-sweep) in the initial conditions. Parameters: κ=0.121, all other

parameters as in Table 2.1.

0 100 200 300 400

Time (25 ps)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

|E|

(a)

0 100 200 300 400

Time (25 ps)

(b)

0 25 50 75 100

Time (25 ps)

(c)

Figure 3.23.: Time trace of the electric ﬁeld amplitude |E| for αg=αq= 4 and (a) Jg= 3, (b)

Jg=4.5, (c) Jg=6.5. Parameters: κ=0.121, all other parameters as in Table 2.1.

(Jg=6.5) the dynamics are similar to 4th order harmonic mode-locked, however

the pulse heights vary non-periodically, as does the electric ﬁeld amplitude between

pulses.

As was discussed in Section 2.2.2 the carrier dependence of the α-factor means

that the values should be diﬀerent in the gain and absorber sections. To investigate

3.2. Solitary mode-locked laser dynamics

0.0

1.0

2.0

3.0

4.0

|E|max

(a)

αg=2,αq=1 up-sweep

0.0

1.0

2.0

3.0

4.0

|E|max

(b)

αg=2,αq=1.5

down-sweep

0.0

1.0

2.0

3.0

4.0

|E|max

(c)

αg=αq=2

0.0

1.0

2.0

3.0

4.0

|E|max

(d)

αg=2,αq=2.5

01234567

Pump Current Jg

0.0

1.0

2.0

3.0

4.0

|E|max

(e)

αg=2,αq=3

Figure 3.24.: Numerical bifurcation diagram showing the pump current dependence of the mode-

locked laser output for various α-factor values. Plotted in blue (red) are electric ﬁeld maxima

|E|max for an up-sweep (down-sweep) in the initial conditions. Parameters: κ=0.121, all other

parameters as in Table 2.1.

the impact of detuning αgand αq, the pump current dependence is shown in

Fig. 3.24 for various αqvalues with αg= 2 ﬁxed. The general trend displayed in

these numerical bifurcation diagrams is that the dynamics are most regular if αqis

equal to, or slightly larger than αg. This can be understood by considering how the

α-factors enter Eq. (2.42). The α-factors enter the exponent in Eq. (2.23)inthe

form αgG−αqQ. Therefore the contributions compensate each other if αgG≈αqQ,

which occurs best when αqαg[VLA05].

Overall, similar dynamics can be exhibited when amplitude-phase coupling is

included and, at least for low α-factor values, the pump current dependence dis-

plays the same qualitative trends as in the absence of amplitude-phase coupling.

However, amplitude-phase coupling causes the pulses to become chirped, and can

lead to a non-periodic modulation of the pulsed dynamics.

3. Mode-locked laser dynamics

3.3. Dynamics induced by feedback from a single

external cavity

Adding optical feedback to the mode-locked laser greatly inﬂuences the dynam-

ics that can be exhibited. This has previously been demonstrated by the theo-

retical studies presented in [AVR09,OTT12a,OTT14,SIM14]. In [AVR09]the

authors used a ﬁnite-diﬀerence travelling-wave model for a linear cavity to study

the inﬂuence of feedback from a short external cavity (feedback delay times up to

≈3T0), whereas in [SIM14] the DDE-model approach is used to study the inﬂu-

ence of long feedback delay times on a laser with a quantum-dot gain medium.

In [OTT12a,OTT14] the same model is used as in this thesis and the dynamics

are investigated in the regimes of short and intermediate feedback delay times.

Also using the DDE-modelling approach, the dynamics arising from coupling to

an external passive cavity, with a roundtrip time lower than that of the laser cav-

ity, were studied in [ARK15a]. In this study the focus was on inducing harmonic

mode-locking via the coupling to the external cavity. We add to the knowledge

provided by these previous studies by presenting an analysis of the dynamics over

a wide range of feedback delay times, and establishing how the dynamics in diﬀer-

ent feedback delay regimes are related. Furthermore, we analyse the bifurcations

of the mode-locked solutions in dependence of the feedback parameters in order to

gain a deeper understanding of the mechanisms leading to various feedback induced

dynamics.

With feedback from one external cavity there are three additional parameters;

the feedback delay time τ=τ1, the feedback strength K=K1and the feedback

phase C=C1(Eqs.(2.42)-(2.44) with N= 1). In this section the dynamics arising

in dependence of these feedback parameters will be investigated. We shall consider

the dynamics in three regimes, short (τ≈1), intermediate (τ≈10) and long

(τ≈100) delay times.

3.3.1. Short delay

We are primarily interested in the inﬂuence of feedback on the fundamentally mode-

locked solution. Since the mode-locked solution is periodic, the dynamics that can

arise with feedback are strongly dependent on the ratio of the period of the mode-

locked solution and the delay time that is introduced by the feedback. The equation

for the electric ﬁeld amplitude, in the single feedback cavity case, is

dt =−γE(t)+γR(t−T)e−iΔΩTE(t−T)

+γKeiCR(t−T−τ)e−iΔΩ(T+τ)E(t−T−τ).(3.7)

3.3. Dynamics induced by feedback from a single external cavity

laser cavity feedback cavity laser cavity feedback cavity

coupling facet coupling facet

Figure 3.25.: Schematic diagrams of the laser cavity coupled to one external feedback cavity, for

external cavity delay times of τ=T0,K (left) and τ=1

2T0,K (right), and sketches of the resulting

pulse trains. The blue line represents the electric ﬁeld and the black line represents the net gain.

