Bifurcation analysis of a semiconductor laser  with two filtered optical feedback loops                                    ...
Abstract    We study the solution structure and dynamics of a semiconductor laser receiving delayedfiltered optical feedbac...
Acknowledgements   First of all, I would like to thank my supervisors Prof. Bernd Krauskopf and Dr. SebastianM. Wieczorek....
“The mathematical description of the world depends on a delicate interplay between discrete and continuous objects. Discre...
Author’s Declaration    I declare that the work in this dissertation was carried out in accordance with the require-ments ...
Contents1   Introduction                                                                                   1    1.1   Mode...
CONTENTS                                                                                       ii4   Overall summary      ...
List of Tables 1.1   System parameters and their values. . . . . . . . . . . . . . . . . . . . . . . .   7 2.1   Notation ...
List of Figures 1.1   Sketch of a 2FOF semiconductor laser realized by coupling to an optical fiber       with two fibre Bra...
LIST OF FIGURES                                                                               v  2.4   Representation of t...
LIST OF FIGURES                                                                            vi  2.10 Boundary curves (orang...
LIST OF FIGURES                                                                            vii  2.15 Saddle transition SN ...
LIST OF FIGURES                                                                             viii  2.21 Additional types of...
LIST OF FIGURES                                                                              ix  2.29 Orange island in the...
LIST OF FIGURES                                                                             x  3.3   Dependence of the sta...
LIST OF FIGURES                                                                               xi  3.9   Projections of the...
xii                                                                           LIST OF FIGURES      3.17 Example of relaxat...
Chapter 1IntroductionSemiconductor lasers are very efficient in transforming electrical energy to coherent light. Co-herent...
2                                                                           Chapter 1. Introductionthe laser produces cons...
3                                                                                                 .              (a)      ...
4                                                                                           Chapter 1. Introduction       ...
1.1. Modelling the 2FOF laser                                                                     5introduced in detail in...
6                                                                       Chapter 1. Introductionrate equations             ...
1.1. Modelling the 2FOF laser                                                                    7       Parameter        ...
8                                                                        Chapter 1. Introductionrotation over any fixed ang...
1.2. Outline of the thesis                                                                      9start by considering, in ...
Chapter 2Classification of EFM structureIn this chapter we perform an extensive study of the external filtered modes (EFMs) ...
12                                                         Chapter 2. Classification of EFM structureof the EFM surface, an...
2.1. External filtered modes                                                                       13                      ...
14                                                    Chapter 2. Classification of EFM structurevariables of the EFMs can b...
2.1. External filtered modes                                                                          15be found as        ...
16                                                    Chapter 2. Classification of EFM structure     For the 2FOF laser the...
2.1. External filtered modes                                                                   172.1.2   EFM components for...
18                                                          Chapter 2. Classification of EFM structure                     ...
2.2. The EFM-surface                                                                             192.2 The EFM-surfaceThe ...
20                                                       Chapter 2. Classification of EFM structure                        ...
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops
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Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops

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The advances in the field of optical communication have been transforming the modern world for over 30 years. Main elements of this revolution are semiconductor lasers and optical fibers. Their performance is the main factor that limits operation and throughput of fiber-optic networks.

In my work I consider a semiconductor laser subject to filtered optical feedback from two filtering elements —- the 2FOF laser for short. The motivation for this study comes from applications where two filters are used to control and stabilise the laser output. I present an analysis of the basic solutions of the 2FOF laser as described by a mathematical model with delay due to the travel time in the two filter loops. In particular, I compute and represent the solutions as surfaces in a suitable space of parameters and phase space variables. This geometric approach allows me to perform a multi-parameter analysis of the 2FOF laser, which in turn provides comprehensive insight into the solution structure and dynamics of the system. As I show, compared to a laser with a single filtered optical feedback loop, the introduction of the second filter significantly influences the solution structure and, therefore, laser operation.

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

  1. 1. Bifurcation analysis of a semiconductor laser with two filtered optical feedback loops ´ Piotr Marek Słowinski 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
  2. 2. Abstract We study the solution structure and dynamics of a semiconductor laser receiving delayedfiltered optical feedback from two filter loops; this system is also referred to 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 and back. 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), and they 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 represent them as anEFM surface in (ωs , dCp , Ns )-space of frequency ωs , filter phase difference dCp , and popu-lation inversion Ns of the laser. The parameter dCp is a measure of the interference betweenthe two filter fields, and it is identified as a key to the EFM structure. To analyse how 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 theory to 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 surface and associated EFMstability regions on all filter and feedback loop parameters. Our theoretical results 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 to understand 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 of understandingthe 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.
  3. 3. Acknowledgements First of all, I would like to thank my supervisors Prof. Bernd Krauskopf and Dr. SebastianM. Wieczorek. Their support and encouragement made my PhD project and stay in Bristol agreat experience, and their wise guidance made an invaluable contribution to my research anddevelopment. Additional thanks go to Dr. Harmut Erzgräber. His published works, as well asprivate communication, helped me to establish my research project. Furthermore, I would liketo 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 WesternResearch Initiative funded my PhD research under studentship number 250, with support fromBookham Technology PLC (now Oclaro Inc.). The Bristol Center for Applied Nonlinear Math-ematics provided support during the write-up of this thesis. The Society for Industrial and Ap-plied Mathematics granted me a SIAM Student Travel Award to attend the SIAM Conferenceon 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 attendanceat the workshop and mini-conference on "Path Following and Boundary Value Problems" inMontréal, 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 researchactivities. During my stay at the Applied Nonlinear Mathematics research group I met manyinspiring people who became my friends and colleagues. For this I am especially grateful.
