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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 ...

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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- 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. 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. 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. “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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
- 37. 2.2. The EFM-surface 21and red curves) arise naturally as intersection curves with planar sections. This is illustrated inﬁgure 2.3 (b) and (c). Panel (b) shows a cutaway through a four 2nπ-translated copies of thefundamental unit of the EFM surface, and illustrates the single red EFM branch for ωs = 0, aswell as 2nπ-translated copies of closed red EFM branches for ωs = −0.06. Panel (c) shows 1with a different cutaway image of the EFM surface how the green EFM branches for ﬁxed Cparise as disjoint intersection curves with a ﬁxed planar section; shown are all the green EFMsbranches from ﬁgure 2.2 (a) that also appear on the EFM surface in ﬁgure 2.3 (a). Note, thatthe two sections in ﬁgure 2.3 (b) and (c) are perpendicular to each other. 1 While the representation of the EFM surface in (ωs , Ns , Cp )-space is quite natural, ithas the disadvantage that an EFM component actually corresponds to a non-closed curve that 1 1runs in the Cp direction over the EFM surface in (ωs , Ns , Cp )-space. To be able to study theEFM components more directly, we now consider the ‘compactiﬁed’ representation of the EFMsurface in (ωs , Ns , dCp )-space; in other words, in this space, the EFM surface is considered asthe one-parameter family (parametrised by dCp ) of actual EFM components themselves, whicharise naturally as closed curves in the (ωs , Ns )-plane by intersection with planar sections givenby dCp = const. This representation of the EFM surface is used in ﬁgure 2.4 for the case of two identicalﬁlters. Shown is one compact fundamental part of the EFM-surface in the (ωs , Ns , dCp )-spacefor dCp ∈ [−π, π], with superimposed red, green and blue EFMs branches from ﬁgure 2.2.The entire EFM surface consists of all 2nπ-translated copies of this compact surface, whichtouch at the points (ωs , Ns , dCp ) = (0, 0, (2n + 1)π) where n ∈ Z. This is shown in panel(b) in projection onto the (ωs , dCp )-plane for dCp ∈ [−3π, 3π], where the EFM-surface isrepresented by grey shading and the coloured curves are the EFMs branches as before. Notethat the green EFM branches are no longer perpendicular to the red EFMs branches. Noticefurther that the EFM surface is not tilted with respect to the dCp -axis. The common points fordCp = (2n + 1)π correspond physically to the situation when both ﬁlter ﬁelds cancel eachother due to destructive interference, so that the the only EFM of the (1.1)–(1.4) is the solitarylaser solution. Three EFM branches pass through these points: the green branch that connectsthree EFMs in ﬁgure 2.2 (a) and two red EFMs branches given by dCp Ns (dCp , ωs = 0) = ±2κ cos cos ΨdCp , (2.14) 2where ∆ ΨdCp = π + tan−1 − tan−1 (α) . (2.15) ΛFigure 2.4 (c) illustrates with a cutaway view how the red EFM branches for ﬁxed ωs arise; thesection for ωs = 0 shows the two cosines from (2.14). Note that the red curves are perpen-dicular to the blue EFM components; the maximal blue EFM component appears as the largest
- 38. 22 Chapter 2. Classiﬁcation of EFM structure . 0.02 (a) Ns 0 1 0 dCp /π -1 0.1 -0.02 ωs 0 -0.1 3 (b) (c) dCp π Ns 0 0 dCp /π 0 -3 ωs. -0.1 0 ωs 0.1Figure 2.4. Representation of the EFM surface of ﬁgure 2.3 in (ωs , Ns , dCp )-space. Panel (a) showsone fundamental 2π interval of the EFM surface (semitransparent grey); superimposed are the EFMbranches from ﬁgure 2.2. The entire EFM surface consists of all 2nπ-translated copies of this compactsurface, which touch at the points (ωs , Ns , dCp ) = (0, 0, (2n + 1)π); panel (b) shows this in projectionof the surface onto the (ωs , dCp )-plane. Panel (c) illustrates how the EFM branches for constant ωs andthe outer-most EFM component for dCp = 2nπ arise as intersection curves with planar sections.closed curve for ﬁxed dCp = 2nπ. Figure 2.5 shows in more detail how the EFM-component depends on dCp . In practice,an EFM component for a given dCp = const is computed by continuation of an EFM in the 1 2 1parameters Cp and ωs , while setting Cp = Cp + dCp . Panel (a) shows the EFM surfacefor dCp ∈ [−π, π] intersected with the two planes given by dCp = 0 and dCp = 0.9π,respectively. The corresponding EFM components arise from the shape of the envelope given
- 39. 2.2. The EFM-surface 23 0.02 . (a) Ns 0 -0.02 -1 dCp π 0 ωs 0.1 0.9 0 -0.1 0.5 (b1) 0.5 (c1)Ω(ωs ) Ω(ωs ) 0 0 −0.5 −0.5 −0.1 0 ωs 0.1 −0.1 0 ωs 0.1 0.02 0.02 (b2) (c2) Ns Ns 0 0 −0.02 −0.02. −0.1 0 ωs 0.1 −0.1 0 ωs 0.1Figure 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 anddCp = 0.9π, respectively. Panels (b1) and (c1) show the corresponding envelope (grey curves) given by 1(2.9). The black 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.by (2.9); it is shown in panels (b1) and (c1) together with the solution curve of (2.3) one obtainswhen conditions (2.14) and (2.15) are satisﬁed. The EFM components themselves, with theseEFMs on them, are shown in panels (b2) and (c2). Figure 2.5 illustrates that changing dCp
- 40. 24 Chapter 2. Classiﬁcation of EFM structureresults in a change in the EFM component, as well as the number of EFMs. For the shown caseof two identical ﬁlters with equal delay times we can say more: here the change of the EFMcomponent is entirely due to the effective feedback rate κeﬀ as described by (2.11). In otherwords, the EFM surface is composed of the EFM components of the corresponding single FOFlaser with κeﬀ as determined by dCp . The EFM component is maximal for the constructive-interference case dCp = 0, and it shrinks when dCp is changed. This also means that fewerEFMs exist; compare ﬁgure 2.5 (b1) and (c1). Finally, as the case of entirely destructiveinterference for dCp = π is approached, the EFM component shrinks down to a point, whichis the degenerate EFM corresponding to the unique solitary laser mode. We conclude from this section that the EFM surface in (ωs , Ns , dCp )-space has emergedas the main object of study. It represents the EFM structure of the 2FOF laser in a convenientgeometric way; in particular, EFM components can easily be obtained as planar slices for ﬁxeddCp .2.3 Classiﬁcation of the EFM surface for dτ = 0So far we have only considered the special case that the two ﬁlters are identical and not detunedfrom the laser; furthermore, the two delay times are equal. We now address the question howthe EFM surface changes as the system is moved away from this special point in the space ofparameters. In this section we consider the case that the two ﬁlter loops have the same delaytime, that is, dτ = 0. The inﬂuence of a difference in delay times is the subject of section 2.5. We start by considering in section 2.3.1 how the EFM components depend on the detunings∆1 and ∆2 for ﬁxed dCp = 0. We then proceed to study how the EFM surface itself changeswith ∆1 and ∆2 . This can be represented for a ﬁxed ﬁlter width Λ of both ﬁlters by an EFMsurface bifurcation diagram in the (∆1 , ∆2 )-plane, where each open region corresponds to adifferent type of EFM surface. In section 2.3.2 we ﬁrst consider the case that the EFM sur-face gives rise to a dCp -independent number of EFM components. We then introduce in sec-tion 2.3.3 ﬁve codimension-one singularity transitions — through extrema and saddle points,and through a cubic tangency (with respect to dCp = const) — that change the EFM surfacein terms of how many EFM components it induces when sliced for ﬁxed dCp . These ﬁve sin-gularity transitions induce a division of the (∆1 , ∆2 )-plane into open regions of different EFMsurface type. We ﬁrst, discuss EFM surface types that arise due to transitions through extremaand saddle points, and next EFM surface types that arise due the cubic tangency. Finally, sec-tion 2.4 shows how the EFM surface bifurcation diagram in the (∆1 , ∆2 )-plane changes withthe ﬁlter width Λ. Throughout this section we make use of the fact that the EFM surface can be representedby its projection onto the (ωs , dCp )-plane. Here we make use of the fact that, due to (2.10),
- 41. 2.3. Classiﬁcation of the EFM surface for dτ = 0 25this surface consists of two sheets in (ωs , dCp , N )-space over the (ωs , dCp )-plane, except atthe boundary of its projection. The boundary itself is given by (real-valued) solutions of Λ2 + (ωs − ∆1 )2 1 Λ2 + (ωs − ∆2 )2 2 dCp = ± cos −1 2κ1 κ2 Λ1 Λ2 2 ωs κ2 Λ2 κ2 Λ2 1 1 2 2 (2.16) × − 2 − 2 (1 + α2 ) Λ1 + (ωs − ∆1 )2 Λ2 + (ωs − ∆2 )2 ωs − ∆ 2 ωs − ∆ 1 − ωs dτ − tan−1 + tan−1 . Λ2 Λ1This equation is derived from (2.9), and it has the advantage that it does not depend on anyof the state variables of (1.1)–(1.4). Hence, in contrast to computing the EFM surface itself,which requires the continuation of EFMs in parameters, its projection onto the (ωs , dCp )-planecan be computed directly from (2.16). Notice also that the projection in the (ωs , dCp )-plane isindependent of the choice of the additional state variable that one chooses for visualisation ofthe EFM-surface.2.3.1 Dependence of the EFM components for ﬁxed dCp = 0 on the detuningsWe now ﬁxed dCp = 0 and consider the detunings ∆1 and ∆2 as free parameters. We ﬁrstconsider an intermediate ﬁxed ﬁlter width Λ = Λ1 = Λ2 = 0.015 of both ﬁlters; moreover,τ1 = τ2 = 500 and the other parameters are as given in Table 1.1. In this situation, onemay ﬁnd one, two or three EFM components in the FOF laser. Because both the top and thebottom part of the envelope given by (2.9) intersect the diagonal, the EFM component arounda solitary laser frequency ωs = 0 is always present. In the presence of the two ﬁlters, one mayﬁnd additional EFM components, which exist around the central frequencies of the ﬁlters asgiven by ∆1 and ∆2 ; note that (2.9) has two obvious extrema for ωs = ∆1 and ωs = ∆2 . Eachadditional EFM component comes with its own pair of saddle-node bifurcation points, givengeometrically by the condition that an extremum of (2.9) intersects the diagonal ωs = Ω(ωs );see ﬁgure 2.1. Figure 2.6 shows EFM components for nonzero detunings and ﬁxed dCp = 0; more specif-ically, we ﬁx ∆1 = 0.2 and increase ∆2 in (a) to (d). Each case is shown by two panels, oneshowing envelope and solution curve, and the other the corresponding EFM-components andEFMs. One can imagine this situation as a single FOF laser with detuning ∆1 , which is subjectto the inﬂuence of the second ﬁlter loop. If the detuning ∆2 of the second ﬁlter is below orabove a critical value then the inﬂuence of the second ﬁlter is negligible and one observes onlytwo EFM-components. However, for intermediate values of ∆2 this is no longer the case. For
- 42. 26 Chapter 2. Classiﬁcation of EFM structure . 0.5 0.5 (a1) (b1)T (ωs ) T (ωs ) 0 0 −0.5 −0.5 −0.2 0 ωs 0.2 −0.2 0 ωs 0.2 0.06 0.06 (a2) (b2) Ns Ns 0 0 −0.05 −0.05 −0.2 0 ωs 0.2 −0.2 0 ωs 0.2 0.5 0.5 (c1) (d1)T (ωs ) T (ωs ) 0 0 −0.5 −0.5 −0.2 0 ωs 0.2 −0.2 0 ωs 0.2 0.06 0.06 (c2) (d2) Ns Ns 0 0 −0.05 −0.05. −0.2 0 ωs 0.2 −0.2 0 ωs 0.2Figure 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, respectively; here Λ1 = Λ2 = 0.015, τ1 = τ2 = 500 andthe other parameters are as given in Table 1.1.∆2 = −0.2 as in ﬁgure 2.6 (a1) and (a2), the two detunings have equal magnitude but oppositesigns. Hence, there are three EFM components, the central one near ωs = 0 being quite small.When ∆2 is increased, the left-most EFM component moves towards larger ωs and then mergeswith the central one; see ﬁgure 2.6 (b1) and (b2) for ∆2 = 0. As ∆1 is increased further so that
- 43. 2.3. Classiﬁcation of the EFM surface for dτ = 0 27it is well in between ωs = 0 and ωs = 0.2, we ﬁnd again a situation with three EFM compo-nents; see ﬁgure 2.6 (c1) and (c2) for ∆2 = 0.158. Finally, for ∆1 = ∆2 = 0.2 we again ﬁndonly two EFM components; see ﬁgure 2.6 (d1) and (d2). Now the amplitudes of envelope andsolution curve are much higher, which is due to maximal constructive interference between thetwo ﬁlter ﬁelds since dCp = 0. Note that the ﬁlters are quite narrow (Λ1 = Λ2 = 0.015), namely much narrower than thedetuning ∆1 = 0.2 between the laser and the ﬁrst ﬁlter. As a result, in ﬁgure 2.6 (a), (c) and (d)the EFM-component around the solitary laser frequency has an elliptical shape, as known fromCOF systems [61, 46]. This is the case because ﬂanks of the both ﬁlters transmittance proﬁlesnears ωs = 0 are quite ﬂat, meaning that all frequencies around the solitary laser frequencyare fed back with approximately the same low feedback strength. Therefore, this situationresembles the effect of weak COF [46]. In ﬁgure 2.6 (b) for ∆2 = 0, on the other hand, theEFM-component around the solitary laser frequency has a ‘bulge’ — much as one ﬁnds inﬁgure 2.2 (a) — which is the result of the frequency selective feedback from the second ﬁlter. For a ﬁxed value of the width Λ = Λ1 = Λ2 of both ﬁlters, one obtains a bifurcationdiagram in the (∆1 , ∆2 )-plane that consists of regions where the 2FOF laser system has one,two or three EFM-components for dCp = 0. The regions are bounded by curves that can becomputed by means of numerical continuation. Namely, the number of the EFM componentschanges when two saddle-node points (black dots in ﬁgure 2.1) come together. This happenswhen the envelope given by Ωe (ωs ) from (2.9) is tangent to the diagonal. Hence, the conditionsthat are continued in ∆1 and ∆2 to obtain the boundary curves are, dΩe (ωs ) Ω(ωs ) = ωs and = 1. dωs Figure 2.7 shows the bifurcation diagram in the (∆1 , ∆2 )-plane for six different values ofΛ. Open regions are labelled with the number of EFM components that one ﬁnds for dCp = 0for the respective values of the detunings ∆1 and ∆2 . Notice the two symmetries of the panelsof ﬁgure 2.7, given by reﬂection across the diagonal ∆1 = ∆2 , and reﬂection across theanti-diagonal ∆1 = −∆2 (which we already encountered in ﬁgure 2.6 (a)). The boundarycurves are coloured orange and grey for presentation purposes only. Figure 2.7 (a) showsthe limiting special case of Λ = 0, which corresponds to an inﬁnitely narrow ﬁlter so thatEFM-components consist of single EFMs. The bifurcation diagram in the (∆1 , ∆2 )-plane forthis case can be obtained analytically by substituting in Eq. (2.9) ωs = ∆1 and ωs = ∆2respectively. The coordinates of the vertical and horizontal lines in ﬁgure 2.7 (a) are given by √ √κ 1 + α2 and the end points at the diagonal by 2κ 1 + α2 ; compare with [28]. Figure 2.7(b) for Λ = 0.001 shows how the limiting case unfolds for Λ > 0. The black parts of curves inﬁgure 2.7 (a) open up to reveal new open regions. As Λ is increased, the bifurcation diagramdeforms, but does initially not change qualitatively; see ﬁgure 2.7 (c) for Λ = 0.01, which is
- 44. 28 Chapter 2. Classiﬁcation of EFM structure . 0.8 (a) 0.8 (b) 1 2 2 1 1 2 1 2 1 2 ∆2 ∆2 ¡ ¡ 2 3 3 2 2 3 2 3¡ 2 3 ¡3 0 0 1 2 2 1 3 3 o 3 2 3 3 2 2 ! ¡ 2 ¡3 2 ¡ ¡ 1 2 1 2 2 1 1 2 1 2 1 −0.8 −0.8 −0.8 0 ∆1 0.8 −0.8 0 ∆1 0.8 0.8 (c) 0.8 (d) 1 2 1 2 1 1 2 1 1 ∆2 3 ∆2 ¡ 2 ©¡ 2 3 2 2 3 2 ¡ 2 0 1 2 1 2 1 0 1 2 1 2 1 2 2 3 2 3 2 ! ¡ 2 ¡ 2 ¡ 2 1 2 1 2 1 1 1 2 1 −0.8 −0.8 −0.8 0 ∆1 0.8 −0.8 0 ∆1 0.8 0.8 (e) 0.8 (f) 2 2 2 2 ∆2 ts ∆2 t t s rrr t 3 0 1 e 0 1 e e e t t kt k t 2 2 2 2 −0.8 −0.8 . −0.8 0 ∆1 0.8 −0.8 0 ∆1 0.8Figure 2.7. Regions in the (∆1 , ∆2 )-plane with a one, two or three EFM components of the 2FOF laserfor dCp = 0. From (a) to (f) Λ takes the values Λ = 0, Λ = 0.001, Λ = 0.01, Λ = 0.06, Λ = 0.12 andΛ = 0.14.the case from ﬁgure 2.8 (a). However, as Λ is increased further, the bifurcation diagram doeschange qualitatively because the different curves move sufﬁciently relative to one another to
- 45. 2.3. Classiﬁcation of the EFM surface for dτ = 0 29‘disentangle’; see ﬁgure 2.7 (d) for Λ = 0.06, where there are now no longer regions with threeEFM components. For larger values of Λ, the curves cease to extend to inﬁnity and are nowconﬁned to a compact region of the (∆1 , ∆2 )-plane; as ﬁgure 2.7 (e) for Λ = 0.12 illustrates,this implies that there is now a single large and connected region with one EFM component.For even larger values of Λ, there are six non-overlapping curves, each bounding a small regionwhere one ﬁnds two EFM components; see ﬁgure 2.7 (f) for Λ = 0.14. When Λ is increasedeven further, the small regions disappear and one ﬁnds a single EFM component for any pointin the (∆1 , ∆2 )-plane. Physically the ﬁlters are now so wide that they do not provide sufﬁcientdifferentiation of the feedback light in frequency; hence, the 2FOF laser is effectively a COFlaser. Figure 2.8 (a) is a three-dimensional bifurcation diagram in (∆1 , ∆2 , Λ)-space that rep-resents the entire transition of the bifurcation diagram in the (∆1 , ∆2 )-plane for dCp = 0 asthe ﬁlter width Λ is changed. Shown are surfaces (coloured orange and grey as in ﬁgure 2.7)that divide this parameter space into regions with one, two or three EFM-components. The bi-furcation diagrams in ﬁgure 2.7 are horizontal cross sections through ﬁgure 2.8 (a); the shown(semitransparent) cross section for Λ = 0.01 yields ﬁgure 2.7 (c). Notice that the grey surfacesin ﬁgure 2.8 (a) extend to higher values of ∆ than the orange surfaces, which can be explainedas follows. For dCp = 0 the two ﬁlter ﬁelds interfere constructively, so that for ∆1 ≈ ∆2the amplitude of the solution curve of (2.3) is larger than that around a single ﬁlter. Hence,a second EFM-component around the central frequencies of both ﬁlters may exist for highervalues of Λ, and the maximum of the grey surfaces is exactly at the diagonal where ∆1 = ∆2 .Above all surfaces (for sufﬁciently large Λ) the 2FOF laser is effectively a COF laser and onlyone EFM-component exists. Figures 2.8 (b) and (c) show that the three-dimensional bifurcation diagram in panel (a)brings out important special cases where the 2FOF laser reduces to the single FOF laser in anontrivial way. Figure 2.8 (b) shows the two-dimensional bifurcation diagram in the (∆1 , Λ)-plane for ∆2 = 0.82. Also shown in light grey is the corresponding bifurcation diagram ofthe single FOF laser (with detuning ∆1 ) that one obtains for the limit that ∆2 = ∞ (when thesecond ﬁlter does not inﬂuence the system any more); compare with [28, Fig. 3(a)]. The close-ness of the two bifurcation diagrams in ﬁgure 2.8 (b) shows that for ∆2 ≥ 0.82 the inﬂuenceof the second ﬁlter is already so small that it does not inﬂuence the number of EFM compo-nents. Figure 2.8 (c) shows the projection onto the (∆1 , Λ)-plane of the diagonal section for∆1 = ∆2 through the surfaces in ﬁgure 2.8 (a). Along the diagonal the 2FOF laser reducesto the single FOF laser with the effective parameters as given by (2.11); in fact, the boundarycurve in ﬁgure 2.8 (c) is exactly that from [28, Fig. 3(a)] for the corresponding effective param-eters, namely κeﬀ = 2κ1 for dCp = 0. Since this curve scales linearly with κ [28], it is exactlytwice the size as the light grey curve in ﬁgure 2.8 (b).
- 46. 30 Chapter 2. Classiﬁcation of EFM structure . 0.21 (a) Λ 0 0.82 0.82 0 0 ∆2 ∆1 -0.82 -0.82 0.13 0.21 (b) (c) Λ Λ 1 2 1 2 1 1 2 1 2 1 0 0. −0.35 0 ∆1 0.35 −0.6 0 ∆1 = ∆2 0.6 Figure 2.8. Panel (a) shows surfaces (orange and grey) that divide the (∆1 , ∆2 , Λ)-space into regions with one, two and three EFM-components of the 2FOF laser for dCp = 0; in the shown (semitranspar- ent) 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). 2.3.2 EFM surface types with dCp -independent number of EFM components We now turn to the question of how the EFM surface itself changes with ∆1 and ∆2 , where we ﬁrst consider the case that the EFM surface is such that the associated number of EFM components is independent of dCp . Figure 2.9 shows two examples of EFM surfaces in (ωs , dCp , Ns )-space for the special case that ∆1 = −∆2 . Panel (a) shows three copies of the fundamental unit of the EFM surface
- 47. 2.3. Classiﬁcation of the EFM surface for dτ = 0 31 . 0.02 (a) 0.04 (b) Ns Ns 0 0 −0.02 −0.04 3 4 dCp 0.1 dCp 0.2 π 0 ωs π 0 ωs 0 0. −3 −0.1 −4 −0.2Figure 2.9. The EFM surface in (ωs , dCp , Ns )-space showing case B for ∆1 = ∆2 = 0 (a), andshowing case B BB for ∆1 = 0.16, ∆2 = −0.16 (b), where Λ = 0.015.for ∆1 = ∆2 = 0. The EFM surface is connected at dCp = π and its integer multiples;hence, it is a single connected component that extends over any 2π interval of dCp , and oneﬁnds a single EFM component for any dCp ; compare with ﬁgure 2.4 (a) and (b). Figure 2.9 (b)is for sufﬁciently large (opposite) detunings when the EFM surface consists of three disjointconnected components that extend over any 2π interval of dCp . Hence, for any dCp one ﬁndsthree EFM components, one around the solitary laser frequency ωs = 0 and the other twoaround the central frequencies ωs = ∆1,2 of the ﬁlters; compare with ﬁgure 2.6 (a). Noticethat, since ∆1 = −∆2 , the central connected component around ωs = 0 is also connected atdCp = π and its integer multiples. We now turn to the question of where in the (∆1 , ∆2 )-plane one can ﬁnd EFM surfacesthat have a dCp -independent number of EFM components. To investigate this question, weconsider how the bifurcation diagram in ﬁgure 2.7 (c), for the representative value of Λ =0.01, with regions of one, two or three EFM components in the (∆1 , ∆2 )-plane for a ﬁxeddCp = 0 changes when dCp is varied over the interval [−π, π]. In the process the boundarycurves between regions move in the (∆1 , ∆2 )-plane and then return to their original positions.Figure 2.10 shows the resulting curves (again in orange and grey) in the (∆1 , ∆2 )-plane forΛ = 0.01, where the dCp -interval [−π, π] is covered in 60 equidistant steps. As a functionof dCp the curves now cover overlapping (orange and grey) regions in the (∆1 , ∆2 )-plane,meaning that in these regions the number of EFM components depends on the value of dCp . By contrast, in the open white regions in the (∆1 , ∆2 )-plane of ﬁgure 2.10 the number ofEFM is independent of the value of dCp . This means that the projection of the EFM surfaceconsists of either one, two or three bands that extend over the entire dCp -interval [−π, π]. Intotal there are four such types (up to symmetry) of EFM surface, and their representatives interms of projections of the EFM surface onto the (ωs , dCp )-plane are shown in ﬁgure 2.11;
- 48. 32 Chapter 2. Classiﬁcation of EFM structure . 0.55 B BB B BB B ∆2 BBB BB B BB BB BB BBB 0 B BB B BB B BB B BB BB B BB BB BB B B BB B BB B −0.55 −0.55 0 ∆1 0.55.Figure 2.10. Boundary curves (orange or grey) in the (∆1 , ∆2 )-plane for Λ = 0.01 for 61 equidistantvalues of dCp from the interval [−π, π]; compare with ﬁgure 2.7 (c). In the white regions the 2FOFlaser has one, two or three EFM components independently of the value of dCp , as is indicated by thelabelling with symbols B and B; representatives of the four types of EFM components can be found inﬁgure 2.11.for the respective values of ∆1 and ∆2 see Table 2.1. Each such band in the projection isrepresented in ﬁgures 2.10 and 2.11 by the letter B. Furthermore, B denotes the band aroundthe frequency of the solitary laser, ωs = 0, and is later referred to as the central band. It playsa special role because it corresponds to a part of the EFM surface that always extends over theentire dCp -interval [−π, π]; moreover, B can be found for any value of detuning Λ of the twoﬁlters, even in the COF limit of an inﬁnitely wide ﬁlter; see the discussion of ﬁgure 2.8 (a) insection 2.3.1.