If one considers the system with instantaneous feedback, i.e. τ= 0 and C=0,

then the system is equivalent to that without feedback, but with modiﬁed internal

losses κ=κ(1+K)2, i.e.

dt =−γE(t)+(1+K)γR(t−T)e−iΔΩTE(t−T).(3.8)

If E0(t) is a mode-locked solution of Eq. (3.8), with period T0,K,5then E0(t) is also

a solution of Eq. (3.7) for all τ=nT0,K,wherenis an integer. This means that for

all feedback delay times that are integer multiples of T0,K, which will be referred

to as resonant feedback delay times, the dynamics are unchanged. For delay times

between these resonant values various dynamics are exhibited. More generally it is

also true that if one considers a solution to Eq. (3.7) with τbetween 0 and T0,K,

and if that solution is periodic, with period T0, then this is also a solution to the

system with the feedback delay time τ+nT0, for integer n[YAN09]. This means

that all periodic solutions that arise for τ∈[0,T

0,K], repeat for larger delay times.6

Therefore, to characterise the type of dynamics that can emerge with feedback, we

ﬁrst study the feedback delay range from zero to T0,K.

τ−Kdependence

In this section the inﬂuence on the dynamics of the feedback strength and feedback

delay time are investigated. Here we consider the inﬂuence of these parameters

separately from phase eﬀects, i.e. for C= 0 and αg=αq= 0 (the inﬂuence of

these parameters is discussed later in this section).

5The subscript Kis included to indicate that the resonant period is Kdependent. From now on

we will refer to the period of the solitary laser (K=0)asT0,S.

6The stability of the solutions can vary with n. This will be discussed in Section 3.3.4.

3. Mode-locked laser dynamics

τ(T0,K)

1.011

1.012

1.013

1.014

1.015

1.016

1.017

1.018

Period T0

1pulse

2pulses

3pulses

4pulses

5pulses

Figure 3.26.: Path continuation: Period of the mode-locked solution (black line) as a function

of the feedback delay time τ, which is given in units of T0,K =1.013. The background colour

indicates the number of pulses in the laser cavity. The green striped regions indicate various

main and higher order resonance locking regions. Parameters: K=0.1, all other parameters as

in Table 2.1.

When τ=nT0,K for integer n, the feedback delay time is resonant with the

dynamics, meaning that pulses from the feedback and laser cavities are coincident

at the coupling facet. This is shown schematically on the left of Fig. 3.25 for

τ=T0,K. Synchronisation eﬀects can also occur for non-integer multiples of T0,K.

For example when τ=1

2T0,K the pulse from the feedback cavity is coupled into the

laser cavity when the main pulse is on the opposite side of the cavity. This induces

a second pulse which is shifted from the main pulse by a time of 1

2T0,K. Feedback

from this second pulse is then synchronised with the main pulse. This scenario

is also depicted in Fig. 3.25. Synchronisation can also occur for other fractional

multiples of T0,K, i.e. when

τ=q

pT0,K for p, q ∈N(3.9)

[OTT12a]. The integer pgives the number of pulses within one T0,K interval, or in

other words the number of pulses in the laser cavity.7This includes the main pulse

and p−1 feedback induced pulses. We shall refer to τvalues which are fractional

multiples of T0,K as higher order resonances, where the order is given by p.

Synchronisation cannot occur for an arbitrarily large number of pulses in the

laser cavity. The number of feedback induced pulses is restricted by the width of

the pulses and length of the laser cavity. The ﬁnite width of the pulses also means

7The exact value of T0,K is dependent on the feedback strength, however we remind the reader

that T0≈T=1.

3.3. Dynamics induced by feedback from a single external cavity

−1

4(a)

τ=0.25

|E| G/2Qtot/2G/2

0.0

0.5

1.0

1.5

2.0

2.5

|E|

(b)

−1

|E|,G/2,Qtot/2,G/2

0.0

0.5

1.0

1.5

2.0

2.5

|E|

(d)

0 5 10 15 20 25 30 35 40

Time (ps)

−1

4(e) τ=0.5

2.5 3.5 4.5

0.0

0.5

1.0

1.5

2.0

2.5

|E|

(f)

Figure 3.27.: Inside resonances: Time traces and phase space portraits of the feedback induced

dynamics for (a)-(b) τ=0.25, (c)-(d) τ=0.35 and (e)-(f) τ=0.5. The time traces (a),(c) and

(d) show the electric ﬁeld amplitude |E| (blue), gain G(green), total losses Qtot =Q−ln (κ)

(red) and the net gain G=G−Qtot (black). The phase space portraits (b), (d) and (f) show

the dynamics in the G−|E|plane. Parameters: (a)-(b) K=0.12, (c)-(f) K=0.15, all other

parameters as in Table 2.1.

that synchronisation does not only occur exactly at the resonances, but rather

extends over a range of τvalues. We will deﬁne the locking ranges of the main

and higher order resonances as the τranges in which the period of the solutions

changes monotonically with τ, the number of pulses in the laser cavity remains

constant and the solutions remain stable. This is illustrated in Fig. 3.26,where

the period of the mode-locked solutions are plotted as a function of the feedback

delay time for K=0.1. The number of pulses in the laser cavity is indicated by

the background colour and the striped green lines indicate the locking regions for

various resonances. In this case the solutions are stable for the entire τrange that

is depicted. As the feedback delay time is changed, the system adapts such that the

pulses in the laser cavity are synchronised with the pulses in the feedback cavity.

However if the delay time is tuned too far from one of the resonances the system

cannot adapt suﬃciently and the pulses become deformed. The locking range for

each resonance, main and higher order, depends on the order of nand p,andon

the feedback strength. The parameter dependence of the main resonance locking

regions, and the change in the period in these regions, will be discussed in more

detail in Section 3.3.4.