  4. 4. “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 asimpler 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
  5. 5. 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 Research Degree Programmesand that it has not been submitted for any other academic award. Except where indicated byspecific reference in the text, the work is the candidate’s own work. Work done in collabora-tion with, or with the assistance of, others, is indicated as such. Any views expressed in thedissertation are those of the author.Signed:Dated:
  6. 6. Contents1 Introduction 1 1.1 Modelling the 2FOF laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 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 for dτ = 0 . . . . . . . . . . . . . . . . . . 24 2.3.1 Dependence of the EFM components for fixed dCp = 0 on the detunings 25 2.3.2 EFM surface types with dCp -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 fixed Λ 41 2.4 Dependence of the EFM surface bifurcation diagram on the filter width Λ . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 EFM stability regions 65 3.1 Dependence of EFM stability on κ . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2 Dependence of EFM stability on Λ . . . . . . . . . . . . . . . . . . . . . . . . 70 3.3 Dependence of EFM stability on ∆ . . . . . . . . . . . . . . . . . . . . . . . . 77 3.3.1 Influence of hole creation on EFM stability . . . . . . . . . . . . . . . 79 3.3.2 Influence of SN -transition on EFM stability . . . . . . . . . . . . . . . 86 3.3.3 Influence of Sω -transition on EFM stability . . . . . . . . . . . . . . . 89 3.3.4 Influence of SC -transition on EFM stability . . . . . . . . . . . . . . . 94 3.4 Dependence of EFM stability on dτ . . . . . . . . . . . . . . . . . . . . . . . 97 3.5 Different types of bifurcating oscillations . . . . . . . . . . . . . . . . . . . . 100 i
  7. 7. CONTENTS ii4 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 . . . . . . . . . . . . . . . . . . . . . . . . 110Bibliography 113Appendices 121A How to construct the EFM surface 121 A.1 Dealing with the S1 -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
  8. 8. 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 number Cmin and the maximal number Cmax of EFM components (for suitable fixed dCp ) of the type; note that in all cases the number of EFM components is independent of dC p . . . . . 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 number Cmin and the maximal number Cmax of EFM components (for suitable fixed dCp ) 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 number Cmin and the maximal number Cmax of EFM components (for suitable fixed dCp ) of the type. . . . . . 48 3.1 Axes ranges for all the panels in figure 3.6 . . . . . . . . . . . . . . . . . . . . 81 iii
  9. 9. 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 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 of Ω(ωs ) with the diagonal; also shown are the intersection 1 2 points (black dots) with the envelope. Here Cp = 0, Cp = π/3, ∆1 = −0.1, ∆2 = 0.05, κ1 = 0.05, κ2 = 0.025, Λ1 = Λ2 = 0.005, τ1 = 500 and τ2 = 400. 13 2.2 1 Projection of EFMs branches onto the (ωs , Ns )-plane (a) and onto the (ωs , Cp )- plane (b). The open circles are the starting points for three different types of branches. The blue branch is the EFM component for dCp = 0, the green 1 branches are for constant Cp , 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. . . . . . . . . . . . . . . . . . . . . 18 2.3 1 Representation of the EFM surface in (ωs , Ns , Cp )-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 figure 2.2. 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 1 (c) show how the EFM branches for constant Cp and for constant ωs , respec- tively, arise as intersection curves of fixed sections with the EFM surface. . . . 20 iv
  10. 10. LIST OF FIGURES v 2.4 Representation of the EFM surface of figure 2.3 in (ωs , Ns , dCp )-space. Panel (a) shows one fundamental 2π 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 constant ωs 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 by dCp = 0 and dCp = 0.9π, respectively. Panels (b1) and (c1) show the corresponding envelope (grey curves) given by (2.9). The black 1 solution curve of (2.3) inside it is for Cp = 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 for dCp = 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 and 0.2, respec- tively; here Λ1 = 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 for dCp = 0. From (a) to (f) Λ takes the values Λ = 0, Λ = 0.001, Λ = 0.01, Λ = 0.06, Λ = 0.12 and Λ = 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 for Λ = 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). . . . . . . . . . . . . . . 30 2.9 The EFM surface in (ωs , dCp , Ns )-space showing case B for ∆1 = ∆2 = 0 (a), and showing case B BB for ∆1 = 0.16, ∆2 = −0.16 (b), where Λ = 0.015. 31
  11. 11. LIST OF FIGURES vi 2.10 Boundary curves (orange or grey) in the (∆1 , ∆2 )-plane for Λ = 0.01 for 61 equidistant values of dCp 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 of dCp , as is indicated by the labelling with symbols B and B; 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 the la- 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] and dCp ∈ [−π, π]. . . . . . . . . . . . . . . . . . . . . . . 33 2.12 Minimax transition M 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 transition M 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. Here Λ = 0.01, ∆1 = 0.4, and ∆2 = 0.28 in (a) and ∆2 = 0.28943 in (b). . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.13 Saddle transition SC 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 transition SC 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). . . . . . . . . . . . . . . . . . . . 37 2.14 Saddle transition Sω 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 the ωs -direction. Panels (a2)– (c2) show the corresponding projection onto the (ωs , dCp )-plane of the entire EFM surface; the local region where the transition Sω 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). . . . . . . . . . . . . . . . . . . 38
  12. 12. LIST OF FIGURES vii 2.15 Saddle transition SN of the EFM-surface in (ωs , dCp , Ns )-space, where two sheets that lie on top of each other in the Ns 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 the N -direction. Panels (a2)–(c2) show the corre- sponding projection onto the (ωs , dCp )-plane of the entire EFM surface; the local region where the transition SN 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 = 0.0.11 in (a), ∆2 = 0.11085 in (b) and ∆2 = 0.1115 in (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.16 Cubic tangency C of the EFM-surface in (ωs , dCp , Ns )-space, where a part of the surface (a1) becomes tangent to a plane {dCp = const} (b1) and then develops a bulge (c1). The unfolding of the cubic tangency into two dCp -folds can be seen clearly in the projections onto the (ωs , dCp )-plane in panels (a2)– (c2). Here Λ = 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 for Λ = 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 transitions M (orange curves), SC (blue curves), Sω (green curves) and SN (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 panels ωs ∈ [−0.3, 0.3] and dCp ∈ [−π, π]. . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.19 Projection of the EFM-surface onto (ωs , dCp )-plane for Λ = 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 of SC transition, (red) curves of SN transition, and (black) curves Ca and (grey) 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. . . . . . . 46
  13. 13. 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 and Λ see Table 2.3. In all panels ωs ∈ [−0.3, 0.3] and dCp ∈ [−π, π]. . . 47 2.22 Global manifestation of local saddle transition SC 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. Here Λ = 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 point DCN C on the curve C of cubic tangency, from which the (red) curve SN and the (blue) curve SC 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 point DCM ω on the curve C of cubic tangency, from which the (orange) curve M and the (green) curve Sω 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 of SC transition, (green) curves of Sω transition, and SC 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 for Λ = 0.01 as figure 2.17, and panel (b) is for Λ = 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 values Λ = 0.01, Λ = 0.015, Λ = 0.06, Λ = 0.098131, Λ = 0.1 and Λ = 0.13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.27 Sketch of EFM surface bifurcation diagram near the boundary {∆2 = −1} of the (∆1 , ∆2 )-square in the transition through Λ = ΛC . Panels (a1)–(a3) show the transition involving the (black) curve Ca 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 for Λ = 0.1 in the (∆1 , ∆2 )-plane with regions of non-banded EFM surface types; compare with figure 2.26 (e). . . . . . . . . . . . . . . . . 56
