Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops

Bifurcation analysis of a semiconductor laser

with two filtered optical feedback loops

Piotr Marek Słowi nski

Department of Engineering Mathematics

University of Bristol

A dissertation submitted to the University of Bristol in

accordance with the requirements of the degree of

Doctor of Philosophy in the Faculty of Engineering.

May 2011

Abstract

We study the solution structure and dynamics of a semiconductor laser receiving delayedfiltered optical feedback from two filter loops; this system is also referredto as the 2FOF laser.The motivation for this study comes from optical communication applications where two filtersare used to control and stabilize the laser output. The overall mathematical model of the 2FOFlaser takes the form of delay differential equations for the (real-valued) inversion of the laser,and the (complex-valued) electric fields of the laser and of the two filters. There are two timedelays that arise from the travel times from the laser to each of the filters andback.

Since, in the optical communication applications the main concern is stable operation ofthe laser source, in our analysis we focus on the continuous-wave solutions of the 2FOF laser.These basic solutions are known as external filtered modes (EFMs), andthey have been studiedfor the case of a laser with only a single filtered optical feedback loop. Nevertheless, comparedto the single FOF laser, the introduction of the second filter significantly influences the structureand stability of the EFMs and, therefore, the laser’s operation.

To analyse the structure and stability of the EFMs we compute and representthem as anEFM surface in(ωs, dCp, Ns)-space of frequencyωs, filter phase differencedCp, and popu-lation inversionNs of the laser. The parameterdCp is a measure of the interference betweenthe two filter fields, and it is identified as a key to the EFM structure. To analysehow the struc-ture and stability of the EFMs depend on all the filter and feedback loop parameters, we makeextensive use of numerical continuation techniques for delay differential equations and asso-ciated transcendental equations. Furthermore, we use singularity theoryto explain changes ofthe EFM surface in terms of the generic transitions through its critical points. Presented in thiswork is a comprehensive picture of the dependence of the EFM surfaceand associated EFMstability regions on all filter and feedback loop parameters. Our theoreticalresults allow us tomake certain predictions about the operation of a real 2FOF laser device.Furthermore, theyshow that many other laser systems subject to optical feedback can be considered as limitingcases of the 2FOF laser.

Overall, the EFM surface is the natural object that one should consider tounderstand dy-namical properties of the 2FOF laser. Our geometric approach allows us to perform a multi-parameter analysis of the 2FOF laser model and provides a compact way ofunderstandingthe EFM solutions. More generally, our study showcases the state-of-the-art of what can beachieved in the study of delay equations with considerable number of parameters with ad-vanced tools of numerical bifurcation analysis.

Acknowledgements

First of all, I would like to thank my supervisors Prof. Bernd Krauskopf and Dr. Sebastian

M. Wieczorek. Their support and encouragement made my PhD project and stay in Bristol a

great experience, and their wise guidance made an invaluable contributionto my research and

development. Additional thanks go to Dr. Harmut Erzgräber. His publishedworks, as well as

private communication, helped me to establish my research project. Furthermore, I would like

to thank Prof. Dirk Roose and Dr. David A. W. Barton, who agreed to review my dissertation.

I greatly appreciate all financial support I received during my PhD. The Great Western

Research Initiative funded my PhD research under studentship number 250, with support from

Bookham Technology PLC (now Oclaro Inc.). The Bristol Center for Applied Nonlinear Math-

ematics provided support during the write-up of this thesis. The Society forIndustrial and Ap-

plied Mathematics granted me a SIAM Student Travel Award to attend the SIAM Conference

on Applications of Dynamical Systems in May 2009 at Snowbird, Utah. The European Com-

mission Marie Curie fellowship supported my attendance at the TC4 SICON event in Lyon,

France, in March 2009. The Centre de Recherches Mathématiques supported my attendance

at the workshop and mini-conference on "Path Following and Boundary Value Problems" in

Montréal, Canada, in July 2007.

Finally, I would like to thank the Department of Engineering Mathematics and the Uni-

versity of Bristol for providing a creative environment and supporting all my other research

activities. During my stay at the Applied Nonlinear Mathematics research group I met many

inspiring people who became my friends and colleagues. For this I am especially grateful.

“The mathematical description of the world depends on a delicate interplay between discrete

and continuous objects. Discrete phenomena are perceived first, but continuous ones have a

simpler description in terms of the traditional calculus. Singularity theory describes the birth

of discrete objects from smooth, continuous sources.

The main lesson of singularity theory is that, while the diversity of general possibilities is

enormous, in most cases only some standard phenomena occur. It is possible and useful to

study those standard phenomena once for all times and recognize them as the elements of

more complicated phenomena, which are combinations of those standard elements.”

V.I. Arnold

Author’s Declaration

I declare that the work in this dissertation was carried out in accordance with the require-

ments of the University’s Regulations and Code of Practice for ResearchDegree Programmes

and that it has not been submitted for any other academic award. Except where indicated by

specific reference in the text, the work is the candidate’s own work. Workdone in collabora-

tion with, or with the assistance of, others, is indicated as such. Any views expressed in the

dissertation are those of the author.

Signed:

Dated:

Contents

1 Introduction 1

1.1 Modelling the 2FOF laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Classification of EFM structure 11

2.1 External filtered modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 EFM components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.2 EFM components for two identical filters . . . . . . . . . . . . . . . . 17

2.2 The EFM-surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Classification of the EFM surface fordτ = 0 . . . . . . . . . . . . . . . . . . 24

2.3.1 Dependence of the EFM components for fixeddCp = 0 on the detunings 25

2.3.2 EFM surface types withdCp-independent number of EFM components 30

2.3.3 Transitions of the EFM surface . . . . . . . . . . . . . . . . . . . . . . 33

2.3.4 The EFM surface bifurcation diagram in the(∆1, ∆2)-plane for fixedΛ 41

2.4 Dependence of the EFM surface bifurcation diagram on the filter widthΛ . . . 52

2.4.1 Unfolding of the bifurcation at infinity . . . . . . . . . . . . . . . . . . 55

2.4.2 Islands of non-banded EFM surface types . . . . . . . . . . . . . . . .56

2.5 The effect of changing the delay difference∆τ . . . . . . . . . . . . . . . . . 59

2.6 Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3 EFM stability regions 65

3.1 Dependence of EFM stability onκ . . . . . . . . . . . . . . . . . . . . . . . . 67

3.2 Dependence of EFM stability onΛ . . . . . . . . . . . . . . . . . . . . . . . . 70

3.3 Dependence of EFM stability on∆ . . . . . . . . . . . . . . . . . . . . . . . . 77

3.3.1 Influence of hole creation on EFM stability . . . . . . . . . . . . . . . 79

3.3.2 Influence ofSN -transition on EFM stability . . . . . . . . . . . . . . . 86

3.3.3 Influence ofSω-transition on EFM stability . . . . . . . . . . . . . . . 89

3.3.4 Influence ofSC-transition on EFM stability . . . . . . . . . . . . . . . 94

3.4 Dependence of EFM stability ondτ . . . . . . . . . . . . . . . . . . . . . . . 97

3.5 Different types of bifurcating oscillations . . . . . . . . . . . . . . . . . . . .100

i

CONTENTS ii

4 Overall summary 105

4.1 Physical relevance of findings . . . . . . . . . . . . . . . . . . . . . . . . . .107

4.1.1 Experimental techniques for the control of parameters . . . . . . . . . 107

4.1.2 Expected experimental results . . . . . . . . . . . . . . . . . . . . . . 108

4.1.3 Existence of multistability . . . . . . . . . . . . . . . . . . . . . . . . 110

Bibliography 113

Appendices 121

A How to construct the EFM surface 121

A.1 Dealing with theS1-symmetry of the 2FOF laser model . . . . . . . . . . . . . 122

A.2 Computation and rendering of the EFM surface . . . . . . . . . . . . . . . . .123

A.3 Determining the stability of EFMs . . . . . . . . . . . . . . . . . . . . . . . . 125

List of Tables

1.1 System parameters and their values. . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Notation and parameter values for the types of EFM-surface in figure 2.11.

The second and third column show the minimal numberCmin and the maximal

numberCmax of EFM components (for suitable fixeddCp) of the type; note

that in all cases the number of EFM components is independent ofdCp . . . . . 33

2.2 Notation and parameter values for the types of EFM-surface in figure 2.18;

the second and third column show the minimal numberCmin and the maximal

numberCmax of EFM components (for suitable fixeddCp) of the type. . . . . . 44

2.3 Notation and parameter values for the types of EFM-surface in figure 2.21;

the second and third column show the minimal numberCmin and the maximal

numberCmax of EFM components (for suitable fixeddCp) of the type. . . . . . 48

3.1 Axes ranges for all the panels in figure 3.6 . . . . . . . . . . . . . . . . . . .. 81

iii

List of Figures

1.1 Sketch of a 2FOF semiconductor laser realized by coupling to an optical fiber

with two fibre Bragg gratings (a), and by two (unidirectional) feedback loops

with Fabry-Pérot filters (b); other optical elements are beam splitters (BS)and

optical isolators (ISO). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Spectrum of light transmitted (left scale) or reflected (right scale) by aFabry-

Pérot filter (black) and by a fibre Bragg grating (grey). The peak is atthe filter’s

central frequency∆, and the filter widthΛ is defined as the full width at half

maximum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 The graph of (2.3) (black curve) oscillates between its envelope (grey curve)

given by (2.9). Frequencies of EFMs (blue dots) are found from intersection

points of the graph ofΩ(ωs) with the diagonal; also shown are the intersection

points (black dots) with the envelope. HereC1p = 0, C2

p = π/3, ∆1 = −0.1,

∆2 = 0.05, κ1 = 0.05, κ2 = 0.025, Λ1 = Λ2 = 0.005, τ1 = 500 andτ2 = 400. 13

2.2 Projection of EFMs branches onto the(ωs, Ns)-plane (a) and onto the(ωs, C1p)-

plane (b). The open circles are the starting points for three different types of

branches. The blue branch is the EFM component fordCp = 0, the green

branches are for constantC1p , and the red branches are for constantωs. Here

∆1 = ∆2 = 0, κ1 = κ2 = 0.05, Λ1 = Λ2 = 0.015, dτ1 = τ2 = 500 and the

other parameters are as given in Table 1.1. . . . . . . . . . . . . . . . . . . . . 18

2.3 Representation of the EFM surface in(ωs, Ns, C1p)-space; compare with fig-

ure 2.2. Panel (a) shows one fundamental element of the EFM surface (semi-

transparent grey); superimposed are the EFM branches from figure2.2. The

entire EFM surface is a single smooth surface that is obtain by connecting all

2nπ-translated copies of the surface element shown in panel (a). Panels (b) and

(c) show how the EFM branches for constantC1p and for constantωs, respec-

tively, arise as intersection curves of fixed sections with the EFM surface. . . . 20

iv

LIST OF FIGURES v

2.4 Representation of the EFM surface of figure 2.3 in(ωs, Ns, dCp)-space. Panel

(a) shows one fundamental2π interval of the EFM surface (semitransparent

grey); superimposed are the EFM branches from figure 2.2. The entireEFM

surface consists of all2nπ-translated copies of this compact surface, which

touch at the points(ωs, Ns, dCp) = (0, 0, (2n + 1)π); panel (b) shows this

in projection of the surface onto the(ωs, dCp)-plane. Panel (c) illustrates how

the EFM branches for constantωs and the outer-most EFM component for

dCp = 2nπ arise as intersection curves with planar sections. . . . . . . . . . . 22

2.5 EFM-components arising as sections through the EFM surface of figure 2.4.

Panel (a) shows the EFM-surface in(ωs, Ns, dCp)-space, intersected with the

planes defined bydCp = 0 anddCp = 0.9π, respectively. Panels (b1) and

(c1) show the corresponding envelope (grey curves) given by (2.9). The black

solution curve of (2.3) inside it is forC1p = 0; it gives rise to the marked blue

EFMs. Panels (b2) and (c2) show the two respective EFM-components and

individual EFMs (blue dots) in the(ωs, dCp)-plane. . . . . . . . . . . . . . . . 23

2.6 Envelope and solution curve fordCp = 0 (a1)-(d1) and the corresponding

EFM-components and EFMs (blue dots) of the 2FOF laser, were∆1 = 0.2 is

fixed and in panels (a)-(d)∆2 takes the values−0.2, 0, 0.158 and0.2, respec-

tively; hereΛ1 = Λ2 = 0.015, τ1 = τ2 = 500 and the other parameters are as

given in Table 1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.7 Regions in the(∆1, ∆2)-plane with a one, two or three EFM components of

the 2FOF laser fordCp = 0. From (a) to (f)Λ takes the valuesΛ = 0, Λ =

0.001, Λ = 0.01, Λ = 0.06, Λ = 0.12 andΛ = 0.14. . . . . . . . . . . . . . . 28

2.8 Panel (a) shows surfaces (orange and grey) that divide the(∆1, ∆2, Λ)-space

into regions with one, two and three EFM-components of the 2FOF laser for

dCp = 0; in the shown (semitransparent) horizontal cross section forΛ = 0.01

one finds the bifurcation diagram from figure 2.7 (c) . Panel (b) showsthe

bifurcation diagram in the(∆1, Λ)-plane for fixed∆2 = 0.82; the light grey

curve is the boundary curve for the limiting single FOF laser for∆2 = ∞.

Panel (c) shows the projection onto the(∆1, Λ)-plane of the section along the

diagonal∆1 = ∆2 through the surfaces in panel (a). . . . . . . . . . . . . . . 30

2.9 The EFM surface in(ωs, dCp, Ns)-space showing caseB for ∆1 = ∆2 = 0

(a), and showing caseBBB for ∆1 = 0.16, ∆2 = −0.16 (b), whereΛ = 0.015. 31

LIST OF FIGURES vi

2.10 Boundary curves (orange or grey) in the(∆1, ∆2)-plane forΛ = 0.01 for

61 equidistant values ofdCp from the interval[−π, π]; compare with fig-

ure 2.7 (c). In the white regions the 2FOF laser has one, two or three EFM

components independently of the value ofdCp, as is indicated by the labelling

with symbolsB andB; representatives of the four types of EFM components

can be found in figure 2.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.11 The four simple banded types of EFM-surface of the 2FOF laser in thela-

belled regions of figure 2.10, represented by the projection (shaded) onto the

(ωs, dCp)-plane; the blue boundary curves are found directly from (2.16). For

notation and the corresponding values of∆1 and∆2 see Table 2.1; in all panels

ωs ∈ [−0.3, 0.3] anddCp ∈ [−π, π]. . . . . . . . . . . . . . . . . . . . . . . 33

2.12 Minimax transitionM of the EFM-surface in (ωs, dCp, Ns)-space, where a

connected component of the EFM surface (a1) shrinks to a point (b1).Panels

(a2) and (b2) show the corresponding projection onto the (ωs, dCp)-plane of

the entire EFM surface; the local region where the transitionM occurs is high-

lighted by dashed lines and the projections of the part of the EFM surface in

panels (a1) and (b1) is shaded grey. HereΛ = 0.01, ∆1 = 0.4, and∆2 = 0.28

in (a) and∆2 = 0.28943 in (b). . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.13 Saddle transitionSC of the EFM-surface in (ωs, dCp, Ns)-space, where lo-

cally the surface changes from a one-sheeted hyperboloid (a1) to a cone aligned

in the dCp-direction (b1) to a two-sheeted hyperboloid (c1). Panels (a2)–

(c2) show the corresponding projection onto the (ωs, dCp)-plane of the entire

EFM surface; the local region where the transitionSC occurs is highlighted

by dashed lines and the projections of the part of the EFM surface in panels

(a1)–(c1) is shaded grey. HereΛ = 0.01, ∆1 = 0.4, and∆2 = 0.23 in (a),

∆2 = 0.232745 in (b) and∆2 = 0.24 in (c). . . . . . . . . . . . . . . . . . . . 37

2.14 Saddle transitionSω of the EFM-surface in (ωs, dCp, Ns)-space, where a con-

nected component (a1) pinches (b1) and then locally disconnects (c1);here the

associated local cone in panel (b1) is aligned in theωs-direction. Panels (a2)–

(c2) show the corresponding projection onto the (ωs, dCp)-plane of the entire

EFM surface; the local region where the transitionSω occurs is highlighted

by dashed lines and the projections of the part of the EFM surface in panels

(a1)–(c1) is shaded grey. HereΛ = 0.01, ∆1 = 0.4, and∆2 = 0.13 in (a),

∆2 = 0.133535 in (b) and∆2 = 0.135 in (c). . . . . . . . . . . . . . . . . . . 38

LIST OF FIGURES vii

2.15 Saddle transitionSN of the EFM-surface in (ωs, dCp, Ns)-space, where two

sheets that lie on top of each other in theNs direction (a1) connect at a point

(b1) and then create a hole in the surface (c1); here the associated local cone

in panel (b1) is aligned in theN -direction. Panels (a2)–(c2) show the corre-

sponding projection onto the (ωs, dCp)-plane of the entire EFM surface; the

local region where the transitionSN occurs is highlighted by dashed lines and

the projections of the part of the EFM surface in panels (a1)–(c1) is shaded

grey. HereΛ = 0.01, ∆1 = 0.4, and∆2 = 0.0.11 in (a),∆2 = 0.11085 in (b)

and∆2 = 0.1115 in (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.16 Cubic tangencyC of the EFM-surface in (ωs, dCp, Ns)-space, where a part

of the surface (a1) becomes tangent to a planedCp = const (b1) and then

develops a bulge (c1). The unfolding of the cubic tangency into twodCp-folds

can be seen clearly in the projections onto the (ωs, dCp)-plane in panels (a2)–

(c2). HereΛ = 0.015, and(∆1, ∆2) = (−0.03,−0.0301) in (a), (∆1, ∆2) =

(−0.04,−0.0401) in (b) and(∆1, ∆2) = (−0.05,−0.051). . . . . . . . . . . . 40

2.17 EFM surface bifurcation diagram in the(∆1, ∆2)-plane forΛ = 0.01 with

regions of different types of the EFM surface; see figure 2.18 for representa-

tives of the labelled types of the EFM surface and Table 2.2 for the notation.

The main boundary curves are the singularity transitionsM (orange curves),

SC (blue curves),Sω (green curves) andSN (red curves). The locus of cu-

bic tangency (black curves) can be found near the diagonal; also shown is the

anti-diagonal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.18 Additional types of EFM-surface of the 2FOF laser in the labelled regions of

figure 2.17, represented by the projection (shaded) onto the (ωs, dCp)-plane;

the blue boundary curves are found directly from (2.16). For notation and

the corresponding values of∆1 and ∆2 see Table 2.2; in all panelsωs ∈[−0.3, 0.3] anddCp ∈ [−π, π]. . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.19 Projection of the EFM-surface onto (ωs, dCp)-plane forΛ = 0.01. Panel (a) is

for ∆1 = −∆2 = 0.003 and panel (b) is for∆1 = −∆2 = 0.08. . . . . . . . . 45

2.20 Enlargement near the center of the(∆1, ∆2)-plane of figure 2.17 with (blue)

curves ofSC transition, (red) curves ofSN transition, and (black) curvesCa

and (grey) curvesCd of cubic tangency; see figure 2.21 for representatives of

the labelled types of the EFM surface and Table 2.3 for the notation. . . . . . .46

LIST OF FIGURES viii

2.21 Additional types of EFM-surface of the 2FOF laser that feature bulges, repre-

sented by the projection (shaded) onto the (ωs, dCp)-plane; the blue boundary

curves are found directly from (2.16). Where necessary, insets show local en-

largements. The corresponding regions in the(∆1, ∆2)-plane can be found in

figures 2.20, 2.25 and 2.29; for notation and the corresponding values of∆1,

∆2 andΛ see Table 2.3. In all panelsωs ∈ [−0.3, 0.3] anddCp ∈ [−π, π]. . . 47

2.22 Global manifestation of local saddle transitionSC of the EFM-surface where

two bulges connect to form a hole. Panels (a1)–(c1) show the relevantpart of

the EFM surface and panels (a2)–(c2) the corresponding projection onto the

(ωs, dCp)-plane. HereΛ = 0.015 and∆2 = −0.02, and∆1 = −0.0248 in

(a),∆1 = −0.02498 in (b) and∆1 = −0.0252 in (c). . . . . . . . . . . . . . . 49

2.23 Sketch of the bifurcation diagram in the(∆1, ∆2)-plane near the (purple)

codimension-two pointDCNC on the curveC of cubic tangency, from which

the (red) curveSN and the (blue) curveSC of saddle transition emanate; com-

pare with figures 2.20 and 2.29 (a) and (b). . . . . . . . . . . . . . . . . . . . .50

2.24 Sketch of the bifurcation diagram in the(∆1, ∆2)-plane near the (golden)

codimension-two pointDCMω on the curveC of cubic tangency, from which

the (orange) curveM and the (green) curveSω of saddle transition emanate;

compare with figures 2.17 and 2.28. . . . . . . . . . . . . . . . . . . . . . . . 50

2.25 Enlargement near the diagonal of the(∆1, ∆2)-plane with (blue) curves ofSC

transition, (green) curves ofSω transition, andSC transition, and (black) curves

Cd of cubic tangency; see figure 2.21 for representatives of the labelled types

of the EFM surface and Table 2.3 for the notation. Panel (a) is forΛ = 0.01 as

figure 2.17, and panel (b) is forΛ = 0.02 . . . . . . . . . . . . . . . . . . . . 52

2.26 EFM surface bifurcation diagram in the compactified(∆1, ∆2)-square,[−1, 1]×[−1, 1], showing regions of band-like EFM surface types; compare with fig-

ure 2.10. The boundary of the square corresponds to∆i = ±∞; from (a) to

(e) Λ takes valuesΛ = 0.01, Λ = 0.015, Λ = 0.06, Λ = 0.098131, Λ = 0.1

andΛ = 0.13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.27 Sketch of EFM surface bifurcation diagram near the boundary∆2 = −1 of

the(∆1, ∆2)-square in the transition throughΛ = ΛC . Panels (a1)–(a3) show

the transition involving the (black) curveCa of cubic tangency that bounds the

orange islands, and panels (b1)–(b3) show the transition involving the (grey)

curve Cd of cubic tangency that bounds the grey islands; compare with fig-

ure 2.26 (c)–(e). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.28 Grey island forΛ = 0.1 in the (∆1, ∆2)-plane with regions of non-banded

EFM surface types; compare with figure 2.26 (e). . . . . . . . . . . . . . . . .56

LIST OF FIGURES ix

2.29 Orange island in the(∆1, ∆2)-plane with regions of non-banded EFM surface

types; the inset in panel (a) shows the details of curves and regions. From (a)

to (d) Λ takes the valuesΛ = 0.1, Λ = 0.145, Λ = 0.166 andΛ = 0.179;

compare panel (a) with figure 2.26 (e). . . . . . . . . . . . . . . . . . . . . . . 58

2.30 The EFM surface (a) forκ = 0.05, ∆1 = ∆2 = 0, Λ = 0.015, andτ1 = 500

andτ2 = 600 so thatdτ = 100, and its intersection with the planes defined by

dCp = 0 anddCp = −π; compare with figure 2.5. Panels (b)–(e) show the

EFM-components fordCp = 0, dCp = −π/2, dCp = −π anddCp = −3π/2,

respectively; the blue dots are the EFMs forC1p as given by (2.12). . . . . . . . 60

2.31 Solution curves of the transcendental equation (2.3) and corresponding EFM

components fordCp = 0, where the dots show the actual EFMs forC1p=0;

hereκ = 0.05, ∆1 = ∆2 = 0, Λ = 0.015, τ1 = 500, anddτ = 200 in panels

(a) anddτ = 300 in panels (b). The inset of panel (b2) shows that the EFM

components are in fact disjoint. . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.32 The EFM surface of typehBB for dτ = 0 (a1) and its EFM components for

dCp = −1.6π (a2), and the corresponding sheared EFM surface fordτ = 230

(b1) and its EFM components fordCp = −1.6π (b2). Hereκ = 0.05, ∆1 =

0.13, ∆2 = −0.1, Λ = 0.01, andτ1 = 500. . . . . . . . . . . . . . . . . . . . 62

3.1 Dependence of the EFM surface on the feedback rateκ (as indicated in the

panels); here∆1 = ∆2 = 0, Λ = 0.015 and dτ = 0. Panels (a1)–(c1)

show the EFM-surface in(ωs, Ns, dCp)-space (semitransparent grey) together

with information about the stability of the EFMs. Panels (a2)–(c2) show cor-

responding projections of the EFM surface onto the(ωs, Ns)-plane and pan-

els (a3)–(c3) onto the(ωs, dCp)-plane. Regions of stable EFMs (green) are

bounded by Hopf bifurcations curves (red) and saddle node bifurcation curves

(blue). In panels (a1)–(c1)ωs ∈ [−0.065, 0.065], dCp/π ∈ [−1, 1] and

Ns ∈ [−0.013, 0.013]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.2 Dependence of the EFM surface on the filter widthΛ = Λ1 = Λ2 (as in-

dicated); here∆1 = ∆2 = 0, κ = 0.01 anddτ = 0. Light grey regions

with black envelopes are projections, for five different values ofΛ, of the EFM

surface onto the(ωs, Ns)-plane. . . . . . . . . . . . . . . . . . . . . . . . . . 70

LIST OF FIGURES x

3.3 Dependence of the stability region on the EFM surface on the common filter

width Λ (as indicated in the panels); here∆1 = ∆2 = 0, κ = 0.01 anddτ = 0.

Panels (a1)–(d1) show the EFM-surface in(ωs, Ns, dCp)-space (semitrans-

parent grey) together with information about the stability of the EFMs. Pan-

els (a2)–(d2) show corresponding projections of the EFM surface onto the

(ωs, Ns)-plane. Black dots indicate codimension-two Bogdanov-Takens bi-

furcation points; curves and regions are coloured as in figure 3.1. In panels

(a1)–(d1)dCp ∈ [−π, π], and the ranges ofNs andωs are as in panels (a2)–(d2). 72

3.4 Projections of the EFM surfaces presented in figure 3.3 (a1)–(d1) onto the

(ωs, dCp)-plane. Black dots indicate codimension-two Bogdanov-Takens bi-

furcation points; curves and regions are coloured as in figure 3.1. . . . .. . . . 73

3.5 Projections of the EFM surface onto the(ωs, Ns)-plane, for increasing filter

width Λ as indicated in the panels; here∆1 = ∆2 = 0, κ = 0.01 anddτ =

0. Black dots indicate codimension-two Bogdanov-Takens bifurcation points;

curves and regions are coloured as in figure 3.1. . . . . . . . . . . . . . . .. . 75

3.6 Influence of local saddle transitionSC , where two bulges connect to form a

hole, on the stability of the EFMs; here∆2 = 0, κ = 0.01, Λ = 0.005 and

dτ = 0. Panels (a1)–(c1) shows two copies of the fundamental2π-interval

of the EFM surface for different values of a detuning∆1, as indicated in the

panels. Panels (a2)–(c2) show enlargements of the region where the hole is

formed. In panels (a1)–(c1) the limit of thedCp-axis corresponds to a planar

section that goes through middle of the hole in panels (a2)–(c2). For the spe-

cific axes ranges see Table 3.1; curves and regions are coloured as infigure 3.1. 80

3.7 Projections with stability information of the EFM surface in figure 3.6 onto

the (ωs, dCp)-plane, shown for increasing filter detuning∆1 = 0.0005 (a),

∆1 = 0.0007 (b) and∆1 = 0.005 (c); here∆2 = 0, κ = 0.01, Λ = 0.005 and

dτ = 0. To illustrate the changes in the EFM surface, panels (a1)–(c1) show

the2π interval of the EFM surface that is shifted byπ with respect to the fun-

damental2π interval of the EFM-surface. Panels (a2)–(c2) show enlargements

of the central part of panels (a1)–(c1). Curves and regions are coloured as in

figure 3.1; dark green colour indicates that there are two stable regions on the

EFM surface that lie above one another in theNs direction. . . . . . . . . . . . 82

3.8 The EFM surface with stability information for filters detunings∆1 = 0.024

and∆2 = 0. Hereκ = 0.01, Λ = 0.005 anddτ = 0; curves and regions are

coloured as in figure 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

LIST OF FIGURES xi

3.9 Projections of the EFM surface with stability information onto the(ωs, dCp)-

plane for increasing filter detuning∆1 = 0.0065 (a),∆1 = 0.0085 (b), ∆1 =

0.0165 (c), ∆1 = 0.0175 (d), ∆1 = 0.0215 (e) and∆1 = 0.024 (f). Here

∆2 = 0, κ = 0.01, Λ = 0.005 anddτ = 0; curves and regions are coloured as

in figure 3.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.10 The EFM surface with stability information for filters detunings∆1 = 0.024

and∆2 = −0.025. Hereκ = 0.01, Λ = 0.005 anddτ = 0; curves and regions

are coloured as in figure 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87


plane for increasing filter detunings∆2 = −0.012 (a), ∆2 = −0.023 (b),

∆2 = −0.024 (c), ∆2 = −0.025 (d). Here∆1 = 0.024, κ = 0.01, Λ = 0.005

anddτ = 0; curves and regions are coloured as in figure 3.7. . . . . . . . . . . 88

3.12 The EFM surface with stability information for∆2 = −0.035 (a), ∆2 =

−0.037 (b). Here∆1 = 0.024, κ = 0.01, Λ = 0.005 anddτ = 0; curves

and regions are coloured as in figure 3.1. . . . . . . . . . . . . . . . . . . . . 90


plane for∆1 = 0.024, ∆2 = −0.036 (a), ∆1 = 0.024, ∆2 = −0.037 (b),

∆1 = 0.026, ∆2 = −0.037 (c), ∆1 = 0.029, ∆2 = −0.037 (d), ∆1 =

0.035, ∆2 = −0.037 (e) and∆1 = 0.036, ∆2 = −0.037 (f). Hereκ = 0.01,

Λ = 0.005 anddτ = 0; curves and regions are coloured as in figure 3.7. . . . . 93

3.14 The EFM surface with stability information for∆1 = 0.044 (a), ∆1 = 0.050

(b). Here∆2 = −0.049, κ = 0.01, Λ = 0.005 anddτ = 0; curves and regions

are coloured as in figure 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 95


plane for∆1 = 0.039, ∆2 = −0.041 (a), ∆1 = 0.039, ∆2 = −0.045 (b),

∆1 = 0.044, ∆2 = −0.049 (c) and∆1 = 0.050, ∆2 = −0.049 (d). Here

κ = 0.01, Λ = 0.005 anddτ = 0; curves and regions are coloured as in

figure 3.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96


plane for increasing delay time in the second feedback loopτ2 = 506 (a),τ2 =

514 (b), τ2 = 562 (c) andτ2 = 750 (d); hereτ1 = 500, κ = 0.01, Λ = 0.015

and∆1 = ∆2 = 0. Black dots indicate codimension-two Bogdanov-Takens

bifurcation points; curves and regions are coloured as in figure 3.7. . . .. . . . 99

xii LIST OF FIGURES

3.17 Example of relaxation oscillations (a) and frequency oscillations (b) found in

the EFM stability diagram in figure 3.7 (c); for∆1 = 0.005, ∆2 = 0, κ = 0.01,

Λ = 0.005 anddτ = 0. RO are found at(ωs, dCp/π) = (0.0035, 1.828), and

FO at(ωs, dCp/π) = (0.0031, 0.802). The different rows show from top to

bottom: the intensityIL and the frequencyφL = dφL/dt of the laser field, the

intensityIF1 and the frequencyφF1 = dφF1/dt of the first filter field, and the

intensityIF2 and the frequencyφF2 = dφF2/dt of the second filter field. Note

the different time scales for ROs and FOs. . . . . . . . . . . . . . . . . . . . . 101

4.1 Regions in the(C1p , dCp)-plane with different numbers of coexisting EFMs,

as indicated by the labelling. Panel (a) shows the regions on a fundamental

2π-interval ofC1p , while panel (b) shows it in the covering space (over several

2π-intervals ofC1p ). Boundaries between regions are saddle-node bifurcation

curves (blue); also shown in panel (b) are periodic copies of the saddle-node

bifurcation curves (light blue). Labels Here∆1 = 0.050, ∆2 = −0.049,

κ = 0.01, Λ = 0.005 anddτ = 0; these parameter values are those for the

EFM surface in figure 3.14 (b). . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.2 EFM-components (grey) in the(ωs, Ns)-plane with stability information. Panel

(a) is fordCp = π, ∆1 = 0.050, ∆2 = −0.049, and panel (b) is fordCp = −π,

∆1 = 0.036, ∆2 = −0.037; furthermore,κ = 0.01, Λ = 0.005 anddτ = 0.

Stable segments of the EFM-components (green) are bounded by the Hopfbi-

furcations (red dots) or by the saddle-node bifurcation (blue dots). The actual

stable EFMs forC1p = 1.03π (a) andC1

p = 0.9π (b) are the black full circles;

open circles are unstable EFMs. The EFM components in panel (a) correspond

to a constantdCp-section through the EFM surface in figure 3.14 (b), and those

in panel (b) to a constantdCp-section through the EFM surface in figure 3.12 (d).112

Chapter 1

Introduction

Semiconductor lasers are very efficient in transforming electrical energy to coherent light. Co-

herent light is created in the laser by recombination of electron-hole pairs,which are generated

by an electrical pump current. The light is reflected by semitransparent mirrors that form the

laser cavity, and is amplified by stimulated emission during multiple passages through an am-

plifying semiconductor medium. The output light exits through one (or both) ofthe semitrans-

parent mirrors [33, 46]. Semiconductor lasers are small (about 1 millimetre long and several

micrometres wide), can easily be mass produced and are used in their millions in every day

applications — most importantly, in optical telecommunication and optical storage systems.

On the down side, semiconductor lasers are known to be very sensitive to optical influences,

especially in the form of external optical feedback from other optical components (such as

mirrors and lenses) and via coupling to other lasers. Depending on the exact situation, optical

feedback may lead to many different kinds of laser dynamics, from increased stability [6, 23]

all the way to complicated dynamics; for example: a period doubling cascade tochaos [73],

torus break-up [70], and a boundary crisis [69] have been identified. See [33, 38] as entry

points to the extensive literature on the possible dynamics of lasers with opticalfeedback.

The simplest and now classical example of optical feedback is conventional optical feed-

back (COF) where light is reflected on a normal mirror and then re-entersthe laser [43]. How-

ever, other types of laser systems with optical feedback have been considered, including lasers

with two COF feedback loops [54], with incoherent feedback [20], with optoelectronic feed-

back [44], with phase-conjugate feedback (PCF) [7, 37] and with filtered optical feedback

(FOF) [14, 28]. In all these cases an external feedback loop, or external cavity, is associated

with a delay timeτ that arises from the travel time of the light before it re-enters the laser.

Due to the fast time scales within a semiconductor laser (on the order of picoseconds), external

optical paths of a few centimetres lead to considerable delay times that cannotbe ignored. As

a consequence, an optical feedback created by an external cavity allows the laser to operate at

various compound-cavity modes; they are referred to as continuous-wave (cw) states because

1

2 Chapter 1. Introduction

the laser produces constant-intensity output at a specific frequency. The cw-states are the sim-

plest nonzero solutions of the system and they form the backbone for understanding the overall

dynamics, even when they are unstable. For example, the typical dynamics of a COF laser

with irregular drop-outs of the power has been attributed to trajectories thatpass closely near

cw-states of saddle type [25, 58].