3. Mode-locked laser dynamics

Examples of the dynamics within the q

p=1

4,q

p=1

3and q

p=1

2locking ranges

are shown in Fig. 3.27. Depicted are time traces of the electric ﬁeld amplitude, the

gain, the total losses and the net gain, where the total losses and net gain are as

deﬁned previously; Qtot =Q−ln (κ)andG=G−Q

tot.8Also shown are phase

portraits of the dynamics projected on the G−|E|plane. Figure 3.27a-b shows

the dynamics arising for τin the q

p=1

4locking region. In this example there are

four pulses within one T0,K interval (≈25ps since T0,K ≈T). The largest pulse

is the main pulse that is generated due to the gain and absorber dynamics. The

three satellite pulses are induced by the feedback and since they do not coincide

with positive net gain windows they are sustained by continual rejection of pulses

from the feedback cavity rather than by ampliﬁcation within the laser cavity. The

amplitude of the satellite pulses decreases within one T0,K interval as each pulse

is the seed for the next. Similar results are seen for τ=0.35 and τ=0.5in

Fig. 3.27c-f, which exhibit dynamics with three and two pulses within one period,

respectively, corresponding to the q

p=1

3and q

p=1

2locking regions. Examples of

the dynamics outside the resonance locking regions are shown in Fig. 3.28.Inthese

regions pulses are deformed or not well separated. Figure 3.28aisneartheedgeof

the zeroth-order main resonance (see Fig. 3.26). For this delay time the pulses from

the feedback cavity are coupled into the laser just after the main pulse has passed

the coupling facet, causing a long trailing edge. This inﬂuences the recovery of the

absorber, meaning that the positive net gain window decays more slowly. Although

New’s stability criterion is still fulﬁlled [NEW74], as shall be discussed further in

Chapter 4, this makes the dynamics less robust against noise perturbations. In

Fig. 3.28bτis chosen to the left of the ﬁrst main resonance locking region. In this

case pulses are coupled into the laser cavity just before the main pulse passes the

coupling facet, leading to a higher net gain before the pulse.

The feedback delay time determines the number of pulses within the feedback

cavity, but the type of dynamics that are exhibited by the system also depends

on the feedback strength. The periodic dynamics with additional feedback in-

duced pulses (Fig. 3.27) occur for low feedback strenghs. For slightly higher feed-

back strengths the system can exhibit dynamics with periodically modulated pulse

heights. Examples of such dynamics are shown in Fig. 3.29. Figure 3.29a shows a

time trace of the electric ﬁeld amplitude for τ=0.25 and K=0.2. A close up of

the dynamics is shown in Fig. 3.29b, where the net gain has also been plotted (black

line). Here it can be seen that there are four pulses in the laser cavity (i.e. within

a time interval of ≈25ps) and that between these pulses there is a competition

for the gain. Within the time interval that is depicted here the net gain window

is transferred from one pulse to the next, and accordingly the pulse amplitudes

change. Physically the situation in this parameter regime is that the pulses that

8The net gain does not include the feedback contributions.

3.3. Dynamics induced by feedback from a single external cavity

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0(a)

τ=0.12

|E| G/2Qtot/2G/2

0.0

0.5

1.0

1.5

2.0

2.5

|E|

(b)

0 5 10 15 20 25 30 35 40

Time (ps)

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

|E|,G/2,Qtot/2,G/2

(c)

τ=0.85

2.5 3.5 4.5

0.0

0.5

1.0

1.5

2.0

2.5

|E|

(d)

Figure 3.28.: Outside resonances: Time traces and phase space portraits of the feedback induced

dynamics for (a)-(b) τ=0.12 and (c)-(d) τ=0.85. The time traces (a) and (c) show the electric

ﬁeld amplitude |E| (blue), gain G(green), total losses Qtot =Q−ln (κ) (red) and the net gain

G=G−Qtot (black). The phase space portraits (b) and (d) show the dynamics in the G−|E|

plane. Parameters: K=0.1, all other parameters as in Table 2.1.

0.0

0.5

1.0

1.5

2.0

2.5

|E|

(a) τ=0.25

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

|E|,G/2

(b) |E| G/2

0.0

0.5

1.0

1.5

2.0

2.5

|E|

(c)

920 930 940 950 960 970 980 990 1000

Time (25 ps)

0.0

0.5

1.0

1.5

2.0

2.5

|E|

(d) τ=0.35

955 956 957 958 959 960

Time (25 ps)

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

|E|,G/2

(e)

2.5 3.5 4.5

0.0

0.5

1.0

1.5

2.0

2.5

|E|

(f)

Figure 3.29.: Quasi-periodic dynamics: Time traces and phase space portraits of the feedback

induced dynamics for (a)-(c) τ=0.25 and (d)-(f) τ=0.35. The time traces (a) and (d) show

the electric ﬁeld amplitude |E| (blue). In (b) and (e) at shorter time interval is depicted and the

net gain G=G−Qtot (black) is also plotted. The phase space portraits (c) and (f) show the

dynamics in the G−|E|plane. Parameters: K=0.2, all other parameters as in Table 2.1.

get coupled back into the laser cavity are large enough to signiﬁcantly inﬂuence the

net gain window, but there is not enough energy in the system to sustain all four

3. Mode-locked laser dynamics

pulses. This leads to competition for the gain between the four pulses in the laser.

The dynamics exhibited for τ=0.35, with K=0.2, are shown in Fig. 3.29d-f.

In this case there are three competing pulses in the laser cavity. In both cases,

τ=0.25 and τ=0.35, the slow modulation period of the pulse amplitudes is

approximately 45 laser cavity roundtrips. This coincides approximately with the

recovery time of the gain section, 1/γg= 40, indicating that the transfer of energy

between the pulses in the laser cavity is mediated by the relatively slow recovery of

the gain section. This slow modulation period is incommensurable with the much

faster pulse repetition period. Consequently the dynamics are quasi-periodic. This

can be seen by looking at the trajectories of the system in its phase space. Projec-

tions on to the G−|E|plane, for τ=0.25 and τ=0.35, are shown in Fig. 3.29c

and Fig. 3.29f, respectively. The projections shown here depict the trajectories for

the time interval shown in subplots (a) and (d) of Fig. 3.29. As the dynamics are

not quite periodic the trajectories ﬁll in more and more of the phase space over

longer time intervals. If the feedback strength is increased further, eventually there

is a transition to harmonic mode-locking for τvalues near the higher-order reso-

nances. This occurs because for higher feedback strengths the losses are reduced

and therefore multiple pulses can be sustained.