- 49. 2.3. Classiﬁcation of the EFM surface for dτ = 0 33 . (a) (b) (c) (d) B BB BBB B BB.Figure 2.11. The four simple banded types of EFM-surface of the 2FOF laser in the labelled regions ofﬁgure 2.10, represented by the projection (shaded) onto the (ωs , dCp )-plane; the blue boundary curvesare 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]. name Cmin Cmax panel ∆1 ∆2 Λ B 1 1 ﬁgure 2.11 (a) 0.080 -0.270 0.01 BB 2 2 ﬁgure 2.11 (b) 0.140 -0.270 0.01 BBB 3 3 ﬁgure 2.11 (c) 0.210 0.130 0.01 B BB 3 3 ﬁgure 2.11 (d) 0.210 -0.130 0.01Table 2.1. Notation and parameter values for the types of EFM-surface in ﬁgure 2.11. The second andthird column show the minimal number Cmin and the maximal number Cmax of EFM components (forsuitable ﬁxed dCp ) of the type; note that in all cases the number of EFM components is independent ofdC p . The notation we use here is more speciﬁc than simply counting the number of EFM com-ponents; for example, it distinguishes the case B BB, where the ﬁlters are detuned to both sidesof the laser frequency ωs = 0, from the cases BBB and BB B (which are related by symme-try), where both ﬁlters are detuned on the same side of the laser frequency. Notice also thattype BB differs physically from type B B in terms of whether the second band B is towardshigher or lower frequencies with respect to the laser frequency (that is, for negative or positiveω). Nevertheless, these two types are related to each other mathematically, because they areeach other’s images under the symmetry transformation (∆1 , ∆2 ) → (−∆1 , −∆2 ). Indeedany type that is not symmetric itself comes as a symmetric pair, and it is sufﬁcient to show onlyone type of such a pair in ﬁgure 2.11. Note that the two EFM surfaces in ﬁgure 2.9 are of typeB and B BB, respectively.2.3.3 Transitions of the EFM surfaceConsider a path in the (∆1 , ∆2 )-plane that takes one from a white region to another whiteregion, where the number EFM components does not depend on the value of dCp . It is clearfrom ﬁgure 2.10 that any such path necessarily leads through (at least one) (grey or orange)
- 50. 34 Chapter 2. Classiﬁcation of EFM structureregion where the EFM surface is such that the number of EFM components does actuallydepend on the value of dCp . For example, a single band B may change into two bands BB,and the question arises what changes of the EFM surface itself are involved in this transition. The point of view we take here is that the classiﬁcation of the EFM surface into differenttypes is generated by ﬁve (local) transitions of codimension one, which we introduce now. Eachsuch transition changes the nature of the associated EFM components one encounters whendCp is changed over [−π, π]. More speciﬁcally, we ﬁnd four generic singularity transitionsthat change the EFM surface topologically as a surface in three-dimensional space; as a result,the number of EFM components changes locally. Furthermore, we consider a cubic tangencyof the EFM surface with respect to a plane dCp = const, which also changes the number ofEFM components locally.The four singularity transitionsThe codimension-one singularity transitions are characterised by the fact that an isolated sin-gularity of the parametrised EFM surface is crossed at an isolated point of a curve in the(∆1 , ∆2 )-plane. To be more speciﬁc, let δ be the bifurcation parameter that parametrises acurve in the (∆1 , ∆2 )-plane, where we assume that the respective bifurcation curve is crossedtransversely at δ = 0. We can then view the EFM surface in (ωs , dCp , Ns )-space locally nearδ = 0 as given by level sets F (ωs , dCp , Ns ) = δ of a function F : R3 → R. The singularityis then given by the condition that grad (F ) = 0; generically, the Hessian at this point is non-singular, which means that the singularity is of codimension one [1, 26, 53]. In this case, thesurface is locally quadratic and has the normal form F (u, v, w) = ±u2 ± v 2 ± w2 = δ, (2.17)where the signs are given by the signs of the eigenvalues of the Hessian at the singularity. Ifall signs are the same then one is dealing with the transition through an extremum, that is, aminimum or a maximum, of the surface; we speak of a minimax transition [41]. Otherwise,the singularity is a saddle. The unfolding of such a saddle on a two-dimensional surface iswell know; see, for example, [2]. It is the transition form a one-sheeted hyperboloid via acone to a two-sheeted hyperboloid; we speak of a saddle transition [41]. Different cases inour context arise depending on how the cone associated with the saddle point is aligned in(ωs , dCp , Ns )-space. There are four distinct singularity transition of the EFM surface in (ωs , dCp , Ns )-space. M the minimax transition through an extremum (a local minimum or maximum). The min- imax transition M of the EFM surface is illustrated in ﬁgure 2.12, where, in terms of
- 51. 2.3. Classiﬁcation of the EFM surface for dτ = 0 35 the projection onto the (ωs , dCp )-plane, it results in the creation or disappearance of an island. The locus of M in the (∆1 , ∆2 )-plane is represented by orange curves in what follows. SC the saddle transitions in the direction of the Cp -axis. The saddle transition SC of the EFM surface is illustrated in ﬁgure 2.13, where, in terms of the projection onto the (ωs , dCp )- plane, it results in a transition between an island and a band. The locus of SC in the (∆1 , ∆2 )-plane is represented by blue curves in what follows. Sω the saddle transitions in the direction of the ωs -axis. The saddle transition Sω of the EFM surface is illustrated in ﬁgure 2.14, where, in terms of the projection onto the (ωs , dCp )- plane, it results for example, in a transition between a band with a hole and two separate bands. The locus of Sω in the (∆1 , ∆2 )-plane is represented by green curves in what follows. SN the saddle transitions in the direction of the Ns -axis. The saddle transition SN of the EFM surface is illustrated in ﬁgure 2.15, where, in terms of the projection onto the (ωs , dCp )-plane, it results in the creation or disappearance of a hole in a band. The locus of SN in the (∆1 , ∆2 )-plane is represented by red curves in what follows. Figures 2.12–2.15 illustrate how the minimax transition M and the three saddle transitionsSC , Sω and SN lead to changes in the EFM-surface. In each of these ﬁgures, we show inthe left column the relevant local part of the EFM surface in (ωs , dCp , Ns )-space before, atand after the bifurcations. The right column shows how the projection of the EFM surfaceonto the (ωs , dCp )-plane (shown over a dCp -interval of 4π) changes accordingly; the localregions where the change occurs are highlighted. In ﬁgures 2.12–2.15 all the presented surfaceshave been rendered from continuations of the EFMs as solutions of equations (1.1)–(1.4); theprojections, on the other hand, were obtained directly from (2.16). Figure 2.12 illustrates the minimax transition M , where a compact piece of the EFM-surface in (ωs , dCp , Ns )-space shrinks to a point. Panel (b1) is very close to the bifurcation;note that after the bifurcation the piece is simply gone, which is why we do not present a sep-arate illustration for this situation. In projection onto the (ωs , dCp )-plane, the local, compactpiece of the EFM-surface is an ‘island’ that shrinks and then disappears in a minimax transi-tion of the projection; note that there are inﬁnitely many such islands due to the translationalsymmetry in dCp ; see panels (a2) and (b2). Figure 2.13 illustrates the saddle transition SC . The local mechanism for this change ofthe EFM-surface in (ωs , dCp , Ns )-space is shown in panels (a1)–(c1). The surface in ﬁg-ure 2.13 (a1) is a one-sheeted hyperboloid. It develops a pinch point and, hence, becomes acone at the moment of bifurcation in ﬁgure 2.13 (b1); note that the cone (or rather its axis of
- 52. 36 Chapter 2. Classiﬁcation of EFM structure . (a1) (a2) 2 1.3Ns 0 dCp π −2 dCp ωs −0.05 0 ωs 0.3 π (b2) 2 (b1) 1.3Ns 0 dCp π −2 dCp ωs −0.05 0 ωs 0.3 . πFigure 2.12. Minimax transition M of the EFM-surface in (ωs , dCp , Ns )-space, where a connectedcomponent of the EFM surface (a1) shrinks to a point (b1). Panels (a2) and (b2) show the correspondingprojection onto the (ωs , dCp )-plane of the entire EFM surface; the local region where the transition Moccurs is highlighted 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).rotation) is aligned with the dCp -axis. After the bifurcation, the EFM-surface is a two-sheetedhyperboloid so that it consists locally of two parts; see ﬁgure 2.13 (c1). As the projections ontothe (ωs , dCp )-plane in panels (a2)–(c2) show, the overall result is the division of a ‘band’ intoa ‘string of islands.’ Note that the saddle transition SC manifests itself as a saddle transition ofthe projection, where the relevant (shaded) part of the surface is aligned with the dCp -axis. Figure 2.14 illustrates the saddle transition Sω . Locally near the point of bifurcation weagain ﬁnd that the EFM-surface in (ωs , dCp , Ns )-space changes from a one-sheeted hyper-boloid in panel (a1), via a cone in panel (b1) to a two-sheeted hyperboloid in panel (c1). How-ever, now the cone is aligned with the ω-axis. As the projections onto the (ωs , dCp )-plane inpanels (a2)–(c2) show, overall we ﬁnd that a single band with a ‘string of holes’ changes intotwo separate bands. The saddle transition Sω manifests itself also as a saddle transition of theprojection, but now the relevant (shaded) part of the surface is aligned with the ωs -axis. Figure 2.15 illustrates the saddle transition SN . In panel (a1) there are two sheets, withdifferent and separate values of Ns , of the EFM-surface in (ωs , dCp , Ns )-space. At the bifur-cation point the two sheets connect locally in a single point. In the process, a ‘hole’ is created
- 53. 2.3. Classiﬁcation of the EFM surface for dτ = 0 37 . (a2) 2 (a1) 0.9Ns 0.2 dCp π −2 ωs −0.05 0 ωs 0.26 dCp /π (b2) 2 (b1) 0.9Ns 0.2 dCp π −2 ωs −0.05 0 ωs 0.26 dCp /π (c2) 2 (c1) 0.9Ns 0.2 dCp π −2 ωs −0.05 0 ωs 0.26. dCp /πFigure 2.13. Saddle transition SC of the EFM-surface in (ωs , dCp , Ns )-space, where locally thesurface changes from a one-sheeted hyperboloid (a1) to a cone aligned in the dCp -direction (b1) to atwo-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 bydashed lines and the projections of the part of the EFM surface in panels (a1)–(c1) is shaded grey. HereΛ = 0.01, ∆1 = 0.4, and ∆2 = 0.23 in (a), ∆2 = 0.232745 in (b) and ∆2 = 0.24 in (c).in the EFM-surface, which then grows in size; see panels (b1) and (c1). If one considers asmall neighbourhood of the emerging hole, then one realises that the transition is locally thatfrom a two-sheeted hyperboloid in panel (a1), via a cone aligned along the Ns -axis in panel(b1) to a one-sheeted hyperboloid in panel (c1). The projections onto the (ωs , dCp )-plane inpanels (a2)–(c2) clearly show how a string of holes appears in the saddle transition SN . Notethat this bifurcation is a minimax transition of the projection but, in contrast to transition M ,
- 54. 38 Chapter 2. Classiﬁcation of EFM structure . (a2) 2 (a1)Ns 0.6 0 −0.6 dCp π dCp −2 ωs −0.05 0 ωs 0.17 π (b2) 2 (b1)Ns 0.6 0 −0.6 dCp π dCp −2 ωs −0.05 0 ωs 0.17 π (c2) 2 (c1)Ns 0.6 0 −0.6 dCp π dCp −2 ωs −0.05 0 ωs 0.17. πFigure 2.14. Saddle transition Sω of the EFM-surface in (ωs , dCp , Ns )-space, where a connected com-ponent (a1) pinches (b1) and then locally disconnects (c1); here the associated local cone in panel (b1)is aligned in 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 bydashed lines and the projections of the part of the EFM surface in panels (a1)–(c1) is shaded grey. HereΛ = 0.01, ∆1 = 0.4, and ∆2 = 0.13 in (a), ∆2 = 0.133535 in (b) and ∆2 = 0.135 in (c).the projection of the surface is now ‘on the outside’ so that locally a hole is created instead ofan island. It is an important realisation that the loci of the four singularity transitions M , SC , Sωand SN can be computed effectively, because it can be expressed as an implicit formula byconsidering a suitable derivative of the envelope equation (2.9) with respect to the parameter inquestion. More speciﬁcally, one follows a fold with respect to dCp of the boundary curve of the
- 55. 2.3. Classiﬁcation of the EFM surface for dτ = 0 39 . (a2) 2 (a1) 1.1 0.8Ns 0 dCp π dCp −2 ωs −0.05 0 ωs 0.15 π (b2) 2 (b1) 1.1 0.8Ns 0 dCp π dCp −2 ωs −0.05 0 ωs 0.15 π (c2) 2 (c1) 1.1 0.8Ns 0 dCp π dCp −2 ωs −0.05 0 ωs 0.15. πFigure 2.15. Saddle transition SN of the EFM-surface in (ωs , dCp , Ns )-space, where two sheets thatlie on top of each other in the Ns direction (a1) connect at a point (b1) and then create a hole in thesurface (c1); here the associated local cone in panel (b1) is aligned in the N -direction. Panels (a2)–(c2)show the corresponding projection onto the (ωs , dCp )-plane of the entire EFM surface; the local regionwhere the transition SN occurs is highlighted by dashed lines and the projections of the part of theEFM 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).projection of the EFM surface onto the (ωs , dCp )-plane. Such a dCp -fold bound an intervalof dCp -values, which is either an island or a hole of the projection. When the dCp -fold iscontinued along a curve (parametrised by δ) in the (∆1 , ∆2 )-plane, say, ∆2 for ﬁxed ∆1 , thena singularity transition corresponds to a fold with respect to the continuation parameter δ. Sucha fold with respect to the parameter δ can be detected and then followed as a boundary curve
- 56. 40 Chapter 2. Classiﬁcation of EFM structure . 1.1 (a2) (a1) dCpNs π dCp 1 ωs −0.075 ωs 0 0.03 π 1.1 (b2) (b1) dCpNs π dCp 1 ωs −0.075 ωs 0 0.03 π 1.1 (c2) (c1) dCpNs π dCp 1 ωs −0.075 ωs 0 0.03 . πFigure 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 unfoldingof 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).in the (∆1 , ∆2 )-plane. Note that this continuation approach makes no difference betweenthe cases M , SC , Sω and SN of singularity transitions. However, which of the singularitytransitions one is dealing with can readily be identiﬁed by checking the (projections of the)EFM surface at nearby parameter point in the (∆1 , ∆2 )-plane. In this way, the loci of thesingularity transitions have been computed numerically to yield the complicated structure ofboundary curves in ﬁgure 2.17.
- 57. 2.3. Classiﬁcation of the EFM surface for dτ = 0 41The cubic tangencyDue to the special role of the parameter dCp we also consider here a ﬁfth local mechanism thatchanges the type of the EFM surface. C the cubic tangency C of the EFM surface, is deﬁned by the condition that the ﬁrst and second derivatives with respect to ωs of equation (2.9) for the envelope of Ω(ωs ) both vanish at an isolated point, and the third derivative is nonzero. The cubic tangency C of the EFM surface is illustrated in ﬁgure 2.16, where, in terms of the projection onto the (ωs , dCp )-plane, it results in the creation of a pair of local extrema of dCp that form a bulge of the EFM surface. The locus of C in the (∆1 , ∆2 )-plane is represented by black and grey curves in what follows. Figure 2.16 illustrates the cubic tangency C. Before the transition the EFM surface in(ωs , dCp , Ns )-space is such that it does not feature dCp -folds of the boundary curve in pro-jection onto the (ωs , dCp )-plane; see panels (a1) and (a2). At the moment of transition theEFM surface is such that the boundary curve of the envelope has a cubic tangency with a curvedCp = const; see panels (b1) and (b2). This cubic tangency of the boundary curve unfolds intoa pair of a local minimum and a local maximum of dCp for nearby values of ωs ; see panels (c1).This pair of extrema corresponds to a ‘bulge’ of the EFM surface; see panels (c1). As a result,there is now an interval of dCp -values where one ﬁnds two distinct EFM components. Notethat the locus C of cubic tangency can be computed by numerical continuation of the conditionthat the ﬁrst two derivatives with respect to ωs of the envelope equation (2.9) are zero.2.3.4 The EFM surface bifurcation diagram in the (∆1 , ∆2 )-plane for ﬁxed ΛFor a ﬁxed value of Λ the ﬁve transitions M , SC , Sω , SN and C of codimension one give riseto boundary curves that divide the (∆1 , ∆2 )-plane into a ﬁnite number of regions. Each suchregion deﬁnes a type of the EFM surface and, overall, we speak of the EFM surface bifurcationdiagram in the (∆1 , ∆2 )-plane. We ﬁrst consider in ﬁgure 2.17 the EFM surface bifurcation diagram for the case Λ = 0.01.As in ﬁgure 2.10, the white regions correspond to the band-like types of the EFM surface fromsection 2.3.2 with a dCp -independent number of EFM components. In the grey regions, on theother hand, one ﬁnds new EFM surface types. In ﬁgure 2.17 we labelled those types that areassociated with transitions between the band-like EFM surface types; there are a total of 15additional types, and their representatives are shown in ﬁgure 2.18; for notation and parametervalues of the individual panels see Table 2.2. As before, the symbol B denotes a connected
- 58. 42 Chapter 2. Classiﬁcation of EFM structure . 0.55 B IB BB hB B Bh BB B B BI BII ∆2 BI I BI B BI hBI BI Bh I t BI e v t e v t e v BI BBI B BB hBB Bh B BB I BB BB BBI BB hh hBhr BBh r B r 33 hh 3 Bh B Bh I Bh B Bh BBB Bh I Bh 0 B IB BB hB Bh BB BI B BB B hBB hB IhB 3 hBI hB hB 33 hhB r rr hB B hBh IB B hhB BB BB B BI BB BhB B Bh B BB IB IB B v e t v e t v e IB t IhB I B I Bh I BB I BI IB II B IB B B BB hB B Bh BB BI B −0.55 M SC Sω SN SN Sω SC M −0.55 0 ∆1 0.55.Figure 2.17. EFM surface bifurcation diagram in the (∆1 , ∆2 )-plane for Λ = 0.01 with regions ofdifferent types of the EFM surface; see ﬁgure 2.18 for representatives of the labelled types of the EFMsurface 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 cubic tangency(black curves) can be found near the diagonal; also shown is the anti-diagonal.component of the EFM surface in the form of a band in projection onto the (ωs , dCp )-planethat extends over the entire Cp -range [−π, π]. There are two noteworthy features of the EFM surface types in ﬁgure 2.18. First of all,there are connected components of the EFM surface that do not extend over the entire Cp -range[−π, π]; we use the symbol I to refer to them because their projection onto the (ωs , dCp )-planeconsists of an ‘island’ when dCp ∈ R/2πZ (inﬁnitely many islands in the covering space whendCp ∈ R). Owing to the underlying symmetry (∆1 , ∆2 ) → (−∆1 , −∆2 ), we again represent
- 59. 2.3. Classiﬁcation of the EFM surface for dτ = 0 43 . (a) (b) (c) BI BII I BI (d) (e) (f) Bh I Bh IhB (g) (h) (i) Bhh hBh BBI (j) (k) (l) (m) B BI Bh B BBh hBB.Figure 2.18. Additional types of EFM-surface of the 2FOF laser in the labelled regions of ﬁgure 2.17,represented by the projection (shaded) onto the (ωs , dCp )-plane; the blue boundary curves are founddirectly from (2.16). For notation and the corresponding values of ∆1 and ∆2 see Table 2.2; in all panelsωs ∈ [−0.3, 0.3] and dCp ∈ [−π, π].in the notation the position of an island with respect to the central band B. As second newfeature is the fact that a band may have up to two (periodically repeated) holes. Similarly to theislands, we reﬂect in the notation the position of a hole with respect to the central frequency ofthe laser (at ωs = 0). Namely, we indicate with left and right subscripts whether a hole is tothe left or to the right of {ωs = 0}; for example, we distinguish the case Bhh from hBh . Weobserve that islands never have holes for any values of the parameters as considered here; thismeans that the symbol I does never have a subscript. In ﬁgure 2.18 we again show only one representative for any pair that is related by sym-metry. Obtaining a representative of the symmetric counterpart corresponds to a reﬂection ofthe respective image in the line ωs = 0; this operation is reﬂected in the notation by reversingthe symbol string representing the EFM surface type. In ﬁgure 2.17 this symmetry operationcorresponds to reﬂection in the antidiagonal of the (∆1 , ∆2 )-plane. Notice also the symmetry
- 60. 44 Chapter 2. Classiﬁcation of EFM structure name Cmin Cmax panel ∆1 ∆2 Λ BI 1 2 ﬁgure 2.18 (a) 0.255 -0.270 0.01 BII 1 3 ﬁgure 2.18 (b) 0.260 0.210 0.01 I BI 1 3 ﬁgure 2.18 (c) 0.254 -0.251 0.01 Bh 1 2 ﬁgure 2.18 (d) 0.100 -0.270 0.01 I Bh 1 3 ﬁgure 2.18 (e) 0.100 -0.250 0.01 IhB 1 3 ﬁgure 2.18 (f) -0.100 0.250 0.01 Bhh 1 3 ﬁgure 2.18 (g) 0.160 0.110 0.01 hBh 1 3 ﬁgure 2.18 (h) 0.095 -0.110 0.01 BBI 2 3 ﬁgure 2.18 (i) 0.250 0.190 0.01 B BI 2 3 ﬁgure 2.18 (j) 0.250 -0.130 0.01 Bh B 2 3 ﬁgure 2.18 (k) 0.210 0.100 0.01 BBh 2 3 ﬁgure 2.18 (l) 0.180 0.130 0.01 hBB 1 3 ﬁgure 2.18 (m) 0.210 -0.100 0.01Table 2.2. Notation and parameter values for the types of EFM-surface in ﬁgure 2.18; the second andthird column show the minimal number Cmin and the maximal number Cmax of EFM components (forsuitable ﬁxed dCp ) of the type.of the EFM surface bifurcation diagram given by reﬂectio in the diagonal; it corresponds to anexchange of the two ﬁlters and, hence, does not change the EFM surface type. The outer part of the EFM surface bifurcation diagram in ﬁgure 2.17, away from the diag-onal, is characterised by grey intersecting strips, which are each bounded by a pair of curvesof singularity transitions. These strips must be crossed in the (∆1 , ∆2 )-plane to move betweendifferent white regions of band-like types of the EFM surface. As an example, consider a suf-ﬁciently large ﬁxed value of one of the detunings, say, of ∆1 , while the other detuning, ∆2 , isallowed to change. The grey strip bounded by the pair of curves M and SC is responsible forthe transition from a single band B to two bands BB via the appearance of a string of islandsthat then merge into the new band B. The pair SN and Sω , on the other hand, also results ina transition from B to BB, but via the appearance of a string of holes in B that then merge toform a gap that splits off the new band B. Note that the illustrations of the singularity transi-tions in ﬁgures 2.12–2.15 are all for ∆1 = 0.4; hence, they also illustrate the transition from Bto BB via SN and Sω and back to B via SC and M as ∆2 is increased from, say, ∆2 = 0. The grey strips in ﬁgure 2.17 are unbounded and extend all the way to inﬁnity. This followsfrom the fact that the limit ∆i → ±∞ reduces to the single FOF laser in a nontrivial way, aswas discussed in section 2.3.1. More speciﬁcally, for the chosen value of Λ = 0.01 the curvein the (∆1 , Λ)-plane of ﬁgure 2.8 (b) is intersected four times, and this accounts for the fourstripes one ﬁnds for ∆2 → ±∞ (and similarly for ∆1 → ±∞).
- 61. 2.3. Classiﬁcation of the EFM surface for dτ = 0 45 . (a) 1 (b) 1 0 0dCp dCp π π −1 −1 −2 −2 −3 −3. −0.12 0 ωs 0.12 −0.12 0 ωs 0.12Figure 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. The anti-diagonal is shown in ﬁgure 2.17 because along it one ﬁnds special, degeneratecases of the EFM surface. There are two different cases, and they are shown in ﬁgure 2.19.Along the red part of the anti-diagonal we ﬁnd a degenerate saddle transition SN . At themoment of transition the upper and lower sheets of the EFM surface touch at a single, isolatedpoint (and its 2π-translates in dCp ); see ﬁgure 2.19 (a). However, as ﬁgure 2.17 shows, oneither side of the red part of the anti-diagonal we ﬁnd the symmetrically related pair hB andBh , which each features a hole. Hence, when the red part of the anti-diagonal is crossed, thehole shrinks and then reappears on the other side of the line ωs = 0; physically, the hole is onthe side of the ﬁlter that is detuned furthest from the solitary laser frequency. From a bifurcationpoint of view, along the red part of the anti-diagonal the EFM surface changes locally from atwo-sheeted hyperboloid to a cone and back to a two-sheeted hyperboloid, rather than to aone-sheeted hyperboloid; compare with ﬁgure 2.15 for the non-degenerate saddle transitionSN . Along the grey part of the anti-diagonal, on the other hand, we ﬁnd a degenerate saddletransition SC . Notice that the EFM surface type on either side of this grey curve is the sameand invariant under the symmetry operation ∆1 → −∆2 . Figure 2.19 (b) shows the momentof transition for the case that the anti-diagonal bounds the two regions of EFM surface typehBh in ﬁgure 2.17. In this transition the two holes (and their 2π-translates in dCp ) touch toform a lemniscate in ﬁgure 2.19 (b). This means that the EFM surface is connected (locally)at isolated points with ωs = 0. We also found this degenerate type of connection of the EFMsurface at such isolated points in ﬁgure 2.9 — for the case that the surface is of type B, andfor the case that it is of type B BB. In effect, along the grey part of the anti-diagonal the EFMsurface changes locally from a one-sheeted hyperboloid to a cone and back to a one-sheetedhyperboloid, rather than to a two-sheeted hyperboloid; compare with ﬁgure 2.13 for the non-degenerate saddle transition SC .