  14. 14. 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 values Λ = 0.1, Λ = 0.145, Λ = 0.166 and Λ = 0.179; compare panel (a) with figure 2.26 (e). . . . . . . . . . . . . . . . . . . . . . . 58 2.30 The EFM surface (a) for κ = 0.05, ∆1 = ∆2 = 0, Λ = 0.015, and τ1 = 500 and τ2 = 600 so that dτ = 100, and its intersection with the planes defined by dCp = 0 and dCp = −π; compare with figure 2.5. Panels (b)–(e) show the EFM-components for dCp = 0, dCp = −π/2, dCp = −π and dCp = −3π/2, 1 respectively; the blue dots are the EFMs for Cp as given by (2.12). . . . . . . . 60 2.31 Solution curves of the transcendental equation (2.3) and corresponding EFM 1 components for dCp = 0, where the dots show the actual EFMs for Cp =0; here κ = 0.05, ∆1 = ∆2 = 0, Λ = 0.015, τ1 = 500, and dτ = 200 in panels (a) and dτ = 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 type hBB for dτ = 0 (a1) and its EFM components for dCp = −1.6π (a2), and the corresponding sheared EFM surface for dτ = 230 (b1) and its EFM components for dCp = −1.6π (b2). Here κ = 0.05, ∆1 = 0.13, ∆2 = −0.1, Λ = 0.01, and τ1 = 500. . . . . . . . . . . . . . . . . . . . 62 3.1 Dependence of the EFM surface on the feedback rate κ (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 width Λ = Λ1 = Λ2 (as in- dicated); here ∆1 = ∆2 = 0, κ = 0.01 and dτ = 0. Light grey regions with black envelopes are projections, for five different values of Λ, of the EFM surface onto the (ωs , Ns )-plane. . . . . . . . . . . . . . . . . . . . . . . . . . 70
  15. 15. 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 and dτ = 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 of Ns and ωs 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 and dτ = 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 transition SC , 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 fundamental 2π-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 the dCp -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 in figure 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 the 2π interval of the EFM surface that is shifted by π with respect to the fun- damental 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 the Ns direction. . . . . . . . . . . . 82 3.8 The EFM surface with stability information for filters detunings ∆1 = 0.024 and ∆2 = 0. Here κ = 0.01, Λ = 0.005 and dτ = 0; curves and regions are coloured as in figure 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
  16. 16. 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 and dτ = 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. Here κ = 0.01, Λ = 0.005 and dτ = 0; curves and regions are coloured as in figure 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.11 Projections of the EFM surface with stability information onto the (ωs , dCp )- 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 and dτ = 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 and dτ = 0; curves and regions are coloured as in figure 3.1. . . . . . . . . . . . . . . . . . . . . 90 3.13 Projections of the EFM surface with stability information onto the (ωs , dCp )- 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). Here κ = 0.01, Λ = 0.005 and dτ = 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 and dτ = 0; curves and regions are coloured as in figure 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.15 Projections of the EFM surface with stability information onto the (ωs , dCp )- 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 and dτ = 0; curves and regions are coloured as in figure 3.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.16 Projections of the EFM surface with stability information onto the (ωs , dCp )- plane for increasing delay time in the second feedback loop τ2 = 506 (a), τ2 = 514 (b), τ2 = 562 (c) and τ2 = 750 (d); here τ1 = 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
  17. 17. 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 and dτ = 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 intensity IL and the frequency φL = dφL /dt of the laser field, the ˙ intensity IF 1 and the frequency φF 1 = dφF 1 /dt of the first filter field, and the ˙ intensity IF 2 and the frequency φF 2 = dφF 2 /dt of the second filter field. Note the different time scales for ROs and FOs. . . . . . . . . . . . . . . . . . . . . 101 4.1 1 Regions in the (Cp , dCp )-plane with different numbers of coexisting EFMs, as indicated by the labelling. Panel (a) shows the regions on a fundamental 1 2π-interval of Cp , while panel (b) shows it in the covering space (over several 1 2π-intervals of Cp ). 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 and dτ = 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 for dCp = π, ∆1 = 0.050, ∆2 = −0.049, and panel (b) is for dCp = −π, ∆1 = 0.036, ∆2 = −0.037; furthermore, κ = 0.01, Λ = 0.005 and dτ = 0. Stable segments of the EFM-components (green) are bounded by the Hopf bi- furcations (red dots) or by the saddle-node bifurcation (blue dots). The actual 1 1 stable EFMs for Cp = 1.03π (a) and Cp = 0.9π (b) are the black full circles; open circles are unstable EFMs. The EFM components in panel (a) correspond to a constant dCp -section through the EFM surface in figure 3.14 (b), and those in panel (b) to a constant dCp -section through the EFM surface in figure 3.12 (d).112
  18. 18. Chapter 1IntroductionSemiconductor 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 generatedby an electrical pump current. The light is reflected by semitransparent mirrors that form thelaser cavity, and is amplified by stimulated emission during multiple passages through an am-plifying semiconductor medium. The output light exits through one (or both) of the semitrans-parent mirrors [33, 46]. Semiconductor lasers are small (about 1 millimetre long and severalmicrometres wide), can easily be mass produced and are used in their millions in every dayapplications — 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 asmirrors and lenses) and via coupling to other lasers. Depending on the exact situation, opticalfeedback 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 to chaos [73],torus break-up [70], and a boundary crisis [69] have been identified. See [33, 38] as entrypoints to the extensive literature on the possible dynamics of lasers with optical feedback. 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-enters the laser [43]. How-ever, other types of laser systems with optical feedback have been considered, including laserswith 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 associatedwith a delay time τ 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), externaloptical paths of a few centimetres lead to considerable delay times that cannot be ignored. Asa consequence, an optical feedback created by an external cavity allows the laser to operate atvarious compound-cavity modes; they are referred to as continuous-wave (cw) states because 1