The type of dynamics arising for delay times that do not correspond to main res-

onances, i.e. periodic dynamics with satellite pulses and quasi-periodic dynamics at

higher feedback strengths, are in agreement with the ﬁndings presented in [AVR09].

In this work the dynamics of a Fabry-Perot passively mode-locking laser subject to

optical feedback was numerically investigated using a travelling-wave model. The

destabilisation of periodic mode-locked dynamics via optical feedback has also been

observed experimentally [GRI09,NIK16]. In [NIK16] feedback induced harmonic

mode-locking was also observed.

A more complete picture of the dependence of the dynamics on τand Kis

shown in Fig. 3.30. Depicted are maps of the dynamics in the τ−Kparameter

plane which are obtained by numerically integrating the DDE system and colour

coding the regions according to the dynamics that are exhibited. White indicates

fundamental mode-locking with only one pulse in the laser cavity, blues indicate

various orders of harmonic mode-locking and grey indicates continuous wave lasing.

In the purple regions the dynamics have the fundamental periodicity but there are

additional feedback induced pulses (see Fig. 3.27) and in the green regions the

dynamics are quasi-periodic (see Fig. 3.29).9To obtain the number of pulses we

only look for maxima in the electric ﬁeld amplitude. This means that no distinction

is made between pulses which are well separated and pulses which have multiple

maxima. Therefore the boundaries of the various dynamical regimes are not the

9The green regions are labelled CD, for complex dynamics, as in later ﬁgures the colour code

does not distinguish between quasi-periodic and chaotic dynamics. However, for the parameters

used here all green regions correspond to quasi-periodic dynamics.

3.3. Dynamics induced by feedback from a single external cavity

Figure 3.30.: Maps of the dynamics in dependence of τand Kobtained for (a) up- and (b) down-

sweeps in K. The feedback delay time is plotted in units of the period for K=0,T0,S =1.014.

The colour code indicates the number of pulses in the laser cavity and the type of dynamics that

are exhibited. White indicates fundamental mode-locking (FML), purples indicate fundamental

mode-locking with feedback induced pulses, greens indicate complex dynamics (in this case

quasi-periodic) with 2-5 pulses in the laser cavity (CD2-CD5) and blues indicate harmonic

mode-locking (HML). The grey regions indicate continuous-wave solutions. Hatching indicates

regions in which diﬀerent results are obtained in the up- and down-sweeps. Parameters: all

other parameters as in Table 2.1.

same as the resonance locking regions. As shown in Fig. 3.26, the locking ranges

are narrower than the regions exhibiting dynamics with the same number of electric

ﬁeld maxima. Figure 3.30a is obtained by preforming an up-sweep in K, meaning

that for each Kvalue the history of the solution for the previous Kvalue is used

for the initial conditions. Results obtained for the down-sweep in Kare shown in

Fig. 3.30b. The hatching indicates the regions where diﬀerent results are obtained

for the up- and down-sweeps. The Krange depicted here is relatively large and

past a certain value more light is being coupled back into the laser cavity than what

is coupled out. Physically this would require the light in the feedback cavity to be

ampliﬁed before it is coupled back into the laser.10 Furthermore, for large feedback

10For single cavity feedback, from Eq. (2.9), the feedback strength is given by

K=(1−r)(rec/r)1/2.(3.10)

In this deﬁnition of the feedback strength a factor of √1−ris included to account for how

much light is coupled from the laser cavity into the feedback cavity and the term is scaled

by 1/√r. The fraction of the out-coupled light that is coupled back into the laser is given

by Kr/(1 −r). This value is limited by the transmittance of the coupling facet √1−r.If

the light is not ampliﬁed in the feedback cavity the maximal feedback strength is given for

rec = 1, which, if we assume an end facet reﬂectivity (or transmittance in the ring cavity case)

of 30%, gives K≈1.3. In experimental studies the feedback strength is typically given in

terms of the percentage of the out-coupled light that is coupled back, therefore to compare

3. Mode-locked laser dynamics

strengths it is no longer valid to neglect contributions from light that has made

multiple roundtrips in the feedback cavity.11 However, we look at such a large range

to gain an understanding of the mathematical boundaries of the dynamics that can

be exhibited by the system under investigation (Eq. (3.7) and Eqs.(2.43)-(2.44)).

The feedback delay time is plotted in units of the period for K=0,T0,S =1.014.

Although the resonant periods T0,K are dependent on K, the variation is small and

the positions of the resonances do not deviate much from the K= 0 case.

For τ≈0andτ≈T0,S there are large regions of fundamental mode-locking.

Within these regions locking to the zeroth and ﬁrst-order main resonances occurs

(τ=nT0,K for n=0,1). At large Kvalues these regions are bounded by continuous

wave lasing. The transition to continuous wave lasing at large Kvalues is similar

to what is observed for large pump currents in the solitary laser case. The electric

ﬁeld becomes so large that the gain and absorber sections are continually saturated

leading to continuous wave emission. Between the main resonances, at low feedback

strengths, there is locking to the higher-order resonances, i.e. τ=1

5T0,K,1

4T0,K, ...

(purple regions in Fig. 3.30). Around each of these higher-order resonances dy-

namics with ppulses are exhibited, where pis the order of the resonance. The

width of the regions with ppulses decreases for increasing p. As can be seen from

Fig. 3.26 the locking regions show the same trend. Furthermore, these regions can

be ordered into a Farey sequence, as was shown in [OTT12a]. At large feedback

strengths, harmonic mode-locking is observed near the higher-order resonances.