- 62. 46 Chapter 2. Classiﬁcation of EFM structure . 0.1 b Bhh B bbb Bb h b Cd Bh B bb B bbb SC ∆2 b b Bh SN B bb b Bhh Ca B bb DCN C bb Bh Bb Bb B bb B Bb Bh 0 . 0 ∆1 0.1Figure 2.20. Enlargement near the center of the (∆1 , ∆2 )-plane of ﬁgure 2.17 with (blue) curvesof SC transition, (red) curves of SN transition, and (black) curves Ca and (grey) curves Cd of cubictangency; see ﬁgure 2.21 for representatives of the labelled types of the EFM surface and Table 2.3 forthe notation.The locus of cubic tangency in the (∆1 , ∆2 )-planeIn ﬁgure 2.17 one ﬁnds (black) curves of cubic tangency near the diagonal in the central regionof the (∆1 , ∆2 )-plane. To understand their role for the EFM surface bifurcation diagram, weshow in ﬁgure 2.20 an enlargement of the (∆1 , ∆2 )-plane near the central (white) region wherethe EFM surface is of type B. Recall that this central region must exist as a perturbation of thespecial case of the EFM surface for ∆1 = ∆2 = 0 in ﬁgure 2.9 (a); from the physical pointof view, this type of EFM surface exists (for κ = 0) as the continuation of the solitary lasermode for κ = 0. Note that for Λ = 0.01, as in ﬁgures 2.17 and 2.20, this central region is quitesmall; however, as we will see in section 2.4, it may grow considerable when the ﬁlter width Λis increased. The central (white) region of EFM surface type B in ﬁgure 2.20 is bounded entirely bycurves of cubic tangency. Therefore, as the ﬁlters are detuned away from the solitary laserfrequency, the ﬁrst transformation of the EFM-surface that gives rise to an additional EFM-component is a cubic tangency. We ﬁnd it convenient to distinguish two different types of
- 63. 2.3. Classiﬁcation of the EFM surface for dτ = 0 47 . (a) (b) (c) Bb B bb b b B (d) (e) (f) (g) B bbb b bb B bb bb B B bbbb (h) (i) (j) (k) BI b BI bb hB b b Bh (l) (m) (n) (o) bb bb b hB Bh BB BB bb.Figure 2.21. Additional types of EFM-surface of the 2FOF laser that feature bulges, represented by theprojection (shaded) onto the (ωs , dCp )-plane; the blue boundary curves are found directly from (2.16).Where necessary, insets show local enlargements. The corresponding regions in the (∆1 , ∆2 )-planecan be found in ﬁgures 2.20, 2.25 and 2.29; for notation and the corresponding values of ∆1 , ∆2 and Λsee Table 2.3. In all panels ωs ∈ [−0.3, 0.3] and dCp ∈ [−π, π].cubic tangency. The (black) curves Ca are invariant under reﬂection in the anti-diagonal; theycorrespond to cusp points of the orange curves in ﬁgures 2.7 and 2.10. The (grey) curves Cd areinvariant under reﬂection in the anti-diagonal; they correspond to cusp points of the grey curvesin ﬁgures 2.7 and 2.10. The (black) cubic tangency locus Ca consists of two elongated and self-intersecting closed curves, one above and one below the diagonal; see ﬁgure 2.17. The (grey)cubic tangency locus Cd also consists of two self-intersecting closed curves, but they cross thediagonal and one lies above and the other below the anti-diagonal; see ﬁgure 2.20 and note thatthe locus Cd is too small to be visible in ﬁgure 2.17. Crossing either of the boundary curves Ca and Cd from inside the central (white) regionlabelled B in ﬁgure 2.20 results in the appearance of a bulge of the EFM surface. We representthis in our notation of this region as B b by a superscript b; as before, whether the superscript
- 64. 48 Chapter 2. Classiﬁcation of EFM structure name Cmin Cmax panel ∆1 ∆2 Λ Bb 1 2 ﬁgure 2.21 (a) -0.0200 0.2000 0.098 B bb 1 2 ﬁgure 2.21 (b) 0.1320 0.1300 0.010 bB b 1 3 ﬁgure 2.21 (c) -0.2400 0.2300 0.098 B bbb 1 3 ﬁgure 2.21 (d) 0.1400 0.1300 0.010 bB bb 1 3 ﬁgure 2.21 (e) -0.2300 0.2170 0.098 bbB bb 1 3 ﬁgure 2.21 (f) -0.2140 0.2150 0.098 B bbbb 1 3 ﬁgure 2.21 (g) -0.2350 0.1780 0.020 BI b 1 3 ﬁgure 2.21 (h) 0.2100 0.1900 0.010 BI bb 1 3 ﬁgure 2.21 (i) 0.2150 0.1850 0.098 b hB 1 3 ﬁgure 2.21 (j) -0.2120 0.2400 0.098 Bhb 1 3 ﬁgure 2.21 (k) 0.1300 0.1161 0.010 bb hB 1 3 ﬁgure 2.21 (l) -0.2120 0.2200 0.098 Bhbb 1 3 ﬁgure 2.21 (m) 0.1337 0.1161 0.010 BB b 2 3 ﬁgure 2.21 (n) 0.2100 0.1600 0.010 BB bb 2 3 ﬁgure 2.21 (o) 0.1530 0.1355 0.010Table 2.3. Notation and parameter values for the types of EFM-surface in ﬁgure 2.21; the second andthird column show the minimal number Cmin and the maximal number Cmax of EFM components (forsuitable ﬁxed dCp ) of the type.appears on the left or on the right of the central band B indicates its position with respect tothe solitary laser frequency given by ωs = 0. As ﬁgure 2.20 shows, we ﬁnd a complicatedEFM surface bifurcation diagram consisting of an interplay of cubic tangency curves Ca and Cdwith saddle-transition curves SC and SN . The four sets of curves divide the (∆1 , ∆2 )-planeinto regions of additional EFM surface types, which are all characterised by a certain numberof bulges as represented in the notation. The corresponding EFM surface types are shown inprojection onto the (ωs , dCp )-plane in ﬁgure 2.21 (a), (b), (d), (k) and (m); for notation andparameter values of the individual panels see Table 2.3. Note that near the central region of the(∆1 , ∆2 )-plane the EFM surface is subject to the interaction of both ﬁlters near the frequencyof the laser, which means that there are no other bands or islands. Figure 2.20 also shows that crossing the saddle-transition curve SC may result in the cre-ation of a hole, which happens, for example, in the transition from B bbb to Bh . This new bmechanism for the creation of a hole is illustrated in ﬁgure 2.22 for the simpler case of a tran-sition from B bb to Bh ; see also ﬁgure 2.21 (b) and ﬁgure 2.18 (d). In ﬁgure 2.22 (a1) and (a2)there are two bulges; one of the bulges is rather small, indicating that it has just been createdin a nearby cubic tangency. At the saddle transition SC in ﬁgure 2.22 (b) we ﬁnd that the twobulges connect locally in the central point of a cone that is aligned in the dCp -direction. In
- 65. 2.3. Classiﬁcation of the EFM surface for dτ = 0 49 . (a2) 1.2 (a1) dCpNs π dCp 0.9 ωs −0.065 ωs 0 0.04 π (b2) 1.2 (b1) dCpNs π dCp 0.9 ωs −0.065 ωs 0 0.04 π (c2) 1.2 (c1) dCpNs π dCp 0.9 ωs −0.065 ωs 0 0.04 . πFigure 2.22. Global manifestation of local saddle transition SC of the EFM-surface where two bulgesconnect 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).contrast to the case shown in Figure 2.13, the geometry of the EFM surface is now such thatthis bifurcation leads to the creation of a hole; see panels (c1) and (c2). This hole can thendisappear again when the (red) curve SN in ﬁgure 2.20 is crossed; for example, this happens inthe transition from Bh to B b . b The connection between cubic tangency and the singularity transitions is given by codimension-two points on the locus of cubic tangency. This feature is prominent in ﬁgure 2.20, wherecurves SN (red) and SC (blue) of saddle transition end at (purple) points on the (grey) curveCd . In ﬁgure 2.23 we present a local unfolding of such a codimension-two point, which is char-
- 66. 50 Chapter 2. Classiﬁcation of EFM structure . 1 1−2 2 2−3 SN SC 4−1 SN 3 C DCN C SC ∆2 1 2 3 4 3−4 C DCN C C 4 . ∆1Figure 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) curveSC of saddle transition emanate; compare with ﬁgures 2.20 and 2.29 (a) and (b). . 1 1−2 2 2−3 M Sω 4−1 M 3 C DCM ω Sω ∆2 1 2 3 4 3−4 C DCM ω C 4 . ∆1Figure 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.
- 67. 2.3. Classiﬁcation of the EFM surface for dτ = 0 51acterised by the fact that the envelope curve given by (2.9) has a cusp point. When making acircle around the codimension-two point DCN C on the curve C, starting at region 1, one ﬁndsthat a hole appears near the boundary of EFM surface in the saddle transition SN ; compare withﬁgure 2.22. The hole then disappears in the saddle transition SC . As a result, there are nowtwo bulges, that is, pairs of local maxima and minima with respect to dCp , which disappear oneafter the other when the cubic tangency curve C is crossed twice to complete the circle back toregion 1. This unfolding can indeed be found locally in ﬁgure 2.20, but note that it involvesthe EFM surface of type B b as that corresponding to region 1 in ﬁgure 2.23; hence, region 2corresponds to Bh , region 3 to B bbb , and region 4 to B bb . b A second case of a codimension-two point on the curve of cubic tangency can be found inﬁgure 2.17, where the curves M (orange) and Sω (green) end at (golden) points on the (black)curve Ca . The local unfolding of such a codimension-two point is presented in ﬁgure 2.24;it is again characterised by a cusp point on the envelope curve given by (2.9), but this timethe cusp points the other way with respect to the EFM surface. When making a circle aroundthe codimension-two point DCN C on the curve C, starting at region 1, an island is createdwhen the minimax transition M is crossed; this island then merges with the remainder ofthe EFM surface in the saddle transition Sω . As a result, there are again two bulges, whichdisappear one after the other when the curve C is crossed twice to complete the circle backto region 1. This unfolding can be found locally in ﬁgure 2.17, where EFM surface of typeBI corresponds to region 1 in ﬁgure 2.23; hence, region 2 corresponds to BII, region 3 toBI bb , and region 4 to BI b . Note that, in terms of the envelope curve (2.9), the unfoldingin ﬁgure 2.24 is topologically equivalent to that in ﬁgure 2.23. However, the two unfoldingsdiffer in where (the projection of) the EFM surface lies with respect to the boundary; hence, therespective panels of ﬁgure 2.23 and ﬁgure 2.24 (where the EFM surface is always on the left)can be transformed into one another by exchanging the colours blue and white in the regions,followed by a reﬂection. Note that ﬁgure 2.21 shows the comprehensive list, in order of increasing complexity, ofEFM surface types that feature bulges — of which there are quite a few more than we identiﬁedin ﬁgure 2.20. Additional EFM surface types can be found near the diagonal of the (∆1 , ∆2 )-plane, but further away from the central point ∆1 = ∆2 = 0. Figure 2.25 (a) shows therespective enlargement of the EFM surface bifurcation diagram from ﬁgure 2.17, which fea-tures an interaction of cubic tangency curves Ca with saddle-transition curves SC and Sω . Weﬁnd ﬁve additional EFM surface types with bulges, representatives of which in the (ωs , dCp )-plane are also shown in ﬁgure 2.21 (h), (i), (k), (n) and (o); for notation and parameter values ofthe individual panels see again Table 2.3. Note, in particular, that crossing the saddle-transitioncurve Sω may lead to secondary bands or islands with bulges. To obtain the remaining cases of EFM surface types with bulges in ﬁgure 2.21 it is nec-essary to change the ﬁlter width parameter Λ. As an example of how new regions of EFM
- 68. 52 Chapter 2. Classiﬁcation of EFM structure .0.17 (a) 0.25 (b) Sω SC Ca Sω SC Ca d d BBh BI bb BI bb bb ∆2 BB bb ∆2 BB BI b BI BI b BI Bhh bbbb B bb BI bb Bh d d BI b BI b b bb Bh B B b B bbb Bh BI bb b b bb Bh BB bb B bbbb b B B B bbb B b bb B bb B bb b bb BBh bb BB bb B Bh Bh Bh Bhh b Bh0.12 0.15 0.12 ∆1 0.17 0.15 ∆1 0.25. Figure 2.25. Enlargement near the diagonal of the (∆1 , ∆2 )-plane with (blue) curves of SC transi- tion, (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 surface types are created in a subtle way with changing Λ, ﬁgure 2.25 (b) shows a similar en- largement as panel (a), but now for Λ = 0.02. Notice that the relative position of the curve Sω has changed in such a way that one ﬁnds a region where the EFM surface is of type B bbbb ; see ﬁgure 2.21 (g). The location in (∆1 , ∆2 , Λ)-space of all EFM surface types in ﬁgure 2.21 can be found in Table 2.3; we will encounter more types in the next section. 2.4 Dependence of the EFM surface bifurcation diagram on the ﬁlter width Λ We now consider more globally how the EFM surface bifurcation diagram in the (∆1 , ∆2 )- plane changes with the common ﬁlter width Λ. Figure 2.7 already indicated that substantial changes to the regions of band-like EFM types must be expected. In particular, for sufﬁciently large Λ the grey and orange curves in ﬁgure 2.7 do not extend to inﬁnity in the (∆1 , ∆2 )-plane any longer. To study this phenomenon we compactify the (∆1 , ∆2 )-plane by the stereographic change of coordinates ∆i ∆i = , η > 0. (2.18) |∆i | + η Note that (2.18) transforms the (∆1 , ∆2 )-plane to the square [−1, 1]×[−1, 1] in the (∆1 , ∆2 )- plane, where ∆i = ±1 corresponds to ∆i = ±∞; we speak of the (∆1 , ∆2 )-square from now
- 69. 2.4. Dependence of the EFM surface bifurcation diagram on the ﬁlter width Λ 53 . (a) (b) BB BB BB BB B B B B B B BBB ∆2 B BB BB ∆2 B BB BB dd d d BB BB BB BB BB BB 0 B B 0 B B BB BB BB d d BB BB d d BB BB B BB BB B BB BB B B B BB BB BB BB B B B B 0 ∆1 0 ∆1 (c) b BB b BB (d) B B B B B Ca ∆2 ∆2 b BB b BB 0 B B 0 B ¡ b BB ¡ b BB ¡ ¡ B B B B B Cd B b BB b BB 0 ∆1 0 ∆1 (e) (f) ∆2 Ca ∆2 Ca 0 B 0 B Cd Cd . 0 ∆1 0 ∆1Figure 2.26. EFM surface bifurcation diagram in the compactiﬁed (∆1 , ∆2 )-square, [−1, 1] × [−1, 1],showing regions of band-like EFM surface types; compare with ﬁgure 2.10. The boundary of the squarecorresponds 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.on. The parameter η is the value of ∆i that is mapped to ∆i = 0.5, and we chose η = 0.4 toensure that the main structure of the EFM surface bifurcation diagram in the (∆1 , ∆2 )-plane
- 70. 54 Chapter 2. Classiﬁcation of EFM structureis represented well in the (∆1 , ∆2 )-square. Figure 2.26 shows the EFM surface bifurcation diagram in the (∆1 , ∆2 )-square in the styleof ﬁgure 2.10, where (orange and grey) boundary curves are shown for 61 equidistant valuesof dCp from the interval [−π, π]. The (white) regions of band-like EFM surface types arelabelled; see Table 1.1. Also shown are the cubic tangency curves Ca (black) and Cd (grey). Figure 2.26 (a) for Λ = 0.01 is simply the compactiﬁed version of ﬁgure 2.10 in the(∆1 , ∆2 )-square. Notice that the (orange and grey) stripes now end at discrete points at thesides of the square. Namely, for ∆i = ±∞ the respective ﬁlter does not inﬂuence the laserany longer, so that the system reduces to single FOF laser on the sides of the (∆1 , ∆2 )-square.Hence, the end points of the stripes are exactly the four intersection points of the line Λ =0.01 with the light grey limiting curve in ﬁgure 2.8 (b). As Λ is increased, the areas of otherEFM surface types covered by (orange and grey) curves expands and four (symmetry-related)smaller regions of band-like EFM surface types BB B and BBB disappear; see ﬁgure 2.26 (b).As Λ is increased further, additional regions of band-like EFM surface types disappear; seeﬁgure 2.26 (b). In the process the pairs of (orange and grey) stripes move closer together,owing to the fact that the four intersection points with the light grey curve in ﬁgure 2.8 (b) dothe same. Moreover, the (black and grey) cubic tangency curves extend over a much largerregion of the (∆1 , ∆2 )-square; hence, the region near ∆1 = ∆2 = 0 where one ﬁnds EFMsurface type B opens up considerably. Figure 2.26 (d) is for Λ = 0.098131, which is the approximate value of Λ where oneﬁnds the cusp points of the grey limiting curve in ﬁgure 2.8 (c). This value can be computedanalytically as 2 ΛC = √ κ 1 + α 2 (2.19) 3 3from the formula in [28] for the single FOF laser. In the context of the EFM surface bifurcationdiagram, this values corresponds to a bifurcation at inﬁnity of the (∆1 , ∆2 )-plane and, hence,a bifurcation at the boundary of the (∆1 , ∆2 )-square. More speciﬁcally, for Λ = ΛC the pairsof orange and grey stripes now end at single points. the cubic tangency curves Ca (black) andCd (grey) extend all the way to the boundary of the square. Notice further that the central regionof EFM surface type B is no longer bounded by curves of cubic tangency but is now joinedup with four, previously separated regions of the same type. For Λ > ΛC the stripes do nolonger extend to the boundary of the square. As a result, one now ﬁnds orange and grey pairsof islands that are bounded almost entirely by (black and grey) cubic tangency curves. Thecomplement of these islands is a single connected (white) region of EFM surface type B; seeﬁgure 2.26 (e). When Λ is increased further, these islands become smaller; see ﬁgure 2.26 (f).
- 71. 2.4. Dependence of the EFM surface bifurcation diagram on the ﬁlter width Λ 55 . Cd b Cd bb Cd B DC bb B NC Sω B Sω ∆2 S ∆2 I B b ∆2 b IB ω b b bb B SC SN M B M B M IB hB B B B B B BB DCM ω (a1) (a2) (a3) ∆1 ∆1 ∆1 Ca DCM ω Ca Ca Bb B bb B bb SC SC ∆2 SC ∆2 ∆2 Sω M Bh Bh B bb SN Bb SN Bb SN Bh BI B B B B B BB DCN C (b1) (b2) (b3) . ∆1 ∆1 ∆1Figure 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 involvingthe (black) curve Ca of cubic tangency that bounds the orange islands, and panels (b1)–(b3) show thetransition involving the (grey) curve Cd of cubic tangency that bounds the grey islands; compare withﬁgure 2.26 (c)–(e).2.4.1 Unfolding of the bifurcation at inﬁnityAs we know from section 2.3.4, the boundaries of the orange and grey regions are formed notonly by the curves Ca and Cd of cubic tangency, but also by the singularity transitions M , SC ,Sω and SN . Figure 2.27 shows how these boundary curves interact in the transition throughΛ = ΛC in a neighbourhood of the respective point on the boundary of the (∆1 , ∆2 )-square.There are two cases: one for the orange regions and one for the grey regions in ﬁgure 2.26. Row (a) of ﬁgure 2.27 shows the transition for the grey regions, which involves the (grey)cubic tangency curve Cd ; labels in the regions indicate the respective EFM surface type. Beforethe bifurcation for Λ < ΛC the grey region extends to two points on the boundary of the(∆1 , ∆2 )-square. As we have seen in ﬁgure 2.17, these points are the limits of the pair ofcurves M and SC and of the the pair of curves Sω and SN , respectively; see ﬁgure 2.27 (a1).Notice further that the singularity transition curves SN and SC end at a (purple) codimension-two point DCN C on the cubic tangency curve Cd ; compare with ﬁgure 2.23. The EFM surfacetypes in the respective regions are also shown. When ΛC is approached, the two limit pointsof the pairs of curves M and SC and Sω and SN approach each other. At the same time,the codimension-two point, and the curve Ca with it, approach the boundary of the (∆1 , ∆2 )-square; as a result, the curves SN and SC become shorter. At the moment of transition at
- 72. 56 Chapter 2. Classiﬁcation of EFM structure . 0.97 DCM ω ∆2 B B M Sω BI Bb B bb DCM ω Bb B Cd 0.26. 0.26 ∆1 0.97Figure 2.28. Grey island for Λ = 0.1 in the (∆1 , ∆2 )-plane with regions of non-banded EFM surfacetypes; compare with ﬁgure 2.26 (e).Λ = ΛC , shown in ﬁgure 2.27 (a2), the grey region is bounded by the minimax transition curveM and by one branch of the cubic tangency curve Cd , which both end at a single point on theboundary. The curve Sω and a second branch of Cd also end at this point on the boundary. ForΛ > ΛC , as in ﬁgure 2.27 (a3), all curves detach from the boundary of the (∆1 , ∆2 )-square;furthermore, the singularity transition curves M and Sω are now attached to the cubic tangencycurve Cd at a (golden) codimension-two point DCM ω ; compare with ﬁgure 2.24. Hence, thegrey island created in this transition, which is invariant under reﬂection in the diagonal, isbounded by the curves Cd and M . Row (b) of ﬁgure 2.27 shows the transition for the orange regions, which involves the(black) cubic tangency curve Ca ; again, the respective EFM surface type are indicated. Thistransition is very similar to that in ﬁgure 2.27 (a), but notice that it now involves a (golden)codimension-two point DCM ω on the curve Ca for Λ < ΛC , and a (purple) codimension-two point DCN C for Λ > ΛC . As a result of this transition the orange island created in thistransition, which is invariant under reﬂection in the anti-diagonal, is bounded by the curves Caand SN .2.4.2 Islands of non-banded EFM surface typesThe grey islands are associated with the diagonal where ∆1 = ∆2 , along which the 2FOF laserreduces to the single FOF laser with effective feedback rate given by (2.11). Hence, the widthof the grey islands along the diagonal is determined by the intersection points of the horizontal
- 73. 2.4. Dependence of the EFM surface bifurcation diagram on the ﬁlter width Λ 57line for the given value of Λ with the curve in ﬁgure 2.8 (c). A grey island for Λ = 0.1 is shownin ﬁgure 2.28. It contains regions of (non-banded) EFM surface types B b , B bb and BI, whichare bounded by the cubic tangency curve Cd and singularity transition curves M (orange) andSω (green); the latter curves end at two (golden) points DCM ω on the curve Cd . When Λ isincreased, the grey islands remain topologically the same and simply shrink down to a point.This happens when Λ has the value of the cusp points of the curve in ﬁgure 2.8 (c), which canagain be computed analytically from the formula in [28] for the single FOF laser as 4 ΛI = 2 ΛC = √ κ 1 + α2 ≈ 0.196261. (2.20) 3 3Furthermore, we can conclude from the single FOF limit along the diagonal that the two sym-metric grey islands disappear at √ 8 2 ∆1 = ∆2 = ∆Id = √ κ 1 + α2 ≈ 0.555111. (2.21) 3 3For Λ > ΛI the EFM surface bifurcation diagram in the (∆1 , ∆2 )-square — and, hence, alsoin the (∆1 , ∆2 )-plane — does no longer contain (grey) islands that are symmetric with respectto the diagonal. The orange islands that exist for Λ > ΛC are associated with the anti-diagonal where∆1 = −∆2 . Figure 2.29 (a) shows an orange island for Λ = 0.1 with regions of (non-banded)EFM surface types; see also the enlargement in the inset panel. Apart from the cubic tangencycurve Ca , we ﬁnd curves SC (blue) and SN (red) of singularity transition. Two curves SCemerge from a boundary point of the island on the anti-diagonal where four branches of Caconnect in a pair of cusps. The two curves SC follow two branches of Ca closely and end attwo (purple) points DCN C on the curve Ca . The two curves SN emerge from a different pointon the anti-diagonal and also follows Ca closely to the same two end points. The (red) sectionof the anti-diagonal in between the two points from which the curves SC and SN emerge,respectively, corresponds to a degenerate saddle transition SN . The remainder of the anti-diagonal corresponds to a degenerate saddle transition SC ; see the discussion in section 2.3.4.Overall, we ﬁnd a quite complicated but consistent structure of the orange island. It features the(non-banded) EFM surface types Bh , hBh , B b , B bb , hB b , bB b , hB bb , bB bb and bbB bb , of whichthe last ﬁve EFM types with bulges are new; compare with ﬁgure 2.21. As Λ is increased, the orange island undergoes topological changes. First, the (purple)end points of the curves SC and SN move across a branch of the cubic tangency locus Ca .As a result, the entire island is now bounded by Ca and the regions Bh and B bb (and theirsymmetric counterparts) disappear; see ﬁgure 2.29 (b). The next qualitative change concernsthe cubic tangency locus Ca , which loses two intersection points, resulting in the loss of two(symmetrically related) regions of EFM types bB and B b ; see ﬁgure 2.29 (c). When Λ is
- 74. 58 Chapter 2. Classiﬁcation of EFM structure . ..0.28 (a) -0.05 (b) Ca Ca B B ∆2 Bb b Bh Bb b b b B hB b SC SC Bh SN DC hBb b B B bb NC bb B b hBh hB bb b SN B B bb hB/ B bb B e e ¢ b DCN C ¢¢ hB DCN C ¢ ¢ B b b B DCN C ¢ b bb B bb bb B B b B-0.97 . -0.45. 0.28 -0.97 0.05 0.45 . .-0.05 -0.15 (c) (d) B B Ca Ca ∆2 Bh Bh SC Bb SC hB hB b Bb B b ¢ B ¢ ¢ B b b B B-0.45. -0.35.. 0.05 ∆1 0.45 0.15 ∆1 0.35Figure 2.29. Orange island in the (∆1 , ∆2 )-plane with regions of non-banded EFM surface types; theinset 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).increased further, two intersection points of curves Ca on the anti-diagonal come together andmerge into a point where four branches of Ca connect. The result is the loss of region bB b ; seeﬁgure 2.29 (d). We found that the orange island does not undergo further qualitative changes,but rather shrinks down to a point and disappears. This happens at Λ =≈ 0.196261, andthis numerical value agrees up to numerical precision with that for ΛI from (2.20). In fact,consideration of (2.9) for ∆1 = −∆2 conﬁrms this observation; furthermore, the position ofwhere the islands disappear can be computed as √ 2 2 ∆1 = −∆2 = ∆Ia = ∆Id /4 = √ κ 1 + α2 ≈ 0.138778. (2.22) 3 3
- 75. 2.5. The effect of changing the delay difference ∆τ 59 Overall we conclude that for Λ > ΛI there are no islands at all, so that the entire (∆1 , ∆2 )-plane consists of a single region of EFM type B. Physically, this means that the two ﬁlters areso wide that they reﬂect light of different frequencies effectively in the same way; hence, thefeedback is no longer frequency selective, and the 2FOF laser is a 2COF laser for sufﬁcientlylarge Λ.2.5 The effect of changing the delay difference ∆τEquation (2.16) for the boundary of the projection of the EFM surface onto the (ωs , dCp )-planeexpresses dCp as a function of ωs . In this equation the delay time difference dτ appears onlyas the coefﬁcient of the linear term of ωs ; hence, a nonzero value of dτ introduces a shearingof the EFM surface with a shear rate of exactly dτ . This shearing of the EFM surface can be made explicit for the special case that ∆1 = ∆2 ,when still considering a single ﬁlter width Λ for both ﬁlters. Namely, then we can deﬁne thecenter line of the projection of the EFM surface as the line through the points where the inversecosine term in (2.16) vanishes. The equation for this center line is then simply dCp (ωs ) = −dτ ωs , (2.23)which is simply the line with slope dτ through the origin of the (ωs , dCp )-plane. Since theωs -range of the EFM surface does not change with dτ , a nonzero dτ indeed leads to a shearwith a shear rate equal to the slope of the center line of the projection of the EFM surface. We conclude that an EFM surface for dτ = 0 can be obtained simply by considering thecorresponding EFM surface for dτ = 0 and shearing it with a shear rate of dτ . As a result ofthis shearing, EFM components may be present for dτ = 0 that are not present for dτ = 0. Figure 2.30 illustrates this effect with the example of the EFM surface for ∆1 = ∆2 = 0and Λ = 0.015 with τ1 = 500 and τ2 = 600, so that the shear rate is dτ = 100. Noticethat this EFM surface is the sheared version of the corresponding EFM surface for dτ = 0 inﬁgure 2.5, which is of type B. The EFM surface in ﬁgure 2.30 still consists of all 2π-translatesof a basic unit, which are connected at the points where dCp = π + 2kπ for k ∈ Z; however,now the basic unit of the EFM surface extends over a dCp -range of more than 2π. Whilethere is always a single EFM component for any value of dCp for the case that dτ = 0, dueto the shear for dτ = 100 we now ﬁnd up to three EFM components in ﬁgure 2.30. Each ofthose EFM components belongs to a different 2π-translated copy of the basic unit of the EFMsurface. Physically, this is due to beating between two frequencies that are associated with thetwo feedback loops, of different delay times; see Eqs. (2.2)–(2.4).