  19. 19. 2 Chapter 1. Introductionthe 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 overalldynamics, even when they are unstable. For example, the typical dynamics of a COF laserwith irregular drop-outs of the power has been attributed to trajectories that pass closely nearcw-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 reflectedlight 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 and the feedbackstrength. Moreover, FOF is a form of coherent feedback, meaning that the phase relationshipbetween outgoing and returning light is also an important parameter. The interest in the FOFlaser is due to the fact that filtering of the reflected light allows additional control over thebehaviour of the output of the system by means of choosing the spectral width of the filter andits 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 stablefrequency tuning of lasers for the telecommunications applications [9]. The single FOF laser system has recently been the subject of a number of experimentaland theoretical studies [15, 18, 22, 23, 27, 28, 31, 51, 64, 74, 76, 77]. Here, we assume that asolitary laser (i.e. without feedback) emits light of constant intensity and frequency Ω0 . It hasbeen shown that single FOF can improve the laser performance [6, 23], but it can also inducea wide range of more complicated dynamics. Its cw-states are called external filtered modes(EFMs) [74], and they lie on closed curves, called EFM-components, in the (ωs , Ns )-plane ofthe lasing frequency ωs (relative to the solitary laser frequency Ω0 ) and inversion Ns of thelaser (the number of electron-hole pairs). The EFM-components are traced out by the EFMs asthe phase of the electric field of the filter (relative to the phase of the laser field), called here thefeedback phase Cp , is varied. An analysis in [28] with dependence on the spectral width of thefilter, 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 thesolitary laser frequency and one around the filter central frequency. A stability and bifurcationanalysis of EFMs in [17] shows that the FOF laser is very sensitive to changes in feedbackphase Cp . Furthermore, the filter parameters Λ and ∆ have a big influence on the possible(non steady-state) dynamics [16, 19]. Importantly, in a FOF laser one can observe not onlythe well-known relaxation oscillations, but also so-called frequency oscillations where only thefrequency of the laser oscillates while its intensity remains practically constant [24]. In lightof the strong amplitude-phase coupling of semiconductor lasers, the existence of frequencyoscillations is somewhat surprising, and they are due to an interaction with the flanks of thefilter transmittance profile [16]. An experimental study of the influence of feedback phase Cpand frequency detuning ∆ on the single FOF laser dynamics can be found in [19]. The limiting
  20. 20. 3 . (a) Laser Optical fiber Grating 1 Grating 2 (b) Laser 2 κ2 F2 (t, τ2 , Cp ) BS 1 κ1 F1 (t, τ1 , Cp ) E(t), N (t) Ω, α, P, T ISO ISO ISO ISO Filter 1 Filter 2 F1 (t) F2 (t) . Λ1 , ∆1 Λ2 , ∆2Figure 1.1. Sketch of a 2FOF semiconductor laser realized by coupling to an optical fiber with twofibre Bragg gratings (a), and by two (unidirectional) feedback loops with Fabry-Pérot filters (b); otheroptical elements are beam splitters (BS) and optical isolators (ISO).cases of small and large Λ and ∆ have been considered in [28, 31, 74]. In all these studies thefilter transmittance has a single maximum defining its central frequency; the FOF problem forperiodic filter transmittance with multiple maxima and minima was considered in [64]. Given the large number of parameters and the transcendental nature of the equations forthe 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 usedto simplify the rate equations of the FOF laser and to investigate the injection laser limit Λ →0. In [28] dependance of the EFM structure on Λ and ∆ is analysed by reduction of thetranscendental equation for their frequencies to a fourth-degree polynomial. Finally, the studypresented in [31] explores the transition of the FOF laser from the injection laser limit to theCOF laser limit, by analysing the limit cases Λ → 0 and Λ → ∞. Due to limitations of theanalytical approaches, the single FOF laser is analysed mainly by means of numerical methods.In particular, popular techniques include: root finding, numerical integration and numericalcontiunation. Root finding is used, for example, in [74, 77], to solve transcendental equationfor the frequencies of EFMs. Numerical integration provides means to compare output ofthe model with experimental time series; see, for example, [30, 25, 49]. Finally, numericalcontinuation allows for very detailed bifurcation analysis of the investigated system; as wasperformed for example, in [17]. In a number of applications, such as the design of pump lasers for optical communication
  21. 21. 4 Chapter 1. Introduction . 100% 0% Fiber Bragg grating T RANSMITTANCE R EFLECTANCE E Λ Fabry–P´ rot filter e 0% 100% . ∆ frequencyFigure 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 filterwidth Λ is defined as the full width at half maximum.systems, the requirements on stable and reliable laser operation at a specific frequency are sostringent (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 is to employ FOF fromtwo 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 ofthe study presented here. The main idea is that the second filter provides extra frequency controlover the laser output. In [4] it has been shown that a second filtered feedback loop may indeedimprove the beam quality. Moreover, in [21] an experimental setup has been realized and it wasshown that the laser may show complicated dynamics as well; however, such dynamics havenot yet been investigated further. With the focus on enhanced stability, industrial pump sourceswith enhanced wavelength and power stability performance due to a 2FOF design are availabletoday [52]. The 2FOF laser has also been considered recently for frequency switching [9] andfor sensor applications [59, 60]. The filtered feedback of the 2FOF laser can be realized in two ways: either by reflectionfrom 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 sense that the overall spectralcharacteristics of the filtering is the same. More specifically, a fibre Bragg grating (FBG) hasa peak in the reflectance at its central frequency, while a Fabry-Pérot filter (FP) has a peak inthe transmittance at its central frequency; see figure 1.2. There are some important practicaldifferences between the two setups in figure 1.1 that are discussed in more detail in section 1.1.Nevertheless, the 2FOF laser in either form can be modelled by rate equations for the complex-valued electric field E inside the laser, for the real-valued population inversion N inside thelaser, and for the complex-valued electric fields F1 and F2 inside the two filters. The 2FOFlaser is hence described by a delay differential equation (DDE) model, equations (1.1)–(1.4)
  22. 22. 1.1. Modelling the 2FOF laser 5introduced 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 times τ1 and τ2 .1.1 Modelling the 2FOF laserThe 2FOF design with fibre Bragg gratings (FBG) as in figure 1.1 (a) is the one that has beenemployed in industrial pump sources [52], and it is also the one considered in [21, 59, 60]; seealso the analysis of FOF from a FBG in [50]. The main advantage is that FBGs are simple andcheap to manufacture in a fiber at desired locations; furthermore, apart from the need to couplethe light into the fiber without any direct reflections (to avoid COF), no additional opticalelements are required so that the device is relatively simple; see figure 1.1 (a). The downside isthat, once it is imprinted into an optical fiber, a FBG cannot be modified. Furthermore, the twofilters are not independent feedback loops. When the two FBGs operate at different frequencies(as in the actual devices) then they are transparent to each other’s central frequency, meaningthat the first FBG only slightly weakens the light reflected from the second FBG and one mayassume that there are no direct interactions between these two filters. However, when bothFBGs operate very close to the same frequency then the feedback from the second FBG isalmost completely blocked, so that the laser receives feedback only from one filter. Anotherissue is that the light reflected from a FBG is due to the interaction with the entire grating, whichmeans 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 frequency and the feedbackphase via changes of the feedback loop length at a sub-wavelength scale. For these reasons it isdifficult to perform controlled experiments with the FBGs setup over large ranges of parametersof interest. The experimental setup in figure 1.1 (b) is less practical in industrial applications, but itallows 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 asindependent parameters. Furthermore, the system can be investigated for any combination offilter 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 loop has been usedsuccessfully in studies of the single FOF laser [18, 19]. In particular, it has been shown that thesystem is modelled very well by a rate equation model where the filters are assumed to have aLorentzian 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