A main concern in practical applications is to achieve stable, and possibly tunable, laser op-

eration. One way of achieving this has been to use filtered optical feedback where the reflected

light is spectrally filtered before it re-enters the laser — one speaks of the(single) FOF laser.

As in any optical feedback system, important parameters are the delay time andthe feedback

strength. Moreover, FOF is a form of coherent feedback, meaning that the phase relationship

between outgoing and returning light is also an important parameter. The interest in the FOF

laser is due to the fact that filtering of the reflected light allows additional control over the

behaviour of the output of the system by means of choosing the spectral width of the filter and

its detuning from the laser frequency. The basic idea is that the FOF laser produces stable out-

put at the central frequency of the filter, which is of interest, for example, for achieving stable

frequency tuning of lasers for the telecommunications applications [9].

The single FOF laser system has recently been the subject of a number of experimental

and theoretical studies [15, 18, 22, 23, 27, 28, 31, 51, 64, 74, 76, 77]. Here, we assume that a

solitary laser (i.e. without feedback) emits light of constant intensity and frequencyΩ0. It has

been shown that single FOF can improve the laser performance [6, 23], but it can also induce

a wide range of more complicated dynamics. Its cw-states are called externalfiltered modes

(EFMs) [74], and they lie on closed curves, called EFM-components, in the (ωs, Ns)-plane of

the lasing frequencyωs (relative to the solitary laser frequencyΩ0) and inversionNs of the

laser (the number of electron-hole pairs). The EFM-components are traced out by the EFMs as

the phase of the electric field of the filter (relative to the phase of the laser field), called here the

feedback phaseCp, is varied. An analysis in [28] with dependence on the spectral width of the

filter, Λ, and the detuning between the filter central frequency and the solitary laser frequency,

∆, showed that there may be at most two EFM-components for the FOF laser: one around the

solitary laser frequency and one around the filter central frequency.A stability and bifurcation

analysis of EFMs in [17] shows that the FOF laser is very sensitive to changes in feedback

phaseCp. Furthermore, the filter parametersΛ and∆ have a big influence on the possible

(non steady-state) dynamics [16, 19]. Importantly, in a FOF laser one canobserve not only

the well-known relaxation oscillations, but also so-called frequency oscillations where only the

frequency of the laser oscillates while its intensity remains practically constant[24]. In light

of the strong amplitude-phase coupling of semiconductor lasers, the existence of frequency

oscillations is somewhat surprising, and they are due to an interaction with the flanks of the

filter transmittance profile [16]. An experimental study of the influence of feedback phaseCp

and frequency detuning∆ on the single FOF laser dynamics can be found in [19]. The limiting

3

(a)Laser Optical fiber

Grating 1 Grating 2

(b)Laser

E(t), N(t)

Ω, α, P, T

κ2F2(t, τ2, C2

p)

κ1F1(t, τ1, C1

p)

BS

ISO ISO ISO ISO

Filter 1

F1(t)

Λ1, ∆1

Filter 2

F2(t)

Λ2, ∆2.

.

Figure 1.1. Sketch of a 2FOF semiconductor laser realized by coupling toan optical fiber with two

fibre Bragg gratings (a), and by two (unidirectional) feedback loops with Fabry-Pérot filters (b); other

optical elements are beam splitters (BS) and optical isolators (ISO).

cases of small and largeΛ and∆ have been considered in [28, 31, 74]. In all these studies the

filter transmittance has a single maximum defining its central frequency; the FOFproblem for

periodic filter transmittance with multiple maxima and minima was considered in [64].

Given the large number of parameters and the transcendental nature of the equations for

the EFMs of the single FOF laser, possibilities for their analytical study are somewhat limited;

examples of such studies are the [15, 28, 31]. In [15] asymptotic expansion methods are used

to simplify the rate equations of the FOF laser and to investigate the injection laser limitΛ →0. In [28] dependance of the EFM structure onΛ and ∆ is analysed by reduction of the

transcendental equation for their frequencies to a fourth-degree polynomial. Finally, the study

presented in [31] explores the transition of the FOF laser from the injection laser limit to the

COF laser limit, by analysing the limit casesΛ → 0 andΛ → ∞. Due to limitations of the

analytical approaches, the single FOF laser is analysed mainly by means of numerical methods.

In particular, popular techniques include: root finding, numerical integration and numerical

contiunation. Root finding is used, for example, in [74, 77], to solve transcendental equation

for the frequencies of EFMs. Numerical integration provides means to compare output of

the model with experimental time series; see, for example, [30, 25, 49]. Finally, numerical

continuation allows for very detailed bifurcation analysis of the investigated system; as was

performed for example, in [17].

In a number of applications, such as the design of pump lasers for optical communication


Fiber Bragg grating

Fabry–Perot filter

TR

AN

SM

ITT

AN

CE

0%

100%

RE

FL

EC

TA

NC

E

100%

0%

∆ frequency

-Λ

.

.

Figure 1.2. Spectrum of light transmitted (left scale) or reflected (right scale) by a Fabry-Pérot filter

(black) and by a fibre Bragg grating (grey). The peak is at the filter’s central frequency∆, and the filter

width Λ is defined as the full width at half maximum.

systems, the requirements on stable and reliable laser operation at a specificfrequency are so

stringent (especially when the device is on the ocean floor as part of a long-range fiber cable)

that other methods of stabilisation have been considered. One approach isto employ FOF from

two filtered feedback loops to stabilise the output of an (edge emitting) semiconductor laser

[4, 9, 21, 52]. This laser system is referred to as the 2FOF laser for short, and it is the subject of

the study presented here. The main idea is that the second filter provides extra frequency control

over the laser output. In [4] it has been shown that a second filtered feedback loop may indeed

improve the beam quality. Moreover, in [21] an experimental setup has been realized and it was

shown that the laser may show complicated dynamics as well; however, such dynamics have

not yet been investigated further. With the focus on enhanced stability, industrial pump sources

with enhanced wavelength and power stability performance due to a 2FOF design are available

today [52]. The 2FOF laser has also been considered recently for frequency switching [9] and

for sensor applications [59, 60].

The filtered feedback of the 2FOF laser can be realized in two ways: eitherby reflection

from an optical fibre with two fibre Bragg gratings (periodic changes of the refractive index)

at given distances, or by transmission through two unidirectional feedback loops with Fabry-

Pérot filters; see figure 1.1. The two setups are equivalent in the sensethat the overall spectral

characteristics of the filtering is the same. More specifically, a fibre Bragg grating (FBG) has

a peak in the reflectance at its central frequency, while a Fabry-Pérotfilter (FP) has a peak in

the transmittance at its central frequency; see figure 1.2. There are someimportant practical

differences between the two setups in figure 1.1 that are discussed in moredetail in section 1.1.

Nevertheless, the 2FOF laser in either form can be modelled by rate equations for the complex-

valued electric fieldE inside the laser, for the real-valued population inversionN inside the

laser, and for the complex-valued electric fieldsF1 andF2 inside the two filters. The 2FOF

laser is hence described by a delay differential equation (DDE) model, equations (1.1)–(1.4)

1.1. Modelling the 2FOF laser 5

introduced in detail in section 1.1 below, which describes the time evolution of seven real-

valued variables and in the presence of two discrete delay timesτ1 andτ2.

1.1 Modelling the 2FOF laser

The 2FOF design with fibre Bragg gratings (FBG) as in figure 1.1 (a) is the one that has been

employed in industrial pump sources [52], and it is also the one consideredin [21, 59, 60]; see

also the analysis of FOF from a FBG in [50]. The main advantage is that FBGsare simple and

cheap to manufacture in a fiber at desired locations; furthermore, apartfrom the need to couple

the light into the fiber without any direct reflections (to avoid COF), no additional optical

elements are required so that the device is relatively simple; see figure 1.1 (a). The downside is

that, once it is imprinted into an optical fiber, a FBG cannot be modified. Furthermore, the two

filters are not independent feedback loops. When the two FBGs operateat different frequencies

(as in the actual devices) then they are transparent to each other’s central frequency, meaning

that the first FBG only slightly weakens the light reflected from the second FBG and one may

assume that there are no direct interactions between these two filters. However, when both

FBGs operate very close to the same frequency then the feedback from the second FBG is

almost completely blocked, so that the laser receives feedback only fromone filter. Another

issue is that the light reflected from a FBG is due to the interaction with the entire grating, which

means that the round-trip time of the reflected light is not so easy to determine. Furthermore,

the optical fiber and the FBGs are susceptible to mechanical strain and to thermal expansion.

Such perturbations result in a modifications of the filter central frequencyand the feedback

phase via changes of the feedback loop length at a sub-wavelength scale. For these reasons it is

difficult to perform controlled experiments with the FBGs setup over large ranges of parameters

of interest.

The experimental setup in figure 1.1 (b) is less practical in industrial applications, but it

allows for exact and independent control of all relevant system parameters. More specifically,

the FOF comes from two independent unidirectional filter loops that do not influence one an-

other. The two delay times and feedback phases can easily be changed in the experiment as

independent parameters. Furthermore, the system can be investigated for any combination of

filter frequencies and widths (but note that every change of the filter properties requires a dif-

ferent FP filter). Finally, this type of experimental setup with a single FOF loophas been used

successfully in studies of the single FOF laser [18, 19]. In particular, it has been shown that the

system is modelled very well by a rate equation model where the filters are assumed to have a

Lorentzian transmittance profile [28, 74].

In spite of the differences in terms of which parameter ranges can be explored in an exper-

iment, both realisations of the 2FOF laser in figure 1.1 can be modelled by the dimensionless


rate equations

dE

dt= (1 + iα)N(t)E(t) + κ1F1(t) + κ2F2(t), (1.1)

TdN

dt= P − N(t) − (1 + 2N(t))|E(t)|2, (1.2)

dF1

dt= Λ1E(t − τ1)e

−iC1p + (i∆1 − Λ1)F1(t), (1.3)

dF2


−iC2p + (i∆2 − Λ2)F2(t). (1.4)

The well established assumptions here are that the delay timesτ1 andτ2 are larger than the

light roundtrip time inside the laser and that the filters have a Lorentzian transmittance profile

(figure 1.2); see [22, 28, 74] for more details. More specifically, one obtains Eqs. (1.1)–(1.4)

as an extension of the rate equations model of the single FOF laser [28, Eqs. (1)–(3)] with an

additional equation for the field of the second filter.

Equation (1.1) describes the time evolution of the complex-valued slowly-varying elec-

tric field amplitudeE(t) = Ex(t) + iEy(t) of the laser. Equation (1.2) describes normalised

population inversionN(t) within the laser active medium. In (1.1)–(1.2) the material prop-

erties of the laser are described by the linewidth enhancement factorα (which quantifies the

amplitude-phase coupling or frequency shift under changes in population inversion [32]), the

ratio T between the population inversion and the photon decay rates, and the dimensionless

pump parameterP . Time is measured in units of the inverse photon decay rate of10−11s.

Throughout, we use values of the semiconductor laser parameters from [17] that are given in

table 1.1.

The two FOF loops enter equation (1.1) as feedback termsκ1F1(t) andκ2F2(t) with nor-

malised feedback strengthsκ1 andκ2 [67, p. 93] of the normalised filters fieldsF1(t) and

F2(t). In general, the presence of a filter in the system gives rise to an integralequation for

the filter field. However, in the case of a Lorentzian transmittance profile as assumed here,

derivation of the respective integral equation yields the description of thefilter fields by DDEs

(1.3) and (1.4); see [74] for more details.

The two filter loops are characterised by a number of parameters. As for any coherent

feedback, we have the feedback strengthκi, the delay timeτi and the feedback phaseCip of the

filter field, which is accumulated by the light during its travel through the feedback loop. Hence,

Cip = Ω0τi. Owing to the large difference in time scales between the optical period2π/Ω0 and

the delay timeτi, one generally considersτi andCip as independent parameters. Namely, as has

been justified experimentally [19, 30], changing the length of the feedbackloop on the optical

1.1. Modelling the 2FOF laser 7

Parameter Meaning Value

LaserP pump parameter 3.5α linewidth enhancement factor 5T inversion decay rate / photon decay rate 100

Feedback loopsκ1 first loop feedback strength from 0.01 to 0.05κ2 second loop feedback strength from 0.01 to 0.05τ1 first loop round-trip time 500τ2 second loop round-trip time 500 to 800C1

p first loop feedback phase 2π-periodicC2

p second loop feedback phase 2π-periodic

Filters∆1 first filter central frequency detuning from -0.82 to 0.82∆2 second filter central frequency detuning from -0.82 to 0.82Λ1 first filter spectral width from 0.0 to 0.5Λ2 second filter spectral width from 0.0 to 0.5

Table 1.1. System parameters and their values.

wavelength scale of nanometres changesCip, but effectively does not changeτi. Two different

strategies for changingCip have been used experimentally: in [19] this is achieved by changing

the length of the feedback loop on the optical wavelength scale with a piezo actuator, and in

[30] Cp is varied indirectly through very small changes in the pump current which, inturn,

affectΩ0.

The optical properties of the filters are given by the detunings∆i of their central frequencies

from the solitary laser frequency, and by their spectral widthsΛi, defined as the frequency

width at half-maximum (FWHM) of the (Lorentzian) transmittance profile. In thisstudy we

consider the filter detunings∆1 and∆2 as independent parameters. Furthermore, we keep both

feedback strengths as well as both filter widths equal, so that throughoutwe useκ := κ1 = κ2

andΛ := Λ1 = Λ2. The values of the feedback parameters are also given in table 1.1.

We remark that system (1.1)–(1.4) contains as limiting cases two alternative setups that

have also been considered for the stabilisation of the laser output. First, when the spectral

width of only one filter is very large then the laser effectively receives feedback from an FOF

loop and from a COF loop; see, for example, [3, 13]. Second, when thespectral widths of

both filters are very large then one is dealing with a laser with two external COFloops; see, for

example, [55, 63] and the discussion in section 2.4.2.

System (1.1)–(1.4) shares symmetry properties with many other systems with coherent

optical feedback. Namely, the system has anS1-symmetry [29, 34, 40] given by simultaneous


rotation over any fixed angle ofE and both filter fieldsF1 andF2. This symmetry can be

expressed by the transformation

(E, N, F1, F2) 7→(Eeiβ , N, F1e

iβ , F2eiβ)

(1.5)

for any0 ≤ β ≤ 2π. In other words, solutions (trajectories) of (1.1)–(1.4) are not isolatedbut

come inS1-families. In particular, the EFMs introduced in the next chapter are grouporbits

under this symmetry, and this fact leads to an additional zero eigenvalue [17, 34], which needs

to be considered for stability analysis. Furthermore, the continuation of EFMs with DDE-

BIFTOOL requires isolated solutions, which can be obtained as follows. After substitution of(Eeibt, N, F1e

ibt, F2eibt)

into (1.1)–(1.4) and dividing through by an exponential factor, the

reference frequencyb becomes an additional free parameter. A suitable choice ofb ensures that

one is considering and computing an isolated solution; see [29, 56] and Appendix A for details.

There is also a rather trivial symmetry property: the feedback phasesCip are2π-periodic

parameters, which means that they are invariant under the translation

Cip 7→ Ci

p + 2π. (1.6)

This property is quite handy, because results can be presented either over a compact fundamen-

tal interval of width2π or on the covering spaceR of Cip; see also [17].

1.2 Outline of the thesis

This thesis is organised in the following way. In chapter 2 we discuss the structure of the

continuous-wave solutions — called the EFMs — of the rate equation model (1.1)–(1.4). We

first introduce in section 2.1 the EFMs and show how they can be uniquely determined by

their frequencies, which are given by solutions of a transcendental equation; furthermore we

show that the envelope of the transcendental equation for the frequencies of the EFMs is key to

understanding the structure of the EFMs. In section 2.1.1 we show the correspondence between

the EFM components of the 2FOF laser and the EFM components known for thesingle FOF

laser. In section 2.2 we introduce the EFM surface, the key object of ourstudies. The EFM

surface provides a geometric approach to the multi-parameter analysis of the2FOF laser. In

this way, it allows for comprehensive insight into the dependence of the EFMs on the feedback

phases of both feedback loops. Moreover, we show that the EFM surface of the 2FOF laser is

a natural generalisation of the EFM component of the single FOF laser.

To study the dependence of the EFM surface on filter and feedback loopparameters in

section 2.3, we introduce a classification of the EFM surface into differenttypes. For clarity

of exposition we successively increase the number of parameters involved in our analysis. We

1.2. Outline of the thesis 9

start by considering, in section 2.3.1, how the EFM components depend on the filter detunings

and filter widths. Next, in sections 2.3.2–2.4 we perform a comprehensive analysis for the

EFM surface. In particular, in section 2.3.2 we introduce the different types of the EFM sur-

face. Next, we show that transitions between different types of the EFM surface correspond to

five codimension-one singularity transitions: through extrema and saddle points, and through

a cubic tangency. We present details of these five singularity transitions in section 2.3.3. Fur-

thermore, in section 2.3.4 we use the loci of the transitions to construct the bifurcation diagram

of the EFM surface in the plane of detunings of the two filters. Finally, in section 2.4 we show

how the EFM surface bifurcation diagram in the plane of filter frequency detunings changes

with the filter widths. We finish our analysis of the dependence of the EFM surface on filter and

feedback loops parameters in section 2.5, where we show that a non-zero difference between

the two delay times has the effect of shearing the EFM surface.

In chapter 3 we present a stability analysis of the EFMs. We start by considering how the

stability of EFMs changes with the common filter and feedback loop parameters and then anal-

yse effects of changing the filter detunings and delay times. In section 3.1 weshow that regions

of stable EFMs are bounded by saddle-node and Hopf bifurcations; moreover, we study how

these stable EFM regions change with the increasing feedback strength. Next, in section 3.2 we

present how the regions of stable EFMs are affected by changing the filter widths. Both these

sections show that, although topologically the EFM surface remains unchanged, the stability of

EFMs changes substantially. In section 3.3 we show in what way the regionsof stable EFMs

are influenced by the singularity transitions of the EFM surface (discussed in section 2.3.3) that

occur as the modulus of detunings of the two filters is increased. The last parameters that we

analyse in section 3.4 in terms of the effect on EFM stability are again the delay times: the

main effect is shearing of the EFM stability regions on the EFM surface. Finally, in section 3.5

we briefly describe what kinds of periodic solutions originate from the Hopf bifurcations that

bound regions of stable EFMs.

An overall summary of the thesis can be found in chapter 4, where we also discuss the

possibility of an experimental confirmation of our findings. Finally, AppendixA gives more

details of how the EFM surface has been rendered from data obtained viacontinuation runs

with the package DDE-BIFTOOL.

Chapter 2

Classification of EFM structure

In this chapter we perform an extensive study of the external filtered modes (EFMs) of the

2FOF laser as modelled by the DDE model (1.1)–(1.4). The analysis shows that the second

filter influences the structure of the EFMs of the laser significantly. As was already mentioned,

in the single FOF laser one may find two (disjoint) EFM components. However, in the 2FOF

system, the number of (disjoint) EFM components depends on the exact phase relationship

between the two filters. When the filter loops have the same delay times, the interference

between the filter fields can give rise to at most three EFM components — one around the

solitary laser frequency and one around the central frequencies of the two filters. However,

when the two delay times are not the same, then the interference between the filter fields may

lead to any number of EFM components.

The EFM structure with dependence on the different system parameters isquite compli-

cated and high-dimensional. To deal with this difficulty we present our results in the form

of EFM surfaces in suitable three-dimensional projection spaces. Thesesurfaces are rendered

from EFM curves in several two-dimensional sections, which are computed with the continu-

ation software DDE-BIFTOOL [11] as steady-state solutions of the DDE model; see the Ap-

pendix for more details. We first present the EFM surface for the case of two identical filter

loops, but with nonzero phase difference between the filters. The properties of the EFM com-

ponents for this special case can be explained by considering slices of the EFM surface for

different values of the filter phase difference. We then consider the influence of other param-

eters on the EFM surface. First, we study the influence of the two filter detunings∆1 and∆2

(from the solitary laser frequencyΩ0) on the number of EFM components (for a representative

value of the spectral width of the filters), which provides a connection with and generalisation

of the single FOF case. The result is a bifurcation diagram in the(∆1, ∆2)-plane whose open

regions correspond to different types of the EFM surface. In the spirit of singularity theory,

we present a classification of the EFM surface where the rationale is to distinguish cases with

different numbers of corresponding EFM components. The boundarycurves in the(∆1, ∆2)-

plane correspond to singularity transitions (for example, through saddle points and extrema)

11

12 Chapter 2. Classification of EFM structure

of the EFM surface, and they can be computed as such. In a next step wealso show how the

bifurcation diagram in the(∆1, ∆2)-plane changes with the spectral width of the filters. This

chapter shows how the 2FOF filter transitions between the two extreme cases of an infinitesi-

mally narrow filter profile, which corresponds to optical injection at a fixed frequency, to that

of an infinitely wide filter profile, which is physically the case of conventional(i.e. unfiltered)

optical feedback. In the final part of the chapter we consider the EFM surface for different

delay times of the filter loops, which yields the geometric result that there may be an arbitrary

number of EFM components.

The EFM surface is the natural object that one should consider to understand dynami-

cal properties of the 2FOF laser. Our analysis reveals a complicated dependence of the EFM

surface on several key parameters and provides a comprehensive and compact way of under-

standing the structure of the EFM solutions.

2.1 External filtered modes

The basic solutions of the 2FOF laser correspond to constant-intensity monochromatic laser

operation with frequencyωs (relative to the solitary laser frequencyΩ0). These solutions are

the external filtered modes. Mathematically, an EFM is a group orbit of (1.1)–(1.4) under the

S1-symmetry (1.5), which means that it takes the form

(E(t), N(t), F1(t), F2(t)) =(Ese

iωst, Ns, F1s ei(ωst+φ1), F 2

s ei(ωst+φ2))

. (2.1)

Here,Es, F 1s andF 2

s are fixed real values of the amplitudes of the laser and filter fields,Ns

is a fixed level of inversion,ωs is a fixed lasing frequency, andφ1, φ2 are fixed phase shifts

between the laser field and the two filter fields. To find the EFMs, we substitute the ansatz (2.1)

into (1.1)–(1.4); separating real and imaginary parts [28, 29] then gives the equation

Ω(ωs) − ωs = 0 (2.2)

where

Ω(ωs) = −√

1 + α2

κ1Λ1 sin

(φ1 + tan−1(α)

)√

Λ12 + (ωs − ∆1)

2+

κ2Λ2 sin(φ2 + tan−1(α)

)√

Λ22 + (ωs − ∆2)

2

, (2.3)

and

φi = ωsτi + Cip + tan−1

(ωs − ∆i

Λi

). (2.4)

Equation (2.2) is a transcendental and implicit equation that allows one to determine all pos-

sible frequenciesωs of the EFMs for a given set of filter parameters. More specifically, the

2.1. External filtered modes 13

−0.2 −0.1 0 0.05 0.1−0.3

0

0.3

Ω(ωs)

ωs.

.

Figure 2.1. The graph of (2.3) (black curve) oscillates between its envelope (grey curve) given by

(2.9). Frequencies of EFMs (blue dots) are found from intersection points of the graph ofΩ(ωs) with

the diagonal; also shown are the intersection points (blackdots) with the envelope. HereC1

p = 0,

C2

p = π/3, ∆1 = −0.1, ∆2 = 0.05, κ1 = 0.05, κ2 = 0.025, Λ1 = Λ2 = 0.005, τ1 = 500 and

τ2 = 400.

sought frequency valuesωs of the 2FOF laser can be determined from (2.2) numerically by

root finding; for example, by Newton’s method in combination with numerical continuation.

The two terms of the sum in the parentheses of (2.3) correspond to the firstand the second

filter, respectively. If one of theκi is set to zero, then (2.2) reduces to the transcendental equa-

tion from [28] for the frequencies of EFMs of the single FOF laser. The advantage of the

formulation of (2.2) is that it has a nice geometric interpretation:Ω(ωs) is a function ofωs that

oscillates between two fixed envelopes. More precisely, whenC1p or C2

p are changed over2π

the graph ofΩ(ωs) sweeps out the area in between the envelopes.

Figure 2.1 shows an example of the solutions of (2.2) as intersection points (blue dots)

between the oscillatory functionΩ(ωs) and the diagonal (the straight line through the origin

with slope 1); see also [74]. Onceωs is known, the corresponding values of the other state


variables of the EFMs can be found from

Ns = −

κ1Λ1 cos(φ1)√

Λ12 + (ωs − ∆1)

2+

κ2Λ2 cos(φ2)√Λ2

2 + (ωs − ∆2)2

, (2.5)

Es =

√P − Ns

1 + 2Ns, (2.6)

F 1s =

EsΛ1√Λ1

2 + (ωs − ∆1)2, (2.7)

F 2s =

EsΛ2√Λ2

2 + (ωs − ∆2)2. (2.8)

This means that an EFM is, in fact, uniquely determined by its value ofωs. Furthermore, it is

useful to consider the envelope ofΩ(ωs) (grey curves) so that Figure 2.1 represents all the rele-

vant geometric information needed to determine and classify EFMs. Notice thatin this specific

example the EFMs are separated into three groups. The diagonal intersects the region bounded

by the envelope in three disjoint intervals where frequenciesωs of EFMs may lie; these in-

tervals correspond to three different EFM components as is discussed insection 2.1.1. As

figure 2.1 suggests, EFMs are created and lost in saddle-node bifurcations when an extremum

of the (black) graph passes through one of the boundary points (blackdots) as a parameter (for

example,C1p ) is changed.

This geometric picture is very similar to that for the single FOF laser [28], but there is

an important difference. The envelope ofΩ(ωs) for the FOF laser is found by considering

the extrema of the sine function (in (2.3) for, say,κ2 = 0). It turns out that the envelope

for the single FOF laser is described by a polynomial of degree four, whose roots are the

boundary points of at most two intervals (or components) with possible EFMs[28]. However,

for the 2FOF laser, considering the extrema of the two sine functions in (2.3)is not sufficient

since they appear in a sum. Hence, we also need to consider mixed terms resulting from the

summation; with the use of standard trigonometric formulae, the equation for the envelope can


be found as

Ωe(ωs) = ±√

1 + α2

[κ2

1Λ21

Λ21 + (ωs − ∆1)

2 +κ2

2Λ22

Λ22 + (ωs − ∆2)

2 +

2κ1κ2Λ1Λ2 cos(C2

p − C1p + ωs (τ2 − τ1) + tan−1

(ωs−∆2

Λ2

)− tan−1

(ωs−∆1

Λ1

))

√Λ2

1 + (ωs − ∆1)2√

Λ22 + (ωs − ∆2)

2

]1/2

.

(2.9)

Indeed, when one of theκi is set to zero then (2.9) reduces to the fourth-order polynomial

describing the envelope of the single FOF laser in [28]. However, for general values of the

parameters, (2.9) is a transcendental equation, and not a polynomial of degree six as one might

have hoped; nevertheless, by means of (2.9) the envelopeΩe(ωs) can be plotted readily.

The transcendental nature of (2.9) means that the study of the EFM structure of the 2FOF

laser is a considerable challenge. As is shown here, the key is to find a suitable geometric

viewpoint that allows one to understand the dependence of the EFMs on thedifferent filter

loop parameters. A first observation is that (2.9) depends on the differences

dCp := C2p − C1

p and dτ := τ2 − τ1,

which we will hence consider as parameters in what follows; note thatdCp is 2π-periodic as

well.

2.1.1 EFM components

It is well-known for the single FOF laser that its EFMs lie on closed curves in the (ωs, Ns)-

plane. These curves are called EFM-components, and they arise as the set of all EFMs found

for different values of feedback phaseCp, whilst the other parameters of the system are fixed.

More specifically, whenCp is changed, EFMs are born in saddle-node bifurcations, then move

over the respective EFM components in the direction of increasingωs, and finally disappear

again in saddle-node bifurcations. From an experimental point of view, EFM components

are quite natural objects that can be measured as groups of EFMs whosefrequencies vary

with the feedback phaseCp; see [19]. For the single FOF laser one finds either one or two

EFM components, depending on the properties of the filter. Intuitively, oneexpects one EFM

component centred around the solitary laser frequency and, if the detuning ∆ is large enough,

a second EFM component around the filter central frequency. As was already mentioned, the

exact dependence on the filter properties can be studied for the single FOF laser by considering

the roots of a polynomial of degree four that arises from the equation forthe envelope of the

EFMs; see [28] for details.


For the 2FOF laser the situation is more complicated. Intuitively, one may think thatnow

up to three EFM components may occur in the(ωs, Ns)-plane: one centred around the solitary

laser frequency and two more around the central frequencies of the twofilters. However, this

intuition is not correct, and we will show that one may in fact have any numberof EFM com-

ponents. Physically, the reason for this vastly more complicated EFM structure of the 2FOF

laser is the interference between the two filter fields, which can be interpreted as giving rise to a

complicated ‘effective’ filter profile. Mathematically, the reason behind the more complicated

EFM structure lies in the transcendental nature of the envelope equation (2.9).

In spite of these underlying difficulties, we now proceed with providing a geometrical rep-

resentation of the EFM structure of the 2FOF laser in dependence on system parameters. Since

the transcendental EFM equation (2.2) is complicated and depends on all system parameters,

its solutions can only be found numerically (except for certain very special choices of the pa-

rameters). From the value of the EFM frequencyωs one can compute the values of the other

EFM quantitiesEs, Ns, F 1s , F 2

s , φ1 andφ2. In particular, the inversionNs can be expressed as

a function ofωs as

N2s + (ωs − αNS)2 =

κ21Λ

21

Λ12 + (ωs − ∆1)

2 +κ2

2Λ22

Λ22 + (ωs − ∆2)

2 +

2κ1κ2Λ1Λ2 cos(dCp + ωs dτ + tan−1

(ωs−∆2

Λ2

)− tan−1

(ωs−∆1

Λ1

))

√Λ1

2 + (ωs − ∆1)2√

Λ22 + (ωs − ∆2)

2.

(2.10)

From this quadratic expression we can conclude that for anyωs there are either no, one or two

solutions forNs. In particular, any EFM component is a smooth closed curve that consists

of two branches, one with a higher and one with a lower value ofNs, which connect at two

points where (2.10) has exactly one solution. EFM components in the(ωs, Ns)-plane can be

computed from the implicit transcendental equations (2.2) and (2.10) by root solving, ideally

in combination with numerical continuation. An alternative approach is to find andthen con-

tinue in parameters EFMs directly as steady-state solutions of the governing system (1.1)–(1.4)

of delay differential equations; this can be achieved with the numerical continuation package

DDE-BIFTOOL [11]. Additionaly, by using DDE-BIFTOOL we can obtain stability informa-

tion on the EFMs.


2.1.2 EFM components for two identical filters

The starting point of our study of the EFM structure is the special case thatthe two filters are

identical, apart from having differing feedback phasesC1p andC2

p . Hence, we now set

κ := κ1 = κ2, ∆1 = ∆2, Λ := Λ1 = Λ2, τ1 = τ2.

The EFMs for this special case are given by the EFMs of a corresponding single FOF laser

with effective feedback strength

κeff = 2κ cos

(dCp

2

)(2.11)

and effective feedback phase

Ceffp =

(C1

p + C2p

)/2.

In other words, we obtain a non-trivial reduction of the 2FOF laser to the FOF laser, where the

feedback phase differencedCp arises as a natural parameter that controls the effective feedback

strengthκeff as a result of interference between the two filter fields. One extreme case isthat

of constructive interference whendCp = 0 so thatκeff = 2κ. The other extreme is the case of

destructive interference whendCp = π andκeff = 0. Hence, by changingdCp we can ‘switch

on’ or ‘switch off’ the overall filter field that the laser sees.

Clearly, which EFMs one finds depends on both feedback phasesC1p andC2

p . Branches

of EFMs are obtained by specifying a single condition onC1p andC2

p , while keeping all other

parameters fixed. The easiest option is to continue EFM curves in, say,C1p while keepingC2

p

constant. Another option is to require that the frequencyωs remains fixed. Note that for the

above choices the feedback phase differencedCp changes along the branch of EFMs.

We now consider EFM components of the 2FOF, which we define as the branches of EFMs

that one finds when the feedback phases,C1p or C2

p are changed while the feedback phase

differencedCp is fixed. This definition is the appropriate generalisation from the single FOF

laser [28]. The underlying idea is that the value ofdCp determines the interference of the light

from the two filtered feedback loops and, hence, an important property of the overall feedback

the laser sees. In the simplest case of two identical feedback loops fixingdCp results in the

fixed effective feedback strengthκeff . However, as we will see, our notion of EFM components

for the 2FOF laser is equally natural for nonidentical filter loops.

Figure 2.2 shows a projection of different branches of EFMs onto the(ωs, Ns)-plane and

onto the(ωs, C1p)-plane, respectively. Here we fixedκ = κ1 = κ2 = 0.05, Λ := Λ1 =

Λ2 = 0.015, τ1 = τ2 = 500, and consider the case where both filters are resonant with the


−0.1 0 0.1−0.02

0

0.02

−0.1 0 0.1−5

0

5(a)

ωs

Ns

(b)

ωs

C1

p

π

.

.

Figure 2.2. Projection of EFMs branches onto the(ωs, Ns)-plane (a) and onto the(ωs, C1

p)-plane (b).

The open circles are the starting points for three differenttypes of branches. The blue branch is the EFM

component fordCp = 0, the green branches are for constantC1

p , and the red branches are for constant

ωs. Here∆1 = ∆2 = 0, κ1 = κ2 = 0.05, Λ1 = Λ2 = 0.015, dτ1 = τ2 = 500 and the other parameters

are as given in Table 1.1.

solitary laser, meaning that∆1 = ∆2 = 0; the other parameters are as given in Table 1.1.

Colour in figure 2.2 distinguishes three types of one-dimensional EFM branches; the three sets

of blue, green and red curves are all clearly visible projection onto the(ωs, Ns)-plane in panel

(a). The continuations were started from the set of EFMs (open circles)that one finds on the

EFM component fordCp = 0 if one insists that one of the EFMs (the top open circle) has a

frequency ofωs = 0.

The outer blue curve in figure 2.2 (a) is a single EFM-component that connects all EFMs; it

is indeed exactly the EFM component of the single FOF laser with a feedback strength ofκeff =

2κ; see [28]. WhenC1p is increased by2π, while keepingdCp = 0, each EFM moves along

the blue EFM component to the position of its left neighbour. Hence, the EFM-component

can be calculated either by the continuation of all EFMs over theC1p -range of[0, 2π], or by the

continuation of a single EFM over several multiples of2π. The green curves are the branches of

EFMs that one obtains by changingC2p while keepingC1

p constant. Notice that green branches

connect an EFM at the top with one at the bottom of the blue EFM-component; theexception

is the branch near the origin of the(ωs, Ns)-plane, which connects three EFMs. WhenC2p

is increased by2π the respective EFMs on the green branch exchange their positions in a

clockwise direction. Finally, the red branches in figure 2.2 (a) are the result of continuation of

EFM solutions for (1.1)–(1.4) inC1p andC2

p while ωs is kept constant; hence, these branches

appear as straight vertical lines that start at the respective EFM. Figure 2.2 (b) shows the exact

same branches but now in projection onto the(ωs, C1p)-plane. In this projection, the blue EFM

branch ‘unwraps’ as a single curve that oscillated inC1p ; this property is characteristic and can

be found, more generally, for lasers with delayed feedback or coupling[28, 34, 61]. In the

(ωs, C1p)-plane the red and the green branches are perpendicular to each other. Furthermore,

the image is invariant under a 2π translation along theC1p -axis.