The order of the harmonic mode-locking is given by pand the number of harmonic

mode-locked regions of each order is given by p−1. For example, between τ=0

and τ=T0,S there are two p= 3 locking regions corresponding to τ=1

3T0,K

and τ=2

3T0,K, and hence two regions of 3rd order harmonic mode-locking. Sta-

ble regions of harmonic mode-locking are also found at the main resonances. For

the τ=1

2T0,K locking region there is a direct transition to 2nd order harmonic

mode-locking from the satellite pulse dynamics (2 pulses). For the locking regions

corresponding to larger p’s the transition is mediated by regions of quasi-periodic

dynamics with the same number of pulses in the laser cavity as the harmonic order.

Also, the onset of harmonic mode-locking occurs at larger Kvalues for larger p’s.

This is because the available net gain has to be distributed over more pulses.

The sequence of dynamics displayed in dependence of the feedback strength in

Fig. 3.30, for the feedback delay time τ=1

2T0,S, is the same as what is reported

with experiments Kshould be scaled by √r/(1 −r). It is also important to note that we are

simulating the dynamics in the laser cavity, whereas in experiments the electric ﬁeld that is

coupled out is observed. Therefore the simulation results will show increased intensities for

larger feedback strengths, whereas experimentally decreased intensities will be observed as less

light is coupled out of the system.

11For example, if K=0.5 then, assuming an end facet reﬂectivity of 30%, the feedback strength

after two roundtrips in the feedback cavity would be K2=0.1.

3.3. Dynamics induced by feedback from a single external cavity

Figure 3.31.: Numerical bifurcation diagrams showing the Kdependence of the mode-locked laser

output for (a) τ=0.004 and (b) τ=0.7. Plotted in blue (red) are electric ﬁeld maxima |E|max for

an up-sweep (down-sweep) in the initial conditions. Labelled are regions of fundamental mode-

locking (FML), harmonic mode-locking (HML) and continuous wave (CW) lasing. Parameters:

all other parameters as in Table 2.1.

in [ARK15a]. In this work the authors use the DDE model to study the inﬂuence

of coupling the laser cavity to a short external cavity. However, in contrast to our

modelling approach for the feedback contribution, in [ARK15a] feedback from the

external cavity is incorporated by including a diﬀerential equation for the electric

ﬁeld in the feedback cavity, which is coupled to the equation for the electric ﬁeld

in the laser cavity. In doing so, feedback contributions from light that has made

multiple roundtrips in the feedback cavity are included. The qualitative agreement

between our results and those of [ARK15a] indicates that, even if contributions to

the feedback from light that has made multiple roundtrips in the external cavity

are neglected, the qualitative trends observed for high feedback strengths are still

the same.

In the main resonant regions (τ≈0andτ≈T0,S), at large feedback strengths,

there is multistability between the fundamentally mode-locked solution and har-

monically mode-locked solutions (hatched regions in Fig. 3.30). Similar results were

obtained for the pump current dependence of the solitary laser (see Fig. 3.17). This

can be understood by considering that the feedback reduces the non-resonant losses

and hence reduces the lasing threshold and the threshold for the onset of mode-

locked dynamics. There are also regions of multistability between mode-locking

and the continuous wave lasing state.

Within the various dynamical regions depicted in Fig. 3.30 the pulse heights

change with τand K. This is illustrated in the one-parameter numerical bifurcation

diagrams in Figs. 3.31 and 3.32. Plotted are maxima in the electric ﬁeld amplitude,

counted over a time span of one hundred cold-cavity roundtrip times, for up- (blue)

3. Mode-locked laser dynamics

0.0

1.0

2.0

3.0

4.0

|E|max

(a) K=0.1down-sweep up-sweep

τ(T0

0.0

1.0

2.0

3.0

4.0

|E|max

(b) K=0.3

2ndHML

3rdHML

Figure 3.32.: Numerical bifurcation diagrams showing the τdependence of the mode-locked laser

output for (a) K=0.1 and (b) K=0.3. Plotted in blue (red) are electric ﬁeld maxima |E|max

for an up-sweep (down-sweep) in the initial conditions. The feedback delay time is plotted in

units of the period for K=0,T0,S =1.014. Parameters: all other parameters as in Table 2.1.

and down-sweeps (red) in the initial conditions. In Fig. 3.31 the results are show

in dependence of the feedback strength for (a) τ=0.004 and (b) τ=0.7. For τ=

0.004 the system is within the locking region of the zeroth-order main resonance.

The pulse heights of the fundamentally and harmonically mode-locked solutions

increase with K, and as already shown in Fig. 3.30 there are regions of multistability

between these solutions. For τ=0.7 the system is in the locking region of the τ=

3T0,K higher-order resonance. Accordingly, for low feedback strengths there are

three maxima with diﬀerent heights. The quasi-periodic region has maxima of many

diﬀerent heights. Near K=0.5 there is a small range of multistability between the

quasi-periodic and 3rd order harmonically mode-locked solutions. Figure 3.32 shows

the τdependence for (a) K=0.1 and (b) K=0.7. The main-resonance locking

regions can be identiﬁed in these plots as the regions near τ= 0 and τ=T0,K where

there is only one maximum. Within these regions the pulse heights are largest at

the exact main resonances, τ= 0 and τ=T0,K. Comparing Figs. 3.32a-b it can

be seen that the maximum in the pulse height is shifted slightly closer to τ=1

for K=0.7. This is because the period T0,K decreases with K. The reasoning

for this is the same as for the pump current dependence of the period (see Section

3.2.3). The pulse heights decrease as τis tuned away from the main resonances

because pulses from the feedback cavity are coupled in at the ﬂanks of the pulse

in the laser cavity, rather than at the center. This also causes the pulse to become

wider. Changes in the pulses shape within the main-resonance locking regions will

be discussed further in Section 3.3.4 (see Figs. 3.63 and 3.66).