  23. 23. 6 Chapter 1. Introductionrate equations dE = (1 + iα)N (t)E(t) + κ1 F1 (t) + κ2 F2 (t), (1.1) dt dN T = P − N (t) − (1 + 2N (t))|E(t)|2 , (1.2) dt dF1 1 = Λ1 E(t − τ1 )e−iCp + (i∆1 − Λ1 )F1 (t), (1.3) dt dF2 2 = Λ2 E(t − τ2 )e−iCp + (i∆2 − Λ2 )F2 (t). (1.4) dtThe well established assumptions here are that the delay times τ1 and τ2 are larger than thelight 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 anadditional equation for the field of the second filter. Equation (1.1) describes the time evolution of the complex-valued slowly-varying elec-tric field amplitude E(t) = Ex (t) + iEy (t) of the laser. Equation (1.2) describes normalisedpopulation inversion N (t) within the laser active medium. In (1.1)–(1.2) the material prop-erties of the laser are described by the linewidth enhancement factor α (which quantifies theamplitude-phase coupling or frequency shift under changes in population inversion [32]), theratio T between the population inversion and the photon decay rates, and the dimensionlesspump parameter P . Time is measured in units of the inverse photon decay rate of 10−11 s.Throughout, we use values of the semiconductor laser parameters from [17] that are given intable 1.1. The two FOF loops enter equation (1.1) as feedback terms κ1 F1 (t) and κ2 F2 (t) with nor-malised feedback strengths κ1 and κ2 [67, p. 93] of the normalised filters fields F1 (t) andF2 (t). In general, the presence of a filter in the system gives rise to an integral equation forthe filter field. However, in the case of a Lorentzian transmittance profile as assumed here,derivation of the respective integral equation yields the description of the filter 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 ifeedback, we have the feedback strength κi , the delay time τi and the feedback phase Cp of thefilter field, which is accumulated by the light during its travel through the feedback loop. Hence, iCp = Ω0 τi . Owing to the large difference in time scales between the optical period 2π/Ω0 and ithe delay time τi , one generally considers τi and Cp as independent parameters. Namely, as hasbeen justified experimentally [19, 30], changing the length of the feedback loop on the optical
  24. 24. 1.1. Modelling the 2FOF laser 7 Parameter Meaning Value Laser P pump parameter 3.5 α linewidth enhancement factor 5 T inversion decay rate / photon decay rate 100 Feedback loops κ1 first loop feedback strength from 0.01 to 0.05 κ2 second loop feedback strength from 0.01 to 0.05 τ1 first loop round-trip time 500 τ2 second loop round-trip time 500 to 800 1 Cp first loop feedback phase 2π-periodic 2 Cp 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 0.82 Λ1 first filter spectral width from 0.0 to 0.5 Λ2 second filter spectral width from 0.0 to 0.5 Table 1.1. System parameters and their values. iwavelength scale of nanometres changes Cp , but effectively does not change τi . Two different istrategies for changing Cp have been used experimentally: in [19] this is achieved by changingthe 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, in turn,affect Ω0 . The optical properties of the filters are given by the detunings ∆i of their central frequenciesfrom the solitary laser frequency, and by their spectral widths Λi , defined as the frequencywidth at half-maximum (FWHM) of the (Lorentzian) transmittance profile. In this study weconsider the filter detunings ∆1 and ∆2 as independent parameters. Furthermore, we keep bothfeedback strengths as well as both filter widths equal, so that throughout we use κ := κ1 = κ2and Λ := 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 thathave also been considered for the stabilisation of the laser output. First, when the spectralwidth of only one filter is very large then the laser effectively receives feedback from an FOFloop and from a COF loop; see, for example, [3, 13]. Second, when the spectral widths ofboth filters are very large then one is dealing with a laser with two external COF loops; see, forexample, [55, 63] and the discussion in section 2.4.2. System (1.1)–(1.4) shares symmetry properties with many other systems with coherentoptical feedback. Namely, the system has an S 1 -symmetry [29, 34, 40] given by simultaneous
  25. 25. 8 Chapter 1. Introductionrotation over any fixed angle of E and both filter fields F1 and F2 . This symmetry can beexpressed by the transformation (E, N, F1 , F2 ) → Eeiβ , N, F1 eiβ , F2 eiβ (1.5)for any 0 ≤ β ≤ 2π. In other words, solutions (trajectories) of (1.1)–(1.4) are not isolated butcome in S 1 -families. In particular, the EFMs introduced in the next chapter are group orbitsunder this symmetry, and this fact leads to an additional zero eigenvalue [17, 34], which needsto 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, F1 eibt , F2 eibt into (1.1)–(1.4) and dividing through by an exponential factor, thereference frequency b becomes an additional free parameter. A suitable choice of b ensures thatone is considering and computing an isolated solution; see [29, 56] and Appendix A for details. i There is also a rather trivial symmetry property: the feedback phases Cp are 2π-periodicparameters, which means that they are invariant under the translation i i Cp → Cp + 2π. (1.6)This property is quite handy, because results can be presented either over a compact fundamen- ital interval of width 2π or on the covering space R of Cp ; see also [17].1.2 Outline of the thesisThis thesis is organised in the following way. In chapter 2 we discuss the structure of thecontinuous-wave solutions — called the EFMs — of the rate equation model (1.1)–(1.4). Wefirst introduce in section 2.1 the EFMs and show how they can be uniquely determined bytheir frequencies, which are given by solutions of a transcendental equation; furthermore weshow that the envelope of the transcendental equation for the frequencies of the EFMs is key tounderstanding the structure of the EFMs. In section 2.1.1 we show the correspondence betweenthe EFM components of the 2FOF laser and the EFM components known for the single FOFlaser. In section 2.2 we introduce the EFM surface, the key object of our studies. The EFMsurface provides a geometric approach to the multi-parameter analysis of the 2FOF laser. Inthis way, it allows for comprehensive insight into the dependence of the EFMs on the feedbackphases of both feedback loops. Moreover, we show that the EFM surface of the 2FOF laser isa natural generalisation of the EFM component of the single FOF laser. To study the dependence of the EFM surface on filter and feedback loop parameters insection 2.3, we introduce a classification of the EFM surface into different types. For clarityof exposition we successively increase the number of parameters involved in our analysis. We
  26. 26. 1.2. Outline of the thesis 9start by considering, in section 2.3.1, how the EFM components depend on the filter detuningsand filter widths. Next, in sections 2.3.2–2.4 we perform a comprehensive analysis for theEFM 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 tofive codimension-one singularity transitions: through extrema and saddle points, and througha 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 diagramof the EFM surface in the plane of detunings of the two filters. Finally, in section 2.4 we showhow the EFM surface bifurcation diagram in the plane of filter frequency detunings changeswith the filter widths. We finish our analysis of the dependence of the EFM surface on filter andfeedback loops parameters in section 2.5, where we show that a non-zero difference betweenthe 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 thestability 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 we show that regionsof stable EFMs are bounded by saddle-node and Hopf bifurcations; moreover, we study howthese stable EFM regions change with the increasing feedback strength. Next, in section 3.2 wepresent how the regions of stable EFMs are affected by changing the filter widths. Both thesesections show that, although topologically the EFM surface remains unchanged, the stability ofEFMs changes substantially. In section 3.3 we show in what way the regions of stable EFMsare influenced by the singularity transitions of the EFM surface (discussed in section 2.3.3) thatoccur as the modulus of detunings of the two filters is increased. The last parameters that weanalyse in section 3.4 in terms of the effect on EFM stability are again the delay times: themain effect is shearing of the EFM stability regions on the EFM surface. Finally, in section 3.5we briefly describe what kinds of periodic solutions originate from the Hopf bifurcations thatbound regions of stable EFMs. An overall summary of the thesis can be found in chapter 4, where we also discuss thepossibility of an experimental confirmation of our findings. Finally, Appendix A gives moredetails of how the EFM surface has been rendered from data obtained via continuation runswith the package DDE-BIFTOOL.