2.2. The EFM-surface 19

2.2 The EFM-surface

The discussion in the previous section shows that the dependence of the EFMs on the feedback

phasesC1p andC2

p (when all other parameters are fixed) gives rise to different one-parameter

families of EFM branches, depending on the conditions one poses. In other words, one is

really dealing with a surface of EFMs in dependence on the two feedback phases, which is

represented by any of the three families of EFM branches we discussed.Motivated by the

question of how many EFM components there are for the 2FOF laser, and in line with the

well-accepted representation of the external cavity mode structure for other laser systems with

delay [34, 61], it is a natural choice to represent this surface by the EFM values ofωs andNs

in combination with one additional parameter.

A first and quite natural choice is to consider the EFM surface in the(ωs, Ns, C1p)-space.

This representation stresses the influence of an individual feedback phase, hereC1p , which is

convenient to make the connection with previous studies in [17, 35]. Figure2.3 shows the EFM

surface for the case of two identical filters in this way. Panel (a) shows agrey semitransparent

two-dimensional object with EFMs branches from figure 2.2 superimposed. This object is the

‘basic’ element of the entire EFM surface, which consists of all infinitely many 2nπ-translated

copies of this basic element. Note that the2nπ-translated copies connect smoothly at the open

ends of the surface element shown in panel (a). The element of the EFM surface was rendered

from computed one-dimensional EFM branches for fixedωs; selection of these branches is

shown as the red curves in figure 2.3 (a). Almost all red EFM branches are closed loops that

connect two points, each on the blue branch. An exception is the central red EFM branch

for ωs = 0, which connects infinitely many points on the infinitely long blue branch. Hence,

this red branch is important for representing the EFM surface properly;it is defined by the

conditions that both sine functions in (2.3) vanish, which means that

C1p = π + tan−1

(∆1

Λ1

)+ tan−1 (α) , (2.12)

C2p = π + tan−1

(∆2

Λ2

)+ tan−1 (α) . (2.13)

The starting points for the calculations of the red EFM branches are taken from the maximal

blue curve, which corresponds to the maximal EFM component fordCp = 0; compare with

figure 2.2 (b). It forms the helix-like curve in(ωs, Ns, C1p)-space that is shown in figure 2.3 (a)

over one2π interval ofC1p . What is more, the shown part of the EFM surface is a fundamental

unit under the translational symmetry ofC1p that contains all the information. This means that

the entire EFM surface is obtained as a single smooth surface from all of the2nπ-translated

copies of the unit in figure 2.3 (a). Notice that the shown part of the EFM surface is tilted in the

C1p direction. More specifically, for negativeωs the red EFM branches are shifted toward higher


(a)

Ns

C1

p/π

ωs

−0.02

0

0.02

15

0

−15−0.1

0

0.1

(b)

Ns

C1

p/π

ωs

0

0

(c)

Ns

C1

p/π

ωs

0

0

.

.

Figure 2.3. Representation of the EFM surface in(ωs, Ns, C1

p)-space; compare with figure 2.2. Panel

(a) shows one fundamental element of the EFM surface (semitransparent grey); superimposed are the

EFM branches from figure 2.2. The entire EFM surface is a single smooth surface that is obtain by

connecting all2nπ-translated copies of the surface element shown in panel (a). Panels (b) and (c) show

how the EFM branches for constantC1

p and for constantωs, respectively, arise as intersection curves of

fixed sections with the EFM surface.

values ofC1p , whereas for positiveωs they are shifted toward lower values ofC1

p ; compare

with the projection onto the(ωs, C1p)-plane in figure 2.2 (b). Notice that for the maximal and

minimal possible values ofωs, the red EFM branches contract to just single points on the blue

EFMs branch.

The representation of the EFM surface in(ωs, Ns, C1p)-space in figure 2.3 is, in effect, the

three-dimensional analogue of the representation of the EFM branches inthe (ωs, C1p)-plane

in figure 2.2 (b). In particular, the two sets of EFM branches for fixedC1p and fixedωs (green


and red curves) arise naturally as intersection curves with planar sections. This is illustrated in

figure 2.3 (b) and (c). Panel (b) shows a cutaway through a four2nπ-translated copies of the

fundamental unit of the EFM surface, and illustrates the single red EFM branch forωs = 0, as

well as2nπ-translated copies of closed red EFM branches forωs = −0.06. Panel (c) shows

with a different cutaway image of the EFM surface how the green EFM branches for fixedC1p

arise as disjoint intersection curves with a fixed planar section; shown areall the green EFMs

branches from figure 2.2 (a) that also appear on the EFM surface in figure 2.3 (a). Note, that

the two sections in figure 2.3 (b) and (c) are perpendicular to each other.

While the representation of the EFM surface in(ωs, Ns, C1p)-space is quite natural, it

has the disadvantage that an EFM component actually corresponds to a non-closed curve that

runs in theC1p direction over the EFM surface in(ωs, Ns, C1

p)-space. To be able to study the

EFM components more directly, we now consider the ‘compactified’ representation of the EFM

surface in(ωs, Ns, dCp)-space; in other words, in this space, the EFM surface is considered as

the one-parameter family (parametrised bydCp) of actual EFM components themselves, which

arise naturally as closed curves in the(ωs, Ns)-plane by intersection with planar sections given

by dCp = const.

This representation of the EFM surface is used in figure 2.4 for the case of two identical

filters. Shown is one compact fundamental part of the EFM-surface in the(ωs, Ns, dCp)-space

for dCp ∈ [−π, π], with superimposed red, green and blue EFMs branches from figure 2.2.

The entire EFM surface consists of all2nπ-translated copies of this compact surface, which

touch at the points(ωs, Ns, dCp) = (0, 0, (2n + 1)π) wheren ∈ Z. This is shown in panel

(b) in projection onto the(ωs, dCp)-plane fordCp ∈ [−3π, 3π], where the EFM-surface is

represented by grey shading and the coloured curves are the EFMs branches as before. Note

that the green EFM branches are no longer perpendicular to the red EFMs branches. Notice

further that the EFM surface is not tilted with respect to thedCp-axis. The common points for

dCp = (2n + 1)π correspond physically to the situation when both filter fields cancel each

other due to destructive interference, so that the the only EFM of the (1.1)–(1.4) is the solitary

laser solution. Three EFM branches pass through these points: the green branch that connects

three EFMs in figure 2.2 (a) and two red EFMs branches given by

Ns(dCp, ωs = 0) = ±2κ cos

(dCp

2

)cos(ΨdCp

), (2.14)

where

ΨdCp= π + tan−1

(∆

Λ

)− tan−1 (α) . (2.15)

Figure 2.4 (c) illustrates with a cutaway view how the red EFM branches for fixedωs arise; the

section forωs = 0 shows the two cosines from (2.14). Note that the red curves are perpen-

dicular to the blue EFM components; the maximal blue EFM component appears asthe largest


(b)

-0.1 0 ωs 0.1-3

0

dCp

π

3(c)

ωs

Ns

dCp/π

0

0

(a)0.02

Ns

0

-0.02-0.1

ωs 00.1-1

dCp/π

0

1

.

.

Figure 2.4. Representation of the EFM surface of figure 2.3 in(ωs, Ns, dCp)-space. Panel (a) shows

one fundamental2π interval of the EFM surface (semitransparent grey); superimposed are the EFM

branches from figure 2.2. The entire EFM surface consists of all 2nπ-translated copies of this compact

surface, which touch at the points(ωs, Ns, dCp) = (0, 0, (2n+1)π); panel (b) shows this in projection

of the surface onto the(ωs, dCp)-plane. Panel (c) illustrates how the EFM branches for constantωs and

the outer-most EFM component fordCp = 2nπ arise as intersection curves with planar sections.

closed curve for fixeddCp = 2nπ.

Figure 2.5 shows in more detail how the EFM-component depends ondCp. In practice,

an EFM component for a givendCp = const is computed by continuation of an EFM in the

parametersC1p and ωs, while settingC2

p = C1p + dCp. Panel (a) shows the EFM surface

for dCp ∈ [−π, π] intersected with the two planes given bydCp = 0 and dCp = 0.9π,

respectively. The corresponding EFM components arise from the shape of the envelope given


−0.1 0 0.1−0.02

0

0.02

−0.1 0 0.1−0.02

0

0.02

−0.1 0 0.1

−0.5

0

0.5

−0.1 0 0.1

−0.5

0

0.5

(a)0.02

Ns

0

-0.02-1

dCp

π 0

0.9-0.1

0ωs

0.1

(b1)

(b2)

(c1)

(c2)

ωs

ωs

ωs

ωs

Ω(ωs) Ω(ωs)

Ns Ns

.

.

Figure 2.5. EFM-components arising as sections through the EFM surfaceof figure 2.4. Panel (a)

shows the EFM-surface in(ωs, Ns, dCp)-space, intersected with the planes defined bydCp = 0 and

dCp = 0.9π, respectively. Panels (b1) and (c1) show the correspondingenvelope (grey curves) given by

(2.9). The black solution curve of (2.3) inside it is forC1

p = 0; it gives rise to the marked blue EFMs.

Panels (b2) and (c2) show the two respective EFM-componentsand individual EFMs (blue dots) in the

(ωs, dCp)-plane.

by (2.9); it is shown in panels (b1) and (c1) together with the solution curveof (2.3) one obtains

when conditions (2.14) and (2.15) are satisfied. The EFM components themselves, with these

EFMs on them, are shown in panels (b2) and (c2). Figure 2.5 illustrates thatchangingdCp


results in a change in the EFM component, as well as the number of EFMs. Forthe shown case

of two identical filters with equal delay times we can say more: here the changeof the EFM

component is entirely due to the effective feedback rateκeff as described by (2.11). In other

words, the EFM surface is composed of the EFM components of the corresponding single FOF

laser withκeff as determined bydCp. The EFM component is maximal for the constructive-

interference casedCp = 0, and it shrinks whendCp is changed. This also means that fewer

EFMs exist; compare figure 2.5 (b1) and (c1). Finally, as the case of entirely destructive

interference fordCp = π is approached, the EFM component shrinks down to a point, which

is the degenerate EFM corresponding to the unique solitary laser mode.

We conclude from this section that the EFM surface in(ωs, Ns, dCp)-space has emerged

as the main object of study. It represents the EFM structure of the 2FOF laser in a convenient

geometric way; in particular, EFM components can easily be obtained as planar slices for fixed

dCp.

2.3 Classification of the EFM surface fordτ = 0

So far we have only considered the special case that the two filters are identical and not detuned

from the laser; furthermore, the two delay times are equal. We now addressthe question how

the EFM surface changes as the system is moved away from this special point in the space of

parameters. In this section we consider the case that the two filter loops havethe same delay

time, that is,dτ = 0. The influence of a difference in delay times is the subject of section 2.5.

We start by considering in section 2.3.1 how the EFM components depend on the detunings

∆1 and∆2 for fixed dCp = 0. We then proceed to study how the EFM surface itself changes

with ∆1 and∆2. This can be represented for a fixed filter widthΛ of both filters by an EFM

surface bifurcation diagram in the(∆1, ∆2)-plane, where each open region corresponds to a

different type of EFM surface. In section 2.3.2 we first consider the case that the EFM sur-

face gives rise to adCp-independent number of EFM components. We then introduce in sec-

tion 2.3.3 five codimension-one singularity transitions — through extrema and saddle points,

and through a cubic tangency (with respect todCp = const) — that change the EFM surface

in terms of how many EFM components it induces when sliced for fixeddCp. These five sin-

gularity transitions induce a division of the(∆1, ∆2)-plane into open regions of different EFM

surface type. We first, discuss EFM surface types that arise due to transitions through extrema

and saddle points, and next EFM surface types that arise due the cubic tangency. Finally, sec-

tion 2.4 shows how the EFM surface bifurcation diagram in the(∆1, ∆2)-plane changes with

the filter widthΛ.

Throughout this section we make use of the fact that the EFM surface canbe represented

by its projection onto the(ωs, dCp)-plane. Here we make use of the fact that, due to (2.10),

2.3. Classification of the EFM surface fordτ = 0 25

this surface consists of two sheets in(ωs, dCp, N)-space over the(ωs, dCp)-plane, except at

the boundary of its projection. The boundary itself is given by (real-valued) solutions of

dCp = ± cos−1

(√Λ2

1 + (ωs − ∆1)2√

Λ22 + (ωs − ∆2)

2

2κ1κ2Λ1Λ2

×(

ω2s

(1 + α2)− κ2

1 Λ21

Λ21 + (ωs − ∆1)

2 − κ22 Λ2

2

Λ22 + (ωs − ∆2)

2

))

− ωs dτ − tan−1

(ωs − ∆2

Λ2

)+ tan−1

(ωs − ∆1

Λ1

).

(2.16)

This equation is derived from (2.9), and it has the advantage that it doesnot depend on any

of the state variables of (1.1)–(1.4). Hence, in contrast to computing the EFM surface itself,

which requires the continuation of EFMs in parameters, its projection onto the(ωs, dCp)-plane

can be computed directly from (2.16). Notice also that the projection in the(ωs, dCp)-plane is

independent of the choice of the additional state variable that one chooses for visualisation of

the EFM-surface.

2.3.1 Dependence of the EFM components for fixeddCp = 0 on the detunings

We now fixeddCp = 0 and consider the detunings∆1 and∆2 as free parameters. We first

consider an intermediate fixed filter widthΛ = Λ1 = Λ2 = 0.015 of both filters; moreover,

τ1 = τ2 = 500 and the other parameters are as given in Table 1.1. In this situation, one

may find one, two or three EFM components in the FOF laser. Because both thetop and the

bottom part of the envelope given by (2.9) intersect the diagonal, the EFMcomponent around

a solitary laser frequencyωs = 0 is always present. In the presence of the two filters, one may

find additional EFM components, which exist around the central frequencies of the filters as

given by∆1 and∆2; note that (2.9) has two obvious extrema forωs = ∆1 andωs = ∆2. Each

additional EFM component comes with its own pair of saddle-node bifurcationpoints, given

geometrically by the condition that an extremum of (2.9) intersects the diagonalωs = Ω(ωs);

see figure 2.1.

Figure 2.6 shows EFM components for nonzero detunings and fixeddCp = 0; more specif-

ically, we fix ∆1 = 0.2 and increase∆2 in (a) to (d). Each case is shown by two panels, one

showing envelope and solution curve, and the other the corresponding EFM-components and

EFMs. One can imagine this situation as a single FOF laser with detuning∆1, which is subject

to the influence of the second filter loop. If the detuning∆2 of the second filter is below or

above a critical value then the influence of the second filter is negligible and one observes only

two EFM-components. However, for intermediate values of∆2 this is no longer the case. For


−0.2 0 0.2−0.05

0

0.06

−0.2 0 0.2−0.05

0

0.06

−0.2 0 0.2−0.5

0

0.5

−0.2 0 0.2−0.5

0

0.5

−0.2 0 0.2−0.05

0

0.06

−0.2 0 0.2−0.05

0

0.06

−0.2 0 0.2−0.5

0

0.5

−0.2 0 0.2−0.5

0

0.5(a1)

(a2)

(c1)

(c2)

(b1)

(b2)

(d1)

(d2)

ωs

ωs

ωs

ωs

ωs

ωs

ωs

ωs

Ns

Ns

Ns

Ns

T (ωs)

T (ωs)

T (ωs)

T (ωs)

.

.

Figure 2.6. Envelope and solution curve fordCp = 0 (a1)-(d1) and the corresponding EFM-

components and EFMs (blue dots) of the 2FOF laser, were∆1 = 0.2 is fixed and in panels (a)-(d)

∆2 takes the values−0.2, 0, 0.158 and0.2, respectively; hereΛ1 = Λ2 = 0.015, τ1 = τ2 = 500 and

the other parameters are as given in Table 1.1.

∆2 = −0.2 as in figure 2.6 (a1) and (a2), the two detunings have equal magnitude butopposite

signs. Hence, there are three EFM components, the central one nearωs = 0 being quite small.

When∆2 is increased, the left-most EFM component moves towards largerωs and then merges

with the central one; see figure 2.6 (b1) and (b2) for∆2 = 0. As∆1 is increased further so that


it is well in betweenωs = 0 andωs = 0.2, we find again a situation with three EFM compo-

nents; see figure 2.6 (c1) and (c2) for∆2 = 0.158. Finally, for∆1 = ∆2 = 0.2 we again find

only two EFM components; see figure 2.6 (d1) and (d2). Now the amplitudes of envelope and

solution curve are much higher, which is due to maximal constructive interference between the

two filter fields sincedCp = 0.

Note that the filters are quite narrow (Λ1 = Λ2 = 0.015), namely much narrower than the

detuning∆1 = 0.2 between the laser and the first filter. As a result, in figure 2.6 (a), (c) and(d)

the EFM-component around the solitary laser frequency has an elliptical shape, as known from

COF systems [61, 46]. This is the case because flanks of the both filters transmittance profiles

nearsωs = 0 are quite flat, meaning that all frequencies around the solitary laser frequency

are fed back with approximately the same low feedback strength. Therefore, this situation

resembles the effect of weak COF [46]. In figure 2.6 (b) for∆2 = 0, on the other hand, the

EFM-component around the solitary laser frequency has a ‘bulge’ — much as one finds in

figure 2.2 (a) — which is the result of the frequency selective feedbackfrom the second filter.

For a fixed value of the widthΛ = Λ1 = Λ2 of both filters, one obtains a bifurcation

diagram in the(∆1, ∆2)-plane that consists of regions where the 2FOF laser system has one,

two or three EFM-components fordCp = 0. The regions are bounded by curves that can be

computed by means of numerical continuation. Namely, the number of the EFM components

changes when two saddle-node points (black dots in figure 2.1) come together. This happens

when the envelope given byΩe(ωs) from (2.9) is tangent to the diagonal. Hence, the conditions

that are continued in∆1 and∆2 to obtain the boundary curves are,

Ω(ωs) = ωs anddΩe(ωs)

dωs= 1.

Figure 2.7 shows the bifurcation diagram in the(∆1, ∆2)-plane for six different values of

Λ. Open regions are labelled with the number of EFM components that one findsfor dCp = 0

for the respective values of the detunings∆1 and∆2. Notice the two symmetries of the panels

of figure 2.7, given by reflection across the diagonal∆1 = ∆2, and reflection across the

anti-diagonal∆1 = −∆2 (which we already encountered in figure 2.6 (a)). The boundary

curves are coloured orange and grey for presentation purposes only. Figure 2.7 (a) shows

the limiting special case ofΛ = 0, which corresponds to an infinitely narrow filter so that

EFM-components consist of single EFMs. The bifurcation diagram in the(∆1, ∆2)-plane for

this case can be obtained analytically by substituting in Eq. (2.9)ωs = ∆1 andωs = ∆2

respectively. The coordinates of the vertical and horizontal lines in figure 2.7 (a) are given by

κ√

1 + α2 and the end points at the diagonal by2κ√

1 + α2; compare with [28]. Figure 2.7

(b) for Λ = 0.001 shows how the limiting case unfolds forΛ > 0. The black parts of curves in

figure 2.7 (a) open up to reveal new open regions. AsΛ is increased, the bifurcation diagram

deforms, but does initially not change qualitatively; see figure 2.7 (c) forΛ = 0.01, which is


−0.8 0 0.8−0.8

0

0.8

2

1

22JJQQs

2JJ]QQk

3AAAAU

HHHY

−0.8 0 0.8−0.8

0

0.8

2

1

22JJ

QQQs

2J

JJ]Q

Qk

−0.8 0 0.8−0.8

0

0.8

1 2 1 2 1

2 2 3 2

1 2 1 2 1

2 3 2 2

1 2 1 2 1

2

3

−0.8 0 0.8−0.8

0

0.8

1 1 2 1

1 1 2 12

2 3 2

2 3 2

2

2

1 2 1 1

−0.8 0 0.8−0.8

0

0.8

1 2 2 1

2 3 3 23

32 3 3 2

1 2 2 1

−0.8 0 0.8−0.8

0

0.8

1 2 1 2 1

2 3 2 3 23

1 2 2 13

2 3 2 3 2

1 2 1 2 1

2

2

SS

SSSo

1

∆1

∆1

∆1

∆1

∆1

∆1

∆2

∆2

∆2

∆2

∆2

∆2

(a)

(c)

(e)

(b)

(d)

(f)

.

.

Figure 2.7. Regions in the(∆1, ∆2)-plane with a one, two or three EFM components of the 2FOF laser

for dCp = 0. From (a) to (f)Λ takes the valuesΛ = 0, Λ = 0.001, Λ = 0.01, Λ = 0.06, Λ = 0.12 and

Λ = 0.14.

the case from figure 2.8 (a). However, asΛ is increased further, the bifurcation diagram does

change qualitatively because the different curves move sufficiently relative to one another to


‘disentangle’; see figure 2.7 (d) forΛ = 0.06, where there are now no longer regions with three

EFM components. For larger values ofΛ, the curves cease to extend to infinity and are now

confined to a compact region of the(∆1, ∆2)-plane; as figure 2.7 (e) forΛ = 0.12 illustrates,

this implies that there is now a single large and connected region with one EFM component.

For even larger values ofΛ, there are six non-overlapping curves, each bounding a small region

where one finds two EFM components; see figure 2.7 (f) forΛ = 0.14. WhenΛ is increased

even further, the small regions disappear and one finds a single EFM component for any point

in the(∆1, ∆2)-plane. Physically the filters are now so wide that they do not provide sufficient

differentiation of the feedback light in frequency; hence, the 2FOF laser is effectively a COF

laser.

Figure 2.8 (a) is a three-dimensional bifurcation diagram in(∆1, ∆2, Λ)-space that rep-

resents the entire transition of the bifurcation diagram in the(∆1, ∆2)-plane fordCp = 0 as

the filter widthΛ is changed. Shown are surfaces (coloured orange and grey as in figure 2.7)

that divide this parameter space into regions with one, two or three EFM-components. The bi-

furcation diagrams in figure 2.7 are horizontal cross sections through figure 2.8 (a); the shown

(semitransparent) cross section forΛ = 0.01 yields figure 2.7 (c). Notice that the grey surfaces

in figure 2.8 (a) extend to higher values of∆ than the orange surfaces, which can be explained

as follows. FordCp = 0 the two filter fields interfere constructively, so that for∆1 ≈ ∆2

the amplitude of the solution curve of (2.3) is larger than that around a single filter. Hence,

a second EFM-component around the central frequencies of both filters may exist for higher

values ofΛ, and the maximum of the grey surfaces is exactly at the diagonal where∆1 = ∆2.

Above all surfaces (for sufficiently largeΛ) the 2FOF laser is effectively a COF laser and only

one EFM-component exists.

Figures 2.8 (b) and (c) show that the three-dimensional bifurcation diagram in panel (a)

brings out important special cases where the 2FOF laser reduces to the single FOF laser in a

nontrivial way. Figure 2.8 (b) shows the two-dimensional bifurcation diagram in the(∆1, Λ)-

plane for∆2 = 0.82. Also shown in light grey is the corresponding bifurcation diagram of

the single FOF laser (with detuning∆1) that one obtains for the limit that∆2 = ∞ (when the

second filter does not influence the system any more); compare with [28, Fig. 3(a)]. The close-

ness of the two bifurcation diagrams in figure 2.8 (b) shows that for∆2 ≥ 0.82 the influence

of the second filter is already so small that it does not influence the number of EFM compo-

nents. Figure 2.8 (c) shows the projection onto the(∆1, Λ)-plane of the diagonal section for

∆1 = ∆2 through the surfaces in figure 2.8 (a). Along the diagonal the 2FOF laser reduces

to the single FOF laser with the effective parameters as given by (2.11); in fact, the boundary

curve in figure 2.8 (c) is exactly that from [28, Fig. 3(a)] for the corresponding effective param-

eters, namelyκeff = 2κ1 for dCp = 0. Since this curve scales linearly withκ [28], it is exactly

twice the size as the light grey curve in figure 2.8 (b).


−0.6 0 0.6 0

0.21

−0.35 0 0.350

0.13

1 2 1 2 1 1 2 1 2 1

∆2 ∆1

Λ

-0.82 -0.82

0.82 0.82

0 0

0

0.21

∆1

Λ Λ

∆1 = ∆2

(a)

(b) (c)

.

.

Figure 2.8. Panel (a) shows surfaces (orange and grey) that divide the(∆1, ∆2, Λ)-space into regions

with one, two and three EFM-components of the 2FOF laser fordCp = 0; in the shown (semitranspar-

ent) horizontal cross section forΛ = 0.01 one finds the bifurcation diagram from figure 2.7 (c) . Panel

(b) shows the bifurcation diagram in the(∆1, Λ)-plane for fixed∆2 = 0.82; the light grey curve is the

boundary curve for the limiting single FOF laser for∆2 = ∞. Panel (c) shows the projection onto the

(∆1, Λ)-plane of the section along the diagonal∆1 = ∆2 through the surfaces in panel (a).

2.3.2 EFM surface types withdCp-independent number of EFM components

We now turn to the question of how the EFM surface itself changes with∆1 and∆2, where

we first consider the case that the EFM surface is such that the associated number of EFM

components is independent ofdCp.

Figure 2.9 shows two examples of EFM surfaces in(ωs, dCp, Ns)-space for the special

case that∆1 = −∆2. Panel (a) shows three copies of the fundamental unit of the EFM surface


−0.1

0

0.1

−3

0

3−0.02

0

0.02

−0.2

0

0.2

−4

0

4−0.04

0

0.04(a) (b)

ωs ωs

dCp

πdCp

π

Ns Ns

.

.

Figure 2.9. The EFM surface in(ωs, dCp, Ns)-space showing caseB for ∆1 = ∆2 = 0 (a), and

showing caseBBB for ∆1 = 0.16, ∆2 = −0.16 (b), whereΛ = 0.015.

for ∆1 = ∆2 = 0. The EFM surface is connected atdCp = π and its integer multiples;

hence, it is a single connected component that extends over any2π interval of dCp, and one

finds a single EFM component for anydCp; compare with figure 2.4 (a) and (b). Figure 2.9 (b)

is for sufficiently large (opposite) detunings when the EFM surface consists of three disjoint

connected components that extend over any2π interval ofdCp. Hence, for anydCp one finds

three EFM components, one around the solitary laser frequencyωs = 0 and the other two

around the central frequenciesωs = ∆1,2 of the filters; compare with figure 2.6 (a). Notice

that, since∆1 = −∆2, the central connected component aroundωs = 0 is also connected at

dCp = π and its integer multiples.

We now turn to the question of where in the(∆1, ∆2)-plane one can find EFM surfaces

that have adCp-independent number of EFM components. To investigate this question, we

consider how the bifurcation diagram in figure 2.7 (c), for the representative value ofΛ =

0.01, with regions of one, two or three EFM components in the(∆1, ∆2)-plane for a fixed

dCp = 0 changes whendCp is varied over the interval[−π, π]. In the process the boundary

curves between regions move in the(∆1, ∆2)-plane and then return to their original positions.

Figure 2.10 shows the resulting curves (again in orange and grey) in the(∆1, ∆2)-plane for

Λ = 0.01, where thedCp-interval [−π, π] is covered in 60 equidistant steps. As a function

of dCp the curves now cover overlapping (orange and grey) regions in the(∆1, ∆2)-plane,

meaning that in these regions the number of EFM components depends on the value ofdCp.

By contrast, in the open white regions in the(∆1, ∆2)-plane of figure 2.10 the number of

EFM is independent of the value ofdCp. This means that the projection of the EFM surface

consists of either one, two or three bands that extend over the entiredCp-interval [−π, π]. In

total there are four such types (up to symmetry) of EFM surface, and theirrepresentatives in

terms of projections of the EFM surface onto the(ωs, dCp)-plane are shown in figure 2.11;


BB

BB

BBB

BBB

BB

BB

BBBB

B BB

BBBB

B BB B BB B

B

BBB

BBB

BB

BBB

BBB

BB

BB

∆1

∆2

−0.55

0

0.55

−0.55 0 0.55

.

.

Figure 2.10. Boundary curves (orange or grey) in the(∆1, ∆2)-plane forΛ = 0.01 for 61 equidistant

values ofdCp from the interval[−π, π]; compare with figure 2.7 (c). In the white regions the 2FOF

laser has one, two or three EFM components independently of the value ofdCp, as is indicated by the

labelling with symbolsB andB; representatives of the four types of EFM components can be found in

figure 2.11.

for the respective values of∆1 and∆2 see Table 2.1. Each such band in the projection is

represented in figures 2.10 and 2.11 by the letterB. Furthermore,B denotes the band around

the frequency of the solitary laser,ωs = 0, and is later referred to as the central band. It plays

a special role because it corresponds to a part of the EFM surface that always extends over the

entiredCp-interval [−π, π]; moreover,B can be found for any value of detuningΛ of the two

filters, even in the COF limit of an infinitely wide filter; see the discussion of figure 2.8 (a) in

section 2.3.1.


(a)

B

(b)

BB

(c)

BBB

(d)

BBB

.

.

Figure 2.11. The four simple banded types of EFM-surface of the 2FOF laserin the labelled regions of

figure 2.10, represented by the projection (shaded) onto the(ωs, dCp)-plane; the blue boundary curves

are found directly from (2.16). For notation and the corresponding values of∆1 and∆2 see Table 2.1;

in all panelsωs ∈ [−0.3, 0.3] anddCp ∈ [−π, π].

name Cmin Cmax panel ∆1 ∆2 Λ

B 1 1 figure 2.11 (a) 0.080 -0.270 0.01

BB 2 2 figure 2.11 (b) 0.140 -0.270 0.01

BBB 3 3 figure 2.11 (c) 0.210 0.130 0.01

BBB 3 3 figure 2.11 (d) 0.210 -0.130 0.01

Table 2.1. Notation and parameter values for the types of EFM-surface in figure 2.11. The second and

third column show the minimal numberCmin and the maximal numberCmax of EFM components (for

suitable fixeddCp) of the type; note that in all cases the number of EFM components is independent of

dCp .

The notation we use here is more specific than simply counting the number of EFMcom-

ponents; for example, it distinguishes the caseBBB, where the filters are detuned to both sides

of the laser frequencyωs = 0, from the casesBBB andBBB (which are related by symme-

try), where both filters are detuned on the same side of the laser frequency. Notice also that

type BB differs physically from typeBB in terms of whether the second bandB is towards

higher or lower frequencies with respect to the laser frequency (that is, for negative or positive

ω). Nevertheless, these two types are related to each other mathematically, because they are

each other’s images under the symmetry transformation(∆1, ∆2) 7→ (−∆1, −∆2). Indeed

any type that is not symmetric itself comes as a symmetric pair, and it is sufficient toshow only

one type of such a pair in figure 2.11. Note that the two EFM surfaces in figure 2.9 are of type

B andBBB, respectively.

2.3.3 Transitions of the EFM surface

Consider a path in the(∆1, ∆2)-plane that takes one from a white region to another white

region, where the number EFM components does not depend on the value of dCp. It is clear

from figure 2.10 that any such path necessarily leads through (at leastone) (grey or orange)


region where the EFM surface is such that the number of EFM components does actually

depend on the value ofdCp. For example, a single bandB may change into two bandsBB,

and the question arises what changes of the EFM surface itself are involved in this transition.

The point of view we take here is that the classification of the EFM surface into different

types is generated by five (local) transitions of codimension one, which we introduce now. Each

such transition changes the nature of the associated EFM components one encounters when

dCp is changed over[−π, π]. More specifically, we find four generic singularity transitions

that change the EFM surface topologically as a surface in three-dimensional space; as a result,

the number of EFM components changes locally. Furthermore, we considera cubic tangency

of the EFM surface with respect to a planedCp = const, which also changes the number of

EFM components locally.

The four singularity transitions

The codimension-one singularity transitions are characterised by the factthat an isolated sin-

gularity of the parametrised EFM surface is crossed at an isolated point ofa curve in the

(∆1, ∆2)-plane. To be more specific, letδ be the bifurcation parameter that parametrises a

curve in the(∆1, ∆2)-plane, where we assume that the respective bifurcation curve is crossed

transversely atδ = 0. We can then view the EFM surface in(ωs, dCp, Ns)-space locally near

δ = 0 as given by level setsF (ωs, dCp, Ns) = δ of a functionF : R3 → R. The singularity

is then given by the condition that grad(F ) = 0; generically, the Hessian at this point is non-

singular, which means that the singularity is of codimension one [1, 26, 53].In this case, the

surface is locally quadratic and has the normal form

F (u, v, w) = ±u2 ± v2 ± w2 = δ, (2.17)

where the signs are given by the signs of the eigenvalues of the Hessian at the singularity. If

all signs are the same then one is dealing with the transition through an extremum, that is, a

minimum or a maximum, of the surface; we speak of a minimax transition [41]. Otherwise,

the singularity is a saddle. The unfolding of such a saddle on a two-dimensional surface is

well know; see, for example, [2]. It is the transition form a one-sheetedhyperboloid via a

cone to a two-sheeted hyperboloid; we speak of a saddle transition [41].Different cases in

our context arise depending on how the cone associated with the saddle point is aligned in

(ωs, dCp, Ns)-space.

There are four distinct singularity transition of the EFM surface in(ωs, dCp, Ns)-space.

M the minimax transition through an extremum (a local minimum or maximum). The min-

imax transitionM of the EFM surface is illustrated in figure 2.12, where, in terms of


the projection onto the (ωs, dCp)-plane, it results in the creation or disappearance of an

island. The locus ofM in the (∆1, ∆2)-plane is represented by orange curves in what

follows.

SC the saddle transitions in the direction of theCp-axis. The saddle transitionSC of the EFM

surface is illustrated in figure 2.13, where, in terms of the projection onto the (ωs, dCp)-

plane, it results in a transition between an island and a band. The locus ofSC in the

(∆1, ∆2)-plane is represented by blue curves in what follows.

Sω the saddle transitions in the direction of theωs-axis. The saddle transitionSω of the EFM

surface is illustrated in figure 2.14, where, in terms of the projection onto the (ωs, dCp)-

plane, it results for example, in a transition between a band with a hole and two separate

bands. The locus ofSω in the (∆1, ∆2)-plane is represented by green curves in what

follows.

SN the saddle transitions in the direction of theNs-axis. The saddle transitionSN of the

EFM surface is illustrated in figure 2.15, where, in terms of the projection ontothe

(ωs, dCp)-plane, it results in the creation or disappearance of a hole in a band. The locus

of SN in the(∆1, ∆2)-plane is represented by red curves in what follows.