- 76. 60 Chapter 2. Classiﬁcation of EFM structure . 0.02 (a) Ns 0 -0.02 3 dCp π 0 -1 0.1 ωs 0 -3 -0.1 0.02 0.02 (b) (c) Ns Ns 0 0 −0.02 −0.02 −0.1 0 ωs 0.1 −0.1 0 ωs 0.1 0.02 0.02 (d) (e) Ns Ns 0 0 −0.02 −0.02. −0.1 0 ωs 0.1 −0.1 0 ωs 0.1Figure 2.30. The EFM surface (a) for κ = 0.05, ∆1 = ∆2 = 0, Λ = 0.015, and τ1 = 500 and τ2 = 600so that dτ = 100, and its intersection with the planes deﬁned by dCp = 0 and dCp = −π; comparewith ﬁgure 2.5. Panels (b)–(e) show the EFM-components for dCp = 0, dCp = −π/2, dCp = −π and 1dCp = −3π/2, respectively; the blue dots are the EFMs for Cp as given by (2.12). As is shown in ﬁgure 2.30 (b)–(e), the exact number of EFM components depends on thevalue of dCp . More speciﬁcally, for dCp = 0 there are three EFM-components, owing to thefact that the corresponding plane in (ωs , dCp , Ns )-space intersects three copies of the basicunit of the EFM surface; see ﬁgure 2.30 (a) and (b). As dCp is changed the EFM componentschange. For dCp = −π/2 as in panel (c), there are still three EFM components (the right and
- 77. 2.5. The effect of changing the delay difference ∆τ 61 . 0.5 0.5 (a1) (b1)T (ωs ) T (ωs ) 0 0 −0.5 −0.5 −0.1 0 ωs 0.1 −0.1 0 ωs 0.1 0.02 0.02 (a2) (b2) Ns Ns 0 0 −0.02 −0.02. −0.1 0 ωs 0.1 −0.1 0 ωs 0.1Figure 2.31. Solution curves of the transcendental equation (2.3) and corresponding EFM components 1for 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 thatthe EFM components are in fact disjoint.central EFM components are not connected), but left of the two outer ones has become smallerand right one larger. When dCp is decreased further, the left EFM component disappears (in aminimax transition when the plane dCp = const passes through the end point of the respectivepart of the EFM surface). For dCp = −π as in ﬁgure 2.30 (d), the two remaining EFMcomponents connect at the origin of the (ωs , Ns )-plane to form a single EFM component in theshape of a ﬁgure eight. We remark that, because for dCp = −π equation (2.9) is equal to 0 atωs = 0, this case for the 2FOF laser corresponds to the situation described in [64] where singleFOF laser is resonant to the minimum of the Fabry-Pérot ﬁlter reﬂectance proﬁle. The singleEFM component then splits up again into two EFM components for dCp < −π. A new EFMcomponent appears (again in a minimax transition) on the right, that is, for positive ωs , so thatthere are again three EFM components; see ﬁgure 2.30 (e) for dCp = −3π/2. Notice that, since the detunings of both ﬁlters are equal, the EFM surface in ﬁgure 2.30 (a)is invariant under the anti-diagonal symmetry operation (ωs , dCp , Ns ) → (−ωs , −dCp , −Ns ).As a result, panel (b) and (d) are invariant under rotation over π of the (ωs , Ns )-plane, whilepanels (c) and (e) are symmetric counterparts. Figure 2.31 shows that with an increased shear of the EFM surface of type B, more EFMcomponents can be found; speciﬁcally, up to ﬁve for dτ = 200 and up to seven for dτ = 300.This is illustrated in ﬁgure 2.31 by the EFM solution curve inside its envelope (top row) and
- 78. 62 Chapter 2. Classiﬁcation of EFM structure . 1.5 0.04 (a1) (a2)dCp π Ns 0 0 −1.6 −3.3 −0.03 −0.15 0 ωs 0.16 −0.15 0 ωs 0.16 1.5 0.04 (b1) (b2)dCp π Ns 0 0 −1.6 −3.3 −0.03. −0.15 0 ωs 0.16 −0.15 0 ωs 0.16Figure 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 fordCp = −1.6π (b2). Here κ = 0.05, ∆1 = 0.13, ∆2 = −0.1, Λ = 0.01, and τ1 = 500.by the respective EFM components in the (ωs , Ns )-plane (bottom row). Figure 2.31 clearlyshows that for sufﬁciently large dτ any number of EFM components can be found, even whenthe EFM surface for dτ = 0 is of the simplest possible type B. Finally, ﬁgure 2.32 shows the effect of shearing on the more complicated EFM surface oftype hBB. As panels (a1) and (a2) show, this EFM surface type for dτ = 0 may give rise toup to three EFM components. When it is sheared with a sheer rate of dτ = 230, however, onemay ﬁnd up to six EFM components. Notice that this increase in the number of possible EFMcomponents is due to the shearing of the strongly undulating boundary of central band, as wellas to the shearing of the holes in it, which now extend over a dCp range of more than 2π.2.6 Conclusion and outlookWe presented a comprehensive study of the EFM structure of the 2FOF in dependence on theparameters of the two ﬁltered feedback loops. The main object of study is the EFM surfacein (ωs , dCp , Ns )-space, which is a generalisation of the EFM components for the single FOFlaser. A combination of analysis and extensive numerical continuations of EFMs allowed us
- 79. 2.6. Conclusion and outlook 63to present the EFM surface bifurcation diagram in the (∆1 , ∆2 )-plane for the special case thatthe delay time difference dτ is zero. More speciﬁcally, we considered ﬁve transitions throughcritical points and a cubic tangency with respect to the phase difference dCp between the twoﬁlter loops, which emerged as a key parameter. These transition give rise to boundary curvesthat divide the (∆1 , ∆2 )-plane into regions of different EFM surface types. We then studiedhow the EFM surface bifurcation diagram in the (∆1 , ∆2 )-plane depends on the width Λ ofthe two ﬁlters. Finally, nonzero dτ was shown to cause a shearing of the corresponding EFMsurface for dτ = 0, and this may give rise to any number of EFM components. Our classiﬁcation of the EFM surface is justiﬁed by physical properties of the 2FOF laser,which translate to speciﬁc properties of the transcendental equations for the EFMs. For exam-ple, there are no islands with holes. What is more, the classiﬁcation of the EFM surface asinduced by the ﬁve singularity transitions considered was motivated by the fact that they couldall be shown to lead to changes of the number of EFM components over substantial ranges ofthe relevant parameters, in particular, dCp . In a way, the inﬂuence of the two ﬁlter loops on the laser can be considered as the feed-back from a single feedback loop with a complicated ﬁlter proﬁle — with several maxima andminima of the transmittance as a function of the frequency of the light. This ﬁlter proﬁle is theresult of interference between the two ﬁlter ﬁelds. This point of view provides a connection tostudies that considered the output of a laser subject to feedback from a non-Lorentzian, morecomplicated ﬁlter proﬁle, such as the periodic ﬁlter proﬁles in [50, 64]. In the case of [64],feedback is considered from a Fabry-Pérot cavity not only close to the transmittance maxi-mum, but over a large spectral range that encompasses several maxima and minima of thetransmittance. In [50], on the other hand, ﬁltered feedback with several maxima and minima ofthe transmittance arise due to side ‘bumps’ of a ﬁbre Bragg grating ﬁlter proﬁle. As we haveobserved, the observations in [50] and [64] correspond in the 2FOF laser simply to a changefrom constructive to destructive interference between the two ﬁlter ﬁelds via a variation of theﬁlter phase difference dCp . A further analysis of the connections between the 2FOF laser andfeedback from other types of ﬁlters is an interesting question for future research. Obviously, an important practical issue is to determine when the EFMs are actually stable.In other words, the next step in the analysis of the 2FOF laser, which is the subject of the nextchapter, is hence the investigation of regions of their stability on the EFM surface.
- 80. Chapter 3EFM stability regionsIn the previous chapter we investigated analytically and numerically the structure of externalﬁltered modes (EFMs) – constant amplitude and frequency solutions to Eqs. (1)–(4). Ouranalysis shows that, in the conveniently chosen (ωs , Ns , dCp )-space, EFMs exist on a compli-cated two-dimensional surface that may consist of up to three disjoint components that are 2πperiodic in the dCp -direction. The EFM surface illustrates how frequency ωs and population inversion Ns of the EFMs 1 2depend on the feedback phases Cp and Cp , with all other parameters ﬁxed. In other words,the EFM surface represents all the possible states that an EFM can take for all the possible 1 2values of Cp and Cp , for given values of the other parameters. The important parameter here 2 1is the feedback phase difference dCp = Cp − Cp , so that it is most convenient to consider thetwo-dimensional EFM surface in (ωs , Ns , dCp )-space; see section 2.2. In this chapter we analyse the stability of the EFMs. In particular, we use bifurcation analy-sis to uncover regions of stable EFMs on the two-dimensional EFM surface in (ωs , Ns , dCp )-space. These regions are bounded by codimension-one saddle-node and Hopf bifurcationcurves, which meet or intersect at codimension-two saddle-node Hopf, Bogdanov-Takens andHopf-Hopf bifurcation points. As is the case for the EFMs themselves, their stability depends on all laser, ﬁlters andfeedback loop parameters. Here we show how the number, shape, and extent of the stabilityregions depend on the ﬁlters and feedback loops parameters. More speciﬁcally, we considertheir dependence on the feedback rates κ1 and κ2 , ﬁlter widths Λ1 and Λ2 , the ﬁlter frequencydetunings ∆1 and ∆2 , and the delay times τ1 and τ2 . We explain the uncovered changes ofthe EFM stability regions in terms of higher codimension bifurcations in conjunction with thechanging geometry of the EFM surface. To analyse the stability of the EFMs we ﬁx all the system parameters except the feedback 1 2phases Cp and Cp . By means of numerical continuation we then calculate the EFM surface 65
- 81. 66 Chapter 3. EFM stability regionstogether with information about stability of the EFMs. This allows us to render the EFM surfacein (ωs , Ns , dCp )-space with stability information on it. Throughout, regions with stable EFMsare marked as hatched green patches on the EFM surface; the remaining grey area representsunstable EFMs. In fact, to indicate the stable EFMs area on three-dimensional image of theEFM surface we use the actual stable parts of the computed EFM branches. In other words,each green line on the semitransparent grey EFM surface is the result of a single numericalcontinuation. To explain and illustrate our results, we also consider the two-dimensional projections ofthe EFM surface onto the (ωs , dCp )-plane and the (ωs , Ns )-plane. In section 2.3 we used the(ωs , dCp )-projections to present our classiﬁcation into different types of the EFM surface. Asbefore, the white regions on the (ωs , Ns )-plane are regions where no EFM exist. The boundaryof the projections of the EFM surface onto the the (ωs , dCp )-plane is found directly from Eq.(2.16) and is now coloured in grey instead of blue. The blue region that represented the insideof the projection in section 2.3 is now divided into regions of stable and unstable EFMs thatare coloured green and grey, respectively. This colouring of the (ωs , dCp )-projections is inagreement with the colouring of the EFM surface itself. For clarity, we use solid green ﬁllingto indicate the stable EFM regions in the projections, instead of hatching. As in chapter 2, both the top and the bottom sheet of the EFM surface are projected onto the(ωs , dCp )-plane; this means that each typical (non-boundary) point in the (ωs , dCp )-projectionof the EFM surface correspond to two EFMs with different Ns values. We show that the EFMstability region extends mostly over the bottom sheet of the EFM surface, which correspondsto higher gain modes. However, for some parameter values stable EFMs can also be foundon the top sheet of the EFM surface. In those rare cases when the two stable EFM regions areseparated by Ns i.e. lay above one another, the corresponding area in the (ωs , dCp )-projectionsis coloured in dark green. The boundary of a region of stable EFMs involves a number of saddle-node bifurcationsand Hopf bifurcation curves that are connected at codimension-two bifurcations points. Thesebifurcations form curves on the EFM surface in (ωs , Ns , dCp )-space. Those curves intersectat Hopf-Hopf and saddle-node Hopf bifurcations. At points where the saddle-node and Hopfbifurcation curves merge, and the Hopf bifurcation curve ends, Bogdanov-Takens bifurcationsare located. Note that for clarity, out of all codimension-two bifurcations, we only label theBogdanov-Takens bifurcations. Throughout this chapter saddle-node bifurcation curves are depicted as blue curves on theEFM surface and the Hopf bifurcation curves are shown as red curves on the EFM surface.Moreover, for the purpose of this analysis we calculate only the bifurcation curves that arepart of the boundary of the EFMs stability regions, or are useful to explain their changes. Allregions and curves in the projections of the EFM surface onto the (ωs , Ns )-plane, are colouredin the same way as in the (ωs , dCp )-projections.
- 82. 3.1. Dependence of EFM stability on κ 67 In chapter 2 we showed that the geometrical and topological properties of the EFM surfacedepend strongly on the ﬁlters and feedback loop parameters. In this chapter we analyse how thestability regions interact with the singularity transitions of the EFM surface in (ωs , Ns , dCp )-space. As a guideline in this challenging task we use the classiﬁcation of the EFM surfacesintroduced in section 2.3. In particular, we are interested in changes of the EFM stabilityregions associated with the saddle transitions of the EFM surface. Clearly, changes of thestable EFM regions must be expected as a result of changes of the EFM surface topology.However, there may also be changes to the EFM stability regions that are due to changes of theEFM surface geometric properties. In what follows we analyse separately the dependence of the EFM surface on the commonfeedback rate κ, the common ﬁlter width Λ, the ﬁlter detunings ∆1 and ∆2 , and the differencedτ between the delay times. To uncover the information on the stability of EFMs, we ﬁrst ﬁxparameters at conveniently chosen values and compute the EFM surface with relevant informa-tion on the EFMs stability. Next, we vary the parameter under investigation, and observe howthe EFM stability regions transform with changes of the EFM surface in (ωs , Ns , dCp )-space. This chapter is structured as follows. In section 3.1 we analyse how regions of stable EFMschange with an increase of the feedback rate κ. Next, in section 3.2 we present how the stableregions are affected by the transition from the 2FOF laser to the COF laser as Λ → ∞. Boththese chapters show that, although topologically the EFM surface remains unchanged of typeB, the stability of EFMs is substantially affected by κ as well as Λ. Moreover, we use theresults from sections 3.1 and 3.2 to choose convenient values of κ and Λ for the followingsections. Section 3.3 shows how the regions of stable EFMs split as the EFM surface graduallytransforms from B via hBh and B BB to I BI with increasing ﬁlter frequency detunings ∆1and ∆2 . In section 3.4 we explain how the EFM stability is affected by shearing of the EFMsurface that is introduced by decreasing dτ . Finally, we brieﬂy describe the kinds of periodicsolutions originating from the Hopf bifurcations in section 3.5, and discuss possible physicalconsequences of the transformations of the EFMs stability regions in section 4.1.3.1 Dependence of EFM stability on κWe start our analysis with the 2FOF laser with two identical ﬁlters with equal delay times. Insection 2.2 we showed that in this case the EFM surface is of the simplest type B, and con-sist of inﬁnitely many dCp -periodic compact units that are connected with each other in thepoints, (ωs , Ns , dCp ) = (0, 0, (2n + 1)π) with n ∈ Z. In this section, we explain the depen-dence of the EFM surface in (ωs , Ns , dCp )-space on the feedback rate κ. While increasing κcauses an expansion of the EFM surface in the ωs and Ns directions, its topology is unaffected.Nevertheless, we demonstrate that the expansion of the EFM surface leads to a change in the
- 83. 68 Chapter 3. EFM stability regionsnumber, extent, and type of boundaries of the EFM stability regions. What is more, we showthat above a particular value of κ, despite further expansion of the EFM surface, the (ωs , Ns )-projection of the EFM stability region remains unchanged. For this study, we ﬁx Λ = 0.015,∆1 = 0, ∆2 = 0, and dτ = 0. Figure 3.1 (a1)–(c1) shows the fundamental 2π interval of the EFM surface for κ = 0.01in (a), κ = 0.015 in (b) and κ = 0.025 in (c). In panels (a2)–(c2) and (a3)–(c3) we showprojections of the stability regions of EFMs onto the (ωs , Ns )-plane and onto the (ωs , dCp )-plane, respectively. In ﬁgure 3.1 (a)–(c) the boundary between regions of stable EFMs and unstable EFMs al-ways consist of seven Hopf bifurcation curves and one saddle-node bifurcation curve. Intersec-tions of those codimension-one bifurcation curves are codimension-two Hopf-Hopf and saddle-node Hopf bifurcation points. Note that for these parameter values considered Bogdanov-Takens bifurcations do not occur. Increasing κ results in an expansion of the EFM surface in the ωs and Ns directions; seepanels (a1)–(c1). To understand the expansion of the EFM surface, it is enough to notice thatNs scales linearly with κ; see Eq. (2.5). Moreover, for ∆1 = ∆2 = 0, a common ﬁlter widthΛ, and dτ = 0, the extrema of ωs are given by ext ωs = ±1/2 −2 Λ2 + 2 Λ Λ2 + 16 κ2 + 16 κ2 α2 . (3.1) Concurrently with the monotonic growth of extrema of ωs , the EFM stability region changes.When the oval shaped Hopf bifurcation curve around the point (ωs , dCp ) = (0, 0) in ﬁgure 3.1(a3) — the one that forms the lower boundary of the stability region in panel (a2) — intersectswith two other Hopf bifurcation curves, then the single region of stable EFMs from panel (a)splits into two; see ﬁgure 3.1 (b). A further increase of κ causes the separation distance of thosetwo regions to grow; compare ﬁgure 3.1 rows (b) and (c). What is more, although the range ofdCp in which stable EFMs exist decreases and the whole EFM surface grows, the projectionof the EFM stability regions onto the (ωs , Ns )-plane in panels (b) and (c) stays unchanged.These results are in agreement with the observation that the 2FOF laser can be reduced to aFOF laser; see also section 2.2. The expansion of the EFM surface in the ωs and Ns directions is the simplest geometricaleffect caused by a change of ﬁlter and feedback loop parameters. We showed that, although thetopology of the EFM surface remains unchanged of type B, increasing κ results in a shrinkingof the stable EFM regions in the (ωs , dCp )-plane. Considering that we are interested in thedependence of the EFMs stability region on parameters, in all following sections we ﬁx thefeedback rate at κ = 0.01 — a representative value at which the region of stable EFMs extendsover the whole fundamental 2π interval of dCp .
- 84. 3.1. Dependence of EFM stability on κ 69 0.008 (a1) (a2) . κ = 0.01 NsNs 00 −0.008 −0.04 0 ωs 0.04 1 dCp (a3) π 0 dCp /π 0 ωs −1 −0.04 0 ωs 0.04 0 0.008 (b1) (b2) κ = 0.015 NsNs 00 −0.008 −0.04 0 ωs 0.04 1 (b3) dCp π 0 dCp /π 0 −1 0 ωs −0.04 0 ωs 0.04 0.008 (c1) (c2) κ = 0.025 NsNs 00 −0.008 −0.04 0 ωs 0.04 1 dCp π (c3) 0 dCp /π 0 ωs −1 ωs. 0 −0.04 0 0.04Figure 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) showcorresponding projections of the EFM surface onto the (ωs , Ns )-plane and panels (a3)–(c3) onto the(ωs , dCp )-plane. Regions of stable EFMs (green) are bounded by Hopf bifurcations curves (red) andsaddle node bifurcation curves (blue). In panels (a1)–(c1) ωs ∈ [−0.065, 0.065], dCp /π ∈ [−1, 1] andNs ∈ [−0.013, 0.013].
- 85. 70 Chapter 3. EFM stability regions . 0.021 Ns 0.01 Λ = 0.005 0 Λ = 0.001 Λ = 0.015 Λ = 0.025 −0.01 Λ = 0.5 −0.021 . −0.11 −0.05 0 0.05 ωs 0.011Figure 3.2. Dependence of the EFM surface on the ﬁlter width Λ = Λ1 = Λ2 (as indicated); here∆1 = ∆2 = 0, κ = 0.01 and dτ = 0. Light grey regions with black envelopes are projections, for ﬁvedifferent values of Λ, of the EFM surface onto the (ωs , Ns )-plane.3.2 Dependence of EFM stability on ΛIn sections 2.3.1 and 2.4 we showed that, as Λ → ∞, the 2FOF laser approaches the COF laserlimit. More speciﬁcally, we showed that increasing Λ leads to the simpliﬁcation of the topologyof the EFM surface — regions corresponding to EFM surface types other then B shrink anddisappear. We now explore the dependence of EFM stability regions on the ﬁlter width Λ.We demonstrate that the fast growth of the EFM surface that leads to the simpliﬁcation of itstopology, results also in changes of the EFM stability. In particular, we show that the stableEFM region expands with increasing ﬁlter width; for sufﬁciently high Λ it is bounded by asingle saddle-node bifurcation curve and a single Hopf bifurcation curve. Throughout thissection we ﬁx κ = 0.01, ∆1 = 0, ∆2 = 0, and dτ = 0. Notice that, as in the previoussection, for ∆1 = 0 and ∆2 = 0 copies of the EFM surface are connected at the points(ωs , Ns , dCp ) = (0, 0, (2n + 1)π) with n ∈ Z. Figure 3.2 illustrates the expansion of the EFM surface with increasing Λ. The black linesare the envelopes of ﬁve projections of the EFM surface (grey) onto the (ωs , Ns )-plane fordifferent values of Λ, as indicated on the ﬁgure. Note that the range of ωs of the EFM surfacefor Λ = 0.5 is ten times larger then the range of the ωs of the EFM surface for Λ = 0.001. Notealso that all the envelopes have two common points at ωs = 0. The difference between the ωs - COF )−range of the envelope for Λ = 0.5 and for the EFM components of COF laser is |max(ωs COF )|−|max(ω Λ=0.5 )−min(ω Λ=0.5 )| = 0.004; here ω Λ=0.5 is given by Eq. (3.1) andmin(ωs s s s √ COF = ±κ 1 + α2 . This shows that already for Λ = 0.5 the 2FOF laser with two identicalωsﬁlters with equal delay times is very close to the COF limit of Λ = ∞.
- 86. 3.2. Dependence of EFM stability on Λ 71 We now investigate how the region of stable EFMs is affected by this expansion of theEFM surface with Λ. Figure 3.3 (a1)–(d1) shows the fundamental 2π interval of the EFMsurface together with the EFM stability information for Λ = 0.001 in (a), Λ = 0.005 in (b),Λ = 0.025 in (c) and Λ = 0.5 in (d). In panels (a2)–(d2) we show corresponding projections ofthe EFM surface onto the (ωs , Ns )-plane. Projections of the EFM surfaces from ﬁgure 3.3 ontothe (ωs , dCp )-plane are shown in ﬁgure 3.4. Black dots indicate codimension-two Bogdanov-Takens bifurcation points; curves and regions are coloured as in ﬁgure 3.1. Given the expansion of the EFM surface, each row in ﬁgure 3.3 and corresponding panelin ﬁgure 3.4 is presented over a different range of the ωs and Ns axes. (This is in contrast toﬁgure 3.2, where the (ωs , Ns )-projections of all the EFM surfaces from Figure 3.3 (a)–(d) areshown together on the same scale.) Note also that, ﬁgure 3.2 also includes the projection of theEFM surface from ﬁgure 3.1 (a) for Λ = 0.015. With increasing Λ the region of stable EFMs and its boundary undergo several complicatedtransitions; see ﬁgures 3.3 (a)–(d) and 3.4 (a)–(d). Figures 3.3 (a) and 3.4 (a), for Λ = 0.001,show the EFM stability region bounded by one saddle-node bifurcation curve (blue) and threeHopf bifurcation curves (red). Moreover, in addition to Hopf-Hopf (HH) and saddle-node Hopf(SH) bifurcation points located at the intersections of the codimension-one bifurcation curves,one can also ﬁnd Bogdanov-Takens (BT) bifurcation points; see ﬁgure 3.3 (a) and (b). Further, in ﬁgures 3.3 (b) and 3.4 (b), for Λ = 0.005, the picture becomes more complicatedand the number of the Hopf bifurcation curves involved in the boundary increases to ﬁve. Next,for sufﬁciently high Λ, the boundary of the stable EFM regions again simpliﬁes, and consistof a single saddle-node bifurcation curve at the top and a single Hopf bifurcation curve at thebottom. Moreover, the region of stable EFMs expands and covers most of the projection of theEFM surface onto the (ωs , dCp )-plane; see ﬁgures 3.3 (c) and 3.4 (c) for Λ = 0.025. Finally, in ﬁgures 3.3 (d) and 3.4 (d), we show that with a further increase of Λ the EFMstability region remains qualitatively unchanged. In comparison to ﬁgures 3.3 (c) and 3.4 (c),we observe a further expansion of the EFM surface and growth of the region of unstable EFMsin the direction of positive ωs .