  27. 27. Chapter 2Classification of EFM structureIn this chapter we perform an extensive study of the external filtered modes (EFMs) of the2FOF laser as modelled by the DDE model (1.1)–(1.4). The analysis shows that the secondfilter 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 2FOFsystem, the number of (disjoint) EFM components depends on the exact phase relationshipbetween the two filters. When the filter loops have the same delay times, the interferencebetween the filter fields can give rise to at most three EFM components — one around thesolitary 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 maylead to any number of EFM components. The EFM structure with dependence on the different system parameters is quite compli-cated and high-dimensional. To deal with this difficulty we present our results in the formof EFM surfaces in suitable three-dimensional projection spaces. These surfaces are renderedfrom 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 filterloops, 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 fordifferent 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 frequency Ω0 ) on the number of EFM components (for a representativevalue of the spectral width of the filters), which provides a connection with and generalisationof the single FOF case. The result is a bifurcation diagram in the (∆1 , ∆2 )-plane whose openregions 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 withdifferent numbers of corresponding EFM components. The boundary curves in the (∆1 , ∆2 )-plane correspond to singularity transitions (for example, through saddle points and extrema) 11
  28. 28. 12 Chapter 2. Classification of EFM structureof the EFM surface, and they can be computed as such. In a next step we also show how thebifurcation diagram in the (∆1 , ∆2 )-plane changes with the spectral width of the filters. Thischapter 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 thatof 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 differentdelay times of the filter loops, which yields the geometric result that there may be an arbitrarynumber 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 EFMsurface on several key parameters and provides a comprehensive and compact way of under-standing the structure of the EFM solutions.2.1 External filtered modesThe basic solutions of the 2FOF laser correspond to constant-intensity monochromatic laseroperation with frequency ωs (relative to the solitary laser frequency Ω0 ). These solutions arethe external filtered modes. Mathematically, an EFM is a group orbit of (1.1)–(1.4) under theS 1 -symmetry (1.5), which means that it takes the form (E(t), N (t), F1 (t), F2 (t)) = Es eiωs t , Ns , Fs ei(ωs t+φ1 ) , Fs ei(ωs t+φ2 ) . 1 2 (2.1) 1 2Here, Es , Fs and Fs are fixed real values of the amplitudes of the laser and filter fields, Nsis a fixed level of inversion, ωs is a fixed lasing frequency, and φ1 , φ2 are fixed phase shiftsbetween 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   κ1 Λ1 sin φ1 + tan−1 (α) κ2 Λ2 sin φ2 + tan−1 (α) Ω(ωs ) = − 1 + α2  +  , (2.3) 2 2 2 2 Λ1 + (ωs − ∆1 ) Λ2 + (ωs − ∆2 )and i ωs − ∆ i φi = ωs τi + Cp + tan−1 . (2.4) ΛiEquation (2.2) is a transcendental and implicit equation that allows one to determine all pos-sible frequencies ωs of the EFMs for a given set of filter parameters. More specifically, the
  29. 29. 2.1. External filtered modes 13 . 0.3 Ω(ωs ) 0 −0.3 . −0.2 ωs −0.1 0 0.05 0.1Figure 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 of Ω(ωs ) with 1the diagonal; also shown are the intersection points (black dots) with the envelope. Here Cp = 0, 2Cp = π/3, ∆1 = −0.1, ∆2 = 0.05, κ1 = 0.05, κ2 = 0.025, Λ1 = Λ2 = 0.005, τ1 = 500 andτ2 = 400.sought frequency values ωs of the 2FOF laser can be determined from (2.2) numerically byroot 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 first and the secondfilter, respectively. If one of the κi 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 theformulation of (2.2) is that it has a nice geometric interpretation: Ω(ωs ) is a function of ωs that 1 2oscillates between two fixed envelopes. More precisely, when Cp or Cp are changed over 2πthe graph of Ω(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 function Ω(ωs ) and the diagonal (the straight line through the originwith slope 1); see also [74]. Once ωs is known, the corresponding values of the other state