Figures 2.12–2.15 illustrate how the minimax transitionM and the three saddle transitions

SC , Sω andSN lead to changes in the EFM-surface. In each of these figures, we showin

the left column the relevant local part of the EFM surface in(ωs, dCp, Ns)-space before, at

and after the bifurcations. The right column shows how the projection of theEFM surface

onto the(ωs, dCp)-plane (shown over adCp-interval of 4π) changes accordingly; the local

regions where the change occurs are highlighted. In figures 2.12–2.15all the presented surfaces

have been rendered from continuations of the EFMs as solutions of equations (1.1)–(1.4); the

projections, on the other hand, were obtained directly from (2.16).

Figure 2.12 illustrates the minimax transitionM , where a compact piece of the EFM-

surface in(ωs, dCp, Ns)-space shrinks to a point. Panel (b1) is very close to the bifurcation;

note that after the bifurcation the piece is simply gone, which is why we do not present a sep-

arate illustration for this situation. In projection onto the(ωs, dCp)-plane, the local, compact

piece of the EFM-surface is an ‘island’ that shrinks and then disappears in a minimax transi-

tion of the projection; note that there are infinitely many such islands due to the translational

symmetry indCp; see panels (a2) and (b2).

Figure 2.13 illustrates the saddle transitionSC . The local mechanism for this change of

the EFM-surface in(ωs, dCp, Ns)-space is shown in panels (a1)–(c1). The surface in fig-

ure 2.13 (a1) is a one-sheeted hyperboloid. It develops a pinch point and, hence, becomes a

cone at the moment of bifurcation in figure 2.13 (b1); note that the cone (orrather its axis of


−0.05 0 0.3−2

0

1.3

2

−0.05 0 0.3−2

0

1.3

2

(a1) (a2)

(b1)(b2)

ωs

ωs

dCp

π

dCp

π

Ns

Ns

ωs

ωs

dCp

π

dCp

π

.

.

Figure 2.12. Minimax transitionM of the EFM-surface in (ωs, dCp, Ns)-space, where a connected

component of the EFM surface (a1) shrinks to a point (b1). Panels (a2) and (b2) show the corresponding

projection onto the (ωs, dCp)-plane of the entire EFM surface; the local region where thetransitionM

occurs is highlighted by dashed lines and the projections ofthe part of the EFM surface in panels (a1)

and (b1) is shaded grey. HereΛ = 0.01, ∆1 = 0.4, and∆2 = 0.28 in (a) and∆2 = 0.28943 in (b).

rotation) is aligned with thedCp-axis. After the bifurcation, the EFM-surface is a two-sheeted

hyperboloid so that it consists locally of two parts; see figure 2.13 (c1). As the projections onto

the(ωs, dCp)-plane in panels (a2)–(c2) show, the overall result is the division of a ‘band’ into

a ‘string of islands.’ Note that the saddle transitionSC manifests itself as a saddle transition of

the projection, where the relevant (shaded) part of the surface is aligned with thedCp-axis.

Figure 2.14 illustrates the saddle transitionSω. Locally near the point of bifurcation we

again find that the EFM-surface in(ωs, dCp, Ns)-space changes from a one-sheeted hyper-

boloid in panel (a1), via a cone in panel (b1) to a two-sheeted hyperboloid in panel (c1). How-

ever, now the cone is aligned with theω-axis. As the projections onto the(ωs, dCp)-plane in

panels (a2)–(c2) show, overall we find that a single band with a ‘string of holes’ changes into

two separate bands. The saddle transitionSω manifests itself also as a saddle transition of the

projection, but now the relevant (shaded) part of the surface is aligned with theωs-axis.

Figure 2.15 illustrates the saddle transitionSN . In panel (a1) there are two sheets, with

different and separate values ofNs, of the EFM-surface in(ωs, dCp, Ns)-space. At the bifur-

cation point the two sheets connect locally in a single point. In the process, a‘hole’ is created


−0.05 0 0.26−2

0.2

0.9

2

−0.05 0 0.26−2

0.2

0.9

2

−0.05 0 0.26−2

0.2

0.9

2

(a1)(a2)

(b1)(b2)

(c1)(c2)

ωs

ωs

ωs

dCp

π

dCp

π

dCp

π

Ns

Ns

Ns

ωs

ωs

ωs

dCp/π

dCp/π

dCp/π.

.

Figure 2.13. Saddle transitionSC of the EFM-surface in (ωs, dCp, Ns)-space, where locally the

surface changes from a one-sheeted hyperboloid (a1) to a cone aligned in thedCp-direction (b1) to a

two-sheeted hyperboloid (c1). Panels (a2)–(c2) show the corresponding projection onto the (ωs, dCp)-

plane of the entire EFM surface; the local region where the transitionSC occurs is highlighted by

dashed lines and the projections of the part of the EFM surface in panels (a1)–(c1) is shaded grey. Here

Λ = 0.01, ∆1 = 0.4, and∆2 = 0.23 in (a),∆2 = 0.232745 in (b) and∆2 = 0.24 in (c).

in the EFM-surface, which then grows in size; see panels (b1) and (c1). If one considers a

small neighbourhood of the emerging hole, then one realises that the transition is locally that

from a two-sheeted hyperboloid in panel (a1), via a cone aligned along the Ns-axis in panel

(b1) to a one-sheeted hyperboloid in panel (c1). The projections onto the (ωs, dCp)-plane in

panels (a2)–(c2) clearly show how a string of holes appears in the saddle transitionSN . Note

that this bifurcation is a minimax transition of the projection but, in contrast to transition M ,


−0.05 0 0.17−2

−0.6

0

0.6

2

−0.05 0 0.17−2

−0.6

0

0.6

2

−0.05 0 0.17−2

−0.6

0

0.6

2

(a1)

(a2)

(b1)

(b2)

(c1)

(c2)

ωs

ωs

ωs

dCp

π

dCp

π

dCp

π

Ns

Ns

Ns

ωs

ωs

ωs

dCp

π

dCp

π

dCp

π

.

.

Figure 2.14. Saddle transitionSω of the EFM-surface in (ωs, dCp, Ns)-space, where a connected com-

ponent (a1) pinches (b1) and then locally disconnects (c1);here the associated local cone in panel (b1)

is aligned in theωs-direction. Panels (a2)–(c2) show the corresponding projection onto the (ωs, dCp)-

plane of the entire EFM surface; the local region where the transitionSω occurs is highlighted by

dashed lines and the projections of the part of the EFM surface in panels (a1)–(c1) is shaded grey. Here

Λ = 0.01, ∆1 = 0.4, and∆2 = 0.13 in (a),∆2 = 0.133535 in (b) and∆2 = 0.135 in (c).

the projection of the surface is now ‘on the outside’ so that locally a hole is created instead of

an island.

It is an important realisation that the loci of the four singularity transitionsM , SC , Sω

andSN can be computed effectively, because it can be expressed as an implicit formula by

considering a suitable derivative of the envelope equation (2.9) with respect to the parameter in

question. More specifically, one follows a fold with respect todCp of the boundary curve of the


−0.05 0 0.15 −2

0

0.81.1

2

−0.05 0 0.15 −2

0

0.81.1

2

−0.05 0 0.15−2

0

0.81.1

2

(a1)

(b1)

(c1)

(a2)

(b2)

(c2)

ωs

ωs

ωs

dCp

π

dCp

π

dCp

π

Ns

Ns

Ns

ωs

ωs

ωs

dCp

π

dCp

π

dCp

π

.

.

Figure 2.15. Saddle transitionSN of the EFM-surface in (ωs, dCp, Ns)-space, where two sheets that

lie on top of each other in theNs direction (a1) connect at a point (b1) and then create a hole in the

surface (c1); here the associated local cone in panel (b1) isaligned in theN -direction. Panels (a2)–(c2)

show the corresponding projection onto the (ωs, dCp)-plane of the entire EFM surface; the local region

where the transitionSN occurs is highlighted by dashed lines and the projections ofthe part of the

EFM surface in panels (a1)–(c1) is shaded grey. HereΛ = 0.01, ∆1 = 0.4, and∆2 = 0.0.11 in (a),

∆2 = 0.11085 in (b) and∆2 = 0.1115 in (c).

projection of the EFM surface onto the(ωs, dCp)-plane. Such adCp-fold bound an interval

of dCp-values, which is either an island or a hole of the projection. When thedCp-fold is

continued along a curve (parametrised byδ) in the(∆1, ∆2)-plane, say,∆2 for fixed∆1, then

a singularity transition corresponds to a fold with respect to the continuation parameterδ. Such

a fold with respect to the parameterδ can be detected and then followed as a boundary curve


−0.075 0 0.031

1.1

−0.075 0 0.031

1.1

−0.075 0 0.031

1.1

(a1)(a2)

(b1)(b2)

(c1)(c2)

ωs

ωs

ωs

dCp

π

dCp

π

dCp

π

Ns

Ns

Ns

ωs

ωs

ωs

dCp

π

dCp

π

dCp

π

.

.

Figure 2.16. Cubic tangencyC of the EFM-surface in (ωs, dCp, Ns)-space, where a part of the surface

(a1) becomes tangent to a planedCp = const (b1) and then develops a bulge (c1). The unfolding

of the cubic tangency into twodCp-folds can be seen clearly in the projections onto the (ωs, dCp)-

plane in panels (a2)–(c2). HereΛ = 0.015, and(∆1,∆2) = (−0.03,−0.0301) in (a), (∆1,∆2) =

(−0.04,−0.0401) in (b) and(∆1,∆2) = (−0.05,−0.051).

in the (∆1, ∆2)-plane. Note that this continuation approach makes no difference between

the casesM , SC , Sω andSN of singularity transitions. However, which of the singularity

transitions one is dealing with can readily be identified by checking the (projections of the)

EFM surface at nearby parameter point in the(∆1, ∆2)-plane. In this way, the loci of the

singularity transitions have been computed numerically to yield the complicated structure of

boundary curves in figure 2.17.


The cubic tangency

Due to the special role of the parameterdCp we also consider here a fifth local mechanism that

changes the type of the EFM surface.

C the cubic tangencyC of the EFM surface, is defined by the condition that the first and

second derivatives with respect toωs of equation (2.9) for the envelope ofΩ(ωs) both

vanish at an isolated point, and the third derivative is nonzero. The cubictangencyC of

the EFM surface is illustrated in figure 2.16, where, in terms of the projection onto the

(ωs, dCp)-plane, it results in the creation of a pair of local extrema ofdCp that form a

bulge of the EFM surface. The locus ofC in the(∆1, ∆2)-plane is represented by black

and grey curves in what follows.

Figure 2.16 illustrates the cubic tangencyC. Before the transition the EFM surface in

(ωs, dCp, Ns)-space is such that it does not featuredCp-folds of the boundary curve in pro-

jection onto the(ωs, dCp)-plane; see panels (a1) and (a2). At the moment of transition the

EFM surface is such that the boundary curve of the envelope has a cubic tangency with a curve

dCp = const; see panels (b1) and (b2). This cubic tangency of the boundary curve unfolds into

a pair of a local minimum and a local maximum ofdCp for nearby values ofωs; see panels (c1).

This pair of extrema corresponds to a ‘bulge’ of the EFM surface; see panels (c1). As a result,

there is now an interval ofdCp-values where one finds two distinct EFM components. Note

that the locusC of cubic tangency can be computed by numerical continuation of the condition

that the first two derivatives with respect toωs of the envelope equation (2.9) are zero.

2.3.4 The EFM surface bifurcation diagram in the(∆1, ∆2)-plane for fixed Λ

For a fixed value ofΛ the five transitionsM , SC , Sω, SN andC of codimension one give rise

to boundary curves that divide the(∆1, ∆2)-plane into a finite number of regions. Each such

region defines a type of the EFM surface and, overall, we speak of the EFM surface bifurcation

diagram in the(∆1, ∆2)-plane.

We first consider in figure 2.17 the EFM surface bifurcation diagram forthe caseΛ = 0.01.

As in figure 2.10, the white regions correspond to the band-like types of theEFM surface from

section 2.3.2 with adCp-independent number of EFM components. In the grey regions, on the

other hand, one finds new EFM surface types. In figure 2.17 we labelledthose types that are

associated with transitions between the band-like EFM surface types; thereare a total of 15

additional types, and their representatives are shown in figure 2.18; for notation and parameter

values of the individual panels see Table 2.2. As before, the symbolB denotes a connected


hB

Bh

hB Bh

hhB

hhBB

hh

Bhh

HHH

BBB

hBh

HHHhBh

BBB

BB

BB

BhB BBh

hBB BhB

hBI BI

LL

LBhI

BBh

BBI

IBB

BBI

BB

BB

BBBB

B

hB

Bh

!!! hBI

BI

aaaBhI

B

hB

Bh

aaaIhB

IB

!!!IBh

JJJIIB

IB

IB IBB

IBB

JJ

JBII

BI

BIBBI

BBI

LLLIhB IB

IBh IBB

AAAIBI

AA

AIBI

IB

IB BBBhB B Bh

IB

B BBhB B Bh

BB BI

BI

BB

BB

BBB

BBB

hBB

BB

BB

hBBBB

BBh

BI

M SC Sω SN SN Sω SC M

∆1

∆2

−0.55

0

0.55

−0.55 0 0.55

.

.

Figure 2.17. EFM surface bifurcation diagram in the(∆1, ∆2)-plane forΛ = 0.01 with regions of

different types of the EFM surface; see figure 2.18 for representatives of the labelled types of the EFM

surface and Table 2.2 for the notation. The main boundary curves are the singularity transitionsM

(orange curves),SC (blue curves),Sω (green curves) andSN (red curves). The locus of cubic tangency

(black curves) can be found near the diagonal; also shown is the anti-diagonal.

component of the EFM surface in the form of a band in projection onto the (ωs, dCp)-plane

that extends over the entireCp-range[−π, π].

There are two noteworthy features of the EFM surface types in figure 2.18. First of all,

there are connected components of the EFM surface that do not extend over the entireCp-range

[−π, π]; we use the symbolI to refer to them because their projection onto the(ωs, dCp)-plane

consists of an ‘island’ whendCp ∈ R/2πZ (infinitely many islands in the covering space when

dCp ∈ R). Owing to the underlying symmetry(∆1, ∆2) 7→ (−∆1, −∆2), we again represent


(a)

BI

(b)

BII

(c)

IBI

(d)

Bh

(e)

IBh

(f)

IhB

(g)

Bhh

(h)

hBh

(i)

BBI

(j)

BBI

(k)

BhB

(l)

BBh

(m)

hBB

.

.

Figure 2.18. Additional types of EFM-surface of the 2FOF laser in the labelled regions of figure 2.17,

represented by the projection (shaded) onto the (ωs, dCp)-plane; the blue boundary curves are found

directly from (2.16). For notation and the corresponding values of∆1 and∆2 see Table 2.2; in all panels

ωs ∈ [−0.3, 0.3] anddCp ∈ [−π, π].

in the notation the position of an island with respect to the central bandB. As second new

feature is the fact that a band may have up to two (periodically repeated) holes. Similarly to the

islands, we reflect in the notation the position of a hole with respect to the central frequency of

the laser (atωs = 0). Namely, we indicate with left and right subscripts whether a hole is to

the left or to the right ofωs = 0; for example, we distinguish the caseBhh from hBh. We

observe that islands never have holes for any values of the parametersas considered here; this

means that the symbolI does never have a subscript.

In figure 2.18 we again show only one representative for any pair that isrelated by sym-

metry. Obtaining a representative of the symmetric counterpart corresponds to a reflection of

the respective image in the lineωs = 0; this operation is reflected in the notation by reversing

the symbol string representing the EFM surface type. In figure 2.17 this symmetry operation

corresponds to reflection in the antidiagonal of the(∆1, ∆2)-plane. Notice also the symmetry



BI 1 2 figure 2.18 (a) 0.255 -0.270 0.01

BII 1 3 figure 2.18 (b) 0.260 0.210 0.01

IBI 1 3 figure 2.18 (c) 0.254 -0.251 0.01

Bh 1 2 figure 2.18 (d) 0.100 -0.270 0.01

IBh 1 3 figure 2.18 (e) 0.100 -0.250 0.01

IhB 1 3 figure 2.18 (f) -0.100 0.250 0.01

Bhh 1 3 figure 2.18 (g) 0.160 0.110 0.01

hBh 1 3 figure 2.18 (h) 0.095 -0.110 0.01

BBI 2 3 figure 2.18 (i) 0.250 0.190 0.01

BBI 2 3 figure 2.18 (j) 0.250 -0.130 0.01

BhB 2 3 figure 2.18 (k) 0.210 0.100 0.01

BBh 2 3 figure 2.18 (l) 0.180 0.130 0.01

hBB 1 3 figure 2.18 (m) 0.210 -0.100 0.01

Table 2.2. Notation and parameter values for the types of EFM-surface in figure 2.18; the second and


suitable fixeddCp) of the type.

of the EFM surface bifurcation diagram given by reflectio in the diagonal;it corresponds to an

exchange of the two filters and, hence, does not change the EFM surface type.

The outer part of the EFM surface bifurcation diagram in figure 2.17, away from the diag-

onal, is characterised by grey intersecting strips, which are each bounded by a pair of curves

of singularity transitions. These strips must be crossed in the(∆1, ∆2)-plane to move between

different white regions of band-like types of the EFM surface. As an example, consider a suf-

ficiently large fixed value of one of the detunings, say, of∆1, while the other detuning,∆2, is

allowed to change. The grey strip bounded by the pair of curvesM andSC is responsible for

the transition from a single bandB to two bandsBB via the appearance of a string of islands

that then merge into the new bandB. The pairSN andSω, on the other hand, also results in

a transition fromB to BB, but via the appearance of a string of holes inB that then merge to

form a gap that splits off the new bandB. Note that the illustrations of the singularity transi-

tions in figures 2.12–2.15 are all for∆1 = 0.4; hence, they also illustrate the transition fromB

to BB via SN andSω and back toB via SC andM as∆2 is increased from, say,∆2 = 0.

The grey strips in figure 2.17 are unbounded and extend all the way to infinity. This follows

from the fact that the limit∆i → ±∞ reduces to the single FOF laser in a nontrivial way, as

was discussed in section 2.3.1. More specifically, for the chosen value ofΛ = 0.01 the curve

in the (∆1, Λ)-plane of figure 2.8 (b) is intersected four times, and this accounts for the four

stripes one finds for∆2 → ±∞ (and similarly for∆1 → ±∞).


−0.12 0 0.12 −3

−2

−1

0

1

−0.12 0 0.12 −3

−2

−1

0

1 (a) (b)

ωs ωs

dCp

πdCp

π

.

.

Figure 2.19. Projection of the EFM-surface onto (ωs, dCp)-plane forΛ = 0.01. Panel (a) is for

∆1 = −∆2 = 0.003 and panel (b) is for∆1 = −∆2 = 0.08.

The anti-diagonal is shown in figure 2.17 because along it one finds special, degenerate

cases of the EFM surface. There are two different cases, and they are shown in figure 2.19.

Along the red part of the anti-diagonal we find a degenerate saddle transition SN . At the

moment of transition the upper and lower sheets of the EFM surface touch ata single, isolated

point (and its2π-translates indCp); see figure 2.19 (a). However, as figure 2.17 shows, on

either side of the red part of the anti-diagonal we find the symmetrically relatedpair hB and

Bh, which each features a hole. Hence, when the red part of the anti-diagonal is crossed, the

hole shrinks and then reappears on the other side of the lineωs = 0; physically, the hole is on

the side of the filter that is detuned furthest from the solitary laser frequency. From a bifurcation

point of view, along the red part of the anti-diagonal the EFM surface changes locally from a

two-sheeted hyperboloid to a cone and back to a two-sheeted hyperboloid, rather than to a

one-sheeted hyperboloid; compare with figure 2.15 for the non-degenerate saddle transition

SN .

Along the grey part of the anti-diagonal, on the other hand, we find a degenerate saddle

transitionSC . Notice that the EFM surface type on either side of this grey curve is the same

and invariant under the symmetry operation∆1 7→ −∆2. Figure 2.19 (b) shows the moment

of transition for the case that the anti-diagonal bounds the two regions of EFM surface type

hBh in figure 2.17. In this transition the two holes (and their2π-translates indCp) touch to

form a lemniscate in figure 2.19 (b). This means that the EFM surface is connected (locally)

at isolated points withωs = 0. We also found this degenerate type of connection of the EFM

surface at such isolated points in figure 2.9 — for the case that the surface is of typeB, and

for the case that it is of typeBBB. In effect, along the grey part of the anti-diagonal the EFM

surface changes locally from a one-sheeted hyperboloid to a cone andback to a one-sheeted

hyperboloid, rather than to a two-sheeted hyperboloid; compare with figure 2.13 for the non-

degenerate saddle transitionSC .


B Bh

Bh

bBhh

bBhh

bBb

h

bBbh

Bbb

Bbb

Bbb

Bbbb

Bbbb

Bb

Bb

Bb

Bbb

Bbbh

Cd

SC

SN

Ca

DCNC

∆1

∆2

0

0.1

0 0.1.

.

Figure 2.20. Enlargement near the center of the(∆1, ∆2)-plane of figure 2.17 with (blue) curves

of SC transition, (red) curves ofSN transition, and (black) curvesCa and (grey) curvesCd of cubic

tangency; see figure 2.21 for representatives of the labelled types of the EFM surface and Table 2.3 for

the notation.

The locus of cubic tangency in the(∆1, ∆2)-plane

In figure 2.17 one finds (black) curves of cubic tangency near the diagonal in the central region

of the(∆1, ∆2)-plane. To understand their role for the EFM surface bifurcation diagram, we

show in figure 2.20 an enlargement of the(∆1, ∆2)-plane near the central (white) region where

the EFM surface is of typeB. Recall that this central region must exist as a perturbation of the

special case of the EFM surface for∆1 = ∆2 = 0 in figure 2.9 (a); from the physical point

of view, this type of EFM surface exists (forκ 6= 0) as the continuation of the solitary laser

mode forκ = 0. Note that forΛ = 0.01, as in figures 2.17 and 2.20, this central region is quite

small; however, as we will see in section 2.4, it may grow considerable when the filter widthΛ

is increased.

The central (white) region of EFM surface typeB in figure 2.20 is bounded entirely by

curves of cubic tangency. Therefore, as the filters are detuned awayfrom the solitary laser

frequency, the first transformation of the EFM-surface that gives rise to an additional EFM-

component is a cubic tangency. We find it convenient to distinguish two different types of


(a)

Bb

(b)

Bbb

(c)bBb

(d)

Bbbb

(e)bBbb

(f)bbBbb

(g)

Bbbbb

(h)

BIb

(i)

BIbb

(j)

hBb

(k)

Bbh

(l)

hBbb

(m)

Bbbh

(n)

BBb

(o)

BBbb

.

.

Figure 2.21. Additional types of EFM-surface of the 2FOF laser that feature bulges, represented by the

projection (shaded) onto the (ωs, dCp)-plane; the blue boundary curves are found directly from (2.16).

Where necessary, insets show local enlargements. The corresponding regions in the(∆1, ∆2)-plane

can be found in figures 2.20, 2.25 and 2.29; for notation and the corresponding values of∆1, ∆2 andΛ

see Table 2.3. In all panelsωs ∈ [−0.3, 0.3] anddCp ∈ [−π, π].

cubic tangency. The (black) curvesCa are invariant under reflection in the anti-diagonal; they

correspond to cusp points of the orange curves in figures 2.7 and 2.10.The (grey) curvesCd are

invariant under reflection in the anti-diagonal; they correspond to cusp points of the grey curves

in figures 2.7 and 2.10. The (black) cubic tangency locusCa consists of two elongated and self-

intersecting closed curves, one above and one below the diagonal; see figure 2.17. The (grey)

cubic tangency locusCd also consists of two self-intersecting closed curves, but they cross the

diagonal and one lies above and the other below the anti-diagonal; see figure 2.20 and note that

the locusCd is too small to be visible in figure 2.17.

Crossing either of the boundary curvesCa andCd from inside the central (white) region

labelledB in figure 2.20 results in the appearance of a bulge of the EFM surface. Werepresent

this in our notation of this region asBb by a superscriptb; as before, whether the superscript



Bb 1 2 figure 2.21 (a) -0.0200 0.2000 0.098

Bbb 1 2 figure 2.21 (b) 0.1320 0.1300 0.010bBb 1 3 figure 2.21 (c) -0.2400 0.2300 0.098

Bbbb 1 3 figure 2.21 (d) 0.1400 0.1300 0.010bBbb 1 3 figure 2.21 (e) -0.2300 0.2170 0.098bbBbb 1 3 figure 2.21 (f) -0.2140 0.2150 0.098

Bbbbb 1 3 figure 2.21 (g) -0.2350 0.1780 0.020

BIb 1 3 figure 2.21 (h) 0.2100 0.1900 0.010

BIbb 1 3 figure 2.21 (i) 0.2150 0.1850 0.098

hBb 1 3 figure 2.21 (j) -0.2120 0.2400 0.098

Bbh 1 3 figure 2.21 (k) 0.1300 0.1161 0.010

hBbb 1 3 figure 2.21 (l) -0.2120 0.2200 0.098

Bbbh 1 3 figure 2.21 (m) 0.1337 0.1161 0.010

BBb 2 3 figure 2.21 (n) 0.2100 0.1600 0.010

BBbb 2 3 figure 2.21 (o) 0.1530 0.1355 0.010

Table 2.3. Notation and parameter values for the types of EFM-surface in figure 2.21; the second and


suitable fixeddCp) of the type.

appears on the left or on the right of the central bandB indicates its position with respect to

the solitary laser frequency given byωs = 0. As figure 2.20 shows, we find a complicated

EFM surface bifurcation diagram consisting of an interplay of cubic tangency curvesCa andCd

with saddle-transition curvesSC andSN . The four sets of curves divide the(∆1, ∆2)-plane

into regions of additional EFM surface types, which are all characterised by a certain number

of bulges as represented in the notation. The corresponding EFM surface types are shown in

projection onto the (ωs, dCp)-plane in figure 2.21 (a), (b), (d), (k) and (m); for notation and

parameter values of the individual panels see Table 2.3. Note that near thecentral region of the

(∆1, ∆2)-plane the EFM surface is subject to the interaction of both filters near the frequency

of the laser, which means that there are no other bands or islands.

Figure 2.20 also shows that crossing the saddle-transition curveSC may result in the cre-

ation of a hole, which happens, for example, in the transition fromBbbb to Bbh. This new

mechanism for the creation of a hole is illustrated in figure 2.22 for the simpler case of a tran-

sition fromBbb to Bh; see also figure 2.21 (b) and figure 2.18 (d). In figure 2.22 (a1) and (a2)

there are two bulges; one of the bulges is rather small, indicating that it has just been created

in a nearby cubic tangency. At the saddle transitionSC in figure 2.22 (b) we find that the two

bulges connect locally in the central point of a cone that is aligned in thedCp-direction. In


−0.065 0 0.040.9

1.2

−0.065 0 0.040.9

1.2

−0.065 0 0.040.9

1.2

(a1)

(b1)

(c1)

(a2)

(b2)

(c2)

ωs

ωs

ωs

dCp

π

dCp

π

dCp

π

Ns

Ns

Ns

ωs

ωs

ωs

dCp

π

dCp

π

dCp

π

.

.

Figure 2.22. Global manifestation of local saddle transitionSC of the EFM-surface where two bulges

connect to form a hole. Panels (a1)–(c1) show the relevant part of the EFM surface and panels (a2)–

(c2) the corresponding projection onto the(ωs, dCp)-plane. HereΛ = 0.015 and∆2 = −0.02, and

∆1 = −0.0248 in (a),∆1 = −0.02498 in (b) and∆1 = −0.0252 in (c).

contrast to the case shown in Figure 2.13, the geometry of the EFM surfaceis now such that

this bifurcation leads to the creation of a hole; see panels (c1) and (c2). This hole can then

disappear again when the (red) curveSN in figure 2.20 is crossed; for example, this happens in

the transition fromBbh to Bb.

The connection between cubic tangency and the singularity transitions is given by codimension-

two points on the locus of cubic tangency. This feature is prominent in figure 2.20, where

curvesSN (red) andSC (blue) of saddle transition end at (purple) points on the (grey) curve

Cd. In figure 2.23 we present a local unfolding of such a codimension-two point, which is char-


4

4 − 1

C

1 1 − 2

SN

2 2 − 3

SC

3

3 − 4

C

SN

SC

CDCNC

1

2

3

4

DCNC

∆1

∆2

.

.

Figure 2.23. Sketch of the bifurcation diagram in the(∆1, ∆2)-plane near the (purple) codimension-

two pointDCNC on the curveC of cubic tangency, from which the (red) curveSN and the (blue) curve

SC of saddle transition emanate; compare with figures 2.20 and 2.29 (a) and (b).

4

4 − 1

C

1 1 − 2

M2 2 − 3

Sω

3

3 − 4

C

M

Sω

CDCMω

1

2

3

4

DCMω

∆1

∆2

.

.

Figure 2.24. Sketch of the bifurcation diagram in the(∆1, ∆2)-plane near the (golden) codimension-

two pointDCMω on the curveC of cubic tangency, from which the (orange) curveM and the (green)

curveSω of saddle transition emanate; compare with figures 2.17 and 2.28.


acterised by the fact that the envelope curve given by (2.9) has a cusppoint. When making a

circle around the codimension-two pointDCNC on the curveC, starting at region 1, one finds

that a hole appears near the boundary of EFM surface in the saddle transitionSN ; compare with

figure 2.22. The hole then disappears in the saddle transitionSC . As a result, there are now

two bulges, that is, pairs of local maxima and minima with respect todCp, which disappear one

after the other when the cubic tangency curveC is crossed twice to complete the circle back to

region 1. This unfolding can indeed be found locally in figure 2.20, but note that it involves

the EFM surface of typeBb as that corresponding to region 1 in figure 2.23; hence, region 2

corresponds toBbh, region 3 toBbbb, and region 4 toBbb.

A second case of a codimension-two point on the curve of cubic tangencycan be found in

figure 2.17, where the curvesM (orange) andSω (green) end at (golden) points on the (black)

curveCa. The local unfolding of such a codimension-two point is presented in figure 2.24;

it is again characterised by a cusp point on the envelope curve given by(2.9), but this time

the cusp points the other way with respect to the EFM surface. When making acircle around

the codimension-two pointDCNC on the curveC, starting at region 1, an island is created

when the minimax transitionM is crossed; this island then merges with the remainder of

the EFM surface in the saddle transitionSω. As a result, there are again two bulges, which

disappear one after the other when the curveC is crossed twice to complete the circle back

to region 1. This unfolding can be found locally in figure 2.17, where EFM surface of type

BI corresponds to region 1 in figure 2.23; hence, region 2 correspondsto BII, region 3 to

BIbb, and region 4 toBIb. Note that, in terms of the envelope curve (2.9), the unfolding

in figure 2.24 is topologically equivalent to that in figure 2.23. However, thetwo unfoldings

differ in where (the projection of) the EFM surface lies with respect to the boundary; hence, the

respective panels of figure 2.23 and figure 2.24 (where the EFM surface is always on the left)

can be transformed into one another by exchanging the colours blue and white in the regions,

followed by a reflection.

Note that figure 2.21 shows the comprehensive list, in order of increasingcomplexity, of

EFM surface types that feature bulges — of which there are quite a few more than we identified

in figure 2.20. Additional EFM surface types can be found near the diagonal of the(∆1, ∆2)-

plane, but further away from the central point∆1 = ∆2 = 0. Figure 2.25 (a) shows the

respective enlargement of the EFM surface bifurcation diagram from figure 2.17, which fea-

tures an interaction of cubic tangency curvesCa with saddle-transition curvesSC andSω. We

find five additional EFM surface types with bulges, representatives of which in the (ωs, dCp)-

plane are also shown in figure 2.21 (h), (i), (k), (n) and (o); for notation and parameter values of

the individual panels see again Table 2.3. Note, in particular, that crossing the saddle-transition

curveSω may lead to secondary bands or islands with bulges.

To obtain the remaining cases of EFM surface types with bulges in figure 2.21it is nec-

essary to change the filter width parameterΛ. As an example of how new regions of EFM


Bbb

BI

Bbh

Bbh

Bbb

b

Bbb

b

Bbbh

Bbbh

Bhh

Bhh

BBb

BBb

BIb

BIb

BBbb

BBbb

BBh

BBh

@@BIbb

@@BIbb

Sω SC Ca

Bbb

BI

BIb

BIb

BIbb

BIbb

Bbbb

Bbbb Bbbbb

Bbbbb

Bbbh

Bbbh

Bbh

Bbh

BBbb

BBbb

Sω SC Ca(a)

∆1

∆2

0.12

0.17

0.12 0.17

(b)

∆1

∆2

0.25

0.150.15 0.25

.

.

Figure 2.25. Enlargement near the diagonal of the(∆1, ∆2)-plane with (blue) curves ofSC transi-

tion, (green) curves ofSω transition, andSC transition, and (black) curvesCd of cubic tangency; see

figure 2.21 for representatives of the labelled types of the EFM surface and Table 2.3 for the notation.

Panel (a) is forΛ = 0.01 as figure 2.17, and panel (b) is forΛ = 0.02

surface types are created in a subtle way with changingΛ, figure 2.25 (b) shows a similar en-

largement as panel (a), but now forΛ = 0.02. Notice that the relative position of the curveSω

has changed in such a way that one finds a region where the EFM surface is of typeBbbbb; see

figure 2.21 (g). The location in(∆1, ∆2, Λ)-space of all EFM surface types in figure 2.21 can

be found in Table 2.3; we will encounter more types in the next section.

2.4 Dependence of the EFM surface bifurcation diagram on the

filter width Λ

We now consider more globally how the EFM surface bifurcation diagram in the (∆1, ∆2)-

plane changes with the common filter widthΛ. Figure 2.7 already indicated that substantial

changes to the regions of band-like EFM types must be expected. In particular, for sufficiently

largeΛ the grey and orange curves in figure 2.7 do not extend to infinity in the(∆1, ∆2)-plane

any longer. To study this phenomenon we compactify the(∆1, ∆2)-plane by the stereographic

change of coordinates

∆i =∆i

|∆i| + η, η > 0. (2.18)

Note that (2.18) transforms the(∆1, ∆2)-plane to the square[−1, 1]×[−1, 1] in the(∆1, ∆2)-

plane, where∆i = ±1 corresponds to∆i = ±∞; we speak of the(∆1, ∆2)-square from now

2.4. Dependence of the EFM surface bifurcation diagram on the filter widthΛ 53

0

0

0

0

0

0

0

0

0

0

0

0

B

BB

B

BB

B

BB

BB

BB

BB

B

B

B

BB

B

BB

B

BB

BBB@@

BB

BB

BBB@@

BB

BBB

BBB

B

BB

B

BB

B

BB

BB

BB

BB

B

B

B

BB

B

BB

B

BB

BBB@@

BB

BB

BBB@@

BB

B

B

B

B

B

B

B

B

B

B bB

bBB

B bB

bBB

B bB bBB

B bB bBB

B

B

B

B

BCa

Cd

B

Ca

Cd

B

Ca

Cd

∆1

∆1

∆1

∆1

∆1

∆1

∆2

∆2

∆2

∆2

∆2

∆2

(a)

(c)

(e)

(b)

(d)

(f)

.

.