- 87. 72 Chapter 3. EFM stability regions 0.005 (a1) (a2) .Ns Λ = 0.001 Ns 0 0 −0.005 dCp /π 0 ωs −0.011 0 ωs 0.011 0 0.005 (b1) (b2)Ns Λ = 0.005 Ns 0 0 −0.005 dCp /π 0 ωs −0.025 0 ωs 0.025 0 0.01 (c1) (c2)Ns Λ = 0.025 Ns 0 0 −0.01 dCp /π 0 ωs −0.05 0 ωs 0.05 0 0.021 (d1) (d2)Ns Λ = 0.5 Ns 0 0 −0.021 . dCp /π 0 ωs −0.11 0 ωs 0.11 0Figure 3.3. Dependence of the stability region on the EFM surface on the common ﬁlter width Λ (asindicated 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 (semitransparent grey) together with information about the stability ofthe EFMs. Panels (a2)–(d2) show corresponding projections of the EFM surface onto the (ωs , Ns )-plane. Black dots indicate codimension-two Bogdanov-Takens bifurcation points; curves and regionsare coloured as in ﬁgure 3.1. In panels (a1)–(d1) dCp ∈ [−π, π], and the ranges of Ns and ωs are as inpanels (a2)–(d2).
- 88. 3.2. Dependence of EFM stability on Λ 73 . 1 1 (a) (b)dCp dCp π π 0 0 −1 −1 −0.011 0 ωs 0.011 −0.025 0 ωs 0.025 1 1 (c) (d)dCp dCp π π 0 0 −1 −1. −0.05 0 ωs 0.05 −0.11 0 ωs 0.11Figure 3.4. Projections of the EFM surfaces presented in ﬁgure 3.3 (a1)–(d1) onto the (ωs , dCp )-plane. Black dots indicate codimension-two Bogdanov-Takens bifurcation points; curves and regionsare coloured as in ﬁgure 3.1. By increasing Λ we move between two limiting cases. For Λ = 0.001 our ﬁndings are inqualitative agreement with results for thlocus semiconductor laser subject to optical injection[72], whose locking region is bounded by the saddle-node curve and two Hopf curves thatoriginate from the BT points. On the other hand, for Λ = 0.5 we are in agreement with theﬁndings for the COF laser [57, 30, 43, 47, 38]. The region of stable EFMs is shifted towards thelower values of the population inversion Ns which correspond to a stable maximum gain modesin the COF laser [47]. Moreover, similarly to the COF laser case, stable EFMs in ﬁgure 3.3 (d)become unstable in Hopf bifurcation that give rise to relaxation oscillations [43, 47, 62]; seesection 3.5. Figures 3.3 and 3.4 showed that the number of the Hopf bifurcation curves that form theboundary of the EFMs stability regions depends strongly on Λ. However, the details of thesechanges are not shown by these ﬁgures. To explain the observed transitions in more detail, wenow consider changes to the number of Hopf bifurcation curves bounding the EFMs stabilityregions, as well as to the number and type of the codimension-two bifurcations points involved
- 89. 74 Chapter 3. EFM stability regionsin the boundary. Figure 3.5 shows how the boundary of the region of stable EFMs changes with an increaseof the ﬁlter width from Λ = 0.001 to Λ = 0.025. Our ﬁndings are presented in the (ωs , Ns )-projections of the EFM surface. Black dots indicate codimension-two BT bifurcation points;curves and regions are coloured as in ﬁgure 3.1. For clarity, the different Hopf bifurcationscurves are labeled H1 to H5 in panels (a)–(b) and H1 to H7 in panels (e)–(j). The number of the Hopf bifurcation curves changes due to transitions of the EFM surface 1for ﬁxed Λ through extrema and saddles of the Hopf surfaces in (Cp , dCp , Λ)-space, whichalso manifest themselves in (ωs , Ns , Λ)-space — called also minimax transitions and saddletransitions, respectively [17]. Moreover, changes of type and number of codimension-twobifurcation points on the boundary of stable EFM region are associated with transitions throughcodimension-three points, as discussed below. In ﬁgure 3.5 (a)–(d) we show how with increasing Λ the number of the Hopf bifurcationscurves that form the boundary of the EFMs stability region increases from three to ﬁve. Ad-ditionally, we show that the change in the number of Hopf bifurcation curves constituting theboundary of EFMs stability regions is followed by the disappearance of four BT bifurcationpoints and the emergence of four HH bifurcation points and four SH bifurcation points. Notethat in the (ωs , Ns )-projections only half of the mentioned codimension-two points is visible;compare with ﬁgure 3.4. Figure 3.5 (a)–(b) are projections of the EFM surface onto the (ωs , Ns )-plane for Λ =0.003 and Λ = 0.004. In comparison to ﬁgure 3.3 (a) we show two additional Hopf bifurcationcurves, H4 and H5 , that become part of the boundary in ﬁgure 3.5 (c) for Λ = 0.005. Note thatH4 and H5 intersect with saddle-node curve at SH points. In ﬁgure 3.5 (a) and (b) the left part of the boundary of the stable EFM region involvesa single Hopf curve H2 that ends at two BT points; only one of which is seen in the view inﬁgure 3.5. In contrary in panel (c) it involves two Hopf curves H2 and H4 that intersect at HHpoints. This change is due to two separate transitions: ﬁrst, the H2 and H4 curves connect at adegenerate HH point and then the BT point moves through a SH point; at the degenerate HHpoint H2 and H4 curves are tangent to each other. Such codimension-three points, where theSH and the BT points meet (also called Bogdanov-Takens-Hopf points), occur also for the FOFlaser, as is mentioned in [17]. Note that, after passing the SH point, the BT points disappear. This happens due to thefollowing transition. As Λ is increased the two BT points move further along the saddle-nodecurve and meet at its end; compare with ﬁgure 3.4 (a) and (b). At this point the BT points atthe two ends of Hopf curve H2 merge, so that H2 is tangent to the saddle-node curve. At suchdegenerate BT point [17, 10] two BT points merge and the Hopf curve H2 detaches from the
- 90. 3.2. Dependence of EFM stability on Λ 75 . 0.005 0.005 (a) (b) Λ = 0.003 Λ = 0.004 Ns H5 Ns H5 H3 H3 0 H4 0 H4 H2 H2 H1 H1 −0.005 −0.005 −0.025 0 ωs 0.025 −0.025 0 ωs 0.025 0.005 0.005 (c) (d) Λ = 0.005 Λ = 0.006 H5 H5 Ns Ns H3 H4 H3 0 H4 0 H2 H 1 H5 H2 H1 H3 −0.005 −0.005. −0.025 0 ωs 0.025 −0.025 0 ωs 0.025Figure 3.5. Projections of the EFM surface onto the (ωs , Ns )-plane, for increasing ﬁlter width Λ asindicated in the panels; here ∆1 = ∆2 = 0, κ = 0.01 and dτ = 0. Black dots indicate codimension-twoBogdanov-Takens bifurcation points; curves and regions are coloured as in ﬁgure 3.1.saddle-node curve; in the (ωs , Ns )-projection in ﬁgure 3.5 only one of the BT points movingalong the saddle-node curve is visible. The transitions through the codimension-three points at the H2 and H4 curves and at theH3 and H4 curves, do not occur concurrently. In the enlargement in ﬁgure 3.5 (c) we show thatthere still exist BT points at the ends of H3 ; compare with ﬁgure 3.4 (b). However, the Hopfbifurcation curves H3 and H5 passed through degenerate HH points, and four new HH pointsappeared; only two of which are seen in the view in ﬁgure 3.5. Figure 3.5 (d) shows the projections of the EFM surface onto the (ωs , Ns )-plane for Λ =0.006. Note that the BT points at the ends of the H3 have merged and disapeared after passingthrough Bogdanov-Takens-Hopf points and later through a degenerate BT point. In this way,the left boundary (H2 and H4 ) and the right boundary (H3 and H5 ) of the stable EFM regionare again qualitatively the same. In Figure 3.5 (e)–(f) we show that with a further increase of Λ, to Λ = 0.015 the number of
- 91. 76 Chapter 3. EFM stability regions 0.01 0.01 (e) (f) Λ = 0.01 Λ = 0.014 H7 H5 H7 H5 H4 H3 H3 0 0 H6 H6 H1 H1 H2 H2 rr H4 −0.01 −0.01 −0.05 0 ωs 0.05 −0.05 0 ωs 0.05 . 0.01 0.01 (g) (h) Λ = 0.017 Λ = 0.02 Ns H7 Ns H7 H5 H5 H3 H3 0 0 H6 H1 H6 H1 H2 H2 rr H4 −0.01 −0.01 −0.05 0 ωs 0.05 −0.05 0 ωs 0.05 0.01 0.01 (i) (j) Λ = 0.023 H7 Λ = 0.025 Ns H5 Ns 0 0 H1 H1 H6 −0.01 −0.01. −0.05 0 ωs 0.05 −0.05 0 ωs 0.05Figure 3.5 (continued). Projections of the EFM surface onto the (ωs , Ns )-plane, for increasing ﬁlterwidth Λ as indicated in the panels; here ∆1 = ∆2 = 0, κ = 0.01 and dτ = 0. Black dots indicatecodimension-two Bogdanov-Takens bifurcation points; curves and regions are coloured as in ﬁgure 3.1.Hopf bifurcation curves that form the boundary of the EFMs stability region increases to seven.Moreover, the number of codimension-two bifurcation points grows to six. In ﬁgure 3.5 (e) forΛ = 0.01, we show two additional Hopf bifurcation curves, H6 and H7 , that become part of theboundary of the stable EFMs region in panel (f) for Λ = 0.014; H6 and H7 become involved
- 92. 3.3. Dependence of EFM stability on ∆ 77in the boundary after they pass through a degenerate HH point. However, while the ﬁlter widthΛ continues to increase, the number of Hopf bifurcation curves forming the boundary of EFMsstability regions decreases and the codimension-two bifurcation points gradually disappear;see ﬁgure 3.5 (g)–(i). In ﬁgure 3.5 (g)–(i) the boundary of the EFMs stability region simpliﬁesin a complicated series of transitions of Hopf bifurcation curves (2)–(7). First, we ﬁnd sad- 1dle and minimax transitions of the Hopf surfaces in (Cp , dCp , Λ)-space, which also manifestthemselves in (ωs , Ns , Λ)-space. Furthermore, there are transitions through degenerate HHpoints at which two different Hopf curves disconnect, and transitions through degenerate SHpoints, at which Hopf bifurcation curves disconnect from saddle-node bifurcation curves; at adegenerate SH point the Hopf bifurcation curve is tangent to the saddle-node bifurcation curve[17, 36]. All the mentioned transitions lead to the region of stable EFMs extending over thewhole ωs -range; it is now bounded at the top by the saddle-node curve and at the bottom byHopf bifurcation curve (1); see ﬁgure 3.5 (j) for Λ = 0.025. Moreover it does not include anycodimension-two bifurcation points. In summary, the investigation of the dependence of the EFM surface and region of stableEFMs on the common ﬁlter width Λ is in qualitative agreement with ﬁndings for both limitcases, Λ → 0 and Λ → ∞. We showed that with increasing Λ, the region of stable EFMsundergoes many complicated transformations. For sufﬁciently high Λ it extends over the wholeωs -range, and is bounded at the top by the saddle-node curve and at the bottom by Hopf curve.We analysed here only the EFM surface of the simplest type B for ∆1 = 0 and ∆2 = 0. To analyse the dependence of EFMs stability on ﬁlter detuning ∆1 and ∆2 in the nextsection we ﬁx the ﬁlter width at a representative value, Λ = 0.005. Namely, the boundary ofthe EFM stability region in ﬁgures 3.3 (b) and 3.4 (b) involves only ﬁve Hopf curves, and itincludes all three kinds of codimension-two bifurcation points. Moreover, from sections 2.3.1and 2.4 we conclude that for this value of Λ it is possible to ﬁnd many different types of theEFM surface.3.3 Dependence of EFM stability on ∆In the two previous sections we only considered the case that the EFM surface is of the sim-plest type B. More speciﬁcally we analysed only EFM stability regions on the EFM surfacecorresponding to the point (∆1 , ∆2 ) = (0, 0) in the EFM surface bifurcation diagram in the(∆1 , ∆2 )-plane presented in sections 2.3.4–2.4. We showed that the regions of stable EFMschange with increasing κ and Λ, even tough the topology of the EFM surface itself remainsunaffected. In particular, we observed a splitting and separation of the stable EFM regionwith increasing κ. Moreover we showed that with increasing Λ the EFM surface expands and
- 93. 78 Chapter 3. EFM stability regionsboundary of the stable EFM region simpliﬁes. These results are in agreement with our ﬁnd-ings in section 2.4, where we show that with increasing Λ the 2FOF laser approaches the COFlaser limit, which results in a diminishing inﬂuence of the ﬁlters detunings on the topology ofthe EFM surface and, hence, on the stability of EFMs. Based on our ﬁndings, throughout thissection we ﬁx κ = 0.01 and Λ = 0.005 — values that are representative for the 2FOF laser,and for which the EFM stability region extends over the whole dCp -range; as before we keepdτ = 0. To analyse the dependence of the stable EFM regions on the ﬁlter detunings ∆1 and ∆2 ,we now calculate the EFM surface with corresponding information on the EFMs stability forseveral representative regions from the EFM surface bifurcation diagram in the (∆1 , ∆2 )-plane. Our goal is to give an overview of changes of EFM stability regions associated withthe singularity transitions of the EFM surface in (ωs , dCp , Ns )-space, as were described insections 2.3.3 and 2.3.4. The starting point of our considerations is the EFM surface in ﬁgure 3.3 (b), for κ =0.01, Λ = 0.005, ∆1 = 0, ∆2 = 0 and dτ = 0; it has a single region of stable EFMs centredaround the solitary laser frequency, represented in here by ωs = 0. We ﬁrst ﬁx ∆2 = 0and increase ∆1 ; in this way we move along the horizontal line ∆2 = 0 in the EFM surfacebifurcation diagram in the (∆1 , ∆2 )-plane. With increasing ∆1 , we ﬁrst observe that the kindof connection between the 2π-periodic fundamental units of the EFM surface change. Next,after passing the point where the curves C and SC meet, a hole in the EFM surface appears andits type changes from B to Bh ; a point of this kind can be found in the enlargement of the EFMsurface bifurcation diagram in the (∆1 , ∆2 )-plane in ﬁgure 2.20. We further increase ∆1 toa value at which, even thought the EFM surface remains unchanged of type Bh , the region ofstable EFMs splits into two stable EFM regions that extend over the whole range of dCp — werefer to such a region as a ‘band of stable EFMs’. This transition is described in section 3.3.1. Next we ﬁx ∆1 = 0.024 which is the maximum value of the ﬁrst transition, and we movevertically down in the EFM surface bifurcation diagram in the (∆1 , ∆2 )-plane by decreasing∆2 to ∆2 = −0.037. Along this path we ﬁrst pass through the SN -transition where a secondhole is created and the EFM surface changes to type hBh . We show that for type hBh aroundeach of the ﬁlter central frequencies a band of stable EFMs exists; see section 3.3.2. With afurther decrease of ∆2 we pass through the Sω -transition where the EFM surface splits intotwo components and changes to type B B. Interestingly, we ﬁnd that the Sω -transition does notaffect the qualitative structure of stable EFMs regions. To ﬁnd a new band of stable EFMs we ﬁx ∆2 = −0.037 and increase ∆1 again. The thirdstable EFM band around the central laser frequency is created just before we pass through thesecond Sω -transition, after which the EFM surface consists of three components and its typechanges to B BB. Changes of the stable EFM regions associated with both Sω -transitions aredescribed in section 3.3.3.
- 94. 3.3. Dependence of EFM stability on ∆ 79 Finally we choose values of ∆1 and ∆2 at which the side components of the EFM surfacesplit into ‘islands’ — dCp periodic disjoint compact objects. In other words, the type of theEFM surface changes from B BB ﬁrst into I BB and then into I BI. The splitting of the sidecomponents of the EFM surface results in a division of the bands of stable EFMs into ‘islands ofstable EFMs’. What is more, the stable EFM regions extend over the whole range of dCp overwhich the side island of the EFM surface exist. On the other hand, the central band of stableEFMs around the free laser frequency expands to ﬁll a larger area of the central component ofthe EFM surface; see section 3.3.4. Overall, we show that detuning the ﬁlters from the solitary laser frequency results inchanges of the EFM surface type, which in turn is coupled to changes of the stability struc-ture of EFMs. From a bifurcation theory point of view, this means that the stability boundariesof the stable EFM regions which consist of saddle-node and Hopf bifurcations curves, mustchange. We show that such changes are due to minimax and saddle transitions of the saddle- 1 1node and Hopf bifurcation surfaces in (Cp , dCp , ∆1 )-space and (Cp , dCp , ∆2 )-space. Insuch transition boundary curves of the EFM stability regions emerge or disappear and con-nect differently, and this results in transformations of the stable EFM regions. Associatedcodimension-two and codimension-three bifurcations points (where the boundary curves alsointeract with each other) are mentioned as necessary, but their detailed discussion is beyond thescope of the analysis presented in this section.3.3.1 Inﬂuence of hole creation on EFM stabilityFor ∆1 = 0 and ∆2 = 0 the EFM surface consist of inﬁnitely many compact objects connectedat the points (ωs , Ns , dCp ) = (0, 0, (2n + 1)π), where n ∈ Z. In this section we increase ∆1for ﬁxed ∆2 = 0. The ﬁrst effect of increasing ∆1 from ∆1 = 0 is a change of the nature ofthe connection between the dCp -periodic units of the EFM surface. A further increase of ∆1result in the transition of the EFM surface from type B to type Bh . Figure 3.6 (a)–(c) illustrates how a hole is created in the EFM surface due to the saddle tran-sition SC ; compare ﬁgure 2.22. Figure 3.6 (a1)–(c1), shows the EFM surface in (ωs , dCp , Ns )-space over a 4π-interval of dCp . Figure 3.6 (a2)–(c2) present enlargements of the narrow con-nection between the fundamental units of the EFM surface. All curves and regions in ﬁgure 3.6(a2)–(c2) are coloured as in ﬁgure 3.1. Note that for clarity panels (a1)–(c1) are shown withoutthe information about the stability of the EFMs. The axes ranges for all the panels can be foundin Table: 3.1. In ﬁgure 3.6 (a), for ∆1 = 0.0005, we show that the fundamental units of the EFM surfaceare no longer connected at single points; instead they are connected along closed loops. Thesmall white curve in the middle of panel (a1) illustrates that connection. In fact, it is a small
- 95. 80 Chapter 3. EFM stability regions . (a1) (a2) ∆1 = 0.0005Ns Ns 0 0 dCp /π 0 0 ωs dCp /π 0 ωs (b1) ∆1 = 0.0007 (b2)Ns Ns 0 0 ωs 0 dCp /π 0 0 ωs dCp /π (c1) ∆1 = 0.005Ns (c2) 0 Ns 0 ωs dCp /π dCp /π 0 0 ωs.Figure 3.6. Inﬂuence of local saddle transition SC , where two bulges connect to form a hole, on thestability of the EFMs; here ∆2 = 0, κ = 0.01, Λ = 0.005 and dτ = 0. Panels (a1)–(c1) showstwo copies of the fundamental 2π-interval of the EFM surface for different values of a detuning ∆1 , asindicated in the panels. Panels (a2)–(c2) show enlargements of the region where the hole is formed. Inpanels (a1)–(c1) the limit of the dCp -axis corresponds to a planar section that goes through middle ofthe hole in panels (a2)–(c2). For the speciﬁc axes ranges see Table 3.1; curves and regions are colouredas in ﬁgure 3.1.
- 96. 3.3. Dependence of EFM stability on ∆ 81 panel min(ωs ) max(ωs ) min(dCp ) max(dCp ) min(Ns ) max(Ns ) ﬁgure 3.6 (a) -0.0250 0.0250 -3.0000 1.0000 -0.0050 0.0050 ﬁgure 3.6 (a1) -0.0050 0.0050 -1.0520 -0.9975 -0.0010 0.0010 ﬁgure 3.6 (b) -0.0250 0.0250 -3.0500 0.9800 -0.0050 0.0050 ﬁgure 3.6 (b1) -0.0048 0.0048 -1.0575 -1.0386 -0.0010 0.0010 ﬁgure 3.6 (c) -0.0230 0.0250 -3.3300 0.7100 -0.0045 0.0050 ﬁgure 3.6 (c1) 0 0.0050 0.6650 0.7460 -0.0008 0.0018 Table 3.1. Axes ranges for all the panels in ﬁgure 3.6closed elliptical curve, which correspond to the narrow neck in the enlargement in ﬁgure 3.6(a2) of the EFM surface in the vicinity of that curve. Figure 3.6 (b), for ∆1 = 0.0007, shows that for only a slightly higher value of ∆1 a holein the EFM surface is created. Therefore, for some range of dCp the EFM surface is connectedalong two disjoint closed loops. This is illustrated by the two white ellipses in the middle ofpanel (b1). In other words we take a constant dCp planar section through the middle of the holeshown in panel (b2). Figure 3.6 (c), for ∆1 = 0.005, shows that increasing ∆1 causes an expansion of the EFMsurface. The expansion is most substantial at the range of dCp at which the constant dCp planarsection through the EFM surface consist of two disjoint closed loops. Note the considerablegrowth of the two closed intersection curves in comparison with panel (b1). The enlargementin ﬁgure 3.6 (c2) shows the hole, through which the constant dCp section in panel (c1) is taken.We again consider only a small range of ωs near the hole in the EFM surface. Note that, due tothe growth of the EFM surface and in contrast to ﬁgure 3.6 (b2), the EFM surface in panel (c2)is not shown over its entire ωs -range. The stability information shown in ﬁgure 3.6 (a2)–(c2), clearly demonstrates that the ap-pearance of the hole strongly affects the stability of the EFMs. For example, a new Hopfbifurcation curve appears in panel (a2); in panel (b2) a region of unstable EFMs emerges at theedge of the hole; and in panel (c2) the whole structure of stable EFMs becomes more compli-cated; compare with ﬁgure 3.3 (b) and ﬁgure 3.4 (b) which show the EFM surface with stabilityinformation before detuning the ﬁlter, ∆1 = 0. We now explain the nature of those stability changes in more detail. Figure 3.7 (a)–(c) showprojections of the EFM surface corresponding to the EFM surfaces presented in the ﬁgure 3.6(a)–(c). Figure 3.7 (a1)–(c1) show the projection of the EFM surface over a [0, 2π] rangeof dCp ; in this way the connection between the fundamental units of the EFM surface andthe hole are presented in the centre of the ﬁgure. Panels (a2)–(b2) show enlargements, near
- 97. 82 Chapter 3. EFM stability regions . 2 1.03 (a1) (a2)dCp dCp π π 1 0 0.94 −0.025 0 ωs 0.025 −0.003 0 ωs 0.003 2 0.98 (b1) (b2)dCp dCp π π 1 0 0.92 −0.025 0 ωs 0.025 −0.004 0 ωs 0.004 2 0.85 (c1) (c2)dCp dCp π π 1 0 0.45. −0.024 0 ωs 0.026 0 ωs 0.005Figure 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 fundamental2π 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 twostable regions on the EFM surface that lie above one another in the Ns direction.