  30. 30. 14 Chapter 2. Classification of EFM structurevariables of the EFMs can be found from   κ1 Λ1 cos(φ1 ) κ2 Λ2 cos(φ2 ) Ns = −  + , (2.5) Λ1 2 + (ωs − ∆1 )2 2 Λ2 + (ωs − ∆2 ) 2 P − Ns Es = , (2.6) 1 + 2Ns 1 Es Λ 1 Fs = , (2.7) Λ1 2 + (ωs − ∆1 )2 2 Es Λ 2 Fs = . (2.8) Λ2 2 + (ωs − ∆2 )2This means that an EFM is, in fact, uniquely determined by its value of ωs . Furthermore, it isuseful to consider the envelope of Ω(ωs ) (grey curves) so that Figure 2.1 represents all the rele-vant geometric information needed to determine and classify EFMs. Notice that in this specificexample the EFMs are separated into three groups. The diagonal intersects the region boundedby the envelope in three disjoint intervals where frequencies ωs of EFMs may lie; these in-tervals correspond to three different EFM components as is discussed in section 2.1.1. Asfigure 2.1 suggests, EFMs are created and lost in saddle-node bifurcations when an extremumof the (black) graph passes through one of the boundary points (black dots) as a parameter (for 1example, Cp ) is changed. This geometric picture is very similar to that for the single FOF laser [28], but there isan important difference. The envelope of Ω(ωs ) for the FOF laser is found by consideringthe extrema of the sine function (in (2.3) for, say, κ2 = 0). It turns out that the envelopefor the single FOF laser is described by a polynomial of degree four, whose roots are theboundary 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 sufficientsince they appear in a sum. Hence, we also need to consider mixed terms resulting from thesummation; with the use of standard trigonometric formulae, the equation for the envelope can
  31. 31. 2.1. External filtered modes 15be found as κ2 Λ2 1 1 κ2 Λ2 2 2Ωe (ωs ) = ± 1 + α2 + 2 + Λ2 + (ωs − ∆1 )2 Λ2 + (ωs − ∆2 )2 1 2 1 ωs −∆2 ωs −∆1 1/2 2κ1 κ2 Λ1 Λ2 cos Cp − Cp + ωs (τ2 − τ1 ) + tan−1 Λ2 − tan−1 Λ1 . 2 2 Λ2 1 + (ωs − ∆1 ) Λ2 2 + (ωs − ∆2 ) (2.9)Indeed, when one of the κi is set to zero then (2.9) reduces to the fourth-order polynomialdescribing the envelope of the single FOF laser in [28]. However, for general values of theparameters, (2.9) is a transcendental equation, and not a polynomial of degree six as one mighthave hoped; nevertheless, by means of (2.9) the envelope Ωe (ωs ) can be plotted readily. The transcendental nature of (2.9) means that the study of the EFM structure of the 2FOFlaser is a considerable challenge. As is shown here, the key is to find a suitable geometricviewpoint that allows one to understand the dependence of the EFMs on the different filterloop parameters. A first observation is that (2.9) depends on the differences 2 1 dCp := Cp − Cp and dτ := τ2 − τ1 ,which we will hence consider as parameters in what follows; note that dCp is 2π-periodic aswell.2.1.1 EFM componentsIt 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 foundfor different values of feedback phase Cp , whilst the other parameters of the system are fixed.More specifically, when Cp is changed, EFMs are born in saddle-node bifurcations, then moveover the respective EFM components in the direction of increasing ωs , and finally disappearagain in saddle-node bifurcations. From an experimental point of view, EFM componentsare quite natural objects that can be measured as groups of EFMs whose frequencies varywith the feedback phase Cp ; see [19]. For the single FOF laser one finds either one or twoEFM components, depending on the properties of the filter. Intuitively, one expects one EFMcomponent 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, theexact dependence on the filter properties can be studied for the single FOF laser by consideringthe roots of a polynomial of degree four that arises from the equation for the envelope of theEFMs; see [28] for details.
  32. 32. 16 Chapter 2. Classification of EFM structure For the 2FOF laser the situation is more complicated. Intuitively, one may think that nowup to three EFM components may occur in the (ωs , Ns )-plane: one centred around the solitarylaser frequency and two more around the central frequencies of the two filters. However, thisintuition is not correct, and we will show that one may in fact have any number of EFM com-ponents. Physically, the reason for this vastly more complicated EFM structure of the 2FOFlaser is the interference between the two filter fields, which can be interpreted as giving rise to acomplicated ‘effective’ filter profile. Mathematically, the reason behind the more complicatedEFM 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. Sincethe 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 frequency ωs one can compute the values of the other 1 2EFM quantities Es , Ns , Fs , Fs , φ1 and φ2 . In particular, the inversion Ns can be expressed asa function of ωs as κ2 Λ2 κ2 Λ2Ns + (ωs − αNS )2 = 2 1 1 + 2 2 2 + Λ1 2 + (ωs − ∆1 )2 Λ2 + (ωs − ∆2 )2 ωs −∆2 ωs −∆1 2κ1 κ2 Λ1 Λ2 cos dCp + ωs dτ + tan−1 Λ2 − tan−1 Λ1 . Λ1 2 + (ωs − ∆1 )2 Λ2 2 + (ωs − ∆2 )2 (2.10)From this quadratic expression we can conclude that for any ωs there are either no, one or twosolutions for Ns . In particular, any EFM component is a smooth closed curve that consistsof two branches, one with a higher and one with a lower value of Ns , which connect at twopoints where (2.10) has exactly one solution. EFM components in the (ωs , Ns )-plane can becomputed from the implicit transcendental equations (2.2) and (2.10) by root solving, ideallyin combination with numerical continuation. An alternative approach is to find and then 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 packageDDE-BIFTOOL [11]. Additionaly, by using DDE-BIFTOOL we can obtain stability informa-tion on the EFMs.