Figure 2.26. EFM surface bifurcation diagram in the compactified(∆1, ∆2)-square,[−1, 1]× [−1, 1],

showing regions of band-like EFM surface types; compare with figure 2.10. The boundary of the square

corresponds to∆i = ±∞; from (a) to (e)Λ takes valuesΛ = 0.01, Λ = 0.015, Λ = 0.06, Λ =

0.098131, Λ = 0.1 andΛ = 0.13.

on. The parameterη is the value of∆i that is mapped to∆i = 0.5, and we choseη = 0.4 to

ensure that the main structure of the EFM surface bifurcation diagram in the(∆1, ∆2)-plane


is represented well in the(∆1, ∆2)-square.

Figure 2.26 shows the EFM surface bifurcation diagram in the(∆1, ∆2)-square in the style

of figure 2.10, where (orange and grey) boundary curves are shown for 61 equidistant values

of dCp from the interval[−π, π]. The (white) regions of band-like EFM surface types are

labelled; see Table 1.1. Also shown are the cubic tangency curvesCa (black) andCd (grey).

Figure 2.26 (a) forΛ = 0.01 is simply the compactified version of figure 2.10 in the

(∆1, ∆2)-square. Notice that the (orange and grey) stripes now end at discretepoints at the

sides of the square. Namely, for∆i = ±∞ the respective filter does not influence the laser

any longer, so that the system reduces to single FOF laser on the sides of the (∆1, ∆2)-square.

Hence, the end points of the stripes are exactly the four intersection points of the line Λ =

0.01 with the light grey limiting curve in figure 2.8 (b). AsΛ is increased, the areas of other

EFM surface types covered by (orange and grey) curves expandsand four (symmetry-related)

smaller regions of band-like EFM surface typesBBB andBBB disappear; see figure 2.26 (b).

As Λ is increased further, additional regions of band-like EFM surface types disappear; see

figure 2.26 (b). In the process the pairs of (orange and grey) stripesmove closer together,

owing to the fact that the four intersection points with the light grey curve in figure 2.8 (b) do

the same. Moreover, the (black and grey) cubic tangency curves extend over a much larger

region of the(∆1, ∆2)-square; hence, the region near∆1 = ∆2 = 0 where one finds EFM

surface typeB opens up considerably.

Figure 2.26 (d) is forΛ = 0.098131, which is the approximate value ofΛ where one

finds the cusp points of the grey limiting curve in figure 2.8 (c). This value canbe computed

analytically as

ΛC =2

3√

3κ√

1 + α2 (2.19)

from the formula in [28] for the single FOF laser. In the context of the EFM surface bifurcation

diagram, this values corresponds to a bifurcation at infinity of the(∆1, ∆2)-plane and, hence,

a bifurcation at the boundary of the(∆1, ∆2)-square. More specifically, forΛ = ΛC the pairs

of orange and grey stripes now end at single points. the cubic tangency curvesCa (black) and

Cd (grey) extend all the way to the boundary of the square. Notice further that the central region

of EFM surface typeB is no longer bounded by curves of cubic tangency but is now joined

up with four, previously separated regions of the same type. ForΛ > ΛC the stripes do no

longer extend to the boundary of the square. As a result, one now finds orange and grey pairs

of islands that are bounded almost entirely by (black and grey) cubic tangency curves. The

complement of these islands is a single connected (white) region of EFM surface typeB; see

figure 2.26 (e). WhenΛ is increased further, these islands become smaller; see figure 2.26 (f).


(a1) (a2) (a3)

(b1) (b2) (b3)

DCNC

DCMω

DCMω

DCNC

SC

SN

SC

SN

SC

SN

Ca CaCa

Sω M

B

Bh

BB

Bbb

Bb

BI

B B

Bh

Bbb

Bb

BB

Bh

Bbb

Bb

B

IB

BB

bbB

bB

hB

B B

I bB

bbB

bB

BB

I bB

bbB

bBSω

M

Sω

M

Sω

M

Cd CdCd

SC SN

∆1 ∆1 ∆1

∆1 ∆1 ∆1

∆2 ∆2 ∆2

∆2 ∆2 ∆2

.

.

Figure 2.27. Sketch of EFM surface bifurcation diagram near the boundary∆2 = −1 of the

(∆1, ∆2)-square in the transition throughΛ = ΛC . Panels (a1)–(a3) show the transition involving

the (black) curveCa of cubic tangency that bounds the orange islands, and panels(b1)–(b3) show the

transition involving the (grey) curveCd of cubic tangency that bounds the grey islands; compare with

figure 2.26 (c)–(e).

2.4.1 Unfolding of the bifurcation at infinity

As we know from section 2.3.4, the boundaries of the orange and grey regions are formed not

only by the curvesCa andCd of cubic tangency, but also by the singularity transitionsM , SC ,

Sω andSN . Figure 2.27 shows how these boundary curves interact in the transition through

Λ = ΛC in a neighbourhood of the respective point on the boundary of the(∆1, ∆2)-square.

There are two cases: one for the orange regions and one for the greyregions in figure 2.26.

Row (a) of figure 2.27 shows the transition for the grey regions, which involves the (grey)

cubic tangency curveCd; labels in the regions indicate the respective EFM surface type. Before

the bifurcation forΛ < ΛC the grey region extends to two points on the boundary of the

(∆1, ∆2)-square. As we have seen in figure 2.17, these points are the limits of the pairof

curvesM andSC and of the the pair of curvesSω andSN , respectively; see figure 2.27 (a1).

Notice further that the singularity transition curvesSN andSC end at a (purple) codimension-

two pointDCNC on the cubic tangency curveCd; compare with figure 2.23. The EFM surface

types in the respective regions are also shown. WhenΛC is approached, the two limit points

of the pairs of curvesM andSC andSω andSN approach each other. At the same time,

the codimension-two point, and the curveCa with it, approach the boundary of the(∆1, ∆2)-

square; as a result, the curvesSN andSC become shorter. At the moment of transition at


B B

BBb

BbbBb

BI

M

Sω

DCMω

DCMω

Cd0.26

0.26

0.97

0.97∆1

∆2

.

.

Figure 2.28. Grey island forΛ = 0.1 in the(∆1, ∆2)-plane with regions of non-banded EFM surface

types; compare with figure 2.26 (e).

Λ = ΛC , shown in figure 2.27 (a2), the grey region is bounded by the minimax transition curve

M and by one branch of the cubic tangency curveCd, which both end at a single point on the

boundary. The curveSω and a second branch ofCd also end at this point on the boundary. For

Λ > ΛC , as in figure 2.27 (a3), all curves detach from the boundary of the(∆1, ∆2)-square;

furthermore, the singularity transition curvesM andSω are now attached to the cubic tangency

curveCd at a (golden) codimension-two pointDCMω; compare with figure 2.24. Hence, the

grey island created in this transition, which is invariant under reflection in thediagonal, is

bounded by the curvesCd andM .

Row (b) of figure 2.27 shows the transition for the orange regions, whichinvolves the

(black) cubic tangency curveCa; again, the respective EFM surface type are indicated. This

transition is very similar to that in figure 2.27 (a), but notice that it now involvesa (golden)

codimension-two pointDCMω on the curveCa for Λ < ΛC , and a (purple) codimension-

two pointDCNC for Λ > ΛC . As a result of this transition the orange island created in this

transition, which is invariant under reflection in the anti-diagonal, is bounded by the curvesCa

andSN .

2.4.2 Islands of non-banded EFM surface types

The grey islands are associated with the diagonal where∆1 = ∆2, along which the 2FOF laser

reduces to the single FOF laser with effective feedback rate given by (2.11). Hence, the width

of the grey islands along the diagonal is determined by the intersection points of the horizontal


line for the given value ofΛ with the curve in figure 2.8 (c). A grey island forΛ = 0.1 is shown

in figure 2.28. It contains regions of (non-banded) EFM surface types Bb, Bbb andBI, which

are bounded by the cubic tangency curveCd and singularity transition curvesM (orange) and

Sω (green); the latter curves end at two (golden) pointsDCMω on the curveCd. WhenΛ is

increased, the grey islands remain topologically the same and simply shrink downto a point.

This happens whenΛ has the value of the cusp points of the curve in figure 2.8 (c), which can

again be computed analytically from the formula in [28] for the single FOF laseras

ΛI = 2 ΛC =4

3√

3κ√

1 + α2 ≈ 0.196261. (2.20)

Furthermore, we can conclude from the single FOF limit along the diagonal that the two sym-

metric grey islands disappear at

∆1 = ∆2 = ∆Id =8√

2

3√

3κ√

1 + α2 ≈ 0.555111. (2.21)

For Λ > ΛI the EFM surface bifurcation diagram in the(∆1, ∆2)-square — and, hence, also

in the(∆1, ∆2)-plane — does no longer contain (grey) islands that are symmetric with respect

to the diagonal.

The orange islands that exist forΛ > ΛC are associated with the anti-diagonal where

∆1 = −∆2. Figure 2.29 (a) shows an orange island forΛ = 0.1 with regions of (non-banded)

EFM surface types; see also the enlargement in the inset panel. Apart from the cubic tangency

curveCa, we find curvesSC (blue) andSN (red) of singularity transition. Two curvesSC

emerge from a boundary point of the island on the anti-diagonal where four branches ofCa

connect in a pair of cusps. The two curvesSC follow two branches ofCa closely and end at

two (purple) pointsDCNC on the curveCa. The two curvesSN emerge from a different point

on the anti-diagonal and also followsCa closely to the same two end points. The (red) section

of the anti-diagonal in between the two points from which the curvesSC andSN emerge,

respectively, corresponds to a degenerate saddle transitionSN . The remainder of the anti-

diagonal corresponds to a degenerate saddle transitionSC ; see the discussion in section 2.3.4.

Overall, we find a quite complicated but consistent structure of the orange island. It features the

(non-banded) EFM surface typesBh, hBh, Bb, Bbb, hBb, bBb, hB

bb, bBbb andbbBbb, of which

the last five EFM types with bulges are new; compare with figure 2.21.

As Λ is increased, the orange island undergoes topological changes. First,the (purple)

end points of the curvesSC andSN move across a branch of the cubic tangency locusCa.

As a result, the entire island is now bounded byCa and the regionsBh and Bbb (and their

symmetric counterparts) disappear; see figure 2.29 (b). The next qualitative change concerns

the cubic tangency locusCa, which loses two intersection points, resulting in the loss of two

(symmetrically related) regions of EFM typesbB and Bb; see figure 2.29 (c). WhenΛ is


B

B

bB

Bb

bbBbBb

bBb

SN DCNC

DCNC

SC

Ca

hB/bbB AA

hBh

bBbb

bbBhB

b

hBbb

ZZbbBbb

bBh

hbB

.

.

B

B

Bb

bB

hbB

bBh

bBb

bB

Bb

SC

Ca

SNDCNC

DCNC

.

.

B

B

Bb

bB

hB

Bh

bBb

Ca

SC

.

.

B

B

Bb

bB

hB

Bh

Ca

SC

.

.

∆1 ∆1

∆2

∆2

(c)

-0.45

-0.05

0.05 0.45

(d)

0.15 0.35-0.35

-0.15

(a)

-0.97

0.28

0.28 -0.97

(b)

0.05 0.45-0.45

-0.05

.

.

Figure 2.29. Orange island in the(∆1, ∆2)-plane with regions of non-banded EFM surface types; the

inset in panel (a) shows the details of curves and regions. From (a) to (d)Λ takes the valuesΛ = 0.1,

Λ = 0.145, Λ = 0.166 andΛ = 0.179; compare panel (a) with figure 2.26 (e).

increased further, two intersection points of curvesCa on the anti-diagonal come together and

merge into a point where four branches ofCa connect. The result is the loss of regionbBb; see

figure 2.29 (d). We found that the orange island does not undergo further qualitative changes,

but rather shrinks down to a point and disappears. This happens atΛ =≈ 0.196261, and

this numerical value agrees up to numerical precision with that forΛI from (2.20). In fact,

consideration of (2.9) for∆1 = −∆2 confirms this observation; furthermore, the position of

where the islands disappear can be computed as

∆1 = −∆2 = ∆Ia = ∆Id/4 =2√

2

3√

3κ√

1 + α2 ≈ 0.138778. (2.22)

2.5. The effect of changing the delay difference∆τ 59

Overall we conclude that forΛ > ΛI there are no islands at all, so that the entire(∆1, ∆2)-

plane consists of a single region of EFM typeB. Physically, this means that the two filters are

so wide that they reflect light of different frequencies effectively in the same way; hence, the

feedback is no longer frequency selective, and the 2FOF laser is a 2COF laser for sufficiently

largeΛ.

2.5 The effect of changing the delay difference∆τ

Equation (2.16) for the boundary of the projection of the EFM surface onto the(ωs, dCp)-plane

expressesdCp as a function ofωs. In this equation the delay time differencedτ appears only

as the coefficient of the linear term ofωs; hence, a nonzero value ofdτ introduces a shearing

of the EFM surface with a shear rate of exactlydτ .

This shearing of the EFM surface can be made explicit for the special case that∆1 = ∆2,

when still considering a single filter widthΛ for both filters. Namely, then we can define the

center line of the projection of the EFM surface as the line through the points where the inverse

cosine term in (2.16) vanishes. The equation for this center line is then simply

dCp(ωs) = −dτωs, (2.23)

which is simply the line with slopedτ through the origin of the(ωs, dCp)-plane. Since the

ωs-range of the EFM surface does not change withdτ , a nonzerodτ indeed leads to a shear

with a shear rate equal to the slope of the center line of the projection of the EFM surface.

We conclude that an EFM surface fordτ 6= 0 can be obtained simply by considering the

corresponding EFM surface fordτ = 0 and shearing it with a shear rate ofdτ . As a result of

this shearing, EFM components may be present fordτ 6= 0 that are not present fordτ = 0.

Figure 2.30 illustrates this effect with the example of the EFM surface for∆1 = ∆2 = 0

andΛ = 0.015 with τ1 = 500 andτ2 = 600, so that the shear rate isdτ = 100. Notice

that this EFM surface is the sheared version of the corresponding EFM surface fordτ = 0 in

figure 2.5, which is of typeB. The EFM surface in figure 2.30 still consists of all2π-translates

of a basic unit, which are connected at the points wheredCp = π + 2kπ for k ∈ Z; however,

now the basic unit of the EFM surface extends over adCp-range of more than2π. While

there is always a single EFM component for any value ofdCp for the case thatdτ = 0, due

to the shear fordτ = 100 we now find up to three EFM components in figure 2.30. Each of

those EFM components belongs to a different 2π-translated copy of the basic unit of the EFM

surface. Physically, this is due to beating between two frequencies that are associated with the

two feedback loops, of different delay times; see Eqs. (2.2)–(2.4).


−0.1 0 0.1−0.02

0

0.02

−0.1 0 0.1−0.02

0

0.02

−0.1 0 0.1−0.02

0

0.02

−0.1 0 0.1−0.02

0

0.02

(a)

0.02

Ns

0

-0.023

dCp

π

0

-1

-3-0.1

0ωs

0.1

(b)

(d)

(c)

(e)

ωs

ωs

ωs

ωs

Ns Ns

Ns Ns

.

.

Figure 2.30. The EFM surface (a) forκ = 0.05, ∆1 = ∆2 = 0, Λ = 0.015, andτ1 = 500 andτ2 = 600

so thatdτ = 100, and its intersection with the planes defined bydCp = 0 anddCp = −π; compare

with figure 2.5. Panels (b)–(e) show the EFM-components fordCp = 0, dCp = −π/2, dCp = −π and

dCp = −3π/2, respectively; the blue dots are the EFMs forC1

p as given by (2.12).

As is shown in figure 2.30 (b)–(e), the exact number of EFM components depends on the

value ofdCp. More specifically, fordCp = 0 there are three EFM-components, owing to the

fact that the corresponding plane in(ωs, dCp, Ns)-space intersects three copies of the basic

unit of the EFM surface; see figure 2.30 (a) and (b). AsdCp is changed the EFM components

change. FordCp = −π/2 as in panel (c), there are still three EFM components (the right and

2.5. The effect of changing the delay difference∆τ 61

−0.1 0 0.1−0.02

0

0.02

−0.1 0 0.1−0.02

0

0.02

−0.1 0 0.1−0.5

0

0.5

−0.1 0 0.1−0.5

0

0.5(a1)

(a2)

(b1)

(b2)

ωs

ωs

ωs

ωs

T (ωs) T (ωs)

Ns Ns

.

.

Figure 2.31. Solution curves of the transcendental equation (2.3) and corresponding EFM components

for dCp = 0, where the dots show the actual EFMs forC1

p=0; hereκ = 0.05, ∆1 = ∆2 = 0, Λ = 0.015,

τ1 = 500, anddτ = 200 in panels (a) anddτ = 300 in panels (b). The inset of panel (b2) shows that

the EFM components are in fact disjoint.

central EFM components are not connected), but left of the two outer ones has become smaller

and right one larger. WhendCp is decreased further, the left EFM component disappears (in a

minimax transition when the planedCp = const passes through the end point of the respective

part of the EFM surface). FordCp = −π as in figure 2.30 (d), the two remaining EFM

components connect at the origin of the(ωs, Ns)-plane to form a single EFM component in the

shape of a figure eight. We remark that, because fordCp = −π equation (2.9) is equal to 0 at

ωs = 0, this case for the 2FOF laser corresponds to the situation described in [64] where single

FOF laser is resonant to the minimum of the Fabry-Pérot filter reflectance profile. The single

EFM component then splits up again into two EFM components fordCp < −π. A new EFM

component appears (again in a minimax transition) on the right, that is, for positive ωs, so that

there are again three EFM components; see figure 2.30 (e) fordCp = −3π/2.

Notice that, since the detunings of both filters are equal, the EFM surface in figure 2.30 (a)

is invariant under the anti-diagonal symmetry operation(ωs, dCp, Ns) 7→ (−ωs, −dCp, −Ns).

As a result, panel (b) and (d) are invariant under rotation overπ of the (ωs, Ns)-plane, while

panels (c) and (e) are symmetric counterparts.

Figure 2.31 shows that with an increased shear of the EFM surface of type B, more EFM

components can be found; specifically, up to five fordτ = 200 and up to seven fordτ = 300.

This is illustrated in figure 2.31 by the EFM solution curve inside its envelope (toprow) and


−0.15 0 0.16−3.3

−1.6

0

1.5

−0.15 0 0.16−0.03

0

0.04

−0.15 0 0.16−3.3

−1.6

0

1.5

−0.15 0 0.16−0.03

0

0.04

(a1) (a2)

(b1) (b2)

ωs

ωs

Ns

Ns

dCp

π

dCp

π

ωs

ωs

.

.

Figure 2.32. The EFM surface of typehBB for dτ = 0 (a1) and its EFM components fordCp = −1.6π

(a2), and the corresponding sheared EFM surface fordτ = 230 (b1) and its EFM components for

dCp = −1.6π (b2). Hereκ = 0.05, ∆1 = 0.13, ∆2 = −0.1, Λ = 0.01, andτ1 = 500.

by the respective EFM components in the(ωs, Ns)-plane (bottom row). Figure 2.31 clearly

shows that for sufficiently largedτ any number of EFM components can be found, even when

the EFM surface fordτ = 0 is of the simplest possible typeB.

Finally, figure 2.32 shows the effect of shearing on the more complicated EFM surface of

typehBB. As panels (a1) and (a2) show, this EFM surface type fordτ = 0 may give rise to

up to three EFM components. When it is sheared with a sheer rate ofdτ = 230, however, one

may find up to six EFM components. Notice that this increase in the number of possible EFM

components is due to the shearing of the strongly undulating boundary of central band, as well

as to the shearing of the holes in it, which now extend over adCp range of more than2π.

2.6 Conclusion and outlook

We presented a comprehensive study of the EFM structure of the 2FOF in dependence on the

parameters of the two filtered feedback loops. The main object of study is theEFM surface

in (ωs, dCp, Ns)-space, which is a generalisation of the EFM components for the single FOF

laser. A combination of analysis and extensive numerical continuations of EFMs allowed us

2.6. Conclusion and outlook 63

to present the EFM surface bifurcation diagram in the(∆1, ∆2)-plane for the special case that

the delay time differencedτ is zero. More specifically, we considered five transitions through

critical points and a cubic tangency with respect to the phase differencedCp between the two

filter loops, which emerged as a key parameter. These transition give rise toboundary curves

that divide the(∆1, ∆2)-plane into regions of different EFM surface types. We then studied

how the EFM surface bifurcation diagram in the(∆1, ∆2)-plane depends on the widthΛ of

the two filters. Finally, nonzerodτ was shown to cause a shearing of the corresponding EFM

surface fordτ = 0, and this may give rise to any number of EFM components.

Our classification of the EFM surface is justified by physical properties ofthe 2FOF laser,

which translate to specific properties of the transcendental equations forthe EFMs. For exam-

ple, there are no islands with holes. What is more, the classification of the EFMsurface as

induced by the five singularity transitions considered was motivated by the fact that they could

all be shown to lead to changes of the number of EFM components over substantial ranges of

the relevant parameters, in particular,dCp.

In a way, the influence of the two filter loops on the laser can be consideredas the feed-

back from a single feedback loop with a complicated filter profile — with several maxima and

minima of the transmittance as a function of the frequency of the light. This filter profile is the

result of interference between the two filter fields. This point of view provides a connection to

studies that considered the output of a laser subject to feedback from anon-Lorentzian, more

complicated filter profile, such as the periodic filter profiles in [50, 64]. In the case of [64],

feedback is considered from a Fabry-Pérot cavity not only close to thetransmittance maxi-

mum, but over a large spectral range that encompasses several maxima and minima of the

transmittance. In [50], on the other hand, filtered feedback with severalmaxima and minima of

the transmittance arise due to side ‘bumps’ of a fibre Bragg grating filter profile. As we have

observed, the observations in [50] and [64] correspond in the 2FOF laser simply to a change

from constructive to destructive interference between the two filter fieldsvia a variation of the

filter phase differencedCp. A further analysis of the connections between the 2FOF laser and

feedback from other types of filters is an interesting question for future research.

Obviously, an important practical issue is to determine when the EFMs are actually stable.

In other words, the next step in the analysis of the 2FOF laser, which is the subject of the next

chapter, is hence the investigation of regions of their stability on the EFM surface.

Chapter 3

EFM stability regions

In the previous chapter we investigated analytically and numerically the structure of external

filtered modes (EFMs) – constant amplitude and frequency solutions to Eqs.(1)–(4). Our

analysis shows that, in the conveniently chosen(ωs, Ns, dCp)-space, EFMs exist on a compli-

cated two-dimensional surface that may consist of up to three disjoint components that are2π

periodic in thedCp-direction.

The EFM surface illustrates how frequencyωs and population inversionNs of the EFMs

depend on the feedback phasesC1p andC2

p , with all other parameters fixed. In other words,

the EFM surface represents all the possible states that an EFM can take for all the possible

values ofC1p andC2

p , for given values of the other parameters. The important parameter here

is the feedback phase differencedCp = C2p − C1

p , so that it is most convenient to consider the

two-dimensional EFM surface in(ωs, Ns, dCp)-space; see section 2.2.

In this chapter we analyse the stability of the EFMs. In particular, we use bifurcation analy-

sis to uncover regions of stable EFMs on the two-dimensional EFM surfacein (ωs, Ns, dCp)-

space. These regions are bounded by codimension-one saddle-nodeand Hopf bifurcation

curves, which meet or intersect at codimension-two saddle-node Hopf,Bogdanov-Takens and

Hopf-Hopf bifurcation points.

As is the case for the EFMs themselves, their stability depends on all laser, filters and

feedback loop parameters. Here we show how the number, shape, and extent of the stability

regions depend on the filters and feedback loops parameters. More specifically, we consider

their dependence on the feedback ratesκ1 andκ2, filter widthsΛ1 andΛ2, the filter frequency

detunings∆1 and∆2, and the delay timesτ1 andτ2. We explain the uncovered changes of

the EFM stability regions in terms of higher codimension bifurcations in conjunction with the

changing geometry of the EFM surface.

To analyse the stability of the EFMs we fix all the system parameters except thefeedback

phasesC1p andC2

p . By means of numerical continuation we then calculate the EFM surface

65

66 Chapter 3. EFM stability regions

together with information about stability of the EFMs. This allows us to render theEFM surface

in (ωs, Ns, dCp)-space with stability information on it. Throughout, regions with stable EFMs

are marked as hatched green patches on the EFM surface; the remaining grey area represents

unstable EFMs. In fact, to indicate the stable EFMs area on three-dimensional image of the

EFM surface we use the actual stable parts of the computed EFM branches. In other words,

each green line on the semitransparent grey EFM surface is the result ofa single numerical

continuation.

To explain and illustrate our results, we also consider the two-dimensional projections of

the EFM surface onto the(ωs, dCp)-plane and the(ωs, Ns)-plane. In section 2.3 we used the

(ωs, dCp)-projections to present our classification into different types of the EFM surface. As

before, the white regions on the(ωs, Ns)-plane are regions where no EFM exist. The boundary

of the projections of the EFM surface onto the the(ωs, dCp)-plane is found directly from Eq.

(2.16) and is now coloured in grey instead of blue. The blue region that represented the inside

of the projection in section 2.3 is now divided into regions of stable and unstable EFMs that

are coloured green and grey, respectively. This colouring of the(ωs, dCp)-projections is in

agreement with the colouring of the EFM surface itself. For clarity, we use solid green filling

to indicate the stable EFM regions in the projections, instead of hatching.

As in chapter 2, both the top and the bottom sheet of the EFM surface are projected onto the

(ωs, dCp)-plane; this means that each typical (non-boundary) point in the(ωs, dCp)-projection

of the EFM surface correspond to two EFMs with differentNs values. We show that the EFM

stability region extends mostly over the bottom sheet of the EFM surface, which corresponds

to higher gain modes. However, for some parameter values stable EFMs canalso be found

on the top sheet of the EFM surface. In those rare cases when the two stable EFM regions are

separated byNs i.e. lay above one another, the corresponding area in the(ωs, dCp)-projections

is coloured in dark green.

The boundary of a region of stable EFMs involves a number of saddle-node bifurcations

and Hopf bifurcation curves that are connected at codimension-two bifurcations points. These

bifurcations form curves on the EFM surface in(ωs, Ns, dCp)-space. Those curves intersect

at Hopf-Hopf and saddle-node Hopf bifurcations. At points where thesaddle-node and Hopf

bifurcation curves merge, and the Hopf bifurcation curve ends, Bogdanov-Takens bifurcations

are located. Note that for clarity, out of all codimension-two bifurcations,we only label the

Bogdanov-Takens bifurcations.

Throughout this chapter saddle-node bifurcation curves are depictedas blue curves on the

EFM surface and the Hopf bifurcation curves are shown as red curves on the EFM surface.

Moreover, for the purpose of this analysis we calculate only the bifurcation curves that are

part of the boundary of the EFMs stability regions, or are useful to explain their changes. All

regions and curves in the projections of the EFM surface onto the(ωs, Ns)-plane, are coloured

in the same way as in the(ωs, dCp)-projections.

3.1. Dependence of EFM stability onκ 67

In chapter 2 we showed that the geometrical and topological properties ofthe EFM surface

depend strongly on the filters and feedback loop parameters. In this chapter we analyse how the

stability regions interact with the singularity transitions of the EFM surface in(ωs, Ns, dCp)-

space. As a guideline in this challenging task we use the classification of the EFM surfaces

introduced in section 2.3. In particular, we are interested in changes of theEFM stability

regions associated with the saddle transitions of the EFM surface. Clearly,changes of the

stable EFM regions must be expected as a result of changes of the EFM surface topology.

However, there may also be changes to the EFM stability regions that are dueto changes of the

EFM surface geometric properties.

In what follows we analyse separately the dependence of the EFM surface on the common

feedback rateκ, the common filter widthΛ, the filter detunings∆1 and∆2, and the difference

dτ between the delay times. To uncover the information on the stability of EFMs, we first fix

parameters at conveniently chosen values and compute the EFM surface with relevant informa-

tion on the EFMs stability. Next, we vary the parameter under investigation, andobserve how

the EFM stability regions transform with changes of the EFM surface in(ωs, Ns, dCp)-space.

This chapter is structured as follows. In section 3.1 we analyse how regions of stable EFMs

change with an increase of the feedback rateκ. Next, in section 3.2 we present how the stable

regions are affected by the transition from the 2FOF laser to the COF laser as Λ → ∞. Both

these chapters show that, although topologically the EFM surface remains unchanged of type

B, the stability of EFMs is substantially affected byκ as well asΛ. Moreover, we use the

results from sections 3.1 and 3.2 to choose convenient values ofκ andΛ for the following

sections. Section 3.3 shows how the regions of stable EFMs split as the EFM surface gradually

transforms fromB via hBh andBBB to IBI with increasing filter frequency detunings∆1

and∆2. In section 3.4 we explain how the EFM stability is affected by shearing of the EFM

surface that is introduced by decreasingdτ . Finally, we briefly describe the kinds of periodic

solutions originating from the Hopf bifurcations in section 3.5, and discuss possible physical

consequences of the transformations of the EFMs stability regions in section4.1.

3.1 Dependence of EFM stability onκ

We start our analysis with the 2FOF laser with two identical filters with equal delay times. In

section 2.2 we showed that in this case the EFM surface is of the simplest typeB, and con-

sist of infinitely manydCp-periodic compact units that are connected with each other in the

points,(ωs, Ns, dCp) = (0, 0, (2n + 1)π) with n ∈ Z. In this section, we explain the depen-

dence of the EFM surface in(ωs, Ns, dCp)-space on the feedback rateκ. While increasingκ

causes an expansion of the EFM surface in theωs andNs directions, its topology is unaffected.

Nevertheless, we demonstrate that the expansion of the EFM surface leads to a change in the


number, extent, and type of boundaries of the EFM stability regions. What ismore, we show

that above a particular value ofκ, despite further expansion of the EFM surface, the(ωs, Ns)-

projection of the EFM stability region remains unchanged. For this study, we fix Λ = 0.015,

∆1 = 0, ∆2 = 0, anddτ = 0.

Figure 3.1 (a1)–(c1) shows the fundamental2π interval of the EFM surface forκ = 0.01

in (a), κ = 0.015 in (b) andκ = 0.025 in (c). In panels (a2)–(c2) and (a3)–(c3) we show

projections of the stability regions of EFMs onto the(ωs, Ns)-plane and onto the(ωs, dCp)-

plane, respectively.

In figure 3.1 (a)–(c) the boundary between regions of stable EFMs andunstable EFMs al-

ways consist of seven Hopf bifurcation curves and one saddle-nodebifurcation curve. Intersec-

tions of those codimension-one bifurcation curves are codimension-two Hopf-Hopf and saddle-

node Hopf bifurcation points. Note that for these parameter values considered Bogdanov-

Takens bifurcations do not occur.

Increasingκ results in an expansion of the EFM surface in theωs andNs directions; see

panels (a1)–(c1). To understand the expansion of the EFM surface,it is enough to notice that

Ns scales linearly withκ; see Eq. (2.5). Moreover, for∆1 = ∆2 = 0, a common filter width

Λ, anddτ = 0, the extrema ofωs are given by

ωexts = ±1/2

√−2 Λ2 + 2 Λ

√Λ2 + 16κ2 + 16κ2α2. (3.1)

Concurrently with the monotonic growth of extrema ofωs, the EFM stability region changes.

When the oval shaped Hopf bifurcation curve around the point(ωs, dCp) = (0, 0) in figure 3.1

(a3) — the one that forms the lower boundary of the stability region in panel (a2) — intersects

with two other Hopf bifurcation curves, then the single region of stable EFMsfrom panel (a)

splits into two; see figure 3.1 (b). A further increase ofκ causes the separation distance of those

two regions to grow; compare figure 3.1 rows (b) and (c). What is more, although the range of

dCp in which stable EFMs exist decreases and the whole EFM surface grows,the projection

of the EFM stability regions onto the(ωs, Ns)-plane in panels (b) and (c) stays unchanged.

These results are in agreement with the observation that the 2FOF laser canbe reduced to a

FOF laser; see also section 2.2.

The expansion of the EFM surface in theωs andNs directions is the simplest geometrical

effect caused by a change of filter and feedback loop parameters. Weshowed that, although the

topology of the EFM surface remains unchanged of typeB, increasingκ results in a shrinking

of the stable EFM regions in the(ωs, dCp)-plane. Considering that we are interested in the

dependence of the EFMs stability region on parameters, in all following sections we fix the

feedback rate atκ = 0.01 — a representative value at which the region of stable EFMs extends

over the whole fundamental2π interval ofdCp.

3.1. Dependence of EFM stability onκ 69

−0.04 0 0.04−0.008

0

0.008

−0.04 0 0.04−1

0

1

−0.04 0 0.04−0.008

0

0.008

−0.04 0 0.04−1

0

1

−0.04 0 0.04 −0.008

0

0.008

−0.04 0 0.04−1

0

1

(a1)

κ = 0.01

(a2)

(a3)

(b1)

κ = 0.015

(b2)

(b3)

(c1)

κ = 0.025

(c2)

(c3)

ωs

ωs

ωs

ωs

ωs

ωs

Ns

Ns

Ns

dCp

π

dCp

π

dCp

π

0

0

0

0

0

0

0

0

0

ωs

ωs

ωs

dCp/π

dCp/π

dCp/π

Ns

Ns

Ns

.

.

Figure 3.1. Dependence of the EFM surface on the feedback rateκ (as indicated in the panels); here

∆1 = ∆2 = 0, Λ = 0.015 anddτ = 0. Panels (a1)–(c1) show the EFM-surface in(ωs, Ns, dCp)-space

(semitransparent grey) together with information about the stability of the EFMs. Panels (a2)–(c2) show

corresponding projections of the EFM surface onto the(ωs, Ns)-plane and panels (a3)–(c3) onto the

(ωs, dCp)-plane. Regions of stable EFMs (green) are bounded by Hopf bifurcations curves (red) and

saddle node bifurcation curves (blue). In panels (a1)–(c1)ωs ∈ [−0.065, 0.065], dCp/π ∈ [−1, 1] and

Ns ∈ [−0.013, 0.013].


−0.11 −0.05 0 0.05 0.011−0.021

−0.01

0

0.01

0.021

Λ = 0.5

Λ = 0.025

Λ = 0.015

Λ = 0.001

Λ = 0.005

ωs

Ns

.

.

Figure 3.2. Dependence of the EFM surface on the filter widthΛ = Λ1 = Λ2 (as indicated); here

∆1 = ∆2 = 0, κ = 0.01 anddτ = 0. Light grey regions with black envelopes are projections, for five

different values ofΛ, of the EFM surface onto the(ωs, Ns)-plane.

3.2 Dependence of EFM stability onΛ

In sections 2.3.1 and 2.4 we showed that, asΛ → ∞, the 2FOF laser approaches the COF laser

limit. More specifically, we showed that increasingΛ leads to the simplification of the topology

of the EFM surface — regions corresponding to EFM surface types other thenB shrink and

disappear. We now explore the dependence of EFM stability regions on thefilter width Λ.

We demonstrate that the fast growth of the EFM surface that leads to the simplification of its

topology, results also in changes of the EFM stability. In particular, we showthat the stable

EFM region expands with increasing filter width; for sufficiently highΛ it is bounded by a

single saddle-node bifurcation curve and a single Hopf bifurcation curve. Throughout this

section we fixκ = 0.01, ∆1 = 0, ∆2 = 0, anddτ = 0. Notice that, as in the previous

section, for∆1 = 0 and ∆2 = 0 copies of the EFM surface are connected at the points

(ωs, Ns, dCp) = (0, 0, (2n + 1)π) with n ∈ Z.