3.3. Dynamics induced by feedback from a single external cavity

τ(T0,S)

0.0

0.5

1.0

1.5

2.0

(a)

γ=30

τ(T0,S)

(b)

γ= 100

FML

2ndHML

3rdHML

4thHML

5thHML

6thHML

2pulses

3pulses

4pulses

5pulses

6pulses

7pulses

CD2

CD3

CD4

CD5

CD6

Figure 3.33.: Inﬂuence of the spectral ﬁlter width: Maps of the dynamics in dependence of τ

and Kobtained for up-sweeps in Kfor γ= 30 and γ= 100. The feedback delay time is plotted

in units of the period for K= 0, (a) T0,S =1.031 and (b) T0,S =1.009. The colour code indicates

the number of pulses in the laser cavity and the type of dynamics that are exhibited. White

indicates fundamental mode-locking (FML), purples indicate fundamental mode-locking with

feedback induced pulses, greens indicate complex dynamics (in this case quasiperiodic) with

2-7 pulses in the laser cavity (CD2-CD6) and blues indicate harmonic mode-locking (HML).

The grey regions indicate continuous-wave solutions. Parameters: all other parameters as in

Table 2.1.

Inﬂuence of the pulse width

As mentioned brieﬂy near the beginning of this section, the number of pulses that

can exist in the laser cavity, and hence number of feedback induced pulses which

can be created, is limited by the pulse widths. This is illustrated in Fig. 3.33,

where τ-Kmaps of the dynamics are shown for ﬁlter widths (a) γ= 30 and (b)

γ= 100 (see Fig. 3.15 for the inﬂuence of spectral ﬁlter width on the pulse width).

With decreased pulse widths (Fig. 3.33b) the maximum order of harmonic mode-

locking is increased (blue regions), as well as the maximum number of feedback

induced satellite pulses (purple regions), and the widths of the multi-pulse regions

are decreased. Otherwise, all features of the map are qualitatively similar to the

γ=66.5 case shown in Fig. 3.30. For very wide pulses (Fig. 3.33a) parameter

regions exhibiting quasi-periodic dynamics are greatly reduced.

The asymmetry of the mode-locked pulses also has an inﬂuence on the feedback

parameter dependence. In the solitary laser case the pulses have a longer trailing

edge compared with the leading edge and the net gain decreases more slowly than

it rises during the fast stage (see Fig. 3.14). As a consequence of this, the system

behaves diﬀerently if pulses are coupled into the laser cavity before or after the

3. Mode-locked laser dynamics

0.0

0.5

1.0

1.5

(a) stable cw

−0.2 0.0 0.2 0.4 0.6 0.8 1.0

−0.5

0.0

0.5

1.0

1.5

(b)

Figure 3.34.: Path continuation: (a) Hopf bifurcations of the maximum-gain continuous-wave

solution in the τ−Kparameter space. In black is the H1Hopf curve. The H2,H3and H4

Hopf curves are depicted in light blue, purple and pink, respectively. (b) Bifurcations of the

fundamental mode-locked solution. Blue lines indicate torus bifurcations and green lines are

saddle-node bifurcations and the red line is a period-halving bifurcation. The black line is

the fundamental Hopf bifurcation curve of the continuous wave solution (H1), with the solid

segment being supercritical and the dashed segments subcritical. The thin grey lines show the

remaining Hopf bifurcations of the continuous-wave solution (compare with (a)). Shaded in

green is the stability region of the fundamental mode-locked solution. In the grey regions the

continuous-wave solution is stable. Parameters: all other parameters as in Table 2.1.

main pulse has passed the coupling facet, leading to asymmetries in the dynamics

for delay times that are tuned from the resonances. This can be clearly seen in

Figs. 3.30 and 3.33, which all show a large region of two pulse dynamics on the

right border of the ﬁrst main resonance region, but not on the left. When the delay

time is larger than the resonant value, then the pulses from the feedback cavity are

coupled in after the main pulse. As the net gain is high on this side of the pulse, the

fed-back pulses experience lower losses, leading to two pulse dynamics at smaller

positives detuning from the main resonance compared with negative detuning.

Path continuation

By numerically integrating the DDE system only the stable solutions can be found.

However, a deeper understanding of the dynamics can be gained through knowl-

edge of the underlying bifurcation structure. As with the solitary laser, the mode-

3.3. Dynamics induced by feedback from a single external cavity

locked solutions that arise with feedback bifurcate from the continuous-wave so-

lution which has the maximum gain (see Section 3.2.1). In Fig. 3.34a Hopf bi-

furcations of the continuous-wave solution are depicted in the τ−Kparameter

space (here we again use our standard value for the spectral ﬁlter width, γ=66.5).

These bifurcation curves were obtained using DDE-BIFTOOL [ENG01]. The H1

Hopf bifurcation which gives rise to the periodic solution with the fundamental

frequency of the laser cavity Ωf=2π/T, i.e. the solution out of which the fun-

damentally mode-locked solution evolves, is plotted in black. The H2,H3and H4

Hopf curves are plotted in light blue, purple and pink, respectively. In the grey

region the continuous-wave solution is stable. The τrange that is depicted includes

negative delay times. The negative τvalues are unphysical, however mathemati-

cally nothing signiﬁcantly changes here since the total delay time in the feedback

term, T+τ, is still positive (Eq. (3.7)). The τdependence of the Hopf curves is

related to their frequency. At τvalues that are integer multiples of 2π/Ωnthere

are peaks in the Kdependence of the HnHopf bifurcations, where Ωn=nΩffor

n=1,2,3,4. In the H4case, the curves form closed ellipses at integer multiples of

2π/Ω4.