  33. 33. 2.1. External filtered modes 172.1.2 EFM components for two identical filtersThe starting point of our study of the EFM structure is the special case that the two filters are 1 2identical, apart from having differing feedback phases Cp and Cp . Hence, we now set κ := κ1 = κ2 , ∆ 1 = ∆2 , Λ := Λ1 = Λ2 , τ1 = τ 2 .The EFMs for this special case are given by the EFMs of a corresponding single FOF laserwith effective feedback strength dCp κeff = 2κ cos (2.11) 2and effective feedback phase eff 1 2 Cp = Cp + Cp /2.In other words, we obtain a non-trivial reduction of the 2FOF laser to the FOF laser, where thefeedback phase difference dCp arises as a natural parameter that controls the effective feedbackstrength κeff as a result of interference between the two filter fields. One extreme case is thatof constructive interference when dCp = 0 so that κeff = 2κ. The other extreme is the case ofdestructive interference when dCp = π and κeff = 0. Hence, by changing dCp we can ‘switchon’ or ‘switch off’ the overall filter field that the laser sees. 1 2 Clearly, which EFMs one finds depends on both feedback phases Cp and Cp . Branches 1 2of EFMs are obtained by specifying a single condition on Cp and Cp , while keeping all other 1 2parameters fixed. The easiest option is to continue EFM curves in, say, Cp while keeping Cpconstant. Another option is to require that the frequency ωs remains fixed. Note that for theabove choices the feedback phase difference dCp changes along the branch of EFMs. We now consider EFM components of the 2FOF, which we define as the branches of EFMs 1 2that one finds when the feedback phases, Cp or Cp are changed while the feedback phasedifference dCp is fixed. This definition is the appropriate generalisation from the single FOFlaser [28]. The underlying idea is that the value of dCp determines the interference of the lightfrom the two filtered feedback loops and, hence, an important property of the overall feedbackthe laser sees. In the simplest case of two identical feedback loops fixing dC p results in thefixed effective feedback strength κeff . However, as we will see, our notion of EFM componentsfor 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 1onto the (ωs , Cp )-plane, respectively. Here we fixed κ = κ1 = κ2 = 0.05, Λ := Λ1 =Λ2 = 0.015, τ1 = τ2 = 500, and consider the case where both filters are resonant with the
  34. 34. 18 Chapter 2. Classification of EFM structure 5 . 0.02 (a) 1 Cp (b) Ns π 0 0−0.02 −5. −0.1 0 ωs 0.1 −0.1 0 ωs 0.1 1Figure 2.2. Projection of EFMs branches onto the (ωs , Ns )-plane (a) and onto the (ωs , Cp )-plane (b).The open circles are the starting points for three different types of branches. The blue branch is the EFM 1component for dCp = 0, the green branches are for constant Cp , 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 parametersare 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 setsof 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 theEFM component for dCp = 0 if one insists that one of the EFMs (the top open circle) has afrequency of ωs = 0. The outer blue curve in figure 2.2 (a) is a single EFM-component that connects all EFMs; itis indeed exactly the EFM component of the single FOF laser with a feedback strength of κeff = 12κ; see [28]. When Cp is increased by 2π, while keeping dCp = 0, each EFM moves alongthe blue EFM component to the position of its left neighbour. Hence, the EFM-component 1can be calculated either by the continuation of all EFMs over the Cp -range of [0, 2π], or by thecontinuation of a single EFM over several multiples of 2π. The green curves are the branches of 2 1EFMs that one obtains by changing Cp while keeping Cp constant. Notice that green branchesconnect an EFM at the top with one at the bottom of the blue EFM-component; the exception 2is the branch near the origin of the (ωs , Ns )-plane, which connects three EFMs. When Cpis increased by 2π the respective EFMs on the green branch exchange their positions in aclockwise direction. Finally, the red branches in figure 2.2 (a) are the result of continuation of 1 2EFM solutions for (1.1)–(1.4) in Cp and Cp while ωs is kept constant; hence, these branchesappear as straight vertical lines that start at the respective EFM. Figure 2.2 (b) shows the exact 1same branches but now in projection onto the (ωs , Cp )-plane. In this projection, the blue EFM 1branch ‘unwraps’ as a single curve that oscillated in Cp ; this property is characteristic and canbe found, more generally, for lasers with delayed feedback or coupling [28, 34, 61]. In the 1(ωs , Cp )-plane the red and the green branches are perpendicular to each other. Furthermore, 1the image is invariant under a 2π translation along the Cp -axis.
  35. 35. 2.2. The EFM-surface 192.2 The EFM-surfaceThe discussion in the previous section shows that the dependence of the EFMs on the feedback 1 2phases Cp and Cp (when all other parameters are fixed) gives rise to different one-parameterfamilies of EFM branches, depending on the conditions one poses. In other words, one isreally dealing with a surface of EFMs in dependence on the two feedback phases, which isrepresented by any of the three families of EFM branches we discussed. Motivated by thequestion of how many EFM components there are for the 2FOF laser, and in line with thewell-accepted representation of the external cavity mode structure for other laser systems withdelay [34, 61], it is a natural choice to represent this surface by the EFM values of ωs and Nsin combination with one additional parameter. 1 A first and quite natural choice is to consider the EFM surface in the (ωs , Ns , Cp )-space. 1This representation stresses the influence of an individual feedback phase, here Cp , which isconvenient to make the connection with previous studies in [17, 35]. Figure 2.3 shows the EFMsurface for the case of two identical filters in this way. Panel (a) shows a grey semitransparenttwo-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-translatedcopies of this basic element. Note that the 2nπ-translated copies connect smoothly at the openends of the surface element shown in panel (a). The element of the EFM surface was renderedfrom computed one-dimensional EFM branches for fixed ωs ; selection of these branches isshown as the red curves in figure 2.3 (a). Almost all red EFM branches are closed loops thatconnect two points, each on the blue branch. An exception is the central red EFM branchfor 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 theconditions that both sine functions in (2.3) vanish, which means that 1 ∆1 Cp = π + tan−1 + tan−1 (α) , (2.12) Λ1 2 ∆2 Cp = π + tan−1 + tan−1 (α) . (2.13) Λ2The starting points for the calculations of the red EFM branches are taken from the maximalblue curve, which corresponds to the maximal EFM component for dCp = 0; compare with 1figure 2.2 (b). It forms the helix-like curve in (ωs , Ns , Cp )-space that is shown in figure 2.3 (a) 1over one 2π interval of Cp . What is more, the shown part of the EFM surface is a fundamental 1unit under the translational symmetry of Cp that contains all the information. This means thatthe entire EFM surface is obtained as a single smooth surface from all of the 2nπ-translatedcopies of the unit in figure 2.3 (a). Notice that the shown part of the EFM surface is tilted in the 1Cp direction. More specifically, for negative ωs the red EFM branches are shifted toward higher
  36. 36. 20 Chapter 2. Classification of EFM structure . 0.02 (a) Ns 0 0.1 ωs −0.02 0 15 1 Cp /π 0 −0.1 −15 (c) ωs 0 (b)Ns Ns 0 0 1 Cp /π ωs 1 Cp /π . 0 1Figure 2.3. Representation of the EFM surface in (ωs , Ns , Cp )-space; compare with figure 2.2. Panel(a) shows one fundamental element of the EFM surface (semitransparent grey); superimposed are theEFM branches from figure 2.2. The entire EFM surface is a single smooth surface that is obtain byconnecting all 2nπ-translated copies of the surface element shown in panel (a). Panels (b) and (c) show 1how the EFM branches for constant Cp and for constant ωs , respectively, arise as intersection curves offixed sections with the EFM surface. 1 1values of Cp , whereas for positive ωs they are shifted toward lower values of Cp ; compare 1with the projection onto the (ωs , Cp )-plane in figure 2.2 (b). Notice that for the maximal andminimal possible values of ωs , the red EFM branches contract to just single points on the blueEFMs branch. 1 The representation of the EFM surface in (ωs , Ns , Cp )-space in figure 2.3 is, in effect, the 1three-dimensional analogue of the representation of the EFM branches in the (ωs , Cp )-plane 1in figure 2.2 (b). In particular, the two sets of EFM branches for fixed Cp and fixed ωs (green

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