Figure 3.2 illustrates the expansion of the EFM surface with increasingΛ. The black lines

are the envelopes of five projections of the EFM surface (grey) onto the(ωs, Ns)-plane for

different values ofΛ, as indicated on the figure. Note that the range ofωs of the EFM surface

for Λ = 0.5 is ten times larger then the range of theωs of the EFM surface forΛ = 0.001. Note

also that all the envelopes have two common points atωs = 0. The difference between theωs-

range of the envelope forΛ = 0.5 and for the EFM components of COF laser is|max(ωCOFs )−

min(ωCOFs )|−|max(ωΛ=0.5

s )−min(ωΛ=0.5s )| = 0.004; hereωΛ=0.5

s is given by Eq. (3.1) and

ωCOFs = ±κ

√1 + α2. This shows that already forΛ = 0.5 the 2FOF laser with two identical

filters with equal delay times is very close to the COF limit ofΛ = ∞.

3.2. Dependence of EFM stability onΛ 71

We now investigate how the region of stable EFMs is affected by this expansion of the

EFM surface withΛ. Figure 3.3 (a1)–(d1) shows the fundamental2π interval of the EFM

surface together with the EFM stability information forΛ = 0.001 in (a), Λ = 0.005 in (b),

Λ = 0.025 in (c) andΛ = 0.5 in (d). In panels (a2)–(d2) we show corresponding projections of

the EFM surface onto the(ωs, Ns)-plane. Projections of the EFM surfaces from figure 3.3 onto

the(ωs, dCp)-plane are shown in figure 3.4. Black dots indicate codimension-two Bogdanov-

Takens bifurcation points; curves and regions are coloured as in figure3.1.

Given the expansion of the EFM surface, each row in figure 3.3 and corresponding panel

in figure 3.4 is presented over a different range of theωs andNs axes. (This is in contrast to

figure 3.2, where the(ωs, Ns)-projections of all the EFM surfaces from Figure 3.3 (a)–(d) are

shown together on the same scale.) Note also that, figure 3.2 also includes the projection of the

EFM surface from figure 3.1 (a) forΛ = 0.015.

With increasingΛ the region of stable EFMs and its boundary undergo several complicated

transitions; see figures 3.3 (a)–(d) and 3.4 (a)–(d). Figures 3.3 (a) and 3.4 (a), forΛ = 0.001,

show the EFM stability region bounded by one saddle-node bifurcation curve (blue) and three

Hopf bifurcation curves (red). Moreover, in addition to Hopf-Hopf (HH) and saddle-node Hopf

(SH) bifurcation points located at the intersections of the codimension-one bifurcation curves,

one can also find Bogdanov-Takens (BT) bifurcation points; see figure3.3 (a) and (b).

Further, in figures 3.3 (b) and 3.4 (b), forΛ = 0.005, the picture becomes more complicated

and the number of the Hopf bifurcation curves involved in the boundary increases to five. Next,

for sufficiently highΛ, the boundary of the stable EFM regions again simplifies, and consist

of a single saddle-node bifurcation curve at the top and a single Hopf bifurcation curve at the

bottom. Moreover, the region of stable EFMs expands and covers most of the projection of the

EFM surface onto the(ωs, dCp)-plane; see figures 3.3 (c) and 3.4 (c) forΛ = 0.025.

Finally, in figures 3.3 (d) and 3.4 (d), we show that with a further increaseof Λ the EFM

stability region remains qualitatively unchanged. In comparison to figures 3.3(c) and 3.4 (c),

we observe a further expansion of the EFM surface and growth of the region of unstable EFMs

in the direction of positiveωs.


−0.011 0 0.011 −0.005

0

0.005

−0.025 0 0.025 −0.005

0

0.005

−0.05 0 0.05−0.01

0

0.01

−0.11 0 0.11−0.021

0

0.021

(a1)

Λ = 0.001

(a2)

(b1)

Λ = 0.005

(b2)

(c1)

Λ = 0.025

(c2)

(d1)

Λ = 0.5

(d2)

ωs

ωs

ωs

ωs

Ns

Ns

Ns

Ns

0

0

0

0

0

0

0

0

0

0

0

0

ωs

ωs

ωs

ωs

dCp/π

dCp/π

dCp/π

dCp/π

Ns

Ns

Ns

Ns

.

.

Figure 3.3. Dependence of the stability region on the EFM surface on the common filter widthΛ (as

indicated in the panels); here∆1 = ∆2 = 0, κ = 0.01 anddτ = 0. Panels (a1)–(d1) show the EFM-

surface in(ωs, Ns, dCp)-space (semitransparent grey) together with information about the stability of

the EFMs. Panels (a2)–(d2) show corresponding projectionsof the EFM surface onto the(ωs, Ns)-

plane. Black dots indicate codimension-two Bogdanov-Takens bifurcation points; curves and regions

are coloured as in figure 3.1. In panels (a1)–(d1)dCp ∈ [−π, π], and the ranges ofNs andωs are as in

panels (a2)–(d2).


−0.011 0 0.011 −1

0

1

−0.025 0 0.025 −1

0

1

−0.05 0 0.05 −1

0

1

−0.11 0 0.11 −1

0

1

(a) (b)

(c) (d)

ωs ωs

ωs ωs

dCp

πdCp

π

dCp

πdCp

π

.

.

Figure 3.4. Projections of the EFM surfaces presented in figure 3.3 (a1)–(d1) onto the(ωs, dCp)-

plane. Black dots indicate codimension-two Bogdanov-Takens bifurcation points; curves and regions

are coloured as in figure 3.1.

By increasingΛ we move between two limiting cases. ForΛ = 0.001 our findings are in

qualitative agreement with results for thlocus semiconductor laser subject tooptical injection

[72], whose locking region is bounded by the saddle-node curve and two Hopf curves that

originate from the BT points. On the other hand, forΛ = 0.5 we are in agreement with the

findings for the COF laser [57, 30, 43, 47, 38]. The region of stable EFMs is shifted towards the

lower values of the population inversionNs which correspond to a stable maximum gain modes

in the COF laser [47]. Moreover, similarly to the COF laser case, stable EFMsin figure 3.3 (d)

become unstable in Hopf bifurcation that give rise to relaxation oscillations [43, 47, 62]; see

section 3.5.

Figures 3.3 and 3.4 showed that the number of the Hopf bifurcation curvesthat form the

boundary of the EFMs stability regions depends strongly onΛ. However, the details of these

changes are not shown by these figures. To explain the observed transitions in more detail, we

now consider changes to the number of Hopf bifurcation curves bounding the EFMs stability

regions, as well as to the number and type of the codimension-two bifurcations points involved


in the boundary.

Figure 3.5 shows how the boundary of the region of stable EFMs changeswith an increase

of the filter width fromΛ = 0.001 to Λ = 0.025. Our findings are presented in the(ωs, Ns)-

projections of the EFM surface. Black dots indicate codimension-two BT bifurcation points;

curves and regions are coloured as in figure 3.1. For clarity, the different Hopf bifurcations

curves are labeledH1 to H5 in panels (a)–(b) andH1 to H7 in panels (e)–(j).

The number of the Hopf bifurcation curves changes due to transitions of the EFM surface

for fixed Λ through extrema and saddles of the Hopf surfaces in(C1p , dCp, Λ)-space, which

also manifest themselves in(ωs, Ns, Λ)-space — called also minimax transitions and saddle

transitions, respectively [17]. Moreover, changes of type and number of codimension-two

bifurcation points on the boundary of stable EFM region are associated withtransitions through

codimension-three points, as discussed below.

In figure 3.5 (a)–(d) we show how with increasingΛ the number of the Hopf bifurcations

curves that form the boundary of the EFMs stability region increases from three to five. Ad-

ditionally, we show that the change in the number of Hopf bifurcation curvesconstituting the

boundary of EFMs stability regions is followed by the disappearance of four BT bifurcation

points and the emergence of four HH bifurcation points and four SH bifurcation points. Note

that in the(ωs, Ns)-projections only half of the mentioned codimension-two points is visible;

compare with figure 3.4.

Figure 3.5 (a)–(b) are projections of the EFM surface onto the(ωs, Ns)-plane forΛ =

0.003 andΛ = 0.004. In comparison to figure 3.3 (a) we show two additional Hopf bifurcation

curves,H4 andH5, that become part of the boundary in figure 3.5 (c) forΛ = 0.005. Note that

H4 andH5 intersect with saddle-node curve at SH points.

In figure 3.5 (a) and (b) the left part of the boundary of the stable EFM region involves

a single Hopf curveH2 that ends at two BT points; only one of which is seen in the view in

figure 3.5. In contrary in panel (c) it involves two Hopf curvesH2 andH4 that intersect at HH

points. This change is due to two separate transitions: first, theH2 andH4 curves connect at a

degenerate HH point and then the BT point moves through a SH point; at the degenerate HH

point H2 andH4 curves are tangent to each other. Such codimension-three points, where the

SH and the BT points meet (also called Bogdanov-Takens-Hopf points), occur also for the FOF

laser, as is mentioned in [17].

Note that, after passing the SH point, the BT points disappear. This happensdue to the

following transition. AsΛ is increased the two BT points move further along the saddle-node

curve and meet at its end; compare with figure 3.4 (a) and (b). At this pointthe BT points at

the two ends of Hopf curveH2 merge, so thatH2 is tangent to the saddle-node curve. At such

degenerate BT point [17, 10] two BT points merge and the Hopf curveH2 detaches from the


−0.025 0 0.025 −0.005

0

0.005

−0.025 0 0.025 −0.005

0

0.005

−0.025 0 0.025 −0.005

0

0.005

−0.025 0 0.025 −0.005

0

0.005

(a)

Λ = 0.003

H1

H4

H2

H3

H5

(b)

Λ = 0.004

H1

H4

H2

H3

H5

(c)

Λ = 0.005

H1

H4

H2

H3

H3

H5

H5

(d)

Λ = 0.006

H1

H4

H2

H3

H5

ωs ωs

ωs ωs

Ns Ns

Ns Ns

.

.

Figure 3.5. Projections of the EFM surface onto the(ωs, Ns)-plane, for increasing filter widthΛ as

indicated in the panels; here∆1 = ∆2 = 0, κ = 0.01 anddτ = 0. Black dots indicate codimension-two

Bogdanov-Takens bifurcation points; curves and regions are coloured as in figure 3.1.

saddle-node curve; in the(ωs, Ns)-projection in figure 3.5 only one of the BT points moving

along the saddle-node curve is visible.

The transitions through the codimension-three points at theH2 andH4 curves and at the

H3 andH4 curves, do not occur concurrently. In the enlargement in figure 3.5 (c) we show that

there still exist BT points at the ends ofH3; compare with figure 3.4 (b). However, the Hopf

bifurcation curvesH3 andH5 passed through degenerate HH points, and four new HH points

appeared; only two of which are seen in the view in figure 3.5.

Figure 3.5 (d) shows the projections of the EFM surface onto the(ωs, Ns)-plane forΛ =

0.006. Note that the BT points at the ends of theH3 have merged and disapeared after passing

through Bogdanov-Takens-Hopf points and later through a degenerate BT point. In this way,

the left boundary (H2 andH4) and the right boundary (H3 andH5) of the stable EFM region

are again qualitatively the same.

In Figure 3.5 (e)–(f) we show that with a further increase ofΛ, toΛ = 0.015 the number of


−0.05 0 0.05−0.01

0

0.01

−0.05 0 0.05−0.01

0

0.01

−0.05 0 0.05−0.01

0

0.01

−0.05 0 0.05−0.01

0

0.01

−0.05 0 0.05−0.01

0

0.01

−0.05 0 0.05−0.01

0

0.01

(e)

Λ = 0.01

H1

H2

H3H4

H5

H6

H7

(f)

Λ = 0.014

H1

H2

H3

HHH4

H5

H6

H7

(g)

Λ = 0.017

H1

H2

H3

HHH4

H5

H6

H7

(h)

Λ = 0.02

H1

H2

H3

H5

H6

H7

(i)

Λ = 0.023

H1

H5

H6

H7

(j)

Λ = 0.025

H1

ωs ωs

ωs ωs

ωs ωs

Ns Ns

Ns Ns

.

.

Figure 3.5 (continued). Projections of the EFM surface onto the(ωs, Ns)-plane, for increasing filter

width Λ as indicated in the panels; here∆1 = ∆2 = 0, κ = 0.01 anddτ = 0. Black dots indicate

codimension-two Bogdanov-Takens bifurcation points; curves and regions are coloured as in figure 3.1.

Hopf bifurcation curves that form the boundary of the EFMs stability region increases to seven.

Moreover, the number of codimension-two bifurcation points grows to six. In figure 3.5 (e) for

Λ = 0.01, we show two additional Hopf bifurcation curves,H6 andH7, that become part of the

boundary of the stable EFMs region in panel (f) forΛ = 0.014; H6 andH7 become involved

3.3. Dependence of EFM stability on∆ 77

in the boundary after they pass through a degenerate HH point. However, while the filter width

Λ continues to increase, the number of Hopf bifurcation curves forming the boundary of EFMs

stability regions decreases and the codimension-two bifurcation points gradually disappear;

see figure 3.5 (g)–(i). In figure 3.5 (g)–(i) the boundary of the EFMs stability region simplifies

in a complicated series of transitions of Hopf bifurcation curves (2)–(7).First, we find sad-

dle and minimax transitions of the Hopf surfaces in(C1p , dCp, Λ)-space, which also manifest

themselves in(ωs, Ns, Λ)-space. Furthermore, there are transitions through degenerate HH

points at which two different Hopf curves disconnect, and transitions through degenerate SH

points, at which Hopf bifurcation curves disconnect from saddle-node bifurcation curves; at a

degenerate SH point the Hopf bifurcation curve is tangent to the saddle-node bifurcation curve

[17, 36]. All the mentioned transitions lead to the region of stable EFMs extending over the

wholeωs-range; it is now bounded at the top by the saddle-node curve and at thebottom by

Hopf bifurcation curve (1); see figure 3.5 (j) forΛ = 0.025. Moreover it does not include any

codimension-two bifurcation points.

In summary, the investigation of the dependence of the EFM surface and region of stable

EFMs on the common filter widthΛ is in qualitative agreement with findings for both limit

cases,Λ → 0 andΛ → ∞. We showed that with increasingΛ, the region of stable EFMs

undergoes many complicated transformations. For sufficiently highΛ it extends over the whole

ωs-range, and is bounded at the top by the saddle-node curve and at the bottom by Hopf curve.

We analysed here only the EFM surface of the simplest typeB for ∆1 = 0 and∆2 = 0.

To analyse the dependence of EFMs stability on filter detuning∆1 and∆2 in the next

section we fix the filter width at a representative value,Λ = 0.005. Namely, the boundary of

the EFM stability region in figures 3.3 (b) and 3.4 (b) involves only five Hopf curves, and it

includes all three kinds of codimension-two bifurcation points. Moreover,from sections 2.3.1

and 2.4 we conclude that for this value ofΛ it is possible to find many different types of the

EFM surface.

3.3 Dependence of EFM stability on∆

In the two previous sections we only considered the case that the EFM surface is of the sim-

plest typeB. More specifically we analysed only EFM stability regions on the EFM surface

corresponding to the point(∆1, ∆2) = (0, 0) in the EFM surface bifurcation diagram in the

(∆1, ∆2)-plane presented in sections 2.3.4–2.4. We showed that the regions of stable EFMs

change with increasingκ andΛ, even tough the topology of the EFM surface itself remains

unaffected. In particular, we observed a splitting and separation of the stable EFM region

with increasingκ. Moreover we showed that with increasingΛ the EFM surface expands and


boundary of the stable EFM region simplifies. These results are in agreement with our find-

ings in section 2.4, where we show that with increasingΛ the 2FOF laser approaches the COF

laser limit, which results in a diminishing influence of the filters detunings on the topology of

the EFM surface and, hence, on the stability of EFMs. Based on our findings, throughout this

section we fixκ = 0.01 andΛ = 0.005 — values that are representative for the 2FOF laser,

and for which the EFM stability region extends over the wholedCp-range; as before we keep

dτ = 0.

To analyse the dependence of the stable EFM regions on the filter detunings∆1 and∆2,

we now calculate the EFM surface with corresponding information on the EFMs stability for

several representative regions from the EFM surface bifurcation diagram in the(∆1, ∆2)-

plane. Our goal is to give an overview of changes of EFM stability regionsassociated with

the singularity transitions of the EFM surface in(ωs, dCp, Ns)-space, as were described in

sections 2.3.3 and 2.3.4.

The starting point of our considerations is the EFM surface in figure 3.3 (b), for κ =

0.01, Λ = 0.005, ∆1 = 0, ∆2 = 0 anddτ = 0; it has a single region of stable EFMs centred

around the solitary laser frequency, represented in here byωs = 0. We first fix ∆2 = 0

and increase∆1; in this way we move along the horizontal line∆2 = 0 in the EFM surface

bifurcation diagram in the(∆1, ∆2)-plane. With increasing∆1, we first observe that the kind

of connection between the 2π-periodic fundamental units of the EFM surface change. Next,

after passing the point where the curvesC andSC meet, a hole in the EFM surface appears and

its type changes fromB to Bh; a point of this kind can be found in the enlargement of the EFM

surface bifurcation diagram in the(∆1, ∆2)-plane in figure 2.20. We further increase∆1 to

a value at which, even thought the EFM surface remains unchanged of typeBh, the region of

stable EFMs splits into two stable EFM regions that extend over the whole rangeof dCp — we

refer to such a region as a ‘band of stable EFMs’. This transition is described in section 3.3.1.

Next we fix∆1 = 0.024 which is the maximum value of the first transition, and we move

vertically down in the EFM surface bifurcation diagram in the(∆1, ∆2)-plane by decreasing

∆2 to ∆2 = −0.037. Along this path we first pass through theSN -transition where a second

hole is created and the EFM surface changes to typehBh. We show that for typehBh around

each of the filter central frequencies a band of stable EFMs exists; see section 3.3.2. With a

further decrease of∆2 we pass through theSω-transition where the EFM surface splits into

two components and changes to typeBB. Interestingly, we find that theSω-transition does not

affect the qualitative structure of stable EFMs regions.

To find a new band of stable EFMs we fix∆2 = −0.037 and increase∆1 again. The third

stable EFM band around the central laser frequency is created just before we pass through the

secondSω-transition, after which the EFM surface consists of three components andits type

changes toBBB. Changes of the stable EFM regions associated with bothSω-transitions are

described in section 3.3.3.


Finally we choose values of∆1 and∆2 at which the side components of the EFM surface

split into ‘islands’ —dCp periodic disjoint compact objects. In other words, the type of the

EFM surface changes fromBBB first into IBB and then intoIBI. The splitting of the side

components of the EFM surface results in a division of the bands of stable EFMs into ‘islands of

stable EFMs’. What is more, the stable EFM regions extend over the whole range ofdCp over

which the side island of the EFM surface exist. On the other hand, the central band of stable

EFMs around the free laser frequency expands to fill a larger area ofthe central component of

the EFM surface; see section 3.3.4.

Overall, we show that detuning the filters from the solitary laser frequencyresults in

changes of the EFM surface type, which in turn is coupled to changes of the stability struc-

ture of EFMs. From a bifurcation theory point of view, this means that the stability boundaries

of the stable EFM regions which consist of saddle-node and Hopf bifurcations curves, must

change. We show that such changes are due to minimax and saddle transitions of the saddle-

node and Hopf bifurcation surfaces in(C1p , dCp, ∆1)-space and(C1

p , dCp, ∆2)-space. In

such transition boundary curves of the EFM stability regions emerge or disappear and con-

nect differently, and this results in transformations of the stable EFM regions. Associated

codimension-two and codimension-three bifurcations points (where the boundary curves also

interact with each other) are mentioned as necessary, but their detailed discussion is beyond the

scope of the analysis presented in this section.

3.3.1 Influence of hole creation on EFM stability

For∆1 = 0 and∆2 = 0 the EFM surface consist of infinitely many compact objects connected

at the points(ωs, Ns, dCp) = (0, 0, (2n + 1)π), wheren ∈ Z. In this section we increase∆1

for fixed ∆2 = 0. The first effect of increasing∆1 from ∆1 = 0 is a change of the nature of

the connection between thedCp-periodic units of the EFM surface. A further increase of∆1

result in the transition of the EFM surface from typeB to typeBh.

Figure 3.6 (a)–(c) illustrates how a hole is created in the EFM surface due tothe saddle tran-

sitionSC ; compare figure 2.22. Figure 3.6 (a1)–(c1), shows the EFM surface in(ωs, dCp, Ns)-

space over a4π-interval ofdCp. Figure 3.6 (a2)–(c2) present enlargements of the narrow con-

nection between the fundamental units of the EFM surface. All curves andregions in figure 3.6

(a2)–(c2) are coloured as in figure 3.1. Note that for clarity panels (a1)–(c1) are shown without

the information about the stability of the EFMs. The axes ranges for all the panels can be found

in Table: 3.1.

In figure 3.6 (a), for∆1 = 0.0005, we show that the fundamental units of the EFM surface

are no longer connected at single points; instead they are connected along closed loops. The

small white curve in the middle of panel (a1) illustrates that connection. In fact, it is a small


(a1)

∆1 = 0.0005

(a2)

(b1)

∆1 = 0.0007 (b2)

(c1)

∆1 = 0.005

(c2)

ωs

ωs

ωs

dCp/π

dCp/π

dCp/π

Ns

Ns

Ns

0

0 0

0

0

0

0 0

0

0

0

0 0

0

Ns

Ns

Ns

dCp/π

dCp/π

dCp/π

ωs

ωs

ωs

.

.

Figure 3.6. Influence of local saddle transitionSC , where two bulges connect to form a hole, on the

stability of the EFMs; here∆2 = 0, κ = 0.01, Λ = 0.005 anddτ = 0. Panels (a1)–(c1) shows

two copies of the fundamental2π-interval of the EFM surface for different values of a detuning∆1, as

indicated in the panels. Panels (a2)–(c2) show enlargements of the region where the hole is formed. In

panels (a1)–(c1) the limit of thedCp-axis corresponds to a planar section that goes through middle of

the hole in panels (a2)–(c2). For the specific axes ranges seeTable 3.1; curves and regions are coloured

as in figure 3.1.


panel min(ωs) max(ωs) min(dCp) max(dCp) min(Ns) max(Ns)

figure 3.6 (a) -0.0250 0.0250 -3.0000 1.0000 -0.0050 0.0050

figure 3.6 (a1) -0.0050 0.0050 -1.0520 -0.9975 -0.0010 0.0010

figure 3.6 (b) -0.0250 0.0250 -3.0500 0.9800 -0.0050 0.0050

figure 3.6 (b1) -0.0048 0.0048 -1.0575 -1.0386 -0.0010 0.0010

figure 3.6 (c) -0.0230 0.0250 -3.3300 0.7100 -0.0045 0.0050

figure 3.6 (c1) 0 0.0050 0.6650 0.7460 -0.0008 0.0018

Table 3.1.Axes ranges for all the panels in figure 3.6

closed elliptical curve, which correspond to the narrow neck in the enlargement in figure 3.6

(a2) of the EFM surface in the vicinity of that curve.

Figure 3.6 (b), for∆1 = 0.0007, shows that for only a slightly higher value of∆1 a hole

in the EFM surface is created. Therefore, for some range ofdCp the EFM surface is connected

along two disjoint closed loops. This is illustrated by the two white ellipses in the middleof

panel (b1). In other words we take a constantdCp planar section through the middle of the hole

shown in panel (b2).

Figure 3.6 (c), for∆1 = 0.005, shows that increasing∆1 causes an expansion of the EFM

surface. The expansion is most substantial at the range ofdCp at which the constantdCp planar

section through the EFM surface consist of two disjoint closed loops. Notethe considerable

growth of the two closed intersection curves in comparison with panel (b1).The enlargement

in figure 3.6 (c2) shows the hole, through which the constantdCp section in panel (c1) is taken.

We again consider only a small range ofωs near the hole in the EFM surface. Note that, due to

the growth of the EFM surface and in contrast to figure 3.6 (b2), the EFM surface in panel (c2)

is not shown over its entireωs-range.

The stability information shown in figure 3.6 (a2)–(c2), clearly demonstratesthat the ap-

pearance of the hole strongly affects the stability of the EFMs. For example,a new Hopf

bifurcation curve appears in panel (a2); in panel (b2) a region of unstable EFMs emerges at the

edge of the hole; and in panel (c2) the whole structure of stable EFMs becomes more compli-

cated; compare with figure 3.3 (b) and figure 3.4 (b) which show the EFM surface with stability

information before detuning the filter,∆1 = 0.

We now explain the nature of those stability changes in more detail. Figure 3.7 (a)–(c) show

projections of the EFM surface corresponding to the EFM surfaces presented in the figure 3.6

(a)–(c). Figure 3.7 (a1)–(c1) show the projection of the EFM surfaceover a [0, 2π] range

of dCp; in this way the connection between the fundamental units of the EFM surfaceand

the hole are presented in the centre of the figure. Panels (a2)–(b2) show enlargements, near


−0.025 0 0.025 0

1

2

−0.003 0 0.003 0.94

1.03

−0.025 0 0.025 0

1

2

−0.004 0 0.004 0.92

0.98

−0.024 0 0.026 0

1

2

0 0.005 0.45

0.85

(a1)(a2)

(b1) (b2)

(c1) (c2)

ωs ωs

ωs ωs

ωs ωs

dCp

πdCp

π

dCp

πdCp

π

dCp

πdCp

π

.

.

Figure 3.7. Projections with stability information of the EFM surface in figure 3.6 onto the(ωs, dCp)-

plane, shown for increasing filter detuning∆1 = 0.0005 (a), ∆1 = 0.0007 (b) and∆1 = 0.005 (c);

here∆2 = 0, κ = 0.01, Λ = 0.005 anddτ = 0. To illustrate the changes in the EFM surface, panels

(a1)–(c1) show the2π interval of the EFM surface that is shifted byπ with respect to the fundamental

2π interval of the EFM-surface. Panels (a2)–(c2) show enlargements of the central part of panels (a1)–

(c1). Curves and regions are coloured as in figure 3.1; dark green colour indicates that there are two

stable regions on the EFM surface that lie above one another in theNs direction.


the connection between the two units of the EFM surface, of the region of the(ωs, dCp)-

projection of the EFM surface from Figure 3.7 (a1)–(c1). Recall that, we colour regions in

dark green where stable EFMs exist on the bottom and on the top sheet of the EFM surface. In

other words, dark green indicates areas, where for a set values ofωs anddCp, two stable EFMs

with different values of population inversionNs exist. The dark green areas can be found in the

following regions of the EFM surface: at the edge of the hole, at the outeredges of the EFM

surface, and at the narrow necks of the central component of the EFMsurface; in the vicinity of

connections between the fundamental units of the EFM surface. The othercurves and regions

in figure 3.7 are coloured as in figure 3.1.

Figure 3.7 (a), for∆1 = 0.0005, shows that with opening the connection between the

fundamental units of the EFM surface, the saddle-node curve underwent a saddle connection

and changed from a closed loop periodically translated along thedCp axis to two infinitely

long 2π-periodic curves that extend along the outer edges of the(ωs, dCp)-projection of the

EFM surface. Note that inside a substantial ‘bulge’ of the right saddle-node curve in figure 3.7

(a2), as well as along the edge of the projection of the EFMs surface, theEFM stability region

exists on both the bottom and the top sheet of the EFM surface. Additionally, inside the bulge

a new Hopf curve ends at two BT points that emerged due to an increase ofthe filter detuning

∆1.

Figure 3.7 (b), for∆1 = 0.0007, shows how the stability of the EFMs changes after the

creation of the hole in the EFM surface. Note that the hole indicates a region inthe(ωs, dCp)-

plane where EFMs do not exist and, hence, also no stable EFMs can be found. Panel (b2) shows

that, as the result of passing through a saddle point of the saddle-node bifurcation surface in

(ωs, dCp, ∆1)-space, the bulge from panel (a2) detaches and forms a new closed saddle-node

curve that surrounds the hole. As a result at the edge of the hole, a region of unstable EFMs

appears.

In figure 3.7 (c), for∆1 = 0.005, we show that increasing∆1 is associated with several

trends in changes of the EFM stability region. First, the connection betweendCp-periodic

units of the EFM surface expands in theωs direction; this results in an expansion of the EFM

stability region in theωs direction. Second, the Hopf curve around the point(ωs, dCp) = (0, 0)

in panels (a1) and (b1) shrinks and shifts away from the point(ωs, dCp) = (0, 0) in panel (c1);

this is the curve that formed the lower boundary of the regions of stable EFMs in section 3.1

and 3.2. Together with that Hopf curve the region of unstable EFMs bounded by it shrinks as

well. Finally, the hole and the region of unstable EFMs around it grows inωs as well asdCp

directions.

Note that, the growth of the hole with increasing∆1 is much slower than the expansion of

the region of unstable EFMs around it. In figure 3.7 (b) the hole is the most important change

to the stable EFM region. The situation changes in figure 3.7 (c): here the stable EFM region


−0.02

0

0.033

−2.38−2

−1

0

11.63

−0.004

0

0.007

ωs

dCp/π

Ns

.

.

Figure 3.8. The EFM surface with stability information for filters detunings∆1 = 0.024 and∆2 = 0.

Hereκ = 0.01, Λ = 0.005 anddτ = 0; curves and regions are coloured as in figure 3.1.

is affected by the expanding region of unstable EFMs; its area is severaltimes larger than the

area of the hole. The expansion of the region of unstable EFMs is associated with a change

of its boundary. In panel (b) it is bounded by a single saddle-node curve. In figure 3.7 (c) the

boundary of the region of unstable EFMs around the hole involves the samesingle saddle-node

bifurcation curve and additionally two Hopf bifurcation curves. One of theHopf curves is the

Hopf curve from panel (b) and the second one appeared in a minimax transition of the Hopf

bifurcation surface.

We now show how, with a further increase of∆1, those trends lead to a division of the

single stable EFM region into two bands of stable EFMs. Figure 3.8, presentsEFM surfaces

in (ωs, dCp, Ns)-space over the4π-interval of dCp. All regions and curves are coloured as

in figure 3.1. (Note that thedCp-range is slightly larger then4π due to surface rendering

requirements.) In figure 3.8 we show that for∆1 = 0.024 (and∆2 = 0) the EFM surface of

type Bh has two bands of stable EFMs; one of the EFM stability bands is centred around the

solitary laser frequency and the other one around the central filter frequency (aroundωs = ∆1).

The two bands of stable EFMs are separated by the hole. There are also very small additional

regions of stable EFMs: one near the left end of the EFM surface (for negativeωs) and one at

the edge of the hole.

The splitting of the single stable EFM region into two regions of stable EFMs that form

bands expanding over the whole range ofdCp is explained in more details in figure 3.9. This


transitions does not involve a topological change of the EFM surface itself, but of the saddle-

node and Hopf curves that bound the EFM stability regions. At such codimension-two points

the Hopf curves bounding EFM stability regions can appear, disappear or can change the way

they are connected.

More specifically, figure 3.9 (a)–(f) shows how stability regions of EFMschange with an

increase of∆1 from ∆1 = 0.065 to ∆1 = 0.024, while the second filter is fixed at∆2 = 0.

Panels (a)–(f) show projections of the EFM surface onto the(ωs, dCp)-plane over[0, 2π] dCp-

range. Changes of the structure of the EFMs stability regions are results of minimax and saddle

transitions of the Hopf curves. In effects the single region of stable EFMsin panel (a) divides

into four regions of stable EFMs in panel (f). Two of them extend over thewhole range ofdCp,

and two exist for very limited ranges ofdCp. Note that for all projections in figure 3.9 (a)–(f)

the EFM surface remains of the typeBh.

In figure 3.9 (a), we show that for∆1 = 0.0065 the closed Hopf curve, visible in the

top half of the panel shrinks significantly in comparison to the same curve for∆1 = 0.005

in figure 3.7 (c1). This closed Hopf curve disappears in a minimax transition of the Hopf

bifurcation surface, for slightly higher value of∆1; see panel (b) for∆1 = 0.0085. Note

also, that figure 3.9 (a) and (b) shows that the region of unstable EFMs surrounding the hole

continues to grow with increasing∆1.

Figure 3.9 (c) to (f) illustrate a division of the single EFM stability region into four regions

of stable EFMs. Each of the partition of the stable EFM region occurring in between panels (c)

to (f) is associated with a saddle transitions of Hopf bifurcation curves. The saddle transition

of the Hopf curves occurring in between panels (c) and (d) results in a separation of the EFM

stability region into two bands of stable EFMs, one around the solitary laser frequencyωs = 0

and another associated with the central filter frequency. Both those regions extend over the

whole range ofdCp. Next in panel (e) there is new small region of stable EFMs that lies on the

edge of the hole; this region detached from the right band of stable EFMs.Although this region

is too small to be detected in experiments, we emphasise its existence because it expands into

a stable EFM band for other values of∆1 and∆2; see section 3.3.3. Finally for∆1 = 0.024

in panel (f) a small region of stable EFMs detaches from the left stable EFMband, which

increases the total number of stable EFM regions to four. Note that, the(ωs, dCp)-projection

in figure 3.9 (f) corresponds to the EFM surface in figure 3.8.


−0.024 0 0.026 0

1

2

−0.024 0 0.026 0

1

2

−0.023 0 0.031 0

1

2

−0.023 0 0.031 0

1

2

−0.022 0 0.032 0

1

2

−0.022 0 0.034 0

1

2

(a) (b)

(c) (d)

(e) (f)

ωs ωs

ωs ωs

ωs ωs

dCp

πdCp

π

dCp

πdCp

π

dCp

πdCp

π

.

.

Figure 3.9. Projections of the EFM surface with stability information onto the(ωs, dCp)-plane for

increasing filter detuning∆1 = 0.0065 (a), ∆1 = 0.0085 (b), ∆1 = 0.0165 (c), ∆1 = 0.0175 (d),

∆1 = 0.0215 (e) and∆1 = 0.024 (f). Here∆2 = 0, κ = 0.01, Λ = 0.005 anddτ = 0; curves and

regions are coloured as in figure 3.7.

3.3.2 Influence ofSN -transition on EFM stability

We now explore how the structure of the stable EFM regions is affected by the change of the

EFM surface from typeBh to type hBh. To reach the parameter range for which the EFM


−0.033

0

0.033

−2.65−2

−1

0

11.41

−0.007

0

0.007

ωs

dCp/π

Ns

.

.

Figure 3.10. The EFM surface with stability information for filters detunings∆1 = 0.024 and∆2 =

−0.025. Hereκ = 0.01, Λ = 0.005 anddτ = 0; curves and regions are coloured as in figure 3.1.

surface is of typehBh we fix ∆1 = 0.024 and decrease∆2 until the second hole in the EFM

surface appears; the second hole is the result of aSN saddle transition; see figure 2.15 in sec-

tion 2.3.3. We find that for the EFM surface of typehBh two bands of stable EFMs exists. The

bands are centred around the central filter frequencies associated with∆1 and∆2. Addition-

ally, we show that at the edges of the holes, a small stable EFM region can befound. This

region is centred around the solitary laser frequency, represented here byωs = 0.