- 98. 3.3. Dependence of EFM stability on ∆ 83the connection between the two units of the EFM surface, of the region of the (ωs , dCp )-projection of the EFM surface from Figure 3.7 (a1)–(c1). Recall that, we colour regions indark green where stable EFMs exist on the bottom and on the top sheet of the EFM surface. Inother words, dark green indicates areas, where for a set values of ωs and dCp , two stable EFMswith different values of population inversion Ns exist. The dark green areas can be found in thefollowing regions of the EFM surface: at the edge of the hole, at the outer edges of the EFMsurface, and at the narrow necks of the central component of the EFM surface; in the vicinity ofconnections between the fundamental units of the EFM surface. The other curves and regionsin ﬁgure 3.7 are coloured as in ﬁgure 3.1. Figure 3.7 (a), for ∆1 = 0.0005, shows that with opening the connection between thefundamental units of the EFM surface, the saddle-node curve underwent a saddle connectionand changed from a closed loop periodically translated along the dCp axis to two inﬁnitelylong 2π-periodic curves that extend along the outer edges of the (ωs , dCp )-projection of theEFM surface. Note that inside a substantial ‘bulge’ of the right saddle-node curve in ﬁgure 3.7(a2), as well as along the edge of the projection of the EFMs surface, the EFM stability regionexists on both the bottom and the top sheet of the EFM surface. Additionally, inside the bulgea new Hopf curve ends at two BT points that emerged due to an increase of the ﬁlter detuning∆1 . Figure 3.7 (b), for ∆1 = 0.0007, shows how the stability of the EFMs changes after thecreation of the hole in the EFM surface. Note that the hole indicates a region in the (ωs , dCp )-plane where EFMs do not exist and, hence, also no stable EFMs can be found. Panel (b2) showsthat, as the result of passing through a saddle point of the saddle-node bifurcation surface in(ωs , dCp , ∆1 )-space, the bulge from panel (a2) detaches and forms a new closed saddle-nodecurve that surrounds the hole. As a result at the edge of the hole, a region of unstable EFMsappears. In ﬁgure 3.7 (c), for ∆1 = 0.005, we show that increasing ∆1 is associated with severaltrends in changes of the EFM stability region. First, the connection between dCp -periodicunits of the EFM surface expands in the ωs direction; this results in an expansion of the EFMstability region in the ωs direction. Second, the Hopf curve around the point (ωs , dCp ) = (0, 0)in panels (a1) and (b1) shrinks and shifts away from the point (ωs , dCp ) = (0, 0) in panel (c1);this is the curve that formed the lower boundary of the regions of stable EFMs in section 3.1and 3.2. Together with that Hopf curve the region of unstable EFMs bounded by it shrinks aswell. Finally, the hole and the region of unstable EFMs around it grows in ωs as well as dCpdirections. Note that, the growth of the hole with increasing ∆1 is much slower than the expansion ofthe region of unstable EFMs around it. In ﬁgure 3.7 (b) the hole is the most important changeto the stable EFM region. The situation changes in ﬁgure 3.7 (c): here the stable EFM region
- 99. 84 Chapter 3. EFM stability regions . 0.007 Ns 0 −0.004 0.033 1.63 ωs 1 dCp /π 0 0 −1 −2 −0.02 −2.38.Figure 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.is affected by the expanding region of unstable EFMs; its area is several times larger than thearea of the hole. The expansion of the region of unstable EFMs is associated with a changeof its boundary. In panel (b) it is bounded by a single saddle-node curve. In ﬁgure 3.7 (c) theboundary of the region of unstable EFMs around the hole involves the same single saddle-nodebifurcation curve and additionally two Hopf bifurcation curves. One of the Hopf curves is theHopf curve from panel (b) and the second one appeared in a minimax transition of the Hopfbifurcation surface. We now show how, with a further increase of ∆1 , those trends lead to a division of thesingle stable EFM region into two bands of stable EFMs. Figure 3.8, presents EFM surfacesin (ωs , dCp , Ns )-space over the 4π-interval of dCp . All regions and curves are coloured asin ﬁgure 3.1. (Note that the dCp -range is slightly larger then 4π due to surface renderingrequirements.) In ﬁgure 3.8 we show that for ∆1 = 0.024 (and ∆2 = 0) the EFM surface oftype Bh has two bands of stable EFMs; one of the EFM stability bands is centred around thesolitary laser frequency and the other one around the central ﬁlter frequency (around ωs = ∆1 ).The two bands of stable EFMs are separated by the hole. There are also very small additionalregions of stable EFMs: one near the left end of the EFM surface (for negative ωs ) and one atthe edge of the hole. The splitting of the single stable EFM region into two regions of stable EFMs that formbands expanding over the whole range of dCp is explained in more details in ﬁgure 3.9. This
- 100. 3.3. Dependence of EFM stability on ∆ 85transitions does not involve a topological change of the EFM surface itself, but of the saddle-node and Hopf curves that bound the EFM stability regions. At such codimension-two pointsthe Hopf curves bounding EFM stability regions can appear, disappear or can change the waythey are connected. More speciﬁcally, ﬁgure 3.9 (a)–(f) shows how stability regions of EFMs change with anincrease of ∆1 from ∆1 = 0.065 to ∆1 = 0.024, while the second ﬁlter is ﬁxed at ∆2 = 0.Panels (a)–(f) show projections of the EFM surface onto the (ωs , dCp )-plane over [0, 2π] dCp -range. Changes of the structure of the EFMs stability regions are results of minimax and saddletransitions of the Hopf curves. In effects the single region of stable EFMs in panel (a) dividesinto four regions of stable EFMs in panel (f). Two of them extend over the whole range of dCp ,and two exist for very limited ranges of dCp . Note that for all projections in ﬁgure 3.9 (a)–(f)the EFM surface remains of the type Bh . In ﬁgure 3.9 (a), we show that for ∆1 = 0.0065 the closed Hopf curve, visible in thetop half of the panel shrinks signiﬁcantly in comparison to the same curve for ∆1 = 0.005in ﬁgure 3.7 (c1). This closed Hopf curve disappears in a minimax transition of the Hopfbifurcation surface, for slightly higher value of ∆1 ; see panel (b) for ∆1 = 0.0085. Notealso, that ﬁgure 3.9 (a) and (b) shows that the region of unstable EFMs surrounding the holecontinues to grow with increasing ∆1 . Figure 3.9 (c) to (f) illustrate a division of the single EFM stability region into four regionsof stable EFMs. Each of the partition of the stable EFM region occurring in between panels (c)to (f) is associated with a saddle transitions of Hopf bifurcation curves. The saddle transitionof the Hopf curves occurring in between panels (c) and (d) results in a separation of the EFMstability region into two bands of stable EFMs, one around the solitary laser frequency ωs = 0and another associated with the central ﬁlter frequency. Both those regions extend over thewhole range of dCp . Next in panel (e) there is new small region of stable EFMs that lies on theedge of the hole; this region detached from the right band of stable EFMs. Although this regionis too small to be detected in experiments, we emphasise its existence because it expands intoa stable EFM band for other values of ∆1 and ∆2 ; see section 3.3.3. Finally for ∆1 = 0.024in panel (f) a small region of stable EFMs detaches from the left stable EFM band, whichincreases the total number of stable EFM regions to four. Note that, the (ωs , dCp )-projectionin ﬁgure 3.9 (f) corresponds to the EFM surface in ﬁgure 3.8.
- 101. 86 Chapter 3. EFM stability regions . 2 2 (a) (b)dCp dCp π π 1 1 0 0 −0.024 0 ωs 0.026 −0.024 0 ωs 0.026 2 2 (c) (d)dCp dCp π π 1 1 0 0 −0.023 0 ωs 0.031 −0.023 0 ωs 0.031 2 2 (e) (f)dCp dCp π π 1 1 0 0. −0.022 0 ωs 0.032 −0.022 0 ωs 0.034Figure 3.9. Projections of the EFM surface with stability information onto the (ωs , dCp )-plane forincreasing 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 andregions are coloured as in ﬁgure 3.7.3.3.2 Inﬂuence of SN -transition on EFM stabilityWe now explore how the structure of the stable EFM regions is affected by the change of theEFM surface from type Bh to type hBh . To reach the parameter range for which the EFM
- 102. 3.3. Dependence of EFM stability on ∆ 87 . 0.007 Ns 0 −0.007 0.033 1.41 ωs 1 dCp /π 0 0 −1 −2 −2.65 −0.033.Figure 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.surface is of type hBh we ﬁx ∆1 = 0.024 and decrease ∆2 until the second hole in the EFMsurface appears; the second hole is the result of a SN saddle transition; see ﬁgure 2.15 in sec-tion 2.3.3. We ﬁnd that for the EFM surface of type hBh two bands of stable EFMs exists. Thebands are centred around the central ﬁlter frequencies associated with ∆1 and ∆2 . Addition-ally, we show that at the edges of the holes, a small stable EFM region can be found. Thisregion is centred around the solitary laser frequency, represented here by ωs = 0. Figure 3.10, for ∆1 = 0.024 and ∆2 = −0.025, shows the EFM surface in (ωs , dCp , Ns )-space over the 4π-interval of dCp . (In fact, the dCp axis range is slightly larger then 4π due tosurface rendering requirements.) All regions and curves are coloured as in ﬁgure 3.1. Figure 3.10 shows the EFM surface of hBh type on which two ‘bands of stable EFMs’exist; both these bands are bounded by Hopf curves. Each band is centred around the centralfrequency of one of the ﬁlters. Additionally, a stable EFM region exists in the centre of theEFM surface at the edges of the holes. This region is bounded by Hopf and saddle-nodebifurcation curves and it is centred around the central laser frequency. We now show thatqualitatively the same structure of the EFM stability regions exists before the second holeappears. Moreover we present the structure of the EFM stability regions for the transitionthrough the anti-diagonal ∆1 = −∆2 in the EFM surface bifurcation diagram in the (∆1 , ∆2 )-plane presented in ﬁgure 2.17. Figure 3.11 (a)–(d) presents how the stability structure of the EFMs change for ﬁxed ∆1 =
- 103. 88 Chapter 3. EFM stability regions . 3 3 (a) (b)dCp dCp π π 2 2 1 1 −0.026 0 ωs 0.033 −0.034 0 ωs 0.033 3 3 (c) (d)dCp dCp π π 2 2 1 1. −0.034 0 ωs 0.033 −0.034 0 ωs 0.033Figure 3.11. Projections of the EFM surface with stability information onto the (ωs , dCp )-plane forincreasing 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 3.7.0.024 and for values of ∆2 in the range ∆2 ∈ [−0.012, −0.037]. In ﬁgure 3.11 (a)–(d) we showthe projection of the EFM surface onto the (ωs , dCp )-plane over the range of dCp ∈ [π, 3π]. In ﬁgure 3.11 (a), for ∆2 = −0.012, the EFM surface is still of the type Bh , the sametype as in ﬁgure 3.9 (a)–(f). There are four regions of stable EFMs in panel (a). Two bands ofstable EFMs centred at the central ﬁlter frequencies and two small EFM stability regions at theedge of the hole relatively close to the solitary laser frequency. Each of stable EFM bands isbounded by a pair of Hopf bifurcation curves. The boundary of the stable EFM regions at theedge of the hole involves single Hopf curve and a single saddle-node curve. In ﬁgure 3.11 (b), for ∆2 = −0.023, a second hole appeared in the EFM surface dueto the SN -transition; the EFM surface is now of type hBh . Concurrently, a new saddle-nodebifurcation curve emerges in the minimax transition; see the small blue loop in the half-planewhere ωs < 0, in ﬁgure 3.11 (b). Note that, while the structure of the bands of stable EFMsassociated with the ﬁlters remains qualitatively unchanged, the two stable EFM regions at theedge of the hole merge and expand; compare with ﬁgure 3.11 (a).
- 104. 3.3. Dependence of EFM stability on ∆ 89 Figure 3.11 (c) shows the (ωs , Ns )-projection of the EFM surface for ∆1 = 0.024 and∆2 = −0.024; that is, on the anti-diagonal ∆1 = −∆2 in the EFM surface bifurcation diagramin the (∆1 , ∆2 )-plane. The mirror symmetry of the ﬁgure 3.11 (c) with respect to ωs = 0is due to ∆1 = −∆2 ; note that the two holes are connected in single points at ωs = 0.Additionally the two saddle-node curves from panel (b) merged into a single curve in the saddletransition; the new saddle-node curve surrounds both holes. The central region of stable EFMsis spread symmetrically along the edges of the holes. The symmetry of the EFM surface for∆1 = −∆2 result in the emergence of a highly degenerated multiple Hopf bifurcation point atthe connection of the two holes; at this point the central Hopf curve intersects itself multipletimes. Decreasing ∆2 to ∆2 = −0.025 results in the unfolding of the multi-Hopf point intoseveral Hopf-Hopf points. Moreover, together with the separation of the holes the saddle-nodebifurcation curve divides into two curves in another transition through the saddle in the saddle- 1node surface in (Cp , dCp , ∆2 )-space; see panel (d). In ﬁgure 3.11 (a)–(d) we investigated how the stability structure of EFMs changes whilethe EFM surface transforms from type Bh to type hBh . We showed that although the topologyof the EFM surface changes signiﬁcantly the structure of the regions of stable EFMs remainsqualitatively unchanged. In ﬁgure 3.11 (a)–(d) we uncovered that there exist three main regionsof stable EFMs. Two associated with the central ﬁlter frequencies forming bands that extendover whole range of dCp , and one associated with the solitary laser frequency that expandswith an increasing detuning of the ﬁlters. The structure of the EFM stability regions that emerged in ﬁgure 3.11 (a)–(d) suggests thatthe EFM stability region centred around the ωs = 0 in ﬁgure 3.9 (f) can be considered as theregion of stable EFMs associated with the second ﬁlter (here ωs = 0 corresponds to ∆2 = 0).Moreover, in a similar way the single region of stable EFMs in ﬁgure 3.7 can be considered asthe overlapping bands of stable EFMs associated with both ﬁlters.3.3.3 Inﬂuence of Sω -transition on EFM stabilityWe now explore other regions of the EFM surface bifurcation diagram in the (∆1 , ∆2 )-planeand investigate how the basic structure of the EFM stability regions uncovered in section 3.3.2changes as the EFM surface undergoes two Sω -transitions. In particular we observe that, as theEFM surface transforms from type hBh through type B Bh to type B BB, the bands of stableEFMs associated with the central ﬁlters frequencies remain qualitatively unchanged. What ismore, we show that, as we increase detuning of the ﬁlters, ∆1 in the positive direction and ∆2in the negative direction, the expansion of the stable EFM region centred around the solitarylaser frequency progresses, and a third band of stable EFMs appears. In ﬁgure 3.12 we present the EFM surfaces of types hB (a), B B (b), B Bh (c) and B BB(d), together with stability information. We show that, despite the splitting of the EFM surface
- 105. 90 Chapter 3. EFM stability regions . 0.007 (a) Ns 0 −0.009 0.033 1.15 1 ωs dCp /π 0 0 −1 −2 −2.9 −0.042 0.007 (b) Ns 0 −0.009 0.033 0.75 ωs 0 dCp /π 0 −1 −2 −3 −0.042 −3.4.Figure 3.12. The EFM surface with stability information for ∆2 = −0.035 (a), ∆2 = −0.037 (b).Here ∆1 = 0.024, κ = 0.01, Λ = 0.005 and dτ = 0; curves and regions are coloured as in ﬁgure 3.1.into up to three components, on each of those components at least one band of stable EFMsexist. Figure 3.12 (a)–(d) shows EFM surfaces in (ωs , dCp , Ns )-space over the 4π-interval ofdCp . (Note that, due to surface rendering requirements, the dCp axes range in ﬁgures 3.12 (a),(b) and (d) is slightly larger then 4π.) All regions and curves are coloured as in ﬁgure 3.1. To observe the ﬁrst Sω -transition, we decrease ∆2 while keeping ∆1 = 0.024 ﬁxed; this is
- 106. 3.3. Dependence of EFM stability on ∆ 91 . 0.009 (c) Ns 0 −0.009 0.043 1.11 1 ωs dCp /π 0 0 −1 −2 −2.89 −0.043 0.009 (d) Ns 0 −0.009 0.043 0.09 0 ωs dCp /π −1 0 −2 −3 −3.93 −0.043.Figure 3.12 (continued). The EFM surface with stability information for ∆1 = 0.035 (c), ∆1 = 0.036(d). Here ∆2 = −0.037, κ = 0.01, Λ = 0.005 and dτ = 0; curves and regions are coloured as inﬁgure 3.1.as in section 3.3.2. In ﬁgure 3.12 (a) and (b), we illustrate the effect of the Sω -transition, wherethe EFM surface changes from type hB to type B B. Note that, although the topology of EFMsurface changes, the structure of the stable EFM regions remains qualitatively the same. Panel (a), for ∆2 = −0.035, presents the EFM surface of type hB with two bands of
- 107. 92 Chapter 3. EFM stability regionsstable EFMs centred at the central ﬁlter frequencies and a stable EFM region at the edge ofthe hole. The structure of the EFM stability region is virtually the same as in ﬁgure 3.10. Inﬁgure 3.12 (b), for ∆2 = −0.037, we show that the Sω -transition does not change the stableEFM regions. Note, however that the region of stable EFMs around the central laser frequencykeeps expanding with decreasing of ∆2 . We now ﬁx ∆2 = −0.037 and increase ∆1 ; in this way, we ﬁnd parameter values at whichthe EFM surface of type B Bh changes to type B BB. In ﬁgure 3.12 (c) and (d), for ∆1 = 0.035and for ∆1 = 0.036, we show that with increasing ∆1 the region of stable EFMs around thecentral laser frequency expands further. In fact, in both panels (c) and (d), three bands of stableEFMs exist. A further increase of ∆1 to the value at which the EFM surface undergoes anotherSω transition and separates into three components does not qualitatively affect the structure ofstable EFM bands. The expansion of the stable EFM region around the central laser frequency can be explainedby the observation that, as the detuning of the ﬁlters increases, the frequencies close to the cen-tral laser frequency are subject to weaker feedback from the ﬂanks of the ﬁlters transmittanceproﬁles. Recall that in section 3.1 we showed that the region of stable EFMs is larger for lowervalues the of feedback rate κ. This observations is also in agreement with the ﬁndings in [17]. To sum up, in ﬁgure 3.12 (a)–(d) we investigated how the stability structure of EFMschanges while the EFM surface transforms from Bh through hBh to B B. In particular, weshowed that, although the Sω -transition strongly affects the topology of the EFM surface, itdoes not change the qualitative structure of the stable EFM regions. Moreover in ﬁgure 3.12(c) and (d) we showed that detuning both ﬁlters further away from central laser frequency resultin the creation of a third band of stable EFMs. Figure 3.13 shows details of how the region of stable EFMs around the central laser fre-quency transforms into a band of stable EFMs. Panels (a)–(f) show that, while the EFM surfacechanges from hB into B BB, the only signiﬁcant change of the stability structure of the EFMs isthe transformation of the central region of stable EFMs into a band that extends over the wholerange of dCp . We ﬁrst further decrease ∆2 in panels (a) and (b). Then we ﬁx ∆2 = −0.037at the value from panel (b), and we increase ∆1 up to ∆1 = 0.036,in panels (c)–(f). Panels(a)–(f) show the (ωs , dCp )-projection of the EFM surface over the range of dCp ∈ [π, 3π]; allregions and curves are coloured as in ﬁgure 3.1. Figure 3.13 (a) shows that, with decreasing ∆2 , one hole from ﬁgure 3.11 (d) disappearstogether with the saddle-node curve surrounding it; simultaneously the other hole expandssubstantially. Concurrently with the hole, the stable EFM region laying at its edge expands aswell. As the EFM surface transforms from type hB to type B B, the periodically shifted loopsof the closed saddle-node curve connect in the saddle transition and transform to two inﬁnitelylong 2π periodic curves. Note that, in panel (b), with decreasing ∆2 the expansion of the stable
- 108. 3.3. Dependence of EFM stability on ∆ 93 . 3 3 (a) (b)dCp dCp π π 2 2 1 1 −0.043 0 ωs 0.033 −0.043 0 ωs 0.033 3 3 (c) (d)dCp dCp π π 2 2 1 1 −0.043 0 ωs 0.035 −0.043 0 ωs 0.037 3 3 (e) (f)dCp dCp π π 2 2 1 1. −0.043 0 ωs 0.041 −0.043 0 ωs 0.043Figure 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.EFM region around ωs = 0 progresses. Figure 3.13 (a) and (b) show the (ωs , dCp )-projectionof the transition depicted in ﬁgure 3.12 (a) and (b). We now ﬁx ∆2 and start increasing ∆1 . Figure 3.13 (c), for ∆1 = 0.026, shows the
- 109. 94 Chapter 3. EFM stability regions(ωs , dCp )-projection of the EFM surface of type B Bh . The SN -transition is again associatedwith the minimax transition of a saddle-node curve and, as before, it does not affect the qualita-tive structure of the stable EFM regions. With a further increase of ∆1 the hole and saddle-nodecurve around it expands. We also observe a further growth of the stable EFM region centred atthe central laser frequency; see panel (d). In ﬁgure 3.13 (e) we show that, after several transitions through degenerate Hopf-Hopfbifurcation points and several transitions through saddle points of the Hopf bifurcation surfacein (ωs , dCp , ∆1 )-space, inﬁnitely many copies of the region of stable EFMs around the laserfrequency connect with themselves along the dCp -axis to form a band of stable EFMs thatextends over the whole range of dCp . As a result, in ﬁgure 3.13 (e) three bands of stableEFMs are present: one around the central laser frequency and one around each of the centralﬁlter frequencies. Note that the closed saddle-node curve surrounding the hole connects, in thesaddle transition, with the saddle-node curve along the edge the EFM surface. Finally, in ﬁgure 3.13 (f) after a further Sω -transition, the EFM surface consist of threedisjoint components; and it is of type B BB. Similarly to the Sω -transition between ﬁgure 3.13(a) and (b), the qualitative structure of the stable EFM regions appears to be unaffected by thistransition. Note that in the saddle transition that accompanied the Sω -transition the saddle-node curve passing through ωs = 0 transformed into loops surrounding points (ωs , dCp ) =(0, (2n + 1)π with n ∈ Z, and inﬁnitely long 2π-periodic curve along the left edge of the right-most component of the EFM surface. Moreover, the closed saddle-node bifurcation curve inﬁgure 3.13 (f) is shifted by π with respect to ﬁgure 3.1. In this section, we showed that although the Sω -transitions change the topology of theEFM surface signiﬁcantly, they do not affect the qualitative structure of the stable EFM re-gions. Moreover, we showed that with increasing detuning of the ﬁlters from the solitary laserfrequency the region of stable EFMs around ωs = 0 expands and transform into an additionalband of stable EFMs. Therefore, on each disjoint component of the EFM surface of type B BBthere exists one band of stable EFMs.3.3.4 Inﬂuence of SC -transition on EFM stabilityWe now show how the structure of stable EFM regions from section 3.3.3 changes with detun-ing the ﬁlters further away from the central laser frequency. In other words, we investigate whathappens to the three bands of stable EFMs as the EFM surface undergoes the SC -transitionwhere its side components split into dCp -periodic islands; the EFM surface type changes fromB BB to I BI in the process. Figure 3.14 shows the EFM surfaces in (ωs , dCp , Ns )-space together with the stabilityinformation over the 4π-interval of dCp . In panel (a), for ∆1 = 0.044 and ∆2 = −0.049, the
- 110. 3.3. Dependence of EFM stability on ∆ 95 . 0.01 (a) Ns 0 −0.011 0.055 0.35 ωs 0 dCp /π −1 0 −2 −3 −3.65 −0.055 0.01 (b) Ns 0 −0.011 0.055 0.64 ωs 0 dCp /π 0 −1 −2 −3 −0.055 −3.38.Figure 3.14. The EFM surface with stability information for ∆1 = 0.044 (a), ∆1 = 0.050 (b). Here∆2 = −0.049, κ = 0.01, Λ = 0.005 and dτ = 0; curves and regions are coloured as in ﬁgure 3.1.EFM surface is of type I BB and in panel (b), for ∆1 = 0.050 and ∆2 = −0.049, it is oftype I BI. (Note that the dCp axes range in panel (b) is slightly larger then 4π due to surfacerendering requirements.) All regions and curves are coloured as in ﬁgure 3.1. In ﬁgure 3.14 we show that, after the outer component of the EFM surface divides intodisjoint islands, stable EFM region still extends over the whole bottom sheet of the islands.
- 111. 96 Chapter 3. EFM stability regions . 2 2 (a) (b)dCp dCp π π 1 1 0 0 −0.05 0 ωs 0.046 −0.051 0 ωs 0.046 2 2 (c) (d)dCp dCp π π 1 1 0 0. −0.053 0 ωs 0.048 −0.054 0 ωs 0.054Figure 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 arecoloured as in ﬁgure 3.7.Moreover, at the narrow necks of the central component of the EFM surface the stable EFMregion extends over the bottom as well as the top sheet of the EFM surface. Similarly to theﬁrst, the second SC -transition has this type of local inﬂuence on the regions of stable EFMs. Inpanel (b) both outer components of the EFM surface have formed islands with the stable EFMregion extending over their whole bottom sheet. Note that the islands at negative ωs -values arevirtually unaffected by the SC -transition of the right-most component of the EFM surface. Figure 3.15 (a)–(d) show how the EFM stability regions change with a further increase ofﬁlters detuning. We present the EFM surface as a projection onto the (ωs , dCp )-plane over therange of dCp ∈ [0, 2π]; all regions and curves are coloured as in ﬁgure 3.7. Panels (a), for ∆1 = 0.039 and ∆2 = −0.041, and (b), for ∆1 = 0.039 and ∆2 =−0.045, show that, with increasing detuning of the ﬁlters, the expansion of the region of stableEFMs continues. The Hopf bifurcation curves that bound regions of unstable EFMs in thecentral component of the EFM surface shrink and disappear in a series of saddle and minimax
- 112. 3.4. Dependence of EFM stability on dτ 97transitions; as a consequence also the regions of unstable EFMs on the bottom sheet of theEFM surface shrink and disappear. In ﬁgure 3.15 (c), for ∆1 = 0.044 and ∆2 = −0.049, theleft component of the EFM surface underwent the SC -transitions and divided into islands. Inother words, the EFMs centred around the left ﬁlter frequency exist only for a limited range ofdCp values. Nevertheless, the region of stable EFMs extends over the whole range of dCp forwhich the EFM island exist. Note that the SC -transition is accompanied by a saddle transitionof saddle-node bifurcation curves. After this transitions the boundary of the EFM stabilityregion on the island involves the closed saddle-node bifurcation curve and two Hopf bifurcationcurves. Finally, the same happens to the right component of the EFM surface in panel (d). Note that the narrow neck of the EFM surface’s central component in ﬁgure 3.15 (d) isshifted almost by π with respect to ﬁgure 3.7 (a). This is an effect of the arctan in Eq. (2.4)approaching the value ±π/2 with increasing ∆1 and decreasing ∆2 . The change in the phaseof the cos terms in Eq. (2.5) results in a shift from π to 0 of the value at which the destructiveinterference occur. To conclude, in section 3.3 we found that for two feedback loops of equal length, dτ = 0and for suitable, and experimentally accessible values of the feedback rate κ = 0.01 and theﬁlter width Λ = 0.005, stable EFM regions have the following structure. There exist two bandsof stable EFMs that are centred around the central ﬁlter frequencies. These bands of stableEFMs overlap each other for low values of the detunings of ﬁlters and, hence, are effectivelyindistinguishable. The structure of the stable EFM bands unfolds with the increasing detuningof the ﬁlters away from the central laser frequency. After the bands are separated, the qualitativestructure of stable EFM regions remains unaffected by the SN and Sω transitions of the EFMsurface. With a further increase of the ﬁlters detuning, the bands of stable EFMs associatedwith the central ﬁlters frequencies split into islands, concurrently with the outer componentsof the EFM surface; this happens at SC -transitions and then the islands and the EFM stabilityregions disappear with them in M -transitions. A third signiﬁcant region of stable EFMs iscentred around solitary laser frequency. The size of this region increases with the detuning ofthe ﬁlters, and for sufﬁciently large ﬁlter detunings a third stable EFM band is created. Thisband of stable EFMs exist even after the side components of the EFM surface disappear in theM -transition.3.4 Dependence of EFM stability on dτSo far in this chapter we discussed the stability of EFMs only for the case of the 2FOF laserwith equal delay times, that is, for dτ = 0. In this section we study how the regions of stableEFMs change with increasing difference dτ = τ2 − τ1 between the two delay times. This is the
- 113. 98 Chapter 3. EFM stability regionsﬁnal part of our analysis of the dependence of EFM stability regions on the ﬁlter and feedbackloop parameters. In section 2.5 we explained in detail that changing dτ causes a shearing of the EFM surfacein (ωs , dCp , Ns )-space along the dCP -axis, with respect to the invariant plane deﬁned byωs = 0. What is more, we showed that the shearing and the other changes of the geometryand topology of the EFM surface are independent of each other. In other words, to uncover theEFM surface for the case of the 2FOF laser with two different delay times, it is sufﬁcient toﬁnd the respective EFM surface for dτ = 0, and shear with the dτ shear rate. Shearing of the EFM surface substantially changes the structure of the intervals of stableEFMs that one ﬁnds for the ﬁxed dCp . However, similarly to the case of the EFM surfaceitself, the overall structure of the EFM stability regions in the (ωs , dCp )-plane can effectivelybe determined by ﬁnding the EFM stability regions for dτ = 0 and shearing them. As a concrete example, we now analyse how the EFM stability regions on the simplest EFMsurface of type B change as we increase dτ . To this end, we ﬁx τ1 = 500 and by changing τ2 inthe interval [506, 750] we increase difference between the delay times to dτ = 250. Throughoutthis section we ﬁx the other ﬁlter parameters at the values corresponding to the EFM surfacein ﬁgure 3.1 (a), that is, to κ = 0.01, Λ = 0.015, and ∆1 = ∆2 = 0. In particular, we showthat even for the simplest EFM surface of type B, the shearing of the EFM surface results inthe possibility of ﬁnding an arbitrary number of bands of stable EFMs in the (ωs , dCp )-plane. Figure 3.16 shows projections of the EFM surface with stability information onto the(ωs , dCp )-plane for increasing dτ . All curves and regions are coloured as in ﬁgure 3.7. Addi-tionally, black dots indicate codimension-two Bogdanov-Takens bifurcation points. Figure 3.16(a)–(d) shows that the stable EFM region is sheared concurrently with the EFM surface. In pan-els (a) and (b), for dτ = 6 and dτ = 14, we show that initially the EFM stability region changesrather subtly. In ﬁgure 3.16 (c), for dτ = 62, the basic unit of the EFM surface starts to extentover the 2π dCp -range. Hence, for some dCp -range, two stable EFM intervals separated byinterval with no EFMs exist; for example two such intervals exist for dCp /π = 0.85. Eachof those intervals belongs to a different 2π-translated copy of the basic EFM surface unit. Inpanel (d), for dτ = 62, the EFM surface is sheared so much that parts of four 2π-translatedcopies of the basic EFM surface unit can be seen over the the 2π dCp -interval. As a result, forexample for dCp /π = 0.5, indicates four intervals of stable EFMs exist. Those stable EFMintervals are separated by intervals with no EFMs as well as by intervals of unstable EFMs. Additionally, the shearing of the stable EFM regions with increasing dτ is associated withminor topological changes of their outer boundary. In ﬁgure 3.16 (a), similarly to ﬁgure 3.1 (a),the outer boundary involves six Hopf bifurcation curves and a single saddle-node bifurcationcurve. However, due to the shearing the structure of this outer boundary changes. Note that,in the bottom right corner of ﬁgure 3.16 (a), the boundary involves only two Hopf bifurcation
- 114. 3.4. Dependence of EFM stability on dτ 99 . 2 2 (a) (b)dCp dCp π π 1 1 0 0 −0.04 0 ωs 0.04 −0.04 0 ωs 0.04 2 2 (c) (d)dCp dCp π π 1 1 0 0. −0.04 0 ωs 0.04 −0.04 0 ωs 0.04Figure 3.16. Projections of the EFM surface with stability information onto the (ωs , dCp )-plane forincreasing delay time in the second feedback loop τ2 = 506 (a), τ2 = 514 (b), τ2 = 562 (c) andτ2 = 750 (d); here τ1 = 500, κ = 0.01, Λ = 0.015 and ∆1 = ∆2 = 0. Black dots indicatecodimension-two Bogdanov-Takens bifurcation points; curves and regions are coloured as in ﬁgure 3.7.curves; the third Hopf curve is no longer involved in the boundary after it passed through theHopf-Hopf point; compare with ﬁgure 3.1 (a). In ﬁgure 3.16 (b), we show that, as the shearing of the stable EFM region progresses, fourBT bifurcation points appear on the saddle-node curve. As those BT bifurcation points movealong the saddle-node curve with increasing dτ , the overall structure of the outer boundary ofthe stable EFM region simpliﬁes. In panel (c), a new Hopf bifurcation curve emerged from thepoint (ωs , dCp ) = (0, 1), when the saddle-node curve passed through a tangency with the linedCp = 1. Additionally, after passing through a saddle point on the saddle-node bifurcationsurface in (ωs , dCp , dτ )-space, the saddle-node curve changed from a closed loop into twoinﬁnitely long curves. In panel (d), for dτ = 250, the boundary of the EFM stability regionsinvolves only Hopf curves and is more complicated than in panel (c). Note that the innerboundary of the EFM stability region is always formed by the closed elliptical Hopf curvesurrounding the points (ωs , dCp ) = (0, 2nπ) with n ∈ Z.