In Fig. 3.34b the bifurcations of the fundamentally mode-locked solution, which is

generated in the H1Hopf bifurcation, are shown (these are shown again in Fig. 3.35

with labels). Torus bifurcations are depicted in blue, saddle-node bifurcations in

green and period-halving in red. The regions of stable fundamental mode-locking

are indicated by the green shading. The positions of the Hopf bifurcations are

indicated by the grey lines and the criticality of the H1curve is indicated by

the dashed (subcritical) and solid (supercritical) segments. The bifurcations of

the fundamental mode-locked solution extend into negative Kregions. Physically

negative feedback strengths mean that the fed-back light has a phase shift of π,

i.e. −Kis equivalent to Keiπ.Theboxnearτ=0.5, K=−0.2 marks a region

where the bifurcations of the fundamental solution could not be continued using

DDE-BIFTOOL.

In order to understand how the periodic solutions evolve out of the H1bifurcation

we consider line scans along Kat various positions in the bifurcation diagram. The

positions of the line scans are indicated by the dotted black lines (1)-(4) in Fig. 3.35

and the results are shown in Fig. 3.36. Plotted in Fig. 3.36 is the period of the

solutions. The stability is indicated by the solid (stable) and dotted (unstable)

segments. In line scan (1) the periodic solution is born in a subcritical section

of the H1curve; the start of the line scan is indicated by the point labelled H1

in Fig. 3.36 (1). As the solution is continued in Kthe period increases and the

system goes through a series of three torus bifurcations (T1,T2and T1). These

torus bifurcations emanate from the Hopf-Hopf points where the H1Hopf branch

crosses the H2,H3and H4branches (see Figs. 3.34 and 3.35). In each of these

torus bifurcations one pair of characteristic exponents enter the unit circle, i.e.

3. Mode-locked laser dynamics

−0.2 0.0 0.2 0.4 0.6 0.8 1.0

−0.5

0.0

0.5

1.0

1.5

(1)

(2)

(3) (4)

(5)

(6)

stable cw

SN1SN6

SN2SN4SN7

SN3

SN8

SN5

T1T2

Figure 3.35.: Path continuation: Bifurcations of the fundamental mode-locked solution. Blue

lines indicate torus bifurcations (T) and green lines are saddle-node bifurcations (SN) and the

red line is a period-halving bifurcation (PH). The black line is the fundamental Hopf bifurcation

curve of the continuous wave solution (H1), with the solid segment being supercritical and the

dashed segments subcritical. The thin grey lines show the remaining Hopf bifurcations of the

continuous-wave solution (compare with Fig. 3.34 (a)). Shaded in green is the stability region

of the fundamental mode-locked solution. In the grey regions the continuous-wave solution is

stable. The dotted black lines numbered (1)-(6) indicate the positions of the line scans that are

shown in Figs. 3.36 and 3.37. Parameters: all other parameters as in Table 2.1.

the number of unstable directions decreases. The solution then becomes stable

after a saddle-node bifurcation (SN1). After this fold in the line scan, the solution

continues to be stable until the second saddle-node bifurcation (SN2)nearK=

−0.4. How the solution continues after the SN2bifurcation is not depicted in

Fig. 3.36 (1), as the remaining section of this solution branch remains unstable after

this point. In the section of the line scan between the two saddle-node bifurcations

the solution for the electric amplitude has the typical mode-locked pulse proﬁle

(see Fig. 3.14) and within this range the period decreases with increasing K. Line

scan (2) (Fig. 3.36 (2)) starts in a supercritical part of H1, however the solution

is initially unstable. The additional unstable directions are related to H3. Line

scan (2) becomes stable after two saddle-node bifurcations (SN3) and then loses

stability again in the third saddle-node bifurcation (SN4). At the τposition of

line scan (3) the H1bifurcation is also supercritical, but there are four additional

unstable multipliers which are related to H2and H3. This branch also goes through

a series of three saddle-node bifurcations, however the solutions remain unstable

3.3. Dynamics induced by feedback from a single external cavity

−0.4−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0.96

0.98

1.00

1.02

Period T0

(1)

SN2

SN1

T2T3

−0.15 −0.1−0.05 0.0 0.05 0.1 0.15 0.2 0.25 0.3

1.01

1.02

1.03

1.04

Period T0

(2)

SN3

SN4

0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

1.012

1.016

1.020

1.024

Period T0

(3)

SN5

SN3

−0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.90

0.94

0.98

1.02

Period T0

(4)

SN6

SN7

Figure 3.36.: Path continuation: Line scans along Kcorresponding to the vertical dotted lines

(1)-(4) plotted in Fig. 3.35. Plotted is the period of the solutions. Solid lines indicate stable

solutions and dotted lines indicate unstable solutions. The bifurcations that occur along the line

scans are labelled in accordance with the labelling in Fig. 3.35. Parameters: (1) τ=−0.008,

(2) τ=0.285, (3) τ=0.390, (4) τ=0.737, all other parameters as in Table 2.1.

(Fig. 3.36 (3)). The line scan terminates in the lower part of the period halving

bifurcation shown in Fig. 3.35 (and Figs. 3.34b). Finally, line scan (4) starts in the

second subcritical region of H1. The solution branch ﬁrst becomes stable after the

saddle-node bifurcation SN6(Fig. 3.36 (4)). Then there is an unstable section of

the line which is bordered by the torus bifurcation branch T4. After the second T4

bifurcation the solution is stable until the saddle-node bifurcation SN7.