Figure 3.10, for∆1 = 0.024 and∆2 = −0.025, shows the EFM surface in(ωs, dCp, Ns)-

space over the4π-interval ofdCp. (In fact, thedCp axis range is slightly larger then4π due to

surface rendering requirements.) All regions and curves are coloured as in figure 3.1.

Figure 3.10 shows the EFM surface ofhBh type on which two ‘bands of stable EFMs’

exist; both these bands are bounded by Hopf curves. Each band is centred around the central

frequency of one of the filters. Additionally, a stable EFM region exists in thecentre of the

EFM surface at the edges of the holes. This region is bounded by Hopf and saddle-node

bifurcation curves and it is centred around the central laser frequency. We now show that

qualitatively the same structure of the EFM stability regions exists before the second hole

appears. Moreover we present the structure of the EFM stability regionsfor the transition

through the anti-diagonal∆1 = −∆2 in the EFM surface bifurcation diagram in the(∆1, ∆2)-

plane presented in figure 2.17.

Figure 3.11 (a)–(d) presents how the stability structure of the EFMs change for fixed∆1 =


−0.026 0 0.033 1

2

3

−0.034 0 0.033 1

2

3

−0.034 0 0.033 1

2

3

−0.034 0 0.033 1

2

3

(a) (b)

(c) (d)

ωs ωs

ωs ωs

dCp

πdCp

π

dCp

πdCp

π

.

.


increasing filter detunings∆2 = −0.012 (a),∆2 = −0.023 (b), ∆2 = −0.024 (c), ∆2 = −0.025 (d).

Here∆1 = 0.024, κ = 0.01, Λ = 0.005 anddτ = 0; curves and regions are coloured as in figure 3.7.

0.024 and for values of∆2 in the range∆2 ∈ [−0.012, −0.037]. In figure 3.11 (a)–(d) we show

the projection of the EFM surface onto the(ωs, dCp)-plane over the range ofdCp ∈ [π, 3π].

In figure 3.11 (a), for∆2 = −0.012, the EFM surface is still of the typeBh, the same

type as in figure 3.9 (a)–(f). There are four regions of stable EFMs in panel (a). Two bands of

stable EFMs centred at the central filter frequencies and two small EFM stability regions at the

edge of the hole relatively close to the solitary laser frequency. Each of stable EFM bands is

bounded by a pair of Hopf bifurcation curves. The boundary of the stable EFM regions at the

edge of the hole involves single Hopf curve and a single saddle-node curve.

In figure 3.11 (b), for∆2 = −0.023, a second hole appeared in the EFM surface due

to theSN -transition; the EFM surface is now of typehBh. Concurrently, a new saddle-node

bifurcation curve emerges in the minimax transition; see the small blue loop in the half-plane

whereωs < 0, in figure 3.11 (b). Note that, while the structure of the bands of stable EFMs

associated with the filters remains qualitatively unchanged, the two stable EFM regions at the

edge of the hole merge and expand; compare with figure 3.11 (a).


Figure 3.11 (c) shows the(ωs, Ns)-projection of the EFM surface for∆1 = 0.024 and

∆2 = −0.024; that is, on the anti-diagonal∆1 = −∆2 in the EFM surface bifurcation diagram

in the (∆1, ∆2)-plane. The mirror symmetry of the figure 3.11 (c) with respect toωs = 0

is due to∆1 = −∆2; note that the two holes are connected in single points atωs = 0.

Additionally the two saddle-node curves from panel (b) merged into a singlecurve in the saddle

transition; the new saddle-node curve surrounds both holes. The central region of stable EFMs

is spread symmetrically along the edges of the holes. The symmetry of the EFM surface for

∆1 = −∆2 result in the emergence of a highly degenerated multiple Hopf bifurcation point at

the connection of the two holes; at this point the central Hopf curve intersects itself multiple

times. Decreasing∆2 to ∆2 = −0.025 results in the unfolding of the multi-Hopf point into

several Hopf-Hopf points. Moreover, together with the separation of the holes the saddle-node

bifurcation curve divides into two curves in another transition through the saddle in the saddle-

node surface in(C1p , dCp, ∆2)-space; see panel (d).

In figure 3.11 (a)–(d) we investigated how the stability structure of EFMs changes while

the EFM surface transforms from typeBh to typehBh. We showed that although the topology

of the EFM surface changes significantly the structure of the regions of stable EFMs remains

qualitatively unchanged. In figure 3.11 (a)–(d) we uncovered that there exist three main regions

of stable EFMs. Two associated with the central filter frequencies forming bands that extend

over whole range ofdCp, and one associated with the solitary laser frequency that expands

with an increasing detuning of the filters.

The structure of the EFM stability regions that emerged in figure 3.11 (a)–(d) suggests that

the EFM stability region centred around theωs = 0 in figure 3.9 (f) can be considered as the

region of stable EFMs associated with the second filter (hereωs = 0 corresponds to∆2 = 0).

Moreover, in a similar way the single region of stable EFMs in figure 3.7 can beconsidered as

the overlapping bands of stable EFMs associated with both filters.

3.3.3 Influence ofSω-transition on EFM stability

We now explore other regions of the EFM surface bifurcation diagram in the (∆1, ∆2)-plane

and investigate how the basic structure of the EFM stability regions uncovered in section 3.3.2

changes as the EFM surface undergoes twoSω-transitions. In particular we observe that, as the

EFM surface transforms from typehBh through typeBBh to typeBBB, the bands of stable

EFMs associated with the central filters frequencies remain qualitatively unchanged. What is

more, we show that, as we increase detuning of the filters,∆1 in the positive direction and∆2

in the negative direction, the expansion of the stable EFM region centred around the solitary

laser frequency progresses, and a third band of stable EFMs appears.

In figure 3.12 we present the EFM surfaces of typeshB (a), BB (b), BBh (c) andBBB

(d), together with stability information. We show that, despite the splitting of the EFMsurface


−0.042

0

0.033

−2.9−2

−1

0

11.15−0.009

0

0.007

−0.042

0

0.033

−3.4−3

−2

−1

00.75

−0.009

0

0.007 (b)

(a)

ωs

ωs

dCp/π

dCp/π

Ns

Ns

.

.

Figure 3.12. The EFM surface with stability information for∆2 = −0.035 (a), ∆2 = −0.037 (b).

Here∆1 = 0.024, κ = 0.01, Λ = 0.005 anddτ = 0; curves and regions are coloured as in figure 3.1.

into up to three components, on each of those components at least one bandof stable EFMs

exist. Figure 3.12 (a)–(d) shows EFM surfaces in(ωs, dCp, Ns)-space over the4π-interval of

dCp. (Note that, due to surface rendering requirements, thedCp axes range in figures 3.12 (a),

(b) and (d) is slightly larger then4π.) All regions and curves are coloured as in figure 3.1.

To observe the firstSω-transition, we decrease∆2 while keeping∆1 = 0.024 fixed; this is


−0.043

0

0.043

−2.89−2

−1

0

11.11−0.009

0

0.009

−0.043

0

0.043

−3.93−3

−2

−1

00.09−0.009

0

0.009 (d)

(c)

ωs

ωs

dCp/π

dCp/π

Ns

Ns

.

.

Figure 3.12 (continued). The EFM surface with stability information for∆1 = 0.035 (c),∆1 = 0.036

(d). Here∆2 = −0.037, κ = 0.01, Λ = 0.005 anddτ = 0; curves and regions are coloured as in

figure 3.1.

as in section 3.3.2. In figure 3.12 (a) and (b), we illustrate the effect of theSω-transition, where

the EFM surface changes from typehB to typeBB. Note that, although the topology of EFM

surface changes, the structure of the stable EFM regions remains qualitatively the same.

Panel (a), for∆2 = −0.035, presents the EFM surface of typehB with two bands of


stable EFMs centred at the central filter frequencies and a stable EFM region at the edge of

the hole. The structure of the EFM stability region is virtually the same as in figure3.10. In

figure 3.12 (b), for∆2 = −0.037, we show that theSω-transition does not change the stable

EFM regions. Note, however that the region of stable EFMs around the central laser frequency

keeps expanding with decreasing of∆2.

We now fix∆2 = −0.037 and increase∆1; in this way, we find parameter values at which

the EFM surface of typeBBh changes to typeBBB. In figure 3.12 (c) and (d), for∆1 = 0.035

and for∆1 = 0.036, we show that with increasing∆1 the region of stable EFMs around the

central laser frequency expands further. In fact, in both panels (c)and (d), three bands of stable

EFMs exist. A further increase of∆1 to the value at which the EFM surface undergoes another

Sω transition and separates into three components does not qualitatively affect the structure of

stable EFM bands.

The expansion of the stable EFM region around the central laser frequency can be explained

by the observation that, as the detuning of the filters increases, the frequencies close to the cen-

tral laser frequency are subject to weaker feedback from the flanksof the filters transmittance

profiles. Recall that in section 3.1 we showed that the region of stable EFMsis larger for lower

values the of feedback rateκ. This observations is also in agreement with the findings in [17].

To sum up, in figure 3.12 (a)–(d) we investigated how the stability structure of EFMs

changes while the EFM surface transforms fromBh throughhBh to BB. In particular, we

showed that, although theSω-transition strongly affects the topology of the EFM surface, it

does not change the qualitative structure of the stable EFM regions. Moreover in figure 3.12

(c) and (d) we showed that detuning both filters further away from central laser frequency result

in the creation of a third band of stable EFMs.

Figure 3.13 shows details of how the region of stable EFMs around the central laser fre-

quency transforms into a band of stable EFMs. Panels (a)–(f) show that,while the EFM surface

changes fromhB intoBBB, the only significant change of the stability structure of the EFMs is

the transformation of the central region of stable EFMs into a band that extends over the whole

range ofdCp. We first further decrease∆2 in panels (a) and (b). Then we fix∆2 = −0.037

at the value from panel (b), and we increase∆1 up to∆1 = 0.036,in panels (c)–(f). Panels

(a)–(f) show the(ωs, dCp)-projection of the EFM surface over the range ofdCp ∈ [π, 3π]; all

regions and curves are coloured as in figure 3.1.

Figure 3.13 (a) shows that, with decreasing∆2, one hole from figure 3.11 (d) disappears

together with the saddle-node curve surrounding it; simultaneously the otherhole expands

substantially. Concurrently with the hole, the stable EFM region laying at its edge expands as

well. As the EFM surface transforms from typehB to typeBB, the periodically shifted loops

of the closed saddle-node curve connect in the saddle transition and transform to two infinitely

long2π periodic curves. Note that, in panel (b), with decreasing∆2 the expansion of the stable


−0.043 0 0.033 1

2

3

−0.043 0 0.033 1

2

3

−0.043 0 0.035 1

2

3

−0.043 0 0.037 1

2

3

−0.043 0 0.041 1

2

3

−0.043 0 0.043 1

2

3

(a) (b)

(c) (d)

(e) (f)

ωs ωs

ωs ωs

ωs ωs

dCp

πdCp

π

dCp

πdCp

π

dCp

πdCp

π

.

.


∆1 = 0.024, ∆2 = −0.036 (a), ∆1 = 0.024, ∆2 = −0.037 (b), ∆1 = 0.026, ∆2 = −0.037 (c),

∆1 = 0.029, ∆2 = −0.037 (d), ∆1 = 0.035, ∆2 = −0.037 (e) and∆1 = 0.036, ∆2 = −0.037 (f).

Hereκ = 0.01, Λ = 0.005 anddτ = 0; curves and regions are coloured as in figure 3.7.

EFM region aroundωs = 0 progresses. Figure 3.13 (a) and (b) show the(ωs, dCp)-projection

of the transition depicted in figure 3.12 (a) and (b).

We now fix ∆2 and start increasing∆1. Figure 3.13 (c), for∆1 = 0.026, shows the


(ωs, dCp)-projection of the EFM surface of typeBBh. TheSN -transition is again associated

with the minimax transition of a saddle-node curve and, as before, it does not affect the qualita-

tive structure of the stable EFM regions. With a further increase of∆1 the hole and saddle-node

curve around it expands. We also observe a further growth of the stable EFM region centred at

the central laser frequency; see panel (d).

In figure 3.13 (e) we show that, after several transitions through degenerate Hopf-Hopf

bifurcation points and several transitions through saddle points of the Hopf bifurcation surface

in (ωs, dCp, ∆1)-space, infinitely many copies of the region of stable EFMs around the laser

frequency connect with themselves along thedCp-axis to form a band of stable EFMs that

extends over the whole range ofdCp. As a result, in figure 3.13 (e) three bands of stable

EFMs are present: one around the central laser frequency and one around each of the central

filter frequencies. Note that the closed saddle-node curve surrounding the hole connects, in the

saddle transition, with the saddle-node curve along the edge the EFM surface.

Finally, in figure 3.13 (f) after a furtherSω-transition, the EFM surface consist of three

disjoint components; and it is of typeBBB. Similarly to theSω-transition between figure 3.13

(a) and (b), the qualitative structure of the stable EFM regions appears tobe unaffected by this

transition. Note that in the saddle transition that accompanied theSω-transition the saddle-

node curve passing throughωs = 0 transformed into loops surrounding points(ωs, dCp) =

(0, (2n+1)π with n ∈ Z, and infinitely long2π-periodic curve along the left edge of the right-

most component of the EFM surface. Moreover, the closed saddle-node bifurcation curve in

figure 3.13 (f) is shifted byπ with respect to figure 3.1.

In this section, we showed that although theSω-transitions change the topology of the

EFM surface significantly, they do not affect the qualitative structure ofthe stable EFM re-

gions. Moreover, we showed that with increasing detuning of the filters from the solitary laser

frequency the region of stable EFMs aroundωs = 0 expands and transform into an additional

band of stable EFMs. Therefore, on each disjoint component of the EFMsurface of typeBBB

there exists one band of stable EFMs.

3.3.4 Influence ofSC-transition on EFM stability

We now show how the structure of stable EFM regions from section 3.3.3 changes with detun-

ing the filters further away from the central laser frequency. In other words, we investigate what

happens to the three bands of stable EFMs as the EFM surface undergoes theSC-transition

where its side components split intodCp-periodic islands; the EFM surface type changes from

BBB to IBI in the process.

Figure 3.14 shows the EFM surfaces in(ωs, dCp, Ns)-space together with the stability

information over the4π-interval ofdCp. In panel (a), for∆1 = 0.044 and∆2 = −0.049, the


−0.055

0

0.055

−3.65−3

−2

−1

00.35−0.011

0

0.01

−0.055

0

0.055

−3.38−3

−2

−1

00.64

−0.011

0

0.01 (b)

(a)

ωs

ωs

dCp/π

dCp/π

Ns

Ns

.

.

Figure 3.14. The EFM surface with stability information for∆1 = 0.044 (a), ∆1 = 0.050 (b). Here

∆2 = −0.049, κ = 0.01, Λ = 0.005 anddτ = 0; curves and regions are coloured as in figure 3.1.

EFM surface is of typeIBB and in panel (b), for∆1 = 0.050 and∆2 = −0.049, it is of

typeIBI. (Note that thedCp axes range in panel (b) is slightly larger then4π due to surface

rendering requirements.) All regions and curves are coloured as in figure 3.1.

In figure 3.14 we show that, after the outer component of the EFM surfacedivides into

disjoint islands, stable EFM region still extends over the whole bottom sheet of the islands.


−0.05 0 0.046 0

1

2

−0.051 0 0.046 0

1

2

−0.053 0 0.048 0

1

2

−0.054 0 0.054 0

1

2

(a) (b)

(c) (d)

ωs ωs

dCp

πdCp

π

ωs ωs

dCp

πdCp

π

.

.


∆1 = 0.039, ∆2 = −0.041 (a), ∆1 = 0.039, ∆2 = −0.045 (b), ∆1 = 0.044, ∆2 = −0.049 (c)

and∆1 = 0.050, ∆2 = −0.049 (d). Hereκ = 0.01, Λ = 0.005 anddτ = 0; curves and regions are

coloured as in figure 3.7.

Moreover, at the narrow necks of the central component of the EFM surface the stable EFM

region extends over the bottom as well as the top sheet of the EFM surface. Similarly to the

first, the secondSC-transition has this type of local influence on the regions of stable EFMs. In

panel (b) both outer components of the EFM surface have formed islandswith the stable EFM

region extending over their whole bottom sheet. Note that the islands at negative ωs-values are

virtually unaffected by theSC-transition of the right-most component of the EFM surface.

Figure 3.15 (a)–(d) show how the EFM stability regions change with a further increase of

filters detuning. We present the EFM surface as a projection onto the(ωs, dCp)-plane over the

range ofdCp ∈ [0, 2π]; all regions and curves are coloured as in figure 3.7.

Panels (a), for∆1 = 0.039 and ∆2 = −0.041, and (b), for∆1 = 0.039 and ∆2 =

−0.045, show that, with increasing detuning of the filters, the expansion of the region of stable

EFMs continues. The Hopf bifurcation curves that bound regions of unstable EFMs in the

central component of the EFM surface shrink and disappear in a seriesof saddle and minimax

3.4. Dependence of EFM stability ondτ 97

transitions; as a consequence also the regions of unstable EFMs on the bottom sheet of the

EFM surface shrink and disappear. In figure 3.15 (c), for∆1 = 0.044 and∆2 = −0.049, the

left component of the EFM surface underwent theSC-transitions and divided into islands. In

other words, the EFMs centred around the left filter frequency exist only for a limited range of

dCp values. Nevertheless, the region of stable EFMs extends over the whole range ofdCp for

which the EFM island exist. Note that theSC-transition is accompanied by a saddle transition

of saddle-node bifurcation curves. After this transitions the boundary ofthe EFM stability

region on the island involves the closed saddle-node bifurcation curve and two Hopf bifurcation

curves. Finally, the same happens to the right component of the EFM surface in panel (d).

Note that the narrow neck of the EFM surface’s central component in figure 3.15 (d) is

shifted almost byπ with respect to figure 3.7 (a). This is an effect of thearctan in Eq. (2.4)

approaching the value±π/2 with increasing∆1 and decreasing∆2. The change in the phase

of thecos terms in Eq. (2.5) results in a shift fromπ to 0 of the value at which the destructive

interference occur.

To conclude, in section 3.3 we found that for two feedback loops of equallength,dτ = 0

and for suitable, and experimentally accessible values of the feedback rate κ = 0.01 and the

filter width Λ = 0.005, stable EFM regions have the following structure. There exist two bands

of stable EFMs that are centred around the central filter frequencies. These bands of stable

EFMs overlap each other for low values of the detunings of filters and, hence, are effectively

indistinguishable. The structure of the stable EFM bands unfolds with the increasing detuning

of the filters away from the central laser frequency. After the bands are separated, the qualitative

structure of stable EFM regions remains unaffected by theSN andSω transitions of the EFM

surface. With a further increase of the filters detuning, the bands of stable EFMs associated

with the central filters frequencies split into islands, concurrently with the outer components

of the EFM surface; this happens atSC-transitions and then the islands and the EFM stability

regions disappear with them inM -transitions. A third significant region of stable EFMs is

centred around solitary laser frequency. The size of this region increases with the detuning of

the filters, and for sufficiently large filter detunings a third stable EFM band iscreated. This

band of stable EFMs exist even after the side components of the EFM surface disappear in the

M -transition.

3.4 Dependence of EFM stability ondτ

So far in this chapter we discussed the stability of EFMs only for the case of the 2FOF laser

with equal delay times, that is, fordτ = 0. In this section we study how the regions of stable

EFMs change with increasing differencedτ = τ2−τ1 between the two delay times. This is the


final part of our analysis of the dependence of EFM stability regions on the filter and feedback

loop parameters.

In section 2.5 we explained in detail that changingdτ causes a shearing of the EFM surface

in (ωs, dCp, Ns)-space along thedCP -axis, with respect to the invariant plane defined by

ωs = 0. What is more, we showed that the shearing and the other changes of the geometry

and topology of the EFM surface are independent of each other. In other words, to uncover the

EFM surface for the case of the 2FOF laser with two different delay times, itis sufficient to

find the respective EFM surface fordτ = 0, and shear with thedτ shear rate.

Shearing of the EFM surface substantially changes the structure of the intervals of stable

EFMs that one finds for the fixeddCp. However, similarly to the case of the EFM surface

itself, the overall structure of the EFM stability regions in the(ωs, dCp)-plane can effectively

be determined by finding the EFM stability regions fordτ = 0 and shearing them.

As a concrete example, we now analyse how the EFM stability regions on the simplest EFM

surface of typeB change as we increasedτ . To this end, we fixτ1 = 500 and by changingτ2 in

the interval[506, 750] we increase difference between the delay times todτ = 250. Throughout

this section we fix the other filter parameters at the values corresponding to the EFM surface

in figure 3.1 (a), that is, toκ = 0.01, Λ = 0.015, and∆1 = ∆2 = 0. In particular, we show

that even for the simplest EFM surface of typeB, the shearing of the EFM surface results in

the possibility of finding an arbitrary number of bands of stable EFMs in the(ωs, dCp)-plane.

Figure 3.16 shows projections of the EFM surface with stability information ontothe

(ωs, dCp)-plane for increasingdτ . All curves and regions are coloured as in figure 3.7. Addi-

tionally, black dots indicate codimension-two Bogdanov-Takens bifurcationpoints. Figure 3.16

(a)–(d) shows that the stable EFM region is sheared concurrently with theEFM surface. In pan-

els (a) and (b), fordτ = 6 anddτ = 14, we show that initially the EFM stability region changes

rather subtly. In figure 3.16 (c), fordτ = 62, the basic unit of the EFM surface starts to extent

over the 2π dCp-range. Hence, for somedCp-range, two stable EFM intervals separated by

interval with no EFMs exist; for example two such intervals exist fordCp/π = 0.85. Each

of those intervals belongs to a different 2π-translated copy of the basic EFM surface unit. In

panel (d), fordτ = 62, the EFM surface is sheared so much that parts of four 2π-translated

copies of the basic EFM surface unit can be seen over the the 2π dCp-interval. As a result, for

example fordCp/π = 0.5, indicates four intervals of stable EFMs exist. Those stable EFM

intervals are separated by intervals with no EFMs as well as by intervals of unstable EFMs.

Additionally, the shearing of the stable EFM regions with increasingdτ is associated with

minor topological changes of their outer boundary. In figure 3.16 (a), similarly to figure 3.1 (a),

the outer boundary involves six Hopf bifurcation curves and a single saddle-node bifurcation

curve. However, due to the shearing the structure of this outer boundary changes. Note that,

in the bottom right corner of figure 3.16 (a), the boundary involves only two Hopf bifurcation

3.4. Dependence of EFM stability ondτ 99

−0.04 0 0.04 0

1

2

−0.04 0 0.04 0

1

2

−0.04 0 0.04 0

1

2

−0.04 0 0.04 0

1

2

(a) (b)

(c) (d)

ωs ωs

ωs ωs

dCp

πdCp

π

dCp

πdCp

π

.

.


increasing delay time in the second feedback loopτ2 = 506 (a), τ2 = 514 (b), τ2 = 562 (c) and

τ2 = 750 (d); hereτ1 = 500, κ = 0.01, Λ = 0.015 and ∆1 = ∆2 = 0. Black dots indicate

codimension-two Bogdanov-Takens bifurcation points; curves and regions are coloured as in figure 3.7.

curves; the third Hopf curve is no longer involved in the boundary after itpassed through the

Hopf-Hopf point; compare with figure 3.1 (a).

In figure 3.16 (b), we show that, as the shearing of the stable EFM region progresses, four

BT bifurcation points appear on the saddle-node curve. As those BT bifurcation points move

along the saddle-node curve with increasingdτ , the overall structure of the outer boundary of

the stable EFM region simplifies. In panel (c), a new Hopf bifurcation curve emerged from the

point (ωs, dCp) = (0, 1), when the saddle-node curve passed through a tangency with the line

dCp = 1. Additionally, after passing through a saddle point on the saddle-node bifurcation

surface in(ωs, dCp, dτ)-space, the saddle-node curve changed from a closed loop into two

infinitely long curves. In panel (d), fordτ = 250, the boundary of the EFM stability regions

involves only Hopf curves and is more complicated than in panel (c). Note that the inner

boundary of the EFM stability region is always formed by the closed elliptical Hopf curve

surrounding the points(ωs, dCp) = (0, 2nπ) with n ∈ Z.


Overall we conclude that the main effect of varyingdτ is the shearing of the EFM stability

regions that can be found fordτ = 0. Additionally, the shearing and the other transformations

of the EFM stability regions — presented in sections 3.1–3.3 — are independent of each other.

In other words, all previous cases of the EFM stability regions fordτ = 0 can be sheared

similarly. As a result of the shearing, even for the EFM surface of the simplest type B, any

number of stable EFM intervals, for fixeddCp can be obtained for sufficiently largedτ , such

stable EFM intervals are separated by intervals where no EFMs exist at all.

3.5 Different types of bifurcating oscillations

The existence of different kinds of intensity oscillations in lasers with opticalfeedback is well

documented; extensive theoretical as well as experimental studies of this subject can be found,

for example in [48, 24, 22, 18]. As in the FOF laser, in the 2FOF system two main types of

oscillations bifurcate from Hopf bifurcation points: relaxation oscillations (ROs), which are

common in semiconductor lasers subject to external perturbation, and frequency oscillations

(FO) which are typical for semiconductor lasers subject to filtered opticalfeedback.

ROs are characterised by a periodic exchange of energy between the optical fieldE and the

population inversionN , which result in oscillations of the laser intensity and (for non-zeroα)

of the laser frequency; a typical RO frequency has a value of several GHz [19]. In FOs, on the

other hand, the laser frequency oscillates with a high amplitude at a frequency on the order of

the external round trip time1/τ , while the intensity of the laser remains almost constant [17,

18]. This is unexpected in a semiconductor laser, in light of strong amplitude-phase coupling

via the linewidth enhancement parameterα [18, 23]. The different mechanisms behind ROs

and FOs manifest themselves in the different time scales of the two kinds of oscillations: the

fast RO arises due to fast processes inside the semiconductor laser, and the slow FO arise due

to interaction of the semiconductor laser with light travelling in the external filtered feedback

loops; time light travels in external loops is orders of magnitude larger then thetimescale of

the processes inside the laser.

Considering that the existence of ROs and FOs in the single FOF laser has been studied

theoretically and confirmed experimentally [17, 19], we expect to find both ROs as well as

FOs in the 2FOF laser. From a bifurcation theory point of view, both types of oscillations are

periodic orbits that originate from Hopf bifurcations [38, 42]. We foundmany Hopf bifurcation

curves in sections 3.1–3.4, from which periodic solution bifurcate. We nowdiscuss which of

those Hopf bifurcations in the(ωs, dCp)-plane give rise to ROs and which to FOs; in this way,

we determine where in the(ωs, dCp)-plane one can expect to find ROs and FOs.

To analyse periodic orbits emanating from Hopf bifurcation points we use again numerical

continuation. We continue the bifurcating periodic solutions with DDE-BIFTOOL [11]. Note

3.5. Different types of bifurcating oscillations 101

0 5 10 15 20 251.7

1.9

2.1

0 100 200 300 400 5001.7

1.9

2.1

0 5 10 15 20 25−2.5

0

2.5

0 100 200 300 400 500−2.5

0

2.5

0 5 10 15 20 251.4

1.9

0 100 200 300 400 5001.4

1.9

0 5 10 15 20 25−2.5

0

2.5

0 100 200 300 400 500−2.5

0

2.5

0 5 10 15 20 251.4

1.9

0 100 200 300 400 5001.4

1.9

0 5 10 15 20 25−2.5

0

2.5

0 100 200 300 400 500−2.5

0

2.5

(a1)

(a2)

(a3)

(a4)

(a5)

(a6)

(b1)

(b2)

(b3)

(b4)

(b5)

(b6)

t t

t t

t t

t t

t t

t t

φF2 φF2

IF2 IF2

φF1 φF1

IF1 IF1

φL φL

IL IL

.

.

Figure 3.17. Example of relaxation oscillations (a) and frequency oscillations (b) found in the EFM

stability diagram in figure 3.7 (c); for∆1 = 0.005, ∆2 = 0, κ = 0.01, Λ = 0.005 anddτ = 0. RO are

found at(ωs, dCp/π) = (0.0035, 1.828), and FO at(ωs, dCp/π) = (0.0031, 0.802). The different

rows show from top to bottom: the intensityIL and the frequencyφL = dφL/dt of the laser field, the

intensityIF1 and the frequencyφF1 = dφF1/dt of the first filter field, and the intensityIF2 and the

frequencyφF2 = dφF2/dt of the second filter field. Note the different time scales for ROs and FOs.

that, the numerical continuation with calculation of stability information for periodicorbits is

computationally much more expensive than the respective calculations for EFMs (which are

steady states). This is why a complete analysis of bifurcating oscillations and their stability

regions is beyond the scope of this work. Rather we show, by continuation of particular bifur-

cating periodic orbits, where generally ROs and FOs can be found for the2FOF laser.

As a starting point, we now investigate the types of periodic solutions bifurcating from

the different types of Hopf bifurcation curves in figure 3.7 (c); for∆1 = 0.005, ∆2 = 0,

κ = 0.01, Λ = 0.005 anddτ = 0. We start our analysis with this case, because beside the

Hopf bifurcation curves that emerged with the appearance of the hole, theboundary of the EFM

stability region in figure 3.7 (c) still includes all the Hopf bifurcation curves that one can find

for the 2FOF laser with two identical filters with equal delay times. To determine what kind


of periodic solutions emanate from the Hopf bifurcation curves in figure 3.7(c), we analyse

frequencies and amplitudes of the periodic solutions that bifurcate from them; at each Hopf

bifurcation curve we start a sufficient number of continuations to be able todetermine the type

of oscillations that bifurcate from it.

Figure 3.17 shows example of ROs (a) and of FOs (b) that can be found for suitable

(ωs, dCp) in figure 3.7 (c); the ROs are for point(ωs, dCp/π) = (0.0035, 1.828), and the

FOs for(ωs, dCp/π) = (0.0031, 0.802). Note that the ROs and FOs exist for the sameωs-

range, but for differentdCp. In respective column of figure 3.17 we show, from top to bottom,

time series of the laser intensityIL, of the laser frequencyφL = dφL/dt, of the first filter

intensityIF1, of the first filter frequencyφF1 = dφF1/dt, of the second filter intensityIF2,

and of the second filter frequencyφF2 = dφF2/dt. Note the difference in time scales between

columns (a) and (b).

In figure 3.17 (a) we show typical ROs: both the laser intensityIL (a1) and the laser

frequencyφL (a2) oscillate at a frequency of 4.2 GHz. At the same time, both filter fields

hardly show any dynamics — the amplitudes of oscillation of the filter fields (a3)–(a6) are at

least one order of magnitude smaller then the amplitude of oscillations of the laserfield. The

fact that the filter fields are practically not involved in the oscillations is consistent with the

general observations that ROs are typical for semiconductor lasers withany type of feedback.

It appears that the role of the feedback loops is merely to decrease the losses, which exicites

the ROs that already exist in damped form in the free-running semiconductor laser.

In figure 3.17 (b) we show an example of the FOs. Here, the laser frequency φL (b2)

oscillates with an amplitude comparable to the oscillations of the laser frequencyφL in the case

of ROs (a2). However, the laser intensityIL (b1) is practically constant — it oscillates with

an amplitude that is two orders of magnitude smaller than the amplitude for ROs in panel (a1).

Moreover, in contrast to ROs both filter fields (b3)–(b6) oscillate with significant amplitudes.

Frequency of the FOs is on the order of1/τ . Note that both filter intensitiesIF1 (b3) and

IF2 (b5) oscillate in antiphase with each other. Moreover, the laser frequencyφL (b2) is in

antiphase with the oscillations of both filter frequenciesφF1 (b4) andφF2 (b6). This shows

that for the 2FOF laser both filter fields are generally involved in FOs.

The boundary of the EFM stability region in figure 3.7 (c) involves seven Hopf bifurcation

curves: one is located in the centre of the upper half, slightly to the left fromωs = 0, two

surround the hole and four form the outer boundary. Most of those Hopf bifurcation curves

give rise to the FOs. In fact, the only Hopf bifurcation curve at which the ROs excite is the

closed Hopf bifurcation curve located in the centre of the upper half of figure 3.7 (c1).

We analysed the Hopf bifurcation curves from figure 3.1 (a), figure 3.3(b) and (d), fig-

ure 3.12 (a) and (c), figure 3.14 (b) and figure 3.16 (d) as well. This allows us to conclude that

3.5. Different types of bifurcating oscillations 103

FOs bifurcate from most of the Hopf bifurcation curves shown in the figures in sections 3.1–

3.4. Similarly to other laser systems with optical feedback, ROs only appear around the solitary

laser frequency — in our case represented byωs = 0 — and only for sufficiently low levels

of the population inversionNs. In particular this fact can be observed in section 3.1, where

we consider the dependence of the EFM stability regions on the common feedback rateκ. In

section 3.1 in figure 3.1 thedCp-periodic copies of RO Hopf bifurcation curves surround the

points(ωs, dCp) = (0, 2n)π with n ∈ Z, and they form the lower boundary of the EFM sta-

bility region in the(ωs, Ns)-projections; see figure 3.1 (a2)–(c2). Note that all the other Hopf

bifurcation curves give rise to FOs. The RO Hopf bifurcation curve onlyappears as a closed

curve in the middle of the EFM stability region for sufficiently largeκ; see figure 3.1 (a3).

Furthermore, the expansion of the RO Hopf bifurcation curve with increasing κ results in the

growth of the region where ROs exists. These observations agree generally with findings for

ROs in other laser systems with optical feedback [17, 22, 38].

To explain how the regions of existence of different periodic solutions depend onΛ we now

consider the Hopf bifurcation curves presented in figures 3.3 and 3.4 in section 3.2. Similarly

to figure 3.1 (a3)–(c3), thedCp-periodic copies of RO Hopf bifurcation curves in figure 3.4

(a)–(d) surround the points(ωs, dCp) = (0, 2nπ) with n ∈ Z; these curves form the lower

boundary of the EFM stability region in the(ωs, Ns)-projections in figure 3.3 (a2)–(d2). All

the other Hopf bifurcation curves give rise to FOs. With increasingΛ, all the Hopf bifurcation

curves that give rise to the FOs gradually disappear; details are described in section 3.2. The

single Hopf bifurcation curve in figure 3.3 (c) and (d) is the one at which ROs bifurcate. This

means that for sufficiently highΛ no FOs exist in the 2FOF system. This result is in good

agreement with the observation that, as theΛ → ∞, the 2FOF laser reduces to the COF laser,

where one does not observe FO [5, 29, 31].