- 115. 100 Chapter 3. EFM stability regions Overall we conclude that the main effect of varying dτ is the shearing of the EFM stabilityregions that can be found for dτ = 0. Additionally, the shearing and the other transformationsof the EFM stability regions — presented in sections 3.1–3.3 — are independent of each other.In other words, all previous cases of the EFM stability regions for dτ = 0 can be shearedsimilarly. As a result of the shearing, even for the EFM surface of the simplest type B, anynumber of stable EFM intervals, for ﬁxed dCp can be obtained for sufﬁciently large dτ , suchstable EFM intervals are separated by intervals where no EFMs exist at all.3.5 Different types of bifurcating oscillationsThe existence of different kinds of intensity oscillations in lasers with optical feedback is welldocumented; extensive theoretical as well as experimental studies of this subject can be found,for example in [48, 24, 22, 18]. As in the FOF laser, in the 2FOF system two main types ofoscillations bifurcate from Hopf bifurcation points: relaxation oscillations (ROs), which arecommon in semiconductor lasers subject to external perturbation, and frequency oscillations(FO) which are typical for semiconductor lasers subject to ﬁltered optical feedback. ROs are characterised by a periodic exchange of energy between the optical ﬁeld E and thepopulation inversion N , which result in oscillations of the laser intensity and (for non-zero α)of the laser frequency; a typical RO frequency has a value of several GHz [19]. In FOs, on theother hand, the laser frequency oscillates with a high amplitude at a frequency on the order ofthe external round trip time 1/τ , while the intensity of the laser remains almost constant [17,18]. This is unexpected in a semiconductor laser, in light of strong amplitude-phase couplingvia the linewidth enhancement parameter α [18, 23]. The different mechanisms behind ROsand FOs manifest themselves in the different time scales of the two kinds of oscillations: thefast RO arises due to fast processes inside the semiconductor laser, and the slow FO arise dueto interaction of the semiconductor laser with light travelling in the external ﬁltered feedbackloops; time light travels in external loops is orders of magnitude larger then the timescale ofthe processes inside the laser. Considering that the existence of ROs and FOs in the single FOF laser has been studiedtheoretically and conﬁrmed experimentally [17, 19], we expect to ﬁnd both ROs as well asFOs in the 2FOF laser. From a bifurcation theory point of view, both types of oscillations areperiodic orbits that originate from Hopf bifurcations [38, 42]. We found many Hopf bifurcationcurves in sections 3.1–3.4, from which periodic solution bifurcate. We now discuss which ofthose Hopf bifurcations in the (ωs , dCp )-plane give rise to ROs and which to FOs; in this way,we determine where in the (ωs , dCp )-plane one can expect to ﬁnd ROs and FOs. To analyse periodic orbits emanating from Hopf bifurcation points we use again numericalcontinuation. We continue the bifurcating periodic solutions with DDE-BIFTOOL [11]. Note
- 116. 3.5. Different types of bifurcating oscillations 101 . 2.1 2.1 IL IL 1.9 1.9 (a1) (b1) 1.7 1.7 0 5 10 15 t 20 25 0 100 200 300 t 400 500 2.5 2.5˙φL ˙ φL 0 0 (a2) (b2) −2.5 −2.5 0 5 10 15 t 20 25 0 100 200 300 t 400 500 1.9 1.9 IF 1 IF 1 (a3) (b3) 1.4 1.4 0 5 10 15 t 20 25 0 100 200 300 t 400 500 2.5 2.5˙φF 1 ˙ φF 1 0 0 (a4) (b4) −2.5 −2.5 0 5 10 15 t 20 25 0 100 200 300 t 400 500 1.9 1.9 (a5) (b5) IF 2 IF 2 1.4 1.4 0 5 10 15 t 20 25 0 100 200 300 t 400 500 2.5 2.5˙φF 2 ˙ φF 2 0 0 (a6) (b6) −2.5 −2.5. 0 5 10 15 t 20 25 0 100 200 300 t 400 500Figure 3.17. Example of relaxation oscillations (a) and frequency oscillations (b) found in the EFMstability diagram in ﬁgure 3.7 (c); for ∆1 = 0.005, ∆2 = 0, κ = 0.01, Λ = 0.005 and dτ = 0. RO arefound 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, theintensity IF 1 and the frequency φ ˙ F 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.that, the numerical continuation with calculation of stability information for periodic orbits iscomputationally much more expensive than the respective calculations for EFMs (which aresteady states). This is why a complete analysis of bifurcating oscillations and their stabilityregions is beyond the scope of this work. Rather we show, by continuation of particular bifur-cating periodic orbits, where generally ROs and FOs can be found for the 2FOF laser. As a starting point, we now investigate the types of periodic solutions bifurcating fromthe different types of Hopf bifurcation curves in ﬁgure 3.7 (c); for ∆1 = 0.005, ∆2 = 0,κ = 0.01, Λ = 0.005 and dτ = 0. We start our analysis with this case, because beside theHopf bifurcation curves that emerged with the appearance of the hole, the boundary of the EFMstability region in ﬁgure 3.7 (c) still includes all the Hopf bifurcation curves that one can ﬁndfor the 2FOF laser with two identical ﬁlters with equal delay times. To determine what kind
- 117. 102 Chapter 3. EFM stability regionsof periodic solutions emanate from the Hopf bifurcation curves in ﬁgure 3.7 (c), we analysefrequencies and amplitudes of the periodic solutions that bifurcate from them; at each Hopfbifurcation curve we start a sufﬁcient number of continuations to be able to determine the typeof oscillations that bifurcate from it. Figure 3.17 shows example of ROs (a) and of FOs (b) that can be found for suitable(ωs , dCp ) in ﬁgure 3.7 (c); the ROs are for point (ωs , dCp /π) = (0.0035, 1.828), and theFOs for (ωs , dCp /π) = (0.0031, 0.802). Note that the ROs and FOs exist for the same ωs -range, but for different dCp . In respective column of ﬁgure 3.17 we show, from top to bottom, ˙time series of the laser intensity IL , of the laser frequency φL = dφL /dt, of the ﬁrst ﬁlter ˙intensity IF 1 , of the ﬁrst ﬁlter frequency φF 1 = dφF 1 /dt, of the second ﬁlter intensity IF 2 , ˙and of the second ﬁlter frequency φF 2 = dφF 2 /dt. Note the difference in time scales betweencolumns (a) and (b). In ﬁgure 3.17 (a) we show typical ROs: both the laser intensity IL (a1) and the laser ˙frequency φL (a2) oscillate at a frequency of 4.2 GHz. At the same time, both ﬁlter ﬁeldshardly show any dynamics — the amplitudes of oscillation of the ﬁlter ﬁelds (a3)–(a6) are atleast one order of magnitude smaller then the amplitude of oscillations of the laser ﬁeld. Thefact that the ﬁlter ﬁelds are practically not involved in the oscillations is consistent with thegeneral observations that ROs are typical for semiconductor lasers with any type of feedback.It appears that the role of the feedback loops is merely to decrease the losses, which exicitesthe ROs that already exist in damped form in the free-running semiconductor laser. In ﬁgure 3.17 (b) we show an example of the FOs. Here, the laser frequency φL (b2) ˙ ˙oscillates with an amplitude comparable to the oscillations of the laser frequency φL in the caseof ROs (a2). However, the laser intensity IL (b1) is practically constant — it oscillates withan amplitude that is two orders of magnitude smaller than the amplitude for ROs in panel (a1).Moreover, in contrast to ROs both ﬁlter ﬁelds (b3)–(b6) oscillate with signiﬁcant amplitudes.Frequency of the FOs is on the order of 1/τ . Note that both ﬁlter intensities IF 1 (b3) and ˙IF 2 (b5) oscillate in antiphase with each other. Moreover, the laser frequency φL (b2) is in ˙ ˙antiphase with the oscillations of both ﬁlter frequencies φF 1 (b4) and φF 2 (b6). This showsthat for the 2FOF laser both ﬁlter ﬁelds are generally involved in FOs. The boundary of the EFM stability region in ﬁgure 3.7 (c) involves seven Hopf bifurcationcurves: one is located in the centre of the upper half, slightly to the left from ωs = 0, twosurround the hole and four form the outer boundary. Most of those Hopf bifurcation curvesgive rise to the FOs. In fact, the only Hopf bifurcation curve at which the ROs excite is theclosed Hopf bifurcation curve located in the centre of the upper half of ﬁgure 3.7 (c1). We analysed the Hopf bifurcation curves from ﬁgure 3.1 (a), ﬁgure 3.3 (b) and (d), ﬁg-ure 3.12 (a) and (c), ﬁgure 3.14 (b) and ﬁgure 3.16 (d) as well. This allows us to conclude that
- 118. 3.5. Different types of bifurcating oscillations 103FOs bifurcate from most of the Hopf bifurcation curves shown in the ﬁgures in sections 3.1–3.4. Similarly to other laser systems with optical feedback, ROs only appear around the solitarylaser frequency — in our case represented by ωs = 0 — and only for sufﬁciently low levelsof the population inversion Ns . In particular this fact can be observed in section 3.1, wherewe consider the dependence of the EFM stability regions on the common feedback rate κ. Insection 3.1 in ﬁgure 3.1 the dCp -periodic copies of RO Hopf bifurcation curves surround thepoints (ωs , dCp ) = (0, 2n)π with n ∈ Z, and they form the lower boundary of the EFM sta-bility region in the (ωs , Ns )-projections; see ﬁgure 3.1 (a2)–(c2). Note that all the other Hopfbifurcation curves give rise to FOs. The RO Hopf bifurcation curve only appears as a closedcurve in the middle of the EFM stability region for sufﬁciently large κ; see ﬁgure 3.1 (a3).Furthermore, the expansion of the RO Hopf bifurcation curve with increasing κ results in thegrowth of the region where ROs exists. These observations agree generally with ﬁndings forROs in other laser systems with optical feedback [17, 22, 38]. To explain how the regions of existence of different periodic solutions depend on Λ we nowconsider the Hopf bifurcation curves presented in ﬁgures 3.3 and 3.4 in section 3.2. Similarlyto ﬁgure 3.1 (a3)–(c3), the dCp -periodic copies of RO Hopf bifurcation curves in ﬁgure 3.4(a)–(d) surround the points (ωs , dCp ) = (0, 2nπ) with n ∈ Z; these curves form the lowerboundary of the EFM stability region in the (ωs , Ns )-projections in ﬁgure 3.3 (a2)–(d2). Allthe other Hopf bifurcation curves give rise to FOs. With increasing Λ, all the Hopf bifurcationcurves that give rise to the FOs gradually disappear; details are described in section 3.2. Thesingle Hopf bifurcation curve in ﬁgure 3.3 (c) and (d) is the one at which ROs bifurcate. Thismeans that for sufﬁciently high Λ no FOs exist in the 2FOF system. This result is in goodagreement with the observation that, as the Λ → ∞, the 2FOF laser reduces to the COF laser,where one does not observe FO [5, 29, 31]. We now explain, with help of ﬁgures from section 3.3, that with increasing modulus of∆1 and ∆2 the RO Hopf bifurcation curve disappears. As for the case of the 2FOF laser withtwo identical ﬁlters with equal delay times presented in sections 3.1 and 3.2, for low values ofﬁlter detuning ∆1 , as in ﬁgure 3.7 (a) and (b), the dCp -periodic copies of RO Hopf bifurcationcurves surround the points (ωs , dCp ) = (0, 2nπ) with n ∈ Z. With increasing ∆1 , thisRO Hopf bifurcation curve shrinks and shifts away from the points (ωs , dCp ) = (0, 2nπ) withn ∈ Z. In ﬁgure 3.7 (c1), for ∆1 = 0.005, only one copy of the whole RO Hopf curve is visibleand it is located in the upper half of the (ωs , dCp )-plane. As the result of a further increaseof ∆1 , the RO Hopf bifurcation curve disappears. All remaining Hopf bifurcation curves giverise to FOs; see ﬁgure 3.9 (a) and (b). What is more, we checked that all periodic solutionsemanating from all the Hopf bifurcation curves involved in EFM stability region boundaries inﬁgures 3.8–3.15 are FOs. To summarise, we found that for moderate values of κ and Λ and ﬁlter detunings ∆1 and∆2 of sufﬁciently large modulus all periodic solutions emanating from the Hopf bifurcation
- 119. 104 Chapter 3. EFM stability regionscurves bounding EFM stability regions are FOs. In contrast, ROs bifurcate from the closedHopf bifurcation curve that exist only if the solitary laser frequency is subject to sufﬁcientlyhigh feedback from the centre of the ﬁlter proﬁle.
- 120. Chapter 4Overall summaryResearch presented in this thesis was motivated by the application of an 2FOF laser as a pumplaser for optical communication systems. Since in such applications the main concern is sta-ble operation of the laser source, in our analysis we focused on structure and stability of theexternal ﬁltered modes (EFMs). The EFMs are solutions of Eqs. (1.1)–(1.4) which physi-cally correspond to constant-intensity monochromatic laser operation. We ﬁrst investigatedthe structure of the EFMs; in other words, we determined how the regions of existence of theEFMs depend on the parameters of the 2FOF laser. We then used the uncovered structure ofthe EFMs as a guideline for the analysis of their stability. What is more, we were able to relatechanges of the EFM stability with changes of the EFM structure. Finally, we brieﬂy studiedthe types of periodic solutions bifurcating from the boundaries of the EFM stability regions.We now present an overall summary of our work and discuss the possibility of an experimentalconﬁrmation of our results. The well established way of analysing the structure of the solutions of the single FOFlaser is the investigation of EFM components — closed curves that are loci of EFMs in the(ωs , Ns )-plane. EFMs trace an EFM component with a changing phase of the feedback ﬁeld.The relevance of EFM components is that they show how many disjoint ωs frequency ranges areavailable for stable laser operation. In contrast to the single FOF laser, where one can observeat most two EFM components, we showed that, due to the transcendental nature of Eq.(2.9)— the envelope of Eq.(2.3) for frequencies ωs of the EFMs — in the 2FOF laser system itis possible to observe an arbitrary number of EFM components. Moreover, the analysis ofEq.(2.9) showed that the number of EFM components in the 2FOF laser depends strongly onthe parameter dCp — the difference between the feedback phases of the two ﬁltered ﬁelds.Therefore, in chapter 2 we introduced the EFM surface in (ωs , dCp , Ns )-space, as a naturalgeneralization of an EFM component for the 2FOF laser. This provided a useful frameworkfor the analysis of the dependence of structure and stability of EFMs on the 2FOF laser’s twofeedback loops and their ﬁlter parameters. Similarly to EFM components, the EFM surface 105
- 121. 106 Chapter 4. Overall summaryprovides information on how the structure of solutions depend on the phase relations of theﬁltered ﬁelds. Note that the feedback phase depends on subwavelength changes of the lengthof the feedback loops and, hence, it is very difﬁcult to control in real-life implementations.Therefore, the feedback phase is the key parameter of interest in analysis of EFMs. By considering the structure of Eq. (2.9) we found out that the dependence of the EFMsurface on the ﬁlter detunings ∆1 and ∆2 and the common ﬁlter width Λ can be analysedseparately from its dependence on the difference dτ between the delay times. Therefore, weﬁrst explored topological changes of the EFM surface with the former ﬁlter parameters. To thisend, we employed singularity theory; in particular, we distinguished ﬁve different mechanismsthrough which the EFM surface can change locally: four singularity transitions and a cubictangency of the EFM surface with respect to a plane dCp = const. What is more, we usedloci of those transitions in the (∆1 , ∆2 )-plane to compute, by numerical continuation, theEFM surface bifurcation diagrams in the (∆1 , ∆2 )-plane. We then analysed, how this diagramdepends on change of the common ﬁlter width Λ. This allowed us to present a completeclassiﬁcation of all possible types of the EFM surface for the case of equal feedback ratesκ = κ1 = κ2 . Furthermore, we used a stereographic change of coordinates to compactify theEFM surface bifurcation diagrams in the (∆1 , ∆2 )-plane to analyse different ways in whichthe 2FOF laser can be reduced in a nontrivial way to the single FOF laser and to the COFlaser. Finally, we showed that a nonzero difference dτ = 0 between the delay times causes ashearing of the EFM surface along the dCp -axis; the topological structure of the single instanceof the EFM surface is preserved under this transformation. It is due to this shearing of theEFM surface that one can ﬁnd an arbitrary number of the EFM components in the 2FOF laser;for dτ = 0 there are at most three EFM components: one exist always around solitary laserfrequency, and the other two can appear around the ﬁlter central frequencies. The ﬁndings presented in chapter 2 also show that the 2FOF laser is a model system, whichcan be used to connect different kinds of laser systems subject to optical feedback. In particular,for Λ → ∞ the 2FOF laser reduces to the 2COF laser. It is also possible to increase the width ofonly one of the ﬁlters, and this will result in a system with FOF and COF branches.Moreover,by setting ∆1 = ∆2 or by shifting one of the ﬁlter central frequencies ∆i to inﬁnity thesystem reduces further to the FOF laser or to the COF laser, respectively. On the other hand,for nonzero dτ one can investigate the transition between a laser subject to feedback from aminimum of ﬁlter proﬁle (as in [64]) and a laser subject to feedback from a ﬁlter maximum, ashere. In chapter 3 we analysed the stability of the EFMs. We ﬁrst investigated how the stabilityof the EFMs depends on the common feedback rate κ and the common ﬁlter width Λ. We nextexamined how it is affected by the geometrical and topological transitions of the EFM surfacewhen the ﬁlter detunings ∆1 and ∆2 are changed. Finally, we showed that, similarly to theEFM surface, changing dτ results in shearing of the EFM stability regions. Analogously, to
- 122. 4.1. Physical relevance of ﬁndings 107other laser system subject to optical feedback, stable EFMs of the 2FOF laser become unstablein Hopf bifurcations; from those Hopf bifurcation two kinds of stable periodic solutions mayarise: the well-known relaxation oscillations, and frequency oscillations which are typical forlasers with ﬁltered feedback. In the next section we discuss our ﬁndings from chapter 3, in conjunction with predictionsconcerning outcomes of possible experiments with the 2FOF laser.4.1 Physical relevance of ﬁndingsDetailed experimental studies of how the dynamics of a semiconductor laser depend on feed-back parameters are very challenging. Nevertheless, multiparameter experimental studies ofthe dynamics of the single FOF lasers were performed successfully [22, 23]. What is more, dueto advance in experimental techniques it is possible today to compare theoretical and experi-mental two-parameter bifurcation diagrams of laser systems; see for example [48, 65, 71]. Eventhough those state-of-the-art studies concern lasers with an optical injection, similar methodsmight be used to analyse the 2FOF lasers. In this section, we ﬁrst discuss how the different parameters of the 2FOF lasers may bechanged in an experiment. Next, we present what features of the EFM structure presentedin Chapters.2 and 3 may be most promising to verify experimentally. Finally, we show twoexamples that demonstrate the high degree of multistability in the 2FOF laser.4.1.1 Experimental techniques for the control of parametersThe 2FOF laser can be realized in experiments either by means of optical ﬁbres with embeddedﬁber Bragg gratings or by two open air (unidirectional) feedback loops with Fabry-Pérot ﬁlters;see ﬁgure 1.1. In both setups, most of the 2FOF laser parameters can be controlled in a veryprecise way. In particular, all the ﬁlter and feedback loop parameters that are analysed in thisthesis, are directly accessible and can be set for an experiment. Some of the parameters, such as 1 2the feedback phases Cp and Cp or feedback rate κ, can be modiﬁed almost continuously duringmeasurements. However, to control some of the other parameters it is necessary to modify theexperimental setup. For example, to change the ﬁlter width Λ or the ﬁlter detunings ∆1 , and∆2 it may be necessary to introduce ﬁltering elements with different characteristics. To explore the EFM surface experimentally — the main object of our studies — it is nec-essary to control the feedback phases in both feedback loops. In previous experiments withthe FOF laser and the COF laser two methods of controlling the feedback phase were used. In[19] the feedback phase is controlled by changing the length of feedback loop on the optical
- 123. 108 Chapter 4. Overall summarywavelength scale (nanometres) with a piezo actuator, and in [30] it is controlled by changingthe laser pump current, which results in shifting the solitary laser frequency; note that [30]concerns the COF laser. It is important to realize that for the 2FOF laser any changes of the solitary laser frequency 1 2affects simultaneously not only both feedback phases Cp and Cp but, more importantly, alsoboth ﬁlter detunings. In other words, scanning through the solitary laser frequency concurrentlychanges the EFM-components and positions of EFMs on them; for each value of the solitarylaser frequency one observes a new set of EFMs that is given by instantaneous ﬁlter detuningsand instantaneous feedback phases that change simultaneously in both feedback loops. There- 1 2fore, to vary the feedback phases Cp and Cp in the 2FOF laser it would be more practical to 1use piezo actuators. Similarly, in an experimental setup with optical ﬁbres the parameters Cp 2and Cp can be controlled via thermal expansion of glass. The ﬁbre temperature of a sectionof an optical ﬁbre can be controlled either by immersing it in a water bath, or by locally heat-ing an optical ﬁbre with a resistance wire heater. Note that to change dCp continuously in an 1 2experiment it is enough to control one of the feedback phases, either Cp or Cp . The other parameters of the 2FOF laser can be changed in the following way: the feedbackrate κ can be controlled with a variable attenuator and a half-wave plate λ/2 [19], or by aneutral density ﬁlter [23]; the ﬁlter width Λ and the ﬁlter detuning ∆1 and ∆2 can be changedby using different Fabry-Pérot resonators with different distances between their mirrors [22];or by exploiting strain and temperature sensitivity of ﬁbre Bragg gratings to control location ofmaximum and width of its reﬂection proﬁle [66]. Finally, changing the feedback loop lengthsdirectly changes the delay times τ1 and τ2 , and in consequence dτ .4.1.2 Expected experimental resultsIn experimental studies of a semiconductor laser with feedback one observes only stable cw-solutions. Therefore, the stability analysis presented in chapter 3 allows us to make somequalitative predictions about possible experimental results. Generally, we showed there thatthe core of the structure of the EFM stability regions, is formed by the two stable EFM bandscentred around the ﬁlters central frequencies. More importantly, we uncovered that those twobands of stable EFMs appear to be quite robust against a wide range of parameter changes.In other words, we showed that for large, experimentally relevant parameters ranges, multipleregions of stable EFMs exist in the 2FOF laser system. Note that, since similarly to the EFMcomponents, the large part of the EFM surface consists of unstable EFMs, the structure of theEFM surface can only be checked indirectly. Nevertheless, although the unstable EFMs cannotbe observed directly in an experiment, it has been shown that they may play a very importantrole in shaping the overall dynamics of the laser output [25, 30, 54, 55, 58].