In Fig. 3.37 line scans in τare shown. These are started from points on line

(1) and their positions in the bifurcation diagram are indicated by the horizontal

dotted lines (5)-(6) in Fig. 3.35. Line scan (5) starts from a stable part of line

(1). Near τ=0.2 the solutions becomes unstable after the torus bifurcation T5.

The unstable solution continues past the period-halving bifurcations depicted in

Fig. 3.35, as these are not bifurcations of this section of the solution branch. Near

τ=0.55 the branch then turns around in a saddle-node bifurcation (SN8)and

3. Mode-locked laser dynamics

0.0 0.1 0.2 0.3 0.4 0.5

1.012

1.016

1.020

1.024

Period T0

(5)

SN8

0.0 0.2 0.4 0.6 0.8 1.0

1.012

1.013

1.014

1.015

1.016

Period T0

(6)

Figure 3.37.: Path continuation: Line scans along τcorresponding to the horizontal dotted lines

(5)-(6) plotted in Fig. 3.35. Plotted is the period of the solutions. Solid lines indicate stable

solutions and dotted lines indicate unstable solutions. The bifurcations that occur along the

line scans are labelled in accordance with the labelling in Fig. 3.35. Parameters: (5) K=0.112,

(6) K=0.054, all other parameters as in Table 2.1.

then terminates on the right side of the period-halving curve. Line scan (6) is at

a slightly low Klevel. This branch of solutions does not undergo any bifurcations

in the depicted τrange (see Fig. 3.37 (6)), i.e. the SN3and PH curves that line

(6) cuts across in Fig. 3.35 are not bifurcations of these solutions. We have shown

line scans that terminate on the lower (line scan (3)) and right (line scan (5)) parts

of the period-halving curve. If a solution were continued upward in Kfrom line

(6), near τ=0.5, then this solution would remain stable until it terminates in

the upper part of the PH curve. Likewise, solutions continued from line (4), for

K≈0.1−0.15, would remain stable until they terminate in the left part of the

PH curve.

We now compare the path continuation results with the maps of the dynamics

obtained by direct numerical integration. To aid with this comparison the bifurca-

tion curves have been plotted with the numerically obtained maps in Figs. 3.38 and

3.39. Looking ﬁrst at Fig. 3.38a, within the main resonance regions (white), the

upper boundaries of fundamental mode-locking are given by the saddle-node bifur-

cations (labelled SN1and SN6in Fig. 3.35). The down-sweep results (Fig. 3.38b)

show that the continuous wave stability boundaries are given by the Hopf curves.12

There are no features in the numerical maps related to the three torus bifurcation

12At the boundaries with harmonic mode-locking there is a slight discrepancy between the Hopf

curves and the numerically obtained boundaries for stable continuous wave lasing. This could

be due the inaccuracies in the path continuation, since the accuracy of the stability calculations

depends on the delay times and for the higher-order Hopf curves the time scale separation

between the delay times and the frequencies of the Hopf bifurcations is greater.

3.3. Dynamics induced by feedback from a single external cavity

τ(T0,S)

0.0

0.5

1.0

1.5

2.0

up-sweep

(a)

τ(T0,S)

(b)

down-sweep

FML

2ndHML

3rdHML

4thHML

5thHML

2pulses

3pulses

4pulses

5pulses

CD2

CD3

CD4

CD5

Figure 3.38.: Comparison of numerics and path continuation: Maps of the dynamics in

dependence of τand Kobtained for (a) up- and (b) down-sweeps in Kand the bifurcations of the

continuous wave and fundamental mode-locked solutions. The feedback delay time is plotted in

units of the period for K=0,T0,S =1.014. Blue lines indicate torus bifurcations and green lines

are saddle-node bifurcations and the red line is a period-halving bifurcation. The black line is the

fundamental Hopf bifurcation curve of the continuous wave solution. The thin grey lines show

the remaining Hopf bifurcations of the continuous-wave solution (compare with Fig. 3.34 (a)).

The colour code indicates the number of pulses in the laser cavity and the type of dynamics that

are exhibited. White indicates fundamental mode-locking (FML), purples indicate fundamental

mode-locking with feedback induced pulses, greens indicate complex dynamics (in this case

quasi-periodic) with 2-5 pulses in the laser cavity (CD2-CD5) and blues indicate harmonic

mode-locking (HML). The grey regions indicate continuous-wave solutions. Parameters: all

other parameters as in Table 2.1.

branches in the main resonance regions. This is because these are bifurcations of un-

stable solutions (see Fig. 3.36 (1)). For comparisons in the low Kregions a smaller

τ-Krange is plotted in Fig. 3.39. Here it can be seen that the boundaries between

the fundamental mode-locked regions (both with (purple) and without (white)

additional feedback induced pulses) are given by torus and saddle-node bifurca-

tions. The torus bifurcations add an extra incommensurable frequency, leading to

the quasi-periodic dynamics. In the τ=1

2T0,K resonance region, the boundary

between the two pulse dynamics (purple) and 2nd order harmonic mode-locking

(blue) is given by the period-halving bifurcation. In this region, starting from low

feedback strengths, the amplitude of the second feedback induced pulse increases

with K, until it has the same amplitude as the main pulse. At this point the

solution is identical to the 2nd order harmonic mode-locked solution and the funda-

mental mode-locked solution ceases to exist. In the low Kregions (K<0.1), the

results of the path continuation show that the stable solutions do not undergo any

bifurcations (Fig. 3.37 (6)). This means that the proﬁle of the solutions changes

continuously between the higher-order resonance regions. Based on these results

3. Mode-locked laser dynamics

τ(T0,S)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

up-sweep

(a)

τ(T0,S)

(b)

down-sweep

FML

2ndHML

3rdHML

4thHML

5thHML

2pulses

3pulses

4pulses

5pulses

CD2

CD3

CD4

CD5

Figure 3.39.: Comparison of numerics and path continuation: Maps of the dynamics in