We now explain, with help of figures from section 3.3, that with increasing modulus of

∆1 and∆2 the RO Hopf bifurcation curve disappears. As for the case of the 2FOF laser with

two identical filters with equal delay times presented in sections 3.1 and 3.2, forlow values of

filter detuning∆1, as in figure 3.7 (a) and (b), thedCp-periodic copies of RO Hopf bifurcation

curves surround the points(ωs, dCp) = (0, 2nπ) with n ∈ Z. With increasing∆1, this

RO Hopf bifurcation curve shrinks and shifts away from the points(ωs, dCp) = (0, 2nπ) with

n ∈ Z. In figure 3.7 (c1), for∆1 = 0.005, only one copy of the whole RO Hopf curve is visible

and it is located in the upper half of the(ωs, dCp)-plane. As the result of a further increase

of ∆1, the RO Hopf bifurcation curve disappears. All remaining Hopf bifurcation curves give

rise to FOs; see figure 3.9 (a) and (b). What is more, we checked that allperiodic solutions

emanating from all the Hopf bifurcation curves involved in EFM stability regionboundaries in

figures 3.8–3.15 are FOs.

To summarise, we found that for moderate values ofκ andΛ and filter detunings∆1 and

∆2 of sufficiently large modulus all periodic solutions emanating from the Hopf bifurcation


curves bounding EFM stability regions are FOs. In contrast, ROs bifurcate from the closed

Hopf bifurcation curve that exist only if the solitary laser frequency is subject to sufficiently

high feedback from the centre of the filter profile.

Chapter 4

Overall summary

Research presented in this thesis was motivated by the application of an 2FOFlaser as a pump

laser for optical communication systems. Since in such applications the main concern is sta-

ble operation of the laser source, in our analysis we focused on structure and stability of the

external filtered modes (EFMs). The EFMs are solutions of Eqs. (1.1)–(1.4) which physi-

cally correspond to constant-intensity monochromatic laser operation. We first investigated

the structure of the EFMs; in other words, we determined how the regions ofexistence of the

EFMs depend on the parameters of the 2FOF laser. We then used the uncovered structure of

the EFMs as a guideline for the analysis of their stability. What is more, we wereable to relate

changes of the EFM stability with changes of the EFM structure. Finally, we briefly studied

the types of periodic solutions bifurcating from the boundaries of the EFM stability regions.

We now present an overall summary of our work and discuss the possibilityof an experimental

confirmation of our results.

The well established way of analysing the structure of the solutions of the single FOF

laser is the investigation of EFM components — closed curves that are loci ofEFMs in the

(ωs, Ns)-plane. EFMs trace an EFM component with a changing phase of the feedback field.

The relevance of EFM components is that they show how many disjointωs frequency ranges are

available for stable laser operation. In contrast to the single FOF laser, where one can observe

at most two EFM components, we showed that, due to the transcendental nature of Eq.(2.9)

— the envelope of Eq.(2.3) for frequenciesωs of the EFMs — in the 2FOF laser system it

is possible to observe an arbitrary number of EFM components. Moreover, the analysis of

Eq.(2.9) showed that the number of EFM components in the 2FOF laser depends strongly on

the parameterdCp — the difference between the feedback phases of the two filtered fields.

Therefore, in chapter 2 we introduced the EFM surface in(ωs, dCp, Ns)-space, as a natural

generalization of an EFM component for the 2FOF laser. This provided a useful framework

for the analysis of the dependence of structure and stability of EFMs on the2FOF laser’s two

feedback loops and their filter parameters. Similarly to EFM components, the EFM surface

105

106 Chapter 4. Overall summary

provides information on how the structure of solutions depend on the phaserelations of the

filtered fields. Note that the feedback phase depends on subwavelengthchanges of the length

of the feedback loops and, hence, it is very difficult to control in real-life implementations.

Therefore, the feedback phase is the key parameter of interest in analysis of EFMs.

By considering the structure of Eq. (2.9) we found out that the dependence of the EFM

surface on the filter detunings∆1 and∆2 and the common filter widthΛ can be analysed

separately from its dependence on the differencedτ between the delay times. Therefore, we

first explored topological changes of the EFM surface with the former filter parameters. To this

end, we employed singularity theory; in particular, we distinguished five different mechanisms

through which the EFM surface can change locally: four singularity transitions and a cubic

tangency of the EFM surface with respect to a planedCp = const. What is more, we used

loci of those transitions in the(∆1, ∆2)-plane to compute, by numerical continuation, the

EFM surface bifurcation diagrams in the(∆1, ∆2)-plane. We then analysed, how this diagram

depends on change of the common filter widthΛ. This allowed us to present a complete

classification of all possible types of the EFM surface for the case of equal feedback rates

κ = κ1 = κ2. Furthermore, we used a stereographic change of coordinates to compactify the

EFM surface bifurcation diagrams in the(∆1, ∆2)-plane to analyse different ways in which

the 2FOF laser can be reduced in a nontrivial way to the single FOF laser and to the COF

laser. Finally, we showed that a nonzero differencedτ 6= 0 between the delay times causes a

shearing of the EFM surface along thedCp-axis; the topological structure of the single instance

of the EFM surface is preserved under this transformation. It is due to thisshearing of the

EFM surface that one can find an arbitrary number of the EFM componentsin the 2FOF laser;

for dτ = 0 there are at most three EFM components: one exist always around solitary laser

frequency, and the other two can appear around the filter central frequencies.

The findings presented in chapter 2 also show that the 2FOF laser is a modelsystem, which

can be used to connect different kinds of laser systems subject to optical feedback. In particular,

for Λ → ∞ the 2FOF laser reduces to the 2COF laser. It is also possible to increase thewidth of

only one of the filters, and this will result in a system with FOF and COF branches.Moreover,

by setting∆1 = ∆2 or by shifting one of the filter central frequencies∆i to infinity the

system reduces further to the FOF laser or to the COF laser, respectively. On the other hand,

for nonzerodτ one can investigate the transition between a laser subject to feedback froma

minimum of filter profile (as in [64]) and a laser subject to feedback from a filter maximum, as

here.

In chapter 3 we analysed the stability of the EFMs. We first investigated how the stability

of the EFMs depends on the common feedback rateκ and the common filter widthΛ. We next

examined how it is affected by the geometrical and topological transitions of the EFM surface

when the filter detunings∆1 and∆2 are changed. Finally, we showed that, similarly to the

EFM surface, changingdτ results in shearing of the EFM stability regions. Analogously, to

4.1. Physical relevance of findings 107

other laser system subject to optical feedback, stable EFMs of the 2FOF laser become unstable

in Hopf bifurcations; from those Hopf bifurcation two kinds of stable periodic solutions may

arise: the well-known relaxation oscillations, and frequency oscillations which are typical for

lasers with filtered feedback.

In the next section we discuss our findings from chapter 3, in conjunctionwith predictions

concerning outcomes of possible experiments with the 2FOF laser.

4.1 Physical relevance of findings

Detailed experimental studies of how the dynamics of a semiconductor laser depend on feed-

back parameters are very challenging. Nevertheless, multiparameter experimental studies of

the dynamics of the single FOF lasers were performed successfully [22, 23]. What is more, due

to advance in experimental techniques it is possible today to compare theoretical and experi-

mental two-parameter bifurcation diagrams of laser systems; see for example[48, 65, 71]. Even

though those state-of-the-art studies concern lasers with an optical injection, similar methods

might be used to analyse the 2FOF lasers.

In this section, we first discuss how the different parameters of the 2FOFlasers may be

changed in an experiment. Next, we present what features of the EFM structure presented

in Chapters.2 and 3 may be most promising to verify experimentally. Finally, we show two

examples that demonstrate the high degree of multistability in the 2FOF laser.

4.1.1 Experimental techniques for the control of parameters

The 2FOF laser can be realized in experiments either by means of optical fibres with embedded

fiber Bragg gratings or by two open air (unidirectional) feedback loops withFabry-Pérot filters;

see figure 1.1. In both setups, most of the 2FOF laser parameters can be controlled in a very

precise way. In particular, all the filter and feedback loop parameters that are analysed in this

thesis, are directly accessible and can be set for an experiment. Some of the parameters, such as

the feedback phasesC1p andC2

p or feedback rateκ, can be modified almost continuously during

measurements. However, to control some of the other parameters it is necessary to modify the

experimental setup. For example, to change the filter widthΛ or the filter detunings∆1, and

∆2 it may be necessary to introduce filtering elements with different characteristics.

To explore the EFM surface experimentally — the main object of our studies — itis nec-

essary to control the feedback phases in both feedback loops. In previous experiments with

the FOF laser and the COF laser two methods of controlling the feedback phase were used. In

[19] the feedback phase is controlled by changing the length of feedback loop on the optical


wavelength scale (nanometres) with a piezo actuator, and in [30] it is controlled by changing

the laser pump current, which results in shifting the solitary laser frequency; note that [30]

concerns the COF laser.

It is important to realize that for the 2FOF laser any changes of the solitary laser frequency

affects simultaneously not only both feedback phasesC1p andC2

p but, more importantly, also

both filter detunings. In other words, scanning through the solitary laser frequency concurrently

changes the EFM-components and positions of EFMs on them; for each value of the solitary

laser frequency one observes a new set of EFMs that is given by instantaneous filter detunings

and instantaneous feedback phases that change simultaneously in both feedback loops. There-

fore, to vary the feedback phasesC1p andC2

p in the 2FOF laser it would be more practical to

use piezo actuators. Similarly, in an experimental setup with optical fibres the parametersC1p

andC2p can be controlled via thermal expansion of glass. The fibre temperature ofa section

of an optical fibre can be controlled either by immersing it in a water bath, or bylocally heat-

ing an optical fibre with a resistance wire heater. Note that to changedCp continuously in an

experiment it is enough to control one of the feedback phases, eitherC1p or C2

p .

The other parameters of the 2FOF laser can be changed in the following way: the feedback

rateκ can be controlled with a variable attenuator and a half-wave plateλ/2 [19], or by a

neutral density filter [23]; the filter widthΛ and the filter detuning∆1 and∆2 can be changed

by using different Fabry-Pérot resonators with different distancesbetween their mirrors [22];

or by exploiting strain and temperature sensitivity of fibre Bragg gratings to control location of

maximum and width of its reflection profile [66]. Finally, changing the feedback loop lengths

directly changes the delay timesτ1 andτ2, and in consequencedτ .

4.1.2 Expected experimental results

In experimental studies of a semiconductor laser with feedback one observes only stable cw-

solutions. Therefore, the stability analysis presented in chapter 3 allows usto make some

qualitative predictions about possible experimental results. Generally, weshowed there that

the core of the structure of the EFM stability regions, is formed by the two stableEFM bands

centred around the filters central frequencies. More importantly, we uncovered that those two

bands of stable EFMs appear to be quite robust against a wide range of parameter changes.

In other words, we showed that for large, experimentally relevant parameters ranges, multiple

regions of stable EFMs exist in the 2FOF laser system. Note that, since similarly tothe EFM

components, the large part of the EFM surface consists of unstable EFMs, the structure of the

EFM surface can only be checked indirectly. Nevertheless, although theunstable EFMs cannot

be observed directly in an experiment, it has been shown that they may play avery important

role in shaping the overall dynamics of the laser output [25, 30, 54, 55, 58].


We now list the features of the CW-states and of the basic periodic solutions of the 2FOF

laser that are most likely to be observe in an experiment; we base our predictions on findings

presented in this thesis and on experimental results for the FOF laser.

A first experimental test of the results from chapter 3 could be a confirmation of the de-

structive interference between the two filter fields. Recall that, for the two identical feedback

loops with equal delay times, the feedback fields cancel each other fordCp = (2n + 1)π with

n ∈ Z. Therefore, for different values ofdCp, and a fixed value ofκ, one should be able to

obtain for the 2FOF laser results as in [22], which illustrate the dependenceof the EFMs of

the single FOF laser on the feedback strength. In other words, one should be able to show

experimentally that the 2FOF laser can be reduced, in a non-trivial way to the single FOF laser

with effective feedback rate controlled bydCp; see section 2.1.2.

A main feature of the stability figures in chapter 3 are bands of stable EFMs centred around

the filter central frequencies. In such a band, stable EFMs exist only for some specific values

of the feedback phasesC1p andC2

p . Therefore, due to fact thatdCp depends on both feedback

phasesC1p andC2

p , each of the stable EFM bands on the EFM surface have to be considered

separately. In particular, to follow a stable EFM on a specific band, one has to fix feedback

phase in the feedback loop associated with this band, and change thedCp by means of the

feedback phase in the other feedback loop. The stable EFMs manifest themselves in an ex-

periment as plateaus of the feedback intensity (different feedback intensities imply different

frequencies of the laser light); see [19, Fig. 3]. Therefore, to confirm the existence of stable

EFM bands, it is first necessary to set the parameters of the experimentalsetup, in such way

that the laser is locked to one of the stable EFMs; then, it can be shown experimentally that

such an EFM exists over several 2π cycles of changingdCp and one is indeed dealing with a

stable EFM band. On the other hand, if the existence of the EFM is limited to some finite range

of the parameterdCp, then one can say that the measured EFM exists on an island of stable

EFMs. Note that, due to the periodic nature of the edges of stable EFM regions, it might be

difficult to distinguish between a narrow stable EFM band and a large stable EFM island.

By combining all the informations about the existence of stable EFM bands andislands,

one might be able to reconstruct the main features of the EFM stability structureas presented in

the chapter 3. More specifically, one should be able to distinguish between the EFM stability

structure involving two wide stable EFM bands as in figures 3.10–3.13; the EFM stability

structure consisting of three stable EFM bands as in figure 3.15 (b); and the EFM stability

structure consisting of combination of stable EFM bands and islands as in figure 3.14 and

figure 3.15 (c) and (d). Furthermore, cases in which the stability structurefrom chapter 3 is

closely aligned with the EFM surface, allow for indirect confirmation of the classification of

the EFM surface into different types presented in chapter 2; an example of such an alignment

is figure 3.14.


Finally, one may also investigate existence of bifurcating FOs and ROs for different pa-

rameter values. A strong confirmation of our results would be to demonstrate the absence of

ROs for moderate values of the common feedback rateκ and the common filter widthΛ and

the filter detunings∆1 and∆2 of sufficiently large modulus; see section 3.5.

4.1.3 Existence of multistability

A next step, which is a necessary to fully describe stability of the EFMs, is theinvestigation

how many stable EFMs coexist for fixed parameters; i.e. for givenC1p andC2

p . Similarly to

the single FOF laser [17, 31], also in the 2FOF laser one can expect to findmultistability for

chosen fixed parameters, that is, including fixed feedback phasesC1p andC2

p . To confirm this

hypothesis, we first show that the total number of coexisting EFMs depends on the parameters

C1p anddCp. We then analyse the stability of individual EFMs, computed for two different sets

of fixed parameters.

Figure 4.1 shows regions in the(C1p , dCp)-plane in which different numbers of EFMs ex-

ist simultaneously; this figure is for∆1 = 0.050, ∆2 = −0.049, κ = 0.01, Λ = 0.005 and

dτ = 0, and the corresponding EFM surface of typeIBI is shown in figure 3.14 (b). Bound-

aries between the regions in figure 4.1 are given by saddle-node bifurcation curves (blue).

Figure 4.1 (a) shows the regions with different numbers of coexisting EFMs over a fundamen-

tal 2π-interval of C1p . The labelling of the regions provides not only information about the

number of EFMs on each EFM component, but also about the number of EFMcomponents

and their relative positions. To emphasize the connection between EFM components and the

EFM surface, we structured the labels in figure 4.1 (a) to be similar to the labelsfor the types

of the EFM surface. More specifically, the hat over a number indicates thenumber of EFMs on

the EFM component centred around the solitary laser frequency. Furthermore, the the numbers

without hats indicate the number of the EFMs on the EFM components centred around the

filter central frequencies; the total number of EFMs for a given parameter values is simply the

sum of numbers of a region label. Note that, the positions of the numbers withouthats indicate

locations of the other EFM components with respect to the central one. In other words, once

the EFM surface is known, it is possible to encode its structure in labels of theregions in the

(C1p , dCp)-plane.

A single EFM, in regions labelled1, corresponds to the solitary laser solution. The total

number of the EFMs on the central EFM component is always odd and new EFMs are formed

in pairs on saddle-node bifurcations curves [31, 38]. We can expectmultistability in all regions

in figure 4.1 (a) with more then one EFM. However, the best regions to look for a high degree

of multistability are the two small regions labelled2 + 1 + 2 and2 + 3 + 2. For the parameter

values in those regions, EFMs exist concurrently on all three EFM components; in the other

regions, EFMs exist at most on two EFM components. Note that, by considering dCp as a


0 1 20

1

2

−5 −4 −3 −2 −1 0 1 20

1

2

2 + b1

2 + b12 + b1

2 + b1

2 + b1 + 2

b1b1

b1 b1

b1

2 + b3

@@2 + b3

@@2 + b3

2 + b3 + 2

2 + b5

2 + b5

b3

b3

b3

b3 + 2

b5b5

b1 + 2

b1 + 2(a)

(b)

C1

p/π

C1

p/π

dCp

π

dCp

π

.

.

Figure 4.1. Regions in the(C1

p , dCp)-plane with different numbers of coexisting EFMs, as indicated

by the labelling. Panel (a) shows the regions on a fundamental 2π-interval ofC1

p , while panel (b) shows

it in the covering space (over several 2π-intervals ofC1

p ). Boundaries between regions are saddle-node

bifurcation curves (blue); also shown in panel (b) are periodic copies of the saddle-node bifurcation

curves (light blue). Labels Here∆1 = 0.050, ∆2 = −0.049, κ = 0.01, Λ = 0.005 anddτ = 0; these

parameter values are those for the EFM surface in figure 3.14 (b).

kind of effective feedback strength (as has been argued in section 2.1.1), it is possible to see

similarities between our findings and results on the number of coexisting EFMs for the single

FOF laser as presented, for example, in [31].

Figure 4.1 (b) illustrates how the regions in the(C1p , dCp)-plane can be related to the

projection of the EFM surface onto the(ωs, dCp)-plane. In figure 4.1 (b) we show the dark

blue saddle-node bifurcation curves from panel (a) over several 2π-intervals ofC1p ; since, each

saddle-node bifurcation curve is presented over different 2π-intervals ofC1p the curves do not


−0.045 0 0.045−0.01

0

0.01

−0.055 0 0.055−0.011

0

0.011

ωs

ωs

Ns

Ns

(a)

(b)

.

.

Figure 4.2. EFM-components (grey) in the(ωs, Ns)-plane with stability information. Panel (a) is for

dCp = π, ∆1 = 0.050, ∆2 = −0.049, and panel (b) is fordCp = −π, ∆1 = 0.036, ∆2 = −0.037;

furthermore,κ = 0.01, Λ = 0.005 anddτ = 0. Stable segments of the EFM-components (green)

are bounded by the Hopf bifurcations (red dots) or by the saddle-node bifurcation (blue dots). The

actual stable EFMs forC1

p = 1.03π (a) andC1

p = 0.9π (b) are the black full circles; open circles are

unstable EFMs. The EFM components in panel (a) correspond toa constantdCp-section through the

EFM surface in figure 3.14 (b), and those in panel (b) to a constantdCp-section through the EFM surface

in figure 3.12 (d).

intersect. More importantly, the relative positions of the dark blue curves in figure 4.1 (b)

correspond to the structure of the saddle-node bifurcation curves in figure 3.15 (d), and hence,

the two figures can easily be related. Note that multiple periodic copies of the saddle-node

bifurcation curves in figure 4.1 (b) are coloured light blue. More generally, figure 4.1 shows

that, as in the case of the single FOF laser, the actual number of coexisting EFMs depends

strongly on the phase relation between the feedback and laser fields.

Figure 4.2 confirms the existence of multistability in the 2FOF system by analysing the

positions of the individual EFMs along the EFM components with stability information, for


two different sets of fixed parameters. Figure 4.2 (a) corresponds to the constantdCp-section,

for dCp = π, through the EFM surface of typeIBI in figure 3.14 (b); figure 4.2 (b) corresponds

to a constantdCp-section fordCp = −π through the EFM surface of typeBBB in figure 3.12

(d). There are stable segments (green) on each of the three EFM components in panels (a)

and (b); they are bounded either by the saddle-node bifurcations (bluedots) or by the Hopf

bifurcations (red dots). The black circles in figure 4.2 are the EFMs forC1p = 1.03π (a) and

for C1p = 0.9π (b); full circles indicate stable EFMs and open circles unstable EFMs.

The EFMs in figure 4.2 (a) are computed for parameter values from the region labelled

2 + 3 + 2 in figure 4.1 (a); in this case four of the EFMs are stable and lie within the respective

three stable segments of the EFM components. Note that the stable EFMs on the outer EFM

components lie very close to the boundaries of the respective stable segments. In fact, as can

be seen in figure 4.1 (a), for slightly different value ofC1p , EFMs exist only on two out of

three EFM components. In figure 4.2 (b) we show a more convincing example of multistabil-

ity, where three of the EFMs are stable and clearly lie well within the respective three stable

segments of the EFM components.

Figure 4.2 suggest that multistability is a rather common feature of the 2FOF laser, so that

it should be possible to observe it in experiments; for example as hysteresis[20]. What is

more, this figure demonstrates that the stability results presented in chapter 3 are indeed an

essential basis for the exploration of multistability in the 2FOF laser. A thoroughanalysis of

multistability of EFMs and investigation of stability of periodic solutions in the 2FOF laser are

a logical next step in the study of the 2FOF laser.

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Appendix A

How to construct the EFM surface

In our work we present structure and stability information of the EFMs by means of showing

the EFM surface in(ωs, dCp, Ns)-space. Since, explicit and easily solvable equations for the

EFMs and their stability do not exist, we find them numerically and we use state-of-the-art

numerical techniques to determine their properties.

The process of constructing the EFM surface can be split into several steps. First, for the

chosen parameter values, we compute a sufficient number of constantdCp sections through

the EFM surface. We then use this computed data to render the EFM surface. Finally, we add

computed stability information to the EFM surface and its two dimensional projections. Each

of the steps is automated as much as is practical; to calculate and render the EFMsurface in an

efficient way, we developed a set of interactive MATLAB scripts and functions.

Recall that, to present the classification of different types of the EFM surfaces in sec-

tion 2.3, it was enough to use Eq. (2.16), which describes the boundary ofthe EFM surface

projection onto the(ωs, dCp)-plane independently of any state variables of the 2FOF laser.

The advantage of this approach is that it allowed for the fast computation ofthe (ωs, dCp)-

projections of the EFM surface for a wide range of the system parameters, in particular, for

different values of the filter detunings∆1 and∆2. However, equation (2.16) for the(ωs, dCp)-

projection boundary does not provide any information on EFMs stability. Furthermore, in

section 2.1 we showed that the EFMs themselves can be calculated as solutionsof the tran-

scendental equations (2.3)–(2.8) and, hence, these could be used to follow EFMs with changing

parameters with any numerical continuation package. However, Eqs. (2.3)–(2.8) also do not

provide the EFM stability information. In fact, to analyse EFM stability it is necessary to con-

tinue the solutions of the full DDE system (1.1)–(1.4); to this end, we use the MATLAB pack-

age DDE-BIFTOOL [11]. The EFM stability information, uncovered with DDE-BIFTOOL,

can be presented either on the EFM surface itself or in any conveniently chosen EFM surface

projection. We now describe our numerical strategy to deal with the mathematical challenges

associated with the analysis of the EFM structure of Eqs. (1.1)–(1.4).

121

122 Appendix A

A.1 Dealing with the S1-symmetry of the 2FOF laser model

In section 1.1 we mentioned that the system (1.1)–(1.4) shares symmetry properties with many

other systems with coherent optical feedback. Namely, system (1.1)–(1.4) has anS1-symmetry,

meaning that the equations are equivariant under any elementb ∈ S1, which is physically a

rotation with a specific phase. This equivariance is the result of the fact that Eq. (1.1) is linear

in E and thatE enters Eq. (1.2) only as its modulus|E| [29, 34, 40]. As a consequence,

solutions of Eqs. (1.1)–(1.4) are group orbits under theS1-symmetry; trajectories in the group

orbit run entirely parallel, and differ only in their phases; see Eq. (1.5).In other words, one can

obtain infinitely many trajectories of the system (1.1)–(1.4) by multiplyingE, F1 andF2 by

any complex number of modulus one; this is true even for chaotic trajectories [40]. Since the

symmetry cannot be simply divided out from the equations, to resolve the phase indeterminacy

of Eqs.(1.1)–(1.4) we consider intersection of the group orbits with the fixed six-dimensional

hyper-half plane

S = (E, N, F1, F2) | Im(E) = 0 and Re(E) ≥ 0 ; (A.1)

these intersections are also called the trace [34, 40]. Note that, the trace ofgroup orbits of Eqs.

(1.1)–(1.4) consist of isolated trajectories which can be studied with numerical continuation.

In practice to analyse the trace of the 2FOF laser by continuation with DDE-BIFTOOL,

it is necessary to transform Eqs.(1.1)–(1.4) into a system in which the phase indeterminacy is

expressed explicitly. To this end, we substitute ansatz

(E, N, F1, F2(t)) =(Eeibt, N, F1e

ibt, F2eibt)

(A.2)

into (1.1)–(1.4) and divide through by an exponential factor, which results in the transformed

system

dE

dt= (1 + iα)N(t)E(t) − ibE(t) + κ1F1(t) + κ2F2(t), (A.3)

TdN

dt= P − N(t) − (1 + 2N(t))|E(t)|2, (A.4)

dF1


−iC1p + (i∆1 − Λ1 − ib)F1(t), (A.5)

dF2


−iC2p + (i∆2 − Λ2 − ib)F2(t). (A.6)

Hereb is an extra free parameter that allows for direct parametrisation of all trajectories in a

group orbit [17, 29, 40, 56]. In other words, each value ofb correspond to one trajectory from

A.2. Computation and rendering of the EFM surface 123

the group orbit. Note that solutions of system (A.3)–(A.6) have the same stability properties as

the solutions of system (1.1)–(1.4). In particular, they have a trivial additional zero eigenvalue,

which must be taken into account for the appropriate stability and bifurcationanalysis of the

solutions [17, 40, 45].

Eqs. (A.3)– (A.6) are of a form in which, by uniquely determining the parameter b, the

phase indeterminacy can be resolved. This can be achieved by requiring, for example, that

Im(E) = 0 in Eq. (A.3); in this way, continuous-wave states of Eqs. (A.3)– (A.6) indeed cor-

respond to the trace of solutions of (1.1)– (1.4). In other words, a one-dimensionalS1-family

of solutions in the seven-dimensional physical space is transformed into a pair consisting of

the one-dimensional trajectory in a six-dimensional sub-space, and associated unique value of

the phase parameterb; for example, a cw-state, which is a circular periodic trajectory of pe-

riod 2π/b in (E, N, F1, F2(t))-space, transforms into an isolated point in the six-dimensional

sub-space with associated frequencyb. Furthermore, by considering cw-states the numerical

strategy gains physical interpretation: the free parameterb is equivalent to the frequencyωs of

an EFM; see section 2.1.

Implementation of the above strategy in DDE-BIFTOOL is straight forward: inaddi-

tion to defining system (A.3)–(A.6) in the form of seven real equations in filesys_rhs.m,

one also needs to specify the condition Im(E) = 0 as an extra equation in the separate file

sys_cond.m. An example of using the same approach to continue solutions of two linearly

coupled oscillators with DDE-BIFTOOL is presented in [12]. More generally, the method

demonstrated here make use of the fact that finding a unique solution of a system of equations

requires that the number of equations and number of unknowns must be equal.

A.2 Computation and rendering of the EFM surface

Generally, to construct the EFM surface in(ωs, dCp, Ns)-space we first compute a number

of sections that are uniformly distributed along the 2π dCp-range considered. Next we use

this computed data to render the EFM surface. Recall, that a constant-dCp section through the

EFM surface in(ωs, dCp, Ns)-space corresponds to the EFM components, which are closed

curves traced by the EFMs with changing feedback phase; see section 2.1.1. To compute the

necessary slices, that is the EFM components, we continue the EFMs with DDE-BIFTOOL

[11]. To ensure thatdCp is fixed, we substituteC2p = C1

p + dCp in Eqs. (A.3)–(A.6). In this

way, by continuation of a single EFM, in the continuation parameterC1p , over several multiples

of 2π, we indeed obtain constantdCp sections through the EFM surface in(ωs, dCp, Ns)-

space. Note that to construct the EFM surface without the stability informationone could also

use Eqs. (2.10) and (2.16).

124 Appendix A

We now present more detailed information on computing and plotting the EFM surface.

First we use the boundary equation (2.16) to find the values ofωs, in dependence ondCp, at

the edges of the bands and holes of the EFM surface. Since, the EFM components are closed

loops, we only need half of the boundary points; for example, those at theleft ends of the EFM

components. We obtain at most three such boundary values ofωs for each value ofdCp. Since,

Eq. (2.16) gives the(ωs, dCp)-projection of a single instance of the EFM surface, even for

dτ 6= 0 there are at most three boundary values ofωs for each value ofdCp. (Recall that, the

extra EFM components that appear in case ofdτ 6= 0 belong to 2π-shifted copies of the basic

instance of the EFM surface.) Typically, to construct the EFM surface, for dτ = 0, in chapter

3, we use 150 values ofdCp that are distributed uniformly along thedCp interval of length 2π.

However, the actual number and distribution of the points along thedCp-axis depends on the

type of the EFM surface and is specified by the user.

After finding ωs anddCp values of the starting points, we setC1p = 0 and use Eq. (2.5)–

(2.8) to compute initial values of the other state variables. SinceC1p = 0 is a guess, we use the

DDE-BIFTOOL correction routine to correct the values ofC1p and the state variables; during

this correction the values ofωs anddCp are kept fixed.

For the purpose of surface rendering, we want all the EFM componentsto be computed in

one direction, e.g. clockwise. Therefore, a second point for the continuation must be specified

in a careful way. We proceed by taking the first point (computed in the previous step), and apply

a small positive perturbation to the corrected value ofC1p . Next, we correct values of the state

variables of the second point to ones that correspond to the new value ofC1p . Having those two

points allows us to define and continue with DDE-BIFTOOL the respective branch of EFMs.

Moreover, all the branches are continued in the same direction — of increasing C1p . As the

result of the above procedure, we now have a set of uniformly distributed constantdCp sections

through the EFM surface. Each of these sections may consist of up to three EFM components.

Typically, the procedure is run from a script in MATLAB; after setting up all the script and

continuation accuracy parameters it takes around one hour to calculate allthe sections needed

to construct one instance of the EFM surface. Note that chosing the continuation accuracy to

be to low may lead to gaps in the computed sections of the EFM surface, or evento erroneous

switching between disjoint EFM components.

The rendering of the EFM surface proceeds as follows. We first assign all the computed

EFM components to the particular bands and island of the EFM surface. In case that band (or

island) of the EFM surface has holes or bulges, we divide the EFM components assigned to

this band (or island) into smaller groups which we further assign to local bands and islands.

To this end, we consider intervals of the 2π dCp-range within which the number of the EFM

components isdCp-independent. Note that this rendering step is directly associated with the

EFM surface type. For example, to construct EFM surface of typeBBh in figure 3.12 (c), we

considered separately the left bandB and and the right band with holeBh. Furthermore, we

A.3. Determining the stability of EFMs 125

divided the EFM components associated with the bandBh into four groups: the first group cor-

responded to a single band before the hole, two other groups corresponded to separate bands on

the left and right side of the hole, and the last group was assigned to a single band past the hole.

We then use these groups of the EFM components to generate separate meshes to represent

each local segment of the EFM surface. To fill gaps between segments ofthe EFM surface ob-

tained in this way, or to close discontinuities at the edges we compute some additional branches

of the EFMs as necessary.

For each of the EFM component group we generate a separate mesh. We first check that

each of them EFM components in the group is a single closed loop, and we trim it otherwise

(since,C1p is 2π-periodic it is possible to go along the EFM component more than once); here

m is the number of EFM components in the group. Next, we rewrite the values ofωs, dCp and

Ns of each point of an EFM component from the DDE-BIFTOOL solution branch structure

into a 3 × k matrix K, wherek is the number of points along the branch. Furthermore, we

apply constant arclength interpolation along each curve defined byK; in result we obtain a

new3 × l matrix L that consist ofl uniformly spaced mesh points. Finally, we split each of

them matricesL into rows, and construct three separatem× l mesh matrices, forωs, dCp and

Ns, which are used to plot the EFM surface; recall that,l is the number of mesh points (after

arclength interpolation) andm is the number of EFM components in the considered group.

On the basis of the above algorithm we developed a MATLAB function that automatically

generates the mesh for the group of the EFM components corresponding toa band or an island

of the EFM surface; the mesh is plotted with built-in MATLAB functions —surf andlight

with appropriate parameters. We process each band or island of the EFM surface separately,

and later plot them one by one in the same figure. A typical size of the mesh representing the

whole EFM surface (over a 4π dCp-range) in chapter 3 is about3 × 300 × 1500 points.

We used these computations to construct all of the EFM surfaces in chapters 2 and 3. To

show the EFM surface near singularity transitions as presented in section 2.3.3 in figures 2.12–

2.15 and in section 2.3.4 in figure 2.16 and figure 2.22, we first selected parameter values most

suitable for the computation of the EFM surface near the singularity transition.To this end, we

analysed the EFM surface bifurcation diagram in the(∆1, ∆2)-plane. To confirm our choice

of parameters we reviewed quite a large number of projections of the EFM surface onto the

(ωs, dCp)-plane near the transition; recall that, the projections are computed quickly with the

projection boundary equation (2.16). We then used the above procedure to calculate and render

the relevant parts of the EFM surface.

A.3 Determining the stability of EFMs

In chapter 3 we also showed figures of the EFM surface with stability information. These were

obtained as follows. We compute the stability of EFMs along EFM components with DDE-

126 Appendix A

BIFTOOL. However, computations of the EFM stability information are much morecomputa-

tionally expensive than computations of the EFMs themselves. More specifically, for optimally

chosen stability computation parameters, it takes around half a second to compute the stability

of a single EFM; that is around 20 times longer than the computation of the EFM itself. There-

fore, we compute information on EFM stability only for half of the EFM components used

to construct the EFM surface, that is, typically 75 over the 2π dCp-interval. Furthermore, we

limit the number of points along an EFM component for which we run computationsto 150.

In spite of these restriction, it takes around three hours to calculate the stability information for

a single instance of the EFM surface. Note that chosing the accuracy parameters for a stability

computation to be to low may lead to errors in the determination of stable EFM regions. To

ensure appropriate accuracy of their computation we checked the alignment of the stable seg-

ments of the EFM components with their known boundaries given by independently computed

saddle-node and Hopf bifurcation curves. Additionally, we also checked the consistency of the

saddle-node and Hopf bifurcation curves themselves.

The saddle-node and Hopf bifurcation curves are computed separatelyas two-parameter

continuations inC1p andC2

p ; starting points are conveniently chosen from saddle-node and Hopf

bifurcations found at the computed EFM components. Note that, to continue thesaddle-node

bifurcations it is necessary to use solutions of the transcendental system(2.3)–(2.8); namely

the transformed system Eqs. (A.3)-(A.6) always has an extra eigenvalue 0. On the other hand,

Hopf bifurcation curves are continued in the full DDE system Eqs. (A.3)-(A.6).

Once the EFM surface is rendered and the stability information is calculated, we overlay the

computed stable segments of the EFM components and the saddle-node and Hopf bifurcation

curves onto the EFM surface. This is done with a set of MATLAB scripts that trim and plot the

bifurcation curves modulo 2π, and with a set of modified DDE-BIFTOOL plotting routines;

these modifications allow us to plot the solution branches in three dimensions.

Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops

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