- 124. 4.1. Physical relevance of ﬁndings 109 We now list the features of the CW-states and of the basic periodic solutions of the 2FOFlaser that are most likely to be observe in an experiment; we base our predictions on ﬁndingspresented in this thesis and on experimental results for the FOF laser. A ﬁrst experimental test of the results from chapter 3 could be a conﬁrmation of the de-structive interference between the two ﬁlter ﬁelds. Recall that, for the two identical feedbackloops with equal delay times, the feedback ﬁelds cancel each other for dCp = (2n + 1)π withn ∈ Z. Therefore, for different values of dCp , and a ﬁxed value of κ, one should be able toobtain for the 2FOF laser results as in [22], which illustrate the dependence of the EFMs ofthe single FOF laser on the feedback strength. In other words, one should be able to showexperimentally that the 2FOF laser can be reduced, in a non-trivial way to the single FOF laserwith effective feedback rate controlled by dCp ; see section 2.1.2. A main feature of the stability ﬁgures in chapter 3 are bands of stable EFMs centred aroundthe ﬁlter central frequencies. In such a band, stable EFMs exist only for some speciﬁc values 1 2of the feedback phases Cp and Cp . Therefore, due to fact that dCp depends on both feedback 1 2phases Cp and Cp , each of the stable EFM bands on the EFM surface have to be consideredseparately. In particular, to follow a stable EFM on a speciﬁc band, one has to ﬁx feedbackphase in the feedback loop associated with this band, and change the dCp by means of thefeedback phase in the other feedback loop. The stable EFMs manifest themselves in an ex-periment as plateaus of the feedback intensity (different feedback intensities imply differentfrequencies of the laser light); see [19, Fig. 3]. Therefore, to conﬁrm the existence of stableEFM bands, it is ﬁrst necessary to set the parameters of the experimental setup, in such waythat the laser is locked to one of the stable EFMs; then, it can be shown experimentally thatsuch an EFM exists over several 2π cycles of changing dCp and one is indeed dealing with astable EFM band. On the other hand, if the existence of the EFM is limited to some ﬁnite rangeof the parameter dCp , then one can say that the measured EFM exists on an island of stableEFMs. Note that, due to the periodic nature of the edges of stable EFM regions, it might bedifﬁcult to distinguish between a narrow stable EFM band and a large stable EFM island. By combining all the informations about the existence of stable EFM bands and islands,one might be able to reconstruct the main features of the EFM stability structure as presented inthe chapter 3. More speciﬁcally, one should be able to distinguish between the EFM stabilitystructure involving two wide stable EFM bands as in ﬁgures 3.10–3.13; the EFM stabilitystructure consisting of three stable EFM bands as in ﬁgure 3.15 (b); and the EFM stabilitystructure consisting of combination of stable EFM bands and islands as in ﬁgure 3.14 andﬁgure 3.15 (c) and (d). Furthermore, cases in which the stability structure from chapter 3 isclosely aligned with the EFM surface, allow for indirect conﬁrmation of the classiﬁcation ofthe EFM surface into different types presented in chapter 2; an example of such an alignmentis ﬁgure 3.14.
- 125. 110 Chapter 4. Overall summary Finally, one may also investigate existence of bifurcating FOs and ROs for different pa-rameter values. A strong conﬁrmation of our results would be to demonstrate the absence ofROs for moderate values of the common feedback rate κ and the common ﬁlter width Λ andthe ﬁlter detunings ∆1 and ∆2 of sufﬁciently large modulus; see section 3.5.4.1.3 Existence of multistabilityA next step, which is a necessary to fully describe stability of the EFMs, is the investigation 1 2how many stable EFMs coexist for ﬁxed parameters; i.e. for given Cp and Cp . Similarly tothe single FOF laser [17, 31], also in the 2FOF laser one can expect to ﬁnd multistability for 1 2chosen ﬁxed parameters, that is, including ﬁxed feedback phases Cp and Cp . To conﬁrm thishypothesis, we ﬁrst show that the total number of coexisting EFMs depends on the parameters 1Cp and dCp . We then analyse the stability of individual EFMs, computed for two different setsof ﬁxed parameters. 1 Figure 4.1 shows regions in the (Cp , dCp )-plane in which different numbers of EFMs ex-ist simultaneously; this ﬁgure is for ∆1 = 0.050, ∆2 = −0.049, κ = 0.01, Λ = 0.005 anddτ = 0, and the corresponding EFM surface of type I BI is shown in ﬁgure 3.14 (b). Bound-aries between the regions in ﬁgure 4.1 are given by saddle-node bifurcation curves (blue).Figure 4.1 (a) shows the regions with different numbers of coexisting EFMs over a fundamen- 1tal 2π-interval of Cp . The labelling of the regions provides not only information about thenumber of EFMs on each EFM component, but also about the number of EFM componentsand their relative positions. To emphasize the connection between EFM components and theEFM surface, we structured the labels in ﬁgure 4.1 (a) to be similar to the labels for the typesof the EFM surface. More speciﬁcally, the hat over a number indicates the number of EFMs onthe EFM component centred around the solitary laser frequency. Furthermore, the the numberswithout hats indicate the number of the EFMs on the EFM components centred around theﬁlter central frequencies; the total number of EFMs for a given parameter values is simply thesum of numbers of a region label. Note that, the positions of the numbers without hats indicatelocations of the other EFM components with respect to the central one. In other words, oncethe EFM surface is known, it is possible to encode its structure in labels of the regions in the 1(Cp , dCp )-plane. A single EFM, in regions labelled 1, corresponds to the solitary laser solution. The totalnumber of the EFMs on the central EFM component is always odd and new EFMs are formedin pairs on saddle-node bifurcations curves [31, 38]. We can expect multistability in all regionsin ﬁgure 4.1 (a) with more then one EFM. However, the best regions to look for a high degreeof multistability are the two small regions labelled 2 + 1 + 2 and 2 + 3 + 2. For the parametervalues in those regions, EFMs exist concurrently on all three EFM components; in the otherregions, EFMs exist at most on two EFM components. Note that, by considering dCp as a
- 126. 4.1. Physical relevance of ﬁndings 111 . 2 (a) b 1 b+2 1 2+b+2 1 2+b 1 b 1 2+b 1 dCp b 3 2+b 3 π 2+b 3 d d 2+b 5 b+2 3 b 1 2+b 5 1 2+b+2 3 b 5 b 5 b 3 b 3 2+b 3 b 1 d d b 1 2+b 1 b+2 1 2+b 1 0 0 1 1 2 . Cp /π 2 (b) dCp π 1 0 −5 −4 −3 −2 −1 0 Cp /π 1 1 2 1Figure 4.1. Regions in the (Cp , dCp )-plane with different numbers of coexisting EFMs, as indicated 1by the labelling. Panel (a) shows the regions on a fundamental 2π-interval of Cp , while panel (b) shows 1it in the covering space (over several 2π-intervals of Cp ). Boundaries between regions are saddle-nodebifurcation curves (blue); also shown in panel (b) are periodic copies of the saddle-node bifurcationcurves (light blue). Labels Here ∆1 = 0.050, ∆2 = −0.049, κ = 0.01, Λ = 0.005 and dτ = 0; theseparameter values are those for the EFM surface in ﬁgure 3.14 (b).kind of effective feedback strength (as has been argued in section 2.1.1), it is possible to seesimilarities between our ﬁndings and results on the number of coexisting EFMs for the singleFOF laser as presented, for example, in [31]. 1 Figure 4.1 (b) illustrates how the regions in the (Cp , dCp )-plane can be related to theprojection of the EFM surface onto the (ωs , dCp )-plane. In ﬁgure 4.1 (b) we show the dark 1blue saddle-node bifurcation curves from panel (a) over several 2π-intervals of Cp ; since, each 1saddle-node bifurcation curve is presented over different 2π-intervals of Cp the curves do not
- 127. 112 Chapter 4. Overall summary 0.011 . (a) Ns 0 −0.011 −0.055 0 ωs 0.055 0.01 (b) Ns 0 −0.01 . −0.045 0 ωs 0.045Figure 4.2. EFM-components (grey) in the (ωs , Ns )-plane with stability information. Panel (a) is fordCp = π, ∆1 = 0.050, ∆2 = −0.049, and panel (b) is 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 bifurcations (red dots) or by the saddle-node bifurcation (blue dots). The 1 1actual stable EFMs for Cp = 1.03π (a) and Cp = 0.9π (b) are the black full circles; open circles areunstable EFMs. The EFM components in panel (a) correspond to a constant dCp -section through theEFM surface in ﬁgure 3.14 (b), and those in panel (b) to a constant dCp -section through the EFM surfacein ﬁgure 3.12 (d).intersect. More importantly, the relative positions of the dark blue curves in ﬁgure 4.1 (b)correspond to the structure of the saddle-node bifurcation curves in ﬁgure 3.15 (d), and hence,the two ﬁgures can easily be related. Note that multiple periodic copies of the saddle-nodebifurcation curves in ﬁgure 4.1 (b) are coloured light blue. More generally, ﬁgure 4.1 showsthat, as in the case of the single FOF laser, the actual number of coexisting EFMs dependsstrongly on the phase relation between the feedback and laser ﬁelds. Figure 4.2 conﬁrms the existence of multistability in the 2FOF system by analysing thepositions of the individual EFMs along the EFM components with stability information, for
- 128. 4.1. Physical relevance of ﬁndings 113two different sets of ﬁxed parameters. Figure 4.2 (a) corresponds to the constant dCp -section,for dCp = π, through the EFM surface of type I BI in ﬁgure 3.14 (b); ﬁgure 4.2 (b) correspondsto a constant dCp -section for dCp = −π through the EFM surface of type B BB in ﬁgure 3.12(d). There are stable segments (green) on each of the three EFM components in panels (a)and (b); they are bounded either by the saddle-node bifurcations (blue dots) or by the Hopf 1bifurcations (red dots). The black circles in ﬁgure 4.2 are the EFMs for Cp = 1.03π (a) and 1for Cp = 0.9π (b); full circles indicate stable EFMs and open circles unstable EFMs. The EFMs in ﬁgure 4.2 (a) are computed for parameter values from the region labelled2 + 3 + 2 in ﬁgure 4.1 (a); in this case four of the EFMs are stable and lie within the respectivethree stable segments of the EFM components. Note that the stable EFMs on the outer EFMcomponents lie very close to the boundaries of the respective stable segments. In fact, as can 1be seen in ﬁgure 4.1 (a), for slightly different value of Cp , EFMs exist only on two out ofthree EFM components. In ﬁgure 4.2 (b) we show a more convincing example of multistabil-ity, where three of the EFMs are stable and clearly lie well within the respective three stablesegments of the EFM components. Figure 4.2 suggest that multistability is a rather common feature of the 2FOF laser, so thatit should be possible to observe it in experiments; for example as hysteresis [20]. What ismore, this ﬁgure demonstrates that the stability results presented in chapter 3 are indeed anessential basis for the exploration of multistability in the 2FOF laser. A thorough analysis ofmultistability of EFMs and investigation of stability of periodic solutions in the 2FOF laser area logical next step in the study of the 2FOF laser.
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- 136. Appendix AHow to construct the EFM surfaceIn our work we present structure and stability information of the EFMs by means of showingthe EFM surface in (ωs , dCp , Ns )-space. Since, explicit and easily solvable equations for theEFMs and their stability do not exist, we ﬁnd them numerically and we use state-of-the-artnumerical techniques to determine their properties. The process of constructing the EFM surface can be split into several steps. First, for thechosen parameter values, we compute a sufﬁcient number of constant dCp sections throughthe EFM surface. We then use this computed data to render the EFM surface. Finally, we addcomputed stability information to the EFM surface and its two dimensional projections. Eachof the steps is automated as much as is practical; to calculate and render the EFM surface in anefﬁcient way, we developed a set of interactive MATLAB scripts and functions. Recall that, to present the classiﬁcation of different types of the EFM surfaces in sec-tion 2.3, it was enough to use Eq. (2.16), which describes the boundary of the EFM surfaceprojection onto the (ωs , dCp )-plane independently of any state variables of the 2FOF laser.The advantage of this approach is that it allowed for the fast computation of the (ωs , dCp )-projections of the EFM surface for a wide range of the system parameters, in particular, fordifferent values of the ﬁlter detunings ∆1 and ∆2 . However, equation (2.16) for the (ωs , dCp )-projection boundary does not provide any information on EFMs stability. Furthermore, insection 2.1 we showed that the EFMs themselves can be calculated as solutions of the tran-scendental equations (2.3)–(2.8) and, hence, these could be used to follow EFMs with changingparameters with any numerical continuation package. However, Eqs. (2.3)–(2.8) also do notprovide the EFM stability information. In fact, to analyse EFM stability it is necessary to con-tinue the solutions of the full DDE system (1.1)–(1.4); to this end, we use the MATLAB pack-age DDE-BIFTOOL [11]. The EFM stability information, uncovered with DDE-BIFTOOL,can be presented either on the EFM surface itself or in any conveniently chosen EFM surfaceprojection. We now describe our numerical strategy to deal with the mathematical challengesassociated with the analysis of the EFM structure of Eqs. (1.1)–(1.4). 121
- 137. 122 Appendix AA.1 Dealing with the S1 -symmetry of the 2FOF laser modelIn section 1.1 we mentioned that the system (1.1)–(1.4) shares symmetry properties with manyother systems with coherent optical feedback. Namely, system (1.1)–(1.4) has an S 1 -symmetry,meaning that the equations are equivariant under any element b ∈ S 1 , which is physically arotation with a speciﬁc phase. This equivariance is the result of the fact that Eq. (1.1) is linearin E and that E enters Eq. (1.2) only as its modulus |E| [29, 34, 40]. As a consequence,solutions of Eqs. (1.1)–(1.4) are group orbits under the S 1 -symmetry; trajectories in the grouporbit run entirely parallel, and differ only in their phases; see Eq. (1.5). In other words, one canobtain inﬁnitely many trajectories of the system (1.1)–(1.4) by multiplying E, F1 and F2 byany complex number of modulus one; this is true even for chaotic trajectories [40]. Since thesymmetry cannot be simply divided out from the equations, to resolve the phase indeterminacyof Eqs.(1.1)–(1.4) we consider intersection of the group orbits with the ﬁxed six-dimensionalhyper-half plane S = {(E, N, F1 , F2 ) | Im(E) = 0 and Re(E) ≥ 0} ; (A.1)these intersections are also called the trace [34, 40]. Note that, the trace of group orbits of Eqs.(1.1)–(1.4) consist of isolated trajectories which can be studied with numerical continuation. In practice to analyse the trace of the 2FOF laser by continuation with DDE-BIFTOOL,it is necessary to transform Eqs.(1.1)–(1.4) into a system in which the phase indeterminacy isexpressed explicitly. To this end, we substitute ansatz (E, N, F1 , F2 (t)) = Eeibt , N, F1 eibt , F2 eibt (A.2)into (1.1)–(1.4) and divide through by an exponential factor, which results in the transformedsystem dE = (1 + iα)N (t)E(t) − ibE(t) + κ1 F1 (t) + κ2 F2 (t), (A.3) dt dN T = P − N (t) − (1 + 2N (t))|E(t)|2 , (A.4) dt dF1 1 = Λ1 E(t − τ1 )e−iCp + (i∆1 − Λ1 − ib)F1 (t), (A.5) dt dF2 2 = Λ2 E(t − τ2 )e−iCp + (i∆2 − Λ2 − ib)F2 (t). (A.6) dtHere b is an extra free parameter that allows for direct parametrisation of all trajectories in agroup orbit [17, 29, 40, 56]. In other words, each value of b correspond to one trajectory from
- 138. A.2. Computation and rendering of the EFM surface 123the group orbit. Note that solutions of system (A.3)–(A.6) have the same stability properties asthe solutions of system (1.1)–(1.4). In particular, they have a trivial additional zero eigenvalue,which must be taken into account for the appropriate stability and bifurcation analysis of thesolutions [17, 40, 45]. Eqs. (A.3)– (A.6) are of a form in which, by uniquely determining the parameter b, thephase indeterminacy can be resolved. This can be achieved by requiring, for example, thatIm(E) = 0 in Eq. (A.3); in this way, continuous-wave states of Eqs. (A.3)– (A.6) indeed cor-respond to the trace of solutions of (1.1)– (1.4). In other words, a one-dimensional S1 -familyof solutions in the seven-dimensional physical space is transformed into a pair consisting ofthe one-dimensional trajectory in a six-dimensional sub-space, and associated unique value ofthe phase parameter b; for example, a cw-state, which is a circular periodic trajectory of pe-riod 2π/b in (E, N, F1 , F2 (t))-space, transforms into an isolated point in the six-dimensionalsub-space with associated frequency b. Furthermore, by considering cw-states the numericalstrategy gains physical interpretation: the free parameter b is equivalent to the frequency ωs ofan EFM; see section 2.1. Implementation of the above strategy in DDE-BIFTOOL is straight forward: in addi-tion to deﬁning system (A.3)–(A.6) in the form of seven real equations in ﬁle sys_rhs.m,one also needs to specify the condition Im(E) = 0 as an extra equation in the separate ﬁlesys_cond.m. An example of using the same approach to continue solutions of two linearlycoupled oscillators with DDE-BIFTOOL is presented in [12]. More generally, the methoddemonstrated here make use of the fact that ﬁnding a unique solution of a system of equationsrequires that the number of equations and number of unknowns must be equal.A.2 Computation and rendering of the EFM surfaceGenerally, to construct the EFM surface in (ωs , dCp , Ns )-space we ﬁrst compute a numberof sections that are uniformly distributed along the 2π dCp -range considered. Next we usethis computed data to render the EFM surface. Recall, that a constant-dCp section through theEFM surface in (ωs , dCp , Ns )-space corresponds to the EFM components, which are closedcurves traced by the EFMs with changing feedback phase; see section 2.1.1. To compute thenecessary slices, that is the EFM components, we continue the EFMs with DDE-BIFTOOL 2 1[11]. To ensure that dCp is ﬁxed, we substitute Cp = Cp + dCp in Eqs. (A.3)–(A.6). In this 1way, by continuation of a single EFM, in the continuation parameter Cp , over several multiplesof 2π, we indeed obtain constant dCp sections through the EFM surface in (ωs , dCp , Ns )-space. Note that to construct the EFM surface without the stability information one could alsouse Eqs. (2.10) and (2.16).
- 139. 124 Appendix A We now present more detailed information on computing and plotting the EFM surface.First we use the boundary equation (2.16) to ﬁnd the values of ωs , in dependence on dCp , atthe edges of the bands and holes of the EFM surface. Since, the EFM components are closedloops, we only need half of the boundary points; for example, those at the left ends of the EFMcomponents. We obtain at most three such boundary values of ωs for each value of dCp . Since,Eq. (2.16) gives the (ωs , dCp )-projection of a single instance of the EFM surface, even fordτ = 0 there are at most three boundary values of ωs for each value of dCp . (Recall that, theextra EFM components that appear in case of dτ = 0 belong to 2π-shifted copies of the basicinstance of the EFM surface.) Typically, to construct the EFM surface, for dτ = 0, in chapter3, we use 150 values of dCp that are distributed uniformly along the dCp interval of length 2π.However, the actual number and distribution of the points along the dCp -axis depends on thetype of the EFM surface and is speciﬁed by the user. 1 After ﬁnding ωs and dCp values of the starting points, we set Cp = 0 and use Eq. (2.5)– 1(2.8) to compute initial values of the other state variables. Since Cp = 0 is a guess, we use the 1DDE-BIFTOOL correction routine to correct the values of Cp and the state variables; duringthis correction the values of ωs and dCp are kept ﬁxed. For the purpose of surface rendering, we want all the EFM components to be computed inone direction, e.g. clockwise. Therefore, a second point for the continuation must be speciﬁedin a careful way. We proceed by taking the ﬁrst point (computed in the previous step), and apply 1a small positive perturbation to the corrected value of Cp . Next, we correct values of the state 1variables of the second point to ones that correspond to the new value of Cp . Having those twopoints allows us to deﬁne and continue with DDE-BIFTOOL the respective branch of EFMs. 1Moreover, all the branches are continued in the same direction — of increasing Cp . As theresult of the above procedure, we now have a set of uniformly distributed constant dCp sectionsthrough the EFM surface. Each of these sections may consist of up to three EFM components.Typically, the procedure is run from a script in MATLAB; after setting up all the script andcontinuation accuracy parameters it takes around one hour to calculate all the sections neededto construct one instance of the EFM surface. Note that chosing the continuation accuracy tobe to low may lead to gaps in the computed sections of the EFM surface, or even to erroneousswitching between disjoint EFM components. The rendering of the EFM surface proceeds as follows. We ﬁrst assign all the computedEFM components to the particular bands and island of the EFM surface. In case that band (orisland) of the EFM surface has holes or bulges, we divide the EFM components assigned tothis band (or island) into smaller groups which we further assign to local bands and islands.To this end, we consider intervals of the 2π dCp -range within which the number of the EFMcomponents is dCp -independent. Note that this rendering step is directly associated with theEFM surface type. For example, to construct EFM surface of type B Bh in ﬁgure 3.12 (c), weconsidered separately the left band B and and the right band with hole Bh . Furthermore, we
- 140. A.3. Determining the stability of EFMs 125divided the EFM components associated with the band Bh into four groups: the ﬁrst group cor-responded to a single band before the hole, two other groups corresponded to separate bands onthe left and right side of the hole, and the last group was assigned to a single band past the hole.We then use these groups of the EFM components to generate separate meshes to representeach local segment of the EFM surface. To ﬁll gaps between segments of the EFM surface ob-tained in this way, or to close discontinuities at the edges we compute some additional branchesof the EFMs as necessary. For each of the EFM component group we generate a separate mesh. We ﬁrst check thateach of the m EFM components in the group is a single closed loop, and we trim it otherwise 1(since, Cp is 2π-periodic it is possible to go along the EFM component more than once); herem is the number of EFM components in the group. Next, we rewrite the values of ωs , dCp andNs of each point of an EFM component from the DDE-BIFTOOL solution branch structureinto a 3 × k matrix K, where k is the number of points along the branch. Furthermore, weapply constant arclength interpolation along each curve deﬁned by K; in result we obtain anew 3 × l matrix L that consist of l uniformly spaced mesh points. Finally, we split each ofthe m matrices L into rows, and construct three separate m × l mesh matrices, for ωs , dCp andNs , which are used to plot the EFM surface; recall that, l is the number of mesh points (afterarclength interpolation) and m is the number of EFM components in the considered group.On the basis of the above algorithm we developed a MATLAB function that automaticallygenerates the mesh for the group of the EFM components corresponding to a band or an islandof the EFM surface; the mesh is plotted with built-in MATLAB functions — surf and lightwith appropriate parameters. We process each band or island of the EFM surface separately,and later plot them one by one in the same ﬁgure. A typical size of the mesh representing thewhole EFM surface (over a 4π dCp -range) in chapter 3 is about 3 × 300 × 1500 points. We used these computations to construct all of the EFM surfaces in chapters 2 and 3. Toshow the EFM surface near singularity transitions as presented in section 2.3.3 in ﬁgures 2.12–2.15 and in section 2.3.4 in ﬁgure 2.16 and ﬁgure 2.22, we ﬁrst selected parameter values mostsuitable for the computation of the EFM surface near the singularity transition. To this end, weanalysed the EFM surface bifurcation diagram in the (∆1 , ∆2 )-plane. To conﬁrm our choiceof parameters we reviewed quite a large number of projections of the EFM surface onto the(ωs , dCp )-plane near the transition; recall that, the projections are computed quickly with theprojection boundary equation (2.16). We then used the above procedure to calculate and renderthe relevant parts of the EFM surface.A.3 Determining the stability of EFMsIn chapter 3 we also showed ﬁgures of the EFM surface with stability information. These wereobtained as follows. We compute the stability of EFMs along EFM components with DDE-
- 141. 126 Appendix ABIFTOOL. However, computations of the EFM stability information are much more computa-tionally expensive than computations of the EFMs themselves. More speciﬁcally, for optimallychosen stability computation parameters, it takes around half a second to compute the stabilityof a single EFM; that is around 20 times longer than the computation of the EFM itself. There-fore, we compute information on EFM stability only for half of the EFM components usedto construct the EFM surface, that is, typically 75 over the 2π dCp -interval. Furthermore, welimit the number of points along an EFM component for which we run computations to 150.In spite of these restriction, it takes around three hours to calculate the stability information fora single instance of the EFM surface. Note that chosing the accuracy parameters for a stabilitycomputation to be to low may lead to errors in the determination of stable EFM regions. Toensure appropriate accuracy of their computation we checked the alignment of the stable seg-ments of the EFM components with their known boundaries given by independently computedsaddle-node and Hopf bifurcation curves. Additionally, we also checked the consistency of thesaddle-node and Hopf bifurcation curves themselves. The saddle-node and Hopf bifurcation curves are computed separately as two-parameter 1 2continuations in Cp and Cp ; starting points are conveniently chosen from saddle-node and Hopfbifurcations found at the computed EFM components. Note that, to continue the saddle-nodebifurcations it is necessary to use solutions of the transcendental system (2.3)–(2.8); namelythe transformed system Eqs. (A.3)-(A.6) always has an extra eigenvalue 0. On the other hand,Hopf bifurcation curves are continued in the full DDE system Eqs. (A.3)-(A.6). Once the EFM surface is rendered and the stability information is calculated, we overlay thecomputed stable segments of the EFM components and the saddle-node and Hopf bifurcationcurves onto the EFM surface. This is done with a set of MATLAB scripts that trim and plot thebifurcation curves modulo 2π, and with a set of modiﬁed DDE-BIFTOOL plotting routines;these modiﬁcations allow us to plot the solution branches in three dimensions.

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